Jasper Goethals
email: jasper.goethals@ugent.be

Denys Nikolayev

Arno Thielens

Gunter Vermeeren

Leen Verloock

Margot Deruyck

Luc Martens

Wout Joseph

Design and Verification of an On-Body Antenna in the Harsh Environment of a Horse Hoof

Keywords: Antenna, animal applications, body area network, GSM900, on-body, PIFA, robustness

This paper describes the design of an antenna embedded in a hoof pad. The operating frequencies are within the GSM900 band. The dimensions are 40 mm x 25 mm x 1.55 mm. During the characterization of the antenna, the dynamic and harsh environment of a horse hoof is taken into account. The antenna impedance is validated numerically and experimentally to be robust to these changes.

I. INTRODUCTION

Owning a horse is not only a serious financial undertaking, but also an emotional investment due to the strong trust bond between a horse and its rider, protecting the health of this valuable asset is thus of utmost importance. The earlier an anomaly is detected, the higher the chances for a full recovery. To monitor the horse's vital parameters, one can put a monitoring device or a wearable on the animal, which will inform the caretaker in an early stage about any possible anomaly [START_REF] Hartmann | Detection of the time of foaling by accelerometer technique in horses ( Equus caballus )-a pilot study[END_REF]. This requires a wireless connection from the device to a gateway, e.g. installed in the stable. Such a wearable must of course not harm the horse (exclude chafing, tendon damage, etc.) or hinder movement, while at the same time the device should be protected from any damage by the horse (e.g. biting). Many existing wearables fail on these requirements by placing the monitoring devices on the head or torso of the horse [START_REF]Equisense Motion[END_REF], [START_REF]Ekico: Leader in horses hooves monitoring tools[END_REF]. In particular, the horse's legs are very delicate due to the many tendons, and, for this reason, no cables nor any other kind of bulky electronics like a battery pack should be attached to them. One way to address this issue is to place a wearable under the hoof inside a thin pad like the sole in a shoe (made of leather or silicone). This pad is attached to the hoof through the nails of the horseshoe and feels naturally for the horse since these pads are today often used to solve orthopedic issues. Fig. 1 shows a horseshoe attached without (left) and with (right) a hoof pad. Note that the frog (V-shape visible in the left figure) is considered to be the hoof's weak part and cannot be covered by any electronics, hence limiting the size of the electronics and antenna.

The goal of this study is to develop a wireless off-body telemetry device that operates from inside the hoof and sends the data to a cellular gateway. To cover sufficiently long Fig. 1. Example of a horseshoe without (left) and with a pad (right) on the hoof [START_REF] Boris | How to Remove a Horseshoe[END_REF], [START_REF]Podi Artist[END_REF] ranges during training and recreational activities and to be able to communicate directly from the horse to a gateway, radio technologies with suitable propagation characteristics need to be used. Lower frequency waves are typically associated with larger achievable ranges. However, lower radio frequencies typically require larger antennas, and limit the maximum achievable throughput, thereby limiting device placement options and the amount of monitoring data, respectively. Here, we chose to use the 900-MHz operating band that allows using Global System for Mobile Communications (GSM) and Long-Term Evolution (LTE) technologies (942 -960 MHz). Conductive objects (such as horseshoe) located in the near field strongly affect the antenna radiation performance [START_REF] Tanghe | Intra-, Inter-, and Extra-Container Path Loss for Shipping Container Monitoring Systems[END_REF]. In addition, proximity of dielectric objects (horse's hoof) impact both the impedance and radiation performances [START_REF] Nikolayev | Optimal Radiation of Body-Implanted Capsules[END_REF], [START_REF] Nikolayev | Impact of Tissue Electromagnetic Properties on Radiation Performance of In-Body Antennas[END_REF]. In this work, we explore an antenna design constrained by the aforementioned factors of operation in the harsh environment of a horse hoof. This paper is organized as follows: Section II focuses on the antenna design considerations and the scenarios incorporating the moving aspects, the modeling approaches, the materials, and the measurement setup on the real horse leg. The results are presented in Section III. Section IV concludes the paper.

II. METHODS

A. Antenna Design

The antenna design has to comply with the spatial constrains of the device. In this work, the antenna is embedded in a large hoof pad with dimensions 135 mm × 150 mm × 4 mm, as depicted in Fig. 1. A hoof pad is usually a thin sheet of leather, plastic or silicon used to prevent rocks or uneven hard surface hurting the bottom of the hoof. A border of around 2 cm has to be kept clear around the edge of the hoof, where the horseshoe will press the pad against the hoof. Between the pad and the hoof, an elastic epoxy resin is applied. This fills the cavity constructed by the pad and the hoof. The epoxy is applied to prevent dirt and rocks to agitate the sensitive hoof area while wearing the pad and horseshoe. The thickness of this layer depends on the horse. For example, a flat footed horse will not form this cavity. The parameters of interest are the reflection coefficient, bandwidth (BW) at |S 11 | = -10 dB, center frequency, the radiation efficiency and gain of the antenna. Based on the geometrical limitations of the space in the hoof pad (< 0.01λ in height), antennas types that are suitable for the considered applications should be of a low profile. Considering the required miniaturization, the choice was made to persue an inverted-F antenna design (IFA). The printed IFA can be easily manufactured in a mechanically robust configuration since it does not require vertical shorting strips or vias that reduce the rigidity of the structure. Fig. 2 shows the antenna layout based on our previous paper [START_REF] Goethals | Combined Antenna-Channel Characterization for Wireless Communication from Horse Hoof to Base Station[END_REF]. The placement of the feed can be used to match the antenna to a 50-Ohm feed on the pad where the RF frontend in located without using discrete components. The dimensions of the board are 40 × 25 × 1.55 mm 3 (corresponding to 0.07λ × 0.12λ × 0.01λ) hence fulfilling the constrains of the device. The antenna itself is 12 × 25 mm 2 . The substrate material for a prototype is FR-4 (ε r = 4, σ = 0.0038 S/m) [START_REF]IS400[END_REF], which is chosen for its low manufacturing cost [START_REF] Coonrod | Understanding When To Use FR-4 Or High Frequency Laminates[END_REF]. However, more durable and shock-resistant substrates with comparable EM properties could be employed for the finalized design. All metallization is modeled as perfect electrical conductor (PEC). The dimensions are denoted on the schematic of the antenna on Fig. 2(right). During the design, the environmental properties and elements were incorporated and studied one by one to quantify their effects on the antenna performance.

B. Application Scenarios

The environment of a horse leg is dynamic. When the horse is in the meadow, it will not only graze, but also walk, trot, gallop, and it might even buck. Since the monitoring device is placed in the hoof pad, the antenna environment will also change dynamically. In this paper, two main scenarios are considered. In the first, the horse has his hoof lifted off the ground. The angle of the front hoof in the air is determined (based on the work done in [START_REF] Dutto | Joint work and power for both the forelimb and hindlimb during trotting in the horse[END_REF]) to be 90 degrees with respect to the ground and lifted 10 cm during a normal walk (so called working step). During this first raised scenario, the leg will be in free space, thus the effect of the soil on the antenna near field could be neglected. In the second scenario, the hoof is on the ground. The soil is modeled by a 500 × 500 × 50 mm 3 block. In this study, we consider the grass and concrete. Grass was assigned a relative permittivity of 8 and a conductivity 0.163 S/m [START_REF] Kim | A new measurement technique for obtaining the complex relative permittivity of terrain surfaces[END_REF], whereas concrete was assigned a relative permittivity of 4.5 and a conductivity of 6.25 mS/m [START_REF] Soutsos | Dielectric properties of concrete and their influence on radar testing[END_REF]. Furthermore, the epoxy layer thickness will vary from horse to horse. To study the dependence of this layer, different thicknesses were simulated.

C. Numerical Model

The antenna is embedded in the pad that consist of polylactic acid (PLA) and epoxy (ε r = 3.5, σ = 0.0023 S/m) [START_REF] Singha | Permittivity and tan delta characteristics of epoxy nanocomposites in the frequency range of 1 MHz-1 GHz[END_REF]. In the simulations, the PLA and the epoxy have the same EM properties. This encapsulation will increase the robustness of the parameters of the antenna to the influence of the horse leg and the ground and is necessary for the mechanical stability of the pad. Fig. 3 shows the complete configuration. A horse's leg is simulated by a 10-cm-long cylinder. The complete setup is built by the four layers: the horseshoe, the pad, the filling epoxy and the leg. We assume that the leg is made out of bone (homogeneous) (ε r = 12, σ = 0.18 S/m) with parameters similar to human bone [START_REF] Gabriel | The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues[END_REF]. However, another simulation with the horse leg out of muscle (ε r = 51, σ = 0.98 S/m) is performed to quantify the effect of the variation of environment EM properties. The horseshoe is modelled as PEC. The antenna is tuned to the scenario where the leg is in the air and is made out of bone, and the filling epoxy is 5 mm thick. The numerical simulation are performed using Sim4Life(S4L), a finite-difference time-domain (FDTD) simulator [17]. For the calculation of the BW and center frequency, a Gaussian signal was excited over the edge source at the feed strip. The center frequency of this Gaussian signal is 1 GHz, and the BW is 1 GHz. The soil block terminates into perfectly matching layers. In the second step, the simulations are performed with a harmonic excitation to calculate the efficiency and the reflection coefficient. The number of cells of the grid were 1.61 MCells and 2.04 MCells for the BW simulations without or with ground, respectively, and 1.43 MCells and 1.64 MCels for the gain pattern simulations without or with ground, respectively.

D. Experimental Setup Using a Real Horse Leg

The produced pad is seen in Fig. 4 (left). A miniature X.FL [START_REF]X.FL Series Connectors -Hirose | DigiKey[END_REF] connection is included to connect the antenna to the test bench with a 50-Ohm feedline. The pad is constructed by 3D-printing a 15-cm diameter cylinder of 1-mm height and a border of 4 mm. A small holder for the 4 × 2.5 cm PCB is also printed with PLA. This pad is then filled with epoxy resin to finish the pad as a solid cylinder with the antenna PCB embedded. To finalize the experimental validation, the previous measurement are repeated on a real horse leg. A leg of a recently deceased horse (< 24 hours) was severed and provided by the faculty of veterinary medicine at Ghent University. The PLA-epoxy hoof pad with the embedded antenna was attached using tape. Next, the hoof iron was attached over this ensemble with more tape. Between the hoof pad and the hoof, a cavity is formed due to the physiology of a horse hoof. This cavity was filled with epoxy Equi-pak [START_REF]Equi-Pak Soft TM 210cc[END_REF]. The farrier usually does this to prevent dirt to enter this cavity and consequently prevent inflammation of the hoof tissue. In the test setup, the horse hoof is at a height of 64 cm from the ground and hold by a wooden trestle. A plastic bucket with horse feed is used as counter weight, as depicted on Fig. 4(right). The measurements were done at a surgical stable at the faculty of veterinary medicine. Besides the free space measurements, different soil types were examined by pressing the hoof gently on the surface by its own weight. Again, grass and concrete were considered.

III. RESULTS

A. Numerical Results

1) Baseline results:

The first configuration is for the lifted hoof. The leg is simulated with bone parameters. Fig. 5 shows the reflection coefficient. Table 1 summarizes the results.

The simulation shows that the bone setup is well matched for the aimed band. Different tissues will shift the resonance to lower frequencies, which is easily explained by the higher permittivity of these materials. Table 1 shows that the efficiency lowers significantly with tissue like muscle that has a higher conductivity in comparison with bone. The low efficiency of <10% is due to the tissue dense environment of the antenna. Furthermore, the thickness of the filling epoxy has a significant influence on the tuning of the antenna. Fig. 5 shows that the antenna is robust against variations in the filling epoxy thickness. The aimed band is still served with deviation of 2 mm in thickness.

2) Influence of the Soil: Fig. 6 shows the influence of the different soil types. We can conclude that the antenna is robust against the change in soil type according to the simulations. Variation of less than 0.01% and only a slight detuning are noted. This is necessary to have to achieve a stable horse hoof to gateway communication.

B. Experimental Results

Fig. 7 shows the results of the horse measurements. A good agreement between the simulations and the experiment is obtained (<3% difference in resonance frequency). The robustness to the different soil types is confirmed. The mismatch is explained by the thickness of the filling epoxy and the unknown parameters of the horse leg. The horse leg physiology (hoof is mostly a bony structure) shows that the thickness of the epoxy will have the highest sensitivity for the antenna performance. The off-set in |S 11 | is due to the loss in the X.Fl wire.

IV. CONCLUSION

In this paper, we have established an antenna in a hoof-pad working in the desired frequency band. We have experimentally validated the simulations with real horse measurements. We found a very good agreement between measurements and simulations in the air (<3% difference in resonance frequency). The antenna is robust against the soil type, but has a dependence on the thickness of the filling epoxy. Future work will be on investigating the robustness and performance by varying tissue parameters of different horses.

Fig. 2 .

 2 Fig. 2. The PCB layout (left) and the dimensions of the antenna (right) with parameters values: W1 = 0.5, W2 = 1.2, Wf = 1, h1 = 3, h2 = 1, L1 = 24, L2 = 17. All dimensions are in mm.

Fig. 3 .

 3 Fig. 3. The complete setup from view(left), perspective view (right).

Fig. 4 .

 4 Fig. 4. The constructed pad (Left), the severed horse leg, covered in a hygienic veterinary glove. The part of the hoof is uncovered from the glove and the hoof pad is attached to it with tape. The cavity created between the hoof pad and the hoof is filled with Equi-pak. (Right)

Fig. 5 .

 5 Fig. 5. |S 11 | plot of the different leg tissues and filling epoxy thickness. The gray band denotes the aimed frequency band.

Fig. 6 .

 6 Fig. 6. Influence of the soil type on the |S 11 | measurement.

Fig. 7 .

 7 Fig. 7. The |S 11 | measurement on the horse leg. VNA label denotes the experimental measurements. S4L label denoted the simulation results for reference.

TABLE I THE

 I SIMULATED RESULTS FOR THE BASIC SETUP FOR DIFFERENT LEG PARAMETERS (BW AND f 0 ARE IN MHZ)

	Tissue	Bone	Muscle	Bone	Bone
	thickness epoxy	5mm	5 mm	3 mm	7 mm
	r /σ[S/m]	12/0.18	51/0.98	12/0.18	12/0.18
	BW	932-1006	921-979	901-983 932-1000
	f 0	968	955	942	976
	η rad	8.8%	2.3%	8.1%	9.8%

ACKNOWLEDGMENT

This work was executed within the imec.icon project Hoof-MATE, a research project bringing together academic researchers and industry partners. The Hoof-MATE project was co-financed by imec and received project support from Flanders Innovation & Entrepreneurship (project nr. HBC.2018.0536). M. Deruyck and A. Thielens are post-doctoral fellows of the FWO-V (Research Foundation -Flanders).


Borja Mercadal

Fabiola Alonso

Denys Nikolayev

Ricardo Salvador

Julien Modolo

Fabrice Bartolomei

Fabrice Wendling

Giulio Ruffini

Evaluating optimal strategies for electric field dosimetry from intracranial electrodes

Keywords: Dosimetry (computational), Static, Work in Progress Presented

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Stereoelectroencephalography (SEEG) is routinely used to identify epileptogenic zones (EZ) in drug-refractory epilepsy through the implantation of intracranial electrodes. Recording of spontaneous activity, and electrical stimulation via SEEG electrodes, are used to determine the 3-dimensional spatiotemporal organization of the epileptic network within the brain, and identify potential targets for surgical resection [START_REF] Bartolomei | Interpretation of SEEG recordings[END_REF]. By electrically stimulating different brain regions, SEEG electrodes can provide a functional map where epileptogenic zones are distinguished from the functional cortex, possibly by triggering symptoms and seizures. Electrical stimulation using intracranial electrodes has also shown the potential to regulate pathological hyperexcitability, despite an incomplete understanding of the underlying mechanisms and anatomical targets. Therefore, evaluating the electric field induced by SEEG electrodes is of considerable interest to develop further and optimize diagnostic and therapeutic applications in drug-refractory epilepsy. Traditionally, numerical methods such as the finite element method (FEM) are used to calculate the electric field in volume conductor problems. The use of realistic head models based on magnetic resonance (MR) images including several tissue types significantly increases the complexity of modeling geometrically accurate electrodes, resulting in models that are computationally expensive and technically challenging to implement. Since a major problem consists is the generation of a proper mesh for the connection between two complex geometries [START_REF] George | Mesh generation and mesh adaptivity: theory and techniques[END_REF], possible alternatives to overcome this problem consist in approximating the electrode's cylindrical shape (3d) by a point source (0d) or a line (1d). Here, using simplified head models, we aimed to evaluate if 0d and 1d current source approximations (point and line source, respectively) are satisfactory to quantify the electric field induced by SEEG electrodes, using a comparison with a realistic model of SEEG electrodes.

Methods

A three-dimensional head model (100 × 100 × 100 mm) with a simplified representation of a sulcus, consisting of concentric cuboids, was built in Comsol Multiphysics v5.6 (Comsol AB, Stockholm, Sweden), and is presented in Figure 1. Each tissue type was represented with a cuboid where the outer one corresponds to the scalp with a conductivity of 0.3 S/m, followed by the skull set to 0.01 S/m, the cerebrospinal fluid (CSF) set to 1.79 S/m, and the grey and white matter with a conductivity of 0.4 and 0.15 S/m respectively. The thickness of the domain representing each tissue is shown in Figure 1.

The realistic geometry of the SEEG electrode and its position is shown in Figure 1. The lead consisted of seven cylindrical contacts, 2 mm long and 0.8 mm in diameter (shown in blue, conductivity of 1000 S/m), separated by 1.5 mm of insulating parts (conductivity of 0.001 S/m). In the point source approximation, points were located at the center of mass of the realistic electrodes. Finally, the line source approximation was built using lines with the same length as contacts and located along their central axis. The three model geometries were meshed in COMSOL resulting in 6701125, 5921464, and 5903782 tetrahedral elements for the realistic, line source, and point source models, respectively.

The electric field magnitude was computed by solving the Laplace equation using a steady-state approach in COMSOL:

where V is the electric potential (V) and is the conductivity (S/m).

In the realistic model, a floating potential boundary condition was imposed at the surface of all contacts, with a zero current in the non-active contacts and 1 mA in the two active contacts. In the case of the line and point source approximations, a line or point source of 1 mA was defined at the active contacts. In order to compare the different models, the results obtained were exported using a 3-D grid with a resolution of 0.25 mm. This data was used to calculate the relative error in the electric field magnitude between the realistic model and the approximations. In order to quantify the performance of the two approximations for different distances and estimate the regions in which they can be used safely, we computed the average relative error as a function of the distance from the lead. For each point of the grid, we computed its distance to the lead axis. These distances were then rounded to millimeters to group the grid points and calculate the mean error at each distance range. We also computed the volumes enclosed by the field iso-surfaces for various field magnitudes. This was performed for all models using COMSOL built-in functions, with the goal of providing an additional tool to compare the different models.

Results

We computed the electric field distribution generated by all consecutive electrode pairs in the lead using the realistic model and the two approximations. Figure 2 presents the results obtained with one of the pairs (see Figure 2a). The resulting electric field distributions for each model are shown in Figure 2b-d. Figure 2e shows the distribution of the relative error of the field magnitude between the realistic model and the two approximations. There is a region with a large error near the lead (1-2 mm from its center), which is seen not only at the active contacts but also near the non-active ones caused by the distortion of the field lines due to the presence of the conductor. The error near the active contacts was lower for the line source model. However, above a certain distance from the lead, the point source approximation provided a better estimation of the field magnitude. The field iso-lines indicate the same (Figure 2f). The iso-lines generated with the line source approximation more closely matched those from the realistic model for the more intense closest iso-line, while the furthest (less intense) iso-line was better approximated by the point source model. Figure 2g presents the mean relative error between the realistic model and the approximations as a function of the orthogonal distance to the lead. The error was consistently lower for the point source approximation regardless of the distance. Using this approach, we quantified the minimum distance at which the relative error decreased below 0.1 and 0.05 for all consecutive pairs. The results are presented in Figure 3a-b. When using the point source model (Figure 3a), except for one pair, at distances of 2 and 4 mm, the mean error was below 0.1 and 0.05, respectively. These distances were larger when considering the line source approximation (Figure 3b). In this case, the minimum distances to obtain an error below 0.1 were generally larger, and in 3 out of the 6 contact pairs simulated, the model yielded an error above 0.05 throughout the entire range analyzed (15 mm).

Finally, we compared the volumes of the electric field iso-surfaces for different magnitudes. Figure 3c shows the volumes obtained with all contact pairs and the three models. The mean relative error between the volumes of the realistic model and the line and point source approximations are displayed in Figure 3d. For all the field magnitudes considered, the point source, on average, produced iso-surface volumes more similar to those obtained with the realistic model.

Discussion

According to our results, the line source approximation better matches the morphology of the electric field distribution obtained with the realistic model near the active contacts. However, for distances of only 1-2 mm, the point source model consistently provides a better estimation of the field magnitude. The geometry of the line source model shapes the electric field more similarly to the realistic one. Therefore, at points nearby the active contacts, the error is lower compared to the point source model. However, the line source model considers a homogeneous current density along the entire line, while in the realistic model, this magnitude varies along the surface of the electrode (especially if the electrode intersects a tissue boundary). Thus, although the field distribution is shaped more similarly to the realistic model, the assumption of a homogeneous current density prevents this approximation from correctly estimating the field magnitude. As a consequence, the electric field magnitude is surprisingly more accurately estimated with the point source model despite being the most simplistic geometrically. Although the values used in this study are not clinically relevant, the field iso-surfaces and their volumes are commonly used to assess the spatial coverage of brain stimulation [START_REF] Collavini | Improvements on spatial coverage and focality of deep brain stimulation in pre-surgical epilepsy mapping[END_REF]. Our results show that these volumes are more accurately estimated by the point source approximation. Similarly, the morphology of the iso-surfaces, except for the largest field values, seem to be better predicted by the point source approximation.

In an attempt to quantitatively assess the performance of the approximations as compared to the realistic model, we computed the average error as a function of the distance from the lead. Based on these results, it appears that the point source approximation can provide an accurate estimate of the field magnitude, in most cases, for distances larger than 4 mm from the lead axis. A relatively accurate estimate can also be expected for distances between 2 and 4 mm. While for shorter distances the error increases drastically. Therefore, unless an accurate estimation of the field near the contacts is needed (~2 mm scale), the point source approximation is a suitable model for estimating the electric fields generated during neuromodulation protocols in patient-specific models.

Figures

Figure 1 .

 1 Figure 1. The three-dimensional head model used in the present study, including five layers of different conductivity matching those measured or estimated in vivo.

  Figure 1. The three-dimensional head model used in the present study, including five layers of different conductivity matching those measured or estimated in vivo.

Figure 2 .

 2 Figure 2. a) Pair of active contacts used in the simulation. b) c) and d): Electric field distribution in a plane containing the lead when injecting a 1 mA current with the contact pair using (b) the realistic model, (c) the point source approximation, and (d) the line source approximation. e) Relative error in a plane containing the lead between the realistic model and the point source approximation (top), and between the realistic model and the line source approximation (bottom). f) Field isolines (for 20, 40, and 100 V/m) in the same plane for the realistic model (black lines), the point source approximation (red dashed lines, top), and the line source approximation (red dashed lines, bottom). g) Mean relative error and standard deviation between the approximations and the realistic model as a function of the orthogonal distance from the lead.

Figure 3 .

 3 Figure 3. a) Distance at which the mean relative error between the point source approximation and the realistic model is below 0.1 and 0.05 for all consecutive contact pairs. If no value is shown for a contact pair, it means that the error was larger throughout the entire range of distances (15 mm). b) Same as a) when comparing the line source approximation and the realistic model. c) Volume of the field iso-surfaces at different field values for all consecutive contact pairs and for the three-electrode models. d) Mean relative error and standard deviation when comparing the volumes shown in c) between the realistic model and the approximations.


Majid Fakhar
email: fakhar@sci.ui.ac.ir

Mohammadreza Khodakhah
email: mrkhodakhah@gmail.com

Ali Mazyaki
email: mazyaki@atu.ac.ir

Antoine Soubeyran
email: antoine.soubeyran@gmail.com

Jafar Zafarani

B Majid Fakhar

Variational rationality, variational principles and the existence of traps in a changing environment

Keywords: Quasi-metric space, The Ekeland variational principle, Variational rationality, Changing environment, Traps Mathematics Subject Classification 47J20, 49J53, 91B50, 91E40

published or not. The documents may come

Variational rationality, variational principles and the existence of traps in a changing environment

Introduction and preliminaries

This paper continues to explore, both in psychology/ behavioral sciences and in mathematics/ variational analysis, the variational rationality approach of stay and change human dynamics initiated by Antoine Soubeyran in psychology and behavioral sciences [START_REF] Soubeyran | Variational rationality: towards a grand theory of motivation driven by worthwhile moves[END_REF][START_REF] Soubeyran | Variational rationality: the concepts of motivation and motivational force[END_REF][START_REF] Soubeyran | Variational rationality: The resolution of goal conflicts via stop and go approach-avoidance dynamics[END_REF][START_REF] Soubeyran | Variational rationality: a general theory of intentions and moving goals as satisficing worthwhile moves[END_REF][START_REF] Soubeyran | Variational rationality and the unsatisfied man: routines and the course pursuit between aspirations, capabilities and beliefs[END_REF][START_REF] Soubeyran | Variational rationality, a theory of individual stability and change: worthwhile and ambidextry behaviors[END_REF] and followed by some researchers in mathematics; see for example [START_REF] Bao | Variational principles, completeness and the existence of traps in behavioral sciences[END_REF][START_REF] Bao | Variational analysis in psychological modeling[END_REF][START_REF] Bao | Fixed points and variational principles with applications to capability theory of wellbeing via variational rationality[END_REF][START_REF] Flores-Bazan | Maximal elements under reference dependent preferences with applications to behavioral traps and games[END_REF][START_REF] Mordukhovich | Variational Analysis and Variational Rationality in Behavioral Sciences: Stationary Traps. Variational Analysis and Set Optimization[END_REF][START_REF] Neto | A generalized proximal linearized algorithm for DC functions with application to the optimal size of the firm problem[END_REF][START_REF] Qiu | Equilibrium versions of variational principles in quasimetric spaces and the robust trap problem[END_REF]. To save space, for a more complete list of applications with other mathematical aspects, see Antoine SOUBEYRAN-Google Sites. Thus, this paper has two complementary aspects, a mathematical aspect and a behavioral aspect.

Mathematical aspect

A quasi-metric q on a nonempty set X is a bifunction q : X × X → R + , where R + denotes the set of non-negative real numbers, such that (i) q(x, y) = 0 if and only if x = y, (ii) q(x, y) ≤ q(x, z) + q(z, y), for all x, y, z ∈ X .

A set X equipped with a quasi-metric q is said to be a quasi-metric space and it is denoted by (X , q). Mathematical aspect starts from the classical Ekeland variational principle (EVP) [START_REF] Ekeland | On the variational principle[END_REF]. EVP problem Let (X , q) be a quasi-metric space, and let f : X → R∪ {+∞} be an objective function. Suppose that -∞ < f * = inf { f (y) : y ∈ X }. Let ε > 0, λ > 0 and a status quo x ∈ X be given such that

(E0) f (x) < f * + ε.
Then, the EVP is the problem of finding x * = y ∈ X such that (E1) q(x, y) ≤ λ; (E2) f (x)f (y) ≥ ε λ q(x, y); (E3) f (y)f (z) < ε λ q(y, z) ∀z = y. For ε > 0, we set E ε = {x ∈ X : condition E 0 is satisfied}, and we consider S E ( f (•), λ, ε, x) the set of all elements y ∈ X which satisfy the conditions (E1), (E2) and (E3); that is, the set of solutions of the EVP problem.

Let (X , q) be a quasi-metric space. A sequence {x n } in X is said to be left convergent to x ∈ X , if lim n→∞ q(x n , x) = 0. The sequence {x n } is said to be left Cauchy if for every ε > 0, there exists N ε ∈ N such that q(x n , x m ) < ε for all m ≥ n ≥ N ε . A quasi-metric space (X , q) is said to be left-complete if every left Cauchy sequence is left convergent. A function f : X → R ∪ {+∞} is said to be sequentially lower monotone (denoted by slm) if for any sequence {x n } left convergent to an element x 0 ∈ X and such that f

(x n+1 ) ≤ f (x n ) we have that f (x 0 ) ≤ lim inf n→∞ f (x n ) = lim n→∞ f (x n ).
Theorem 1 [START_REF] Qiu | Equilibrium versions of variational principles in quasimetric spaces and the robust trap problem[END_REF]Theorem 3.1] Let (X , q) be left-complete quasi-metric space, Y be a real linear space, K ⊂ Y be a convex cone and k 0 ∈ K \ -vcl(K ), such that K is k 0 -closed. Let F : X × X → 2 Y \ {∅} be a set-valued bimap satisfying the following conditions:

(i) ∃x 0 ∈ X such that F(x 0 , x 0 ) ⊂ -K ; (ii) ∃α ∈ R, ∃s 0 ∈ (-∞, +∞)k 0 -K such that F(x 0 , X ) ∩ (-s 0 + αk 0 -K ) = ∅; (iii) F(x, z) ⊂ F(x, y) + F(y, z) -K , ∀x, y, z ∈ X ; (iv) F is left K -slm in (X , q). Assume that ψ is F-decreasing (that is ψ(x) ≤ ψ(x ) whenever F(x, x ) ⊂-k). Then, there exists x¯ ∈ X such that (a) F(x 0 , x) + ψ(x 0 )q(x 0 , x)k 0 ⊂ -K ; (b) ∀x ∈ X \ { x}, F( x, x) + ψ( x)q( x, x)k 0 ⊂ -K .

According to the above theorem we have the following result which shows the set of solutions of the EVP problem is nonempty.

Theorem 2 (Replication of EVP) Let (X , q) be a left-complete quasi-metric space, and let f : X → R ∪ {+∞} be proper, bounded from below and slm. For ε > 0, λ > 0 and x

∈ E ε , we have S E ( f , λ, ε, x) = ∅. Proof Assume that F : X × X → 2 R \ {∅} is defined by F(u, v) := { f (v) -f (u)} and we consider K := [0, ∞). Suppose that ψ(x) := 1
λ for all x ∈ X and k 0 = ε. Then, conditions (iii), (iv) of Theorem 1 are satisfied and ψ is F-decreasing. By choosing s 0 = 0, x 0 = x and α = -1, it is clear that conditions (i) and (ii) of Theorem 1 are satisfied too. Therefore, by Theorem 1 there exists x ∈ X such that

(a) f ( x) -f (x 0 ) + 1 λ q(x 0 , x)ε ∈ -K ; (b) ∀z ∈ X \ { x}, f (z) -f ( x) + 1 λ q( x, z)ε / ∈ -K .
If we put x = y, then conditions (a) and (b) are equivalent to conditions (E2) and (E3), respectively. Since f (x) < f * + ε and by (E2) we have

ε = ε + f * -f * > f (x) -f (y) ≥ ε λ q(x, y).
Hence, q(x, y) ≤ λ.

At the mathematical level, the goal of this paper is to extend the EVP as follows. Let λ : X x -→ λ(x) ∈ R ++ (R ++ denotes the set of positive real numbers) be a nonconstant, and positive resource function, ε > 0 and a status quo x ∈ X be given such that x ∈ E ε . Then, the generalized Ekeland variational principle (GEVP) problem is to find x * = y ∈ X such that

(VR1) q(x, y) ≤ λ(x); (VR2) f (x) -f (y) ≥ ε λ(x) q(x, y); (VR3) f (y) -f (z) < ε
λ(y) q(y, z) for all z = y. We also set

S G E ( f (•), λ(•), , x) := {y ∈ X : conditions (VR1)-(VR3) are satisfied}, that is, S G E ( f (•), λ(•), ε, x)
is the set of solutions of the GEVP. The main problem in this paper, at the mathematical level, is to give sufficient conditions for the existence of solution of the GEVP problem. This generalization is important because it gives conditions of existence of solution for the GEVP problem and its equivalent formulations when there are several perturbation functions p x (x, y) = ε λ(x) q(x, y) and p y (y, z) = ε λ(y) q(y, z) that do not satisfy the triangle inequality. Related literature Several papers dealing with the EVP already considered various types of perturbations related to the generalized metric q of the underlying space; see for example [START_REF] Bao | Variational analysis in cone pseudo-quasimetric spaces and applications to group dynamics[END_REF][START_REF] Borwein | A smooth variational principle with applications to subdifferentiability and differentiability of convex functions[END_REF][START_REF] Chen | Vector Ekeland's variational principle in an F-type topological space[END_REF][START_REF] Cobzas | Ekeland variational principle in asymmetric locally convex spaces[END_REF][START_REF] Hai | Variants of the Ekeland variational principle for approximate proper solutions of vector equilibrium problems[END_REF][START_REF] Hamel | Phelps' lemma, Danes' drop theorem, and Ekeland's principle in locally convex spaces[END_REF][START_REF] Khanh | Versions of Ekeland's variational principle involving set perturbations[END_REF][START_REF] Kruger | Borwein-Preiss variational principle revisited[END_REF][START_REF] Lin | Variational inclusions problems with applications to Ekeland's variational principle, fixed point and optimization problems[END_REF]. But none are status quo dependent as ours.

Behavioral aspect

The behavioral aspect motivates our interest for the GEVP. At the behavioral level this paper gives conditions for the existence of variational traps, rather easy to reach and difficult to leave, in a changing environment, when available resources are changing each period. Thus, this paper is a continuation of the recent (VR) variational rationality approach of stay (stop) and change (go) human dynamics driven by changing needs and moving goals in a changing environment.

The VR approach This approach [START_REF] Soubeyran | Variational rationality: towards a grand theory of motivation driven by worthwhile moves[END_REF][START_REF] Soubeyran | Variational rationality: the concepts of motivation and motivational force[END_REF][START_REF] Soubeyran | Variational rationality: The resolution of goal conflicts via stop and go approach-avoidance dynamics[END_REF][START_REF] Soubeyran | Variational rationality: a general theory of intentions and moving goals as satisficing worthwhile moves[END_REF][START_REF] Soubeyran | Variational rationality and the unsatisfied man: routines and the course pursuit between aspirations, capabilities and beliefs[END_REF][START_REF] Soubeyran | Variational rationality, a theory of individual stability and change: worthwhile and ambidextry behaviors[END_REF] is highly multidisciplinary because it can model and unify a lot of stay and change human dynamics in behavioral sciences. See [START_REF] Soubeyran | Variational rationality: towards a grand theory of motivation driven by worthwhile moves[END_REF] for different aspects in psychology, economics, management sciences, sociology, political sciences, decision theory, game theory, artificial intelligence and mathematics. For example, stays refer to exploitation phases, temporary repetitions of the same action, temporary habits, routines, rules and norms, while changes refer to exploration phases, learning and innovations processes, forming and breaking habits and routines, changing doings (actions), havings and beings. The spatial and unifying VR "traveler perspective" The VR approach models human dynamics in a locomotion space where individuals change or stay for a while. It models each individual as a traveler and a position as a city. Traveler's behavior means "doing a bundle of situated activities", which represents "living in a city". Doing the same situated bundle in the previous and current periods is like "living in a city for two periods". Changing a behavior means moving to another city. His/Her short-term problem is to choose, at the beginning of each period, between to live again in the status quo city or to live in a new city. The longterm problem is to reach a final (desired) city. The presence of costs of moving generates inconveniences (hence resistances) to change. Then, it may be not worthwhile to move even if there are advantages of moving. This defines a variational trap " rather worthwhile to reach, but not worthwhile to quit". Thus, the VR approach focuses the attention on the existence of traps which prevent individuals from reaching their desired ends (desires). The VR approach is unifying. It can provide a general theory of motivation, moving goals and intentions as well as a general theory of self regulation. Its traveler perspective helps to get closer to the Lewin's dream of "Topological psychology" (see [START_REF] Lewin | Principles of Topological Psychology[END_REF][START_REF] Lewin | The Conceptual Representation and Measurement of Psychological Forces[END_REF]). Moreover, its formulation exhibits a striking and an almost one to one correspondence between famous principles in psychology and most of the famous variational principles and leading optimizing algorithms in the recent developments of variational analysis, see [START_REF] Soubeyran | Variational rationality: towards a grand theory of motivation driven by worthwhile moves[END_REF].

Main topics of the VR approach

The VR approach starts with the idea that all things are changing at two levels. First, in our internal environment (body and head), our needs and the goals, thoughts, feelings, emotions, preferences and utilities that help us to choose how much of each unsatisfied need must be satisfied, how and when. Second, in our external environment, all the related means (things, objects, peoples, and landscapes). The difficulty is that, most of the time, internal and external environments are changing endogenously, because doing something changes the environment where this thing has been done. The main topic of the VR approach: stop (go)-go (stop) human dynamics The VR approach has modeled life as a succession of stays and changes, "stop and go", "go and stop" dynamics where, each period, individuals, i) stay, i.e., continue doing some situated activities and, ii) change, i.e., stop and start doing other situated activities. This is in accordance with Lewin's view of human behaviors (see [START_REF] Lewin | Principles of Topological Psychology[END_REF][START_REF] Lewin | The Conceptual Representation and Measurement of Psychological Forces[END_REF]), one of the giants that can be compared to Freud, among all the psychologists of the last century. He saw life as "a constant interplay between completing old situations and opening up new ones".

The main VR question

The VR approach is driven by the following question: when to start, continue, and stop changing in a changing internal and external environment, for an individual, an organization or several interacting individuals (games)? As such, it is a theory of starts, ends and transitions, where ends can be desired or not (desires or traps). More concretely, the VR approach considers the following main questions that each individual can pose: i) I am there, in an undesirable position, ii) I want to be elsewhere, sooner or later, in a better position, iii) I am unable to reach in one move this momentary desired position within the current period. Hence, iv), I must accept to follow a transition, that is, a succession of acceptable moves (changes or stays) over a succession of periods. The way the VR approach poses this question provides the main verbal and spatial model that drives the VR approach. This basic model must be decomposed in two submodels: i) a model and its variants for proximal (short run) dynamics, and, ii) a model for distal (long run) dynamics, including also a lot of variants.

Generalized variational principles

In this section, we study the existence of solutions for the GEVP problem, where the positive resource factor λ : X

x -→ λ(x) ∈ R ++ is not constant. Then, perturbation p can be redefined as p(x, y) = 1 λ(x) q(x, y), where q is a quasi-metric. This apparently small change does not provide a generalization if p itself is a quasi-metric. A quasi-metric space (X , q) is called bounded if for some (equivalently for all) x ∈ X , sup{q(x, z) : z ∈ X } < ∞.

In the following result, we show that if λ is not constant, then p does not have the triangle property.

Proposition 1 Suppose that (X , q) is a quasi-metric space, q is unbounded and λ(•) is a positive function on X . If p(x, y) := 1 λ(x) q(x, y), for every x, y ∈ X and p satisfies the triangle inequality, then λ(•) is a constant function.

Proof Let x, y ∈ X and x = y. Since q is unbounded, there exist k ∈ X and a sequence {z n } in X such that lim n→∞ q(k, z n ) = +∞. By triangle inequality we have

lim n→∞ q(x, z n ) = lim n→∞ q(y, z n ) = +∞. Since q(x,z n ) q(y,z n ) ≤ q(x,y) q(y,z n ) + 1, then lim sup n→∞ q(x,z n ) q(y,z n ) ≤ 1. Similarly, lim sup n→∞ q(y,z n ) q(x,z n ) ≤ 1, so that lim n→∞ q(x, z n ) q(y, z n ) = lim n→∞ q(y, z n ) q(x, z n ) = 1,
because, for any sequence {α n } of positive real numbers lim sup

n 1 α n = 1 lim inf n α n .
From the triangle inequality property of p, we get p(y, z n ) ≤ p(y, x) + p(x, z n ), for all n ∈ N, and so

1 λ(y) q(y, z n ) q(x, z n ) ≤ 1 λ(y) q(y, x) q(x, z n ) + 1 λ(x) .
Letting n → ∞, one obtains, 1 λ(y) ≤ 1 λ(x) , or equivalently, λ(y) ≥ λ(x). Since x, y are arbitrary points, λ(x) ≥ λ(y), and so λ(x) = λ(y). Hence, λ(•) is a constant function.

In continuation of our work, we discuss the necessity of investigating the GEVP is independent of the EVP. In fact, we show that we may have S G E = ∅ and S E = ∅ simultaneously; while on the other hand, S G E = ∅ and S E = ∅ may be true at the same time. The following example shows that S G E for a suitable quasi-metric space (X , q), a positive function λ(•) and a real valued function f can be nonempty, but S E is a empty set.

Example 1 Let X = R + and f : X → R + be defined as follow:

f (x) := √ x x > 0, 1 x = 0.
We have f * = 0 so that the following conditions must be satisfied by x and y:

(E0) 0 < ε < 1 2 , 0 < √ x < ε; (E1) |x -y| ≤ 1; (E2) f (x) -f (y) ≥ ε|x -y|; (E3) f (y) -f (z) < ε|y -z| ∀z ∈ X \ {y}.
We distinguish 3 cases. Case I. Suppose that y = 0. Then, (E3) is equivalent to

εz + √ z -1 > 0 ∀z = 0,
which obviously fails for sufficiently small z.

Case II. Suppose that y = x. Then, by (E3)

ε √ x > 1 -ε √ z, ∀z ∈ (0, x), which implies √ x ≥ 1 ε > 2 in contradiction to (E0
). Case III. Assume that y ∈ (0, ∞) \ {x}. Then, (E2) implies 0 < y < x and

ε( √ x + √ y) ≤ 1. (1)
On the other hand, by (E3), we have

ε √ y > 1 -ε √ z ∀z ∈ (0, y).
Therefore, ε √ y ≥ 1, and so ε( √ x + √ y) > 1 which contradicts to (1). Hence,

S E ( f (•), 1, ε, x) = ∅.
Let now λ : X → R ++ be given by

λ(x) := x x > 0, 1 x = 0.
The conditions that must be satisfied are

(E0) ε > 0, 0 < f (x) < ε; (VR1) |x -y| ≤ λ(x); (VR2) f (x) -f (y) ≥ ε λ(x) |x -y|; (VR3) f (y) -f (z) < ε λ(y) |y -z| for all z ∈ X \ {y}. Case I. If x = 0, then ε > 1. For y > 0, (VR2) is equivalent to εy + √ y -1 ≤ 0.
The inequalities 2 .

y > 0 and εy + √ y -1 ≤ 0 hold if and only if 0 < √ y ≤ -1 + √ 1 + 4ε 2ε , or equivalently 0 < y ≤ 1 + 2ε - √ 1 + 4ε 2ε
We show that (VR3) holds for all these y. If z > 0 and z = y, (VR3) becomes

√ y - √ z < ε y |y -z|.
If z > y, then this inequality is trivially true. If 0 < z < y, then it is equivalent to

y -ε √ y < ε √ z. Since √ y ≤ -1+ √ 1+4ε 2ε 2 < ε, it follows y -ε √ y = √ y( √ y -ε) < 0 < ε √ z. If z = 0, then (VR3) becomes √ y -1 < ε, which is true, because √ y < ε < 1 + ε.
If y = 0, then by (VR3) we must have εz + √ z -1 > 0, for all z > 0, which fails for sufficiently small z. Consequently

S G E ( f (•), λ(•), ε, 0) = {y ∈ R : 0 < y ≤ 1 + 2ε - √ 1 + 4ε 2ε 2 }.
Case II. If x > 0, then 0 < √ x < ε. In this case, take y = x. Then (VR1), (VR2) are obviously satisfied.

If z = 0, then (VR3) is equivalent to √ x -1 < ε, which is true, because √ x < ε. If 0 < z < x, then it is equivalent to x -ε √ x < ε √ z, which holds, because x -ε √ x = √ x( √ x -ε) < 0. If z > x, then (VR3) is trivially true.
In what follows we assume that (X , q) is a quasi-metric space, f : X → R ∪ {+∞} is a proper bounded from below function and λ(•) is a positive function on X . We also set arg min f := {x ∈ X : f (x) = inf y∈X f (y)}. We consider B[a, r ] := {x ∈ X : q(a, x) ≤ r }, B(a, r ) := {x ∈ X : q(a, x) < r }, where a ∈ X and r > 0. We now obtain relation among the sets

S G E ( f (•), λ(•), ε, x) and S E ( f (•), λ(x), ε, x). It is easy to see that if ε > 0, x ∈ E ε and f (•) or λ(•) is constant, then S E ( f (•), λ(x), ε, x) = S G E ( f (•), λ(•), ε, x). Theorem 3 Suppose that ε > 0, x ∈ E ε , S E ( f (•), λ(x), ε, x) = ∅ and the following condi- tion holds λ(x) ≥ λ(y), ∀y ∈ S E ( f (•), λ(x), ε).
(2)

Then S E ( f (•), λ(x), ε, x) ⊂ S G E ( f (•), λ(•), ε, x), and so S G E ( f (•), λ(•), ε, x) = ∅.
Proof Since S E ( f (•), λ(x), ε, x) = ∅, then conditions (VR1) and (VR2) hold. From condition (E2), we have f (y) ≤ f (x) for all y ∈ S E ( f (•), λ(x), ε, x). Therefore, from (2) we deduce that λ(x) ≥ λ(y). Hence, to obtain condition (VR3), it is enough to replace λ(x) in condition (E3) by λ(y). Thus,

S E ( f (•), λ(x), ε, x) ⊂ S G E ( f (•), λ(•), ε, x). The function λ(•) is said to be f -increasing iff λ(x 1 ) ≤ λ(x 2 ) if f (x 1 ) ≤ f (x 2 )
, where x 1 , x 2 ∈ X ; see [START_REF] Qiu | Equilibrium versions of variational principles in quasimetric spaces and the robust trap problem[END_REF]. Motivated by this notion, we introduce the concept of f -proper increasing which is weaker than the notion of f -increasing.

Definition 1 The function λ(•) is said to be f -proper increasing iff λ(x 1 ) ≤ λ(x 2 ) if f (x 1 ) ≤ f (x 2 ) < +∞, where x 1 , x 2 ∈ X .
It is clear that if λ(•) is f -increasing, then it is f -proper increasing, but the following example shows that the converse of this fact does not necessarily hold in general.

Example 2 Let f (x) := e x x ∈ Z, +∞ x / ∈ Z. If λ(x) := e x , then it is clear that λ is f -proper increasing, but it is not f -increasing.
As a consequence of Theorem 3, we obtain the following result.

Corollary 1 Suppose that ε > 0, x ∈ E ε and S E ( f , λ(x), ε, x) = ∅. If λ(•) is f -proper increasing, then S G E ( f (•), λ(•), ε, x) = ∅.
Proof It is enough to show that condition (2) holds. For this purpose, suppose that y ∈ S E ( f (•), λ(x), ε, x). From condition (E2), we have f (y) ≤ f (x). On the other hand x ∈ E ε , thus f (x) < +∞. Therefore, λ(y) ≤ λ(x), and so condition (2) in Theorem 3 holds.

Remark 1 (a) If (X , q) is left-complete and f is slm, then by Theorem 2 we have

S E ( f (•), λ(x), ε, x) = ∅. (b)
Corollary 1 is a similar result to Theorem 1 for real valued functions.

The following example shows that condition (2) is weaker than the f -proper increasing condition relative to λ(•).

Example 3 Suppose that k ≥ 1, f (x) := e x and λ(x) := k 1 e x for all x ∈ R. It is clear that λ is not f -proper increasing and S E ( f (•), λ(x), ε, x) ={x + },for every ε>0and x ∈ E ε . Hence, condition (2) is satisfied.
We now skim conditions under which the GEVP does not hold. However, according to Theorem 2, one can see

that if λ(•) is a positive constant function, ε>0and x ∈ E ε ,then S GE ( f (•), λ(•), ε, x) = S E ( f (•), λ(x), ε, x) =∅.
Proposition 2 Suppose that ε>0,x ∈ E ε and inf X f < inf A f , for every bounded subset A of X. Then there exists λ :

X → R ++ such that S GE ( f (•), λ(•), ε, x) =∅. Proof Since the set A := {x} is bounded, then inf X f < inf A f = f (x). Therefore, there exists x 1 ∈ X such that f (x 1 ) < f (x). If we define λ(x) := 2εq(x, x 1 ) f (x) -f (x 1 ) ,
we have

f (x 1 ) + ε λ(x) q(x, x 1 ) < f (x 1 ) + 2ε λ(x) q(x, x 1 ) = f (x),
and so

x / ∈ S G E ( f (•), λ(•), ε, x). On the other hand, from inf X f < inf B[x,λ(x)] f , there exists x 2 ∈ X such that f (x 2 ) < inf B[x,λ(x)] f
. Now, for z = x, we consider λ(•) as follows:

λ(z) := 2εq(z,x 2 ) inf B[x,λ(x)] f -f (x 2 ) z ∈ B[x, λ(x)] and z = x 1 o.w. Hence, f (x 2 ) + ε λ(z) q(z, x 2 ) < f (z) for any x = z ∈ B[x, λ(x)], and so z / ∈ S G E ( f (•), λ(•), ε, x). Therefore, S G E ( f (•), λ(•), ε, x) = ∅.
Remark 2 (a) It is clear that for every bounded subset A of X , inf X f < inf A f iff there is no bounded minimizing sequence. Furthermore, there exists a bounded subset A of X such that inf X f = inf A f , whenever arg min f = ∅. (b) If arg min f = ∅ and X is a finite dimensional normed space, then inf X f < inf A f for every bounded subset A of X . Suppose that A ⊆ X is bounded, then cl A (the closure of A) is compact and so there exists x ∈ cl A such that

f ( x) = inf cl A f . Hence, inf X f < f ( x) = inf cl A f ≤ inf A f .
(c) It is well known that the closure of a bounded subset A of a reflexive Banach space X with respect to the weak topology is weakly compact. Hence, by the same proof as that of the part (b), one can obtain that inf X f < inf A f , for every bounded subset A of X , in the case where f is lower semicontinuous with respect to the weak topology and arg min f = ∅.

The following example shows that condition inf X f < inf A f , for every bounded subset A of X , in the above result cannot be omitted.

Example 4 Suppose that f : R → R is defined as follows:

f (x) :=| x | ∀x ∈ R. Then inf R f = inf [-a,a] f for each a ∈ R + . We shall show that S G E ( f (•), λ(•), ε, x) = ∅, for every positive function λ(•), ε > 0 and for each x ∈ E ε . If x ∈ E ε and x = 0, then it is clear that x ∈ S G E ( f (•), λ(•), ε, x). So, we suppose that x = 0. If ε λ(x) < 1, then we have | 0 | + ε λ(x) | x -0 |≤| x |.
In this case it is easy to see that

y = 0 ∈ S G E ( f (•), λ(•), ε, x). If ε λ(x) ≥ 1, then x ∈ S G E ( f (•), λ(•), ε, x).
In order to characterize the non-emptiness of S G E ( f (.), λ(.), ε, x), where x ∈ E ε , we need the following concept. A quasi-metric space (X , q) is said to have the property (P), if for each u = v and for every positive real number r with q(u, v) < r , there exists w ∈ X such that r = q(u, w) = q(u, v) + q(v, w). Now, we show that the quasi-metric induced by an asymmetric norm on a real vector space (see [START_REF] Cobzaş | Functional Analysis in Asymmetric Normed Spaces[END_REF]) has the property (P). An asymmetric norm on a real vector space E is a functional p : E → [0, +∞) satisfying the following conditions: (AN1) p(u) > 0, whenever u = 0 (AN2) p(αu) = α p(u), for all u ∈ E and α ≥ 0;

(AN3) p(u + v) ≤ p(u) + p(v) for all u, v ∈ E.
Proposition 3 Let p be an asymmetric norm on a real vector space E and q p be a quasi metric on E defined by q p (u, v) = p(vu) for all u, v ∈ E. Then, (E, q p ) has the property (P).

Proof Let u, v ∈ E, u = v and r be a positive real number with

q p (u, v) < r . If w = r q p (u,v) (v -u) + u, then q p (u, w) = p(w -u) = p( r p(v -u) (v -u)) = r , q p (v, w) = p(w -v) = p( r -p(v -u) p(v -u) (v -u)) = r -p(v -u).
Hence, q p (u, w) = q p (u, v) + q p (v, w).

Theorem 4 Let (X , q) be a left-complete quasi-metric space, then the following statements hold.

(i) If f is slm, ε > 0 and x ∈ E ε ∩ arg max B[x,λ(x)] λ, then S G E ( f (•), λ(•), ε, x) = ∅.
(ii) If (X , q) has the property (P) and x / ∈ arg max B(x,λ(x)) λ, then for every ε > 0 there exists a slm function f on X such that x ∈ E ε and S G E ( f (•), λ(•), ε, x) = ∅. Proof (i) Suppose that ε > 0, f is slm and x ∈ arg max B[x,λ(x)] λ ∩ E ε . From Theorem 2, there exists y ∈ S E ( f , λ(x), ε, x). Hence, conditions (VR1), (VR2) hold. By condition (E3) we have f (z) + ε λ(x) q(y, z) > f (y), for all z = y. Since y ∈ B[x, λ(x)], then λ(y) ≤ λ(x), and so

f (z) + ε λ(y) q(y, z) ≥ f (z) + ε λ(x) q(y, z) > f (y), ∀z = y. Thus, y ∈ S G E ( f (•), λ(•), ε, x) and S E ( f (•), λ(x), ε, x) ⊂ S G E ( f (•), λ(•), ε, x). (ii) If x / ∈ arg max B(x,λ(x))
λ, then there exists y ∈ B(x, λ(x)) with 0 < λ(x) < λ(y). For a given ε > 0, we choose n ∈ N sufficiently large such that

k 1 := λ(x) λ(y) + q(x, y) λ(x) - q(x, y) λ(y) ε + 3 n < ε,
and let

k 2 := λ(x) λ(y) - q(x, y) λ(y) ε + 2 n
, and

k 3 := 1 n . It is clear that 0 < k 3 < k 2 < k 1 < ε. Now, we define f (t) := ⎧ ⎨ ⎩ k 1 (1 + q(x, t)) t ∈ B(x, λ(x)) and t = y, k 2 t = y, k 3 exp(λ(x) -q(x, t)) o.w.
It is obvious that f is slm and x ∈ E ε . We shall show that S GE ( f (•), λ(•), ε, x) =∅.Bythe property (P), there exists w ∈ X such that λ(x) = q(x,w) = q(x, y) + q(y,w). Hence, we

have f (x) = k 1 , f (y) = k 2 and f (s) = k 3 , for each s ∈ {t ∈ X : q(x, t) = λ(x)}. Form conditions (VR1), (VR2) and x ∈ E ε , we have S G E ( f (•), λ(•), ε, x) ⊂ B[x, λ(x)] ∩ {t ∈ X : f (t) ≤ f (x)}. Hence, S G E ( f (•), λ(•), ε, x) ⊂ {x, y} ∪ {t ∈ X : q(x, t) = λ(x)}. If x ∈ S G E ( f (•), λ(•), ε, x), condition (VR3)(for z = y) gives k 2 + ε λ(x) q(x, y) ≥ k 1 .
Thus,

k 1 -k 2 = ε λ(x) q(x, y) + 1 n ≤ ε λ(x) q(x, y). But this is impossible. If y ∈ S G E ( f (•), λ(•), ε, x), condition (VR3)(for z = w) gives k 3 + ε λ(y) q(y, w) ≥ k 2 .
Therefore,

k 2 -k 3 = λ(x) λ(y) - q(x, y) λ(y) ε + 1 n ≤ ε λ(y) q(y, w) = ε λ(y) (λ(x) -q(x, y)), which is absurd. If s ∈ {t ∈ X : q(x, = ∩ S G E ( f (•), λ(•), ε, x), then from condition (VR2) we have ε = ε λ(x) q(x, s) ≤ f (x) -f (s) = k 1 -k 3 < k 1 < ε, which is a contradiction.
In what follows we give two examples of a positive function λ(•) such that x ∈ arg max B[x,λ(x)] λ.

Example 5 (I) Suppose that a, b > 0 with a = b and consider the following quasi-metric on

R + . q(x, y) := a(y -x) if y ≥ x, b(x -y) if y < x.
Then, the set

B[x, λ(x)] = y ∈ R + : q(x, y) ≤ λ(x) is the closed interval [x-λ(x) b , x+ λ(x)
a ]. Suppose that λ : R + → R ++ is defined as follows:

λ(x) := 2-| x -1 | x ∈ [0, 2], 1 o.w.
Since λ(1) ≥ λ(x) for all x ∈ R + , then 1 ∈ arg max B[1,λ(1)] λ. (II) Let p be an asymmetric norm on a real vector space E, and let λ : E → R ++ be defined as follows:

λ(x) := 2 -p(x) p(x) < 2, 1 p(x) ≥ 2.
Then λ(0) = 2 ≥ λ(x) for all x ∈ E, and so 0 ∈ arg max B[0,λ(0)] λ.

In the sequel, from the GEVP we deduce two general forms of Caristi's theorem [START_REF] Caristi | Fixed point theorem for mappings satisfying inwardness conditions[END_REF] and Takahashi's theorem [START_REF] Takahashi | Existence theorems generalizing fixed point theorems for multivalued mappings[END_REF]. We also show that these results are equivalent.

Theorem 5 (Caristi) Suppose that S G E ( f (•), λ(•), ε, x) = ∅ for every ε > 0 and x ∈ E ε , then the following statements hold.

(i) If T : X ⇒ X is a set-valued mapping with nonempty values satisfying

1 λ(u) q(u, v) ≤ f (u) -f (v), ∀u ∈ X , ∀v ∈ T (u), ( 3
)
then there exists w ∈ X such that T (w) = {w}.

(ii) If T : X ⇒ X is a set-valued mapping with nonempty values satisfying ∀u ∈ X , ∃v ∈ T (u) such that 1 λ(u) q(u, v) ≤ f (u) -f (v), ( 4
)
then T has a fixed point in X .

Proof Let ε = 1 and y ∈ S G E ( f (•), λ(•), 1, x).
Then, by condition (VR3) we have

f (y) -f (z) < 1 λ(y) q(y, z) ∀z = y. ( 5
)
(i) We show that T (y) = {y}. Assume on the contrary that there exists y = z ∈ T (y). Then by (3), we get

1 λ(y) q(y, z) ≤ f (y) -f (z),
which contradicts with [START_REF] Bao | Fixed points and variational principles with applications to capability theory of wellbeing via variational rationality[END_REF].

(ii) We show that y ∈ T (y). Assume on the contrary that y / ∈ T (y). Hence, from (4) there exists y = z ∈ T (y) such that 1 λ(y) q(y, z) ≤ f (y)f (z), which contradicts [START_REF] Bao | Fixed points and variational principles with applications to capability theory of wellbeing via variational rationality[END_REF].

Theorem 6 (Takahashi) Assume that S G E ( f (•), λ(•), ε, x) = ∅ and the following condition holds.

∀x ∈ X with inf

X f < f (x) ∃u = x : f (u) + ε λ(x) q(x, u) ≤ f (x). ( 6
)
Then arg min f = ∅.

Proof Suppose that y ∈ S G E ( f (•), λ(•), ε, x), then by (VR3) we have f (y) -f (z) < ε λ(y) q(y, z) ∀z = y (7)
We show that y ∈ arg min f . If y / ∈ arg min f , then from (6) there exists u = y such that

f (u) + ε q(y, u) ≤ f (y). ( 8

) λ(y)

From Eqs. ( 7) and( 8), we have f (y)< f (y) which is a contradiction. Now, we show that Caristi's theorem and Takahashi' theorem are equivalent.

Theorem 7 Part (i) of Theorem 5 and the following statements are equivalent.

(EV) There exists y ∈ X such f (y) < f (z) + 1 λ(y) q(y, z), ∀z = y. (TA) If for every x ∈ X with inf X f < f (x) there exists u = x such that

f (u) + 1 λ(x) q(x, u) ≤ f (x),
then arg min f = ∅.

Proof By Theorems 5 and 6, condition (EV) imply Theorem 5(i) and (TA), respectively. (TA)⇒(EV). If (EV) does not hold, then

∀y ∈ X ∃y = z ∈ X suchthat f (z) + 1 λ(y) q(y, z) ≤ f (y). (9)
Therefore, by condition (TA) there is a ȳ ∈ X such that f ( ȳ) = inf X f . On the other hand, by ( 9) there exists ȳ = z ∈ X such that

f (z) + 1 λ( ȳ) q( ȳ, z) ≤ f ( ȳ).
Hence,

1 λ( ȳ) q( ȳ, z) ≤ f ( ȳ) -f (z) ≤ 0.
Thus, q( ȳ, z) = 0, and so ȳ = z which is a contradiction. Theorem 5(i)⇒ (EV). Let T : X ⇒ X be defined as follows:

T (u) = {v ∈ X : f (v) + 1 λ(u) q(u, v) ≤ f (u)}.
Hence, 1 λ(u) q(u, v) ≤ f (u)f (v) for any u ∈ X and v ∈ T (u). Part (i) of Theorem 5 implies that there exists y ∈ X such that T (y) = {y}. Therefore, if z ∈ X and z = y, then z / ∈ T (y), and so (EV) holds.

3 Application: how the size of a monopoly changes in a changing environment

The VR approach: a short summary

To save space we present very briefly the VR approach [START_REF] Soubeyran | Variational rationality: towards a grand theory of motivation driven by worthwhile moves[END_REF][START_REF] Soubeyran | Variational rationality: the concepts of motivation and motivational force[END_REF][START_REF] Soubeyran | Variational rationality: The resolution of goal conflicts via stop and go approach-avoidance dynamics[END_REF][START_REF] Soubeyran | Variational rationality: a general theory of intentions and moving goals as satisficing worthwhile moves[END_REF][START_REF] Soubeyran | Variational rationality and the unsatisfied man: routines and the course pursuit between aspirations, capabilities and beliefs[END_REF][START_REF] Soubeyran | Variational rationality, a theory of individual stability and change: worthwhile and ambidextry behaviors[END_REF] on a leading example relative to the limit size of a monopoly. This reminder is mandatory if we want the reader to understand the interest of the mathematical part of the paper.

The need for change The following example is borrowed from [START_REF] Soubeyran | Variational rationality: towards a grand theory of motivation driven by worthwhile moves[END_REF]. It is one of the leading models that drive the VR approach in economics and management sciences. Consider the simplest model of an organization, that is, a monopoly that produces, within a given period, ψ(x) = x ∈ R + units of a final good, using x ∈ X = R + workers. This monopoly sells x ∈ R + units of a final good at the unit price p = p s (x) = abx ≥ 0, a, b > 0 and pays a wage w > a to each worker. Thus, production costs are wage costs wx. Let s be the environment of the monopoly in the current period (see below its content). Then, the per period profit of this firm is g s (x) = p(x)x -wx. That is, g s (x) = (a -w)xbx 2 . The size of this monopoly is, either the number of workers x it uses or, how much units x of the final good it produces and sells. Let g * s = sup {g s (y) : y ∈ X } < +∞ be the highest profit the entrepreneur can hope to satisfy within a period. Then, the optimal size x * of this monopoly solves the equation g s (x * ) = g * s . This example gives its ideal size x * s = (a -w)/2b and the ideal profit level g * s = (a -w) 2 /4b. Starting from the status quo x, the discrepancy f s (x) = g * sg s (x) ≥ 0 models the unrealized profit of the monopoly when its size is x at the status quo. When it exists, an unrealized profit f s (x)>0 in the previous period generates an unsatisfaction/frustration feeling that pushs the monopoly, in the current period, to change from having employed x workers in the previous period to employ y = x workers in the current period. Thus, moving means, either, i) to change, i.e., to hire yx > 0workers,or to fire xy > 0 workers or, ii) to stay, stopping to hire or to fire workers if y = x. The environment Let s = (x,λ(x), ε, a, b,w, α + ,α ) be the environment of the monopoly in the current period. In this simple example it includes the status quo x,thesizeλ(x)>0 of a pool of workers that the monopoly can hire or fire, ε>0, the level a and the elasticity of the inverse demand b, the per worker wage w, and the unit cost to fire or to hire a worker α + > 0,α > 0. These unit costs have material, financial, psychological and social aspects. Notice that ε is a upper bound of the unrealized profit: 0 ≤ f s (x) ≤ ε. Advantages to move (change rather than stay) The entrepreneur will have some advantages to move from x to y, i.e., A s (y/x) = g s (y)g s (x) = f s (x)f s (y)>0, if he can improve his profit from g s (x) to g s (y)> g s (x) or decrease his frustration feelings from f s (x) to f s (y)< f s (x). In the example, A s (y/x) = g s (y)g s (x) = (a -w)(yx)b(y 2x 2 ). Inconveniences to move from the status quo To save space, we limit the complex definition of inconveniences to move I s (y/x) to the monopoly example. In this case they represent costs to hire yx ≥ 0workers I s + (y/x) = α + (yx), if y ≥ x, or costs to fire xy > 0 workers I s (y/x) = α (xy), if y < x. For the general case; see [START_REF] Soubeyran | Variational rationality: towards a grand theory of motivation driven by worthwhile moves[END_REF]. Thus, we will suppose that limited knowledge relative to the inverse demand function p(x) = abx,and limited time and resources forbid the entrepreneur to reach the optimal size of the firm x * within the current period. Motivation and resistance to move Motivation to move (change rather than stay) M s (y/x) = U s [A s (y/x)] is the utility of advantages to move. Resistance to move R s (y/x) = D s [I s (y/x)] is the disutility of inconveniences to move. The utility and disutility functions U s [.] : X -→ R + , D s [.] : X -→ R + are, i) zero at zero, i.e., U s [0] = 0 = D s [0] and, ii) strictly increasing. The balance B s (y/x) = M s (y/x) -ξ s R s (y/x) between motivation M s (y/x) and resistance to move R s (y/x) weights the difference between them. The weight ξ s > 0 models how much an individual gives importance to resistance to move compared to motivation. This is a way to model the status quo bias. Worthwhile moves A move m = x y is worthwhile if the balance B s (y/x) = M s (y/x)ξ s R s (y/x) ≥ 0 between motivation and resistance to move (change rather than stay) is non negative. That is, if motivation to move is high enough relative to resistance to move, i.e., if M s (y/x) ≥ ξ s R s (y/x). Desires A position x * ∈ X is a desire (desired end) if A s (z/x * ) ≤ 0forall z ∈ X. In the example the equilibrium condition A s (z/x * ) = g s (z)g s (x * ) ≤ 0forall z ∈ X defines a maximum of the profit function g s (.). Stationary traps The position x * ∈ X is a (strict) stationary trap if, starting from the status quo x * , there is no way to find a worthwhile change x [.] is invertible, it is equivalent to the resource constraint I s (y/x) ≤ λ s . In the monopoly example λ s = λ(x) > 0 models the size of a pool of workers that the entrepreneur can hire or fire. Variational traps in a changing environment The VR approach [START_REF] Soubeyran | Variational rationality: towards a grand theory of motivation driven by worthwhile moves[END_REF] makes a clear distinction between a proximal dynamic, which runs over a small number of periods, and a distal (longuer) dynamic. More precisely, it models a proximal dynamic as a three periods model, including the previous period, the current period (first effective period) and the next period (second effective period). One of its main discovery has been to show how the EVP is a specific instance of a VR proximal dynamic, where an entity (for example, a monopoly), i) period 1: starts moving, making a worthwhile move x y = x * in the initial environment s:

* z = x * . That is, if B s (z/x * ) = M s (z/x * ) -ξ s R s (z/x * )<0, i.e., if M s (z/x * )<ξ s R s (z/x * ) for all z ∈ X {x * } . The equilibrium condition B s (z/x * )< 0forall z ∈ X, z = x * ,
M s (y/x) ≥ ξ s R s (y/x) with 0 ≤ R s (y/x) ≤ λ(x);
ii) period 2: stops moving, having reached a stationary trap, in a new (or the same) environment s :

M s (z/y) < ξ s R s (z/y) forall z = y = x * .
This worthwhile proximal dynamic defines a variational trap, worthwhile to reach (period 1), but not worthwhile to leave (period 2).

Interpretations of the results: when to stop changing when the environment changes

Behavioral results

This paper shows the existence of traps in a changing environment in a specific context. This context considers resources (a pool of workers) as changing from λ s = λ(x) to λ s = λ(y), when the size of the monopoly changes from x to y. That is, A) In the first period, the monopoly starts moving, making a worthwhile move. The environment of the monopoly in the first period is s = (x, λ(x), ε, a, b, w, α + , α ). To be concrete, consider the following environment in the first period:

s = (x, λ(x), ε > 0, a = 2, b = 1/2, w = 1, α + = 1, α = 2).
Then, in the previous period, the profit function at x was g s (x) = g(x) = x -(1/2)x 2 , the optimal size of the monopoly was x * s = x * = 1, the highest profit was g * s = g * = 1/2 and unrealized profits at x has been

f s (x) = f (x) = g * -g(x).
In this first period advantages to move are

A s (y/x) = A(y/x) = g(y) -g(x) = f (x) - f (y) = (y -x) -(1/2)(y 2 -x 2
). Inconveniences to move are I s (y/x) = I (y/x), where I (y/x) = yx if y ≥ x and I (y/x) = 2(xy), if y < x. Then, q : X × X (x, y) -→ q(x, y) = I (y/x) ∈ R + is a quasi-metric. The resource constraint in the first period is I (y/x) = q(x, y) ≤ λ(x). We consider a linear formulation of motivation and resistance to move, M s = U s [A s ] = A s and R s = D s [I s ] = δ s I s where δ s = ε λ(x) > 0 and ε > 0. Then, a worthwhile move x y from the previous period to the current (first) period satisfies condition(VR1) g(y)g(x) = f (x)f (y) ≥ ε λ(x) q(x, y). Notice that the formulation R s = D s [I s ] = δ s I s with δ s = ε λ(x) > 0 means that when the pool of workers λ(x) increases, resistance to move R s decreases because it is easier to hire or to fire workers. B) In the second period, the monopoly stops moving because it reaches a stationary trap. Condition (VR3) g(z)g(y) = f (y)f (z) < ε λ(y) q(y, z) for all z = y, means that, in the second period, the monopoly has reached a variational trap, worthwhile to reach, but not worthwhile to leave. At the same time the environment moved from s = (x, λ(x), ε, a, b, w, α + , α ) in the current period (first period) to s = (y, λ(y), ε, a, b, w, α + , α ) in the next period (second period). In this context, the resource constraint q(y, z) ≤ λ(y) plays no effective role in the second period. Interpretation of Proposition 2 We remind that, in the monopoly example, i) g(x) is the profit of the monopolist when he produces x units of the final good, that is, when he employs x workers, ii) g * = g * X = sup {g(z) : z ∈ X } < +∞ is his aspiration level, that is, the highest profit he can hope to get and, iii) f (x) = g *g(x) ≥ 0 is his unrealized profit at the status quo x. Then, inf X f = g *g * = 0. Let g * A = sup {g(z) : z ∈ X } ≤ g * X be the highest profit the monopolist can hope to reach if he produces a bounded quantity x ∈ A ⊂ X = R + of the final good (a mild hypothesis !). Then, condition inf X f < inf A f for each bounded subset A of X is equivalent to g * A < g * X for each bounded set A ⊂ X . Thus, Proposition 2 means that a monopolist will not reach his idealistic size g * X as long as he produces a bounded number of units of the final good. This is not the case in our example where the optimal size 0 < x * = (a -w)/2b < +∞ of the monopoly is finite and exists.

Interpretation of Theorem 4 Condition

x ∈ arg max B[x,λ(x)] λ = max {λ(y) : y ∈ X , q(x, y) ≤ λ(x)} , [START_REF] Cobzaş | Functional Analysis in Asymmetric Normed Spaces[END_REF] means that, away from the status quo, for each y ∈ B[x, λ(x)] that satisfies the resource constraint q(x, y) ≤ λ(x) the resource constraint at y will be more severe than at the status quo x, that is, λ(y) ≤ λ(x). Theorem 4 shows that under this condition (and the given other ones), starting from the status quo x, a variational trap will exist. This condition is beautiful and intuitive, because it supposes that resistance to change will be not lower at any y ∈ B[x, λ(x)] than at x. Thus, this is in favor of the existence of a trap at some y ∈ B[x, λ(x)].

The behavioral interpretations of the Caristi theorem [START_REF] Caristi | Fixed point theorem for mappings satisfying inwardness conditions[END_REF] and the Takahashi theorem [START_REF] Takahashi | Existence theorems generalizing fixed point theorems for multivalued mappings[END_REF] follow easily. But their interpretations require too much space and other examples. They will be examined elsewhere.

  defines a (strict) maximum x * of the worthwhile balance function B s (./x * ) on X. Resistance to change constraint It looks like 0 ≤ R s (y/x) ≤ R s < +∞. It comes from a lot of different reasons; for example resource constraints. Given that R s (y/x) = D s [I s (y/x)] and if D s

Acknowledgements

The authors would like to thank the associate editor and anonymous reviewer for their constructive comments, which helped us to improve the paper.