On solvability of general nonlinear variational like inequalities in reflexive Banach spaces

VARIATIONAL-LIKE INEQUALITIES

IN REFLEXIVE BANACH SPACES

ZEQING LIU, JUHE SUN, SOO HAK SHIM, AND SHIN MIN KANG

Received 28 December 2004

We introduce and study a new class of general nonlinear variational-like inequalities in reflexive Banach spaces. By applying a minimax inequality, we establish two existence and uniqueness theorems of solutions for the general nonlinear variational-like inequality.

1. Introduction

Recently, variational inequality theory has been extended and applied in various direc-tions, see [6,7,8,9,10] and the references therein. In particular, Ding [1,2,3], and Ding and Tan [4] studied the existence of solutions for several nonlinear variational-like inequalities in reflexive Banach spaces.

In this paper, a new class of general nonlinear variational-like inequalities in reflexive Banach spaces are introduced. Utilizing a minimax inequality due to Ding and Tan [4], we provide some efficient conditions, which ensure the existence and uniqueness of solutions for the general nonlinear variational-like inequality. Our results improve and generalize many known results in the literature.

2. Preliminaries

LetDbe a nonempty convex subset of a reflexive Banach spaceBwith dual spaceB∗and letu,vbe the dual pairing betweenu∈B∗andv∈B. LetT,A:D→B∗,N:B∗×B∗→

B∗, andη:D×D→B be mappings. Suppose thata:B×B→(−∞, +∞) is a coercive continuous bilinear form, that is, there exist positive constantscanddsuch that

(c1)a(u,v)≥cv2_{for allv∈B;} (c2)a(u,v)≤duvfor allu,v∈B. Clearly,c≤d.

Let f :D→(−∞, +∞] be a real functional and letz∗∈B∗. We consider the following general nonlinear variational-like inequality problem: findu∈Dsuch that

N(Tu,Au)−z∗,η(v,u)+a(u,v−u)≥f(u)−f(v), ∀v∈D. (2.1)

We have the following special cases.

Copyright©2005 Hindawi Publishing Corporation

(A) IfN(Tu,Au)=Tu−Au,a(u,v)≡0 for allu,v∈D, andz∗=0, then the general nonlinear variational-like inequality (2.1) reduces to

Tu−Au,η(v,u)≥f(u)−f(v), ∀v∈D, (2.2)

which was introduced and studied by Ding [1].

(B) IfN(Tu,Au)=Tu−Auanda(u,v)≡0 for allu,v∈D, then the general nonlinear variational-like inequality (2.1) reduces to

Tu−Au−z∗,η(v,u)≥ f(u)−f(v), ∀v∈D, (2.3)

which was studied by Yao [10] in Hilbert spaces.

Definition 2.1. LetDbe a nonempty subset of a reflexive Banach spaceBwith dual space

B∗. LetT:D→B∗,N:B∗×B∗→B∗, andη:D×D→Bbe mappings.

(1)Tis said to beLipschitz continuouswith constantαif there exists a constantα >0 such that

Tu−Tv ≤αu−v, ∀u,v∈D. (2.4)

(2)N is said to beη-relaxed monotonewith constantγwith respect toTin the first argument if there exists a constantγ >0 such that

N(Tu,w)−N(Tv,w),η(u,v)≥ −γu−v2_{, ∀}

u,v∈D,w∈B∗. (2.5)

(3)Nis said to beη-strongly monotonewith constantξwith respect toTin the first argument if there exists a constantξ >0 such that

N(Tu,w)−N(Tv,w),η(u,v)≥ξu−v2_{, ∀}

u,v∈D,w∈B∗. (2.6)

(4)Nis said to beη-monotonewith respect toAin the second argument if

N(w,Au)−N(w,Av),η(u,v)≥0, ∀u,v∈D,w∈B∗. (2.7)

(5)ηis said to be Lipschitz continuous with constantδif there exists a constantδ >0 such that

_{η(u,v)≤δu−v, ∀u,v∈D. (2.8)}

(6)ηis said to be strongly monotone with constantτif there exists a constantτ >0 such that

u−v,η(u,v)≥τu−v2_{, ∀u,v∈D. (2.9)}

Definition 2.2. LetDbe a nonempty convex subset of a reflexive Banach spaceBand let

f :D→(−∞, +∞] be a real functional. (1) f is said to beconvexif

(2) f is said to belower semicontinuousonDif for eachα∈(−∞, +∞], the set{u∈

D:f(u)≤α}is closed inD.

Definition 2.3. Let D be a nonempty subset of a reflexive Banach space B with dual spaceB∗and letT:D→B∗andη:D×D→Bbe two mappings.T andηare said to have 0-diagonally concave relationwith respect toz∗∈B∗if the functionalϕ:D×D→ (−∞, +∞] defined byϕ(u,v)= Tu−z∗,η(u,v)is 0-diagonally concaveinv, that is, for any finite set{v1,...,vm} ∈Dand for anyu=m

i=1λiviwithλi≥0 and

_m

i=1λi=1,

m

i=1

λiϕ

u,vi

≤0. (2.11)

Remark 2.4. It is easy to see that, if for eachu∈D,η(u,u)=0 and the functionalv→ Tu−z∗,η(u,v)is concave, then the mappingsTandηhave the 0-diagonally concave relation with respect toz∗onD.

Lemma2.5 [4]. Let Dbe a nonempty convex subset of a topological vector space and let ϕ:D×D→[−∞, +∞]be such that

(a)for each u∈D,v→ϕ(u,v)is lower semicontinuous on each nonempty compact subset ofD,

(b)for each nonempty finite set{v1,...,vm} ∈Dand for anyu=

_m

i=1λiviwithλi≥0 andm_i=1λi=1,min1≤i≤mϕ(u,vi)≤0,

(c)there exist a nonempty compact convex subsetX0ofDand a nonempty compact sub-setKofDsuch that for eachv∈D−K, there isu∈co(X0∪ {v})withϕ(u,v)>0. Then there existsv∈Ksuch thatϕ(u,v)≤0for allu∈D.

3. Existence and uniqueness theorems

In this section, we use the minimax inequality technique due to Ding and Tan [4] to prove the existence and uniqueness theorems of solutions for the general nonlinear variational-like inequality (2.1).

Theorem 3.1. Let Dbe a nonempty closed convex subset of a reflexive Banach space B with dual spaceB∗. Assume thatη:D×D→Bis Lipschitz continuous with constantδ, for eachv∈D,η(·,v)is continuous onD, andη(v,u)= −η(u,v)for allv,u∈D. Suppose thata:B×B→(−∞, +∞)is a coercive continuous bilinear form and f :D→(−∞, +∞] is a proper convex lower semicontinuous functional withint(domf)∩D= ∅. LetT,A:

D→B∗ andN:B∗×B∗→B∗ be continuous mappings, let N be η-strongly monotone with constantαwith respect toT in the first argument andη-monotone with respect toA in the second argument. Assume thatNandηhave the 0-diagonally concave relation with respect toz∗∈B∗. Then the general nonlinear variational-like inequality (2.1) has a unique solutionu∈D.

Proof. Define a functionalϕ:D×D→(−∞, +∞] by

SinceT,A,N,a, andηare continuous and f is lower semicontinuous, it follows that for each v∈D, the functionalu→ϕ(v,u) is weakly lower semicontinuous on D. We claim that ϕsatisfies the condition (b) of Lemma 2.5. If it is false, there exist a finite set{v1,...,vm} ⊂Dandu=m

i=1λiviwithλi≥0 and

_m

i=1λi=1 such thatϕ(vi,u)>0 for alli=1,...,m, that is,

z∗−N(Tu,Au),η(vi,u)−au,vi+ f(u)−fvi>0 (3.2)

for alli=1,...,m. It follows that

m

i=1

λi

z∗−N(Tu,Au),ηvi,u

> m

i=1

λia

u,vi

−f(u) +

m

i=1

λif

vi

≥0, (3.3)

which contradicts the condition thatNandηhave the 0-diagonally concave relation with respect toz∗. Therefore, the condition (b) ofLemma 2.5holds. Since f is proper convex lower semicontinuous, it follows from [5] that its subdifferential∂ f(v)= ∅for allv∈ int(domf). It is easy to see that f(u)≥ f(v∗) +r,u−v∗for allv∗∈int(domf)∩D,

r∈∂ f(v∗), andu∈B. For any fixedv∗∈int(domf)∩D, set

Q=(α+c)−1 δNTv∗,Av∗+z∗+dv∗+r (3.4)

andK= {u∈D:u−v∗ ≤Q}. ThenKandD0= {v∗}are both weakly compact con-vex subsets ofD. It follows that for eachu∈D−K,

ϕv∗,u=z∗−N(Tu,Au),ηv∗,u−au,v∗−u+f(u)−fv∗

≥z∗,ηv∗,u+N(Tu,Au)−NTv∗,Au,ηu,v∗

+NTv∗,Au−NFv∗,Gv∗,ηu,v∗

−NFv∗,Gv∗,ηv∗,u

+av∗−u,v∗−u−av∗,v∗−u+r,u−v∗

≥u−v∗ (α+c)u−v∗−δNTv∗,Av∗+z∗−dv∗− r>0, (3.5)

that is, the condition (c) ofLemma 2.5holds. ByLemma 2.5, there existsu∈Dsuch that

ϕ(v,u)≤0 for allv∈D, that is,

N(Tu,Au)−z∗,η(v,u)+a(u,v−u)≥f(u)−f(v), ∀v∈D. (3.6)

Now we prove thatu is a unique solution of the general nonlinear variational-like inequality (2.1). Suppose thatu1andu2are two solutions of the general nonlinear vari-ational-like inequality (2.1). It follows that

NFu1,Gu1−z∗,ηu2,u1+au1,u2−u1≥fu1−fu2,

NFu2,Gu2

−z∗,ηu1,u2

+au2,u1−u2

≥ fu2

−fu1

Usingη(u,v)= −η(v,u) for alluandv∈D, (3.7), we deduce that

0≥NTu1,Au1

−NTu2,Au2

,ηu1,u2

+au1−u2,u1−u2

≥(α+c)u1−u2 2

≥0, (3.8)

which means thatu1=u2 anduis the unique solution of the general nonlinear vari-ational-like inequality (2.1). This completes the proof. Theorem3.2. LetD,B,B∗,η,a, f,T, and N be as inTheorem 3.1, letA:D→B∗be Lipschitz continuous with constantγ, and letN:B∗×B∗→B∗beη-relaxed monotone with constantαwith respect toTin the first argument and Lipschitz continuous with constantβ in the second argument. Assume thatNandηhave the0-diagonally concave relation with respect toz∗∈B∗andc > α+βγδ. Then the general nonlinear variational-like inequality (2.1) has a unique solutionu∈D.

Proof. Let

Q=(c−α−βγδ)−1 δNTv∗,Av∗+z∗+dv∗+r (3.9)

andK= {u∈D:u−v∗ ≤Q}.It follows from the proof ofTheorem 3.1that

ϕv∗,u=z∗,ηv∗,u+N(Tu,Au)−NTv∗,Au,ηu,v∗

−NTv∗,Au−NTv∗,Av∗,ηv∗,u

−NTv∗,Av∗,ηv∗,u

+av∗−u,v∗−u−av∗,v∗−u+r,u−v∗

≥u−v∗ (c−α−βγδ)u−v∗

−δNTv∗,Av∗+z∗−dv∗− r>0.

(3.10)

The rest of the proof follows precisely as in the proof ofTheorem 3.1. This completes the

proof.

Remark 3.3. Theorems3.1and3.2improve Ding’s [1, Theorems 3.1 and 3.2] and Yao’s [10, Theorem 3.1].

Acknowledgments

The authors would like to thank the referee for a critical reading of the manuscript and many valuable suggestions. This work was supported by the Science Research Founda-tion of EducaFounda-tional Department of Liaoning Province (2004C063) and Korea Research Foundation Grant (KRF-2003-005-C00013).

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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, Jinju 660-701, Korea

E-mail address:[email protected]

Shin Min Kang: Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Korea