#### Abstract

We introduce two new concepts of weakly relaxed monotone mappings and weakly relaxed semimonotone mappings. Using the KKM technique, the existence of solutions for variational-like problems with weakly relaxed monotone mapping in reflexive Banach spaces is established. Also, we obtain the existence of solution for variational-like problems with weakly relaxed semimonotone mappings in arbitrary Banach spaces by using the Kakutani-Fan-Glicksberg fixed-point theorem.

#### 1. Introduction

The variational inequality theory provides us with a simple, natural, unified, and elegant framework to study a wide class of linear and nonlinear problems arising in many fields, such as mechanics, engineering sciences, elasticity, optimization, control, programming, economics, transportation, oceanography, and regional. Because of their wide applicability, various extensions and generalizations of the classical variational inequality problem have been proposed and studied in recent years. Variational-like inequalities problems is one of cornerstone in this field. Some special case of generalized variational inequalities and variational-like inequalities have been studied by several authors including Bai et al. [1], Chang et al. [2], dos Santos et al. [3], Xiao and Huang [4], Zhao and Xia [5, 6], and references therein.

It is well known that the monotonicity and generalized monotonicity play an important role of the study in variational inequality theory. In recent years, a substantial number of papers on existence results for solving variational inequality problems and variational-like inequality problems based on different generalization of monotonicity such as pseudomonotonicity, quasimonotonicity, relaxed monotonicity, semimonotonicity, and -monotonicity (see [7–15]) appeared. In [16], Fang and Huang introduced a new concept of relaxed monotonicity and obtained the existence of solutions for variational-like inequalities with relaxed monotone mappings in reflexive Banach spaces. Recently, Sintunavarat [17] established the existence of solution of mixed equilibrium problem with the weakly relaxed -monotone bi-function in Banach spaces.

Inspired and motivated by the work of Fang and Huang [16] and Sintunavarat [17], in this paper we introduce the two new concepts of weakly relaxed monotone mappings and weakly relaxed semimonotone mappings as well as two classes of variational-like inequalities with weakly relaxed monotone mappings and weakly relaxed semimonotone mappings. By using the KKM technique, we study some existence of solutions for variational-like inequalities with weakly relaxed monotone mappings in reflexive Banach spaces. We also obtain the solvability of variational-like inequalities with weakly relaxed semimonotone mappings in arbitrary Banach spaces by using the Kakutani-Fan-Glicksberg fixed-point theorem. Our results in this paper extend and improve the results of Fang and Huang [16] and many results in the literature.

#### 2. Preliminaries

In this paper, unless otherwise specified, is a nonempty closed convex subset of a real reflexive Banach space with dual space . The following basic knowledge will be useful in our work.

*Definition 1 (see [18]). *Let and be two mappings. A mapping is said to be -hemicontinuous if, for any fixed , the mapping defined by
is continuous at .

*Definition 2 (see [16]). *Let and be two mappings. A mapping is said to be -coercive with respect to a proper function if there exists such that
whenever is large enough.

*Remark 3. *If , where is the indicator function of , then Definition 2 coincides with the definition of -coercivity in the sense of Yang and Chen [15].

*Definition 4 (see [16]). *A mapping is said to be relaxed monotone if there exist a function and with for all and such that
where is a constant.

*Remark 5. * (1) If for all , then (3) reduces to
and is said to be -monotone.

(2) If for all and , where and , then (3) becomes
and is said to be -monotone (see in [14, 19, 20]).

(3) Every monotone mapping is relaxed monotone with for all and .

*Definition 6. *Let be a set-valued mapping. Then, is said to be KKM mapping if, for any finite subset of , we have , where denotes the convex hull of .

*Remark 7. *Let . If is KKM mapping and for all , then is also KKM mapping.

Lemma 8 (see [21]). *Let be a nonempty subset of a Hausdorff topological vector space , and let be a KKM mapping. If is closed in for all and compact for some , then .*

#### 3. Variational-Like Inequalities Problems with Weakly Relaxed Monotone Mapping

In this section, we introduce the new class of mapping which generalizes several classes. Using KKM technique, we study and prove the existence of solutions for variational-like inequalities problems with mapping in such class in Banach spaces.

*Definition 9. *A mapping is said to be weakly relaxed monotone if there exist a function and with
for all and such that

*Remark 10. *We obtain that the relaxed monotonicity implies weakly relaxed monotonicity. So, class of relaxed monotone mapping is subclass of weakly relaxed monotone mapping class. Also, we get that classes of relaxed monotone mapping, -monotone mapping, and monotone mapping are subclass of weakly relaxed monotone mapping class.

Theorem 11. *Let be an -hemicontinuous and weakly relaxed monotone and let be a proper convex function. Suppose that the following conditions hold: *(a)* for all ; *(b)*for any fixed , the mapping is convex.**Then, the following problems (9) and (10) are equivalent:
*

*Proof. *Suppose that (9) has a solution. So, there exists such that
Since is weakly relaxed monotone, we have
and then
Therefore, is a solution of problem (10).

Conversely, suppose that is a solution of problem (10) and is any point in with . For , we let . From (10), we get that . It follows from being convex that . From (10), we have
and thus
The convexity of implies that
From (15) and (16), we get
By the assumption (b), we have
It follows from (17) and (18) that
that is
for all . Taking in the previous inequality and using -hemicontinuity of , we get
From (6), we get is indeterminate form. Using L’ Hôpital’s rule, we obtain that
By property (7) of weakly relaxed monotone of , we have
for all with . In case of , the relation
is trivial. Therefore, is also a solution of problem (9).

Theorem 12. *Let be a nonempty bounded closed convex subset of a real reflexive Banach space , and let be the dual space of . Suppose that is an -hemicontinuous and weakly relaxed monotone mapping and is a proper convex lower semicontinuous function. If the following conditions hold: *(a)* for all ,*(b)* for any fixed , the mapping is convex and lower semicontinuous function, *(c)* is weakly lower semicontinuous; that is, for any net , we have that converges to in implies that ,**
then the problem (9) is solvable.*

*Proof. *Consider the set valued mapping defined by
for all .

It is easy to see that solves the problem (9); that is,
if and only if . Thus, it is sufficient to prove that .

Next, we show that is a KKM mapping. Assume the contrary, then there exists such that . This implies that there exists such that , where , and , but .

From (25), we have
By (b) and (25), we obtain that
which is a contradiction. Therefore, is a KKM mapping.

Now, we define another set-valued mapping by
for all .

Next, we will claim that for all . For each , let ; then,
From the weakly relaxed monotonicity of , we get
This implies that and hence for all . So, is also a KKM mapping.

By assumption, and are two convex lower-semicontinuous functions. Then it is easy to see that they are both weakly lower semicontinuous. From the definition of and the weakly lower semicontinuity of , we get that is weakly closed for all . Since is closed bounded and convex, it also is weakly compact, and then is weakly compact in for each . From Lemma 8 and Theorem 11, we obtain that
So, there exists , such that
that is, problem (9) has a solution. This completes the proof.

We know that the relaxed monotonicity implies the weakly relaxed monotonicity. Therefore, Theorem 12 can be deduced to the following corollary.

Corollary 13. *Let be a nonempty bounded closed convex subset of a real reflexive Banach space , and let be the dual space of . Suppose that is an -hemicontinuous and relaxed monotone mapping and is a proper convex lower semicontinuous function. If the following conditions hold: *(a)* for all , *(b)* for any fixed , the mapping is convex and lower semicontinuous function, *(c)* is weakly lower semicontinuous, **
then the problem (9) is solvable.*

*Remark 14. *Since the monotonicity, -monotonicity, and relaxed -monotonicity imply relaxed monotonicity, we can be applying Corollary 13 to the other problems for the mapping satisfies these property.

Next, we study and prove that result for the case of is unbounded set.

Theorem 15. *Let be a nonempty unbounded closed convex subset of a real reflexive Banach space , and let be the dual space of . Suppose that is an -hemicontinuous and weakly relaxed monotone mapping and is a proper convex lower semicontinuous function. If the following conditions hold: *(a)* for all , *(b)* for any fixed , the mapping is convex and lower semicontinuous function, *(c)* is weakly lower semicontinuous, *(d)* is weakly -coercive with respect to ; that is, there exists such that
* *whenever and large enough, **
then the problem (9) is solvable.*

*Proof. *For , define . Consider the following problem:

Since is bounded, by Theorem 12, we get that the problem (35) has at least one solution .

For in the weakly -coercivity condition (d), we fixed . From (35), we can find that such that
Since , we have . If for all , we may choose large enough so that the weakly -coercivity condition (d) implies that
which contradicts (36). Therefore, there exists such that .

For each , we can choose such that . From (35) and the fact that , we have
By the above inequality and the convexity of and mapping in (b), we get
for all . This implies that
for all . Therefore, is a solution of the problem (9). This completes the proof.

It is easy to see that the relaxed monotonicity implies the weakly relaxed monotonicity. So, Theorem 15 can be deduced to the following corollary.

Corollary 16. *Let be a nonempty unbounded closed convex subset of a real reflexive Banach space , and let be the dual space of . Suppose that is an -hemicontinuous and relaxed monotone mapping and is a proper convex lower semicontinuous function. If the following conditions hold: *(a)* for all , *(b)* for any fixed , the mapping is convex and lower semicontinuous function, *(c)* is weakly lower semicontinuous, *(d)* is weakly -coercive with respect to , **
then the problem (9) is solvable.*

Since the -coercivity with respect to implies that the weakly -coercivity with respect to , we can utilize Corollary 16 to the result of Fang and Huang [16].

Corollary 17 (Theorem 2.3 [16]). *Let be a nonempty unbounded closed convex subset of a real reflexive Banach space , and let be the dual space of . Suppose that is an -hemicontinuous and relaxed monotone mapping and is a proper convex lower semicontinuous function. If the following conditions hold: *(a)* for all , *(b)* for any fixed , the mapping is convex and lower semicontinuous function, *(c)* is weakly lower semicontinuous, *(d)* is -coercive with respect to , **
then the problem (9) is solvable.*

*Remark 18. *Theorems 11, 12, and 15 are improvement of the results of Fang and Huang [16] from the corresponding results of variational-like inequality problems for relaxed monotone mapping to weakly relaxed monotone mapping. Also, these results are extension of the known results of Hartman and Stampacchia [22] and corresponding results in [14, 19, 23].

#### 4. Variational-Like Inequalities Problems with Weakly Relaxed Semimonotone Mapping

Through this section, let be an arbitrary Banach space with its dual space , let denote the dual space of , and let be a nonempty closed convex subset of .

*Definition 19. *A mapping is said to be weakly relaxed semimonotone if the following conditions hold: (a) for each fixed , is weakly relaxed monotone; that is, there exist mappings and such that for all and and
(b) for each fixed , is completely continuous; that is, for any net with converges to in , we have that converges to in the norm topology of .

*Remark 20. *We obtain that relaxed semimonotonicity due to Fang and Huang [16] implies weakly relaxed semimonotonicity. Therefore, the class of relaxed semimonotone mappings is subclass of the class of weakly relaxed semimonotone mappings.

Let and be two mappings and is a proper convex lower-semicontinuous function. We consider the following problem:

Theorem 21. *Let be a real Banach space and let be a nonempty bounded closed convex set. Suppose that is a weakly relaxed semimonotone mapping and let be a proper convex lower semicontinuous function. If the following conditions hold: *(a)* for all , *(b)* for any fixed , the mapping is convex and lower semicontinuous function, *(c)* for each , is finite-dimensional continuous; that is, for any finite-dimensional subspace , is continuous, *(d)* is convex lower semicontinuous, **
then the problem (43) is solvable.*

*Proof. *Let be a finite-dimensional subspace with . For each , consider the following problem:
It follows from being bounded closed and convex and being continuous on and weakly relaxed monotone for each fixed that from Theorem 12, we obtain that problem (44) has a solution .

Next, define a set-valued mapping as follows:
By Theorem 11, we get that, for each ,

It is known that every convex lower-semicontinuous function in Banach spaces is weakly lower semicontinuous. Therefore, condition (b) and the proper convex lower semicontinuity of and implies that has nonempty bounded closed and convex values. By the complete continuity of , we have that is upper semicontinuous. Using the Kakutani-Fan-Glicksberg fixed-point theorem, we obtain that has a fixed-point and thus

Now, define
From (47), using Theorem 11, we obtain that is nonempty and bounded. Here, we denote by the -closure of in and thus is -compact in .

It is known that, for any , , we have . Therefore, has the finite intersection property; that is,

Next, we show that, for ,
Indeed, for , let be such that and . Then, there exists a net in such that converges to in . From the definition of , we have
By the complete continuity of and the proper convex lower semicontinuity of and , we get
Again, using Theorem 11, we conclude that
This implies that is a solution of the problem (43). This completes the proof.

Theorem 22. *Let be a real Banach space and let be a nonempty unbounded closed convex set. Suppose that is a weakly relaxed semimonotone mapping and let be a proper convex lower semicontinuous function. If the following conditions hold: *(a)* for all , *(b)* for any fixed , the mapping is convex and lower semicontinuous function, *(c)* for each , is finite-dimensional continuous, *(d)* is convex lower semicontinuous, *(e)* there exists a point such that , **then the problem (43) is solvable.*

*Proof. *For , we denote by the closed ball with radius and center at in . By Theorem 21, the problem
has a solution .

Let be large enough such that . Therefore,
By condition (e), we get that is bounded. So, we may suppose that converges to in as . It follows from Theorem 11 that
Letting , we have
Again from Theorem 11, we get
This show that is a solution of the problem (43). This completes the proof.

*Remark 23. *Theorems 21 and 22 extend and improve Theorems 3.1 and 3.2 of Fang and Huang [16] and Theorems 2.1 to 2.6 of Chen [7].

#### Acknowledgments

Marwan Amin Kutbi gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research. Wutiphol Sintunavarat gratefully acknowledges Thammasat University for the support of this research.