FOR GENERALIZED NONLINEAR
VARIATIONAL-LIKE INEQUALITIES
ZEQING LIU, JUHE SUN, SOO HAK SHIM, AND SHIN MIN KANG
Received 28 December 2004 and in revised form 9 June 2005
We introduce and study a new class of generalized nonlinear variational-like inequalities. Under suitable conditions, we prove the existence of solutions for the class of generalized nonlinear variational-like inequalities. A new iterative algorithm for finding the approx-imate solutions of the generalized nonlinear variational-like inequality is given and the convergence of the algorithm is also proved. The results presented in this paper improve and generalize some results in recent literature.
1. Introduction
Variational-like inequalities are a useful and important generalization of variational in-equalities [3,8,26]. They have potential and significant applications in optimization the-ory, structural analysis, and economics, see [3,4,5,6,7,8,9,10,11,12,13,14,15,16, 17,18,19,20,21,22,23,24,25,26,27,28,29]. Some mixed variational-like inequalities have been studied by Parida and Sen [26], Tian [27], and Yao [29] by using the Berge maximum theorem in finite- and infinite-dimensional spaces. Huang and Deng [10] ex-tended the auxiliary principle technique to study the existence of solutions for a class of generalized strongly nonlinear mixed variational-like inequalities. By using the mini-max inequality technique, Ding [5,6] studied some classes of nonlinear variational-like inequalities in reflexive Banach spaces.
The purpose of this paper is to introduce and study a new class of generalized nonlin-ear variational-like inequalities, which includes several kinds of variational-like inequal-ities as special cases. A few existence results of solutions for the generalized nonlinear variational-like inequality are established. We construct an iterative algorithm for find-ing the approximate solutions of the generalized nonlinear variational-like inequality and obtain the convergence of the algorithm under certain conditions.
2. Preliminaries
LetH be a real Hilbert space endowed with an inner product·,·and norm · , re-spectively. LetKbe a nonempty closed convex subset ofH, letA,C,F:K→H,N:H× H→H, andη:K×K→H be mappings, and let f :K→(−∞,∞] be a real functional.
Copyright©2005 Hindawi Publishing Corporation
Suppose thata:H×H→(−∞,∞) is a coercive continuous bilinear form, that is, there exist positive constantscanddsuch that
(C1)a(v,v)≥cv2, for allv∈H; (C2)a(u,v)≤duv, for allu,v∈H. Clearly,c≤d.
We consider the following generalized nonlinear variational-like inequality problem. Findu∈Ksuch that
a(u,v−u) +f(v)−f(u)≥N(Au,Cu) +Fu,η(v,u), ∀v∈K. (2.1)
Special cases. (A) IfN(Au,Cu)=Au−Cu,a(u,v)=0 andFu=0 for allu,v∈K, then the generalized nonlinear variational-like inequality problem (2.1) is equivalent to finding u∈Ksuch that
Cu−Au,η(v,u)≥f(u)−f(v), ∀v∈K, (2.2)
which was introduced and studied by Ding [5].
(B) IfN(Au,Cu)=Au−Cu,a(u,v)=0 andη(u,v)=gu−gvfor allu,v∈K, then the generalized nonlinear variational-like inequality problem (2.1) is equivalent to finding u∈Ksuch that
Cu−Au,gv−gu≥f(u)−f(v), ∀v∈K, (2.3)
which was studied by Yao [29].
Definition 2.1. LetA,C:K→H,N:H×H→Handη:K×K→Hbe mappings. (1)Ais said to beLipschitz continuouswith constantαif there exists a constantα >0 such that
Au−Av ≤αu−v, ∀u,v∈K. (2.4)
(2)N is said to beLipschitz continuouswith constantβin the first argument if there exists a constantβ >0 such that
N(u,w)−N(v,w)≤βu−v, ∀u,v,w∈H. (2.5)
(3)Nis said to beη-antimonotonewith respect toAin the first argument if
N(Au,w)−N(Av,w),η(u,v)≤0, ∀u,v∈K,w∈H. (2.6)
(4)Nis said to beη-relaxed Lipschitzwith constantγwith respect toCin the second argument if there exists a constantγ >0 such that
N(w,Cu)−N(w,Cv),η(u,v)≤ −γu−v2, ∀u,v∈K,w∈H. (2.7)
(5)ηis said to beLipschitz continuouswith constantδif there exists a constantδ >0 such that
Similarly, we can define the Lipschitz continuity ofNin the second argument.
Definition 2.2. LetK be a nonempty closed convex subset of a Hilbert spaceH and f : K→(−∞,∞] be a real functional.
(1) f is said to beconvexif for anyu,v∈Kand for anyα∈[0, 1],
fαu+ (1−α)v≤α f(u) + (1−α)f(v). (2.9)
(2) f is said to belower semicontinuousonKif for eachα∈(−∞,∞], the set{u∈K: f(u)≤α}is closed inK.
Lemma2.3 [1,2]. LetXbe a nonempty closed convex subset of a Hausdorff linear topolog-ical spaceE, and letφ,ψ:K×K→Rbe mappings satisfying the following conditions:
(a)ψ(x,y)≤φ(x,y), for allx,y∈X, andψ(x,x)≥0, for allx∈X; (b)for eachx∈X,φ(x,y)is upper semicontinuous with respect toy; (c)for eachy∈X, the set{x∈X:ψ(x,y)<0}is a convex set;
(d)there exists a nonempty compact setK⊂Xandx0∈Ksuch thatψ(x0,y)<0, for all y∈X\K.
Then there existsy∈Ksuch thatφ(x,y)≥0, for allx∈X.
3. Existence theorems
In this section, we give four existence theorems of solutions for the generalized nonlinear variational-like inequality (2.1).
Theorem3.1. LetKbe a nonempty closed convex subset of a Hilbert spaceH. Leta:H× H→(−∞,∞)be a coercive continuous bilinear form with (C1) and (C2) and let f :K→ (−∞,∞]be a proper convex lower semicontinuous functional with int(domf)∩K= ∅. Suppose thatA,C:K→H andN:H×H→Hare continuous mappings,η:K×K→H is Lipschitz continuous with constantδ, for eachv∈K,η(·,v)is continuous andη(v,u)= −η(u,v)for allu,v∈K. Assume that N isη-antimonotone with respect toAin the first argument andη-relaxed Lipschitz with constantξwith respect toCin the second argument. Suppose that for givenx,y∈H andv∈K, the mappingu→ N(x,y),η(u,v)is concave and upper semicontinuous. IfF:K→His completely continuous, then the generalized non-linear variational-like inequality (2.1) has a solutionu∈K.
Proof. We first prove that for each fixedu∈K, there exists a uniquew∈Ksuch that
a(w,v −w) + f(v)−f(w) ≥N(Aw,C w) + Fu,η(v, w) , ∀v∈K. (3.1)
Letube inK. Define the functionalsφandψ:K×K→Rby
φ(v,w)=a(v,v−w) +f(v)−f(w)−N(Av,Cv) +Fu,η(v, w),
ψ(v,w)=a(w,v−w) +f(v)−f(w)−N(Aw,Cw) +Fu,η(v,w) (3.2)
We check that the functionalsφandψ satisfy all the conditions ofLemma 2.3in the weak topology. It follows from the definitions ofφandψthat for allv,w∈K,
φ(v,w)−ψ(v,w)=a(v−w,v−w)−N(Av,Cv)−N(Aw,Cv),η(v,w) −N(Aw,Cv)−N(Aw,Cw),η(v,w)
≥(c+ξ)v−w2≥0,
(3.3)
which means thatφ andψ satisfy the condition (a) ofLemma 2.3. Notice that f is a convex lower semicontinuous functional and for givenx,y∈H,v∈K, the mapping u→ N(x,y),η(u,v) is concave and upper semicontinuous. It follows thatφ(v,w) is weakly upper semicontinuous with respect towand the set{v∈K:ψ(v,w)<0}is con-vex for eachw∈K. Therefore, the conditions (b) and (c) ofLemma 2.3hold. Since f is proper convex lower semicontinuous, for eachv∈int(domf),∂ f(v)= ∅, see Ekeland and Temam [9]. Letv∗be in int(domf)∩K. It follows that
f(u)≥f(v∗) +r,u−v∗, ∀r∈∂ f(v∗),u∈K. (3.4)
Put
D=(c+ξ)−1r+δNAv∗,Cv∗+δFu,
T=w∈K:w−v∗≤D. (3.5)
Obviously,Tis a weakly compact subset ofKand for anyw∈K\T,
ψv∗,w=aw−v∗,v∗−w+fv∗−f(w)−N(Aw,Cw) +Fu,η v∗,w ≤ −aw−v∗,w−v∗−r,w−v∗
+N(Aw,Cw)−NAv∗,Cw,ηw,v∗
+NAv∗,Cw−NAv∗,Cv∗,ηw,v∗
+NAv∗,Cv∗,ηw,v∗+Fu, ηw,v∗
≤ −w−v∗ (c+ξ)w−v∗− r −δNAv∗,Cv∗−δFu<0, (3.6)
which yields that the condition (d) ofLemma 2.3holds. ThusLemma 2.3ensures that there exists aw∈Ksuch thatφ(v,w) ≥0 for allv∈K, that is,
a(v,v−w) + f(v)−f(w) ≥N(Av,Cv) +Fu,η(v,w) , ∀v∈K. (3.7)
Lettbe in (0, 1] andvbe inK. Replacingvbyvt=tv+ (1−t)win (3.7), we see that
avt,t(v−w)
+f(vt)−f(w) ≥
NAvt,Cvt
+Fu,η vt,w
, ∀v∈K. (3.8)
Note thatais bilinear and f is convex. From (3.8) we deduce that
t avt,v−w
+f(v)−f(w) ≥tNAvt,Cvt
which implies that
avt,v−w+f(v)−f(w) ≥NAvt,Cvt+Fu,η(v, w) , ∀v∈K. (3.10)
Lettingt→0+in the above inequality, we conclude that
a(w, v−w) + f(v)−f(w) ≥N(Aw,C w) + Fu,η(v, w) , ∀v∈K. (3.11)
That is,w is a solution of (3.1). Now we prove the uniqueness. For any two solutions w1,w2∈Kof (3.1), we know that
aw1,w2−w1
+fw2
−fw1
≥NAw1,Cw1
+Fu,η w2,w1
, aw2,w1−w2
+fw1
−fw2
≥NAw2,Cw2
+Fu,η w1,w2
. (3.12)
Adding these inequalities, we deduce that
cw1−w2 2
≤aw1−w2,w1−w2
≤NAw1,Cw1
−NAw2,Cw1
,ηw1,w2
+NAw2,Cw1
−NAw2,Cw2
,ηw1,w2
≤ −ξw1−w22,
(3.13)
which yields thatw1=w2. That is,wis a unique solution of (3.1). This means that there exists a mappingG:K→KsatisfyingG(u) =w, where wis the unique solution of (3.1) for eachu∈K.
Next we show thatGis a completely continuous mapping. Letu1andu2be arbitrary elements inK. Using (3.1), we get that
aGu1,Gu2−Gu1
+fGu2
−fGu1
≥NAGu1
,CGu1
+Fu1,η
Gu2,Gu1
, aGu2,Gu1−Gu2
+fGu1
−fGu2
≥NAGu2
,CGu2
+Fu2,η
Gu1,Gu2
. (3.14)
Adding (3.14), we arrive at
cGu1−Gu22≤aGu1−Gu2,Gu1−Gu2 ≤NAGu1
,CGu1
−NAGu2
,CGu1
,ηGu1,Gu2
+NAGu2
,CGu1
−NAGu2
,CGu2
,ηGu1,Gu2
+Fu1−Fu2,η
Gu1,Gu2
≤ −ξGu1−Gu2 2
+δFu1−Fu2Gu1−Gu2,
(3.15)
that is,
Gu1−Gu2≤ δ
SinceFis completely continuous, it follows from (3.16) thatG:K→K is a completely continuous mapping. Hence the Schauder fixed point theorem guarantees thatGhas a fixed point u∈K, which means that u is a solution of the generalized nonlinear
variational-like inequality (2.1). This completes the proof.
Theorem3.2. Leta, f,C,N,F, andηbe as inTheorem 3.1and letNbe Lipschitz contin-uous with constantζin the first argument. Suppose thatA:K→His Lipschitz continuous with constantρ. Ifc+ξ > δζρ, then the generalized nonlinear variational-like inequality (2.1) has a solutionu∈K.
Proof. Put
D=(c+ξ−δζρ)−1r+δNAv∗,Cv∗+δFu,
T=w∈K:w−v∗≤D. (3.17)
As in the proof ofTheorem 3.1, we conclude that
ψv∗,w≤ −aw−v∗,w−v∗−r,w−v∗
+N(Aw,Cw)−NAv∗,Cw,ηw,v∗ +NAv∗,Cw−NAv∗,Cv∗,ηw,v∗
+NAv∗,Cv∗,ηw,v∗+Fu,η w,v∗ ≤ −w−v∗ (c+ξ−δζρ)w−v∗
− r −δNAv∗,Cv∗−δFu<0
(3.18)
for anyw∈K\T. The rest of the argument is now essentially the same as in the proof of
Theorem 3.1and therefore is omitted.
Theorem3.3. Leta,f,A,C,N, andηbe as inTheorem 3.1. Suppose thatF:K→His Lips-chitz continuous with constantl. Ifδl/(c+ξ)<1, then the generalized nonlinear variational-like inequality (2.1) has a unique solutionu∈K.
Proof. Let u1 and u2 be arbitrary elements inK. As in the proof of Theorem 3.1, we deduce that
Gu1−Gu2≤ δ
c+ξFu1−Fu2≤ δl
c+ξu1−u2, ∀u1,u2∈K, (3.19)
which yields thatG:K→K is a contraction mapping and hence it has a unique fixed pointu∈K, which is a unique solution of the generalized nonlinear variational-like
in-equality (2.1). This completes the proof.
The following theorem follows from the arguments of Theorems3.1,3.2and,3.3.
4. Algorithm and convergence theorems
Based onTheorem 3.1, we suggest the following iterative algorithm.
Algorithm4.1. LetA,C,F:K→H,N:H×H→H, andη:K×K→H be mappings, and let f :K →(−∞,∞]be a real functional. For any given u0∈K, compute sequence {un}n≥0by the iterative scheme
aun+1,v−un+1
+f(v)−fun+1
≥NAun+1,Cun+1
+Fun,η
v,un+1
, (4.1)
for allv∈Kandn≥0.
Theorem4.2. Let a, f,F,N,A,C, andηbe as inTheorem 3.3. If δl/(c+ξ)<1, then the generalized nonlinear variational-like inequality (2.1) possesses a unique solution and the iterative sequence{un}n≥0generated byAlgorithm 4.1converges strongly to the unique solution.
Proof. UsingAlgorithm 4.1, we obtain that
aun+1,un−un+1
+fun
−fun+1
≥NAun+1,Cun+1
+Fun,η
un,un+1
, aun,un+1−un
+ fun+1
−fun
≥NAun,Cun
+Fun−1,η
un+1,un
, (4.2)
for alln≥1. Adding (4.2), we get that
cun+1−un2≤aun+1−un,un+1−un
≤NAun+1,Cun+1−NAun,Cun+1,ηun+1,un +NAun,Cun+1
−NAun,Cun
,ηun+1,un
+Fun−Fun−1,η
un+1,un
≤ −ξun+1−un 2
+δlun−un−1un+1−un,
(4.3)
that is,
un+1−un≤ δl
c+ξun−un−1, ∀n≥1, (4.4)
which yields that{un}n≥0is a Cauchy sequence byδl/(c+ξ)<1. Consequently,{un}n≥0 converges to some elementuinK. Lettingn→ ∞in (4.1), we infer that
a(u,v−u) +f(v)−f(u)≥N(Au,Cu) +Fu,η(v,u), ∀v∈K. (4.5)
Hence uis a solution of the generalized nonlinear variational-like inequality (2.1). It follows from Theorem 3.3 that u is the unique solution of the generalized nonlinear
Similarly we have the following result.
Theorem4.3. Leta,f,F,N,A,C, andηbe as inTheorem 3.4. If0< δl/(c+ξ−δζρ)<1, then the generalized nonlinear variational-like inequality (2.1) possesses a unique solution and the iterative sequence {un}n≥0 generated by Algorithm 4.1 converges strongly to the unique solution.
Acknowledgments
The authors thank the referees for their valuable suggestions for the improvement of the paper. This work was supported by the Science Research Foundation of Educational Department of Liaoning Province (2004C063) and Korea Research Foundation Grant (KRF-2003-005-C00013).
References
[1] S. S. Chang,Variational Inequalitity and Complementarity Theory with Applications, Shanghai Scientific Technology and Literature Press, Shanghai, 1991.
[2] ,On the existence of solutions for a class of quasi-bilinear variational inequalities, J. Sys-tems Sci. Math. Sci.16(1996), 136–140 (Chinese).
[3] M. S. R. Chowdhury and K.-K. Tan,Generalization of Ky Fan’s minimax inequality with appli-cations to generalized variational inequalities for pseudo-monotone operators and fixed point theorems, J. Math. Anal. Appl.204(1996), no. 3, 910–929.
[4] P. Cubiotti,Existence of solutions for lower semicontinuous quasi-equilibrium problems, Comput. Math. Appl.30(1995), no. 12, 11–22.
[5] X. P. Ding,Algorithm of solutions for mixed-nonlinear variational-like inequalities in reflexive Banach space, Appl. Math. Mech.19(1998), no. 6, 489–496 (Chinese).
[6] ,Existence and algorithm of solutions for nonlinear mixed variational-like inequalities in Banach spaces, J. Comput. Appl. Math.157(2003), no. 2, 419–434.
[7] X. P. Ding and K.-K. Tan,A minimax inequality with applications to existence of equilibrium point and fixed point theorems, Colloq. Math.63(1992), no. 2, 233–247.
[8] X. P. Ding and E. Tarafdar,Generalized variational-like inequalities with pseudomonotone set-valued mappings, Arch. Math. (Basel)74(2000), no. 4, 302–313.
[9] I. Ekeland and R. Temam,Convex Analysis and Variational Problems, North-Holland, Amster-dam, 1976.
[10] N.-J. Huang and C.-X. Deng,Auxiliary principle and iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities, J. Math. Anal. Appl.256(2001), no. 2, 345–359.
[11] Z. Liu, L. Debnath, S. M. Kang, and J. S. Ume,Completely generalized multivalued nonlinear quasi-variational inclusions, Int. J. Math. Math. Sci.30(2002), no. 10, 593–604.
[12] ,On the generalized nonlinear quasivariational inclusions, Acta Math. Inform. Univ. Os-traviensis11(2003), no. 1, 81–90.
[13] ,Sensitivity analysis for parametric completely generalized nonlinear implicit quasivaria-tional inclusions, J. Math. Anal. Appl.277(2003), no. 1, 142–154.
[14] ,Generalized mixed quasivariational inclusions and generalized mixed resolvent equations for fuzzy mappings, Appl. Math. Comput.149(2004), no. 3, 879–891.
[16] ,Convergence and stability of perturbed three-step iterative algorithm for completely gen-eralized nonlinear quasivariational inequalities, Appl. Math. Comput.149(2004), no. 1, 245–258.
[17] Z. Liu, S. M. Kang, and J. S. Ume,On general variational inclusions with noncompact valued mappings, Adv. Nonlinear Var. Inequal.5(2002), no. 2, 11–25.
[18] ,Completely generalized multivalued strongly quasivariational inequalities, Publ. Math. Debrecen62(2003), no. 1-2, 187–204.
[19] ,Generalized variational inclusions for fuzzy mappings, Adv. Nonlinear Var. Inequal.6 (2003), no. 1, 31–40.
[20] ,The solvability of a class of quasivariational inequalities, Adv. Nonlinear Var. Inequal.6 (2003), no. 2, 69–78.
[21] Z. Liu, J. S. Ume, and S. M. Kang,General strongly nonlinear quasivariational inequalities with relaxed Lipschitz and relaxed monotone mappings, J. Optim. Theory Appl.114(2002), no. 3, 639–656.
[22] ,Resolvent equations technique for general variational inclusions, Proc. Japan Acad. Ser. A Math. Sci.78(2002), no. 10, 188–193.
[23] , Nonlinear variational inequalities on reflexive Banach spaces and topological vector spaces, Int. J. Math. Math. Sci.2003(2003), no. 4, 199–207.
[24] , Completely generalized quasivariational inequalities, Adv. Nonlinear Var. Inequal.7 (2004), no. 1, 35–46.
[25] P. D. Panagiotopoulos and G. E. Stavroulakis,New types of variational principles based on the notion of quasidifferentiability, Acta Mech.94(1992), no. 3-4, 171–194.
[26] J. Parida and A. Sen,A variational-like inequality for multifunctions with applications, J. Math. Anal. Appl.124(1987), no. 1, 73–81.
[27] G. Q. Tian,Generalized quasi-variational-like inequality problem, Math. Oper. Res.18(1993), no. 3, 752–764.
[28] J. C. Yao,The generalized quasi-variational inequality problem with applications, J. Math. Anal. Appl.158(1991), no. 1, 139–160.
[29] ,Existence of generalized variational inequalities, Oper. Res. Lett.15(1994), no. 1, 35– 40.
Zeqing Liu: Department of Mathematics, Liaoning Normal University, P.O. Box 200, Dalian, Liaoning 116029, China
E-mail address:[email protected]
Juhe Sun: Department of Mathematics, Liaoning Normal University, P.O. Box 200, Dalian, Liaon-ing 116029, China
E-mail address:[email protected]
Soo Hak Shim: Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Chinju 660-701, Korea
E-mail address:[email protected]
Shin Min Kang: Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Chinju 660-701, Korea
