A Nash type solution for hemivariational inequality systems
aa r X i v : . [ m a t h . A P ] F e b A Nash type solution for hemivariational inequality systems Systems
Duˇsan Repovˇs and Csaba Varga c Faculty of Mathematics and Physics, and Faculty of Education, University of Ljubljana,P. O. B. 2964, Ljubljana, Slovenia 1001Babe¸s-Bolyai University, Faculty of Mathematics and Computer Sciences, RO-400084 Cluj-Napoca,Romania
Email address : [email protected] and [email protected]
Abstract
In this paper we prove an existence result for a general class of hemivariational inequalitiessystems using the Ky Fan version of the KKM theorem (1984) or the Tarafdar fixed pointtheorem (1987). As application we give an infinite dimensional version for existence result ofNash generalized derivative points introduced recently by Krist´aly (2010) and also we give anapplication to a general hemivariational inequalities system.
Key words:
Fixed point theory, nonsmooth functions, hemivariational inequality systems.
In the last years many papers have been dedicated to the of study the existence and multiplic-ity of solutions for hemivariational inequality systems or differential inclusion systems defined onbounded or unbounded domain, see [2],[3], [11], [12], [13], [16]. In these papers the authors usecritical points theory for locally Lipschitz functions, combined with the
Principle of SymmetricCriticality and different topological methods. For a comprehensive treatment of hemivariationalinequality and hemivariational inequalities systems on bounded domains using critical point the-ory for nonsmooth functionals we refer to the monographs of D. Motreanu and V. R˘adulescu [17]and D. Motreanu and P. D. Panagiotopoulos [18]. For very recent results concerning variationalinequalities and elliptic systems using critical points theory and different variational methods seealso the book of Krist´aly, R˘adulescu and Varga [15].The aim of this paper is to prove that the existence of at least one solution for a general classof hemivariational inequalities systems on a closed and convex set (either bounded or unbounded)without using critical point theory. We apply a version of the well-known theorem of Knaster–Kuratowski–Mazurkiweicz due to Ky Fan [7] or the Tarafdar fixed point theorem [23]. We startthe paper by giving in Section 2 the assumptions and formulating the hemivariational inequalitiessystem problem which we study. The main results concerning the existence of at least one solutionfor the hemivariational inequalities systems which we study are given in Section 3. Section 4contains applications to Nash and Nash general derivative points and existence results for someabstract class of hemivariational inequalities systems.1
Assumptions and Formulation of the Problem
Let X , X , . . . , X n be reflexive Banach spaces and Y , Y , . . . , Y n , Z , . . . , Z n Banach spaces suchthat there exist linear operators T i : X i → Y i , T i : X i → Z i for i ∈ { , . . . , n } . We suppose thatthe following condition hold: (TS) T i : X i → Y i and S i : X i → Z i are compact for i = 1 , n .We denote by X ∗ i the topological dual of X i and h· , ·i i denotes the duality pairing between X ∗ i , whereas X i for i = 1 , n . Also, let K i ⊂ X i be closed, convex sets for i = 1 , n and weconsider A i : Y × · · · × Y i × · · · × Y n → R the continuous functions which are locally Lipschitzin the i th variable and we denote by A ◦ i ( u , . . . , u i , . . . , u n ; v i ) the partial Clarke derivative in thedirectional derivative in the i th variable, i.e. the Clarke derivative of the locally Lipschitz function A i ( u , . . . , u i , . . . , u n ) at the point u i ∈ Y i in the direction v i ∈ Y i , that is A ◦ i ( u , . . . , u i , . . . u n ; v i ) = lim sup w → u i τ ց A i ( u , . . . , w + τ v i , . . . u n ) − A i ( u , . . . , w, . . . u n ) τ We suppose that for every i = 1 , n the following condition holds: (A) the functions A ◦ i : Y × · · · × Y n × Y i → R are upper semi-continuous.We also consider the following nonlinear operators F i : K × · · · × K i × · · · K n → X ⋆i , i = 1 , n .We suppose that the operators F i satisfy the following condition: (F) the functions ( u , . . . , u n )
7→ h F i ( u , . . . , u n ) , v i i i are weakly upper semi-continuous for every v i ∈ X i and i = 1 , n . Definition 2.1 [see [5]]
Let Z be a Banach space and j : Z → R a locally Lipschitz function. Wesay that j is regular at u ∈ Z if for all v ∈ Z the one-sided directional derivative j ′ ( u ; v ) exists and j ′ ( u ; v ) = j ◦ ( u ; v ) . If j is regular at every point u ∈ Z we say that j is regular. We have the following elementary result.
Proposition 2.1
Let J : Z × · · · × Z n → R be a regular, locally Lipschitz function. Then thefollowing assertions hold:a) ∂J ( u , . . . , u n ) ⊆ ∂ J ( u , . . . , u n ) × · · · × ∂ n J ( u , . . . , u n ) (see [5], Proposition 2.3.15) , where ∂ i J, i = 1 , n denotes the Clarke subdifferential in the i th variable;b) J ◦ ( u , . . . , u n ; v , . . . , v n ) ≤ n X i =1 J ◦ i ( u , . . . , u n ; v i ) , where J ◦ i denotes the Clarke derivative inthe i th variable; andc) J ◦ ( u , . . . , u n ; 0 , . . . , v i , . . . , ≤ J ◦ i ( u , . . . , u n ; v i ) . We introduce the following notations: • K = K × · · · × K n , • u = ( u , . . . , u n ) • T u = ( T u , . . . , T n u n ) • Su = ( S u , . . . , S n u n ) • A ( T u, T v − T u ) = n X i =1 A ◦ i ( T u, T i v i − T i u i ) • F ( u, v − u ) = n X i =1 h F i u, v i − u i i i . 2n this paper we study the following problem: Find u = ( u , . . . , u n ) ∈ K × . . . × K n such that for all v = ( v , . . . , v n ) ∈ K × . . . × K n and i ∈ { , . . . , n } we have: ( QHS ) A ◦ i ( T u ; T v i − T u i ) + h F i ( u ) , v i − u i i i + J ◦ i ( Su ; S i v i − S i u i ) ≥ . In this case we say that u = ( u , . . . , u n ) is a Nash equilibrum point for the system (QHS) .To prove our main result we use the FKKM theorem due to Ky Fan [7] and the Tarafdar fixedpoint theorem [23]. Definition 2.2
Suppose that X is a vector space and E ⊂ X . A set-valued mapping G : E → X is called a KKM mapping, if for any x , . . . , x n ∈ E the following holds conv { x , . . . , x n } ⊂ n [ i =1 G ( x i ) . The following version of the KKM theorem is due to Ky Fan [7].
Theorem 2.1
Suppose that X is a locally convex Hausdorff space, E ⊂ X and that G : E → X isa closed-valued KKM map. If there exists x ∈ E such that G ( x ) is compact, then \ x ∈ E G ( x ) = ∅ . Theorem 2.2
Let K be a nonempty, convex subset of a Hausdorff topological vector space X. Let G : K ֒ → K be a setvalued map such thati) for each u ∈ K , G ( u ) is a nonemty convex subset of K ;ii) for each v ∈ K , G − ( v ) = { u ∈ K : v ∈ G ( u ) } contains an open set O v which may beempty;iii) ∪ v ∈ K O v = K ; andiv) there exists a nonemty set K contained in a compact convex subset K of K such that D = ∩ v ∈ K O cv is either empty or compact (where O cv is the complement of O v in K ).Then there exists a point u ∈ K such that u ∈ G ( u ) . E. Tarafdar in [23] proved the equivalence of Theorems 2.1 and 2.2.
Theorem 3.1
Let K i ⊂ X i , i = 1 , n be nonempty, bounded, closed and convex sets. Let A i : Y × · · · × Y i × · · · × Y n → R be a locally Lipschitz function in the i th variable for all i ∈ { , . . . , n } satisfying condition (A) . We suppose that the operators T i : X i → Y i , S i : X i → Z i and F i : K × . . . × K n → X ⋆i ( i = 1 , n ) satisfy the condition (TS) respectively (F) . Final we consider theregular locally Lipschitz function J : Z × · · ·× Z n → R . Under these conditions the problem (QHS)admits at least one solution. Before proving Theorem 3.1, we make two remarks.
Remark 3.1
We observe that for every v ∈ K the function u A ( T u, T v − T u ) + F ( u, v − u ) + J ◦ ( Su ; Sv − Su ) is weakly upper semi-continuous. Indeed, from the condition (A) and from the fact that the opera-tors T i are compact follows that A ( T u, T v − T u ) is weakly upper semi-continuous. From (F) it fol-lows that F ( u, v − u ) is weakly upper semi-continuous. The third term, i.e. J ◦ ( Su ; Sv − Su ) is weaklyupper semi-continuous, because J ◦ ( · ; · ) is upper semi-continuous and the operators S i : X i → Z i are compact. emark 3.2 If there exists u ∈ K , such that for every v ∈ K we have: (3.1) A ( T u, T v − T u ) + F ( u, v − u ) + J ◦ ( Su ; Sv − Su ) ≥ , then u ∈ K is a solution of the problem (QHS). Indeed, if we fix an i = { , . . . , n } and put v j := u j , j = i in the above inequality and using iii) Proposition 2.1 we get that ( QHS ) , A ◦ i ( T u ; T v i − T u i ) + h F i ( u i ) , v i − u i i i + J ◦ i ( Su ; S i v i − S i v i ) ≥ . for all i ∈ { , . . . , n } . In the sequel we give two proofs, using Theorems 2.1 and 2.2.
First Proof:
Let G : K ֒ → K be the set-valued map defined by G ( v ) = { u ∈ K : A ( T u, T v − T u ) + F ( u, v − u ) + J ◦ ( Su ; Sv − Su ) ≥ } . For every v ∈ K , we have G ( v ) = ∅ because v ∈ G ( v ) and taking into account that the function u A ( T u, T v − T u ) + F ( u, v − u ) + J ◦ ( Su ; Sv − Su )is weakly upper semi-continuous, it follows that the set G ( v ) is weakly closed. Now we prove that G is a KKM mapping. We argue by contradiction, let v , . . . , v k ∈ K and w ∈ conv { v , . . . , v k } such that w / ∈ ∪ ki =1 G ( v i ). From this it follows that(3.2) A ( T w, T v i − T w ) + F ( w, v i − w ) + J ◦ ( Sw ; Sv i − Sw ) < , for all i = { , . . . , k } . Because of w ∈ conv { v , . . . , v k } the existence of λ , . . . , λ k ∈ [0 ,
1] with k X i =1 λ i = 1 such that w = k X i =1 λ i v i follows . If we multiply the inequalities (3.2) with λ i and addingfor i = { , . . . , k } we obtain(3.3) A ( T w, T w − T w ) + F ( w, w − w ) + J ◦ ( Sw ; Sw − Sw ) < A ( · , · ) , F ( · , · ) and J ◦ ( · , · ) are positive homogeneous and convex in the secondvariable. From inequality (3.3) it follows that 0 = A ( T w, T w − T w ) + F ( w, w − w ) + J ◦ ( Sw ; Sw − Sw ) <
0, which is a contradiction. Because the set K is bounded, convex and closed, it followsthat it is weakly closed and by the Eberlein-Smulian theorem we have is weakly compact. Because G ( v ) ⊂ K is weakly closed, we have that G ( v ) is weakly compact and from Theorem 2.1 it followsthat ∩ v ∈ K G ( v ) = ∅ , therefore from Remark 3.2 it follows that the problem (QHS) has a solution. Second Proof.
Using Remark 3.2 we prove the existence of an element u ∈ K such that for every v ∈ K we have A ( T u, T v − T u ) + F ( u, v − u ) + J ◦ ( Su ; Sv − Su ) ≥ . In this case u ∈ K will be the solution of systems (QHS).We argue by contradiction. Let us assume that for each u ∈ K , there exists v ∈ K such that(3.4) A ( T u, T v − T u ) + F ( u, v − u ) + J ◦ ( Su ; Sv − Su ) < . Now, we define the set-valued mapping G : K ֒ → K by(3.5) G ( u ) = { v ∈ K : A ( T u, T v − T u ) + F ( u, v − u ) + J ◦ ( Su ; Sv − Su ) < } . G ( u ) = ∅ for every u ∈ K . Because the function A ( · , · ) + F ( · , · ) + J ◦ ( · ; · ) is convex in the seconde variable, we get that G ( u ) is a convex set. Now, we prove that forevery v ∈ K , the set G − ( v ) = { u ∈ K : v ∈ G ( u ) } is weakly open. Indeed, from weakly uppersemicontinuity of the function u A ( T u, T v − T u ) + F ( u, v − u ) + J ◦ ( Su ; Sv − Su )it follows that[ G − ( v )] c = { u ∈ K : A ( T u, T v − T u ) + F ( u, v − u ) + J ◦ ( Su ; Sv − Su ) ≥ } is weakly closed, therefore G − ( v ) is weakly open.Now we verify iii) from Theorem 2.2, i.e. [ v ∈ K G − ( v ) = K . Because for every v ∈ K we have G − ( v ) ⊂ K , it follows that [ v ∈ K G − ( v ) ⊂ K . Conversely, let u ∈ K be fixed. Since G ( u ) = ∅ there exists v ∈ K such that v ∈ G ( u ). In the next step we verify iv) Theorem 2.2. We assertthat D = \ v ∈ K [ G − ( v )] c is empty or weakly compact. Indeed, if D = ∅ , then D is a weakly closedset of K since it is the intersection of weakly closed sets. But K is weakly compact hence we getthat D is weakly compact. Taking O v = G − ( v ) and K = K = K we can apply Theorem 2.2 toconclude that there exists u ∈ K such that u ∈ G ( u ). This give0 = A ( T u , T u − T u ) + F ( u , u − u ) + J ( Su ; Su − Su ) < , which is a contradiction.Therefore the system (QHS) has a solution. Remark 3.3
If in Theorem 3.1 the sets K i , i = 1 , n are only convex and closed but not boundedwe impose the following coercivity condition. (CC) there exist K i ⊂ K i compact sets and v i ∈ K i such that for all v = ( v , . . . , v n ) ∈ K × . . . × K n \ K × . . . × K n we have A ( T v, T v − T v ) + F ( v, v − v ) + n X i =1 J i ( Sv, S i v i − S i v i ) < , where v = ( v , . . . , v n ) . In this case the problem (QHS) has a solution. In this section we are concerned with two applications. In the first application we study the relationbetween Nash equlibrum and Nash generalized derivative equilibrum points for a hemivariationalinequalities system and in the second application we give an existence result for an abstract classof hemivariational inequalities systems.Let X , . . . , X n be Banach spaces and K i ⊂ X i and the functions f i : K ×· · ·× K i ×· · ·× K n → R for i ∈ { , . . . , n } . The following notion was introduced by J. Nash [19], [20]: Definition 4.1
An element ( u , . . . , u n ) ∈ K × · · · × K n is Nash equlibrum point of functions f , . . . , f n if for each i ∈ { , . . . , n } and ( u , . . . , u n ) ∈ K × · · · K n we have f i ( u , . . . , u i , . . . , u n ) ≥ f i ( u , . . . , u i , . . . , u n ) . Now let D i ⊂ X i be open sets such that K i ⊂ D i for all i ∈ { , . . . , n } . We consider thefunction f i : K × · · · × D i × · · · K n → R which are continuous and locally Lipschitz in the i th variable. The next notion was introduced recently by A. Krist´aly [13] and is a little bit differentform for functions defined on Riemannian manifolds.5 efinition 4.2 If ( u , . . . , u n ) ∈ K × · · · × K n is an element such that f i ( u , . . . , u n ; u i − u i ) ≥ , for every i = { , . . . , n } and ( u , . . . , u n ) ∈ K × · · · × K n we say that ( u , . . . , u n ) is a Nashgeneralized derivative points for the functions f , . . . , f n . Remark 4.1
If the functions f i , i ∈ { , . . . , n } are differentiable in the i th variable, then the abovenotion coincides with the Nash stationary point introduced in [9] . Remark 4.2
Is is easy to observe that any Nash equlibrum point is a Nash generalized derivativepoint.
The following result is an existence result for Nash generalized derivative points and is aninfinite-dimensional version of a result from the paper [13]. Therefore, if in Theorem 3.1 we choose F i = 0 , i ∈ { , . . . , n } and J = 0 we obtain the following result. Theorem 4.1 (i) Let Y , Y , . . . , Y n and X , X , . . . , X n , be a reflexive Banach spaces and T i : X i → Y i compact, linear operators. We consider the closed, convex, bounded sets K i ⊂ X i and thefunctions A i : Y × · · · × Y n → R , i = 1 , . . . , n which are locally Lipschitz in the i th variable andsatisfies the condition (A) . In these conditions, there exists ( u , . . . , u i , . . . , u n ) ∈ K × · · · × K i ×· · · × K n such that for all i ∈ { , . . . , n } and ( u , . . . , u i , . . . , u n ) ∈ K × · · · × K i × · · · × K n wehave A i (( T u , . . . , T i u i , . . . , T n u n ); T i u i − T i u i ) ≥ , i.e. ( u , . . . , u i , . . . , u n ) is a Nash generalized derivative points for the function A i , i ∈ { , . . . , n } .(ii) If the sets K i , i = { , . . . , n } are only closed and convex we suppose that there exists thebounded, closed sets K i ⊂ K i and v i ∈ K i , i = { , . . . , n } such that for every ( u , . . . , u n ) ∈ K × · · · × K n \ K × · · · × K n we have A ( T u, T v − T u ) < . Then there exist u = ( u , . . . , u i , . . . , u n ) ∈ K × · · ·× K i × · · ·× K n such that for all i ∈ { , . . . , n } and u = ( u , . . . , u i , . . . , u n ) ∈ K × · · · × K i × · · · × K n we have A i ( T u ; T i u i − T i u i ) ≥ , i.e. u = ( u , . . . , u i , . . . , u n ) is a Nash generalized derivative points for the functions A i , i ∈{ , . . . , n } . In the next step we give an existence result for a general system of hemivariational inequalities.In this case in Theorem 3.1 we choose Y i = Z i , i ∈ { , . . . , n } and we suppose that the functions A i : Y × · · · × Y i × · · · × Y n → R are differentiable in the i th variable for i ∈ { , . . . , n } . In thiscase we suppose that the functions A ′ i : Y × · · · × Y i × · · · × Y n × Y i → R are continuous for i ∈ { , . . . , n } . Let also J : Y × · · · × Y i × · · · × Y n → R a locally Lipschitz regular function.Under these conditions we have the following result. Corollary 4.1
Let
J, A i : Y × · · · × Y i × · · · × Y n → R be the function as above and supposethat the condition (TS) holds and let K i ⊂ X i , i = { , . . . , n } be bounded, closed and convex sets.Under these conditions there exist an element u = ( u , . . . , u n ) ∈ K × · · · × K n such that forevery u = ( u , . . . , u n ) ∈ K × · · · × K n and i ∈ { , . . . , n } we have: A ′ i ( T u ; T i u i − T i u i ) + J i ( T u ; T i u i − T i u i ) ≥ .
6f in Theorem 3.1 we take A i = 0 then we obtain the following existence result for a generalclass of hemivariational inequalities systems. Corollary 4.2
Let K i ⊂ X i bounded, closed and convex subsets of the reflexive Banach spaces X i for i ∈ { , . . . , n } . We suppose that F i : K × · · · × K n → X ⋆i satisfies the condition (F) and J : Z × · · · × Z n → R is a regular locally Lipschitz function and the condition (TS) holds.Then there exists u = ( u , . . . , u i , . . . , u n ) ∈ K × · · · × K i × · · · K n such that for every u =( u , . . . , u i , . . . , u n ) ∈ K × · · · × K i × · · · K n and i ∈ { , . . . , n } we have h F i ( u ); u i − u i i i + J i ( Su ; S i u i − S i u i ) ≥ . The above result generalize the main result from the paper of A. Krist´aly [10].Indeed, let Ω ⊂ R N be a bounded, open subset. Let j : Ω × R k × · · · R k | {z } n → R a Carath´eodoryfunction such that j ( x, · , . . . , · ) is locally Lipschitz for every x ∈ Ω and satisfies the followingassumptions for all i ∈ { , . . . , n } :( j i ) there exists h i ∈ L pp − (Ω , R + ) and h i ∈ L ∞ (Ω , R + ) such that | z i | ≤ h i ( x ) + h i ( x ) | y | p − R kn for almost x ∈ Ω and every y = ( y , . . . , y n ) ∈ R k × · · · × R k | {z } n and z i ∈ ∂ i j ( x, y , . . . , y n ).In this case let S = ( S , . . . , S n ) : X × · · · × X n → L p (Ω , R k ) × · · · L p (Ω , R k ) and J ◦ S : K × · · · × K n → R is defined by J ( Su ) = Z Ω j ( x, S u ( x ) , . . . S n u n ( x )) dx. Using a result from Clarke [5] we have:(I) J i ( Su ; S i v i ) ≤ Z Ω j i ( x, S u ( x ) , . . . S n u x ; S i v i ( x )) dx, for every i ∈ { , . . . , n } and v i ∈ X i .Therefore we have the following existence result obtained by Krist´aly [10]. Corollary 4.3
Let K i ⊂ X i bounded, closed and convex subsets of the reflexive Banach spaces X i for i ∈ { , . . . , n } . We suppose that F i : K × · · · × K n → X ⋆i satisfies the condition (F) and j : Ω × R k × · · · R k | {z } n → R a Carath´eodory function such that j ( x, · , . . . , · ) is a regular, locallyLipschitz function satisfying condition ( j i ) and the condition (TS) holds. Then there exists u =( u , . . . , u i , . . . , u n ) ∈ K × · · · × K i × · · · K n such that for every u = ( u , . . . , u i , . . . , u n ) ∈ K ×· · · × K i × · · · K n and i ∈ { , . . . , n } we have h F i ( u ); u i − u i i i + Z Ω j i ( x, S u ( x ) , . . . , S n u n ( x ); S i u i ( x ) − S i u i ( x )) dx ≥ . Remark 4.3 If n = 1 we obtain a similar result from the paper of Panagiotopoulos, Fundo andR˘adulescu [8] . Remark 4.4
If the Banach spaces X i , i ∈ { , . . . , n } are separable and the domain Ω ⊂ R N isunbounded then a similar inequality to (I) was proved in the paper D´alyai and Varga [6]. Therefore,we can state a similar result as Corollary 4.3 in the case when Ω ⊂ R N is an unbounded domain. Acknowledgement:
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