P Delépine

C Guillaume

† T Montier

J C Clément

J J Yaouanc

H Des Abbayes

F Berthou

A Le Pape

C Férec

Biodistribution study of phosphonolipids: a class of non-viral vectors efficient in mice lung-directed gene transfer

Keywords: lipoplexes, phosphonolipids, biodistribution, imaging, small animal model

Background A multitude of cationic lipids have been synthesized since they were first proposed for use in gene therapy. Cationic lipids are able to efficiently transfect cells both in vitro and in vivo. Whereas most research groups have focused their investigations on the toxicity of these molecules, and on the location of expression of the DNA transferred by these vectors, little has been done to determine their biodistribution and elimination pathways. Our group has developed a family of cationic lipids termed phosphonolipids. Following a large in vitro screening experiment, we have selected several molecules for in vivo testing, with some of these phosphonolipids forming lipoplexes efficient in transfecting mouse lungs. It was thus of interest to study their fate after intravenous injection.

Methods

The respective biodistributions of both the GLB43 phosphonolipid and plasmid DNA were investigated and compared with DNA expression sites. Using the optimal conditions determined for phosphonolipids, we followed the gene transfer agent and plasmid DNA distributions versus time by radiolabeling them with 14 Ca n d 32 P, respectively. Otherwise, we performed imaging by radiolabeling plasmid DNA with 99m Tc.

Results

The lipoplexes appear to be directly located in the lung after administration. Secondly, the plasmid is released mainly into the lungs and the phosphonolipid vector is rapidly degraded. The hydrophilic moiety of the phosphonolipid is eliminated in the urine, as is the free plasmid.

Conclusions

This study reveals that there are slight differences in the observed results depending on the technique used to label the DNA; secondly, results show that the residence time of phosphonolipids in the mouse body is related to the DNA binding time.

Introduction

The concept of gene therapy consists of the transfer of a gene into a cell with the aim of treating hereditary or acquired diseases. Because of electrostatic repulsion between DNA and cell surfaces, naked DNA is effective only in transfecting cells of particular tissues, for example, muscle [START_REF] Manthorpe | G e n e therapy by intramuscular injection of plasmid DNA: studies on firefly luciferase gene expression in mice[END_REF][START_REF] Doh | Spatial-temporal patterns of gene expression in mouse skeletal muscle after injection of lacZ plasmid DNA[END_REF][START_REF] Brown | Gene delivery with synthetic (non-viral) carriers[END_REF]. Thus, strategies for assisting DNA transfer have been developed. A variety of vectors including liposomes, viruses and cationic lipids have been tested [START_REF] Felgner | Lipofection: a highly efficient, lipid-mediated DNA-transfection procedure[END_REF][START_REF] Griesenbach | Gene therapy progress and prospects: cystic fibrosis[END_REF] but, at present, no one vector can be considered as the perfect gene transfer agent. Although gene transfer using viruses occurs at high levels, viruses also induce inflammatory and immunological responses that, following repeated administration, decrease transfection efficiency [START_REF] Griesenbach | Gene therapy progress and prospects: cystic fibrosis[END_REF][START_REF] Smith | Gene therapy -where are we?[END_REF]. Transfection levels with cationic lipids are lower than those obtained with viruses; however, they are well tolerated and no decreases in transfection efficiency have been reported after repeated administration. All of the above-mentioned vectors have been tested for gene therapy in various diseases, with some methods used in clinical trials.

Since the CFTR gene was initially cloned, cystic fibrosis (CF) has become an ideal candidate disease for the use of genetic therapy. As such, various research groups have tried to develop and adapt these vectors to achieve a sufficient level of CFTR gene transfection in the lung to correct the CF disease. It is now clear that the type of vector required is dependent on the disease being treated, and thus on the type of tissue and cells to be transfected [START_REF] Friedmann | Overcoming the obstacles to gene therapy[END_REF]. Two major kinds of vectors are used in the treatment of CF: (1) viruses, most notably adenoviruses and adeno-associated viruses, because of their natural respiratory tract specificity, and (2) cationic lipids.

The use of cationic lipids is of interest because they have few side effects and are highly efficacious in transfecting the lung cells of rodents when delivered via intravenous injection [START_REF] Brown | Gene delivery with synthetic (non-viral) carriers[END_REF][START_REF] Zhun | Systemicgeneexpressionafter intravenous DNA delivery into adult mice[END_REF][START_REF] Floch | Systemic administration of cationic lipids/DNA complexes and the relationship between formulation and lung transfection efficiency[END_REF]. However, this method of delivery can potentially result in systemic, rather than localized, cell transfection. To this end it is important to study plasmid DNA and lipid vector distribution. Although some groups have already reported such investigations, very few studies have been focused on the biodistribution of both components simultaneously.

Our group has previously shown the efficacy of some cationic phosphonolipids in the transfection of mice lungs via intravenous injection [START_REF] Delépine | Cationic phosphonolipids as nonviral vectors: in vitro and in vivo applications[END_REF] and has studied the plasmid DNA distribution after lipoplex administration [START_REF] Delépine | Visualization of the transgene distribution according to the administration route allows prediction of the transfection efficacy and validation of the results obtained[END_REF]. This report deals with the biodistribution of a lipoplex solution composed of GLB43, the most efficient of our ammonium polar head phosphonolipids, after systemic administration. To perform these experiments, both plasmid and lipid were radiolabeled with 32 Pa n d 14 C, respectively; their localization was observed at various times after injection to determine which organs had been reached by the lipoplexes. In parallel, the plasmid was radiolabeled by chelation with 99m Tc and an imaging sequence was performed to follow the plasmid distribution. We report here results that show the fast biodegradation of the lipoplexes and elimination of the split transfer agent following the release of the plasmid, mainly in the lungs.

Materials and methods

Cationic phosphonolipids

The phosphonolipid GLB43 (Figure 1) was synthesized by our group as previously described [START_REF] Le Bolc'h | C a t i o n i c phosphonolipids as non-viral vectors for DNA transfection[END_REF]. GLB43 was 14 C-radiolabeled by methylation with 14 C-methyl iodide (specific activity 55 mCi/mmol; Amersham, UK) during the final step of synthesis. The output of the last stage of the synthesis is 100%, giving GLB43 a specific activity of 55 mCi/mmol.

Plasmid DNA

Two different plasmids were used: pCMVLacZ and pCMVLuc, which encode β-galactosidase and luciferase proteins, respectively. Plasmids were produced in our laboratory and were transformed into the DH 5α strain of E. coli. DNA was isolated from bacterial culture using a Qiagen endofree GigaKit and its concentration determined by measurement at 260 nm on a spectrophotometer. The purity was checked using the A260/A280 ratio. Only the plasmids with a ratio value between 1.8 and 2.00 were used.

Radiolabeling of the plasmid DNA

The pCMV LacZ plasmid DNA was labeled with [α-P32] dCTP using a nick translation technique [START_REF] Maniatis | Molecular Cloning. A Laboratory Manual[END_REF]. Residual unincorporated nucleotides were removed using a G50 Sephadex nick-column (Amersham Pharmacia). The specific activity of the probes was 20 µCi/µg. The [α-P32] plasmid was used within 24 h of labeling.

For imaging studies, only pCMVLuc was labeled by chelation with 99m Tc. Its specific activity was 5 µCi/µg.

Lipoplex preparation

Cationic phosphonolipid was dissolved in chloroform in glass vials. To inject 5 µCi of 14 C per mouse, the solution of radiolabeled GLB43 was diluted with unlabeled solution to give a final concentration of 17.5% of 14 C-labeled GLB43. Chloroform was then evaporated by vacuum, resulting in a dry lipid film. Sterile pyrogen-free distilled water was added in appropriate amounts to the lipid. The vials were then sealed and stored overnight at 4 • C. The resulting solution was sonicated for 10 min in a bath sonicator (Prolabo, Paris, France). The plasmid and lipid were then combined and incubated for 30 min at room temperature to form the lipoplexes. The lipid-to-DNA charge ratio used was 4 as this ratio had been previously shown to be the optimal for GLB43 [START_REF] Floch | Systemic administration of cationic lipids/DNA complexes and the relationship between formulation and lung transfection efficiency[END_REF]. Each mouse received 50 µgofplasmidDNA,with5µCi of each component (lipid and plasmid) in the preparation.

Lipid stability

The phosphonolipids were expected to degrade at low temperatures due to their thermal sensitivity. In addition, lipid stability at animal body temperature was also assessed. Thin-layer chromatography (TLC) on silica gel (Merck, Germany) was performed after various temperature treatments to evaluate the stability of GLB43. Untreated GLB43 powder was dissolved in chloroform and applied as the first spot to provide a reference, as this is the usual way the phosphonolipids are prepared. The second spot corresponded to a sample of the unlabeled sonicated lipid solution used for animal testing. For the third spot we applied the 14 C-GLB43 sonicated solution. The synthesis intermediates obtained just before the adjunction of methyl iodide were also applied to the plate (fourth spot), and, finally, sonicated lipid solution that had been incubated for 15 min at 40 • C(slightlyover mouse body temperature) constituted the fifth spot.

The chromatogram was developed to 10 cm with ethanol. For visualization, the plate was sprayed with sulfuric acid, dried, then sprayed with absolute ethanol and immediately burnt.

Distribution of the lipoplexes in mice

All the animal experiments were conducted in accordance with the ''Principles of laboratory animal care'' (NIH publication #85-23 revised 1985) and with the agreement of the regional veterinary services (authorization A29019-3, April 28th, 2000). Five-week-old female Swiss mice were used. Once anesthetized by ether inhalation, each mouse received 200 µl of the lipoplex solution as an intravenous bolus in the tail vein. Mice were then transferred to metabolism cages. For these experiments, the 32 P-labeled plasmid was diluted with cold plasmid solution and then mixed with the phosphonolipid. Five µCi of each component (lipid and plasmid) were injected into each mouse.

Sample collection

A time-course experiment (5, 15 and 30 min, 1, 6 and 24 h) was performed using 2-5 animals, which were anesthetized and euthanized by exsanguination (cardiac puncture). It should be noted that at 6 and 24 h, the animals received only 14 C-radiolabeled lipoplexes. Their lungs, liver, spleen, heart, kidney, brain, skin, viscera, tail and carcass were then harvested. Urine was collected in the metabolism cages. Each organ and the carcass were then crushed in mortars cooled by liquid nitrogen. The total radioactivity of each pulverized organ was measured by liquid scintillation in a 1600 CA Tricarb liquid scintillation analyzer (Packard). The 32 P measurements were performed without scintillation liquid, the 14 C measurements after addition of scintillation liquid.

Imaging studies

Six-week-old female Swiss mice were anesthetized by ether inhalation. Mice then received 200 µlo fs a l i n e solution (NaCl 0.9%) containing either a lipoplex solution or plasmid solution that corresponded to 50 µgo f 99m Tc-radiolabeled plasmid DNA via the tail vein. Each preparation was tested once on five mice. The distribution of radioactivity was evaluated by means of a specially designed high-resolution imaging device (Gamma imager, Biospace; France). Images were recorded just after injection and then 30 min, 3 and 6 h post-injection. The scintigraphic technique used was based on the one described by Lerondel et al. [START_REF] Lerondel | Radioisotopic imaging allows optimization of adenovirus lung deposition for cystic fibrosis gene therapy[END_REF].

Luciferase expression

We decided to explore gene expression in the three organs that exhibited the highest levels of radioactivity: the lungs, the liver and the spleen. Mice were injected intravenously as described above with unlabeled lipoplexes containing 50 µg of plasmid. Luciferase activity was assayed at 24, 48, 72 h and 7 days post-administration using a 'luciferase assay system' (Promega). Five mice were used for each time point. After collection, each tissue was crushed in a mortar cooled by liquid nitrogen and incubated with lysis buffer for 30 min (Promega). Centrifugation at 10 000 g was performed and supernatant luciferase activity was assayed using an MLX microtiter plate luminometer (Dynex Technologies). The RLU values were converted into picogrammes of luciferase by means of a standard curve established for each type of tissue. The total protein concentration of these supernatants was measured with a Coomassie Plus protein assay kit (Pierce). Results are also expressed in picogrammes of luciferase per mg of total proteins.

Results and discussion

In a previous study we investigated the ability of some cationic phosphonolipids to efficiently transfect mouse lung. Using our best phosphonolipid, GLB43, the optimal conditions for lung transfection via intravenous injection were obtained with a lipid-to-DNA charge ratio of 4 without co-lipid. Under these experimental conditions, 24 h after injection, the maximum level of expression is 500-fold higher than that obtained with naked DNA [START_REF] Floch | Systemic administration of cationic lipids/DNA complexes and the relationship between formulation and lung transfection efficiency[END_REF]. Moreover, we have shown that this kind of lipoplex is able to transfect lung epithelial cells (particularly type I pneumocytes) after an intratracheal administration [START_REF] Guillaume-Gable | C a t i o n i c phosphonolipids as nonviral gene transfer agents in the lungs of mice[END_REF]. At this stage of the work, we felt that it was important to determine the biodistribution of the lipoplexes in the mice.

Expression levels

We performed an expression study in the lungs, the l i v e ra n dt h es p l e e no ft h em i c e .T h el u n g sw e r ec h o s e n because they concentrate the main part of the radioactivity and because they are our organs of interest. Liver and spleen were chosen because they are major organs of the reticulo-endothelial system (RES) and they may capture lipoplexes.

As depicted in Figure 2, the expression levels obtained under the above-mentioned conditions are notable only in the lungs. This organ specificity was noted, despite some liver localization, with gene expression driven by the CMV promoter expected to result in ubiquitous expression of the transgene. This expected ubiquitous expression can be discussed because some expression has been reported in the nose with a construction using a KRT18 promoter whereas the same construction with a CMV promoter did not exhibit any [START_REF] Koehler | Targeting transgene expression for cystic fibrosis gene therapy[END_REF].

Osaka et al. have already reported such discordance (localization/expression) and correlated them with an accumulation of DNA essentially in the lungs and RES tissues [START_REF] Osaka | Pharmacokinetics, tissue distribution, and expression efficiency of plasmid [33P]DNA following intravenous administration of DNA/cationic lipid complexes in mice: use of a novel radionuclide approach[END_REF]. In the investigations reported here we attempted to better understand this phenomenon by following the biodistribution of the lipoplex components, i.e. both the lipid and the DNA.

Thin-layer chromatography

In parallel to the experiments on biodistribution, we studied the stability of GLB43, which is expected to be thermosensitive. Confirmation of the identity of radiolabeled and unlabeled GLB43 as well as their lipid stability were checked by TLC. Figure 3 shows the chromatogram obtained after detection. One should note that the migration of the sonicated and unsonicated lipidic solutions is similar; therefore, sonication did not alter GLB43. The same observation was performed for 14 Clabeled GLB43, and unlabeled GLB43 spots. However, after migration, the occurrence of a second spot that did not exhibit any radioactivity is noticeable. This corresponds to about 5% of unlabeled lipid (synthesis intermediate).

This experiment also revealed that a 15-min incubation time at 40 • C caused a partial degradation of GLB43, which consisted of the loss of one acyl chain. Indeed, three spots corresponding to intact GLB43, GLB43 with only one acyl chain, and a free fatty acyl chain are also observed on the chromatogram.

Biodistribution

Some biodistribution studies have already been performed but, in most cases, they have dealt only with the distribution of DNA [START_REF] Thierry | S y s t e m i cg e n e therapy: biodistribution and long-term expression of a transgene in mice[END_REF][START_REF] Bogdanov | DNA binding chelates for nonviral gene delivery imaging[END_REF][START_REF]Prolonged organ retention and safety of plasmid DNA administered in polyethylenimine complexes[END_REF] or lipid vectors alone [START_REF] Parker | Tissue distribution of the cytofectin component of plasmid-DNA/cationic lipid complexes following intravenous administration in mice[END_REF][START_REF] Bragonzi | Biodistribution and transgene expression with nonviral cationic vector/DNA complexes in the lungs[END_REF]. The studies that deal with both factors at the same time are usually performed by labeling them with an external marker such as 3 H-cholesteryl hexadecyl ether for the lipid, or by using 14 C-inulin to represent the DNA [START_REF] Ishiwata | Characteristics and biodistribution of cationic liposomes and their DNA complexes[END_REF]. Rhodamine was also used to follow either DNA or lipid distribution in the lungs [START_REF] Uyechi | Mechanism of lipoplex gene delivery in mouse lung: binding and internalization of fluorescent lipid and DNA components[END_REF]. These approaches may not reflect the true situation: most researchers in this field are indeed aware of the importance of co-lipid adjunction and the fact that the addition of products to the Moreover, products like inulin do not bear anionic charges like DNA, and thus the resulting complex may not have the same chemical properties with regard to interactions, stability and reactivity as those of the true lipoplex.

As such, we chose to follow plasmid DNA and GLB43 by radiolabeling them. This technique does not alter the integrity of the lipoplex, and thus neither its behavior nor fate. To observe the distribution of the lipoplexes, the release of the DNA from the lipoplexes and the elimination pathway of its different components, we radiolabeled DNA and GLB43 with two different markers: the lipid was labeled with 14 C and the plasmid with 32 P( T a b l e s1 and 2). Five minutes after injection of the GLB43/DNA lipoplex, radioactivity was no longer detectable in the blood. The radioactivity from 14 Ca n d 32 P were mainly located in two organs: the lungs and the liver, and, to a lesser extent, in the urine. Some radioactivity was also found in the tail and likely resulted from the diffusion of the lipoplexes through the capillary endothelium during the injection. As shown in Tables 1 and2, a slight increase is noted in lung radioactivity values for the lipid and the plasmid at 15 min in comparison with the values at 5min.

Beyond 15 min, 14 C-lipid distribution significantly changed: in parallel with a moderate rise in the liver and a large increase in the urine, a fast and significant (p < 0.05) decrease was observed in the lungs. At 30 min, the respective levels of radioactivity in the urine and the organs fell by 10-30% of the injected dose.

After 15 min the plasmid distribution [ 32 P] changed as seen for the lipid: a decrease was also observed in the lungs, which, however, remained the most radioactive organ. A significant (p < 0.05) rise in the plasmid content of the liver and a less marked one in the spleen were also observed. The radioactivity in the urine stayed low. Therefore, within 5-15 min after injection into the mice, the plasmid and the cationic lipid vector were colocated and mainly found in the lungs. After 15 min, the biodistributions of 32 Pa n d 14 C components were the same. Later, one could note differences in that their biodistribution was no longer concomitant. This suggests that in the period between the injection time and 15 min, GLB43 phosphonolipid and plasmid are still associated and thus the integrity of the lipoplexes is conserved. They are mainly located in the lungs, which constitute the first microvascular bed encountered after a tail vein injection. These results are in agreement with those previously reported by Li et al. [START_REF] Li | Dynamic changes in the characteristics of cationic lipidic vectors after exposure to mouse serum: implications for intravenous lipofection[END_REF]. These authors showed that when lipoplexes are exposed to serum in vitro, they aggregate first and then release DNA and lipid, the latter being further degraded. With GLB43, the first step occurs in the first 15 min.

Concerning the lipoplex degradation, after 15 min, the amount of phosphonolipid released from the lungs is far greater than that of the plasmid. Three different hypotheses can be put forward to explain this phenomenon: (1) when lipoplexes are internalized via an endocytic pathway, the plasmid DNA is released from the lipoplexes and stays in the cell (cytosol or nucleus), while the phosphonolipid, or at least its hydrophilic moiety, is rapidly excreted from the cell; (2) the DNA alone penetrates the cell and the phosphonolipid is eroded by blood flux in the vessels; (3) lipoplex aggregates are progressively eroded, which results in the fast elimination of phosphonolipids, whereas the degradation of plasmid DNA is much slower. According to some preliminary studies conducted within our group, DNA decompaction in vitro takes place in the cell only after endocytosis (Montier et al., submitted). This observation supports the first hypothesis suggested above.

At the 30-min time point, 32 Pa n d 14 C studies indicate that about 30% of the lipid and DNA are located in the liver. This suggests either a secondary aggregation after pulmonary release or a metabolic route for the lipoplexes. At the same time, a further third of the 14 C radioactivity is detected in the urine, and 1 h post-injection it accounts for about 70%. This suggests degradation of GLB43 and the elimination of its radiolabeled hydrophilic moiety via urinary excretion. This latter process takes place after the release of the DNA because no significant 32 P radioactivity is detected in the urine. The results obtained with GLB43 also show that the release of the DNA in the lungs from the lipoplexes occurs in the first 60 min after injection, as already reported for other cationic lipids [START_REF] Barron | Lipoplex-mediated gene delivery to the lung occurs within 60 minutes of intravenous administration[END_REF]. According to Osaka et al. [START_REF] Osaka | Pharmacokinetics, tissue distribution, and expression efficiency of plasmid [33P]DNA following intravenous administration of DNA/cationic lipid complexes in mice: use of a novel radionuclide approach[END_REF],whousedadifferentlipidvector,this release can occur later than 1 h.

We have also observed that after 30 min about 30% of the DNA is located in the liver. However, despite gene control under a CMV promoter, no gene expression was seen in this organ (Figure 2). One can explain this observation as follows: (1) either the plasmid DNA is released from the lipoplexes in the lungs and partially degraded during its travel in blood stream before reaching the liver; or (2) some lipoplexes that have reached the liver after lung aggregation are caught by the RES cells before being directed mainly to the lysosomes.

Imaging studies

Through the use of specialized imaging techniques, we followed the distribution of the 99m Tc-labeled plasmid. First, it is of note that the plasmid alone is directly eliminated in the urine (Figure 4a). This may explain why plasmid alone is not able to transfect mice when intravenously injected. When the complexes are administered intravenously, the radioactivity is immediately located to the lung and more generally in the chest (Figure 4b). At 30 min, the area showing radioactivity had increased and included the whole chest. A small amount is also visible in the liver. The most important change, however, is the large quantity of radioactivity seen in the bladder. At 1 h, the lung radioactivity is still predominant, but a slight increase in the liver signal is also noticed (data not shown). At 3 h, the signal intensity in the lungs progressively attenuates with the liver having a negligible radioactive signal. The radioactivity seen in the bladder also decreases by elimination in the urine. At 6 h, the residual signal remains essentially located in the lung. The liver radioactivity has also largely decreased with the radiolabeled elements eliminated in the urine.

This observation was not made in the 32 Ps t u d y , probably because of the short time over which the experiment was conducted (30 min). Indeed, the use of a chelate formed by associating 99m Tc and plasmid is a common technique [START_REF] Winnard | Preparation and use of NHS-MAG3 for technetium-99m labeling of DNA[END_REF]. So, whereas it has been shown that the nature of the chelator influences the biodistribution [START_REF] Zhang | Influence of different chelators (HYNIC, MAG3 and DTPA) on tumor cell accumulation and mouse biodistribution of technetium-99m labeled to antisense DNA[END_REF], the chelate is expected to be stable enough to remain intact in the blood stream and thus to reflect the plasmid DNA movement over the course of a few hours. After this time, the interaction between the chelates and albumin can interfere significantly with the results by exchange of 99m Tc. This phenomenon impedes imaging beyond 6 h after intravenous administration.

Otherwise, during the first 30 min of the imaging study, no elimination of 99m Tc nor of plasmid DNA occurred (the total radioactivity only decreased as a result of the short half-life of the technetium) but a large quantity is present in the bladder. Some signal is also present in the liver. With these results, a difference was noticed between the two ways of labeling plasmid DNA. Indeed, the 32 P study revealed hardly any plasmid DNA presence in the urine 30 min after injection, contrary to the 99m Tc imaging. This could suggest a degradation of the 99m Tcplasmid chelate followed by urinary elimination of the 99m Tc. However, if this is the case, 99m Tc would have been bound by serum albumin and its urinary presence is unjustified. This difference has not yet been explained. Surprisingly, this urinary location seen by imaging is also observed with free DNA chelates (not associated as a lipoplex) but, in this case, nearly all the signal is located in the bladder (Figure 4a). This eliminates the hypothesis of a poor chelating efficiency but reinforces the idea of free plasmid that is not coupled in a lipoplex, despite the theoretical excess of phosphonolipids (charge ratio =±4). The distribution also suggests a progressive release of DNA from the lipoplexes. The two hypotheses discussed above may also coexist with a primary tendency towards the release of DNA from lipoplexes, eliminated in the urine, and to a lesser extent a relocation of some lipoplexes from the lung to the liver. The signal trace in the liver provides proof for this hypothesis.

Conclusions

The use of two different techniques to follow lipoplex biodistribution reveals a slight difference in biodistribution pattern according to the method used, i.e. internal ( 32 P) or external ( 99m Tc) labeling of the plasmid DNA. Thus, it demonstrates how the technique used to study the distribution influences the final result. Such differences have been reported by Bogdanov [START_REF] Bogdanov | DNA binding chelates for nonviral gene delivery imaging[END_REF], who compared 32 P-and 99m Tc-peptide-based labeling. Indeed, the slight modification in the composition has an important impact on the properties of the lipoplexes (size and zeta-potential importance).

Taking into account the desirability of our molecules compared with others, their fast elimination after DNA release is a major benefit. Whereas in mouse lungs the decrease in both DNA and other cationic lipids was similar [START_REF] Barron | Lipoplex-mediated gene delivery to the lung occurs within 60 minutes of intravenous administration[END_REF], GLB43 levels decrease more rapidly than that of the plasmid DNA. Thus, it appears that the GLB43 in excess is quickly eliminated after DNA release. This is of great interest because it limits the damage due to an accumulation of exogene molecules. Moreover, whereas a rapid release from the lipoplexes is usually considered as inducing low efficiency, GLB43 results in attractive transfection levels in the lungs.

In conclusion, the structure of GLB43-plasmid DNA lipoplexes is maintained for approximately 15 min. After this time period, GLB43 is rapidly altered and its hydrophilic moiety, which is reactive, is eliminated in the urine. This is desirable because the shorter the amount of time this reactive moiety spends in the body after DNA release, the better it is for the organism. In addition, the lipidic component very likely has an hepatic metabolism similar to the natural fatty acyl chain. Thirty minutes after injection, following its release from the GLB43 complex, the plasmid DNA is still principally located in the target tissue, the lungs, and also in the liver. Therefore, the relative instability of GLB43 in the blood stream could be one of its main assets. These results are very informative but, although the levels of both DNA and lipid parts significantly differ within time when using ANOVA or Student's t-test, we have to keep in mind they were performed only on small size samples. Moreover, further studies dealing with DNA biodistribution should be carried out for longer periods of time. It would also be of interest to study the acyl chain distribution and to correlate it with the observed hepatic toxicity.

Figure 1 .

 1 Figure 1. Structure of GLB43. * C indicates the labeled carbon atom when the final synthesis step used 14 C-methyl iodide

Figure 2 .

 2 Figure 2. Luciferase expression levels in the lungs, liver and spleen, the three organs predominantly accumulating the plasmid DNA. Each animal received an intravenous injection of a preparation containing 50 µgo fp l a s m i dD N A .E v a l u a t i o n of the transfection was performed 24, 48, 72 h and 7 days after injection. (Mean ± SD)

Figure 3 .

 3 Figure 3. Chromatogram obtained after migration of GLB43 samples differentially treated. Each sample was applied onto a silica gel plate. The chromatogram was developed to 10 cm using ethanol, and then revealed

Figure 4 .

 4 Figure 4. Kinetic study by scintigraphy. (a) Scintigraphic shot of a mouse 30 min after an intravenous injection of 50 µg of plasmid DNA labeled with 99m Tc. (b) Mice were injected intravenously with a GLB43 lipoplex solution in which plasmid DNA was labeled with 99m Tc (see Materials and Methods). Scintigraphy of each animal was performed at injection times, and at 30 min, 3 and 6 h post-injection

Table1. Percentageof 14 C radioactivity present in organs of injected mice. 14 C-Radiolabeled lipoplexes were injected into the mice via the tail vein. Radioactivity in each organ was counted at different times after injection, as indicated. At 30 min and beyond, the total 14 C radioactivity was no longer 100% due to the collection difficulties after urinary elimination

	skin	1.25 ± 1.17	0.58 ± 0.28	1.94 ± 1.63	0.13 ± 0.05
	tail	15.12 ± 18.52	12.09 ± 10.62	3.91 ± 3.09	5.29 ± 4.60
	carcass	2.66 ± 1.69	1.65 ± 0.76	3.73 ± 0.25	0.53 ± 0.26
	viscera	1.87 ± 1.16	1.40 ± 0.50	2.81 ± 1.40	0.58 ± 0.15
	urine	8.14 ± 5.40	2.28 ± 0.56	29.25 ± 4.26	22.37 ± 3.54
	spleen	1.38 ± 0.87	0.65 ± 0.34	2.05 ± 1.42	1.78 ± 1.01	0.52 ± 0.22
	liver	15.80 ± 10.71	11.76 ± 1.49	12.25 ± 12.45	2.50 ± 1.60	2.14 ± 1.43
	lungs	52.9 ± 39.54	69.27 ± 7.55	9.42 ± 8.29	2.00 ± 0.97	0.56 ± 0.26
	heart	0.61 ± 0.18	0.20 ± 0.04	0.31 ± 0.28	0.15 ± 0.06	0.02 ± 0.01
	brain	0.04 ± 0.02	0.04 ± 0.02	0.05 ± 0.02	0.05 ± 0.01	-
	blood	0.22 ± 0.18	0.08 ± 0.08	0.07 ± 0.06	0.02 ± 0.01	0.01 ± 0.00
		5m i n a	15 min a	30 min c	1h c	6h b

Table 2 . Percentage of 32 P radioactivity present in organs of injected mice. 32 P-Radiolabeled lipoplexes were injected into the mice via the tail vein. Radioactivity in each organ was counted at different times after injection, as shown. Values have been corrected for the radioactive decay of 32

 2

	P

Acknowledgements

The authors thank Virginie Scotet for the statistical analysis and Anne-Cecile Hervé, Erwan Guenin, Marie-Paule Friocourt, Andrew T. Braun and Yvonne Dréano for their technical assistance. This work was supported by the 'conseil régional de Bretagne' (PRIR), and the French associations ''Vaincre la mucoviscidose'' and ''Association pour la Recherche sur le Cancer'' (ARC).


Rong Hu

Mircea Sofonea

Yi-Bin Xiao

A Tykhonov-type well-posedness concept for elliptic hemivariational inequalities

Keywords: Mathematics Subject Classification. 35M86, 47J40, 49J52, 74K10, 74M15 Hemivariational inequality, Tykhonov well-posedness, Convergence results, Contact problem, Spring-rod system

In this paper, we introduce a new Tykhonov-type well-posedness concept for elliptic hemivariational inequalities, governed by an approximating function h. We characterize the well-posedness in terms of the metric properties of the family of approximating sets, under various assumptions on h. Then, we use the well-posedness properties in order to obtain convergence results of the solution with respect to the data. The proofs are based on arguments of monotonicity combined with the properties of the Clarke directional derivative. Our results provide mathematical tools in the study of a large number of static problems in Contact Mechanics. To provide an example, we consider a mathematical model which describes the equilibrium of a rod-spring system with unilateral constraints. We prove the unique weak solvability of the model, and then we illustrate our abstract convergence results in the study of this contact problem and provide the corresponding mechanical interpretations.

Introduction

Everywhere in this paper, unless stated otherwise, (X, • X ) is a real Banach space, •, • denotes the duality pairing between X and its dual X * , K is a nonempty subset of X, A : X → X * , j : X → R is a locally Lipschitz function and f ∈ X * . We denote by j 0 (u; v) the generalized directional derivative of j at the point u in the direction v, see Definition 2. With these notation, we consider the hemivariational inequality

u ∈ K, Au, v -u + j 0 (u; v -u) ≥ f, v -u ∀ v ∈ K. (1.1)
Such kind of inequalities arise in Contact Mechanics. They model the equilibrium of elastic bodies acted upon the body forces and surface tractions, in frictional or frictionless contact with an obstacle. References in the field are [START_REF] Naniewicz | Mathematical Theory of Hemivariational Inequalities and Applications[END_REF][START_REF] Panagiotopoulos | Hemivariational Inequalities[END_REF] and, more recently [START_REF] Cai | Convergence results for elliptic variational-hemivariational inequalities[END_REF][START_REF] Migórski | Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems[END_REF][START_REF] Sofonea | Variational-Hemivariational Inequalities with Applications[END_REF][START_REF] Sofonea | Well-posedness of minimization problems in contact mechanics[END_REF][START_REF] Xiao | On the optimal control of variational-hemivariational inequalities[END_REF][START_REF] Xiao | Generalized penalty method for elliptic variational-hemivariational inequalities[END_REF]. There, existence and uniqueness results for inequality problems of the form (1.1) can be found, under various assumptions on the data. A convergence result for such inequalities was provided in [START_REF] Zeng | On convergence of solutions to variational-hemivariational inequalities[END_REF] and general results on their numerical analysis of such inequalities can be found in [START_REF] Han | Numerical analysis of stationary variational-hemivariational inequalities with applications in contact mechanics[END_REF][START_REF] Han | Numerical analysis of hemivariational inequalities in contact mechanics[END_REF][START_REF] Han | Numerical analysis of stationary variational-hemivariational inequalities[END_REF]. Results on the Tykhonov regularization for hemivariational inequalities can be found in the recent paper [START_REF] Tang | On the existence of solutions and Tykhonov regularization of hemivariational inequality problems[END_REF].

The current paper was inspired by three types of studies related to the hemivariational inequality (1.1): the well-posedness in the sense of Tykhonov, the continuous dependence of the solution with respect to the data, and the perturbation with a convex function, under specific assumptions. We briefly describe in what follows each of these approaches.

First, the concept of well-posedness in the sense of Tykhonov was introduced for minimization problems in [START_REF] Tykhonov | On the stability of functional optimization problems[END_REF]. Later, it was extended to variational inequalities in [START_REF] Lucchetti | A characterization of Tykhonov well-posedness for minimum problems with applications to variational inequalities[END_REF][START_REF] Lucchetti | Some properties of "wellposedness" variational inequalities governed by linear operators[END_REF] and to a particular class of hemivariational inequalities in [START_REF] Goeleven | Well-posed hemivariational inequalities[END_REF]. References in the field include [8-10, 20, 21, 25, 27]. For an inequality of the form (1.1), the Tykhonov well-posedness is based on the following two ingredients: First, it is assumed that inequality (1.1) has a unique solution; second, this solution represents the limit in X,a sε → 0, of solution sequence, called approximating sequence, to the following problem:

u ∈ K, Au, v -u + j 0 (u; v -u)+ε v -u X ≥ f, v -u ∀ v ∈ K. (1.2)
Hemivariational inequalities of the form (1.2) have been considered in [START_REF] Hu | Equivalence results of well-posedness for split variational-hemivariational inequalities[END_REF][START_REF] Sofonea | Tykhonov well-posedness of elliptic variational-hemivariational inequalities[END_REF][START_REF] Xiao | Well-posedness of hemivariational inequalities and inclusion problems[END_REF], among others. There, necessary and sufficient condition for the well-posedness of inequality (1.1) are provided, under various assumptions on the nonlinear operator A.

Next, we focus on the continuous dependence of the solution with respect to the data and, for simplicity, we restrict to the dependence of the solution with respect to f . To this end, we consider a perturbation f ε ∈ X * of f , together with the hemivariational inequality

u ∈ K, Au, v -u + j 0 (u; v -u) ≥ f ε ,v-u ∀ v ∈ K. (1.3)
Then, it is easy to see that the solution of (1.3) satisfies the inequality

u ∈ K, Au, v -u + j 0 (u; v -u)+ f ε -f X * v -u X ≥ f, v -u ∀ v ∈ K,
and using the notation h(ε)= f εf X * , we deduce that

u ∈ K, Au, v -u + j 0 (u; v -u)+h(ε) v -u X ≥ f, v -u ∀ v ∈ K. (1.4)
It follows from here that the continuous dependence of the solution of (1.1) with respect to f can be deduced from a convergence result for the solutions of (1.4) to the solution of (1.1).

Finally, consider a perturbed version of inequality (1.

1)o ft h ef o r m u ∈ K, Au, v -u + ϕ(u, v) -ϕ(u, u)+j 0 (u; v -u) ≥ f, v -u ∀ v ∈ K (1.5) where ϕ : X × X → R is a functional. Assume now that ϕ(u, •)=ε ϕ(u, •)f o re a c hu ∈ X
where ε is a positive parameter converging to zero and ϕ(u, •):X → R is a continuous seminorm. Then, there exists a function h :

X → R such that ϕ(u, v) -ϕ(u, u)=ε ϕ(u, v) -ε ϕ(u, u) ≤ ε ϕ(u, v -u) ≤ εh(u) v -u X
for all u, v ∈ X. This implies that the solution of (1.5) satisfies the inequality

u ∈ K, Au, v -u + j 0 (u; v -u)+εh(u) v -u X ≥ f, v -u ∀ v ∈ K. (1.6)
Therefore, the convergence of the solution of inequality (1.5)a sε → 0 can be deduced by studying the convergence of the solution of (1.6)a sε → 0. Such kind of situations arise in Contact Mechanics, as explained in [START_REF] Migórski | Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems[END_REF][START_REF] Sofonea | Variational-Hemivariational Inequalities with Applications[END_REF]. There, ε is either a friction coefficient or a stiffness coefficient, and establishing such type of convergence results allows us to establish the link between various contact models and to provide various mechanical interpretations.

A careful analysis reveals that inequalities (1.2), (1.4), (1.6) are of the form

u ∈ K, Au, v -u + j 0 (u; v -u)+h(ε, u) v -u X ≥ f, v -u ∀ v ∈ K (1.7)
with a convenient choice of the function h(ε, u), defined for ε>0a n du ∈ X. Moreover, both the three situations described above require to establish convergence results for the solution of (1.7) to the solution of (1.1), as ε → 0. It follows from here that the study of the perturbed hemivariational inequalities of the form (1.7) is useful in the study of the hemivariational inequality (1.1), since it could provide its well-posedness, the continuous dependence of the solution with respect to the data and other convergence results.

Motivated by the previous remarks, in this paper we study the link between the solutions of inequalities (1.1) and (1.7), under various assumptions on the data. Our aim is to introduce a new Tykhonov-type well-posedness concept for inequality (1.1), to derive convergence results and to provide mathematical tools useful in the study of mathematical models which describe the contact of deformable bodies and structures.

The rest of the paper is structured as follows. In Sect. 2, we recall some preliminary material which is needed in the rest of the paper. In Sect. 3, we introduce and study a new Tykhonov-type concept of well-posedness which extends that in ( [START_REF] Goeleven | Well-posed hemivariational inequalities[END_REF][START_REF] Sofonea | Tykhonov well-posedness of elliptic variational-hemivariational inequalities[END_REF]). In Sect. 4, we use the results in Sect. 3 in order to establish new convergence results of the solution to inequality (1.1) with respect to the data K, A, j and f . Finally, in Sect. 5, we illustrate the results in Sect. 4 in the study of a rod-spring system with unilateral constraints.

Preliminaries

Everywhere below, we use • X for the norm of space X. Unless stated otherwise, all the limits, upper limits and lower limits below are considered as n →∞ , even if we do not mention it explicitly. The symbols "⇀" and "→" denote the weak and the strong convergence in the space X.

We start with some definitions related to the operator A and function j.

Definition 1. An operator A : X → X * is said to be: (a) monotone, if for all u, v ∈ X,w eh a v e Au -Av, uv ≥0;

(b) strongly monotone, if there exists m A > 0 such that

Av 1 -Av 2 ,v 1 -v 2 ≥m A v 1 -v 2 2 X ∀ v 1 ,v 2 ∈ X; (2.1) (c) bounded, if A maps bounded sets of X into bounded sets of X * ; (d) pseudomonotone, if it is bounded and u n ⇀uin X with lim sup Au n ,u n -u ≤0 implies lim inf Au n ,u n -v ≥ Au, u -v for all v ∈ X.
Definition 2. The generalized (Clarke) directional derivative of a locally Lipschitz function j : X → R at the point u ∈ X in the direction v ∈ X is defined by

j 0 (u; v) = lim sup x→u, λ↓0 j(x + λv) -j(x) λ .
The generalized (Clarke) gradient (subdifferential) of j at u is a subset of the dual space X * given by

∂j(u)={ ξ ∈ X * | j 0 (u; v) ≥ ξ, v ∀ v ∈ X }.
Definition 3. A locally Lipschitz function j : X → R is said: (a) to be regular (in the sense of Clarke) at the point u ∈ X if for all v ∈ X the one-sided directional derivative j ′ (u; v)e x i s t sa n dj 0 (u; v)=j ′ (u; v); (b) to satisfy the relaxed monotonicity condition if there exists α j > 0 such that

ξ 1 -ξ 2 ,u 1 -u 2 ≥-α j u 1 -u 2 2 X ∀ u i ∈ X, ξ i ∈ ∂j(u i ),i=1, 2.
We now recall the following properties related to the directional derivative and Clarke subdifferential. Proposition 4. Assume that j : X → R is a locally Lipschitz function. Then the following holds.

(a) For every u ∈ X, the function X ∋ v → j 0 (u; v) ∈ R is positively homogeneous and subadditive, i.e., j 0 (u; λv)=λj 0 (u; v) for all λ ≥ 0, v ∈ X and j 0 (u;

v 1 + v 2 ) ≤ j 0 (u; v 1 )+j 0 (u; v 2 ) for all v 1 , v 2 ∈ X, respectively. (b) For every u, v ∈ X, we have j 0 (u; v) = max { ξ, v |ξ ∈ ∂j(u) }. (c) The function X × X ∋ (u, v) → j 0 (u; v) ∈ R is upper semi-continuous, i.e.
, for all u, v ∈ X, {u n }, {v n }⊂X such that u n → u and v n → v in X, we have lim sup j 0 (u n ; v n ) ≤ j 0 (u; v).

For more details on the definitions and properties above, we refer to the books [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF][START_REF] Migórski | Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems[END_REF][START_REF] Naniewicz | Mathematical Theory of Hemivariational Inequalities and Applications[END_REF], for instance. In the study of inequality (1.1), we consider the following assumptions.

(K 1 ) K is a nonempty closed convex subset of X.

(A 1 ) A : X → X * is a pseudomonotone operator. (A 2 ) A : X → X * is a strongly monotone operator with constant m A > 0.

(j 1 ) j : X → R is a locally Lipschitz function. (j 2 ) There exists α j ≥ 0 such that

j 0 (v 1 ; v 2 -v 1 )+j 0 (v 2 ; v 1 -v 2 ) ≤ α j v 1 -v 2 2 X ∀ v 1 ,v 2 ∈ X. (j 3 ) α j <m A . (j 4 ) There exist c 0 ,c 1 ≥ 0 such that ξ X * ≤ c 0 + c 1 v X ∀ v ∈ X, ξ ∈ ∂j(v). (f ) f ∈ X * .
It can be proved that for a locally Lipschitz function j : X → R, condition (j 2 ) is equivalent to the relaxed monotonicity condition introduced in Definition 3(b). A proof of the statement can be found in, e.g., [START_REF] Migórski | Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems[END_REF]. Note also that if j : X → R is a convex function, then condition (j 2 ) holds with α j =0 , since it reduces to the monotonicity of the (convex) subdifferential. Examples of functions which satisfy conditions (j 1 ), (j 2 ) and (j 4 ) can be found in [START_REF] Migórski | Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems[END_REF][START_REF] Migórski | A class of variational-hemivariational inequalities in reflexive Banach spaces[END_REF], for instance. Nevertheless, for the convenience of the reader, we provide below such an example.

Example 5.

Let Ω be a bounded domain of R d with a smooth boundary Γ and let Γ 0 be a measurable part of Γ such that meas Γ 0 > 0. Let X = H 1 (Ω) be the Sobolev space endowed with its usual Hilbertian structure. Assume that

⎧ ⎨ ⎩ p : R → R is such that (a) |p(r 1 ) -p(r 2 )|≤L p |r 1 -r 2 | for all r 1 ,r 2 ∈ R, with L p > 0; (b) p(0) = 0. (2.2)
Next, consider the functions q : R → R and j : X → R defined by

q(r)= r 0 p(s)ds ∀ r ∈ R, (2.3)
j(v)= Γ0 q(v) dΓ ∀ v ∈ X. (2.4)
Note that here and below, we write v for the trace of the function v ∈ X to Γ. Using standard arguments ([13, Lemma 3.50 (iii)], for instance), it follows that

q 0 (r; s)=p(r) s ∀ r, s ∈ R, (2.5)
where q 0 (r; s) denotes the generalized directional derivative of q at the point r in the direction s. Therefore, using (2.2) and [START_REF] Migórski | Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems[END_REF]Corollary 4.15] it follows that j satisfies conditions (j 1 ) and (j 4 ) and, in addition,

j 0 (u; v)= Γ0 p(u)vdΓ ∀ u, v ∈ X. (2.6) Hence, given v 1 ,v 2 ∈ X we have j 0 (v 1 ; v 2 -v 1 )+j 0 (v 2 ; v 1 -v 2 )= Γ0 (p(v 1 ) -p(v 2 ))(v 2 -v 1 ) dΓ.
Moreover, using assumption (2.2) we obtain that

j 0 (v 1 ; v 2 -v 1 )+j 0 (v 2 ; v 1 -v 2 ) ≤ L p γ 2 v 1 -v 2 2 X ,
where, here and below, γ denotes the norm of the trace operator defined on X with values in L 2 (Γ 0 ). This inequality shows that j satisfies condition (j 2 ) with α j = L p γ 2 .

We now recall the following existence and uniqueness result.

Theorem 6. Assume that X is a reflexive space and (K 1 ), (A 1 ), (A 2 ), (j 1 ), (j 2 ), (j 3 ), (j 4 ), (f ) hold.

Then, there exists a unique solution u to the hemivariational inequality (1.1).

A proof of Theorem 6 can be found in [START_REF] Sofonea | Variational-Hemivariational Inequalities with Applications[END_REF]Chapter 5]. It is carried out in several steps, by using the properties of the subdifferential, a surjectivity result for pseudomonotone multivalued operators and the Banach fixed point argument.

Well-posedness results

In this section, we introduce and study a general concept of well-posedness for the hemivariational inequality (1.1). To this end, for each ε>0, we consider the set Ω(ε) defined as follows:

Ω(ε)={ u ∈ K : Au, v -u + j 0 (u; v -u)+h(ε, u) v -u X ≥ f, v -u ∀ v ∈ K } (3.1)
where, recall, h :( 0 , +∞) × X → R is a given function. We refer to the family of sets {Ω(ε)} ε>0 as the family of approximating sets. Moreover, we denote by S the set of solutions of inequality (1.1), i.e.,

S = { u ∈ K : Au, v -u + j 0 (u; v -u) ≥ f, v -u ∀ v ∈ K } (3.2)
and we recall that S is said to be a singleton if S has a unique element.

Next, we proceed with the following definitions.

Definition 7. A sequence {u n }⊂X is called an approximating sequence for the hemivariational inequality (1.1) if there exists a sequence {ε n }⊂R such that 0

<ε n → 0a n du n ∈ Ω(ε n ), for each n ∈ N.
Definition 8. The hemivariational inequality (1.1) is said to be well-posed if it has a unique solution and every approximating sequence for (1.1) converges in X to its solution.

Note that this concept of well-posedness above extends that used in [START_REF] Sofonea | Tykhonov well-posedness of elliptic variational-hemivariational inequalities[END_REF][START_REF] Xiao | Well-posedness of hemivariational inequalities and inclusion problems[END_REF]. Indeed, the later can be recovered in the particular case when

h(ε, u)=ε ∀ ε>0,u∈ X. (3.3)
Moreover, this concept is quite different from that introduced in [START_REF] Goeleven | Well-posed hemivariational inequalities[END_REF] for hemivariational inequalities with constraints.

Our aim in what follows is to characterize the well-posedness of hemivariational inequality (1.1)i n terms of the metric properties of the approximating sets {Ω(ε)} ε>0 and to indicate sufficient conditions on the data which guarantee this well-posedness. To this end, we recall the following definition.

Definition 9.

Let Ω be a nonempty subset of X. Then the diameter of Ω, denoted diam(Ω), is defined by equality

diam(Ω) = sup a, b∈Ω a -b X .
Moreover, we consider the following additional assumptions.

(K 2 ) K is a nonempty closed subset of X. (h 1 ) h(ε, u) ≥ 0 ∀ u ∈ X, ε > 0. (h 2 ) h(ε n ,u n ) → 0 whenever 0 <ε n → 0a n d{u n }⊂X is bounded. (h 3 )0<ε 1 <ε 2 =⇒ h(ε 1 ,u) ≤ h(ε 2 ,u) ∀ u ∈ X. (h 4 ) There exists L h :]0, +∞[→ R such that (a) |h(ε, u) -h(ε, v)|≤L h (ε) u -v X ∀ u, v ∈ X, ε > 0, (b) L h (ε) → 0a sε → 0.
Our main result in this section is the following. Theorem 10. Let X be a Banach space. The following statements hold.

(a) Under assumption (j 1 ), (h 1 ) and (f ), the hemivariational inequality (1.1) is well-posed if and only if its set of solution S is nonempty and diam(Ω(ε)) → 0 as ε → 0.

(b) Under assumptions (K 2 ), (A 1 ), (j 1 ), (h 1 ), (h 2 ), (h 3 ) and (f ), the hemivariational inequality (1.1) is well-posed if and only if the set Ω(ε) is nonempty for each ε>0 and diam(Ω(ε)) → 0 as ε → 0.

(c) Under assumptions (A 2 ), (j 1 ), (j 2 ), (j 3 ), (h 1 ), (h 2 ), (h 4 ) and (f ), the hemivariational inequality (1.1) is well-posed if and only if the set S is a singleton.

(d) If X is a reflexive space, then under assumptions (K 1 ), (A 1 ), (A 2 ), (j 1 ), (j 2 ), (j 3 ), (j 4 ), (h 1 ), (h 2 ), (h 4 ) and (f ), the hemivariational inequality (1.1) is well-posed.

Proof. (a) We work under the assumption (j 1 ), (h 1 ) and (f ) and note that, in this case, for each ε>0 we have S⊂Ω(ε).

(3.4)

Assume that (1.1) is well-posed. Then, by definition, S is a singleton and, therefore, S = ∅. Arguing by contradiction, we assume that diam(Ω(ε)) → 0a sε → 0. Then, there exist δ 0 ≥ 0, a sequence {ε n }⊂R and two sequences {u n }, {v n }⊂X such that 0

<ε n → 0, u n ,v n ∈ Ω(ε n )a n d u n -v n X ≥ δ 0 2 ∀ n ∈ N. (3.5)
Now, since both {u n } and {v n } are approximating sequences for the hemivariational inequality (1.1), the well-posedness of (1.1) implies that u n → u and v n → u in X where u denotes the unique element of S. This is in contradiction with (3.5). We conclude from here that diam(Ω(ε)) → 0a sε → 0. Conversely, assume that S is nonempty and diam(Ω(ε)) → 0asε → 0. We claim that S is a singleton. Indeed, let u, u ′ ∈S and let {u n } be an approximating sequence for (1.1). Then there exists a sequence {ε n }⊂R such that 0 <ε n → 0a n du n ∈ Ω(ε n ) for all n ∈ N. Using (3.4) and Definition 9,w eh a v e

u -u ′ X ≤ u -u n X + u ′ -u n X ≤ 2 diam(Ω(ε n ))
→ 0, which implies that u = u ′ and proves the claim. Moreover, for any approximating sequence {u n },w e have uu n X ≤ diam(Ω(ε n )) → 0, which implies that u n → u in X and, therefore, (1.1) is well-posed.

(b) We work under the assumptions (K 2 ), (A 1 ), (j 1 ), (h 1 ), (h 2 ), (h 3 ) and (f ). Assume that (1.1) is well-posed. Then, we use the part (a) of the theorem and inclusion (3.4) to see that the set Ω(ε)i s nonempty for each ε>0 and diam(Ω(ε)) → 0a sε → 0.

Conversely, assume that the set Ω(ε) is nonempty for each ε>0 and diam(Ω(ε)) → 0asε → 0. Then, using (3.4), again, and Definition 9 we deduce that the hemivariational inequalities (1.1) admits at most one solution. Let {u n } be an approximating sequence for (1.1). Then there exists a sequence {ε n }⊂R such that 0 <ε n → 0a n du n ∈ Ω(ε n ) for all n ∈ N. Since diam(Ω(ε)) → 0, for any δ>0 there exists a positive integer

N δ such that diam(Ω(ε n )) ≤ δ ∀ n ≥ N δ . (3.6)
Let n, m ∈ N be such that n, m ≥ N δ and assume that ε m ≤ ε n . Then using assumption (h 3 )w eh a v e h(ε m ,u m ) ≤ h(ε n ,u m ) and, therefore (3.1) implies that u m ∈ Ω(ε n ). On the other hand, Definition 7 guarantees that u n ∈ Ω(ε n ), too. Therefore, (3.6) implies that

u n -u m X ≤ δ.
This inequality holds if ε m >ε n , too, since in this case u n ,u m ∈ Ω(ε m ). We conclude from here that {u n } is a Cauchy sequence in X and, since X is assumed to be a Banach space, there exists u ∈ X such that

u n → u in X. (3.7)
This convergence combined with assumption (K 2 ) yields

u ∈ K. (3.8)
We now prove that u solves the hemivariational inequality (1.1) and, to this end, we use a pseudomonotonicity argument. First, the inclusion

u n ∈ Ω(ε n ) implies that Au n ,u n -v ≤j 0 (u n ; v -u n )+ f, u n -v + h(ε n ,u n ) u n -v X ∀ v ∈ K, n ∈ N.
Next, we pass to the upper limit as n →∞in this inequality and use the convergence (3.7), Proposition 4 (c) and assumption (h 2 ) to deduce that

lim sup Au n ,u n -v ≤j 0 (u; v -u)+ f, u -v ∀ v ∈ K. (3.9)
On the other hand, regularity (3.8) allows us to test with v = u in (3.9) to find that lim sup Au n ,u nu ≤0.

Therefore, by the pseudomonotonicity of the operator A, guaranteed by assumption (A 1 ), we obtain

lim inf Au n ,u n -v ≥ Au, u -v ∀ v ∈ X. (3.10)
We now combine (3.8), (3.9) and (3.10) to see that u is a solution to the hemivariational inequality (1.1), which implies that S is a singleton. This together with (3.7) indicates that any approximating sequence of (1.1) converges to the unique element of S. It follows from here that the hemivariational inequality (1.1) is well-posed, which concludes the proof of (b). (c) We work under the assumptions (A 2 ), (j 1 ), (j 2 ), (j 3 ), (h 1 ), (h 2 ), (h 3 ) and (f ). Assume that inequality (1.1) is well-posed. Then, by Definition 7 it follows that S is a singleton.

Conversely, assume that S is a singleton and denote by u ∈ K the unique solution of (1.1). Let {u n }⊂X be an approximating sequence for the hemivariational inequality (1.1). Then there exists a sequence {ε n }⊂R such that 0 <ε n → 0a n d

Au n ,v-u n + j 0 (u n ; v -u n )+h(ε n ,u n ) v -u n X ≥ f, v -u n ∀ v ∈ K.
(3.11)

Letting v = u n in inequality (1.1)a n dv = u in inequality (3.11), we add the resulting inequalities to see that

Au n -Au, u n -u ≤j 0 (u n ; u -u n )+j 0 (u; u n -u)+h(ε n ,u n ) u n -u X .
We now use assumptions (A 2 ) and (j 2 ) to obtain that

(m A -α j ) u n -u X ≤ h(ε n ,u n ). (3.12)
Next, we use assumption (h 4 )(a) to write

h(ε n ,u n )=h(ε n ,u n ) -h(ε n ,u)+h(ε n ,u) ≤ L(ε n ) u n -u X + h(ε n ,u)
and, therefore, (3.12) yields

(m A -α j -L(ε n )) u n -u X ≤ h(ε n ,u). (3.13)
Note that Definition 7 guarantees that ε n → 0. Therefore, passing to the upper limit in (3.13) and using conditions (h 4 )(b), (h 2 ) and the smallness condition (j 3 ) yield lim sup u nu X ≤ 0.

We deduce from here that u n → u in X and conclude the proof of (c).

(d) We work under the assumptions (K 1 ), (A 1 ), (A 2 ), (j 1 ), (j 2 ), (j 3 ), (j 4 ), (h 1 ), (h 2 ), (h 3 ) and (f ). Moreover, we assume that the space X is reflexive. We use Theorem 6 to deduce that inequality (1.1) has a unique solution. Then we apply part (c) of the theorem to conclude the proof.

We end this section with two remarks. First, the statement (a)-(c) in Theorem 10 provides equivalence results. They do not guarantee the well-posedness of the hemivariational inequality (1.1). In contrast, sufficient conditions which guarantee its well-posedness are provided by the statement (d) of the theorem. Second, the function h :( 0 , ∞) × X → R defined by (3.3) satisfies assumptions (h 1 )-(h 4 ). Therefore, Theorem 10 works in this particular case and allows us to recover part of the result obtained in [START_REF] Sofonea | Tykhonov well-posedness of elliptic variational-hemivariational inequalities[END_REF][START_REF] Xiao | Well-posedness of hemivariational inequalities and inclusion problems[END_REF].

Convergence results

In this section, we use Theorem 10 in order to prove two convergence results for the solution of inequality (1.1). To this end, for each ε>0, we consider an operator A ε : X → X * , a function j ε : X → R and an element f ε ∈ X * . With these data we consider the following perturbed version of inequality (1.1):

u ∈ K, A ε u, v -u + j 0 ε (u; v -u) ≥ f ε ,v-u ∀ v ∈ K. (4.1)

Consider now the following assumptions. (h A ) There exists h

A :]0, +∞[×X → R such that (a) A ε u -Au X * ≤ h A (ε, u) ∀ u ∈ X, ε > 0, (b) h A satisfies assumptions (h 1 ), (h 2 ) and (h 4 ) with L hA :]0, +∞[→ R. (j ε ) j ε : X → R is a locally Lipschitz function. (h j ) There exists h j :]0, +∞[×X → R such that (a) j 0 ε (u; v) -j 0 (u; v) ≤ h j (ε, u) v X ∀ u, v ∈ X, ε > 0, (b) h j satisfies assumptions (h 1 ), (h 2 ) and (h 4 ) with L hj :]0, +∞[→ R. (f ε ) f ε → f in X * as ε → 0.
We complete these assumptions with the following two examples which will be useful in the next section. Example 11. An example of operator A ε : X → X * which satisfies assumption (h A ) is given by A ε u = Au + εT u for all u ∈ X, ε>0, where T : X → X * is a Lipschitz continuous operator. Indeed, it is easy to see that in this case condition (h A ) is satisfied with h A (ε, u)=ε Tu X * and L hA (ε)=εL T , L T being the Lipschitz constant of the operator T . Example 12. An example of functions j ε , j which satisfy assumption (h j ) can be constructed by using the notation presented in Example 5 in the one-dimensional case. Let L>0, Ω = (0,L) and let

X = { v ∈ H 1 (0,L) | v(0) = 0 } (4.2)
which is a real Hilbert space with the inner product

(u, v) X = L 0 u ′ v ′ dx ∀ u, v ∈ X. (4.3)
and the associated norm • X . Here and below the prime denotes the derivative with respect to x ∈ (0,L), i.e., u ′ = du dx . Assume that p is a function which satisfies condition (2.2)a n dl e tΓ 0 = {L}. Then, using (2.4), we deduce that j(v)=q(v(L)) for all v ∈ X where, recall, q is given by (2.3). Moreover, (2.6) implies that

j 0 (u; v)=p(u(L))v(L) ∀ u, v ∈ X. (4.4)
Let ε>0 and replace the function p by the function

p ε (r)=p(r)+εr ∀ r ∈ R. (4.5)
Then, it is easy to see that the function p ε satisfies condition (2.2). Moreover, the corresponding function j ε is given by j ε (v)=q ε (v(L)) for all v ∈ X where

q ε (r)= r 0 p ε (s)ds ∀ r ∈ R. (4.6)
In addition,

j 0 ε (u; v)=p ε (u(L))v(L) ∀ u, v ∈ X. (4.7)
Finally, an elementary calculation shows that

|v(L)|≤ √ L v X ∀ v ∈ X. (4.8)
We now use (4.4), (4.7) and (4.5), (4.8) to see that the functions j ε , j satisfy conditiontion (h j ) with h j (ε, u)=Lε u X and L hj (ε)=Lε.

We have the following existence, uniqueness and convergence result.

Theorem 13. Assume that X is a reflexive space and (K 1 ), (A 1 ), (A 2 ), (j 1 ), (j 2 ), (j 3 ), (j 4 ), (f ), (h A ), (h j ), (j ε ) and (f ε ) hold. Moreover, assume that for each ε>0 the element u ε is a solution to inequality (4.1) and let be u the solution of inequality (1.1) provided in Theorem 6.T h e n

u ε → u in X as ε → 0. (4.9)
Proof. The proof is structured in three steps, as follows.

(i) The perturbed hemivariational inequality. Let ε>0. We claim that if w is a solution to inequality (4.1) then w satisfies the inequality of form (1.7) with h :]0, +∞) × X → R given by

h(ε, u)=h A (ε, u)+h j (ε, u)+ f ε -f X * ∀ u ∈ X, ε > 0. (4.10)
Indeed, assume that w is a solution of (4.1) and let v ∈ K. Then,

Aw, v -w + A ε w -Aw, v -w + j 0 (w; v -w) +j 0 ε (w; v -w) -j 0 (w; v -w)+ f -f ε ,v-w ≥ f, v -w , which imples that Aw, v -w + A ε w -Aw X * v -w X + j 0 (w; v -w) +j 0 ε (w; v -w) -j 0 (w; v -w)+ f -f ε X * v -w X ≥ f, v -w .
We now use assumptions (h A )(a) and (h j )(a) to see that

Aw, v -w + h A (ε, w) v -w X + j 0 (w; v -w) +h j (ε, w) v -w X + f -f ε X * v -w X ≥ f, v -w
and, combining this inequality with the definition (4.10) of the function h we obtain that w is a solution of (1.7), as claimed.

(ii) Properties of the function h. We claim that the function h defined by (4.10) satisfies conditions (h 1 ), (h 2 ) and (h 4 ). Indeed, this statement is a direct consequence of the definition of h combined with assumptions (h A )(b), (h j )(b) and (f ε ).

(iii) End of proof. In this step, we assume that for each ε>0 the element u ε is a solution to inequality (4.1) and show that u ε → u in X as ε → 0. In fact, let Ω(ε) be the set defined by (3.1) with the function (4.10). Then, the step (i) shows that u ε ∈ Ω(ε). (4.11) Assume now that {ε n } is a sequence of positive numbers such that ε n → 0. It follows from inclusion (4.11) that u εn ∈ Ω(ε n )foreac hn ∈ N and, therefore, Definition 7 shows that {u εn } is an approximating sequence for the inequality (1.1). On the other hand, step (ii) guarantees that all the assumptions of Theorem 10 (d) are satisfied and, therefore, we deduce that inequality (1.1) is well-posed with respect to the family {Ω(ε)} ε>0 . We now use Definition 8 to see that u εn → u in X as n →∞which proves the convergence (4.9).

We now move to a second convergence result and, to this end, for each ε>0 we consider a set K ε together with the following perturbed version of inequality (1.1):

u ∈ K ε Au, v -u + j 0 (u; v -u) ≥ f ε ,v-u ∀ v ∈ K ε . (4.12)
Consider now the following assumptions. ( A) There exists

L A > 0 such that Au -Av X * ≤ L A u -v X ∀ u, v ∈ X.
( j) There exists L j > 0 such that

j 0 (u; w) -j 0 (v; w) ≤ L j u -v X w X ∀ u, v, w ∈ X.
( K) There exists θ ∈ X and for each ε>0 there exist

c ε > 0, d ε ∈ R such that (a) K ε = c ε K + d ε θ, (b) c ε → 1, d ε → 0a s ε → 0.
Our second result in this section is the following.

Theorem 14. Assume that X is a reflexive space and (K 1 ), ( K), ( A), (A 2 ), (j 1 ), (j 2 ), (j 3 ), (j 4 ), (f ), ( j) hold. Then the hemivariational inequality (1.1) has a unique solution u ∈ K and, for each ε>0 the hemivariational inequality (4.12) has a unique solution

u ε ∈ K ε . Moreover, u ε → u in X as ε → 0.
Proof. It is well known that any monotone Lipschitz continuous operator is pseudomonotone, and therefore, assumptions ( A), (A 2 ) imply that A satisfies condition (A 1 ). On the other hand, it is easy to check that assumptions (K 1 ), ( K)(a), imply that for any ε>0 the set K ε is non empty, closed and convex, i.e., it satisfies condition (K 1 ). The existence and uniqueness of the solution to the hemivariational inequalities (1.1) and (4.12) is now a direct consequence of the Theorem 6. Assume now that ε>0 is fixed and denote by u ε the solution of inequality (4.12). Then, using assumption ( K), it follows that there exists u ε ∈ K such that

u ε = c ε u ε + d ε θ. (4.13)
Therefore, since v ∈ K implies that c ε v + d ε θ ∈ K ε , inequality (4.12) combined with Proposition 4 (a) yield

u ε ∈ K, A(c ε u ε + d ε θ),v-u ε + j 0 (c ε u ε + d ε θ; v -u ε ) ≥ f, v -u ε ∀ v ∈ K. (4.14)
We now introduce the operator A ε : X → X * , the function j ε : X → R and the element f ε defined by

A ε v = c ε A(c ε v + d ε θ),j ε (v)=j(c ε v + d ε θ) ∀ v ∈ X (4.15)
f ε = c ε f. (4.16)
Note that the function j ε is locally Lipschitz and, therefore, condition (j ε ) is satisfied. Moreover, an elementary calculation based on Definition 2 implies that

j 0 ε (u; v)=c ε j 0 (c ε u + d ε θ; v) ∀ u, v ∈ X. (4.17)
Next, we multiply inequality (4.14) with c ε > 0, then use equalities (4.15)-(4.17) to deduce that

u ε ∈ K, A ε u ε ,v-u ε + j 0 ε ( u ε ; v -u ε ) ≥ f ε ,v-u ε ∀ v ∈ K. (4.18)
Our aim in what follows is to use Theorem 13 to prove that

u ε → u in X as ε → 0 (4.19)
and, to this end, we prove in what follows the validity of conditions (h A ), (h j ) and (f ε ). Below in the proof, we assume that u, v ∈ X and ε>0 are given. First, we use (4.15) to see that

A ε v -Av =(c ε -1)A(c ε v + d ε θ)+A(c ε v + d ε θ) -Av
and, using assumption ( A), we deduce that

A ε v -Av X * ≤|c ε -1| A(c ε v + d ε θ) X * + L A (c ε -1)v + d ε θ X .
This proves that condition (h A )(a) holds with function

h A (ε, v)=|c ε -1| A(c ε v + d ε θ) X * + L A (c ε -1)v + d ε θ X (4.20)
which clearly satisfies condition (h 1 ). Since A is a Lipschitz continuous operator, using assumption ( K)(b) it is easy to see that the function (4.20) satisfies condition (h 2 ), too. Finally, using assumption ( A), again, it follows that

|h A (ε, u) -h A (ε, v)|≤L A |c ε -1|(c ε +1) u -v X .
This inequality combined with the convergence c ε → 1 shows that the function h A defined by (4.18) satisfies condition (h 4 ) with L hA = L A |c ε -1|(c ε + 1). We conclude from above that condition (h A )i s satisfied.

Next, we use (4.17) to write

j 0 ε (u; v) -j 0 (u; v)=(c ε -1)j 0 (c ε u + d ε θ; v)+j 0 (c ε u + d ε θ; v) -j 0 (u; v)
and, using assumption ( j), we find that

j 0 ε (u; v) -j 0 (u; v) ≤|c ε -1||j 0 (c ε u + d ε θ; v)| + L j (c ε -1)u + d ε θ X v X . (4.21)
On the other hand, by Proposition 4 (b) and assumption (j 4 ), we have

|j 0 (c ε u + d ε θ; v)|≤|max{ ξ, v |ξ ∈ ∂j(c ε u + d ε θ)}| ≤ (c 0 + c 1 c ε u + d ε θ X ) v X ,
and substituting this inequality in (4.21), we deduce that condition (h j ) holds with function

h j (ε, u)=(c 0 + c 1 c ε u + d ε θ X )|c ε -1| + L j (c ε -1)u + d ε θ X , (4.22)
which satisfies condition (h 1 ) and (h 2 ). Note that

|h j (ε, u) -h j (ε, v)|≤(c 1 c ε + L j )|c ε -1| u -v X ,
which shows that the function h j defined by (4.22) satisfies condition (h 4 ) with L hj (ε)=(c 1 c ε +L j )|c ε -1|.

Finally, since c ε → 1a n df ε = c ε f , condition (f ε ) is obviously satisfied. We are now in a position to use Theorem 13 in order to see that the convergence (4.19) holds. We now use (4.13), (4.19)a n d assumption ( K)(b) to see that u ε → u in X as ε → 0, which concludes the proof. The abstract results we present in this paper are useful in the study of various mathematical models which describe the equilibrium of elastic bodies in frictional contact with a foundation. In this section, we present a simple example which illustrate the applicability of these results. The boundary value problem under consideration is the following.

Problem 15. Find a displacement field u :[ 0 ,L] → R and a stress field σ :[ 0 ,L] → R such that σ(x)=F(x, u ′ (x)) for x ∈ (0,L), (5.1)

σ ′ (x)+f 0 (x)=0 forx ∈ (0,L), (5.2)
u(0) = 0, (5.3)
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ g 1 ≤ u(L) ≤ g 2 , -σ(L)=p(u(L)) if g 1 <u(L) <g 2 , -σ(L) ≤ p(u(L)) if u(L)=g 1 , -σ(L) ≥ p(u(L)) if u(L)=g 2 .
(5.4) Problem 15 represents a mathematical model which describes the equilibrium of a rod-spring system with unilateral constraints, submitted to the action of body forces. In the reference configuration the rod occupies the interval [0,L]o nt h eOx axis, L being a given positive constant. The physical setting is depicted in Fig. 1. A brief description of the equations and boundary conditions in this problem is the following.

First, equation (5.1) represents the elastic constitutive law in which F denotes a nonlinear constitutive function. Here and everywhere below the prime denotes the derivative with respect to x, i.e., u ′ = du dx . Concrete examples of nonlinear elastic constitutive laws of the form (5.1) can be found in [START_REF] Sofonea | Variational-Hemivariational Inequalities with Applications[END_REF], for instance. Equation (5.2) is the equilibrium equation in which f 0 denotes the density of body forces acting on the rod. Condition (5.3) represents the displacement condition. We use it here since the rod is assumed to be fixed at the end x =0.

Finally, conditions (5.4) represent the boundary conditions in which g 1 and g 2 are given bounds and p is a real-valued function which will be described below. Such conditions model the physical setting in which the extremity x = L of the rod is attached to a spring which prevents its motion. The spring has an elastic behaviour, as far as the displacement of the point x = L, denoted by u(L), belongs to the open interval (g 1 ,g 2 ). Its behavior is described with the stiffness function p, assumed to be positive for a positive argument and negative for a negative one. This property of p shows that the spring push the rod when it is in compression and pull it when it is in extension. When u(L)=g 2 the spring is completely compressed and when u(L)=g 1 the spring is completely extended. In both these cases, it behaves like a rigid and, therefore, it does not allow further extension or compression of the rod, respectively. Mathematical models which describe the equilibrium of spring-rod systems in similar physical settings can be found in [START_REF] Migórski | A class of variational-hemivariational inequalities in reflexive Banach spaces[END_REF][START_REF] Sofonea | Convergence results for elliptic quasivariational inequalities[END_REF], together with various mechanical interpretations.

Moreover, for any ε>0 define the operator A ε : X → X * , the function j ε : X → R and the element f ε ∈ X * by equalities

A ε u, v = L 0 Fu ′ v ′ dx + ε L 0 (u ′ -P 0 u ′ )v ′ dx for all u, v ∈ X,
(5.17)

j ε (v)=q(v(L)) + ε 2 (v(L)) 2 for all v ∈ X, (5.18)
f ε ,v = L 0 (f 0 + ε)v dx for all v ∈ X. (5.19)
With these notation, we consider the following perturbation of Problem 16.

Problem 17. Find a displacement field u such that the inequality below holds:

u ∈ K, (A ε u, v -u) X + j 0 ε (u; v -u) ≥ (f ε ,v-u) X ∀ v ∈ K. (5.20)
Our main result in this section is the following. Then Problem 16 has a unique solution u and, for each ε which satisfies the smallness condition

0 <ε< m F L -L p , (5.22)
Problem 17 has a unique solution, denoted u ε . In addition, u ε → u in X as ε → 0.

Proof. For the existence and uniqueness part we use Theorem 6 on the space X given by (4.2). To this end we use assumption (5.8) to see that the set K given by (5.9) is a nonempty closed convex subset of X, and therefore, it satisfies assumption (K 1 ). Next, we use assumption (5.5) to see that the operator A defined by (5.10) satisfies the inequalities

(Au -Av, u -v) X ≥ m F u -v 2 X ∀ u, v ∈ X, (5.23
)

Au -Av X * ≤ L F u -v X ∀ u, v ∈ X. (5.24)
This show that A is strongly monotone and Lipshitz continuous and, therefore, conditions (A 1 ) and (A 2 ) hold, the second one with m A = m F . Next, the results presented in Example 5 show that the function j defined by (5.11) satisfies the properties (j 1 )( j 2 ) and (j 4 ) with α j = L p γ 2 . On the other hand, using inequality (4.8) it is easy to see that γ ≤ √ L and, therefore, assumption (5.21) guarantees that condition (j 3 ) is satisfied, too. We are now in a position to use Theorem 6 to obtain the existence of a unique solution to Problem 16.

The unique solvability of Problem 17 follows from similar arguments. In this case, an elementary calculus shows that for each ε>0 the operator A ε is Lipschitz continuous with Lipschitz constant L F +2ε and strongly monotone with constant m Aε = m F . In addition, since j ε (v)=q ε (v(L)) for all v ∈ X where q ε is defined by (4.6), it it easy to see that the function j ε satisfies conditions (j 1 ), (j 2 ) and (j 4 ) with α jε =( L p + ε)L. Therefore, if (5.22) holds we deduce that α jε <m Aε which shows that condition (j 3 ) holds in this case, too.

For the convergence part, we use Theorem 13. To this end, we define the operator T : X → X * be equality

Tu,v = L 0 (u ′ -P 0 u ′ )v ′ dx for all u, v ∈ X, (5.25)
and use the properties of the projection map to see that T is a Lipschitz continuous operator. We now use equalities (5.10), (5.25) and (5.17) to see that we are in the framework of Example 11. We conclude from here that condition (h A ) holds. On the other hand, Example 12 guarantees that assumption (h j ) holds, too. Finally, condition (f ε ) is a consequence of equalities (5.12) and (5.19). We are now in a position to use Theorem 13 to see that u ε → u in X as ε → 0, which concludes the proof.

Once the displacement field u is known, the stress field σ can be easily obtained by using the constitutive law (5.1). A function u which satisfies (5.1) is called a weak solution to the contact Problem 15. We conclude by Theorem 10 that the Problem 15 has a unique weak solution.

We now illustrate the convergence result in Theorem 14, and to this end, we consider the set

K ε = { v ∈ X | g 1ε ≤ v(L) ≤ g 2ε }, (5.26)
together with the following assumptions g 1ε < 0 <g 2ε , (5.27) g 1ε → g 1 and g 2ε → g 2 as ε → 0.

(5.28)

We also consider the following perturbation of Problem 16.

Problem 19. Find a displacement field u ∈ K ε such that

u ∈ K ε , (Au, v -u) X + j 0 (u; v -u) ≥ (f, v -u) X ∀ v ∈ K ε .
(5.29)

We have the following existence, uniqueness and convergence result. Proof. Let ε>0. We note that assumption (5.8) allows us to define the constants c ε and d ε , by equalities c ε = g 1εg 2ε g 1g 2 and d ε = g 2ε g 1g 1ε g 2 g 1g 2 .

(5.30) Also, let θ be a function such that θ ∈ X and θ(L)=1.

(5.31)

Assume now that u and v are two elements of X such that v = g 1εg 2ε g 1g 2 u + g 2ε g 1g 1ε g 2 g 1g 2 θ.

Then, using (5.31) it is easy to check that g 1 ≤ u(L) ≤ g 2 if and only if g 1ε ≤ v(L) ≤ g 2ε and, therefore, equalities (5.30) show that u ∈ K if and only if c ε u + d ε θ ∈ K ε . We conclude from here that

K ε = c ε K + d ε θ.
(5.32)

On the other hand, assumption (5.28) shows that c ε → 1a n d d ε → 0a sε → 0, (5.33) It follows from (5.32) and (5.33) that condition ( K) is satisfied. Theorem 20 is now a direct consequence of Theorem 14.

We end this section with some mechanical interpretation of our convergence results. First, we note that Problem 17 represents the variational formulation of a problem similar to Problem 15, in which the constitutive law (5.1) was replaced by the constitutive law σ(x)=F(x, u ′ (x)) + ε(u ′ (x) -P 0 u ′ (x)), the stiffness function r → p(r) was replaced by the function r → p(r)+εr and the density of the body forces f 0 was replaced by f 0 + ε. Thus, the convergence result in Theorem 18 shows that the weak solution of equilibrium problem of the spring-rod system can be approached as close as one wish by the weak solution of the equilibrium problem of the spring-rod system with the above perturbed constitutive law, stiffness function and body forces, for a small parameter ε. In addition, the convergence result in Theorem 18 shows that small perturbations on the data g 1 and g 2 , lead to small perturbation of the weak solution of Problem 16.

Fig. 1 .

 1 Fig. 1. The rod-spring system with unilateral constraints

Theorem 18 .

 18 Assume (5.5)-(5.8),(5.14), and moreover, assume that L p L<m F .(5.21)

Theorem 20 .

 20 Assume (5.5)-(5.8), (5.21), (5.22), (5.27) and (5.28). Then Problem 16 has a unique solution u ∈ K and, for each ε>0, Problem 19 has a unique solution u ε ∈ K ε Moreover, u ε → u in X as ε → 0.

Acknowledgements

This research was supported by the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No 823731 CONMECH, the National Natural Science Foundation of China (11771067), the Applied Basic Project of Sichuan Province (2019YJ0204) and the Fundamental Research Funds for the Central Universities (ZYGX2019J095).

We now turn to the variational formulation of Problem 15, and to this end, we assume that the constitutive function F and the stiffness function p satisfy the following conditions.

for anyε ∈ R. (e)The mappingx →F(x, 0) belongs to L 2 (0,L).

(5.5) p : R → R satisfies condition (2.2) and, moreover, p(r) > 0i fr>0a n dp(r) ≤ 0i fr<0.

(5.6)

We also assume that the density of body force has the regularity

and, finally,

(5.8)

We use the space (4.2) which is a real Hilbert space with the canonical inner product (4.3) and the associated norm • X . We denote by X * and •, • the dual of X and the duality pairing between X * and X, respectively and by q : R → R the function defined by (2.3). We also define the set K, the operator A : X → X * , the function j : X → R and the element f ∈ X * by equalities

(5.9)

(5.12)

Then, the variational formulation of Problem 15, obtained by using standard arguments, is as follows.

Problem 16. Find a displacement field u such that the inequality below holds:

and denote by P 0 : R → K 0 the projection operator on K 0 , that is (5.16)


On June 7 th 2021, the American Food and Drug Administration (FDA) publicly announced that it would grant accelerated approval for aducanumab in Alzheimer's disease (AD) based on the "evidence that Aduhelm™ reduces amyloid-beta plaques in the brain and that the reduction in these plaques is reasonably likely to predict important benefits to patients". For the first time in the field of AD, a governmental agency has used a surrogate endpoint for drug approval, i.e. an endpoint "that is thought to predict clinical benefit but is not itself a measure of clinical benefit". The FDA considered reduction of amyloid load, as measured with amyloid positron emission tomography (PET), to be a valid surrogate endpoint of clinical benefit in AD. This position is unprecedented and raises numerous controversies.

What is a surrogate endpoint? In the late 1980's, methodologists defined operational criteria for using surrogate endpoints in clinical trials [START_REF] Heller | Statistical controversies in clinical research: An initial evaluation of a surrogate end point using a single randomized clinical trial and the Prentice criteria[END_REF] . The definition of surrogacy implies a causal relationship: the surrogate marker must mediate the treatment effect on the "true" clinical outcome. In the case of AD, the application of these operational criteria to establish amyloid-load measured with PET as a surrogate endpoint would require that: (1) the treatment has an effect on cognitive decline; (2) the treatment effect on slowing cognitive decline is captured by amyloid load; (3) the treatment has an effect on amyloid load; and (4) amyloid load is associated with cognitive decline (Figure 1). In neurology, this approach accelerated for instance the development of disease-modifying therapies in relapsing-remitting multiple sclerosis. New T2-lesions on brain MRI were first validated as surrogates of clinical relapses using seminal clinical trials of beta-interferons and glatiramer-acetate (positive on clinical outcomes), and then these MRI surrogate markers were successfully implemented in subsequent phase II trials to develop second-generation drugs. With this definition of surrogacy in mind, what do we know about the relationships between anti-amyloid therapies, amyloid load, and potential clinical outcomes in AD? First, in order to use amyloid load as a surrogate endpoint, there needs to be a preceding clinical trial that demonstrates that anti-amyloid therapies reduce cognitive decline. This is a pre-requisite to being able to assess a surrogate marker within the framework of this positive trial, which can then be used as an endpoint in a subsequent, separate trial. Otherwise, how can we expect that a treatment may have an effect on cognitive decline when we are relying on surrogate endpoints? However, to date, no phase III clinical trial using an anti-amyloid therapy has yet proven to be positive without controversies. One could argue that the recent positive donanemab 2 and lecanemab [START_REF] Swanson | A randomized, double-blind, phase 2b proof-of-concept clinical trial in early Alzheimer's disease with lecanemab, an anti-Aβ protofibril antibody[END_REF] phase II trials, combined to the aducanumab's EMERGE trial, could be used to fulfill this criterion but this would ignore the negative aducanumab's ENGAGE trial and the other negative antiamyloid clinical trials (which could arguably be the consequence of lower pharmacodynamic potential, the use of lower doses or the wrong target population).

Second, even under the hypothesis that the donanemab, lecanemab and aducanumab EMERGE trials provide evidence for a positive effect of the treatment on cognitive decline, amyloid load should be mediating the treatment effect on cognitive decline. In other words, anti-amyloid therapies should reduce cognitive decline by reducing an individual's amyloid load and not by reducing another, more causal and direct, surrogate marker 1 . This can be tested statistically (mediation analyses), but, to the best of our knowledge, Biogen did not provide this information with the aducanumab trials (and the publicly available FDA statistician's report suggests the opposite at the individual level). While there is currently a lack of data regarding the mediation effect of amyloid load reduction between the treatment effect and cognitive decline, some examples from previous anti-amyloid clinical trials underline that there might not be such a direct mediation effect. On the one hand, previous results from active and passive immunotherapies have failed to demonstrate a clinical effect despite a significant reduction of amyloid load [START_REF] Liu | Amyloid-β 11C-PiB-PET imaging results from 2 randomized bapineuzumab phase 3 AD trials[END_REF][START_REF] Holmes | Long-term effects of Aβ42 immunisation in Alzheimer's disease: follow-up of a randomised, placebo-controlled phase I trial[END_REF] . On the other hand, anti-amyloid immunotherapies targeting specifically soluble ABeta species may also have an impact on cognition, that would not be mediated by amyloid plaques removal (amongst the published phase III studies, the results from the solanezumab EXPEDITION3 trial were actually the closest to be significant on clinical outcomes, without any effect on amyloid load) [START_REF] Honig | Trial of solanezumab for mild dementia due to Alzheimer's disease[END_REF] .

Third and fourth, while aducanumab has convincingly demonstrated a reduction of amyloid load in every trial, observational studies have previously shown that amyloid load is poorly directly associated with cognition. Furthermore, longitudinal cohorts have also shown that cognitively unimpaired individuals above 60 years old with in vivo evidence of amyloid pathology have a lifetime risk of progression to symptomatic AD below 50%, and show no or only minimal acceleration of tau-PET tracer uptake in the following 1 or 2 years [START_REF] Dubois | Clinical diagnosis of Alzheimer's disease: recommendations of the International Working Group[END_REF] . Using multi-modal imaging and mediation statistical analyses, amyloid load has proven to be only indirectly related to cognitive decline .

The methodological points discussed here lead to an obvious conclusion: to date, there is no valid surrogate marker of clinical outcome in AD therapeutic trials. In addition, the decision to use a surrogate endpoint should be decided a priori, before the beginning of the study. But the FDA decision raises a good question: what could be a good surrogate endpoint for future clinical trials in AD? In this regard, measurements of neurodegeneration or tau pathology are known to be more direct correlates of cognitive decline than amyloid load [START_REF] Hanseeuw | Association of Amyloid and Tau with Cognition in Preclinical Alzheimer Disease: A Longitudinal Study[END_REF] . Small subgroup analyses from aducanumab, lecanemab and donanemab trials suggest through heterogeneous findings that these drugs have an impact on tau pathology (measured with PET or CSF phospho-tau) and on neurodegeneration (measured with CSF total tau or neurofilament light chain -NfL) [START_REF] Mintun | Donanemab in Early Alzheimer's Disease[END_REF][START_REF] Swanson | A randomized, double-blind, phase 2b proof-of-concept clinical trial in early Alzheimer's disease with lecanemab, an anti-Aβ protofibril antibody[END_REF] . Thus, the potential surrogacy of all these promising biomarkers should now be properly investigated in the ongoing aducanumab, lecanemab, and donanemab trials with larger samples so that they could be used in future trials as valid surrogate endpoints. Indeed, after the learning curve the field has undergone in anti-amyloid therapies, the development of surrogate endpoints, based on a strict methodology, is of utmost importance to speed up the development of disease-modifying therapies in AD, as it did in the past speed up the development and management of disease-modifying therapies in multiple sclerosis.

1 US

 1 Food and Drug Administration Approval of Aducanumab-Is Amyloid Load a Valid Surrogate End Point for Alzheimer Disease Clinical Trials? Vincent Planche 1,2 , MD, PhD and Nicolas Villain 3,4* , MD, PhD. 1. Univ. Bordeaux, CNRS, IMN, UMR 5293, F-33000 Bordeaux, France. 2. Centre Mémoire Ressources Recherches, Pôle de Neurosciences Cliniques, CHU de Bordeaux, F-33000 Bordeaux, France 3. AP-HP Sorbonne Université, Hôpital Pitié-Salpêtrière, Department of Neurology, Institute of Memory and Alzheimer's Disease, Paris, France 4. Institut du Cerveau -ICM, Sorbonne Université, INSERM U1127, CNRS 7225, Paris, France * Corresponding author: Institut de la Mémoire et de la Maladie d'Alzheimer Département de Neurologie Groupe Hospitalier Pitié-Salpêtrière 47-83 boulevard de l'Hôpital 75651 PARIS Cedex 13, France nicolas.villain@aphp.fr tel: 33 1 42 16 18 37

Figure 1 .

 1 Figure 1. Graphical representation of the criteria demonstrating surrogacy (known as Prentice

Acknowledgments (disclosure of interests): During the past three years, VP was a local unpaid investigator or subinvestigator for the following trials: Evoke and Evoke+ (NCT04777396, NCT04777409, NovoNordisk), Tango and Embark (NCT03352527, NCT04241068, Biogen that developed aducanumab), Lucidity (NCT03446001, TauRx Pharmaceuticals), Autonomy (NCT04619420, Janssen). He received Research grants from Fondation Bettencourt Schueller (CCA-Inserm-Bettencourt). Independent of this work, NV receives research support from Fondation Bettencourt-Schueller, Fondation Servier, Union Nationale pour les Intérêts de la médecine (UNIM) and Fondation pour la Recherche sur l'Alzheimer; travel grants from the Movement Disorders Society, Merz-Pharma and GE Healthcare SAS; is an unpaid sub-investigator in NCT04241068 (aducanumab, Biogen that developed aducanumab), NCT04437511 (donanemab, Eli-Lilly), NCT04592341 (gantenerumab, Roche), NCT03887455 (lecanemab, Eisai that developed aducanumab), NCT03352557 (gosuranemab, Biogen that developed aducanumab), NCT03828747 and NCT03289143 (semorinemab, Roche), NCT04619420 (JNJ-63733657, Janssen -Johnson & Johnson), NCT04374136 (AL001, Alector), NCT04592874 (AL002, Alector); and has given unpaid lectures in symposia organized by Eisai (that developed aducanumab).