We introduce and study variational-like inequalities for generalized pseudomonotone set-valued mappings in Banach spaces. By using KKM technique, we obtain the existence of solutions for variational-like inequalities for generalized pseudomonotone set-valued mappings in reflexive Banach spaces. The results presented in this paper are generalizations and improvements of the several well-known results in the literature.
1. Introduction and Preliminaries
Variational inequality theory plays an important role in many fields, such as optimal control, mechanics, economics, transportation equilibrium, and engineering science. It is well known that monotonicity plays an important role in the study of variational inequality theory. In recent years, a number of authors have proposed many generalizations of monotonicity such as pseudomonotonicity, relaxed monotonicity, relaxed η-α monotonicity, quasimonotonicity, and semimonotonicity, p-monotonicity. For details, refer to [1–11] and the references therein.
Verma [11] studied a class of variational inequalities with relaxed monotone operators. In 2003, Fang and Huang [4] introduced a new concept of relaxed η-α monotonicity and obtained the existence of solution for variational-like inequalities in reflexive Banach spaces. B. S. Lee and B. D. Lee [12] defined weakly relaxed α-semipseudomonotone set-valued variational-like inequalities and generalize the result of Fang and Huang [4]. Bai et al. [1] defined relaxed η-α-pseudomonotone concepts for single valued mappings. For set-valued mappings, Kang et al. [13] defined relaxed η-α pseudomonotone concepts which generalize monotone concepts for single valued mapping in Fang and Huang [4] and Bai et al. [1]. Recently Sintunavarat [14] established the existence of solution of mixed equilibrium problem with the weakly relaxed α-monotone bifunction in Banach spaces.
In 2013, Kutbi and Sintunavarat [15] introduce two new concepts of weakly relaxed η-α monotone mappings and weakly relaxed η-α semimonotone mappings and obtained the existence of solution for variational-like inequality problems in reflexive Banach spaces.
Inspired and motivated by the results of Fang and Huang [4] and Kutbi and Sintunavarat [15], in this paper, we introduce the concept of weakly relaxed η-α pseudomonotone mapping and by using Knaster Kuratowski Mazurkiewicz (KKM) technique [16], we study some existence of solution for variational-like inequality for set-valued pseudomonotone mapping.
In this paper, we suppose that E is a reflexive Banach space with dual space E∗, and 〈·,·〉 denotes the pairing between E and E∗. Let K be a nonempty closed convex subset of E and 2E denote the family of all the nonempty subset of E.
The following definitions and results will be useful in our work.
Definition 1.
A mapping T:K→2E∗ is said to be weakly relaxed η-α pseudomonotone if there exist a mapping η:K×K→E and functions ϕ:K×K→R∪{+∞}, α:E→R with α(t,z)=k(t)α(z) for z∈E, where k:(0,∞)→(0,∞) is a function with limt→0(k(t)/t)=0, such that (1)u,ηx,y+ϕy,x-ϕx,x≥0,implying (2)v,ηx,y+ϕy,x-ϕx,x≥αx-yfor x,y∈E,u∈Tx,v∈Ty.
Remark 2.
(i) If ϕ(x,x)=0 in Definition 1 then we have the following pseudomonotone concept defined in Kang et al. [13]: (3)u,ηy,x+ϕy,x≥0∀u∈Tx,implying (4)v,ηy,x+ϕy,x≥αy-x,∀v∈Ty.
(ii) If T is single valued mappings, ϕ(x,x)=0, and k(t)=tp for p>1, then we have the following relaxed η-α monotone concepts defined in Fang and Huang [4] and the following η-α pseudomonotone concepts, defined in Bai et al. [1]:
For any x,y∈K(5)Tx-Ty,ηx,y≥αx-y.
For any x,y∈K(6)Ty,ηx,y≥0implies Tx,ηx,y≥αx-y.
Definition 3 (see [17]).
Let T:K→2E∗ and η:K×K→K be two mappings; T is said to be η-hemicontinuous for any x,y∈K, if the mapping defined by ϕ:[0,1]→R defined by(7)ϕt=Tty+1-tx,ηy,xis continuous at 0+.
Definition 4 (see [4]).
Let T:K→2E∗ and η:K×K→K be two mappings and ϕ:K×K→R⋃{+∞} be a proper functional. Then T is said to be η-coercive with respect to first argument of ϕ, if there exists x0∈K such that (8)u-u0,ηx,x0+ϕx,x0-ϕx0,x0ηx,x0⟶∞,whenever x→∞, for all u∈T(x),u0∈T(x0).
If ϕ(·,·)=ϕ(·) then there exists x0∈K such that (9)u-u0,ηx,x0+ϕx-ϕx0ηx,x0⟶∞,whenever x→∞, for all u∈T(x),u0∈T(x0).
If ϕ=δK, where δK is the indicator function of K, then Definition 4 coincides with the definition of η-coercivity in the sense of Yang and Chen [18].
Definition 5.
A multivalued mapping T:K→2E∗ is said to be relaxed η-α monotone if there exists a function η:K×K→E and α:E→E with α(tz)=tpα(z) for all t>0,p>1, and z∈E such that (10)u-v,ηx,y≥αx-y,∀x,y∈K,u∈Tx,v∈Ty.
Remark 6.
(i) If T is single valued then (10) becomes (11)Tx-Ty,ηx,y≥αx-y,∀x,y∈K,and then T is said to be relaxed η-α monotone [4].
(ii) If T is single valued and η(x,y)=x-y then (10) becomes (12)Tx-Ty,x-y≥αx-y,∀x,y∈K,and then T is called relaxed α-monotone [4].
(iii) If η(x,y)=x-y for all x,y∈K and α(z)=kzp, where k>0 and p>1, then (12) becomes (13)Tx-Ty,x-y≥kx-yp,∀x,y∈K,and T is called p-monotone [5, 10, 11].
Definition 7.
A mapping T:K→2E is said to be weakly relaxed η-α monotone if there exists a function η:K×K→E and α:E→R with∗limt→0+αtx=0,∗∗limt→0+ddtαtx=0for all t>0 and x∈E such that (14)u-v,ηx,y≥αx-y,∀x,y∈E,u∈Tx,v∈Ty.
Remark 8.
If T is single valued mapping then Definition 7 reduces to Definition 9 of [15].
Remark 9.
If T is weakly relaxed η-α monotone, then T is weakly relaxed η-α pseudomonotone mapping but the converse is not true.
Definition 10 (see [16]).
A mapping F:K→2E is said to be KKM mapping if, for any {x1,…,xn}⊂K, co{x1,…,xn}⊂⋃i=1nF(xi), where co{x1,…,xn} denote the convex hull of x1,…,xn.
Lemma 11 (see [19]).
Let M be a nonempty subset of a Hausdorff topological vector space X and let F:M→2X be a KKM mapping. If F(x) is closed in X for all x∈M and compact for some x∈M, then (15)⋂x∈MFx≠ϕ.
2. Existence Results
In this section, we discuss the existence of the following variational-like inequality: VLIFind x∈K such thatu,ηy,x+ϕy,x-ϕx,x≥0,∀y∈K,u∈Tx,where K is a nonempty closed convex subset of a reflexive Banach space E.
Theorem 12.
Suppose that T:K→2E∗ is η-hemicontinuous and weakly relaxed η-α pseudomonotone mapping. Let ϕ:K×K→R⋃{+∞} be a proper convex function and η:K×K→E be a mapping. Suppose that the following conditions hold:
ηx,x=0,∀x∈K.
For any fixed y∈K, u∈T(x), the mapping x→〈u,η(x,y)〉 is convex.
x→η(x,·) and x→f(x,·) are convex.
Then problems (16) and (17) are equivalent as follows:(16)Findx∈Ksuchthatu,ηy,x+ϕy,x-ϕx,x≥0,∀y∈K,u∈Tx,(17)Findx∈Ksuchthatv,ηy,x+ϕy,x-ϕx,x≥αy-x,∀y∈K,v∈Ty.
Proof.
Suppose that (16) has a solution. So there exist x∈K(18)u,ηy,x+ϕy,x-ϕx,x≥0,∀y∈K,u∈Tx.Since T is weakly relaxed η-α pseudomonotone, we have (19)v,ηy,x+ϕy,x-ϕx,x≥αy-x,∀y∈K,v∈Ty.Therefore x∈K is a solution of (17).
Conversely, suppose that x∈K is a solution of (17) and y∈K is any point with ϕ(y,y)<∞. From (17) we know that ϕ(x,x)<∞. For t∈(0,1) let yt=(1-t)x+ty,t∈(0,1); then we have yt∈K. Since x∈K is a solution of problem (17), it follows that (20)vt,ηyt,x+ϕyt,x-ϕx,x≥αyt-x=αty-x=ktαy-x.The convexity of ϕ and condition (ii) of Theorem 12 imply that (21)ϕyt,x-ϕx,x≥ϕ1-tx+ty,x-ϕx,x≤tϕy,x-ϕx,x,vt,ηyt,x=vt,η1-tx+ty,x≤1-tvt,ηx,x+tvt,ηy,x=tvt,ηy,x.It follows from (21) that (22)vt,ηy,x+ϕy,x-ϕx,x≥αty-xt=kttαy-xfor all y∈K and vt∈T(yt). Taking t→0+ in the previous inequality and using η-hemicontinuity of T, we get (23)u,ηy,x+ϕy,x-ϕx,x≥0,for all y∈K and u∈T(x) with ϕ(y,y)<∞. In case of ϕ(y,y)=∞ the relation (24)u,ηy,x+ϕy,x-ϕx,x≥0is trivial. Therefore x∈K is solution of (16).
Theorem 13.
Let K be a nonempty bounded closed convex subset of a real reflexive Banach space E and E∗ the dual space of E. Suppose that T:K→2E∗ is an η-hemicontinuous and weakly relaxed η-α pseudomonotone mapping. Let ϕ:K×K→R⋃{+∞} be a proper convex lower semicontinuous function and η:K×K→E be a mapping. Assume that
η(x,y)+η(y,x)=0,∀x∈K,
for any fixed y∈K and u∈T(x) the mapping x↦〈u,η(y,x)〉 is convex and lower semicontinuous function,
x→η(x,·) and x→f(x,·) are convex and lower semicontinuous,
α:E→R is weakly lower semicontinuous; that is, for any net {xβ}, xβ converges to x in σ(E,E∗) implying that α(x)≤liminfα(xβ).
Then problem (17) is solvable.
Proof.
Define two set-valued mappings F,G:K→2E as follows: (25)Fy=x∈K:∃u∈Tx,u,ηy,x+ϕy,x-ϕx,x≥0∀y∈K,(26)Gy=x∈K:∃v∈Ty,v,ηy,x+ϕy,x-ϕx,x≥αy-x∀y∈K.We claim that F is a KKM mapping. If F is not a KKM mapping, then there exist {y1,y2,…,yn}⊂K such that co{y1,y2,…,yn}⊈⋃i=1nF(yi). This implies that there exist y0∈co{y1,y2,…,yn} such that y=∑i=1ntiyi, where ti≥0,i=1,2,…,n, and ∑i=1nti=1, but y∉⋃i=1nF(yi). From the definition of F, we have (27)v,ηyi,y+ϕyi,y-ϕy,y<0,for i=1,2,…,n,and it follows that (28)0=v,ηy,y=v,η∑i=1ntiyi,y≤∑i=1ntiv,ηyi,y<∑i=1ntiϕy,y-ϕyi,y=ϕy,y-∑i=1ntiϕyi,y≤ϕy,y-ϕy,y=0,which is a contradiction. This implies that F is a KKM mapping. Now we prove that F(y)⊂G(y), for all y∈K.
For any given y∈K and letting x∈F(y), we have (29)u,ηy,x+ϕy,x-ϕx,x≥0.Since T is weakly relaxed η-α pseudomonotone, we get (30)v,ηy,x+ϕy,x-ϕx,x≥αy-x.It follows that x∈G(y) and so F(y)⊂G(y).
This implies that G is also KKM mapping. From the assumption, we know that G(y) is weakly closed for all y∈K. In fact since x↦〈v,η(y,x)〉 and ϕ are two convex lower semicontinuous functions, from the definition of G and weakly semilower continuity of α, it is easy to see that G(y) is weakly closed for all y∈K. Since K is bounded closed and convex, we know that K is weakly compact and so G(y) is weakly compact in K for each y in K. From Lemma 11 and Theorem 12, we obtain that (31)⋂y∈KFy=⋂y∈KGy≠ϕ.Hence, there exists x∈K such that (32)u,ηy,x+ϕy,x-ϕx,x≥0,∀x∈K,u∈Tx;that is, problem (16) has a solution.
Theorem 14.
Let K be a nonempty unbounded closed convex subset of a real reflexive Banach space E and E∗ the dual space of E. Suppose that T:K→2E∗ is an η-hemicontinuous and weakly relaxed η-α pseudomonotone mapping. Let ϕ:K×K→R⋃{+∞} be a proper convex lower semicontinuous function and η:K×K→E be a mapping. Assume that
η(x,y)+η(y,x)=0,∀x∈K,
for any fixed y∈K and u∈T(x) the mapping x↦〈u,η(y,x)〉 is convex and lower semicontinuous function,
x→η(x,·) and x→f(x,·) are convex and lower semicontinuous,
α:E→R is weakly lower semicontinuous,
T is η-coercive with respect to ϕ; that is, there exist x0∈K such that (33)u-u0,ηx,x0+ϕx,x0-ϕx0,x0ηx,x0⟶+∞,
whenever x→∞.
Then problem (16) is solvable.
Proof.
Let(34)Br=y∈E:y≤r.Consider the following problem, xr∈K∩Br, such that (35)ur,ηy,xr+ϕy,xr-ϕxr,xr≥0,∀y∈K such that y∈K∩Br.By Theorem 13, we know that (26) has a solution xr∈K∩Br; choose r>x0 with x0 as in the coercivity conditions. Then we have (36)ur,ηx0,xr+ϕx0,xr-ϕxr,xr≥0.Moreover(37)ur,ηx0,xr+ϕx0,xr-ϕxr,xr=-ur-u0,ηxr,x0+ϕx0,xr-ϕxr,xr+u0,ηx0,xr≤-ur-u0,ηxr,x0+ϕx0,xr-ϕxr,xr+u0ηx0,xr≤ηxr,x0ur-u0,ηxr,x0+ϕxr,x0-ϕx0,x0ηx,x0+u0.Now if x=r for all r, we may choose r large enough so that the above inequality and η-coercivity of T with respect to ϕ imply that (38)ur,ηx0,xr+ϕx0,xr-ϕxr,xr<0,which contradicts (39)ur,ηx0,xr+ϕx0,xr-ϕxr,xr≥0.Hence there exist r such that xr<r. For any y∈K, we can choose ϵ>0 small enough so that ϵ∈(0,1) and ur+ϵ(y-xr)∈K∩Br. It follows from (ii) that (40)ur,ηxr+ϵy-xr,xr+ϕxr+ϵy-xr,xr-ϕxr,xr≥0,∀y∈K,∀ur∈Txr.By the assumption of η, we have (41)ur,ηy,xr+ϕy,xr-ϕxr,xr≥0,∀y∈K,∀ur∈Txr.So xr∈K is a solution of (16).
It is easy to see that weakly relaxed η-α monotonicity implies weakly relaxed η-α pseudomonotonicity. So Theorems 12, 13, and 14 are deduced to the following corollaries.
Corollary 15.
Suppose that T:K→2E∗ is η-hemicontinuous and weakly relaxed η-α monotone mapping. Let ϕ:K×K→R⋃{+∞} be a proper convex function and η:K×K→E be a mapping. Suppose that the following conditions hold:
ηx,x=0,∀x∈K.
For any fixed y∈K, u∈T(x), the mapping x→〈u,η(x,y)〉 is convex.
x→η(x,·) and x→f(x,·) are convex.
Then problems ∗∗∗ and ∗∗∗∗ are equivalent as follows:∗∗∗Findx∈Ksuchthatu,ηy,x+ϕy,x-ϕx,x≥0,∀y∈K,u∈Tx,∗∗∗∗Findx∈Ksuchthatv,ηy,x+ϕy,x-ϕx,x≥αy-x,∀y∈K,v∈Ty.
Corollary 16.
Let K be a nonempty bounded closed convex subset of a real reflexive Banach space E and E∗ the dual space of E. Suppose that T:K→2E∗ is an η-hemicontinuous and weakly relaxed η-α monotone mapping. Let ϕ:K×K→R⋃{+∞} be a proper convex lower semicontinuous function and η:K×K→E be a mapping. Assume that
η(x,y)+η(y,x)=0,∀x∈K,
for any fixed y∈K and u∈T(x) the mapping x↦〈u,η(y,x)〉 is convex and lower semicontinuous function,
x→η(x,·) and x→f(x,·) are convex and lower semicontinuous,
α:E→R is weakly lower semicontinuous; that is, for any net {xβ}, xβ converges to x in σ(E,E∗) implying that αx≤liminfα(xβ).
Then problem (17) is solvable.
Corollary 17.
Let K be a nonempty unbounded closed convex subset of a real reflexive Banach space E and E∗ the dual space of E. Suppose that T:K→2E∗ is an η-hemicontinuous and weakly relaxed η-α monotone mapping. Let ϕ:K×K→R⋃{+∞} be a proper convex lower semicontinuous function and η:K×K→E be a mapping. Assume that
η(x,y)+η(y,x)=0,∀x∈K,
for any fixed y∈K and u∈T(x) the mapping x↦〈u,η(y,x)〉 is convex and lower semicontinuous function,
x→η(x,·) and x→f(x,·) are convex and lower semicontinuous,
α:E→R is weakly lower semicontinuous,
T is η-coercive with respect to ϕ; that is, there exist x0∈K such that (42)u-u0,ηx,x0+ϕx,x0-ϕx0,x0ηx,x0⟶+∞,
whenever x→∞.
Then problem (16) is solvable.
Remark 18.
Theorems 12, 13, and 14, improve Theorems 11, 12, and 15 of results of Kutbi and Sintunavarat [15] and also Fang and Huang [4]. These results are also the extensions of the known results of Bai et al. [1] and Hartman and Stampacchia [7] and corresponding results of Goeleven and Motreanu [5], B. S. Lee and B. D. Lee [12], Siddiqi et al. [8], and Verma [9].
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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