IREGE, Favre-Bonté Véronique

Keywords: Internet, Small Businesses, Tourism, Websites. 1. Introduction

Small and Medium-sized Entreprises (SMEs) represent a significant share of firms in industrialized countries. However, little work is available concerning Small Businesses (SB) that have less than 50 employees (according to the majority of official definitions). ICT are considered a contributing factor in the renewal of competitiveness, in particular by reducing constraints that bear on SMEs. In this context, one can ask under what conditions the implementation of a website may change the strategic positioning of small accommodation businesses in the tourism sector.

Small and Medium-sized Entreprises 1 (SMEs) represent a significant share of firms in industrialized countries. For example, they accounted for nearly 99% of companies in the European Union (Eurostat 2 ). Many works have focused their attention on the issues that are specific to such businesses, whether on their business model, their management or strategy [START_REF] Julien | Les PME, bilan et perspectives[END_REF]. However, little work is available concerning Small Businesses (SB) that have less than 50 employees (according to the majority of official definitions). In 2003, they represented nearly 92% of European companies according to the Observatory of European SMEs [START_REF] Harindranath | ICT Adoption and Use in UK SMEs: A Failure of Initiatives?[END_REF]. Moreover, OECD member countries play a leading role in international tourism, representing 66% of global arrivals while EU member countries accounted for 50.2% in 2010. Tourism sector also contributes 4.4% of GDP for EU members 3 . In France tourism plays a major economic role with a large number of SMEs and VSEs (235,000 in 2009, in France). This sector is quite advanced in the use of ICT, where information is at the heart of services [START_REF] Aldebert | Repérer les compétences pour mieux comprendre le processus d'innovation. Le cas d'une TPE touristique[END_REF]. In 2010, e-tourism weighed more than 8 billion Euros (excluding business travel), up 19% compared to 2009. After growing 6% in 2009, e-tourism benefits in full from the pick-up in the overall market. Also, according CCM Benchmark, in 2010, e-tourism represented 46.1% of all e-commerce B-to-C in France. This suggests a strong use of ICT for sales and marketing, a hypothesis supported in several works and studies. [START_REF] Bonn | Using the Internet as a Pleasure Travel Planning Tool: an Examination of the Sociodemographic and Behavioral Characteristics Among Internet Users and Nonusers[END_REF], show that Internet marketing is suitable for tourism products and services. Some research suggests developing a better understanding of the profile and behaviour of Internet surfers and shoppers [START_REF] Sigala | Reviewing the profile and behaviour of Internet users: Research directions and opportunities in tourism and hospitality[END_REF]). Yet, in France, its use by SMEs remains low: only 12% have a website that offers online transactions (Eurostat 2010).

1 SMEs are defined by the European Commission as having less than 250 persons employed. They should also have an annual turnover of up to EUR 50 million, or a balance sheet total of no more than EUR 43 million according to Commission Recommendation of 6 May 2003. 2 http://epp.eurostat.ec.europa.eu/statistics_explained/index.php/Small_and_medium-sized_enterprises 3 http://www.oecd.org/cfe/tourism/oecdtourismtrendsandpolicies2012.htm Although SMEs are aware of the main benefits of ICT, 46% of SMEs in France remain sceptic about its ability to develop their turnover (France Numérique 2012, p. 56). One current concern for governments and institutions lies in the use and contribution of technology to business development. Indeed, ICT are considered a contributing factor in the renewal of competitiveness, in particular by reducing constraints that bear on SMEs [START_REF] Sullivan | Economics of Information Technology[END_REF][START_REF] Raymond | The assimilation of E-business in manufacturing SMEs: determinants and effects on growth and internationalization[END_REF][START_REF] Amabille | Les usages d'Internet, facteurs de compétitivité de la « PME réseau » ?[END_REF]Qiang et al 2006[START_REF] Matthews | ICT Assimitation and SME Expansion[END_REF][START_REF] Bellaaj | Effets modérateurs des capacités complémentaires dans le e-commerce. Application au contexte des PME[END_REF]. In this context, one can ask under what conditions the implementation of a website may change the strategic positioning of small accommodation businesses in the tourism sector.

Given the heterogeneity of activities in the tourism industry, we focused our attention on hospitality businesses in general (campsites, hotels, spas), which represented more than 20% of companies in France in 2007 (INSEE, EAE services 2007) and are often at the heart of the tourist activity (this is the second activity in the sector in terms of turnover after catering).

Our analysis focuses on Websites because in the Technology Acceptance Model (TAM), it is one of the ebusiness adoption constructs [START_REF] Heijden | Factors influencing the usage of Websites: The case of a generic portal in the Nether-lands[END_REF][START_REF] Cass | Web retailing adoption: Exploring the nature of internet users Web retailing behaviour[END_REF]. Our problematic is therefore to understand "under what conditions can the implementation of a website change the strategic positioning of small accommodation businesses in the tourism sector?".

The objective of this paper is twofold. Firstly, it aims to show that despite their small size, companies are able to use the Internet for their marketing and to develop sales. It also aims to show that, from a certain point and under certain conditions, the Internet can also change the strategic positioning of SMEs. In the first part of the article, we will review the role that the Internet plays in the development of SMEs. In the second part, after having exposed our methodological choices and the case studies, we will show the impact for accommodation businesses of implementing a website, but also several implications with respect to their managers.

The role of ICT and the Internet in the development of SMEs

Despite the constraints inherent to their small size, SMEs can use ICT to maintain or enhance their competitive advantage within an increasingly difficult economic environment (increased competition, globalization...). However, ICTs alone are not designed to create value.

It is only when they are combined with other resources or implemented in a particular context that they actually create value. We will see how and under what conditions ICT and Internet in particular can play a role in marketing (1.1), but also a strategic role (1.2) in SMEs. We will also mention the managers of such businesses who, depending on their characteristics and involvement, can have a "facilitating" impact on the creation of value, thus representing a condition for the development of the SME (1.3).

Internet as a lever for marketing and sales in SMEs ...

In general, research shows that the use of ICT in SMEs is mainly focused on operational aspects with very little strategic orientation [START_REF] Dyerson | National Survey of SMEs' Use of IT in Four Sectors[END_REF]. According to [START_REF] Lee | A comparative study of web site performance[END_REF], a web presence offers several major advantages: cost reduction, revenue growth, targeted marketing, improved customer satisfaction, quality improvement, and response to customer needs. For many it is difficult to assess the precise results of Internet use for business. The approach for research therefore often focuses on perceived benefits, including sales and marketing dimensions, which remain a prime motivation among managers of SMEs [START_REF] Mehrtens | A model of Internet adoption by SMEs[END_REF][START_REF] Dyerson | National Survey of SMEs' Use of IT in Four Sectors[END_REF][START_REF] Galloway | Rural small firms' use of the internet: From global to local[END_REF][START_REF] Sanders | Investigating the Relationship Between Scottish Rural Small Firms' Social Networks, Extra-local Market Diversification and Internet Usage[END_REF].

One of the most important aspects concerns the development of e-commerce for businesses [START_REF] Raymond | The assimilation of E-business in manufacturing SMEs: determinants and effects on growth and internationalization[END_REF][START_REF] Bellaaj | Effets modérateurs des capacités complémentaires dans le e-commerce. Application au contexte des PME[END_REF][START_REF] Dyerson | National Survey of SMEs' Use of IT in Four Sectors[END_REF], but with very few companies offering the possibility for online order and payment [START_REF] Lucchetti | The Adoption of ICT among SMEs: Evidence from an Italian Survey[END_REF]. Although it remains a marginal channel, SMEs also use ICT to develop a sophisticated pricing in real time, thus making it possible to use yield management techniques and dynamic pricing [START_REF] Aldebert | Repérer les compétences pour mieux comprendre le processus d'innovation. Le cas d'une TPE touristique[END_REF]. However, often, only part of the sales process is dematerialized, in particular upstream of the order (communication, catalogues...). This is partly explained by the lack of financial and human resources. Investment in appropriate software solutions represents a non-negligible cost for a SME. To basic equipment, the cost of training and maintenance also come on top of this [START_REF] Mcdonagh | Euroclicking and the Irish SME: Prepared for ecommerce and the single currency[END_REF][START_REF] Montazemi | How They Manage IT: SMES in Canada and the U.S[END_REF][START_REF] Matthews | ICT Assimitation and SME Expansion[END_REF][START_REF] Dyerson | National Survey of SMEs' Use of IT in Four Sectors[END_REF].

When considering the commercial dimension, ICT and the Internet in particular also offer a means to improve customer relations by maintaining more regular and interactive links with some consumer segments and providing targeted information [START_REF] Levenburg | Does Size Matter? Small Firms' Use of E-Business Tools in the Supply Chain[END_REF][START_REF] Qureshil | Information Technology Interventions for Growth and Competitiveness in Micro-Enterprise[END_REF][START_REF] Mattson | Developing a strategic abstraction tool for service innovation[END_REF]. The visibility of the company (to enhance its reputation) is also one of the aspects that has been highlighted in some works [START_REF] Mehrtens | A model of Internet adoption by SMEs[END_REF]chetti-Luc and Sterlacchini, 2004). The development of websites and the reduced costs associated with their implementation offers the opportunity to create a showcase for companies with otherwise limited geographic reach and resources. Finally, improvement of communication remains one of the main motivations (see Questions 1, 2, 3 in our framework Table 1).

In SMEs, the concept of competitive advantage is often restricted to the commercial aspect with a more customer focused dimension (Amabille and Gadille, 2003;[START_REF] Montazemi | How They Manage IT: SMES in Canada and the U.S[END_REF]. Some companies however do engage in a real strategic thinking through the implementation of their website, especially when they have reached a stage of maturity and mastery of functionalities and services (the conclusion of the transaction and online payment is one of the most advanced phases). Through the Internet SMEs seek to achieve market differentiation. This differentiation consists of focusing not on lower prices but on a differential value perceived by the customer [START_REF] Aldebert | Technologies de l'information et de la communication : un outil stratégique pour les petites et moyennes entreprises touristiques[END_REF][START_REF] Saarijärvi | The mechanisms of value co-creation[END_REF].

… Internet can also be a strategic positioning tool for SMEs.

We have seen that ICT, particularly Internet can allow small companies to expand their sales and marketing strategy (dynamic pricing, visibility with new clients, etc). However, the more the smaller businesses use the Internet as a channel of communication and relationship with their customers, the more the way they are perceived changes and improves compared to their direct competitors and larger competitors [START_REF] Morrison | Marketing small hotels on the World Wide Web[END_REF]. This is especially true for SMEs that are at an advanced stage of Internet use.

The Internet is thus no longer a marketing tool, but also a trigger that can initiate strategic thinking [START_REF] Longé | Intégrer les sites web dans les stratégies[END_REF]. These authors mention that websites represent carriers for organizational and strategic resources that are only revealed through the congruence between ICT management and strategic choices. This congruence occurs for businesses that have an "advanced use" of the Internet. [START_REF] Scaglione | Website adoption and sales performance in Valais' hospitality industry[END_REF] also show that the income growth rate of hotels that have their own domain name is higher than for those that have an Internet presence via regional portals: Internet becomes a lever for strategic positioning for companies that are at an advanced stage in their use of ICT. In previous research the Internet has been found to play a pivotal role in the creation of relationships in firms and, in most cases, is a mechanism for the creation of international opportunities in SMEs [START_REF] Glavas | How international entrepreneurship characteristics influence Internet capabilities for the international business processes of the firm[END_REF]. It is therefore necessary to identify these companies. To this effect, two typologies seem relevant and relatively convergent in their conclusions: the typology of [START_REF] Daniel | Adoption of e-commerce by SMEs in the UK, towards a stage model[END_REF] and that of [START_REF] Aldebert | Technologies de l'information et de la communication : un outil stratégique pour les petites et moyennes entreprises touristiques[END_REF] 4 .

In the typology of [START_REF] Daniel | Adoption of e-commerce by SMEs in the UK, towards a stage model[END_REF], the authors identify four groups corresponding to different stages in the use of ICT: companies developing their first e-commerce services (stage 1), and those that use email to communicate with customers, suppliers and employees (stage 2). Those that have created a website and developed means to order online (stage 3) and finally, the "earlier adopters" (stage 4) that have introduced online ordering and develop means for online payment (last column of the table in Annex 1). This stage 4 corresponds to the pioneers (typology of [START_REF] Aldebert | Technologies de l'information et de la communication : un outil stratégique pour les petites et moyennes entreprises touristiques[END_REF], they feature specific aspects: firstly, they view sales efforts not merely as distribution activities, but as an opportunity to create value. Also, strategically, their managers seek to acquire market segments through differentiation or specialisation and they are engaged in innovation, both in products, services, processes, types of customers, and in the internal structure of the company. This enables companies not only to become better known but also to be recognized. Finally, these companies anticipate on the market and adopt a pro-active behavior. 4 Even though the categories of [START_REF] Daniel | Adoption of e-commerce by SMEs in the UK, towards a stage model[END_REF] are older than a decade, these categories are still relevant because recent literature confirms the four categories [START_REF] Aldebert | Technologies de l'information et de la communication : un outil stratégique pour les petites et moyennes entreprises touristiques[END_REF][START_REF] Hamad | Antecedents of business-to-business e-commerce adoption and its effect on competitive advantage in manufacturing small and medium-sized enterprises: A comparative study of United States of America and Egypt[END_REF]. In fact, even if the pace of change in the ICT sector is important, there are still businesses that are ahead in the use of technology and some that are still followers. Of course the characteristics intra-categories have changed somewhat (for example, nowadays the companies the least developed in terms of IT master IT better than 10 years ago, but the differences between categories remain the same: there are still groups much more in advanced than others).

SMEs are therefore able to compete with larger players thanks to a clear and differentiating positioning, although they have much more limited means. They are then in a position to change consumers' perception using their website which becomes a showcase for their business and is thereby an important component of positioning (even if a website can be expensive, its value can be very important) [START_REF] Morrison | Marketing small hotels on the World Wide Web[END_REF]. In general, ICT has no direct effect on the positioning companies within their market or on company performance. However, through organizational processes and facilitating conditions, ICT will have an impact (both strategic and financial). Moreover, it has also been suggested that the Internet has enhanced the firm's ability to interact with international customers, suppliers and business partners through multiple, inexpensive integrated Internet technologies [START_REF] Glavas | How international entrepreneurship characteristics influence Internet capabilities for the international business processes of the firm[END_REF]. In general, regardless of the size of the company, ICT entails changes in terms of organization and strategic alignment of resources, technology and strategy [START_REF] Henderson | Strategic Alignment: leveraging Information Technology for Transforming Organizations[END_REF]. Thus, according to [START_REF] Bergeron | Comportements stratégiques, choix et gestion des systèmes d'information : contribution à la performance[END_REF], ICT investment strategies must be co-aligned with the strategic behavior of companies in order to have a positive effect on performance (see Questions 4, 5 in our framework Table 1). It is therefore important to take into consideration the managers as they play an important role in this strategic alignment via the diffusion of technology and Internet use, especially in small firms.

The role of small firms managers in the use of ICT

Different characteristics make up the organizational profile of SMEs [START_REF] Marchesnay | La petite entreprise : sortir de l'ignorance[END_REF][START_REF] Julien | Les PME, bilan et perspectives[END_REF]: a strong centralization of management and decision-making, an implicit shortterm strategy and low breakdown of tasks favouring versatility rather than specialization.

These characteristics have been identified in research on small businesses as they highlight not only the importance of the owner-manager in decision-making, especially regarding the choice of information technologies [START_REF] Bergeron | Comportements stratégiques, choix et gestion des systèmes d'information : contribution à la performance[END_REF][START_REF] Cragg | IT alignment and firm performance in small manufacturing firms[END_REF]Monnoyer-Bordered, 2003, Ghobakhloo, M. and[START_REF] Ghobakhloo | The role of owner/manager in adoption of electronic commerce in small businesses: The case of developing countries[END_REF], but also the importance of the managers' perception of the choices made [START_REF] Bergeron | Comportements stratégiques, choix et gestion des systèmes d'information : contribution à la performance[END_REF] and their interest for technology (Aldebert and Guegen, 2009;[START_REF] Spencer | A hierarchical model of technology adoption for small owner-managed travel firms: An organizational decision-making and leadership perspective[END_REF]. Their interest in ICT is driven by practical results on process improvement and performance [START_REF] Uwizeyemungu | Information technology adoption and assimilation: towards an adapted framework for service sector SMEs[END_REF][START_REF] Connolly | Research methods: A guide to using the case study method to explore hospitality information technology phenomena[END_REF].

According to the literature [START_REF] Mehrtens | A model of Internet adoption by SMEs[END_REF][START_REF] Beckinsale | Exploring Internet Adoption Drivers in SMEs[END_REF][START_REF] Glavas | How international entrepreneurship characteristics influence Internet capabilities for the international business processes of the firm[END_REF], the first element explaining the use of ICT concerns the characteristics of the managers and their profile. This is also in line with the decision makers' level of perceived benefits of ICT [START_REF] Pollard | The changing faces of information systems issues in small firms[END_REF][START_REF] Mcdonagh | Euroclicking and the Irish SME: Prepared for ecommerce and the single currency[END_REF], especially when the way of conducting business is considered satisfactory internally. Conversely, psychological barriers and fear of change can be explanatory factors for the low level of use of ICT in SMEs [START_REF] Acar | Use of information and communication technologies by small and medium-sized enterprises (SMEs) in building construction[END_REF] and a fortiori in SMEs (see Question 6 in our framework Table 1).

Finally, lack of skills and technical and managerial expertise [START_REF] Caldeira | Understanding the successful adoption and use of IS/IT in SMEs: an explanation from Portuguese manufacturing industries[END_REF][START_REF] Matthews | ICT Assimitation and SME Expansion[END_REF][START_REF] Dyerson | National Survey of SMEs' Use of IT in Four Sectors[END_REF][START_REF] Qureshil | Information Technology Interventions for Growth and Competitiveness in Micro-Enterprise[END_REF]) may be accentuated by the risk of dependence on publishers of software solutions who are in a privileged position through their technical expertise.

It is noteworthy however that the skills required for implementing ICT may be held by a person other than the manager. In general, the presence of an "expert" user, whether executive or another member of the company is likely to reduce the technological expertise barrier and facilitate the adoption of ICT tools. The availability of staff with technology-oriented skills is often a condition for their adoption in SMEs. The role of stakeholders close to the company, namely the customers and suppliers, as well as other informal influencers (local authorities, professional associations ...) is also among the explanatory factors, [START_REF] Burgess | Managing Information Technology in small business: Challenges and solutions[END_REF]. Media and software solution providers may also encourage managers to equip themselves with ICT.

As a conclusion to this first part, it appears that Internet applications are customer-relationship-management oriented both upstream (search for new customers, offering discounts, etc.) but also downstream (exchange of information, relationship marketing...). It also appears that the Internet can be a tool for strategic positioning, especially when the managers are involved and integrate the tool into their vision. Even if they do not possess the technical and managerial skills specific to ICT, thanks to the perceived benefits and network features, they are in a position to develop a marketing strategy and commercial activities on the Internet (cf.

Table 1).

[Insert Table 1]

Empirical study of four SMEs in the field of hospitality

In this second part, we start with a presentation of the methodology used and the characteristics of SMEs studied (2.1). We then discuss the role of the Internet in a logical servuction 5showing that the SMEs studied use their website not only in a commercial and marketing perspective (2.2) but also to strengthen and develop their competitive positioning, particularly when the manager displays specific characteristics (2.3).

Choice of qualitative methodology and description of the cases studied

We deliberately focused our attention on SMEs that had a website so as to ensure there would be significant use of ICT. In France, although 50% of small businesses have a website, only 12% use it to manage their reservations ("most advanced types" category mentioned in Part 1). To choose the relevant methodology we follow recommendations of [START_REF] Aldebert | TIC et performance : rôle du dirigeant de PME touristiques[END_REF] who argued that as hospitality is an emerging field, it is more useful to use case studies as a qualitative research method to understand complex ICT phenomena.

Multiple case studies, using non-probability purposive selection criteria was deemed to be appropriate for this research methodology. The use of multiple case studies gives a holistic perspective of a complex phenomenon. In-depth 2½-3 hour case interviews were conducted with owner/founders from four French tourism SMEs (fewer than 50 employees) in the Poitou-Charentes and Haute-Savoie regions, two famous tourism areas in France 6 . An interview protocol was developed to guide the interviews (see Frame 1).

FRAME 1: CONTENT OF INTERVIEWS

The semi-structured interviews with the managers were designed to understand the main uses, barriers and impacts of ICT in general and the Internet in particular in these SMEs. Thus, the interviews started by identifying their background (training, experience...) and that of his company (size, creation date, job ...). Then, we addressed the use of the Internet and the incentives that have driven the company to invest in equipment (type of ICT use, implementation strategy, means implemented ...). Finally, we concluded the interviews by seeking to identify the impact that introducing a website had on the activities of the company and the perceived benefits. We sought to fully understand the managers' motivations in relation to the specificities of the tourism sector, in which activity is cyclical and subject to strong competition, and occasionally involves much larger players. 6 The concept of replication logic is used to select cases. Both literal replication and theoretical replication provides a foundation for outcomes through confirmation and analytical generalisability to the research [START_REF] Yin | Case study research: Design and methods[END_REF], through a theory building process [START_REF] Eisenhardt | Building theories from case study research[END_REF]. That is, cases were chosen to give literal replication (predicted similarities in the results) and theoretical replication (contrasting results but for predicted reasons) so as to generate similarities and differences in the results from cases. For example, within the selection criteria all four firms are French SMEs. Similarities in the predicted responses (literal replication) were expected as these cases are all SMEs, at an advanced stage in their use of Internet and from the same tourism industry, while diversity in responses (theoretical replication) was also expected across small firms in comparison with medium firms. Theoretical replication was also expected across each case due to differences in geographical location (sea versus mountain).

Our research question: "Under what conditions may the implementation of a website change the strategic positioning of small accommodation businesses in the tourism sector?" is characterized by both description and interpretation. We opted for a qualitative methodology, which is particularly relevant when the research goal is to explore, understand or generate ideas on why and how. Based on the literature reviewed in the first part of the article we defined an analytical framework that addressed several key issues (marketing dimension on one hand and on the other hand strategic dimensions).

We used this framework to conduct semi-structured interviews with the managers of the SMEs. We reformulated some of the questions several times during the interviews in order to check the consistency of answers. We systematically crossed the interview data in order to identify inferences with our questions and when there was any doubt, we contacted the managers concerned to deepen certain points. We went back to see them at least once after the first interview. Then, requests for clarification were made via email or by phone. Finally, case study methodology has a unique strength in its ability to deal with a full variety of evidence, such as documents, artefacts, observations and interviews [START_REF] Eisenhardt | Building theories from case study research[END_REF][START_REF] Yin | Case study research: Design and methods[END_REF]. In this research marketing material, company documents, company website content and interviews with informal influencers (see Frame 2) were utilised to triangulate the results.

The use of various sources of evidence in case studies allows the investigator to address a broader range of behavioural issues and aims at corroborating the same fact or phenomenon [START_REF] Patton | Qualitative evaluation and research methods[END_REF].

FRAME 2: COMPLEMENTARY INFORMATION & SECONDARY DATA

The interviews were complemented by inner (internal memos, website, etc.) and external secondary data (press clippings). We crossed the managers' statements with the websites functionalities to avoid too much subjectivity, as well as with sales brochures and contributions by informal influencers. We also conducted six interviews with informal influencers and public and private organizations: Tourism advisor for the Chamber of Commerce of La Rochelle, sales manager of the group Hotels & Chalets de Tradition, Commercial Director of RESO 7 France and three members of RESO 7374. These players enabled us to have a broader and better understanding of the importance of the environment for SMEs, thus complementing the managers' statements. We were able to better understand the characteristics and peculiarities of activities in the tourism industry. The informal influencers and various organizations we encountered also made it easier for us to contact managers of SMEs and generated trust during the interviews.

Each of the SMEs had a website as well as other applications (software, CRM, etc.), constituting the businesses' information system, even if it was not necessarily very sophisticated.

The website was the most important tool for all the managers interviewed. Therefore, we focused our attention on the website of small businesses with their features. To make the analysis of the use of websites by SMEs more concise, we present our results in the form of tables featuring the items of our analytical framework. We then comment on these tables showing each of the dimensions (Table 2: Marketing and sales dimensions; Table 3:

Strategic dimension).

Use of the Internet in a servuction approach

The first observation is that the cases studied, which are businesses that are at an advanced stage in their use of Internet, use significant features of their website to develop sales and marketing dimensions. This is in accordance with the literature. The first three items of our interview grid confirmed this, even though there was some disparity between the SMEs.

[Insert Table 2]

The increase in notoriety and customer base acquired via the website (Q1) is cited by all managers. Thus, the main motivation for implementing a website was the possibility for businesses to expand their customer base in France and abroad. Very quickly, beyond the notoriety that their well-controlled interactive presence on Internet can provide, the managers realized the possibilities that a software solution and a dynamic website could offer, especially when translated into several languages. For example, the manager of the hotel The Macchi recently had their website translated into Russian in order to address this customer segment (who started coming to the hotel) and strengthen the establishment's reputation in Russia:

"the website was first perceived as a need to attract customers but soon we perceived a potential for development and an axis of improvement". The manager of the campsite The Cormorants on the Ré Island explained that their website was translated into six languages and generates 30% of direct sales and over 50% of customer contacts (some customers finalize the transaction other than via the website). He also considers that the ratio of foreign customers has increased. Finally, the thalassotherapy The "3 Iles" generates 25% of its sales through its website (purchase of services, products and gift vouchers). These observations show that internationalization is becoming increasingly facilitated through the use of the internet (what [START_REF] Etemad | Internetization as the necessary condition for internationalization in the newly emerging economy[END_REF] called "internatization" 8 ) However, although the use of websites in SMEs makes it possible to implement new business techniques (Q2), these are not aggressive. The websites do not really serve to smooth out business activity because, even though the establishments have two seasonal peaks in winter and summer, the managers do not want to use marketing techniques that "devaluate" the services they offer. We can cite the example of The Cormorans campsite that has a website proposing promotions via banners for last minute offers that can adjust to the competition and booking capabilities. This strategy is not a pricing strategy and, although occupancy statistics provided by the tool could make it possible to propose discounts for the low season, SMEs do not wish to implement this. Yield management is thus used with caution because the keywords are "quality, service, satisfaction" and the managers focus on their image first:

"tools should be in the service of our values (friendliness, etc.)" Hotel The Macchi. This result differs in part from the results of [START_REF] Dyerson | National Survey of SMEs' Use of IT in Four Sectors[END_REF] which suggested that the main motivation lay more in cost reduction than in value creation.

Our case study also confirms the impact of certain website features on customer loyalty (Q3).

Our SMEs took a step further than those studied by [START_REF] Lucchetti | The Adoption of ICT among SMEs: Evidence from an Italian Survey[END_REF]. For example, in order to personalize the relationship with customers, Le Dahu created a private club where customers can exchange photos, ideas, etc. with other customers after their stay.

This club tells the life of the hotel throughout the year and maintains a link. The manager of this hotel also created a customer database (one sheet per customer with a hundred features):

"I think continuously at improving services via technology" Hotel Le Dahu. This database, regularly updated and coupled to a CRM, enables them to send personalized offers and therefore strengthen customer loyalty (50% of clients return regularly to Le Dahu). The manager of the Cormorants campsite on the Ré Island also reported that regular customers visit their website to keep in touch with new developments (a newsletter is also being developed). These different features proposed via the websites are designed to consolidate customer loyalty and increase sales for SMEs as these are very sensitive to small variations in their economic activity and cash flow. It also helps to maintain a personalized link that is important given the close relationship established between the managers and their clients. Some customers of both hotels were even considered as family.

However, the SMEs of our case studies faced difficulties that have already been identified in the literature [START_REF] Montazemi | How They Manage IT: SMES in Canada and the U.S[END_REF]: firstly, they must absorb the cost of software or building a dynamic and interactive website, including for example Content Management System... Especially regarding the website, the cost of referencing on search engines is very important (3 K€ per year for operating the website for The Cormorants campsite, 18 K€ per year for referencing on Google for the Thalassotherapy establishment). However, this cost is fully integrated into the budget and is regarded as a necessary expense to develop business.

Internet used to strengthen the competitive position of small businesses, in particular through the role of the managers

The work of [START_REF] Morrison | Marketing small hotels on the World Wide Web[END_REF] has shown that the advantages of ICT and Internet can be beneficial from a strategic point of view: cost reduction, strategy of differentiation, diversification, search for new income sources, or improving the performance of the organization. In the SMEs studied here, from a certain level of expertise, on the sales and marketing dimensions, the websites can lead to improved competitive positioning. The website helps to enhance differentiation but also the differential value perceived by the customer [START_REF] Aldebert | Technologies de l'information et de la communication : un outil stratégique pour les petites et moyennes entreprises touristiques[END_REF]. This evolution in strategic positioning is not necessarily originally intended by the management, but if the perceived benefits are significant, they will gradually integrate the website as a component of the strategy. Thus, the Macchi Hotel gradually added services to anticipate requests from foreign customers who were acquired via the Internet: "if you stop providing new things, different experiences, you're dead !" Through a ripple effect, the website pushed the strategic positioning towards the upscale. Managers therefore have a role in the decision of aligning their organization to adopt a more qualitative competitive positioning. Another example is the case of Les Cormorants campsite. The creation of a website accompanied the upscale move of the campsite, which improved by proposing more qualitative services (Internet connection for customers, booking facilities, new mobile homes, etc).

The manager told us during the interview: "my firm is one of the first movers in the ICT use regarding competitors and we have to keep this position". For the thalassotherapy resort, the website helped strengthen the establishment thanks to a better visibility compared to its local competitors. The manager reported that the website and its functionalities allowed the implementation of a differentiation strategy especially "for offers considered as static."

[Insert Table 3]

Our observations show that the use of ICT in general improves the management of SMEs and requires changes in their organization, which starts with the back office using data from customers and prospects. For example, thanks to the synchronization of their various internal databases and access to data in real time, all of the businesses studied observed an improvement in their management. The data flowing through the website can be used for back office management, leading to an alignment of the IS components and optimised management (Q4).

This fact is obvious for the two hotels: "I [the manager] do not regret the time when, every night I had to "synchronize" the data manually for each computer"(Hotel Le Dahu). They also reported time-saving and increased reliability thanks to the online booking feature. However, connecting different systems together requires a more significant investment and can entail a certain level of complexity, involving higher risks for managers. Implementing electronic solutions for the back and front office processes is often a gradual undertaking. Some establishments such as Les Cormorans campsite have taken a step further and have started to interconnect their systems in order to introduce remote management. The alignment between IS and the organization is based on a very pragmatic approach and a financial dimension as regarding the firm's size, as confirmed by the manager of Les Cormorans campsite ("We have a real payback with Internet website").

Consequently, the competitive advantage gained by using websites enables small businesses to strengthen their differentiation (offering new services, targeting their sales offerings) and create close or even privileged relationships with their client [START_REF] Bellaaj | Effets modérateurs des capacités complémentaires dans le e-commerce. Application au contexte des PME[END_REF]. This is a commercial and marketing approach that is favoured by managers and which often contributes to a quality-oriented and more upscale positioning (Q5). This result is also in line with [START_REF] Qureshil | Information Technology Interventions for Growth and Competitiveness in Micro-Enterprise[END_REF] and confirms the idea that a website is a showcase for companies but also a strategic positioning element in its own right. Our research shows that small businesses can use their website as a tool for differentiation with respect to their competitors, including, but not limited to, dimensions related to relationships with their customers.

Success in the use of a website also depends on a carefully thought out choice of technology with respect to the positioning of the company and the nature of its activities. The features proposed on the website must also stem from careful consideration as to the core business (and/or mission) because this will determine the services that should be offered and will influence consumers and prospects' perception. Thus, hotel managers must determine whether they offer accommodation, create emotions, etc. Various interviews with managers and informal influencers also mentioned the importance of monitoring the features and services in order to adapt them where necessary, taking into account not only changes in consumer behaviour but also the progressive establishment of competitors. For example, the two SMEs in the Charente-Maritime region planned to develop geolocation applications for mobile phones connected with their websites. This will enable them to offer services that are typically proposed by larger or more upscale establishments. The manager of Les Cormorants campsite even considers a complementarity approach with local shops in order to influence the prospects' choices.

The managers are not necessarily technophiles in that they do not necessarily buy the latest technology, etc. In our cases, they are in their forties or older and have not received any IT training. Rather, it is their ability to perceive the potential offered by technology as well as their innovative and risk-taking profile that seems to emerge (Q6). The managers we interviewed monitor the websites of their direct competitors (design, services, and occupancy); one of them, the manager of the thalassotherapy resort, used Netvibes9 as a web monitoring tool and told us: "If I could, I would bet everything on the internet to disseminate as much information as possible". This finding is in line with [START_REF] Mehrtens | A model of Internet adoption by SMEs[END_REF] and [START_REF] Spencer | A hierarchical model of technology adoption for small owner-managed travel firms: An organizational decision-making and leadership perspective[END_REF]: despite a lack of knowledge to pilot the implementation of software or run a website, managers will endeavour to master the tools by spending time to understand how these work (eg. Hotel Le Dahu) or in a first step will hire personnel with the necessary technical skills (eg. Hotel Le Macchi). The manager of the thalassotherapy resort confirms that "Recruiting a yield manager is a priority to develop real pricing strategy with the website". This corresponds to an alignment of the establishment and resources on ICT.

As for external stakeholders (suppliers, customers, etc.), these cannot be considered as indirect instigators: it seems as if, taken separately, these actors had no actual weight, however their aggregation is a motivator for ICT use. Informal influencers (Chambers of Commerce, clubs, etc.) are viewed as actors on which the manager can rely for guidance or rather to confirm choices. Thus, the sales manager at HCT can don many hats: "I am at the same time the initiator, a provider of technological solutions, an advisor. My role is to act as a "contact person" that enables hotel managers to rely on someone if needed. My presence reassures them and gives them confidence when it comes to taking new technology on board". His awareness-enhancing and monitoring role with respect to ICT is very important for managers of small businesses that have limited time and resources. In addition, informal influencers play an important liaising role between the businesses and the technical service providers that propose solutions. However, this proactive role refers back to the question of differentiation and one can ask whether in the long term, informal influencers might not cancel the differentiation asset of the "early adopters" and thus their competitive advantage by trying to disseminate ICT to all companies in the sector. However, triggers (strongly driven by informal influencers) are intimately interrelated and converge on the manager who seems to play a central role in the end.

Conclusion

Our research confirms that tourism and hospitality industries have adopted ICT to enhance operational efficiency, and most importantly to improve service quality and customer experience [START_REF] Law | Information technology applications in hospitality and tourism: A review of publications from 2005 to 2007[END_REF]. For small businesses in the tourist accommodation sector, implementing and using ICT in general and the Internet in particular makes it possible to achieve better visibility with their customers. It also encourages them to better structure their offering (improved readability) and better meet their clients' expectations: a virtuous circle is established. Gradually, the optimization of the sales and marketing dimensions encourages them to adopt a more global strategic thinking and eventually enables them to clarify their competitive positioning. This represents an asset in terms of competition but also with respect to potential partners, making it easier for example to obtain funding and/or support. The performance of tourism firms can be improved by introducing new ICT, but positive impacts are especially obtained where ICT is combined with other strategic and managerial measures such as competence building and HRM [START_REF] Blake | CGE Tourism Analysis and Policy Modelling[END_REF]. Managers have to deal with future ICT-related issues by integrating them into the company's strategic management and business mission. A way to achieve this goal is to constantly upgrade the ICT knowledge and skills of staff, as in this way overall technical proficiency can be assured. [START_REF] Law | Information technology applications in hospitality and tourism: A review of publications from 2005 to 2007[END_REF].

In terms of managerial implications, our research showed that:

-The Internet can be used as a means to innovate in the delivery of services thanks to upscale differentiation or niche targeting (personalization of offering, strengthening customer relationship, improving visibility versus larger competitors...);

-The website is the first representation of the positioning of the SME, allowing it to compete with larger companies, provided they offer the most advanced features; -It is not necessary for the managers to be "early adopters," what matters is their awareness with respect to ICT and the importance of implementing such technology.

Stakeholders, partners and personal/professional network can play an instigator or "support" for the managers: this environment helps them overcome the costs (human, financial, etc.) related to the installation and maintenance of a website in particular and ICT in general;

-Finally, technology must serve a strategy, especially for small companies that have limited resources and cannot afford to go wrong. ICT alone do not create a competitive advantage but when combined with other resources they serve to support strategy and create value.

Our research naturally has limitations inherent to the qualitative approach, including the small number of respondents, which do not allow drawing general conclusions for small businesses in. However, in the Charente Maritime and Haute-Savoie regions of France, small businesses in the tourism industry that are advanced in terms of ICT use are few, which made it somewhat difficult to identifying and obtaining their agreement for an interview. Finally, the choice of the tourism sector can be considered a limitation because it is a sector that is ahead of others in terms of ICT adoption as mentioned in our introduction.

This exploratory study opens several perspectives. For instance, a quantitative study could be undertaken on a larger sample of small businesses, extending the scope to include three other types defined by [START_REF] Daniel | Adoption of e-commerce by SMEs in the UK, towards a stage model[END_REF] in order to empirically test our results. The contribution of quantitative data would allow us to measure the contribution of ICT to the development of companies based on standard indicators (turnover, size ...). Finally, it could prove interesting to rethink existing typologies and to test whether it is necessary to define additional categories in connection with innovation linked to ICT and the Internet (LBS 10 applications, non-contact payment, social networks, etc.) in order to better identify the businesses strategic choices, including small enterprises.

10 LBS : Location Based Services Scale from the interviews and verbatims (the number of + indicates the impacts of the website) for Q4 and Q5: + : weak impact of the website on processes and organization (website as a tool among others) ++ : middle impact of the website on processes and organization (changes in tasks nature, adjustment of processes or tools, etc.) +++ : high impact of the website on processes and organization (creation of new tools, elaboration of new marketing strategy, development or add new skills in the organization, etc.) For the question 6, the scale is based on the traditional profile of innovation early adopter (technical skills, relationships in their networks, adoption of new technologies, etc). Early adopters (+++) appropriate innovations faster than other people and they are also more involve in product development as compared to the early majority (++). We precise that small firms are not innovators in the Rogers'classification (1962) but early adopters. [START_REF] Aldebert | TIC et performance : rôle du dirigeant de PME touristiques[END_REF] demonstrate in a recent study that small business manager's interest and perception promotes a great use of ICT in his firm.

Table 1 : Analytical framework used for our case studies 1

 1 Does the website allow increasing the visibility of SMEs and the customer base? 2 Does the website allow implementing new marketing techniques? 3 Does the website allow retaining and maintaining the customer base? 4 Does the website require alignment / modification of the organization? 5 Does the website allow to initiate a reflection on the creation of value and improve competitive positioning? 6 Does the manager play a major role in the creation and use of the website, particularly in terms of profile and network?

Table 2 : Development of sales and marketing dimension via websites

 2 Scale from the interviews and verbatims (the number of + indicates the importance of the website in the marketing strategy and its place compared to others traditional marketing tools): + : website considered as a simple tool regarding tools and techniques used by SMEs ++ : website considered as a complementary tool regarding tools and techniques used by SMEs +++ : website considered as a new and strategic tool to create added value for the SMEs

			Les 3 Iles	Les Cor-mo-	Hôtel	Hôtel
				rans	Le Dahu	Le Macchi
	1	Does the website allow in-
		creasing of the visibility of SMEs and the customer	++	+++	+++	+++
		base?
	2	Does the website allow the
		implementation of new mar-	++	+	++	+
		keting techniques?
	3	Does the website allow to
		retain and maintain the cus-	++	++	+++	++
		tomer base?

Table 3 : Evolution of strategic positioning

 3

			Les 3 Iles	Les	Hôtel	Hôtel
				Cormorans	Le Dahu	Le Macchi
	4	Does the website require
		alignment / modification of	+	++	+++	++
		the organization?
	5	Does the website allow a re-
		flection on the creation of value and improve competi-	++	+++	++	++
		tive positioning?
	6	Does the manager play a major role in the creation and	+++	+++	+++	+++

Neologism used by[START_REF] Eiglier | Servuction, le marketing des services, Collection Stratégie et management[END_REF] to describe the production of a service.

"Internetization" refers to the process of increasing adoption, diffusion, and deployment of internet-based technologies and processes that increasingly serve as the back bone of internationalization.

Netvibes may be defined as a free web portal customizable by each user based on their choice of RSS, API, etc.

Note de l'auteur

The contribution of the Internet to the strategic positioning of Small Businesses in the tourism industry


Christian Tanguay

Introduction

En juin 2017, lorsque j'ai rencontré les membres des archives LGBT du centre LGBT de Montréal, j'ai mesuré à quel point la France était en retard sur ce sujet. À écouter ces militant.e.s, il apparaissait évident que la question des « droits culturels » passait aussi par un droit d'accès à la culture et, pour ce qui concerne les contre-cultures ou les cultures subalternes, un droit à les répertorier, les archiver, les collecter. Les droits culturels résonnent alors comme un rempart à l'oubli. En France la question des archives LGBT -comme celle du féminisme d'ailleurs -bat son plein depuis quelques années et, depuis l'été 2017, cette question commence même à trouver un écho du côté des prises de positions politiques. La mairie de Paris, ou tout du moins certain.e.s élu.e.s, milite à côté des associations pour qu'un lieu puisse enfin accueillir ces archives. Ce sont en réalité des milliers de coupures de presses, de vidéos, de témoignages, de lettres, d'histoires individuelles ou associatives qui se perdent au fur et à mesure que leurs propriétaires décèdent ou, le plus souvent, s'en débarrassent, faute de lieu pour les héberger. Et, à chaque fois, c'est quelque part une partie de la culture LGBTI qui s'efface ou se disperse. Les droits culturels se lient à cet égard aux questions mémorielles : pour se rappeler du passé, encore faut-il pouvoir matériellement l'accueillir.

Il y a de cela plus de vingt ans, un jeune garçon se questionnait sur l'attirance qu'il pouvait éprouver pour l'un de ses camarades de classe. Le tout dans le plus grand des silences puisqu'il ne s'agissait, à ce moment-là, que d'interrogations intérieures. Pourquoi alarmer tout son entourage, se disait-il ? Qui sait, peut-être se trompait-il et que cette attirance était sans lendemain. Mais une chose était certaine pour lui : depuis le premier jour où il avait mis les pieds dans une cour de récréation, le fait de paraître homo était un sujet d'invectives et de moqueries par ses pairs. Après avoir tenté d'inhiber tout sentiment, de s'être censuré au maximum, il a alors entrepris une longue quête dans le silence le plus complet puisqu'il n'y avait pas de lieu à l'époque où il pouvait se sentir suffisamment en sécurité pour se questionner et en discuter dans les établissements d'enseignement qu'il a fréquentés. Ce n'est qu'au cégep, soit à la dixième année de scolarisation dans le système québécois, qu'il a pu débuter ses réflexions et bénéficier enfin d'un peu plus de liberté. En feuilletant son manuel de philosophie, son regard s'est porté sur une image présentant des militants homosexuels défendant leurs droits civils. Ce n'était qu'une seule photo, accompagnée d'un maigre texte de trois lignes, mais ce moment a bouleversé la vie du jeune homme.

Cette histoire, c'est la mienne. C'est le parcours d'un jeune homme qui deviendra le directeur général du Centre communautaire LGBTQ+ de Montréal et de la Bibliothèque À Livres Ouverts (BALO), une bibliothèque spécialisée sur la thématique de la diversité sexuelle et l'expression du genre. Ce parcours atypique est pavé de rencontres fortuites et d'un travail acharné, lequel nous a menés à recevoir en février 2017 le prestigieux prix Newlen-Symons de l'American Library Association (ALA) à Chicago pour nos services en bibliothèque auprès des communautés LGBTQ+.

Prenons le temps de décrire la bibliothèque et sa collection. Il s'agit d'un espace, d'un lieu qui, d'une part, témoigne de notre existence et permet de croire en la possibilité d'un monde meilleur et plus inclusif, puis, d'autre part, permet de célébrer, de se rassembler, et s'engage, avant tout, dans la protection et dans la mise en valeur des richesses du patrimoine culturel LGBTQ+. Le mariage d'un centre communautaire et d'une bibliothèque dans un seul et même espace constitue une force. La Bibliothèque À Livres Ouverts (BALO) contient plus de 20 000 documents de tous les genres (des essais, des romans, des BD, de la poésie, des périodiques et plus encore) et de toutes les époques, qui célèbrent la diversité des orientations sexuelles et des expressions du genre. La richesse de cette collection incroyable ne nous empêche cependant pas de poursuivre le travail, en incorporant constamment de nouveaux titres pour tenir compte des avancées tant sur le plan humain que sur les savoirs scientifiques.

Revenons à l'histoire de ce jeune homme dans un établissement d'enseignement public, sensible à la notion du droit à la culture. Ce jeune homme a pris conscience, bien des années plus tard, qu'il vivait dans un environnement binaire et hétéronormatif. Tout au long de son parcours, il lui a été dit qu'il était un homme, qu'il vivrait avec une femme, qu'il aurait fort probablement des enfants et qu'il pourrait à son tour reproduire ce modèle souvent présenté comme « le cycle normal de la vie ». Inconsciemment, toutes les histoires présentées au cours du programme pédagogique s'ancraient dans cette logique. L'ensemble des oeuvres qui lui a été donné à lire renforçait cette quasi unique conception du monde, régi par un modèle hégémonique hétérosexuel à l'intérieur duquel la seule division possible était la division entre les hommes et les femmes. De façon ponctuelle, certes, il avait pu voir aux téléjournaux, dans une émission de variétés ou dans une série télé des personnes non hétérosexuelles ou non cisgenres, mais cela restait anecdotique.

Son prochain rendez-vous fut la bibliothèque de son cégep (équivalent québécois d'un lycée), mais arrêtons-nous un instant au mot-clé de recherche « homosexualité » au moment d'interroger un catalogue d'oeuvres. Le concept de l'homosexualité en soi réfère à la sexualité et focalise indirectement sur le corps et sur la génitalité. Rappelons-nous que l'histoire n'est pas neutre. À l'époque où Dewey a ajouté l'homosexualité dans son système de classification décimale, il s'agissait alors d'une maladie et d'un comportement criminel. Il y a eu aussi durant une longue période un mouvement d'autocensure et de mise à l'index des ouvrages traitant des questions LGBTQ+, où il n'était possible de parler de l'homosexualité que dans un contexte tragique, puisque celle-ci était jugée contre-nature et faisait l'objet de sévères réprimandes par l'État et par les religions. S'il y avait un nombre très limité d'oeuvres à thématique LGBTQ+ qui peuplaient les tablettes de ce cégep, la grande majorité de celles-ci traitait du sujet comme étant quelque chose de foncièrement négatif ou de tragique. Rien pour rassurer ce jeune homme donc. Si le nombre d'oeuvres que nous pourrions qualifier de problématiques quant au traitement du sujet est aujourd'hui moins grand au sein des établissements d'enseignement, un retard persiste toutefois dans le développement des collections LGBTQ+ et ce, dans la majorité des lieux, tant dans les librairies généralistes que dans les bibliothèques publiques ou les établissements d'enseignement.

Malgré le travail des bibliothécaires, les investissements culturels peinent encore aujourd'hui à rétablir l'équité dans ces lieux de démocratie du savoir. Une simple recherche en surface en consultant les catalogues de cinq établissements d'enseignement collégial au Québec nous permet de fournir quelques proportions quant à l'espace qu'occupent les oeuvres touchant aux thématiques LGBTQ+. En général, nous y retrouvons moins de 150 titres qui répondent au terme de recherche « homosexualité » alors que la collection complète d'une bibliothèque de cégep comporte en moyenne plus de 80 000 documents. De plus, seulement 30 titres se réfèrent au concept « lesbienne », 10 titres apparaissent sous le terme de recherche « trans et moins de 5 sous le terme « asexualité » . Cette recherche sommaire s'est limitée à l'interrogation du catalogue web de cinq établissements d'enseignement collégial : deux dans la région métropolitaine, deux en région éloignée des grands centres et un dans une ville de taille moyenne. Le nombre de résultats chute drastiquement si nous limitons les résultats aux livres publiés lors des cinq dernières années. Loin de vouloir accuser les bibliothécaires, notre expérience de terrain nous permet de réaliser que les difficultés rencontrées ont des origines diverses : un nombre restreint de publications en français, qui s'explique notamment par des enjeux de traduction, un tirage limité de documents en français qui se retrouvent rapidement épuisés, une difficulté à mettre les oeuvres LGBTQ+ de l'avant au moment de la mise en marché, toutes ces difficultés sont autant de facteurs permettant de contextualiser les défis rencontrés.

L'enjeu de l'accès aux oeuvres à thématique LGBTQ+ est vraiment un élément d'intérêt central. Si l'on estime la population de personnes homosexuelles, bisexuelles ou ayant des relations sexuelles avec une personne du même sexe biologique à 10 % de la population, nous nous retrouvons dans une situation où l'hétérosexisme domine totalement la sphère du milieu documentaire. Dans le cas d'un établissement d'enseignement collégial type, nous nous retrouvons dans une situation où moins de 0,2 % des livres traitent de cette diversité qui compose une partie importante du tissu social de notre société. De surcroît, cette situation d'invisibilité renvoie aussi le message d'un manque d'intérêt, qui contribue au sentiment de marginalité et de honte dont les communautés LGBTQ+ sont marquées. Elle prive aussi les proches et parents des personnes LGBTQ+ de documents et ressources qui leur permettraient de mieux comprendre leurs cheminements et lutter contre les stéréotypes et préjugés.

Dans ce contexte, la pertinence d'une bibliothèque qui se spécialise sur les thématiques LGBTQ+ est d'autant plus grande et importante. Nous recevons à la BALO par le canal du don de nombreux documents rares, voire épuisés. Au moment de leur acquisition, la valeur économique de ces documents est souvent plus élevée en raison du faible tirage ou des coûts de production. De plus, plusieurs petites maisons d'édition en France ne sont pas distribuées au Québec. Malgré tout, nous parvenons à recueillir certains titres grâce à la générosité des éditrices/éditeurs, auteur.e.s ou de donatrices/donateurs. La présence de chercheuses/chercheurs et de grandes universités sur le territoire montréalais, tant francophones qu'anglophones, stimulent la présence d'oeuvres scientifiques qui trouvent le chemin de notre institution. Nous nous retrouvons ainsi dans une situation bien particulière : un très grand pan de la mémoire de nos communautés se retrouve à l'intérieur de nos murs, mais le devoir de diffusion de la culture va bien au-delà de ceux-ci pour s'étendre aux quatre coins de la province.

Le portrait que nous faisons de la situation démontre une situation d'iniquité, fruit d'un contexte bien particulier. Le patrimoine documentaire des minorités sexuelles et de genres est le résultat d'une lente évolution où nous avons dû prouver que nous étions des citoyennes/citoyens à part entière et non des criminel.le.s, des personnes en possession de leurs moyens et non des malades mentales/mentaux, des gens sensés et non des âmes damnées. Suivant les avancées en termes de droits juridiques, nous sommes arrivé.e.s au moment où nous, comme communautés, avons la responsabilité de nous assurer que notre mémoire, notre imaginaire et notre histoire puissent rejoindre le plus grand nombre. C'est une question d'équité sociale et de réparation du préjudice historique subi, un devoir de réparation. Autant des bibliothèques spécialisées comme la nôtre sont importantes, autant l'accès à un ensemble documentaire riche et adapté aux différents types d'établissements, qu'ils soient des bibliothèques nationales, municipales, en milieu rural ou d'établissements scolaires, est essentiel.

Au moment d'écrire cet article, la première ébauche de la révision de la politique culturelle québécoise a été dévoilée. Malgré les présentations que nous (la BALO) avons faites auprès du ministère de la Culture et des Communications de Québec, nous avons constaté que cette situation d'invisibilité risquait fortement de se poursuivre. À moins d'un revirement de situation, le Québec risque de manquer ce rendez-vous. Dans un contexte où la classe politique du Québec se désole des politiques homophobes de la Russie et de la répression en Tchétchénie envers les personnes LGBTQ+, cette dernière a la possibilité de poser un geste réel et concret, de donner l'exemple d'une démarche d'ouverture en permettant à l'ensemble des bibliothèques d'être des lieux d'inclusion pour des personnes de la diversité. Plus qu'une possibilité, il s'agit même d'un devoir que cette classe politique a envers nous. En tant que citoyennes/citoyens, nous avons le droit d'avoir accès à des oeuvres documentaires et fictives qui nous permettent de nous instruire, de connaître notre histoire, de nous divertir, de nous émanciper.

Pour ce jeune homme et pour toutes les autres personnes des communautés LGBTQ+ présentes et futures, nous n'avons qu'un seul souhait, c'est qu'ils/qu'elles/qu'illes puissent se sentir le/la bienvenu.e au sein d'un ensemble de bibliothèques auxquelles ils/elles/illes auront accès tout au long de leur vie. Que dans celles-ci, ils/elles/illes puissent trouver les ouvrages qui sauront les transformer et leur permettre de se réaliser en tant qu'individu à part entière tant sur le plan personnel que social.

« POUR CE JEUNE HOMME ET POUR TOUTES LES AUTRES PERSONNES DES COMMUNAUTES LGBTQ+ PRESENTES ET FUTURES, NOUS N'AVONS

  QU'UN SEUL SOUHAIT, C'EST QU'ILS/QU'ELLES/QU'ILLES PUISSENT SE SENTIR LE/LA BIENVENU.E AU SEIN D'UN ENSEMBLE DE BIBLIOTHEQUES AUXQUELLES ILS/ELLES/ILLES AURONT ACCES TOUT AU LONG DE LEUR VIE. QUE DANS CELLES-CI, ILS/ELLES/ILLES PUISSENT TROUVER LES OUVRAGES QUI SAURONT LES TRANSFORMER ET LEUR PERMETTRE DE SE REALISER EN TANT QU'INDIVIDU A PART ENTIERE TANT SUR LE PLAN PERSONNEL QUE SOCIAL. »

	CHRISTIAN TANGUAY

Cet article est paru dans le premier HS des Cahiers de la LCD (« Droits culturels et lutte contre les discriminations »,

2018), pp : 110-115. 2 Christian Tanguay est responsable du centre communautaire LGBT de Montréal.


Namig J Guliyev
email: njguliyev@yahoo.com

Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary conditions

Inverse problems of recovering the coefficients of Sturm-Liouville problems with the eigenvalue parameter linearly contained in one of the boundary conditions are studied: 1) from the sequences of eigenvalues and norming constants; 2) from two spectra.

Necessary and sufficient conditions for the solvability of these inverse problems are obtained.

Introduction

In this paper we consider inverse eigenvalue problems for the equation ℓy := -y ′′ (x) + q(x)y(x) = λy(x), x ∈ [0, π]

(1.1)

with the boundary conditions y ′ (0) -hy(0) = 0, (1.2)

λ(y ′ (π) + Hy(π)) = H 1 y ′ (π) + H 2 y(π), (1.3)
where q(x) ∈ L 2 (0, π) is a real-valued function, h, H, H 1 , H 2 ∈ R and

ρ := HH 1 -H 2 > 0. (1.4)
Let us denote this problem by P(q, h, H, H 1 , H 2 ). Problems with the eigenvalue parameter linearly contained in the boundary conditions have been studied extensively. In [START_REF] Fulton | Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions[END_REF][START_REF] Walter | Regular eigenvalue problems with eigenvalue parameter in the boundary condition[END_REF] an operator-theoretic formulation of the problems of the form (1.1)-(1.3) has been given. It has been shown that one can associate a self-adjoint operator in adequate Hilbert space with such problems whenever the condition (1.4) holds. Oscillation and comparison results have been obtained in [START_REF] Binding | Oscillation theory for indefinite Sturm-Liouville problems with eigenparameter-dependent boundary conditions[END_REF][START_REF] Binding | Sturm-Liouville problems with eigenparameter dependent boundary conditions[END_REF][START_REF] Kapustin | Oscillation properties of solutions to a nonselfadjoint spectral problem with spectral parameter in the boundary condition Differential[END_REF]. Basis properties and eigenfunction expansions have been considered in [START_REF] Kapustin | Spectral problems with the spectral parameter in the boundary condition Differential[END_REF][START_REF] Kapustin | A remark on the convergence problem for spectral expansions corresponding to a classical problem with spectral parameter in the boundary condition Differential[END_REF][START_REF] Kerimov | On the basis properties of one spectral problem with a spectral parameter in a boundary condition Siberian[END_REF][START_REF] Wray | Absolutely convergent expansions associated with a boundary-value problem with the eigenvalue parameter contained in one boundary condition[END_REF]. Problems with various singularities have been analyzed in [START_REF] Altinisik | Eigenvalues and eigenfunctions of discontinuous Sturm-Liouville problems with eigenparameter-dependent boundary conditions[END_REF][START_REF] Fulton | Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions[END_REF]. In the case ρ < 0 the problem (1.1)-(1.3) can be associated with a self-adjoint operator in Pontryagin space and not all eigenvalues are necessarily real (see [START_REF] Amara | Asymptotics of eigenvalues and eigenfunctions of the Sturm-Liouville problem with a small parameter and a spectral parameter in the boundary condition Math[END_REF][START_REF] Ben | The Sturm-Liouville problem with physical and spectral parameters in the boundary condition Math[END_REF][START_REF] Binding | Application of two parameter eigencurves to Sturm-Liouville problems with eigenparameter-dependent boundary conditions[END_REF]).

Inverse problems involving linear dependence on the spectral parameter in the boundary conditions have also been investigated. In [START_REF] Mamedov | Determination of a second order differential equation with respect to two spectra with a spectral parameter entering into the boundary conditions (Russian) Izv[END_REF] sufficient conditions for two sequences of real numbers to be the spectra of the problems P(q, h, 0, H 1 , H 2 ) and P(q, h, 0, H 1 , H 2 ) are provided, where H 1 H 2 = H 1 H 2 , H 2 , H 2 > 0 and H 1 = H 1 . Various uniqueness theorems are proved in [START_REF] Binding | Inverse spectral problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions[END_REF][START_REF] Browne | A uniqueness theorem for inverse eigenparameter dependent Sturm-Liouville problems[END_REF][START_REF] Mccarthy | Eigenparameter dependent inverse Sturm-Liouville problems[END_REF]26]. We shall provide in this paper another proof of the unique solvability. Numerical techniques are discussed in [START_REF] Mccarthy | Eigenparameter dependent inverse Sturm-Liouville problems[END_REF]. In [START_REF] Binding | A hierarchy of Sturm-Liouville problems Math[END_REF][START_REF] Binding | Transformations between Sturm-Liouville problems with eigenvalue dependent and independent boundary conditions[END_REF] so called "almost isospectral" transformations (i.e., transformations preserving all but finitely many eigenvalues) are studied and using these transformations many direct and inverse results for problems with the spectral parameter in one of the boundary conditions are derived from those for classical Sturm-Liouville problems.

The present paper is devoted to the study of inverse problems by (i) one spectrum and a sequence of norming constants; (ii) two spectra. We obtain necessary conditions for eigenvalues and norming constants in Section 2. In Section 3 we prove that the kernel of the operator transforming the function cos √ λx to the corresponding solution of the equation (1.1) satisfies the Gel'fand-Levitan-Marchenko type integral equation. In Section 4 we show that the boundary-value problem (1.1)-(1.3) can be uniquely determined from its spectrum and norming constants. Reconstruction of the coefficients of the problem from these spectral characteristics is realized in Section 5 using the method analogous to that of Gel'fand and Levitan [START_REF] Gel | On the determination of a differential equation from its spectral function[END_REF] (see also [START_REF] Marchenko | The Sturm-Liouville operators and their applications (Russian) (Kiev: Naukova Dumka) English transl[END_REF][START_REF] Yurko | Inverse spectral problems and their applications (Russian) (Saratov: Izd-vo Saratovskogo pedinstituta) See also: Freiling G and Yurko V A 2001 Inverse Sturm-Liouville problems and their applications[END_REF]).

Sections 6 and 7 are devoted to the study of inverse problems by two spectra. In Section 6 we consider the problems P(q, h, H, H 1 , H 2 ) and P(q, h, H, H 1 , H 2 ) with h = h. It's proved that the eigenvalues of two such problems interlace and norming constants of first problem are expressed by these eigenvalues. We use these expressions in Section 7 to solve the inverse problem by two spectra, similarly to the work of Gasymov and Levitan [START_REF] Levitan | Determination of a differential equation by two of its spectra (Russian) Uspekhi Mat[END_REF] for the classical Sturm-Liouville problems.

Preliminaries

Let ϕ(x, λ) and ψ(x, λ) be the solutions of (1.1) satisfying the initial conditions

ϕ(0, λ) = 1, ϕ ′ (0, λ) = h, ψ(π, λ) = -λ + H 1 , ψ ′ (π, λ) = λH -H 2 .
(2.1)

We define

χ(λ) := ϕ(x, λ)ψ ′ (x, λ) -ϕ ′ (x, λ)ψ(x, λ), which is independent of x ∈ [0, π].
The function χ(λ) is entire and has zeros at the eigenvalues of the problem (1.1)- (1.3). The set of eigenvalues is countable, consists of real numbers and for each eigenvalue λ n there exists such a number k n that

ψ(x, λ n ) = k n ϕ(x, λ n ), k n = 0. (2.2)
In the Hilbert space H = L 2 (0, π) ⊕ C let an inner product be defined by

(F, G) := π 0 F 1 (x)G 1 (x)dx + 1 ρ F 2 G 2 for F = F 1 (x) F 2 , G = G 1 (x) G 2 ∈ H.
We define operator (see [START_REF] Fulton | Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions[END_REF])

A(F ) := -F ′′ 1 (x) + q(x)F 1 (x) H 1 F ′ 1 (π) + H 2 F 1 (π) with D(A) = {F ∈ H|F 1 (x), F ′ 1 (x) ∈ A C [0, π], ℓF 1 ∈ L 2 (0, π), F ′ 1 (0) -hF 1 (0) = 0, F 2 = F ′ 1 (π) + HF 1 (π)} . Then Φ n := ϕ(x, λ n ) ϕ ′ (π, λ n ) + Hϕ(π, λ n ) are orthogonal eigenelements of A: (Φ n , Φ m ) = 0, n = m.
We also define norming constants by

γ n := Φ n 2 = π 0 ϕ 2 (x, λ n )dx + (ϕ ′ (π, λ n ) + Hϕ(π, λ n )) 2 ρ .
The numbers {λ n , γ n } n≥0 are called the spectral data of the problem (1.1)-(1.3).

Lemma 2.1. The following equality holds:

χ(λ n ) = k n γ n , (2.3)
where χ(λ) = d dλ χ(λ). Proof. Using (2.1) and (2.2) in the equality

(λ -λ n ) π 0 ψ(x, λ)ϕ(x, λ n )dx = (ψ(x, λ)ϕ ′ (x, λ n ) -ψ ′ (x, λ)ϕ(x, λ n ))| π 0 we obtain: χ(λ) λ -λ n = π 0 ψ(x, λ)ϕ(x, λ n )dx + ρ k n .
As λ → λ n this equality leads to (2.3).

Remark. Simplicity of the eigenvalues of (1.1)-(1.3) also follows from this lemma.

Theorem 2.1. Following asymptotics hold:

s n := λ n = n -1 + ω nπ + ζ n n , {ζ n } ∈ l 2 , (2.4
)

γ n = π 2 + ζ ′ n n , {ζ ′ n } ∈ l 2 , (2.5)
where

ω = h + H + 1 2 π 0 q(x)dx.
Proof. We denote s := √ λ. Then from the asymptotic estimates (see [START_REF] Marchenko | The Sturm-Liouville operators and their applications (Russian) (Kiev: Naukova Dumka) English transl[END_REF][START_REF] Yurko | Inverse spectral problems and their applications (Russian) (Saratov: Izd-vo Saratovskogo pedinstituta) See also: Freiling G and Yurko V A 2001 Inverse Sturm-Liouville problems and their applications[END_REF])

ϕ(x, λ) = cos sx + h + 1 2 x 0 q(t)dt sin sx s + 1 2 x 0 q(t) sin s(x -2t) s dt +O e | Im sx| |s| 2 , ϕ ′ (x, λ) = -s sin sx + h + 1 2
x 0 q(t)dt cos sx + 1 2

x 0 q(t) cos s(x -2t)dt +O e | Im sx| |s| using (2.1) we have:

χ(λ) = -s 3 sin sπ + h + H + 1 2 π 0 q(x)dx s 2 cos sπ + I(s)s 2 , (2.6)
where

I(s) = 1 2 π 0 q(t) cos s(π -2t)dt + O e | Im sπ| |s| .
Now using Bessel's inequality it's easy to obtain (2.4) and (2.5).

Since the function χ(λ) is entire of order 1/2, from Hadamard's theorem(see [18, Section 4.2]), using (2.6) we obtain:

χ(λ) = -π(λ -λ 0 )(λ -λ 1 ) ∞ n=2 λ n -λ (n -1) 2 .

Main Equation

Theorem 3.1. Let f (x) ∈ A C [0, π]. Then f (x) = ∞ n=0 1 γ n π 0 f (t)ϕ(t, λ n )dt ϕ(x, λ n ) with uniform convergence in [0, π]. Proof. We denote G(x, t, λ) := 1 χ(λ) ϕ(x, λ)ψ(t, λ), 0 ≤ x ≤ t ≤ π ψ(x, λ)ϕ(t, λ), 0 ≤ t ≤ x ≤ π
and consider the function

Y (x, λ) := π 0 G(x, t, λ)f (t)dt = 1 χ(λ) ψ(x, λ) x 0 ϕ(t, λ)f (t)dt + ϕ(x, λ) π x ψ(t, λ)f (t)dt .
Using (2.2) and (2.3) we obtain:

Res λ=λn Y (x, λ) = 1 χ(λ n ) ψ(x, λ n ) x 0 ϕ(t, λ n )f (t)dt + ϕ(x, λ n ) π x ψ(t, λ n )f (t)dt = k n χ(λ n ) ϕ(x, λ n ) π 0 ϕ(t, λ n )f (t)dt = 1 γ n ϕ(x, λ n ) π 0 ϕ(t, λ n )f (t)dt.
Noting that ϕ(x, λ) and ψ(x, λ) are solutions of (1.1) and integrating by parts we can write:

Y (x, λ) = f (x) λ + Z(x, λ) λ ,
where

Z(x, λ) = 1 χ(λ) ψ(x, λ) x 0 ϕ ′ (t, λ)f ′ (t)dt + ϕ(x, λ) π x ψ ′ (t, λ)f ′ (t)dt +hf (0)ψ(x, λ) -(λH -H 2 )f (π)ϕ(x, λ) + ψ(x, λ) x 0 ϕ(t, λ)q(t)f (t)dt + ϕ(x, λ) π x ψ(t, λ)q(t)f (t)dt .
Using asymptotic estimates for the functions ϕ(x, λ), ψ(x, λ) and χ(λ) the following equality can be proved:

lim |s|→∞ s∈G δ max 0≤x≤π |Z(x, λ)| = 0,
where G δ = {s : |s -n| ≥ δ, n = 0, ±1, ±2, . . .} for some small fixed δ > 0. Now consider the contour integral

I N (x) = 1 2πi C N Y (x, λ)dλ,
where

C N = {λ : |λ| = (N -1/2) 2 }.
From the above equalities we have:

I N (x) = f (x) + ε N (x), lim N →∞ max 0≤x≤π |ε N (x)| = 0.
On the other hand, using the residue calculus we obtain:

I N (x) = N n=0 1 γ n π 0 f (t)ϕ(t, λ n )dt ϕ(x, λ n ).
From last two equalities we obtain the statement of the theorem.

a(x) := ∞ n=1 cos s n x γ n - cos(n -1)x α 0 n-1
, where

α 0 n = π 2 , n ≥ 1, π, n = 0. Then a(x) ∈ W 1 2 (0, 2π).
We denote

F (x, t) = cos s 0 x cos s 0 t γ 0 + ∞ n=1 cos s n x cos s n t γ n - cos(n -1)x cos(n -1)t α 0 n-1 . (3.1) Since F (x, t) = cos s 0 x cos s 0 t γ 0 + a(x + t) + a(x -t) 2 , Lemma 3.1 implies that F (x, t) is continuous and d dx F (x, x) ∈ L 2 (0, π).
Using the transformation operators ( [START_REF] Marchenko | The Sturm-Liouville operators and their applications (Russian) (Kiev: Naukova Dumka) English transl[END_REF][START_REF] Yurko | Inverse spectral problems and their applications (Russian) (Saratov: Izd-vo Saratovskogo pedinstituta) See also: Freiling G and Yurko V A 2001 Inverse Sturm-Liouville problems and their applications[END_REF]), we can write equalities

ϕ(x, λ) = cos sx + x 0 K(x, t) cos stdt, (3.2)
cos sx = ϕ(x, λ) + x 0 H(x, t)ϕ(t, λ)dt, (3.3)
where K(x, t) and H(x, t) are real-valued continuous functions and

K(x, x) = h + 1 2 x 0 q(t)dt. (3.4)
Theorem 3.2. For each fixed x ∈ (0, π] the kernel K(x, t) satisfies the following equation:

F (x, t) + K(x, t) + x 0 K(x, τ )F (τ, t)dτ = 0, 0 < t < x. (3.5)
Proof. Using equalities (3.2) and (3.3) we obtain:

N n=0 ϕ(x, λ n ) cos s n t γ n = N n=0 cos s n x cos s n t γ n + cos s n t γ n x 0 K(x, τ ) cos s n τ dτ , N n=0 ϕ(x, λ n ) cos s n t γ n = N n=0 ϕ(x, λ n )ϕ(t, λ n ) γ n + ϕ(x, λ n ) γ n t 0 H(t, τ )ϕ(τ, λ n )dτ .
Therefore we can write:

Φ N (x, t) = I N (x, t) + I ′ N (x, t) + I ′′ N (x, t) + I ′′′ N (x, t), where Φ N (x, t) = N n=0 ϕ(x, λ n )ϕ(t, λ n ) γ n - N -1 n=0 cos nx cos nt α 0 n , I N (x, t) = N n=0 cos s n x cos s n t γ n - N -1 n=0 cos nx cos nt α 0 n , I ′ N (x, t) = N -1 n=0 cos nt α 0 n x 0 K(x, τ ) cos nτ dτ, I ′′ N (x, t) = x 0 K(x, τ ) N n=0 cos s n t cos s n τ γ n - N -1 n=0 cos nt cos nτ α 0 n dτ, I ′′′ N (x, t) = - N n=0 ϕ(x, λ n ) γ n t 0 H(t, τ )ϕ(τ, λ n )dτ.
Let f (x) be an absolutely continuous function. Then using Theorem 3.1 we obtain (uniformly on x ∈ [0, π]):

lim N →∞ π 0 f (t)Φ N (x, t)dt = 0, lim N →∞ π 0 f (t)I N (x, t)dt = π 0 f (t)F (x, t)dt, lim N →∞ π 0 f (t)I ′ N (x, t)dt = x 0 f (t)K(x, t)dt, lim N →∞ π 0 f (t)I ′′ N (x, t)dt = π 0 f (t) x 0 K(x, τ )F (τ, t)dτ dt, lim N →∞ π 0 f (t)I ′′′ N (x, t)dt = - π x f (t)H(t, x)dt,
We put K(x, t) = H(x, t) = 0 for x < t. Since f (x) can be chosen arbitrarily, we have

F (x, t) + K(x, t) + x 0 K(x, τ )F (τ, t)dτ -H(t, x) = 0.
When t < x this equation implies (3.5).

Uniqueness

Lemma 4.1. For each fixed x ∈ (0, π] equation (3.5) has a unique solution K(x, t) ∈ L 2 (0, x).

Proof. It suffices to prove that homogeneous equation

g(t) + x 0 F (τ, t)g(τ )dτ = 0
has only trivial solution g(t) = 0. Let g(t) be a solution of the above equation and g(t) = 0 for t ∈ (x, π). Then

y(t, α) + b a A(t, τ, α)y(τ, α)dτ = f (t, α), a ≤ t ≤ b, (4.1)
where A(t, τ, α) and f (t, α) are continuous functions. Assume that, for some fixed α = α 0 the homogeneous equation

z(t) + b a A 0 (t, τ )z(τ )dτ = 0, A 0 (t, τ ) := A(t, τ, α 0 )
has only trivial solution. Then in some neighbourhood of the point α = α 0 the solution y(t, α) of the equation (4.1) is continuous on t and α. Moreover, the function y(t, α) has the same smoothness as A(t, τ, α) and f (t, α).

Theorem 4.1. Let P(q, h, H, H 1 , H 2 ) and P( q, h, H, H 1 , H 2 ) be two boundary-value problems with one boundary condition depending linearly on the spectral parameter and

λ n = λ n , γ n = γ n , n ≥ 0.
Then q(x) = q(x) a.e. on (0, π), h

= h, H = H, H 1 = H 1 , H 2 = H 2 .
Proof. According to the formula (3.1) F (x, t) = F (x, t). Then from the main equation (3.5) we obtain K(x, t) = K(x, t). Equality (3.4) implies that h = h and q(x) = q(x) a.e. on (0, π). From (3.2) we have ϕ(x, λ n ) = ϕ(x, λ n ). In consideration of (2.6) we obtain χ(λ) ≡ χ(λ) and k n = k n . Finally, by using (2.1) and (2.2) the remaining part of the theorem can be proved.

Reconstruction by spectral data

Let two sequences of real numbers {λ n } and {γ n } (n ∈ Z + ) with the following properties be given:

s n = λ n = n -1 + ω nπ + ζ n n , γ n = π 2 + ζ ′ n n , {ζ n }, {ζ ′ n } ∈ l 2 , (5.1)
λ n = λ m , n = m, γ n > 0, n ∈ Z + . (5.2)
Using these numbers we construct F (x, t) by the formula (3.1) and determine K(x, t) from (3.5). Substituting t → tx, τ → τ x in (3.5) we obtain:

F (x, xt) + K(x, xt) + x 1 0 K(x, xτ )F (xτ, xt)dτ = 0, 0 ≤ t ≤ 1.
According to this equation and Lemmas 4.1 and 4.2 K(x, t) is determined uniquely and d dx K(x, x) ∈ L 2 (0, π). Now, let's construct the functions q(x), ϕ(x, λ), χ(λ) and the number h by q(x) := 2 d dx K(x, x), h := K(0, 0),

ϕ(x, λ) := cos sx + x 0 K(x, t) cos stdt, (5.3)
χ(λ) = -π(λ -λ 0 )(λ -λ 1 ) ∞ n=2 λ n -λ (n -1) 2
and put

k n := χ(λ n ) γ n .
From (5.2) we have: k n = 0.

u(z) = sin πz + Aπ 4z 4z 2 -1 cos πz + f (z) z , v(z) = cos πz -Bπ sin πz z + g(z) z ,
where

f (z) = π 0 f(t) cos ztdt, f (t) ∈ L 2 [0, π], π 0 f (t)dt = 0, g(z) = π 0 g(t) sin ztdt, g(t) ∈ L 2 [0, π],
it is necessary and sufficient to have the form

u(z) = πz ∞ n=1 n -2 (u 2 n -z 2 ), u n = n - A n + α n n , v(z) = ∞ n=1 n - 1 2 -2 (v 2 n -z 2 ), v n = n - 1 2 - B n + β n n ,
where α n and β n are arbitrary sequences that satisfy conditions

∞ n=1 |α n | 2 < ∞, ∞ n=1 |β n | 2 < ∞.
Lemma 5.2. The following equality holds:

∞ n=0 ϕ(x, λ n ) k n γ n = 0 (5.4)
Proof. Using the residue calculus we get:

N n=0 ϕ(x, λ n ) k n γ n = N n=0 ϕ(x, λ n ) χ(λ n ) = N n=0 Res λ=λn ϕ(x, λ) χ(λ) = 1 2πi C N ϕ(x, λ) χ(λ) dλ,
where

C N = {λ : |λ| = (N -1/2) 2 }. Lemma 5.1 implies that χ(λ) = -s 3 sin sπ + O |s| 2 e | Im sπ| .
We denote G δ = {s : |s -n| ≥ δ, n = 0, ±1, ±2, . . .} for some small fixed δ > 0 and recall that (see e.g. [25, p. 15])

| sin sπ| ≥ C δ e | Im sπ| , s ∈ G δ ,
where C δ does not depend on s. Therefore we obtain

|χ(λ)| ≥ C δ |s| 3 e | Im sπ| , s ∈ G δ , |s| ≥ s δ for sufficiently large s δ . Since |ϕ(x, λ)| = O e | Im sπ| , x ∈ [0, π],
we obtain (5.4).

As in the theory of classical Sturm-Liouville problems (see [25, Lemma 1.5.8 and Corollary 1.5.1]) the following lemmas can be proved:

Lemma 5.3. The following relations hold:

-ϕ ′′ (x, λ) + q(x)ϕ(x, λ) = λϕ(x, λ), ϕ(0, λ) = 1, ϕ ′ (0, λ) = h.

Lemma 5.4. For any f (x), g(x) ∈ L 2 (0, π) the following equality holds:

π 0 f (x)g(x)dx = ∞ n=0 1 γ n π 0 f (t)ϕ(t, λ n )dt π 0 g(t)ϕ(t, λ n )dt . Proposition 5.1. ([25, Proposition 1.8.6]) Let numbers {ρ n } n≥0 , ρ 2 n = ρ 2 k (n = k) of the form ρ n = n + a n + ξ n n , {ξ n } ∈ l 2 , a ∈ C
be given. Then the sequence {cos ρ n x} n≥0 forms a Riesz basis in the space L 2 (0, π).

Corollary 5.1. For any fixed n 0 ∈ Z + the system {ϕ(x, λ n )}(n = n 0 ) forms a Riesz basis in the space L 2 (0, π).

Proof. According to Lemma 5.3 we can write representation of the form (3.3). Therefore, there is one-to-one correspondence between expansions in {cos s n x}(n = n 0 ) and {ϕ(x, λ n )}(n = n 0 ).

Lemma 5.5. For any f (x) ∈ W 2 2 (0, π), the expansion

f (x) = ∞ n=0 1 γ n π 0 f (t)ϕ(t, λ n )dt ϕ(x, λ n ) (5.5)
holds.

Proof. Consider the series

f * (x) = ∞ n=0 c n ϕ(x, λ n ), (5.6)
where

c n := 1 γ n π 0 f (t)ϕ(t, λ n )dt.
Using Lemma 5.3 and integrating by parts we obtain:

c n = 1 γ n λ n π 0 f (t) (-ϕ ′′ (t, λ n ) + q(t)ϕ(t, λ n )) dt = 1 γ n λ n (hf (0) -f ′ (0) + f ′ (π)ϕ(π, λ n ) -f (π)ϕ ′ (π, λ n )) + 1 γ n λ n π 0 ϕ(t, λ n )(-f ′′ (t) + q(t)f (t))dt.
We can easily prove that as n → ∞

c n = O 1 n 2 , ϕ(t, λ n ) = O(1)
uniformly on t ∈ [0, π]. Therefore, the series (5.6) converges absolutely and uniformly on x ∈ [0, π]. According to Lemma 5.4

π 0 f (x)g(x)dx = ∞ n=0 c n π 0 g(t)ϕ(t, λ n )dt = π 0 g(t) ∞ n=0 c n ϕ(t, λ n )dt = π 0 g(t)f * (t)dt.
Since g(x) can be chosen arbitrarily, we conclude that f * (x) = f (x).

We can write (5.4) as

ϕ(x, λ n 0 ) γ n 0 = - n =n 0 k n 0 ϕ(x, λ n ) k n γ n
for any n 0 ∈ Z + . Let m = n 0 be any fixed number and f (x) = ϕ(x, λ m ). Then using the above equality in (5.5), we have:

ϕ(x, λ m ) = n =n 0 c mn ϕ(x, λ n ),
where

c mn = 1 γ n π 0 ϕ(t, λ m ) ϕ(t, λ n ) - k n 0 k n ϕ(t, λ n 0 ) dt.
Corollary 5.1 implies c mn = δ mn . Here δ mn is the Kronecker delta. In other words, denoting a mn := π 0 ϕ(t, λ m )ϕ(t, λ n )dt, we have:

a mm - k n k m a mn = γ m , m = n.
It's clear (from definition) that a mn = a nm . Using these relations we calculate:

k 2 m (γ m -a mm ) = -k m k n a mn = -k n k m a nm = k 2 n (γ n -a nn ), m = n. Therefore k 2 n (γ n -a nn ) = const.
Let's denote this constant by ρ. Then we have:

π 0 ϕ 2 (t, λ n )dt = γ n - ρ k 2 n , π 0 ϕ(t, λ m )ϕ(t, λ n )dt = - ρ k m k n , m = n.
Now using the equality

ϕ(π, λ)ϕ ′ (π, µ) -ϕ ′ (π, λ)ϕ(π, µ) = (λ -µ) π 0 ϕ(t, λ)ϕ(t, µ)dt
we write:

k n ϕ(π, λ n )k m ϕ ′ (π, λ m ) -k n ϕ ′ (π, λ n )k m ϕ(π, λ m ) λ n -λ m = -ρ, n = m. Denoting A n := k n ϕ(π, λ n ), B n := k n ϕ ′ (π, λ n )
we can write the above equality as

A n B m -B n A m = ρ(λ m -λ n ), n = m. (5.7)
Let i, j, m and n be pairwise distinct nonnegative integers. By summing the equalities

A n B m -B n A m = ρ(λ m -λ n ), A m B i -B m A i = ρ(λ i -λ m ), A i B n -B i A n = ρ(λ n -λ i ),
we have:

A n (B m -B i ) + B n (A i -A m ) = B m A i -A m B i .
Writing this equality again, but this time with n replaced by j and subtracting them we finally obtain:

(A n -A j )(B m -B i ) = (A m -A i )(B n -B j ).
If B n = B j for some n, j ∈ Z + , then B n = const. In this case (5.7) implies A n = κ 1 λ n + κ 2 with some constants κ 1 and κ 2 . Continuing this procedure for the case

B n = B j we obtain A n = κ 1 λ n + κ 2 and B n = κ 3 λ n + κ 4 .
So in both cases

k n ϕ(π, λ n ) = κ 1 λ n + κ 2 , k n ϕ ′ (π, λ n ) = κ 3 λ n + κ 4 .
Using (5.1), (5.3) and Lemma 5.1 we calculate:

k n = (-1) n n 2 + O(n), ϕ(π, λ n ) = (-1) n-1 + O( 1 n ), λ n = n 2 + O(n).
Therefore κ 1 = -1. Denoting H 1 := κ 2 , H := -κ 3 , H 2 := κ 4 we obtain:

λ n (ϕ ′ (π, λ n ) + Hϕ(π, λ n )) = H 1 ϕ ′ (π, λ n ) + H 2 ϕ(π, λ n ), n ∈ Z + (5.8)
for some constants H, H 1 and H 2 . From (5.7) we have:

HH 1 -H 2 = ρ.
Hence we have proved

λ 0 = 0, λ 1 = 1 4 , λ n = (n -1) 2 , n ≥ 2, γ 0 = π, γ n = π 2 , n ≥ 1.
Then from (3.1) we have:

F (x, t) = 2 π cos x 2 cos t 2 .
Solving the equation (3.5) and then using the relation (3.4) we obtain:

K(x, t) = - 2 cos x 2 cos t 2 π + x + sin x , q(x) = 2(π + x) sin x + 4(1 + cos x) (π + x + sin x) 2 , h = - 2 π .
In order to reconstruct the second boundary condition, we construct the solution ϕ(x, λ) using (5.3):

ϕ(x, λ) =        cos sx - 4s sin sx(1 + cos x) -cos sx sin x (4s 2 -1)(π + x + sin x) , λ = 1 4 , π cos x 2 π + x + sin x , λ = 1 4 .
Then from (5.8) we have:

8πHn 4 -(2πH + 8πH 2 + 1)n 2 + H 1 + 2πH 2 = 0, n = 0, 1, . . . .
From these equalities we finally calculate the coefficients of the second boundary condition:

H = 0, H 1 = 1 4 , H 2 = - 1 8π .

On two problems with common parameter dependent boundary condition

Consider two eigenvalue problems for the equation -y ′′ (x) + q(x)y(x) = λy(x) (6.1)

with boundary conditions

y ′ (0) -hy(0) = 0, λ(y ′ (π) + Hy(π)) = H 1 y ′ (π) + H 2 y(π), (6.2)
y ′ (0) -hy(0) = 0, λ(y ′ (π) + Hy(π)) = H 1 y ′ (π) + H 2 y(π), (6.3)
where q(x) ∈ L 2 (0, π) is a real-valued function, h, h, H, H 1 , H 2 ∈ R and

ρ := HH 1 -H 2 > 0.
We can assume without loss of generality that h < h. Denote by λ 0 < λ 1 < λ 2 < . . . and µ 0 < µ 1 < µ 2 < . . . the eigenvalues of the problems (6.1), (6.2) and (6.1), (6.3), respectively. Let ϕ(x, λ) and ψ(x, λ) be the solutions of equation (6.1) satisfying

ϕ(0, λ) = 1, ϕ ′ (0, λ) = h, ψ(0, λ) = 1, ψ ′ (0, λ) = h.
Eigenvalues of the problems (6.1), (6.2) and (6.1), (6.3) coincide with the zeros of the functions

Φ(λ) := λ(ϕ ′ (π, λ) + Hϕ(π, λ)) -H 1 ϕ ′ (π, λ) -H 2 ϕ(π, λ), Ψ(λ) := λ(ψ ′ (π, λ) + Hψ(π, λ)) -H 1 ψ ′ (π, λ) -H 2 ψ(π, λ),
respectively. We denote f (x, λ) = ψ(x, λ) + m(λ)ϕ(x, λ) and choose m(λ) such that

λ(f ′ (π, λ) + Hf (π, λ)) -H 1 f ′ (π, λ) -H 2 f (π, λ) = 0. Then m(λ) = - Ψ(λ) Φ(λ) .
Using Green's formula, we can write

π 0 f (x, λ)f (x, µ)dx = - ρf (π, λ)f (π, µ) (H 1 -λ)(H 1 -µ) + (h -h) m(λ) -m(µ) λ -µ .
From here when µ → λ it follows that:

π 0 f 2 (x, λ)dx + ρf 2 (π, λ) (H 1 -λ) 2 = (h -h) ṁ(λ).
Since the left-hand side of the last equality is always positive, the function m(λ) monotonically decreases in the set R \ {λ n |n ∈ Z + }. Therefore, zeros and poles of m(λ) interlace and according to (2.4) we obtain: λ 0 < µ 0 < λ 1 < µ 1 < λ 2 < µ 2 < . . . . Now we use Green's formula again:

(λ -λ n ) π 0 f (x, λ)ϕ(x, λ n )dx = (f (x, λ)ϕ ′ (x, λ n ) -f ′ (x, λ)ϕ(x, λ n ))| π 0 = - ρ(λ -λ n )f (π, λ)ϕ(π, λ n ) (H 1 -λ n )(H 1 -λ) -(h -h) = - ρ(λ -λ n )ψ(π, λ)ϕ(π, λ n ) (H 1 -λ n )(H 1 -λ) + ρ(λ -λ n )ϕ(π, λ)ϕ(π, λ n ) (H 1 -λ n )(H 1 -λ) Ψ(λ) Φ(λ) -(h -h).
On the other hand According to Theorem 5.1 we can uniquely construct such a real-valued function q(x) ∈ L 2 (0, π) and numbers h, H, H 1 , H 2 ∈ R that the numbers {λ n } and {γ n } are the eigenvalues and the norming constants of problem P(q, h, H, H 1 , H 2 ). We put: h := h + σ and denote by τ n the eigenvalues of problem P(q, h, H, H 1 , H 2 ). Let ϕ(x, λ) and ψ(x, λ) be the solutions of equation ( 6

Lemma 3 . 1 .

 31 (cf.[START_REF] Yurko | Inverse spectral problems and their applications (Russian) (Saratov: Izd-vo Saratovskogo pedinstituta) See also: Freiling G and Yurko V A 2001 Inverse Sturm-Liouville problems and their applications[END_REF] Lemma 1.5.4]) Assume that numbers {s n , γ n } n≥0 satisfying the conditions (2.4), (2.5) and γ n = 0 are given and denote

F

  (τ, t)g(τ )g(t)dτ dt = ) cos ntdt and noting that γ n > 0 we obtain:x 0 g(t) cos s n tdt = 0, n ≥ 0.The system {cos s n t} is complete in L 2 (0, π) (see[START_REF] Yurko | Inverse spectral problems and their applications (Russian) (Saratov: Izd-vo Saratovskogo pedinstituta) See also: Freiling G and Yurko V A 2001 Inverse Sturm-Liouville problems and their applications[END_REF] Proposition 1.8.6]). Therefore g(t) = 0. Lemma 4.2. ([25, Lemma 1.5.2]) Consider an integral equation

Lemma 5 . 1 .

 51 [START_REF] Marchenko | The Sturm-Liouville operators and their applications (Russian) (Kiev: Naukova Dumka) English transl[END_REF] Lemma 3.4.2]) For the functions u(z), v(z) to be represented in the form

Theorem 5 . 1 .Example 5 . 1 .

 5151 For the sequences {λ n } and {γ n } (n ∈ Z + ) to be the spectral data of a problem of the type (1.1)-(1.3) it is necessary and sufficient to satisfy conditions (5.1)-(5.2). Assume that the spectral data of some eigenvalue problem of the form (1.1)-(1.3) is the following:

(λ -λ n ) π 0 f

 0 (x, λ)ϕ(x, λ n )dx = (λ -λ n ) π 0 ψ(x, λ)ϕ(x, λ n )dx -(λ -λ n )Ψ(λ) Φ(λ) π 0 ϕ(x, λ)ϕ(x, λ n )dx. Ψ(λ n ) = (-1) n+1 (n -1) n } ∈ l 2 .

. 1 )Theorem 7 . 1 .

 171 satisfying the conditionsϕ(0, λ) = 1, ϕ ′ (0, λ) = h, ψ(0, λ) = 1, ψ ′ (0, λ) = h.Then zeros and poles of the functionm(λ) := -λ(ψ ′ (π, λ) + Hψ(π, λ)) -H 1 ψ ′ (π, λ) -H 2 ψ(π, λ) λ(ϕ ′ (π, λ) + Hϕ(π, λ)) -H 1 ϕ ′ (π, λ) -H 2 ϕ(π, λ)coincide with {λ n } and {τ n }, respectively. The numbers {λ n } and{τ n } interlace, lim λ→-∞ m(λ) =λ -λ n ) ,i.e. m(λ) ≡ m(λ) and therefore zeros of m(λ) and m(λ) coincide: τ n = µ n . Thus we have proved For the sequences {λ n } and {µ n } (n ∈ Z + ) to be the spectra of two problems of the type (1.1)-(1.3) with common parameter dependent boundary condition it is necessary and sufficient to satisfy conditions (7.1)-(7.2) and interlace.

[26] Yurko V A 2002 Method of spectral mappings in the inverse problem theory (Utrecht: VSP)

Acknowledgments

The author is grateful to Professor H. M. Huseynov for the problem statement and for useful discussions.

When λ n → λ from these two equalities we obtain:

Thus we expressed norming constants by two spectra. We shall use this expression to solve the problem of reconstruction of differential operator by two of its spectra. Using Theorem 2.1 and Lemma 5.1 from the last equality we obtain the following asymptotic estimation:

Reconstruction by two spectra

Let two sequences of real numbers {λ n } and {µ n } (n ∈ Z + ) with the following properties be given:

We define functions

and put:

Since zeros of Φ(λ) are simple, we get:

Using Lemma 5.1 again we calculate: