Elliptic and evolutionary variational-hemivariational inequalities with applications

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(1)Jagiellonian U niversity in Krakow F aculty o f M a th em a tics and C o m p u ter S cien ce. JA G IE L L O N IA N U N IV E R S IT Y IN K R A K Ó W. Biao Zeng E lliptic and E volutionary V ariational-H em ivariational Inequalities w ith A pplications P h D T h esis. D octoral dissertation w ritten under supervision of prof. dr hab. Stanisław Migórski. K rakow 2018.

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(3) The author would like to th an k Professor Stanislaw Migórski for his imm easurable help during the last four years, Professors A nna Ochal and Krzysztof Bartosz for their advices and inspirations, and my wife who always believes in me.. This research was partially supported by the N ational Science Center of Poland under M aestro Project no. U M O -2012/06/A /ST1/00262..

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(5) A bstract The dissertation consists of a series of five articles whose common features are problems described by elliptic and evolutionary variational-hem ivariational inequali ties. The m ain objective of the thesis is to present various results on existence, unique ness, regularity, continuous dependence of solution on the d ata for inverse and optim al control problems, quasi mixed equilibrium problems and evolutionary subdifferential inclusions. The applicability of results is illustrated by the several models which ap pear in applications: interior and boundary sem iperm eability model, quasistatic fric tional contact problem w ith unilateral constraint, and non-stationary Navier-Stokes equation w ith nonmonotone and m ultivalued frictional boundary condition..

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(7) Streszczenie Rozprawa doktorska sklada się z jednotem atycznego cyklu pieciu artykulow poswieconych zagadnieniom opisywanym przez eliptyczne i ewolucyjne nierownosci wariacyjno-hemiwariacyjne. Głównym celem rozprawy jest przedstawienie rezultatów o istnieniu, jednoznacznosci, regularnosci i ciaglej zaleznosci rozwiazania od danych dla zagadnien odwrotnych i problemow sterowania optymalnego, mieszanych zagadnien równowagi i ewolucyjnych inkluzji zawierajacych operator subrozniczki. Zastosowania otrzym anych rezultatów sa ilustrowane przez kilka modeli: model wewnetrznej i brze gowej polprzepuszczalnosci, quasistatyczny model kontaktowy z tarciem i ograniczeni ami jednostronnym i oraz niestacjonarne rownanie N aviera-Stokesa z niemonotonicznym i wielowartosciowym warunkiem brzegowym opisujacym zjawisko tarcia..

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(9) C on ten ts 1. I n tr o d u c tio n a n d m o tiv a tio n o f t h e re s e a rc h. 1. 2. M a in r e s u lts o f t h e th e s is 2.1 Continuous dependence on the d a ta for variational-hem ivariational in equality ............................................................................................................... 2.2 Inverse problem for variational-hem ivariational in e q u a lity .................... 2.3 Quasi mixed equilibrium problem w ith optim al c o n t r o l ........................ 2.4 Evolutionary problem with weakly continuous o p e r a t o r ......................... 5. 3. 5 7 10 11. M o d e ls m o tiv a tin g th e s tu d ie s 3.1 Interior and boundary sem iperm eability p ro b le m ..................................... 3.2 Q uasistatic frictional contact problem w ith unilateral constraint . . . 3.3 N on-stationary Navier-Stokes equation w ith nonm onotone multivalued frictional boundary condition ........................................................................ 13 13 15. 4. R e fe re n c e s. 21. 5. L is t o f p u b lic a tio n s o f B ia o Z e n g. 28. 6. S ta te m e n ts o f th e c o -a u th o rs. 18. 141.

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(11) 1. In tro d u ctio n and m o tiv a tio n o f th e research. The study of various variational inequalities, hem ivariational inequalities and equi librium problems has a long history in Applied M athem atics. They are very of ten related to problems in m athem atical physics, partial differential equations, eco nomics, engineering, mechanics and problems in the theory of control and optim iza tion, see [76, 77]. The theory of hem ivariational inequalities has been initiated in early 1980s with the pioneering works of Panagiotopoulos, see [60, 61, 62, 63] and the references therein. As a generalization of variational inequalities, hem ivariational inequalities are varia tional descriptions of physical phenom ena th a t include nonconvex, nondifferentiable and locally Lipschitz functions. In the past decades, the theory of hem ivariational inequalities has emerged as one of the m ost promising branches of pure, applied and industrial m athem atics. This theory gives a powerful m athem atical apparatus for studying a wide range of problems arising in diverse fields such as structural mechan ics, elasticity, economics, optim ization, contact problems, and others, see [11, 54]. The literature on hem ivariational inequalities and their applications is today very extensive and still growing. It is well known th a t such inequalities have often an equivalent for m ulation as operator subgradient inclusions. Elliptic and evolutionary hem ivariational inequalities have been studied in e.g. [27, 44, 49, 50, 51, 52] by using m ethods based on surjectivity results for several classes of monotone operators, in e.g. [29, 30, 31] by ex ploiting the Rothe approxim ation m ethods, and in e.g. [37, 71, 75, 78] by introducing and applying some concepts of quasim onotonicity and KKM technique. As another im portant and useful generalization of variational inequalities, the theory of equilibrium problems provides a unified and general framework to study a wide class of m athem atical problems involving equilibria such as optim ization, opti mal control, convex analysis, transp o rtatio n , economics, network and noncooperative games among others, see, for instance, [4, 7, 8, 12, 18, 32, 36, 58, 59, 64, 73]. There fore, since the equilibrium problem unifies at least all the above mentioned problems in a common form ulation and many techniques and m ethods established in order to solve one of them may be extended, it is w orth to find out suitable ways to solve them efficiently. For more details, we refer to [1, 2, 3, 6, 9, 33]. V ariational-hem ivariational inequalities represent a special class of inequalities which involve b oth convex and nonconvex functions. Various classes of such inequali ties have been recently investigated, for instance, in [25, 36, 45, 54, 55, 67, 70]. They play an im portant role in a description of many mechanical problems arising in solid and fluid mechanics. Nonetheless, in applications, many im portant models have mo tivated the study of optim al control for variational-hem ivariational problems. O ptim al control problems, identification problems and inverse problems for equa tions, variational inequalities, hem ivariational inequalities and equilibrium problems 1.

(12) belong to an expanding and vibrant branch of applied m athem atics th a t has found numerous applications, for instance, in [7, 10, 19, 23, 24, 26, 28, 32, 34, 36, 39, 40, 41, 42, 43, 46, 47, 48, 53, 65, 74]. A lthough the theory and com putational techniques for optim al control for equations and variational inequalities have been studied for quite some tim e now, it seems th a t there are still m any questions to be answered. Moreover, despite the fact th a t there are many im portant models leading to hem ivariational in equalities, the corresponding optim al control problem is largely unexplored and this fact is the m otivation of the present work. The present thesis is m otivated by several classes of variational inequalities, hemivariational inequalities and equilibrium problems studied in last decades. We under line th a t the m athem atical theory of variational-hem ivariational inequalities is not currently completely developed. O ur experience during the last years shows th a t deep results from Functional Analysis, Nonlinear Analysis and N onsm ooth Analysis might give an impulse for further research in this field. The m ain objective of the thesis is to provide system atic approaches to study the existence, uniqueness, convergence and optim al control for elliptic and evolutionary variational-hem ivariational inequalities. Our overall research effort in the thesis consists of the following steps th a t are closely related to each other: a) to present the existence and uniqueness of solutions to variational-hem ivariational inequalities, b) to deliver the convergence of solutions to variational-hem ivariational inequali ties, c) to study inverse problems for variational-hem ivariational inequalities, d) to apply the above theories to various models in C ontact Mechanics. In addition to the above listed targets, in the thesis we employ the theory of monotone and pseudomonotone multivalued operators in Banach spaces, the Banach fixed contraction principle, the KKM theorem , and th e finite element m ethod. The thesis consists of the following five papers dealing w ith elliptic and evolution ary variational-hem ivariational inequalities and their applications (in chronological order). (I) B. Zeng, Z.H. Liu, S. Migorski, On Convergence of Solutions to VariationalHemivariational Inequalities, Zeitschrift fü r angewandte M athem atik und Physik (2018) 69:87, 20 pages, doi: 10.1007/s00033-018-0980-3. IF: 1.687. (II) B. Zeng, S. Migorski, V ariational-Hem ivariational Inverse Problem s for Unilate ral Frictional C ontact, Applicable Analysis (2018), 21 pages, doi: 10.1080/00036 811.2018.1491037. IF: 0.963. (III) S. Migorski, B. Zeng, Convergence of Solutions to Inverse Problem s for a Class 2.

(13) of V ariational-Hem ivariational Inequalities, Discrete & C ontinuous Dynamical System s-B (2018), 22 pages, doi:10.3934/dcdsb.2018172. IF: 0.972. (IV) Z.H. Liu, S. Migorski, B. Zeng, Existence Results and O ptim al Control for a Class of Quasi Mixed Equilibrium Problem s Involving the (f, g, h)-Quasimonotonicity, Applied M athematics and Optimization (2017), 21 pages, doi: 10.1007/s00245-017-9431-3. IF: 1.236. (V) B. Zeng, S. Migorski, Evolutionary Subgradient Inclusions w ith Nonlinear W eak ly Continuous O perators and Applications, C om puters and M athem atics with Applications 75 (2018), 89-104. IF: 1.531. The m ain feature of all papers in the thesis is the m athem atical and nonsm ooth analysis of the problems. In paper (I) we investigate the convergence of the sequence of solutions to the tim e-dependent variational-hem ivariational inequalities under the presence of perturbations in the data. First, we give an existence and uniqueness re sult for the problem, and then, deliver a continuous dependence result when all the d ata are subjected to perturbations. A sem iperm eability problem is given to illustrate our m ain results. In paper (II) we study inverse problems for a class of nonlinear el liptic variational-hem ivariational inequalities. We prove results on the well posedness of a variational-hem ivariational inequality and on existence of solution to inverse problems. We illustrate our findings by an inverse problem for a frictional unilateral contact problem in nonlinear elasticity. Paper (III) is a continuation of paper (II). The solution of the variational-hem ivariational inequality is approxim ated by its penal ized version. We prove existence of solutions to inverse problems for b o th the initial inequality problem and the penalized problem. We show th a t optim al solutions to the inverse problem for the penalized problem converge, up to a subsequence, when the penalty param eter tends to zero, to an optim al solution of the inverse problem for the initial variational-hem ivariational inequality. In paper (IV), by introducing a new concept of the (f, g, h)-quasim onotonicity and applying the maximal monotonic ity of bifunctions and the KKM technique, we show the existence results for quasi mixed equilibrium problems when the constraint set is compact, bounded and un bounded, respectively, which extend and improve several well-known results in many respects. Next, we also obtain a result on existence to an optim al control problem. O ur m ain results can be applied to evolution equations, differential inclusions and hem ivariational inequalities. In paper (V), we consider the first order evolutionary inclusions w ith nonlinear weakly continuous operators and a multivalued term which involves the Clarke subgradient of a locally Lipschitz function. First, we provide a surjectivity result for stationary inclusion w ith weakly-weakly upper semicontinuous multifunction. Then, we use this result to prove the existence of solutions to the Rothe sequence and the evolutionary subgradient inclusion. Finally, we apply our results to 3.

(14) the non-stationary Navier-Stokes equation w ith nonm onotone and multivalued fric tional boundary conditions. The outline of the thesis is as follows. In Section 2, we provide a short description of m ain results of the papers (I)-(V ). In Section 3, we present some m otivating models to illustrate the results of Section 2. Sections 4 and 5 contain a list of papers related to the thesis, and a list of publications of the author, respectively. In Section 6, statem ents of the co-authors of the papers (I)-(V) are attached.. 4.

(15) 2. M ain resu lts o f th e th e sis. The aim of this section is to present a short description of the m ain results obtained in papers (I)-(V ). Let (X, II ■||x ) be a Banach space. We denote by X * its dual space and by (■, -)x the duality pairing between X * and X . The following two crucial notions of the convex subdifferential and the Clarke generalized gradient are frequently used in the thesis. D e f i n i t i o n 1 Let p : X ^ R U { + œ } be a proper, convex and l.s.c. function. The mapping dcp : X ^ 2X* defined by dcp(u) = { u* G X * | (u*,v — u)x < p(v) — p (u) for all v G X } for u G X , is called the subdifferential of p. A n element u* G dcp(u) is called a subgradient of p in u. D e f i n i t i o n 2 Given a locally Lipschitz function p : X ^ R, we denote by p 0(u; v) the (Clarke) generalized directional derivative of p at the point u G X in the direction v G X defined by p(Z + Xv) — p(Z ) 0 p (u; v) = lim sup ----------. A^0+,Ç— yu X The generalized gradient of p at u G X , denoted by dp(u), is a subset of X * given by dp(u) = { u* G X * | p 0(u; v) > (u*,v)x. for all v G X }.. One of the problems studied in the thesis is the following inequality problem: find u G K and (Au — f , v — u)x + p (u ,v ) — p (u ,u ) + j 0(u; v —u) > 0 for all v G K, where K is a nonempty, closed and convex subset of a reflexive Banach space X , A : X ^ X * and p : K x K ^ R are given maps to be specified later, j : X ^ R is a locally Lipschitz function, and f G X * is fixed. The above problem is called a variational-hem ivariational inequality w ith constraint since it involves b o th the function p which is convex and lower semicontinuous in the second variable and the function j which is assumed to be locally Lipschitz.. 2.1. C o n tin u o u s d e p e n d e n c e on th e d a ta for v a ria tio n a l-h e m iv a ria tio n a l in e q u a lity. In paper (I), we tre a t the following tim e-dependent variational-hem ivariational in equality. 5.

(16) P r o b le m 3 Find a function u : R+ = [0, + œ ) ^ u(t) G K and. X such that, fo r all t G R+,. {Au(t) — f (t), v —u (t))x + p (u (t),v ) — p (u (t),u (t)). (1). + j 0(u(t); v —u (t)) > 0 for all v G K. The goal of this p art is to study the convergence of solutions to the variationalhem ivariational inequality (1) when the d ata A, f , p, j and K are subjected to perturbations. The dependence of solutions to elliptic variational-hem ivariational inequalities on the d ata has been studied only recently. For such inequalities the dependence with respect to functions p and j was first investigated in [55], where A and K were not sub jected to perturbations. A result on the dependence of solutions to elliptic variational inequality w ith respect to perturbations of the set K of a special form was studied in [66]. There, the d ata A, p and f were independent of perturbations. For a class of elliptic history-dependent variational-hem ivariational inequalities studied in [69], the convergence result was obtained in a case when p depends on a history-dependent operator, and A does not depend on perturbations. A result on th e convergence with respect to the set of constraint K was also obtained for elliptic quasivariational in equalities in [5] and a result on the dependence of solution to evolution second order hem ivariational inequality w ith respect to perturbations of operators can be found in [35]. In all aformentioned papers the convergence results were applied to various m athem atical models describing deformable bodies in contact mechanics. Further more, it is well know th a t the continuous dependence results are of im portance in optim al control and identification problems, see e.g. [10, 36, 74]. The perturbed problem corresponding to Problem 3 reads as follows. Let p > 0 be a param eter. P r o b le m 4 Find a function u p : R+ ^ X such that, fo r all t G R+, up(t) G K p and {Apu p(t) — f p(t),v p —u p(t))x + pp(up(t),v p) — pp(up(t) ,u p(t)). (2). + j°(u p (t); vp —up(t)) > 0 for all vp G Kp. The following notion of the Mosco convergence of sets is useful in this p a rt of the thesis. For the definitions, properties and other modes of set convergence, we refer to [13, C hapter 4.7] and [56]. D e f i n i t i o n 5 Let (X, || ■||) be a normed space and {K p}p>0 c 2x \{ 0 } . We say that K p converge to K in the Mosco sense, as p ^ 0, denoted by K p —^ K if and only if the two conditions hold 6.

(17) (i) fo r each x G K , there exists {xp}p>0 such that x p G K p and x p ^ x in X , (ii) fo r each subsequence {xp}p>0 such that x p G K p and x p ^ x weakly in X , we have x G K . The aim of this p a rt to consider the class of abstract tim e-dependent variationalhem ivariational inequalities of the form ( 1 ) for which we study the dependence of the solution w ith respect to the d a ta A, f , p, j and K . Here, the set of constraint is of a general form and for its convergence we use the Mosco convergence recalled in Definition 5. The applicability of the convergence result can be found in the study of a sem iperm eability problem, see Section 3.1. The m ain results in this p art are following (i) for each p > 0, Problem 4 has the unique solution u p G C(R+; K p), (ii) for each t G R+, there is a subsequence {up(t)} such th a t u p(t) ^ u (t) in X , as p ^ 0, where u G C (R+; K ) is the unique solution to Problem 3.. 2.2. In v erse p ro b lem for v a ria tio n a l-h e m iv a r ia tio n a l in eq u a li ty. The variational-hem ivariational inequality studied in papers (II) and (III) can be form ulated as follows. Let K be a nonempty, closed and convex set of constraints in a reflexive Banach space X , and P be a normed space of param eters. Given a nonlinear pseudomonotone operator A : P x X ^ X *, a convex functional p : P x K x K ^ R, a locally Lipschitz (in general nonconvex) functional j : P x X ^ R, and a map f : P ^ X * w ith some properties to be specified later, the abstract variationalhem ivariational inequality has the form: P r o b le m 6 (P rob/em ( D P )). Given p G P , find u = u(p) G K such that. {A(P ,u) — f (P),v —u )x + p (p ,u ,v ) — p (p ,u ,u ). (3). + j 0 (p, u; v —u) > 0 for all v G K. We recall the following notion of the penalty operator, see [14]. D e f i n i t i o n 7 A n operator P : X ^ X * is said to be a penalty operator of a set K c X if and only if P is bounded, demicontinuous, m onotone and K = {x G X | P x = 0x *}. Assume th a t P : P x X ^ X * is such th a t P(p, ■) is a penalty operator of K for each p G P . For every fixed A > 0 and p G P , we consider the following penalized problem associated with Problem 6 . 7.

(18) P r o b le m 8 (P rob/em (D P )A) Find an element u A = u A(p) G X such that {A(p, u A) — f (p), v —u A)x + A {P (p, u A) , v —u A)x. (4). + p (p , u A,v) — p(p, u A,u A) + j° ( p ,u A; v —u A) > 0 for all v G X. In many applied problems, identification consists in the determ ination of the un known param eter p in the direct problem from various m easurem ents (observations) of the data. For instance, let H b e a subset of R d which is occupied by an elastic body. Consider, see [39] for details, an inverse problem in which the aim is to m atch the final shape of the body w ith the desired shape U by controlling the Young modulus E . In other words, given a target shape U in a suitable space, we are interested in finding the Young modulus for which the target is attained, i.e., we minimize the quadratic cost functional F (u(E )) = 1 / (u — U )2 dx 2 Jn over E in the following set of admissible param eters P ad = {E G L 2(H) | e0 < E (x ) < ei for a.e. x G H} w ith e0, ei > 0. There are many possible choices of classes of admissible param eters P ad c P and of cost functionals, see, for instance, [26, 40, 41, 42]. The above example of inverse problem enters the framework considered in paper (II). Since the direct problem is a variational-hem ivariational inequality, we are able to incorporate in this setting vari ous complicated physical phenom ena modeled by nonmonotone and nondifferentiable potentials which are m et in applications. On the other hand, the contact problem under consideration offers some nontrivial m athem atical interest. We also note th a t the techniques and results discussed in paper (II) could be also used in many other param eter identification problems in mechanics. In the first p art of paper (II), we provide results on the well posedness of Prob lem 6, which is called the direct problem. The existence and uniqueness of solution to Problem 6 has been recently obtained in [55]. We provide a slightly different proof of existence and uniqueness for Problem 6 and show a new result which says th a t the m apping P 9 p m u(p) G K is Lipschitz continuous. This continuous dependence result is fundam ental to obtain the existence of solution to an inverse problem in which we minimize an appropriate cost functional F defined on the space of admissi ble param eters P ad c P . The second p a rt of paper (II) is devoted to study an inverse problem for the penalized variational-hem ivariational inequality in which a penalty operator is introduced. The m ain result in this p art is to establish the existence of optim al solutions for the penalized direct problem. 8.

(19) Problem (D P )\ . Direct problem is a penalized variational-hemivariational inequality w ith solution u\(p) E X .. Problem ( I P )^. Inverse problem for ( D P ) \ w ith an optim al solution (P*x,ux (p*x ))._______________. Problem ( D P ). Direct problem is a ------ > variational-hem ivariational inequality w ith solution u(p) E K .. a^o ^. Problem ( I P ). Inverse problem for (D P ) w ith an optim al solution (p * ,u (p*))._________________. Figure 1: Outline of paper (III) Finally, we study in paper (III) the convergence of optim al solutions of the pe nalized problem to an optim al solution of the initial problem, as th e positive penalty param eter A tends to zero. It allows to approxim ate optim al solutions to inverse prob lem for variational-hem ivariational inequality by an optim al solution to a simpler direct problem in which the unilateral constraint is replaced by a problem w ithout constraints. We refer to Section 3.2 for a class of inverse problems for a frictional contact problem from theory of elasticity to which the results of paper (III) can be applied. The outline of paper (III) is depicted on Figure 1 and is as follows. (1) First we study the variational-hem ivariational inequality, called the direct problem ( D P ), see Problem 6, for every fixed param eter p. For this problem, we state a result on its unique solvability and on the continuous dependence of solution w ith respect to a param eter. (2) Next, we investigate a penalized variational-hem ivariational inequality, called the penalized direct problem ( D P )^, see Problem 8, for every fixed penalty param eter A > 0 and a fixed param eter p. For every fixed A and p, we prove existence and uniqueness of solution u \ = u\(p) E X and show th a t u \ ^ u, as A ^ 0, where u = u(p) is the unique solution to the problem ( D P ). (3) Then, we study an inverse problem ( I P )A associated w ith the problem ( D P )A. We prove a result on existence of optim al solutions (p*\,u(p*x)) to inverse problem (IP )a for every fixed A. (4) Finally, we prove the m ain result on the convergence of optim al solutions: we can find a subsequence of |(p ^ , u(p^))} th a t converges to an element (p*,u(p*)) which is an optim al solution to the inverse problem ( I P ) for ( D P ).. 9.

(20) 2.3. Q u asi m ix e d eq u ilib riu m p ro b lem w ith o p tim a l co n tro l. The aim of paper (IV) is to study the existence of solution to a generalized problem of quasi mixed equilibrium in a reflexive Banach space. We introduce a new concept of stable (f, g, h)-quasimonotonicity, and use properties of the maximal monotonicity of bifunctions and Knaster-Kuratowski-M azurkiewicz (KKM) technique. O ur results extend and improve some investigations presented in [9, 36, 71, 75, 78] in many aspects. We recall the concept of (f, g, h)-quasim onotonicity which is useful to establish existence theorems. For more details and a discussion of this concept, we refer to [71, 75, 78]. Let E be a real Banach space w ith the norm denoted by || ■||E and the dual space E *. Let (■, -)E be the duality pairing between E * and E . Let K be a nonempty, closed and convex subset of E . Let F : K ^ E *, f : K x K m E , g, h : K x K m R U { ± œ } and D(g) = {x G K | g (x ,y ) = —œ for all y G K } w ith D(g) n D (h) = 0. D e f i n i t i o n 9 A multivalued operator F is said to be (i) (f, g, h)-pseudomonotone, if fo r each x , y G K , (x*, f ( x , y )) + g (x ,y ) > h(y ,x). =^. (y*, f ( x , y )) + g (x ,y ) > h(y ,x). fo r all x* G F (x) and y* G F (y), (ii) (f, g, h )-quasimonotone, if fo r each x , y G K , (^ f ( x y )) + g (x y ) > h ^ x). =^. (y*, f ( x y )) + g ( x ,y) >. h(y , x) (5). fo r all x* G F (x) and y* G F (y), (iii) stably (f, g, h)-pseudom onotone with respect to a set Z C E *, if F and F (■) —Z are (f, g, h)-pseudomonotone fo r every Z G Z , (iv) stably (f, g, h )-quasimonotone with respect to a set Z C E *, if F and F (■) — Z are (f, g, h )-quasimonotone fo r every Z G Z . In paper (IV), we first discuss the following quasi mixed equilibrium problem involving a multivalued operator F: f find x G D(g) n D (h) and x* G F (x) such th a t I (x*, f (x ,y ))E + g(x, y) + h(x, y) > 0 10. for all y G K..

(21) Then, we focus on the optim al control problem for (6) w ith h = 0 which is form ulated below. Let V be a reflexive Banach space of controls and U C V be a set of admissible controls. We suppose th a t U is nonempty, closed and convex. Given a control u G U , we consider the following optim al control problem find x G IXo) C K and x* € F (x) such th a t. {. (7). (x* — B ( u ) ,f (x,y)) + g (x ,y ) > 0 for all y G K,. where B : V m E * is a compact mapping. We denote by Sol(u) C K the set of all solutions of (7) corresponding to a control u. Given another Banach space W (the so-called observation space), a compact m ap ping £ : E m W , and a target element Y G W , we consider the following cost function H (u, x) = ||£ (x ) — Y 12 + e ||u ||2, where e > 0. The optim al control problem studied in this work consists in finding an optim al pair (u, x) G U x K th a t solves the following m inim ization problem min{ H (u, x) | u G U, x G Sol(u) }.. (8). In paper (IV), under the stable (f, g, h)-quasim onotonicity w ith respect to a set of the multivalued operator F , some conditions on operators B, f , and functions g and h, we show the existence of solutions in the case when the constraint set K is compact, bounded and unbounded, respectively, and prove a result on existence of the optim al solutions to the control problem (8).. 2 .4. E v o lu tio n a r y p ro b lem w ith w ea k ly c o n tin u o u s o p era to r. In paper (V) we study the first order evolutionary inclusion w ith a nonlinear weakly continuous, bounded and coercive operator and a multivalued term involving the Clarke subgradient of a locally Lipschitz function. This study is closely related to a first order evolutionary hem ivariational inequality. In paper (V) we formulate the inclusion problem in the framework of evolution triple of spaces. Recall th a t V C H C V * is an evolution triple of spaces, if V is a reflexive and separable Banach space, H is a separable Hilbert space, and the em bed ding V C H is dense, continuous and compact. Given 0 < T < + œ , we introduce the Bochner spaces V = L 2(0 ,T ; V ) and W = {v G V | v' G V*}, where the time derivative w' = d w /d t is understood in the sense of vector-valued distributions. By reflexivity of V , it follows th a t bo th V and its dual space V * = L 2(0, T ; V *) are reflex ive Banach spaces. Moreover, the space W endowed w ith the graph norm ||w ||w = 11.

(22) ||w||v + Hw'llv* is a separable and reflexive Banach space. Let H = L 2(0 ,T ; H ). Iden tifying H = L 2 (0 , T ; H ) w ith its dual, we obtain the continuous embeddings W c V c H c V *. Let U be a reflexive Banach space, y : V m U be a linear, continuous and compact operator, and U = L 2 (0 ,T ; U ). The Nemytskii operator 7 : V M U corresponding to Y is defined by (Yv)(t) = y (v(t)) for v G V , a.e. t G (0 ,T ). Recall the following definition of a weakly continuous operator. D e f i n i t i o n 10 ([21]) Let X be a reflexive Banach space. A n operator F : X m X * is said to be weakly continuous, if fo r any sequence {un}n>i C X with un M u weakly in X , then F u n M F u weakly in X *. The first existence result proved in paper (V) reads as follows. T h e o r e m 11 Let X be a reflexive and separable Banach space, and F : X M 2X be a weakly-weakly upper semicontinuous and coercive multivalued operator with nonempty, bounded, closed and convex values. Then, F is surjective in the sense that fo r any f G X * the inclusion F (u) 9 f has a solution u G X . Given a weakly continuous operator A : V M V *, a locally Lipschitz functional J : U M R, f G V* and u 0 G H , we consider the evolutionary inclusion of the following form. P r o b le m 12 Find u G W such that u(0) = u 0 and u '(t) + A u(t) + y *d J(Y u (t)) 9 f (t) for a.e. t G (0 ,T ).. (9). An element u G W is called a solution to Problem 12 if there exists n G U * such th a t u '(t) + A u(t) + y *n(t) = f (t) a.e. t G (0 ,T ), n(t) G d J (y (t)) a.e. t G (0 ,T ) and u ( 0 ) = u 0. The second existence result of paper (V) concerns Problem 12 and it was obtained by using the Rothe m ethod and Theorem 11. There are several novelties of paper (V). First, the existence result for Problem 12 obtained under the hypotheses of weak continuity, boundedness and coercivity of operator A, extends a theorem in [21 ] proved for elliptic equations. Second, in contrast to [30, 31], the existence proof for Problem 12 uses the Rothe m ethod applied to evolutionary subgradient inclusion w ith weakly continuous operator. Third, we provide a new application of Problem 12 and obtain a result on the existence of solution to non-stationary Navier-Stokes problem w ith multivalued frictional boundary condition, see Section 3.3. 12.

(23) In the future, we plan to study more general form of evolutionary variationalhem ivariational inequality of parabolic type, which is form ulated as follows. Let V and Y be reflexive separable Banach spaces, 0 < T < œ and V = L 2(0, T ; V ) w ith its dual V*. Given a nonempty, closed and convex subset K of V , the so-called constraint set, operators A : V ^ V * and M : V ^ Y , functions ^ : K x K ^ R and j : Y ^ R, and f G V*, we consider the following problem. Find u G V such th a t u(t) G K for a.e. t G (0 ,T ), u' G V* and (u'(t) + A u(t) — f (t), v —u (t ) )v + ^ (u (t), v) — <p(u(t),u(t)). {. + j 0(M u(t); M v — M u ( t) ) > 0 for all v G K , a.e. t G (0 ,T ). (10). u(0) = 0.. Note th a t Problem (10) reduces, in particular cases, to several im portant inequalities studied in the literature. For instance, if j = p = 0, then (10) is a parabolic variational inequality of the first kind. If j = 0, K = V and ^ is independent of the first variable, then problem (10) is a parabolic variational inequality of the second kind.. 3. M o d els m o tiv a tin g th e stu d ies. In this section we present a collection of models to which the results of Section 2 can be applied. For details, we refer to papers (I)-(V ) and the references therein.. 3.1 In terio r an d b o u n d a r y se m ip e r m e a b ility p ro b lem Let H be a bounded domain of R d w ith Lipschitz continuous boundary d ^ = r which consists of two disjoint m easurable p arts Ti and r 2 such th a t m ( r i) > 0. Recall th a t R+ = [0, + œ ) . Consider the classical model of the heat conduction described by the following boundary value problem. P r o b le m 13 Find a temperature u : H x R+ ^ R such that —d iv a(x , V u) = f ( t , u ). in. H x R+,. (11). f ( t ,u ) = fi(t) + f 2 (u), —f 2 (u) G d h ( x , u ). in. H x R+,. (12). for. t G R+,. (13). u (t) G U u = 0 du —- — G k(u)dcg ( x , u ). 13. on r 1 x R+,. (14). on r 2 x R+.. (15).

(24) We shortly describe the equations and relations in problem (11)-(15). Equation (11) is the stationary heat equation related to the nonlinear operator in divergence form w ith the tem perature and tim e dependent heat source f = f ( t ,u ) . In (12), the function f adm its an additive decomposition on f 1 = f 1(t) which is prescribed and independent of the tem perature u, and f 2 = f 2(u) which is a m ultivalued function of u in the Clarke subgradient term . Here h = h ( x ,r ) is a locally Lipschitz function in the second argum ent function. Condition (13) introduces an additional constraint for the tem perature (or the pressure of the fluid in a fluid flow model). The tem perature u is constrained to belong to a convex, closed set U . For instance, the set U can be used to introduce a bilateral obstacle which means th a t we look for the tem perature w ithin prescribed bounds in the domain H. The homogeneous (for simplicity) Dirichlet boundary condition is supposed in (14). In boundary condition (15) the expression d u = a(æ, V u) ■v represents the heat flux through the p art r 2, where v denotes the outw ard unit norm al on r . Here, g = g ( x ,r ) is a prescribed function, convex in its second argum ent, dcg stands for its convex subdifferential, and a given function k depends on the tem perature and it is assumed to be positive. Note th a t in (15) we deal w ith the nonlinearity which is determ ined by a law of the form kôcg. The m otivation to study such sem iperm eability problem comes from [17, C hap ter I] (where the m onotone sem iperm eability relations were considered), and [60, C hapter 5.5.3] and [61] (where nonm onotone relations were studied). In Problem 13 we have introduced constraints given by b o th the interior and boundary semiper m eability relations (12) and (15), respectively. The heat conduction problem w ith such relations leads to the weak form ulation which is a variational-hem ivariational inequality. We provide this form ulation below. We introduce the following spaces V = { v G H 1(H) | v = 0 on r },. H = L 2(H).. Since m ( r 1) > 0, on V we can consider the norm ||v||V = ||V v ||L2(n) for v G V which is equivalent on V to the H 1(H) norm. By using the Green formula, we can obtain the variational form ulation of Problem 13. P r o b le m 14 Find u : R+ m U such that fo r all t G R+ / a (x , V u(t)) ■V (v —u(t)) dx + (k (u (t))g (x , v) — k (u (t))g (x , u (t))) d r Jn Jr 2 + n. h0(x ,u (t); v — u (t)) dx > n. f 1(t)(v —u(t)) dx for all v G U.. For Problem 14 we can apply the abstract results obtained in paper (I) and recalled in Section 2.1 w ith the following functional framework: X = V , K = U , f (t) = f 1(t) 14.

(25) for all t G R+ and A : V ^ V *,. (Au,v)v =. a(x, V u ) ■V v d x. for u ,v G V,. Jn p : V x V ^ R,. p(u,v)=. k(u)g(v) d r. for u ,v G V,. Jr j : V ^ R,. j (v) =. h(v) dx. for v G V.. n Moreover, we also consider the dependence of solution to Problem 14 on the per tu rbation of the m apping a, functions k, g, h and f i, and the set U . P r o b le m 15 Find u p : R+ ^ Up such that for all t G R+ / ap(x, V up(t)) ■V(vp —up(t)) dx + / (kp(up(t))gp(x,vp) — kp(up(t))gp(x,up(t))) d r Jn Jr2 + / h°p( x , u p(t); vp — u p(t)) dx > n n. f ip(t)(vp — u p(t)) dx for all vp GUp.. The m ain results for the semiperm eability problem are following: (i) for each p > 0, Problem 15 has a unique solution u p G C (R+; Up), (ii) for each t G R+, there is a subsequence of {up(t)}, not relabelled, such th a t up(t) ^ u(t) in V , as p ^ 0, where u G C (R+; U ) is the unique solution to Problem 14. Finally, we note th a t similar semiperm eability problems can be form ulated in electrostatics and in flow problems through porous media, where the semiperm eability relations are realized by natural and artificial m embranes of various types, see [17, 60, 62, 63]. Our convergence result is also applicable to various problems in contact mechanics like a nonlinear elastic contact problem w ith norm al compliance condition w ith unilateral constraint, and a contact problem w ith the Coulomb friction law in which the friction bound is supposed to depend on the norm al displacement, see e.g. [5, 55, 66].. 3.2. Q u a sista tic fric tio n a l c o n ta c t p ro b lem w ith u n ila te r a l c o n stra in t. An elastic body occupies an open, bounded and connected set ^ C R d, d = 2, 3 in applications. The boundary of ^ is denoted by r = d ^ and it is assumed to be Lipschitz continuous. We denote by v the outw ard unit norm al at r . We suppose th a t r consists of three m utually disjoint and m easurable parts r i , r 2 and r 3 such th a t 15.

(26) meas ( ^ ) > 0. We denote by Sd th e space of sym metric real d x d matrices. On R d and Sd we use the standard inner products and norms defined by 1 ■n = CiVi, ll^l = ( | ■|)1/2 a ■T = aijTij , ||a || = ( a ■a ) 1/2. for all for all. 1 = (&), n = (m ) E Rd , a = ( j ij ), t = (rij ) E S d,. respectively. Given a vector field | , the notation and | T represent its norm al and tangential components on the boundary defined by = | ■v and | T = | — v . For a m atrix a , the symbols j v and a T denote its norm al and tangential components on the boundary, av = ( a v ) ■v and a T = a v — a vv . We denote by P a normed space of param eters. We consider the following qua sistatic frictional contact problem w ith Signorini condition of unilateral constraint, see e.g. [68]. Its classical form ulation is the following. P r o b le m 16 Given p E P , find a displacement field u : Q ^ a : Q ^ Sd and an interface force n : r 3 ^ R such that a = A(p, e (u )) D iv a + /o (p ) = 0. u v < g,. (Jv + n < 0,. ||aTII <. (uv — g)((7v + n) = 0,. uT Fb(p,uv), —aT = Fb(p, uv) ^ - ^ l|uTI. R d, a stress field. in Q,. (16). in. (17). Q,. u = 0. on r 1,. (18). a v = f 2 (p). on r 2 ,. (19). n E d j v (p, u v). on r 3 ,. (20). if ||U tI = 0. on r 3.. (21). In the above problem, equation (16) represents the elastic constitutive law in which A is the elasticity operator and e (u ) denotes the linearized strain tensor defined by e (u ) =. 1 (Sij(u)), £ij(u) = ^(ui,j + uj,i),. dui ui,j = d j. in Q,. where u = (u^_,. . . , ud) and i, j = 1 , . . . ,d. Equation (17) is the equation of equilibrium, where f 0 denotes the density of the body forces, (18) is the displacement homogeneous boundary condition which means th a t the body is fixed on r 1, and (19) represents the traction boundary condition with surface tractions of density f 2 acting on r 2. Finally, conditions (20) and (21) given on the contact surface r 3, represent the contact and the friction law, respectively. Here, g > 0 denotes the thickness of the elastic layer covering the rigid foundation, Fb is the friction bound and d j v (p, ■) represents the Clarke subdifferential of a given 16.

(27) locally Lioschitz function j„(p, ■). More details on physical settings, interpretation of boundary conditions, and formulations of static contact models for elastic m aterials can be found in [54, 57, 68]. We use the spaces V and H defined by V = { V G H 1(H; R d) | V = 0 on r },. H = L 2(H; Sd).. On the space V we consider the inner product and the corresponding norm given by (u,. v )v. = ( e (u ),e (v ) )H, ||v ||v = ||e (v )||H for all u ,. v. G V.. Recall, see e.g. [54], th a t, since m e a s ( r 1) > 0, it follows th a t V is a Hilbert space. The space H is a Hilbert space endowed w ith the inner product (7 , t )h =. a j( x ) T j ( x ) dx for all 7 , t g H Jn. and the associated norm || ■||%. We introduce the set of admissible displacement fields U defined by U = { v G V | v„ < g on r 3 }. (22) Moreover, for all p G P , we define an element f (p) G V * by ( f (p ^ v ) V = ( f. 0(P ) , V) L2(n;Rd). + ( f 2 (p ) , V) L2(r2;Rd). fo r a ll. v. V.. G. (2 3 ). Using (16)-(21), by a standard argum ent, see papers (II) and (III), we derive the following variational form ulation of Problem 16. P r o b le m 17 Given p G P , find u G U such that (A(p, e (u )), e(v ) —e ( u ) ) w + / Ffe( p ,u „ ) ( | v t || — H u I) d r Jr3 +. j°(p , u„; v„ —u „) d r > ( f (p), v — u )v for all Jr3. v. G U.. Problem 17 represents a variational-hem ivariational inequality. To solve it, we can apply the abstract results of papers (II) and (III) stated in Section 2.2 w ith the following functional framework: X = V , K = U and A : P x V M V *, p : P x V x V M R, j : P x V M R,. (A(p, u ),. v. p(p, u ,. ) = / Fb(p,u„ )|| v t | d r for u , J r :i. v. ) = (A(p, e (u )), e (v ))H for u ,. j (p, v ) = / j„ (p, v„) d r Jr3 17. for. v. GV. v. v. G V,. G V,.

(28) for all p G P . We also define the operator P : V M V * by (P u ,. v. )=. (uv —g)+vv d r for all u ,. v. G V.. (24). Jr 3 It follows from e.g. [54, Theorem 2.21] th a t P is an example of the penalty operator of the set U defined by (22). This operator is independent of the param eter p. The penalized variational-hem ivariational inequality corresponding to Problem 17 reads as follows. P r o b le m 18 Given p G P and A > 0, find u A = u A(p) G V such that (A(p, e ( u A)), e(v ) — e ( u A))w + / F ,(p ,u Av)(|| V t || — ||u At ||) d r Jr3 + I j 0(p, uav; Vv —uav) d r + 1 I (uav —g)+(vv —uav) d r •/r3 A ./ r3 > ( f (p), V —u A)V. for all v G V.. The following are the m ain results for Problem s 17 and 18. (i) The unique solvability of Problem s 17 and 18, and th e convergence of the sequence of solutions to Problem 18 to the unique solution to Problem 17, as the penalty param eter A tends to zero. (ii) The continuous dependence of the solution u A = u A(p) to Problem 18 on the param eter p G P . (iii) The existence of solution to an inverse problem for the penalized variationalhem ivariational inequality in Problem 18, and the convergence of the sequence of optim al solutions of the corresponding inverse problems.. 3 .3. N o n -sta tio n a r y N a v ier—S to k es e q u a tio n w ith n o n m o n o to n e m u ltiv a lu ed fric tio n a l b o u n d a r y c o n d itio n. In this p art of the thesis, we provide b o th classical and weak formulations of the non-stationary Navier-Stokes equation to illustrate the applicability of the results of paper (V) which are recalled in Section 2.4. Let Q be a bounded open and connected domain in R d w ith d = 2 , 3 . The boundary r = d Q is supposed to be Lipschitz continuous and it is composed of two measurable parts r D and r C with disjoint relatively open sets r D and r C such th a t m e a s ( r D) > 0. We denote by v the unit outw ard norm al vector on r and by x G Q the position vector. For 0 < T < œ , let Q = Q x (0 ,T ), = r D x (0 ,T ) and = r C x (0 ,T ). We are concerned w ith the following parabolic problem which classical form ulation reads as follows. 18.

(29) P r o b le m 19 Find a flow velocity field u = u ( x , t ) and a pressure p = p ( x , t ) such that u ' — v0 A u + (u ■V )u + V p = f. in. Q,. (25). div u = 0. in. Q,. (26). u = 0. on. ,. (27). uv = 0. on. ,. (28). —S T G d j ( u T). on. ,. (29). u (0) = u 0. in. H.. (30). Problem 19 describes the non-stationary flow of incompressible viscous liquid occupying the volume H subjected to a given external volume forces of density / = f (x ,t). Equation (25) is the conservation law, where the expression (u ■V )v = ( E d = 1u j f x r ) .. denotes the nonlinear convective term and v0 > 0 is a viscosity. constant of the fluid. The solenoidal (divergence free) condition (26) states th a t the m otion of the fluid is incompressible. Here, the to tal stress tensor in the fluid is given by < j = —p I + S in Q, where I denotes the identity m atrix and S : H x ^ is the extra (viscous) p art of the stress tensor. The sym metric p art of the velocity gradient D :H ^ is given by D ( u ) = | (V u + V u T). We assume th a t the extra stress tensor S is related w ith the symm etric p a rt of the velocity gradient D by means of the constitutive law S = 2 v0 D ( u ) in Q. Let us pass to the boundary conditions. Condition (27) represents the adherence boundary condition boundary conditions on th e p art (since the fluid is viscous). On the p art , we decompose the velocity vector into the norm al and tangential parts. For an extra stress tensor field S , we define its norm al and tangential com ponents by S v = ( S v ) ■v and S T = S v — S vv , respectively. We assume th a t there is no flux condition through r C, so th a t the norm al component of the velocity van ishes on this p a rt of the boundary, cf. (28). The tangential components of the stress tensor S T and the velocity u T are assumed to satisfy the m ultivalued friction law (29). The condition (29) is called the nonmonotone and multivalued frictional bound ary condition of friction type. Examples of convex and nonconvex function j are provided in paper (V). In condition (30), u 0 denotes the initial velocity. We refer to [15, 16, 19, 20, 22, 38, 44, 47, 72] and the references therein for more details on a m athem atical analysis and physical interpretation of the problem. To provide the weak form ulation of Problem 19, we introduce the following spaces F = { v G Cœ (H; R d) | div v = 0 in H, v = 0 on r D, vv = 0 on r C}, V = closure of F in H 1(H; R d) 19. (31).

(30) and H = { v G Cœ (Q; R d) | div v = 0 in Q, vv = 0 on r C}, H = closure of H in L 2(Q; R d).. (32). The space V is equipped w ith the norm ||v|| = ||v ||Hi(n;Rd) for v G V . On V we introduce also the norm given by ||v ||v = ||D (v )||L2(n;Sd) for v G V . From the Korn inequality c k I M I ^ n R ) < ||D (v )||L2(n;sd) for v G V w ith ck > 0, see e.g. [16, Theorem 4], it follows th a t || ■ ||Hi(n;Rd) and || ■ ||v are the equivalent norms on V . Moreover, V is a reflexive separable Banach space, H is a separable Hilbert space, the embedding V C H is continuous, compact and V is dense in H . This means th a t (V, H, V *) forms an evolution triple of spaces. We also need the following spaces V = L 2(0 ,T ; V ), V * = L 2(0, T, V *), H = L 2(0 ,T ; H ), W = {v G V | v' G V *} and U = L 2( r c ; R d). Using a standard procedure, see e.g. [15, 16, 19, 20], we obtain the following varia tional form ulation of Problem 19 which is a hem ivariational inequality of parabolic type. P r o b le m 20 Find a velocity field u G W such that (u'(t), v )v + v 0. Jn. V u (t) ■V v dx +. Jn. (( u ■V )u (t)) ■v dx. + I j 0(u T(t); v T) d r > f (t) ■v dx for all v G V, a.e. t G (0 ,T ), J rC Jn u(0) = u 0 in Q. For Problem 20 we are able to apply results described in Section 2.4. We use the following functional framework (A u, v ) v = a(u, v) + b(u, u, v) J (v) =. j (vT(x)) d r. for u , v G V,. for v G U,. J rc where a : V x V ^ R,. a(u, v) = v 0. b : V x V x V ^ R,. b(u, v, w ) =. Jn. V u ■V v dx, (( u ■V )v ) ■w dx. n for u , v, w G V. In a conclusion, we derive a result on the solvability of Problem 20, see Theorem 18 in paper (V) for more details. 20.

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(38) 5. List o f p u b lica tio n s o f B ia o Zeng (I) B. Zeng, Z.H. Liu, S. Migorski, On Convergence of Solutions to VariationalHemivariational Inequalities, Zeitschrift fü r angewandte M athem atik und Physik (2018) 69:87, 20 pages, doi: 10.1007/s00033-018-0980-3. IF: 1.687.. (II) B. Zeng, S. Migorski, V ariational-Hem ivariational Inverse problems for Unilate ral Frictional C ontact, Applicable Analysis (2018), 21 pages, doi: 10.1080/00036 811.2018.1491037. IF: 0.963. (III) S. Migórski, B. Zeng, Convergence of Solutions to Inverse Problem s for a Class of V ariational-Hem ivariational Inequalities, Discrete & Continuous Dynamical System s-B (2018), doi:10.3934/dcdsb.2018172. IF: 0.994. (IV) Z.H. Liu, S. Migorski, B. Zeng, Existence Results and O ptim al Control for a Class of Quasi Mixed Equilibrium Problem s Involving the (f, g, h)-Quasimonotonicity, Applied M athematics and O ptimization (2017), doi: 10.1007/s00245017-9431-3. IF: 1.236. (V) B. Zeng, S. Migórski, Evolutionary Subgradient Inclusions w ith Nonlinear W eak ly Continuous O perators and applications, Computers and M athematics with Applications 75 (2018), 89-104. IF: 1.531. (VI) Z.H. Liu, S. Migorski, B. Zeng, O ptim al Feedback Control and Controllability for Hyperbolic Evolution Inclusions of Clarke’s Subdifferential Type, Computers and M athem atics with Applications 74 (2017), 3183-3194. IF: 1.531. (VII) Y. Huang, Z.H. Liu, B. Zeng, O ptim al Control of Feedback Control Systems Governed by Hemivariational Inequalities, Computers and M athem atics with Applications 70 (2015), 2125-2136. IF: 1.531. (VIII) Y. Huang, B. Zeng, J. Zhao, On Stochastic Subdifferential Systems Driven by Standard Brownian Motion, Miskolc M athematical Notes 17 (2016), 327-338. IF: 0.388. (IX) Z.H. Liu, B. Zeng, Existence Results for a Class of Hemivariational Inequa lities Involving the Stable (g, f , a)-Q uasim onotonicity, Topological Methods in Nonlinear Analysis 47 (1) (2016), 195-217. IF: 0.667. (X) Z.H. Liu, S.D. Zeng, B. Zeng, Well-Posedness for Mixed Quasi-Variational Hemi variational Inequalities, Topological Methods in Nonlinear Analysis 4 7 (2) (2016), 561-578. IF: 0.667.. 28.

(39) (XI) Z.H. Liu, X.M. Li, B. Zeng, O ptim al Feedback Control for Fractional N eutral Dynamical Systems, O ptimization 67 (2018), 549-564. IF: 0.943. Publications (I)-(V ) form the PhD dissertation.. 29.

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(41) (I) B. Zeng, Z.H. Liu, S. Migóorski On Convergence of Solutions to Variational-hem ivariational Inequalities.

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(43) Z. Angew. Math. Phys. (2018) 69:87 © 2018 The Author(s) https://doi.org/10.1007/s00033-018-0980-3. Z eitsch rift für a n g e w a n d te M a th e m a tik un d P h y sik Z A M P CrossMark. On convergence of solutions to variational-hemivariational inequalities Biao Zeng, Zhenhai Liu and Stanisław Migórski. In this paper we investigate the convergence behavior of the solutions to the time-dependent variationalhemivariational inequalities with respect to the data. First, we give an existence and uniqueness result for the problem, and then, deliver a continuous dependence result when all the data are subjected to perturbations. A semipermeability problem is given to illustrate our main results. Abstract.. Mathematics Subject Classification. Keywords.. 47J20, 49J40, 49J45, 74M10, 74M15.. Variational-hemivariational inequality, Mosco convergence, Semipermeability problem, Pseudomonotone.. 1. Introduction V ariational-hem ivariational inequalities represent a special class of inequalities which involve both con vex and nonconvex functions. Elliptic hem ivariational and variational-hem ivariational inequalities were introduced by Panagiotopoulos in the 1980s and studied in m any contributions, see [15, 17] and the references therein. Various classes of such inequalities have been recently investigated, for instance, in [7, 9, 10, 12, 20, 22]. They play an im portant role in describing m any mechanical problems arising in solid and fluid mechanics. In this paper we study the following tim e-dependent variational-hem ivariational inequality: find u : R+ = [0, + œ ) ^ X such th a t, for all t G R + , u(t) G K and (Au(t) - f (t), v - u ( t) ) x + <p(u(t),v) - ip(u(t), u(t)) + j 0 (u(t); v —u(t)) > 0. for all v G K ,. (1). where K is a nonempty, closed and convex subset of a reflexive Banach space X , A : X ^ X * and ^ : K x K ^ R are given m aps to be specified later, j : X ^ R is a locally Lipschitz function, and f : R+ ^ X * is fixed. The notation j 0 (u; v) stands for the generalized directional derivative of j at point u G X in the direction v G X . The goal of the paper is to study the convergence of solution of the variational-hem ivariational inequality ( 1 ) when the d a ta A, f , ^ , j and K are subjected to perturbations. The dependence of solutions to elliptic variational-hem ivariational inequalities on the d a ta has been studied only recently. For such inequalities the dependence w ith respect to functions ^ and j was investi gated in [13], where A and K were not subjected to perturbations. A result on the dependence of solutions to elliptic variational inequalities w ith respect to perturbations of the set K of a special form was studied Project supported by the National Science Center of Poland under Maestro Project No. UM0-2012/06/A/ST1/00262, and Special Funds of Guangxi Distinguished Experts Construction Engineering, Guangxi, P.R. China. It is also supported by the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under Grant No. 3792/GGPJ/H2020/2017/0. Published online: 04 June 2018 Birkhäuser.

(44) 87. P a g e 2 of 20. B. Zeng, Z. Liu and S. Migórski. ZAM P. in [19]. There, the d a ta A, p and f were independent of perturbations. For a class of elliptic historydependent variational-hem ivariational inequalities studied in [21], the convergence result was obtained in a case when p depends on a history-dependent operator, and A does not depend on perturbations. A result on the convergence w ith respect to the set of constraints K were studied for elliptic quasivariational inequalities in [1]. In ah aforementioned papers the convergence results were applied to various m athem atical models of deform able bodies in contact mechanics. Note th a t a result on th e dependence of solutions to evolution second order hem ivariational inequalities w ith respect to perturbations of the operators can be found in [8]. Furtherm ore, it is well known th a t th e continuous dependence results are of im portance in optim al control and identification problems, see, e.g., [2, 9, 23]. The aim of the paper is twofold. F irst, we consider the class of abstract tim e-dependent variationalhem ivariational inequalities of the form ( 1) for which we study the dependence of the solution with respect to the d a ta A, f , p, j and K . O ur hypotheses on p and j are different th an the one used in the aforementioned papers. Moreover, the set of constraints is of a more general form. Second, we illustrate the applicability of the convergence results in the study of a sem iperm eability problem. Sem iperm eability problems were first considered in [5] for convex potentials (which lead to m onotone relations) and, later, in [11, 16, 17] for nonconvex superpotentials (leading to nonm onotone relations). They concern the treatm en t of semiperm eable m em branes either in the interior or on the boundary of the body and arise, for instance, in flow problems through porous m edia and heat conduction problems. In the current paper we study a sem iperm eability problem involving sim ultaneously both m onotone and nonm onotone relations. Its weak form ulation is a variational-hem ivariational inequality. Note th a t the convergence results for sem iperm eability problems are provided here for the first tim e. Finally, we underline th a t our convergence results of Sect. 3 are also applicable to various problems in contact mechanics like a nonlinear elastic contact problem w ith norm al compliance condition with unilateral constraint, and a contact problem w ith the Coulomb friction law in which the friction bound is supposed to depend on the norm al displacem ent, studied in, e.g., [1, 6, 13, 19]. The rest of this paper is organized as follows. In Sect. 2, we will introduce some necessary prelim inary m aterials. Section 3 is devoted to th e proofs of convergence results for the elliptic variationalhem ivariational inequality and its tim e-dependent counterpart. In Sect. 4, we apply the results to a sem iperm eability problem.. 2. Preliminaries In this section we recall notation, basic definitions and a result on unique solvability of a variationalhem ivariational inequality. Let (X, II ■||x ) be a Banach space. We denote by X * its dual space and by (■, -)x the duality pairing between X * and X . The strong and weak convergences in X are denoted by “ and “ respectively. Let C (R+; X ) be th e space of continuous functions defined on interval R + = [0, + œ ) w ith values in X . For a subset K c X the symbol C (R+; K ) denotes the set of continuous functions on R + w ith values in K . We also recall th a t the convergence of a sequence {xn }n> 1 to the element x, in the space C (R +; X ), can be described as follows .x n ^ x in C (R +; X ), as n if and only if. {. m ax ||xn (t) —x ( t) ||x _ >0,. as n. oo,. for all. k G N.. (2). t€[0,fc]. We recall th e definitions of th e convex subdifferential, the (Clarke) generalized gradient and the pseudom onotone single-valued operators, see [3,4]. Definition 1. A function f : X ^ R is said to be lower semicontinuous (l.s.c.) at u, if for any sequence {un }n> 1 C X w ith un ^ u, we have f (u) < lim inf f (un ). A function f is said to be l.s.c. on X , if f is l.s.c. a t every u G X ..

(45) ZAM P. O n c o n v e rg e n c e o f s o lu tio n s t o v a r ia tio n a l- h e m iv a r ia tio n a l in e q u a litie s P a g e 3 o f 20. 87. Definition 2. Let ^ : X ^ RU | + ^ } be a proper, convex and l.s.c. function. The m apping 8(pc : X ^ 2x defined by d(pc(u) = { u* G X * | (u*,v — u )X < ^(v) —^ (u ). for all. v GX }. for u G X , is called the subdifferential of <p. An element u* G 3c^ (u ) is called a subgradient of <p in u. D efinition 3. Given a locally Lipschitz function ^ : X ^ R, we denote by ^ °(u ; v) the (Clarke) generalized directional derivative of <p at the point u G X in the direction v G X defined by ^ or (u; v)\ = °'u;. vlim sup + *v) sup ^(C ^ —^ (C) . . — ----A—°+ ,z ,Z—u —— u a—°+ **. The generalized gradient of <p at u G GX X , denoted by d<p(u), is a subset < of X * given by 3<p(u) = { u* G X * | ^ °(u ; v) > (u*, v )x. for all. v G X }.. Furtherm ore, a locally Lipschitz function ^ : X ^ R is said to be regular (in the sense of Clarke) a t u G X , if for all v G X the directional derivative ^ ( u ; v) exists, and for all v G X , we have ^ ( u ; v) = ^ °(u ; v). The function is regular (in the sense of Clarke) on X if it is regular at every point in X . Definition 4. A single-valued operator F : X ^ X * is said to be pseudomonotone, if it is bounded (sends bounded sets into bounded sets) and satisfies the inequality ( F u ,u —v ) < l i m i n f (F u n , un —v )x. for all. v G X,. where u n ^ u in X w ith lim s u p (F u n , u n —u )x < 0. The following result provides a useful characterization of a pseudom onotone operator. Lemma 5. (see [12, Proposition 1.3.66]) Let X be a reflexive Banach space and F : X ^ X * be a single valued operator. The operator F is pseudomonotone if and only i f F is bounded and satisfies the following condition: if un ^ u in X and lim s u p (F u n , un —u )x < 0, then F u n ^ F u in X * and lim (F u n , un —u )x = 0. The following notion of the Mosco convergence of sets will be useful in the next sections. For the definitions, properties and other modes of set convergence, we refer to [4 , C hapter 4.7] and [14]. D efinition 6 . Let (X, || ■||) be a norm ed space and {K p}p>° c 2x \{0}. We say th a t K p converge to K in the Mosco sense, p ^ 0, denoted by K p —^. K if and only if the two conditions hold. (m1) for each x G K , there exists {xp}p>° such th a t x p G K p and x p ^ x in X , (m2) for each subsequence {xp}p>° such th a t x p G K p and x p ^ x in X , we have x GK . For the following properties of the Mosco convergence, we refer to [14, p. 520]. R em ark 7. Let K p —^ K . Then, K = 0 implies K p = 0 and the opposite is not true. Also, if K p is a closed and convex set for all p > 0, then K is also closed and convex. Finally, we recall a result on existence and uniqueness of solution to the following variationalhem ivariational inequality. Problem. 8.. F ind u G K such that (Au —f , v — u )x + ^ (u , v) — ^ (u , u) + j° (u ; v — u) > 0. for all. v G K,. (3). Problem 8 was studied in [13] where results on its unique solvability, continuous dependence on the d a ta and a penalty m ethod were provided. We need the following hypotheses on the d a ta of Problem 8 ..