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Applied Mathematics, 2010, 1, 81-86 doi:10.4236/am.2010.11010 Published Online May 2010 (http://www.SciRP.o rg/journal/am) Copyright © 2010 SciRes. AM Nonzero Solutions of Generalized Variational Inequalities* Jue Li, Yisheng Lai Department of Information and Computer Science, Zhejiang Gongshang University, Hangzhou, China E-mail: juelee@gmail.com Received March 17, 2010; revised April 22, 2010; accepted April 29, 2010 Abstract The existence of nonzero solutions for a class of generalized variational inequalities is studied by ﬁxed point index approach for multivalued mappings in ﬁnite dimensional spaces and reﬂexive Banach spaces. Some new existence theorems of nonzero solutions for this class of generalized variational inequalities are estab- lished. Keywords: Variational Inequality, Fixed Point Index of Multivalued Mappings, Nonzero Solution 1. Introduction Variational inequality theory with applications are an important part of nonlinear analysis and have been ap- plied intensively to mechanics, diﬀerential equation, cybernetics, quantitative economics, optimization theory and nonlinear programming etc. (see [1-4]). Variational inequalities, generalized variational in- equalities and generalized quasivariational inequalities were studied intensively in the last 30 years with topo- logical method, variational method, semi-ordering me- thod, ﬁxed point method, minimax theorem of Ky Fan and KKM technique ([1-4]). In 1998, motivated by the paper [5], Zhu [6] studied a system of variational inequa- lities involving the linear operators in reﬂexive Banach spaces by using the coincidence degree theory due to Mawhin [7]. Some existence results of positive solutions for this system of variational inequalities in reﬂexive Banach spaces were proved. Let X be a real Banach space, * X its dual and (·, ·) the pair between * X and X . Suppose that K is a nonempty closed convex subset of X . Find uK∈ , 0u= /, and () w gu∈ such that (,)(,),Auvuwvuv K− ≥−∀∈ (1) where mapping * :AK X→ is nonlinear and * :2 X gK→ is a multi-valued mapping. The existence of nonzero solutions for variational in- equalities is an important topic of variational inequality theory. [8] discussed the variational inequality (1) when A is coercive or monotone and g is set-contractive or upper semi-continuous. [9] considered the variational inequa l i t y (1) when A is single-valued continuous and g is set-contractive. On the other hand, recently, under some different con- ditions, [10,11] obtained some existence theorems of nonzero solutions for a class of generalized variational inequalities by ﬁxed point index approach for mul- ti-valued mappings in reﬂexive Banach space. Based on the importance of studying the existence of nonzero solutions for variational inequalities, and motivated and inspired by recent research works in this field, in this paper, we discuss the existence of nonzero solutions for a class of generalized variational inequalities as follows: Find ,0u Ku∈≠ such that ( ,)()() ((), )(,), Au vuj vju guvufvuv K −+ − ≥− +−∀∈ (2) where * ,:Ag KX→ are two nonlinear mapping and * fX∈ . A mapping * :AX X→ is called hemicontinuous at 0 xX ∈ if for each yX∈ , 00 () n Aty Awxx ∗ + when 0 n t→+ . A multivalued mapping :T * () 2 X DT X⊂→ is said to be locally bounded in v if there exists a neigh- bourhood V of x for each xX∈ such that the set (( ))TV DT∩ is bounded in * X . Suppose that K is a closed convex subset of X with 0K∈ . For such K , the recession cone rcK of K is defined by rcK = {:, } wXv wKvK∈+ ∈∀∈. It is easily seen that the recession cone is indeed a cone and we have that rcK = ∅ /. For a proper lower semicontinuous convex functional : {}jX R→ ∪∞ with (0) 0j= and ()jK⊂ [0, )R + = +∞ , in the virtue of [12], the limit 1 lim( ) tj tw t →+∞ = * This work was supported by The Zhejiang Provincial Natural Science Foundation(No.Y7080068) and the Foundation of Depart- ment of Education of Zhejiang Province(No.20070628) J. LI ET AL. Copyright © 2010 SciRes. AM 82 ()jw ∞ exists in {}R∪∞ for every wX∈ and j ∞ is also a lower semicontinuous convex functional with (0) 0j∞= and with the property that ()ju v+≤ ()( ),,juj vuvX ∞ + ∀∈ . Suppose that K is a closed convex subset of X and U is an open subset of X with K U UK=∩= / ∅ . The closure and boundary of K U relative to K are denoted by K U and () K U∂ respectively. Assume that :2 K K TU → is an upper semicontinuous mapping with nonempty compact convex values and T is also condensing, i.e., (()) ()TS S αα < where α is the Kuratowski measure of noncompactness on X . If ()x Tx∉ for () K xU∈∂ , then the fixed point index, (, ) K i TU , is well defined (see[13]). Proposition 1[13] Let K be a nonempty closed convex subset of real Banach space X and U be an open subset of X . Suppose that :2 K K TU → is an upper semicontinuous mapping with nonempty compact convex values and () x Tx∉ for () K xU ∈∂ . Then the index, (, ) K i TU , has the following properties: 1) If (, )0 K iTU = / , then T has a fixed point; 2) For mapping 0 X with constant value 0 {}x , if 0K xU∈ , then 0 ( ,)1 K i XU= ; 3) Let 12 , UU be two open subsets of X with 12 UU∩=∅ . If ()x Tx∉ when 12 (())(()) KK xU U∈∂ ∪∂ , then 1212 (,)(, )(,) K KK i TUUi TUi TU∪= + ; 4) Let : [0,1]2 K K HU×→ be an upper semicontinu- ous mapping with nonempty compact convex values and (([0,1]))( )H QQ αα ×< whenever ( )0,K Q QU α = ⊂ / . If (, )xHtx∉ for every [0,1],( ) K t xU∈ ∈∂ , then ( ) ( 1,,)((0,),) KK iHU iHU⋅= ⋅ . For every * qX∈ , let () Uq be the set of solutions in K of the following variational inequality ( ,)()() (,)(,), Au vuj vju qvufvuvK −+ − ≥− +−∀∈ (3) Define a mapping * :2 K A KX→ by * (): (),. A K qUqqX= ∈ Obvio usl y, () A Kq= ∅ if and only if the variational inequa l i t y (3) has no solution in K . 2. Nonzero Solutions in n R Lemma 1 Let X be a separable reflexive Banach space. Suppose that * :AX X→ is a bounded mono- tone hemicontinuous mapping (i.e., for every bounded subset D of X , ()AD is bounded) and :(, ]jK→−∞ +∞ is a proper convex lower semic o n- tinuous functional. Assume that there exists 0 vK∈ sa- tisfying 00 , inf [(,lim )()()]0 uuKAu uvj uj v →+∞ ∈−+ −> ‖‖ (4) Then for any given * fX∈ there exists uX∈ such that ( ,)()() ( ,),. Au vuj vj u fv uvX −+ − ≥− ∀∈ (5) Proof. Without loss of generality, assume that 0f= , otherwise, set ()()( ,)jvjvf v= − . Let {: r K xX= ∈ }xr≤‖ ‖. Because X is a separable reflexive Banach space, for given r , there exists a closed convex sets sequences ,1, 2,, m Km=… satisfying the following con- ditions: )a 1 ,1, 2,; r mm K KKm + ⊂⊂ =… )b , mm KX⊂ m X is m -dimensional subspace of X ; )c 1m m K ∞ = is dense in r K . First, we shall verify that for each m , there exists mm uK ∈ such that (,)()()0,. mm m m Auvujvj uvK−+−≥ ∀∈ (6) Because m X is a finite dimensional subspace (deno- ted its inner product by [.,.] ), there exists a linear conti- nuous mapping * : m XX π → such that (, )g ω = [ ,]g πω for all m K ω ∈. Thus inequality (6) can be written [() ,)] ( )(),. m Auuuvu jvjuv K π −+− − ≤ −∀∈ (7) Define a function ( ):(,] mm JvX→−∞ +∞ by ( ), () , \. m m mm jvv K Jv vX K ∈ =+∞ ∈ Then inequality (7) can be written [() ,)] ( )(), mm m Auuuvu Jv JuvK π −+− − ≤ −∀∈ (8) which is equivalent to the equality () m J uPAu u π =−+ (9) by [2,3], where m J P is an approximate mapping of m J. J. LI ET AL. Copyright © 2010 SciRes. AM 83 Obviously, ( ): m J mm PAI KK π −+ → is continuous. According to Brouwer's fixed point theorem (see [2,3]), there exists mm uK∈ satisfying the equality (9), that is, m u is a solution of the variational inequality (6). Second, we shall verify that for each r , there exists r r uK∈ such that (,)( )()0,. r rr r Auvujvj uvK−+−≥ ∀∈ (10) In fact, r m KK⊂ and A is a bounded mapping, whi ch implies that there constant C such that m Au C≤‖‖ for 1, 2,m=… . Since X is a reflexive and r K is weakly closed, there exists a subsequence {}{} m uu µ ⊂ such that r w uu µ → and r r uK∈ . Because 1m m K ∞ = is dense in r K , for any given 0 ε > , there exists 01m m uK ∞ = ∈ such that 0 . r uu ε −≤‖‖ It then follows from (6) that 00 (,)()().Au uujuju µµ µ −≤ − (11) when µ is sufficiently large. Thus we have 00 0 sup( , sup( ,)s l up( ,) s im ) lim lim up( ()())lim . r r Au uu Au uuAu uu ju juC µµ µ µµ µ µµ µ µ ε − ≤−+ − ≤− +⋅ Since j is a lower semicontinuous function and ε is an arbitrary positive number, we have sup(,) 0.lim r Au uu µµ µ −≤ (12) This together with A being a monotone hemiconti- nuous mapping impl ies that inf (,) ( ,),. lim r rr Au uv Au uvvK µµ µ − ≥− ∀∈ (13) If 1m m vK ∞ = ∈ , it then follows from (6) that (,)()()Auuvj vj u µµ µ −≤ − (14) when µ is sufficiently large. It thus follows from (13) that 1 lim lim (,)inf (,) inf (()()) ( )(),. rr rm m Au uvAuuv jv ju jv juvK µµ µ µ µ ∞ = −≤ − ≤− ≤ −∀∈ (15) Because 1m m K ∞ = is dense in r K , the above in- equality holds for all r vK∈ . therefore r u is a solution of the variational inequality (10). New we shall verify that the variational inequality (5) has a solution. Taking 0 vv= in (10), we have 00 (,)()() 0 rr r Auuvj uj v−+−≤ (16) and so it then follows from condition (4) that there ex- ists constant C > 0 such that r uC≤‖‖ . Taking r > C then r ur < ‖ ‖ and so r u is an inner point of r B . Thus for any given X ω ∈ , we have (1 ) r tu−+ r tB ω ∈by taking (0,1)t∈ small enough. Let (1) r vtu t ω =−+ in (10), then we obtain (,)(()())0 rr r tAuutjj u ωω −+− ≥ by j being a convex lower semicontinuous function. Thus (,)( )()0,. rr r AuujjuX ωω ω −+−≥∀∈ Therefore r u is a solution of the variational inequalit y (5). Theorem 1 Let K be a nonempty unbounded closed convex set in n XR= with 0K∈ . Suppose that * XX → is a bounded monotone hemicontinuous map- pingwith (, )0()Au uuK≥ ∀∈and :jK→ (,] −∞ +∞ is a bounded proper convex lowersemicontinuous functional with (0) 0j= (i.e., for every bounded subset D of K , ()jD is bounded). Give a continuous mapping * :gK X→ and * fX∈ . Assume a) 0 ( ,)() lim u Au uj u u → += +∞ ‖‖ ‖‖ ; b) there exists constant 0 α ≥ such that 1 ( ,)()() inf suplimli ()m uu Au uj ug uuK uu αα + →+∞ →+∞ +>∈ ‖‖ ‖‖ ‖‖ ‖‖‖‖ ; c) there exists a point 0\ {0}u rcK∈ such that 0 (,) 0fu = / Then (2) has a nonzero solution. Proof. It is easy to see from condition (b) and Lemma 1 that the variational inequality (3) has a solution in K for every * qX∈ . Define a mapping :2 K A KgK → by ()( ):(( )), AA K guKguuK=∈ Then A Kg is an upper semi-continuous mapping with nonempty compact convex values by [10, Lemma 1]. Let {: } R KxK x R=∈≤‖‖ . We shall verify that (, )1 R KA iK gK= for large enough R and ( ,) r KA iK gK 0= for small enough r . J. LI ET AL. Copyright © 2010 SciRes. AM 84 Firstly, define a mapping by : [0,1]2, RK HK×→ ( ,)(()) A Ht utKg u= . It is easily seen that (, )Htu is an upper semicontinuous mapping with nonempty compact convex values. We claim that there exists large enough R such that (, )u Htu∉ for all (0,1),t∈ () R uK∈∂ . Otherwise, there exist two sequences { },{},[0,1], n nn t ut∈ 0, nn tu=→ +∞ /‖‖ such that n u∈ (,)( ()) n nnAn Ht utKgu= or ( ()) n An n uK gu t∈. Thus ((),) () () ((), )(, ), nn n nn n nn n nn uu u Avjvj tt t uu guvf vuK tt −+ + ≥− +−∀∈ (17) Letting 0v= and denoting n n n u zu =‖‖ in (17), we obtain from (17) that 11 ( )((),)( )() () (,)( )(,) nnnnn nnn nn nn nn n n n tuutu Aj utt ut gu t t zfz u u αα αα α ++ + ≤+ ‖‖‖‖ ‖‖ ‖‖ (18) Denote n n n u yK t =∈ . Then n y→ +∞‖‖ .We can ob- tain from (18) that 1 (, )()() () . nn nn n n nn n nn Ayyjyguf t y uy gu f uy α α αα αα + +≤+ ≤+ ‖‖ ‖‖‖‖‖‖ ‖‖ ‖‖‖‖ ‖‖ ‖‖ (19) Hence we have 1 ( ,)()() inf suliml pim uu Au uj ugu uu αα + →+∞ →+∞ +≤ ‖‖ ‖‖ ‖‖ ‖‖‖‖ which contradicts to condition (b). Therefore ( ,) ((1,),) ((0, ),) ˆ (0,) 1 RR KAK R K R K iKgKiHK iH K iK = ⋅ = ⋅ = = (20) by Proposition 1(4) and (2). Secondly, we shall verify that (, )0 r KA iK gK= for small enough r ( 1r< ). In fact, there exist consta nt s 12 ,, 0 CC M> fr om the boundedness of j , locally boundedness of A and condition (b) such that for all 1 uK ∈, we have 012 | ()()|(),ju ujuCguC+− ≤≤‖‖, 0 20 00 ((),) |, |(, )| g uuCuAuM Au uMu ≤≤ ≤ ‖‖‖,‖ ‖‖ (21) Since 0 (,) 0fu = / , let 0 (,)0fu < . Take N large enough such that 0 120 (1)( ,)()NfuCCM u−>+ + (22) Define a mapping by [0,1]2(,, ) rK H KHtu×→ = (() ) A Kg utNf−. Then H is an upper semi- continuous mapping with nonempty compact convex values. We claim that there exists a small enough r such that (, )u Htu∉ for all ( ),[0,1] r u Kt∈∂ ∈ . Otherwise, there exist sequences { },{},[0,1], n nn t ut∈ ( ),0 r nn u Ku∈∂→‖‖ such that (,) n nn uHt u∈= (( )) A nn Kg utNf−. Thus (,)()() (() ,)(,), nn n nn nn Auvuj vj u guNtfvufv uvK −+ − ≥−− +−∀∈ Taking 0, n n n u vz u = =‖‖ , we have () 1(,) ((),)(1)(, ) n nn nn nnn n ju Au u uu guztNf z + ≤ +− ‖‖‖‖ Since (,) () nn n n Auuj u u +→ +∞ ‖‖ and 2 ((), )(1)(, ) ( )(1) (1 ), nnn n n guztNf z guN f C Nf +− ≤ ++ ≤ ++ ‖‖‖‖ ‖‖ we obtain a contradiction. Therefore ( ,) r KA iK gK= ((0,),)((1,),) rr KK iHK iHK⋅= ⋅ by Proposition 1 (4). If ((1, ),)0 r K iH K⋅= / , then the mapping (1,):2 K HK⋅→ has a fixed point u in r K by Proposition 1(1), i.e., (1,)(( )) A uHuKg uNf∈= − . Thus ( ,)()() (() ,)(,), Au vuj vj u guNfv ufv uvK −+ − ≥−− +−∀∈ Taking 0 vuu= + , we have 00 00 (, )()() ((), )(1)(, ) Auujuuju guuNf u + +− ≥+− (23) Hence 0 00 0 0120 201 (1)( ,) (, )()()((), ) () N fu Auujuujugu u MuCCu CMuC − ≤+ +−− ≤++=++‖‖‖‖‖‖ J. LI ET AL. Copyright © 2010 SciRes. AM 85 by (21) and (23). That contradicts to (22). Therefore, ((1, ),)0 r K iH K⋅= and then (, )0 r KA K gKi= . It follows from Proposition 1(3) that (, KA i Kg \ )1 Rr KK= . Therefore there exists a fixed point \ Rr uK K∈ which is a nonzero solution of (2). 3. Nonzero Solutions in Reﬂexive Banach Spaces Theorem 2 Let X be a reflexive Banach space and KX⊂ a nonempty unbounded closed convex set with 0K∈ . Suppose that * :AX X→ is a bounded mono- tone hemicontinuous mapping with (,) 0Au u≥ for uK∈ and :(, ]jK→−∞ +∞ is a bounded convex lower semicontinuous functional with (0) 0j=. Assume that * :gK X→ is continuous from the weak topology on X to the strong topology on * X . Give * fX∈ . The following conditions are assumed to be satisfied (a) 0 (,) 0fu = / for some 0 \ {0}u rcK∈ ; (b) there constant 0 α ≥ such that 1 ( ,)()() inf suplimli ()m uu Au uj ug uuK uu αα + →+∞ → +∞ +>∈ ‖‖ ‖‖ ‖‖ ‖‖‖‖ ; (c) 0 infim ()0lw s s u ju → > . Then (2) has a nonzero solution. Proof. It is easily seen that 0 (, )() lim u Auu ju u → += ‖‖ ‖‖ +∞ by the condition (c). Let FX⊂ be a finite di- mensional subspace containing 0 u. We shall show that all conditions in Theorem 1 are satisfied on space F . Denote F K KF= ∩ which is a nonempty un- bounded closed convex set. Let : F jF X→ be an in- jective mapping and ** * : F jX F→ its dual mapping. Denote * ** (|):,( |): FFFFF F AjAFFF gjgKK= →= * F→. We know that * , F FF Aj Aj= * F FF gj gj=. Then, , FF Ag are hemicontinuous and continuous respective- l y. For 12 ,F xx K∈, we have 1212 ** 1212 1 212 121 2 (( )(),) (( )(),) (,()) (, )0 FF FF F AxAx xx j Axj Axxx AxAx j xx AxAx xx −− =−− =−− = −−≥ by the monotony of A . This means that F A is mono- tone. On the other hand, ** F jf F∈ and * 0 ( ,) F j fu= 00 (,)(,) 0 F f jufu= = / . Similarly, we have 1 (,) ()() inf sup ( lim lim ). FF uu F Auujugu uu uK αα + →+∞ →+∞ +> ∈ ‖‖ ‖‖ ‖‖ ‖‖‖‖ Therefore all conditions in Theorem 1 are satisfied on space F and so there exists , 0 F FF u Ku ∈= / such that * ( (),)()() ((), )(, ), FF FF FFF FFF A uvujvju guvujfvuvK −+ − ≥−+−∀∈ It yields that ((),) () () ((), )(, ), FF F FFF F Auv ujvju guvufvuvK −+ − ≥− +−∀∈ Taking 0v= , we get (,) ()((),)(,) FFFF FF Auuj uguufu +≤+. Hence 1 (,) ()(). FF FF F FF Auuj uguf u uu α αα + +≤+ ‖‖ ‖‖ ‖‖‖‖‖‖ This together with condition (b) implies that there ex- ists a constant M > 0 such that F uM ≤ ‖ ‖ for all finite dimensional subspace F containing 0 u. Since X is reflexive and K is weakly closed, with a similar argu- ment to that in the proof of Theorem 2 in [10] (also see [8]), we shall show that there exists uK ′∈ such that for every finite dimensional subspace F containing 0 u, u′ is in the weak closure of the set 1 1 {} FF FF Vu ⊂ = where 1 F is a finite dimensional subspace in X . In fact, since F V is bounded, we know that () w F V (the weak closure of the set F V ) is weakly compact. On the other hand, let 12 , ,, m FF F be finite dimen- siona l subspaces containing 0 u. Define () : m F = 12 {, ,,} m span FFF . Then ()m F containing 0 u is a finite dimensional subspace. Hence, 1 i m F i V = = 11 () 11 1 ( {}){} im m FF iFFFF uu =⊂⊂ == ∅ / , then () w F F V= / ∅ . T hat is to say, there exists uK ′∈ such that for every finite dimensional subspace F containing 0 u, u′ is in the weak closure of the set 11 {} FFF F Vu ⊂ = ∪ . Now let vK∈ and F′ a finite dimensional subspace of X which contains 0 u and v . Since u′ belongs to J. LI ET AL. Copyright © 2010 SciRes. AM 86 the weak closure of the set 1 1 {} FF FF Vu ′ ′⊂ = . We may find a sequence {} F u α in F V ′ such that w F uu α →′ . However, F u α satisfies the following inequality (,)()( ) ((),)(,) FF F FF F Auvuj vj u gu vufvu αα α ααα −+− ≥−+− (24) The monotony of A implies that ( ,)()() ((),)(,) FF FF F Av vujvj u gu vufvu αα ααα −+− ≥−+− Letting w F uu α ′ → yields tha t (,)()() ((), )(, ), Av vujvj u guvufvuv K ′′ −+ − ′′′ ≥−+−∀∈ Thus (,)()() ((), )(, ), Auvuj vju guvufvuv K ′′ ′ −+ − ′′ ′ ≥− +−∀∈ by Minty’s Theorem [2,3]. We claim that 0u′= / . Otherwise, 0 w F u α → . Taking 0v= in (24) yields that () (,)((),)(,) ((),)(,) FFFFFF FFF j uAuug uufu guuf u αααα αα αα α ≤− ++ ≤+ The right side of the above inequality tends to 0, which contradicts to the condition (c). Therefore u′ is a nonzero solution of (2). 4. References [1] G. X. Z. Yuan, “KKM Theory and Applications in Non- linear Analysis,” Marcel Dekker, New York, 1999. [2] D. Kinderlehrer and G. Stampacchia, “An Introduction to Variational Inequalities and Their Applications,” Acade- mic Press, New York, 1980. [3] S. S. 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