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Variational-Like Inequalities for Weakly Relaxed [eta]-[alpha] Pseudomonotone Set-Valued Mappings in Banach Space.

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 [eta]-[alpha] 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 [eta]-[alpha] 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 [alpha]-semipseudomonotone set-valued variational-like inequalities and generalize the result of Fang and Huang [4]. Bai et al. [1] defined relaxed [eta]-pseudomonotone concepts for single valued mappings. For set-valued mappings, Kang et al. [13] defined relaxed [eta]-[alpha] 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 a-monotone bifunction in Banach spaces.

In 2013, Kutbi and Sintunavarat [15] introduce two new concepts of weakly relaxed [eta]-[alpha] monotone mappings and weakly relaxed [eta]-[alpha] 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 [eta]-[alpha] 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.sup.*], and <*, *> denotes the pairing between E and [E.sup.*]. Let K be a nonempty closed convex subset of E and [2.sup.E] 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 : [mathematical expression not reproducible] is said to be weakly relaxed [eta]-[alpha] pseudomonotone if there exist a mapping [eta] : K x K [right arrow] E and functions [phi] : K x K [right arrow] R [union] {+[infinity]} [alpha] : E [right arrow] R with [alpha](t, z) = k(t)[alpha](z) for z [member of] E, where k : (0, [infinity]) [right arrow] (0, [infinity]) is a function with [lim.sub.t[right arrow]0] (k(t)/t) = 0, such that

<u, [eta] (x, y)> + [phi] (y, x) - [phi] (x, x) [greater than or equal to] 0, (1)

implying

<v, [eta] (x, y)> + [phi] (y, x) - [phi] (x, x) [greater than or equal to] [alpha] (x - y) for x, y [member of] E, u [member of] T (x), v [member of] T (y). (2)

Remark 2. (i) If [phi](x, x) = 0 in Definition 1 then we have the following pseudomonotone concept defined in Kang et al. [13]:

<u, [eta](y, x)> + [phi](y, x) [greater than or equal to] 0 [for all]u [member of] T(x), (3)

implying

<v, [eta](y, x)> + [phi](y, x) [greater than or equal to] [alpha](y - x), [for all]v [member of] T(y). (4)

(ii) If T is single valued mappings, [phi](x, x) = 0, and k(t) = [t.sup.p] for p > 1, then we have the following relaxed [eta]-[alpha] monotone concepts defined in Fang and Huang [4] and the following [eta]-[alpha] pseudomonotone concepts, defined in Bai et al. [1]:

(a) For any x, y [member of] K

<Tx - Ty, [eta] (x, y)> [greater than or equal to] [alpha](x - y). (5)

(b) For any x, y [member of] K

<Ty, [eta] (x, y)> [greater than or equal to] 0 implies <Tx, [eta](x, y)> [greater than or equal to] [alpha](x - y). (6)

Definition 3 (see [17]). Let [mathematical expression not reproducible] be two mappings; T is said to be [eta]-hemicontinuous for any x, y [member of] K, if the mapping defined by [phi] : [0, 1] [right arrow] R defined by

[phi](t) = <T(ty + (1 - t)x), [eta](y, x)> (7)

is continuous at [0.sup.+].

Definition 4 (see [4]). Let [mathematical expression not reproducible] and q : K x K [right arrow] K be two mappings and [phi] : K x K [right arrow] R [union] {+[infinity]} be a proper functional. Then T is said to be [eta]-coercive with respect to first argument of [phi], if there exists [x.sub.0] [member of] K such that

<u - [u.sub.0], [eta] (x, [x.sub.0])> + [phi] (x, [x.sub.0]) - [phi] ([x.sub.0], [x.sub.0])/[parallel][eta](x, [x.sub.0])[parallel] [right arrow] [infinity], (8)

whenever [parallel]x[parallel] [right arrow] [infinity], for all u [member of] T(x), [u.sub.0] [member of] T([x.sub.0]).

If [phi](*, *) = [phi]* then there exists [x.sub.0] [member of] K such that

<u - [u.sub.0], [eta] (x, [x.sub.0])> + [phi](x) - [phi] ([x.sub.0])/[parallel][eta](x, [x.sub.0])[parallel] [right arrow] [infinity], (9)

whenever [parallel]x[parallel] [right arrow] [infinity], for all u [member of] T(x), [u.sub.0] [member of] T([x.sub.0]).

If [phi] = [[delta].sub.K], where [[delta].sub.K] is the indicator function of K, then Definition 4 coincides with the definition of [eta]-coercivity in the sense of Yang and Chen [18].

Definition 5. A multivalued mapping [mathematical expression not reproducible] is said to be relaxed [eta]-[alpha] monotone if there exists a function [eta] : K x K [right arrow] E and [alpha] : E [right arrow] E with [alpha](tz) = [t.sup.p][alpha](z) for all t > 0, p > 1, and z [member of] E such that

<u - v, [eta] (x, y)> [greater than or equal to] [alpha](x - y), [for all]x, y [member of] K, u [member of] T(x), v [member of] T(y). (10)

Remark 6. (i) If T is single valued then (10) becomes

<Tx - Ty, [eta] (x, y)> [greater than or equal to] [alpha](x - y), [for all]x, y [member of] K, (11)

and then T is said to be relaxed [eta]-[alpha] monotone [4].

(ii) If T is single valued and [eta](x, y) = x - y then (10) becomes

<Tx -Ty, x - y> [greater than or equal to] [alpha](x - y), [for all]x, y [member of] K, (12)

and then T is called relaxed [alpha]-monotone [4].

(iii) If [eta](x, y) = x - y for all x, y [member of] K and [alpha](z) = k[[parallel]z[parallel].sup.p], where k > 0 and p > 1, then (12) becomes

<Tx - Ty, x - y> [greater than or equal to] k [[parallel]x - y[parallel].sup.p], [for all]x, y [member of] K, (13)

and T is called p-monotone [5,10,11].

Definition 7. A mapping [mathematical expression not reproducible] is said to be weakly relaxed [eta]-[alpha] monotone if there exists a function [eta] : K x K [right arrow] E and [alpha] : E [right arrow] R with

[mathematical expression not reproducible], *

[mathematical expression not reproducible] **

for all t > 0 and x [member of] E such that

<u - v, [eta] (x, y)> [greater than or equal to] [alpha](x - y), [for all]x, y [member of] E, u [member of] T(x), v [member of] T(y). (14)

Remark 8. If T is single valued mapping then Definition 7 reduces to Definition 9 of [15].

Remark 9. If T is weakly relaxed [eta]-[alpha] monotone, then T is weakly relaxed [eta]-[alpha] pseudomonotone mapping but the converse is not true.

Definition 10 (see [16]). A mapping F : K [right arrow] [2.sup.E] is said to be KKM mapping if, for any {[x.sub.1], ..., [x.sub.n]} [subset] K, co{[x.sub.1], ..., [x.sub.n]} [subset] [[union].sup.n.sub.i=1] F([x.sub.i]), where co{[x.sub.1], ..., [x.sub.n]} denote the convex hull of [x.sub.1], ..., [x.sub.n].

Lemma 11 (see [19]). Let M be a nonempty subset of a Hausdorff topological vector space X and let F : M [right arrow] [2.sup.X] be a KKM mapping. If F(x) is closed in X for all x [member of] M and compact for some x [member of] M, then

[mathematical expression not reproducible]. (15)

2. Existence Results

In this section, we discuss the existence of the following variational-like inequality:

[mathematical expression not reproducible], (VLI)

where K is a nonempty closed convex subset of a reflexive Banach space E.

Theorem 12. Suppose that [mathematical expression not reproducible] is [eta]-hemicontinuous and weakly relaxed [eta]-[alpha] pseudomonotone mapping. Let [phi] : K x K [right arrow] R [union] {+[infinity]} be a proper convex function and [eta] : K x K [right arrow] E be a mapping. Suppose that the following conditions hold:

(i) [eta](x, x) = 0, [for all]x [member of] K.

(ii) For any fixed y [member of] K, u [member of] T(x), the mapping x [right arrow] <u, [eta](x, y)> is convex.

(iii) x [right arrow] [eta](x, *) and x [right arrow] f(x, *) are convex.

Then problems (16) and (17) are equivalent as follows:

Find x [member of] K such that <u, [eta] (y, x)> + [phi] (y, x) - [phi] (x, x) [greater than or equal to] 0, [for all]y [member of] K, u [member of] T(x), (16)

Find x [member of] K such that <v, [eta] (y, x)> + [phi] (y, x) - [phi] (x, x) [greater than or equal to] [alpha](y - x), [for all]y [member of] K, v [member of] T(y). (17)

Proof. Suppose that (16) has a solution. So there exist x [member of] K

<u, [eta] (y, x)> + [phi] (y, x) - [phi] (x, x) [greater than or equal to] 0, [for all]y [member of] K, u [member of] T(x). (18)

Since T is weakly relaxed [eta]-[alpha] pseudomonotone, we have

<v, [eta] (y, x)> + [phi] (y, x) - [phi] (x, x) [greater than or equal to] [alpha](y - x), [for all]y [member of] K, v [member of] T(y). (19)

Therefore x [member of] K is a solution of (17).

Conversely, suppose that x [member of] K is a solution of (17) and y [member of] K is any point with [phi](y, y) < [infinity]. From (17) we know that [phi](x, x) < [infinity]. For t [member of] (0, 1) let [y.sub.t] = (1 - t)x + ty, t [member of] (0, 1); then we have [y.sub.t] [member of] K. Since x [member of] K is a solution of problem (17), it follows that

<[v.sub.t], [eta] ([y.sub.t], x)> + [phi] ([y.sub.t], x) - [phi] (x, x) [greater than or equal to] [alpha] ([y.sub.t] - x) = [alpha](t(y - x)) = k(t)[alpha](y - x). (20)

The convexity of [phi] and condition (ii) of Theorem 12 imply that

[mathematical expression not reproducible]. (21)

It follows from (21) that

<[v.sub.t], [eta] (y, x)> + [phi](y, x) - [phi] (x, x) [greater than or equal to] [alpha](t(y - x))/t = [k(t)/t] [alpha](y - x) (22)

for all y [member of] K and [v.sub.t] [member of] T([y.sub.t]). Taking t [right arrow] [0.sup.+] in the previous inequality and using [eta]-hemicontinuity of T, we get

<u, [eta](y, x)> + [phi](y, x) - [phi](x, x) [greater than or equal to] 0, (23)

for all y [member of] K and u [member of] T(x) with [phi](y, y) < [infinity]. In case of [phi](y, y) = [infinity] the relation

<u, [eta](y, x)> + [phi](y, x) - [phi](x, x) [greater than or equal to] 0 (24)

is trivial. Therefore x [member of] 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.sup.*] the dual space of E. Suppose that [mathematical expression not reproducible] is an [eta]-hemicontinuous and weakly relaxed [eta]-[alpha] pseudomonotone mapping. Let [eta] : K x K [right arrow] R [union] {+[infinity]} be a proper convex lower semicontinuous function and [eta] : K x K [right arrow] E be a mapping. Assume that

(i) [eta](x, y) + [eta](y, x) = 0, [for all]x [member of] K,

(ii) for any fixed y [member of] K and u [member of] T(x) the mapping x [??] <u, [eta](y, x)> is convex and lower semicontinuous function,

(iii) x [right arrow] [eta](x, *) and x [right arrow] f(x, *) are convex and lower semicontinuous,

(iv) [alpha] : E [right arrow] R is weakly lower semicontinuous; that is, for any net {[x.sub.[beta]]}, [x.sub.[beta]] converges to x in [sigma](E, [E.sup.*]) implying that [alpha](x) [less than or equal to] lim inf [alpha]([x.sub.[beta]]).

Then problem (17) is solvable.

Proof. Define two set-valued mappings F,G : K [right arrow] [2.sup.E] as follows:

F (y) = {x [member of] K : [there exists]u [member of] T (x), <u, [eta] (y, x)> + [phi] (y, x) - [phi] (x, x) [greater than or equal to] 0} [for all]y [member of] K, (25)

G (y) = {x [member of] K : [there exists]v [member of] T (y), <v, [eta] (y, x)> + [phi] (y, x) - [phi](x, x) [greater than or equal to] [alpha](y - x)} [for all]y [member of] K. (26)

We claim that F is a KKM mapping. If F is not a KKM mapping, then there exist {[y.sub.1], [y.sub.2], ..., [y.sub.n]} [subset] K such that co{[y.sub.1], [y.sub.2], ..., [y.sub.n]} [not subset or equal to] [U.sup.n.sub.i=1] F([y.sub.i]) This implies that there exist [y.sub.0] [member of] co{[y.sub.1], [y.sub.2], ..., [y.sub.n]} such that [[summation].sup.n.sub.i=1] [t.sub.i][y.sub.i], where [t.sub.i] [greater than or equal to] 0, i = 1, 2, ..., n, and [[summation].sup.n.sub.i=1] [t.sub.i] = 1, but y [not member of] [U.sup.n.sub.i=1] F([y.sub.i]). From the definition of F, we have

<v, [eta] ([y.sub.i], y)> + [phi] ([y.sub.i], y) - [phi] (y, y) < 0, for i = 1, 2, ..., n, (27)

and it follows that

[mathematical expression not reproducible], (28)

which is a contradiction. This implies that F is a KKM mapping. Now we prove that F(y) [subset] G(y), for all y [member of] K.

For any given y [member of] K and letting x [member of] F(y), we have

<u, [eta](y, x)> + [phi](y, x) - [phi](x, x) [greater than or equal to] 0. (29)

Since T is weakly relaxed [eta]-[alpha] pseudomonotone, we get

<v, [eta] (y, x)> + [phi] (y, x) - [phi] (x, x) [greater than or equal to] [alpha](y - x). (30)

It follows that x [member of] G(y) and so F(y) [subset] G(y).

This implies that G is also KKM mapping. From the assumption, we know that G(y) is weakly closed for all y [member of] K. In fact since x [??] <v, [eta] (y, x)> and [phi] are two convex lower semicontinuous functions, from the definition of G and weakly semilower continuity of [alpha], it is easy to see that G(y) is weakly closed for all y [member of] 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

[mathematical expression not reproducible]. (31)

Hence, there exists x [member of] K such that

<u, [eta] (y, x)> + [phi] (y, x) - [phi] (x, x) [greater than or equal to] 0, [for all]x [member of] K, u [member of] T(x); (32)

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 [mathematical expression not reproducible] is an [eta]-hemicontinuous and weakly relaxed [eta]-[alpha]pseudomonotone mapping. Let [phi] : K x K [right arrow] R [union] {+[infinity]} be a proper convex lower semicontinuous function and [eta] : K x K [right arrow] E be a mapping. Assume that

(i) [eta](x, y) + [eta](y, x) = 0, [for all]x [member of] K,

(ii) for any fixed y [member of] K and u [member of] T(x) the mapping x [??] <u, [eta](y, x)> is convex and lower semicontinuous function,

(iii) x [right arrow] [eta] (x, *) and x [right arrow] f(x, *) are convex and lower semicontinuous,

(iv) [alpha] : E [right arrow] R is weakly lower semicontinuous,

(v) T is [eta]-coercive with respect to [phi]; that is, there exist [x.sub.0] [member of] K such that

<u - [u.sub.0], [eta] (x, [x.sub.0])> + [phi] (x, [x.sub.0]) - [phi] ([x.sub.0], [x.sub.0])/[parallel][eta](x, [x.sub.0]) + [infinity],

whenever [parallel]x[parallel] [right arrow] [infinity].

Then problem (16) is solvable.

Proof. Let

[B.sub.r] = [y [member of] E: [parallel]y[parallel] [less than or equal to] r}. (34)

Consider the following problem, [x.sub.r] [member of] K [intersection] [B.sub.r], such that

<[u.sub.r], [eta] (y, [x.sub.r])> + [phi] (y, [x.sub.r]) - [phi] ([x.sub.r], [x.sub.r]) [greater than or equal to] 0, [for all]y [member of] K such that y [member of] K [intersection] [B.sub.r]. (35)

By Theorem 13, we know that (26) has a solution [x.sub.r] [member of] K [intersection] [B.sub.r]; choose r > [parallel][x.sub.0][parallel] with [x.sub.0] as in the coercivity conditions. Then we have

<[u.sub.r], [eta] ([x.sub.0], xr)> + [phi] ([x.sub.0], [x.sub.r]) - [phi]([x.sub.r], [x.sub.r]) [greater than or equal to] 0. (36)

Moreover

[mathematical expression not reproducible]. (37)

Now if [parallel]x[parallel] = r for all r, we may choose r large enough so that the above inequality and [eta]-coercivity of T with respect to [phi] imply that

<[u.sub.r], [eta] ([x.sub.0], [x.sub.r])> + [phi] ([x.sub.0], [x.sub.r]) - [phi]([x.sub.r], [x.sub.r]) < 0, (38)

which contradicts

<[u.sub.r], [eta] ([x.sub.0], [x.sub.r])> + [phi] ([x.sub.0], [x.sub.r]) - [phi]([x.sub.r], [x.sub.r]) [greater than or equal to] 0. (39)

Hence there exist r such that [parallel][x.sub.r][parallel] < r. For any y [member of] K, we can choose [epsilon] > 0 small enough so that [epsilon] [member of] (0,1) and [u.sub.r] + [epsilon](y - [x.sub.r]) [member of] K [intersection] [B.sub.r]. It follows from (ii) that

[mathematical expression not reproducible]. (40)

By the assumption [eta], of we have

<[u.sub.r], [eta] (y, [x.sub.r])> + [phi] (y, [x.sub.r]) - [phi] ([x.sub.r], [x.sub.r]) [greater than or equal to] 0, [for all]y [member of] K, [for all][u.sub.r] [member of] T ([x.sub.r]). (41)

So [x.sub.r] [member of] K is a solution of (16).

It is easy to see that weakly relaxed [eta]-[alpha] monotonicity implies weakly relaxed [eta]-[alpha] pseudomonotonicity. So Theorems 12, 13, and 14 are deduced to the following corollaries.

Corollary 15. Suppose that [mathematical expression not reproducible] is [eta]-hemicontinuous and weakly relaxed [eta]-[alpha] monotone mapping. Let [phi] : K x K [right arrow] R [union] {+[infinity]} be a proper convex function and [eta] : K x K [right arrow] E be a mapping. Suppose that the following conditions hold:

(i) [eta](x, x) = 0, [for all]x [member of] K.

(ii) For any fixed y [member of] K, u [member of] T(x), the mapping x [right arrow] <u, [eta](x, y)> is convex.

(iii) x [right arrow] [eta](x, *) and x [right arrow] f(x, *) are convex.

Then problems *** and **** are equivalent as follows:

[mathematical expression not reproducible], ***

[mathematical expression not reproducible]. ****

Corollary 16. Let K be a nonempty bounded closed convex subset of a real reflexive Banach space E and [E.sup.*] the dual space of E. Suppose that [mathematical expression not reproducible] is an [eta]-hemicontinuous and weakly relaxed [eta]-[alpha] monotone mapping. Let [phi] : K x K [right arrow] R [union]{+[infinity]} be a proper convex lower semicontinuous function and [eta] : K x K [right arrow] E be a mapping. Assume that

(i) [eta](x, y) + [eta](y, x) = 0, [for all]x [member of] K,

(ii) for any fixed y [member of] K and u [member of] T(x) the mapping x [??] <u, [eta](y, x)> is convex and lower semicontinuous function,

(iii) x [right arrow] [eta](x, *) and x [right arrow] f(x, *) are convex and lower semicontinuous,

(iv) [alpha] : E [right arrow] R is weakly lower semicontinuous; that is, for any net {[x.sub.[beta]]}, [x.sub.[beta]] converges to x in [sigma](E, [E.sup.*]) implying that [alpha](x) [less than or equal to] lim inf [alpha]([x.sub.[beta]]).

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.sup.*] the dual space of E. Suppose that [mathematical expression not reproducible] is an [eta]-hemicontinuous and weakly relaxed [eta]-[alpha] monotone mapping. Let [phi] : K x K [right arrow] R [union]{+[infinity]} be a proper convex lower semicontinuous function and [eta] : K x K [right arrow] E be a mapping. Assume that

(i) [eta](x, y) + [eta](y, x) = 0, [for all]x [member of] K,

(ii) for any fixed y [member of] K and u [member of] T(x) the mapping x [??] <u, [eta](y, x)> is convex and lower semicontinuous function,

(iii) x [right arrow] [eta](x, *) and x [right arrow] f(x, *) are convex and lower semicontinuous,

(iv) [alpha] : E [right arrow] R is weakly lower semicontinuous,

(v) T is q-coercive with respect to <p; that is, there exist [x.sub.0] [member of] K such that

[mathematical expression not reproducible], (42)

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].

http://dx.doi.org/10.1155/2016/4760839

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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Syed Shakaib Irfan and Mohammad Firdosh Khan

College of Engineering, Qassim University, P.O. Box 6677, Buraidah, Al-Qassim 51452, Saudi Arabia

Correspondence should be addressed to Syed Shakaib Irfan; shakaib@qec.edu.sa

Received 1 July 2016; Accepted 18 August 2016

Academic Editor: Julien Salomon
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Title Annotation:Research Article
Author:Irfan, Syed Shakaib; Khan, Mohammad Firdosh
Publication:International Journal of Analysis
Article Type:Report
Date:Jan 1, 2016
Words:4393
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