#### Abstract

We introduce the notion of relaxed (*ρ*-*θ*)-*η*-invariant pseudomonotone mappings, which is weaker than invariant pseudomonotone maps. Using the KKM technique, we establish the existence of solutions for variational-like inequality problems with relaxed (*ρ*-*θ*)-*η*-invariant pseudomonotone mappings in reflexive Banach spaces. We also introduce the concept of (*ρ*-*θ*)-pseudomonotonicity for bifunctions, and we consider some examples to show that (*ρ*-*θ*)-pseudomonotonicity generalizes both monotonicity and strong pseudomonotonicity. The existence of solution for equilibrium problem with (*ρ*-*θ*)-pseudomonotone mappings in reflexive Banach spaces are demonstrated by using the KKM technique.

#### 1. Introduction

Let be a nonempty subset of a real reflexive Banach space , and let be the dual space of . Consider the operator and the bifunction . Then the variational-like inequality problem (in short, VLIP) is to find , such that where denote the pairing between and .

If we take , then (1.1) becomes to find , such that which is classical variational inequality problems (VIPs). These problems have been studied in both finite and infinite dimensional spaces by many authors [1–3]. VIP has numerous applications in optimization, nonlinear analysis, and engineering sciences.

In the study of VLIP and VIP, monotonicity is the most common assumption for the operator . Recently many authors established the existence of solutions for (VIP) and VLIP under generalized monotonicity assumptions, such as quasimonotonicity, relaxed monotonicity, densely pseudomonotonicity, relaxed --monotonicity, and relaxed --pseudomonotonicity (see [1, 3–6] and the references therein). In 2008 [7], Behera et al. defined various concepts of generalized (-)--invariant monotonicities which are proper generalization of generalized invariant monotonicity introduced by Yang et al. [8]. Chen [9] defined semimonotonicity and studied semimonotone scalar variational inequalities problems in Banach spaces. Fang and Huang [3] obtained the existence of solution for VLIP using relaxed --monotone mappings in the reflexive Banach spaces. In [1], Bai et al. extended the results of [3] with relaxed --pseudomonotone mappings and provided the existence of solution of the variational-like inequalities problems in reflexive Banach spaces. Bai et al. [10] studied variational inequalities problems with the setting of densely relaxed -pseudomonotone operators and relaxed -quasimonotone operators, respectively.

Inspired and motivated by [1, 3, 10], we introduce the concept of relaxed ()--invariant pseudomonotone mappings. Using the KKM technique, we establish the existence of solutions for Variational-like inequality problems with relaxed ()--invariant pseudomonotone mappings. We also introduce the notion of ()-pseudomonotonicity for bifunctions, and study some examples to show that -pseudomonotonicity is proper generalization of monotonicity and the strong pseudomonotonicity. The existence of solutions of equilibrium problem with -pseudomonotone mappings in reflexive Banach spaces are demonstrated, by using the KKM technique.

#### 2. Preliminaries

Let be a real reflexive Banach space and be a nonempty subset of , and be the space of all continuous linear functionals on . Consider the functions , and and .

*Definition 2.1. *The operator is said to be relaxed invariant pseudomonotone mapping with respect to and *, *if for any pair of distinct points *, *one has

*Remark 2.2. *(i) If we take then from (2.1) it follows that, here is said to be invariant pseudomonotone, see [8].

(ii) If we take , and , then (2.1) reduces to , and is said to be pseudomonotone map.

(iii) If , , , that is, let (where ). Then (2.1) follows that , and is called relaxed -pseudomonotone mapping [10].

It is obvious that every invariant pseudomonotone mapping is relaxed -invariant pseudomonotone. However, the converse is not true in general, which is illustrated by the following counterexample.

*Example 2.3. *Let and be defined by
Let the functions and be defined by
Now,

Take , then
Therefore is relaxed -invariant pseudomonotone mapping with respect to and . But is not invariant pseudomonotone mapping with respect to the same . In fact, if we take and . Therefore we have
However,

*Definition 2.4. *The operator is said to be relaxed *-*invariant quasimonotone mapping with respect to and *, *if for any pair of distinct points *, *one has

Next, we will show that relaxed -invariant quasimonotonicity and relaxed (-)--invariant pseudomonotonicity coincide under some conditions. For this we need the following -hemicontinuity definition.

*Definition 2.5 (see [3]). *Let and *. * is said to be -hemicontinuous if for any fixed *, *the mapping defined by is continuous at *. *

Lemma 2.6. * Let be an -hemicontinuous and relaxed (-)--invariant quasimonotone on . Assume that the mapping is concave and is hemicontinuous in the second argument. Then for every with one has either or .*

*Proof. *Suppose there exists some such that . Then we have to prove that .

Let . Then
Since is relaxed (-)--invariant quasimonotone on , we have
Since is -hemicontinuous and is hemicontinuous in the first variable and letting , we have

Denote , and

*Definition 2.7. *A point is said to be a -positive point of on if *, *either or there exists such that .

Let denotes the set of all -positive points of on . Now we give a characterization of relaxed (-)--invariant pseudomonotone.

Lemma 2.8. * Let be -hemicontinuous and relaxed (-)--invariant quasimonotone on with . Assume that the mapping is concave and is hemicontinuous. Then is relaxed (-)--invariant pseudomonotone on .*

*Proof. *Let with . Therefore by the previous Lemma, we have either
Now we will show that the second inequality in (2.12) is impossible. In fact, since and , there exists such that , which shows that the second inequality in (2.12) is impossible. Therefore,
hence is relaxed (-)--invariant pseudomonotone on .

*Remark 2.9. * Lemmas 2.6 and 2.8 generalize Bai et al. [10] results [Lemma 2.1 and Proposition 2.1] from the case of relaxed -quasimonotone operators to relaxed (-)--invariant quasimonotone operators.

*Definition 2.10. *Let be a set-valued mapping. Then is said to be KKM mapping if for any of one has , where denotes the convex hull of *. *

Lemma 2.11 (see [11]). * 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 .*

Theorem 2.12 (see [12]). *Bounded, closed, convex subset of a reflexive Banach space is weakly compact.*

#### 3. VLIP with Relaxed (-)--Invariant Pseudomonotonicity

In this section, we establish the existence of the solution for VLIP, using relaxed (-)--invariant pseudomonotone mappings in reflexive Banach spaces.

Theorem 3.1. * is -hemicontinuous and relaxed (-)--invariant pseudomonotone mapping. Let the following hold: *(i)*, and ;*(ii)* is convex in second argument and concave in first argument;*(iii)*for a fixed , the mapping is convex.**Then the following problems (a) and (b) are equivalent: *(a)*find , , ;*(b)*find , , .*

* Proof. *Assume that is a solution of (a). Therefore, (b) follows from the definition of relaxed (-)--invariant pseudomonotonicity of .

Conversely, suppose that there exists an satisfying (b), that is,
Choose any point and consider , then .*Case I. *When .

Therefore from (3.1) we have
Now,
From (3.2) and (3.3) we have
Since is -hemicontinuous in the first argument and taking we get
*Case II. *When , let .

From (3.1) we have
From (ii), (iii), (3.3), and (3.6) we get,
Since is -hemicontinuous and taking we have
*Case III. *When , let .

From (3.1) we have

From (i), (ii), (iii),(3.3), and (3.9) we get,
Since is -hemicontinuous and taking we have

Theorem 3.2. * Let be a nonempty bounded closed convex subset of a real reflexive Banach space . is -hemicontinuous and relaxed (-)--invariant pseudomonotone mapping. Let the following hold: *(i)*, and ;*(ii)* is convex in second argument and concave in first argument, and lower semicontinuous in the first argument;*(iii)*for a fixed , the mapping is convex and lower semicontinuous.** Then the problem (1.1) has a solution.*

*Proof. *Consider the set valued mappings and such that
It is easy to see that solves the VLIP if and only if . Thus it suffices to prove . To prove this, first we will show that is a KKM mapping.

If possible let not be a KKM mapping. Then there exists such that, that means there exists a , where , , , but .

Hence, ; for .

From (i) and (iii) it follows that
which is a contradiction. Hence is a KKM mapping.

From the relaxed -invariant pseudomonotonicity of it follows that . Therefore is also a KKM mapping.

Since is closed bounded and convex, it is weakly compact. From the assumptions, we know that is weakly closed for all . In fact, because and are lower semicontinuous. Therefore, is weakly compact in . Therefore from Lemma 2.11 and Theorem 3.1 it follows that .So there exists such that , that is, (1.1) has a solution.

Theorem 3.3. * Let be a nonempty unbounded closed convex subset of a real reflexive Banach space . is -hemicontinuous and relaxed (-)--invariant pseudomonotone mapping. Let the following hold: *(i)*, and ;*(ii)* is convex in second argument and concave in first argument, and lower semicontinuous in the first argument;*(iii)*for a fixed , the mapping is convex and lower semicontinuous;*(iv)* is weakly -coercive, that is, there exits such that , whenever and .** Then (1.1) is solvable.*

*Proof. *For , assume .

Consider the problem: find such that

By Theorem 3.2 we know that problem (3.14) has at least one solution .

Choose with as in condition (iv). Then and
From (i) we get,
If for all , we may choose large enough so that by the assumption (iv) and (3.16) imply that , which contradicts (3.15).

Therefore there exists such that . For any , we can choose small enough such that .

From (3.14) it follows that
Hence .

#### 4. Equilibrium Problem with (-)-Pseudomonotone Mappings

Let be a nonempty subset of a real reflexive Banach space , and consider the bifunction . Then the equilibrium problem (in short, EP) is to find , such that Equilibrium problem was first introduced and studied by Blum and Oettli [2] in 1994. EP has many applications in nonlinear analysis, optimization, and game theory. The EP contains many problems as particular cases for examples, mathematical programming problems, complementary problems, Nash equilibrium problems in noncooperative games, variational inequality problems, fixed point problems, and minimax inequality problems.

Next we describe a number of particular cases of EP to explain our interest in EP, which have been discussed in [2]. (i) Optimization problem: let , and consider minimization problem(M) find . If we set . Then problems EP and (M) are equivalent.(ii) Variational inequality problem: if we define where is a given mapping, where denotes the space of all continuous linear maps on . Then EP collapses into the classical VIP which states the following,(VIP) find such that , with .(iii) Fixed point problem: let be a Hilbert space, and is a nonempty closed convex subset of . Let be a given mapping. Then the fixed point problem is to(FPP) find such that . Set . Then solves EP if and only if is a solution of FPP.

The purpose of this section is to establish the existence of solution for equilibrium problems with -pseudomonotone mappings in the reflexive Banach spaces. We first introduce the notion of (-)-monotone mappings and (-)-pseudomonotone mappings. We also provide some examples to justify that (-)-monotone mapping generalizes weakly monotone maps, and (-)-pseudomonotone mapping generalizes pseudomonotone, weakly pseudomonotone maps.

Let be a nonempty subset of a real reflexive Banach space . Consider the function and and .

*Definition 4.1. *The function is said to be monotone with respect to if*, *for all *, *one has

When

(i) and , is weakly monotone;(ii), is monotone;(iii) and , is strongly monotone. We now give an example to show that (-) monotonicity is a generalization of both monotonicity and weakly monotonicity.

*Example 4.2. *Let . Let the functions and be defined by
Then

Therefore is (-)-monotone with respect to .

There exists no constant such that . As if we assume and to be such that their difference is very small, then right-hand side of the inequality tends to zero and left-hand side is always greater than 2. Hence is not weakly monotone. Again since is positive valued, is not monotone.

*Definition 4.3. *The function is said to be pseudomonotone with respect to if for any pair of distinct points *, *one has

Every (-)-monotone mapping is a -pseudomonotone with respect to the same and . However, the converse is not true in general, which follows from the following counterexample.

*Example 4.4. * Let the functions and be defined by
Take . We have to show

Now,
Hence is (-)-pseudomonotone mapping with respect to . But is not (-)-monotone mapping with respect to the same . In fact,

Note that in the above example, is neither a monotone nor pseudomonotone mapping.

*Definition 4.5. *The function is said to be (*- *)*-*quasimonotone with respect to if for any pair of distinct points *, *one has

Next, we will show that (-)-quasimonotonicity and (-)-pseudomonotonicity are equivalent under certain conditions.

Lemma 4.6. * Let be hemicontinuous and (-)-quasimonotone on . Assume that is concave in the second argument and is hemicontinuous in the second argument. Then for every with one has either or .*

*Proof. *Suppose there exists some such that . Then we have to prove that .

Let . Then
Since is relaxed (-)-quasimonotone on , it implies that
Now letting , we have
This completes the proof.

Theorem 4.7. * Let be a nonempty convex subset of a real reflexive Banach space . Suppose is (-)-pseudomonotone with respect to and is hemicontinuous in the first argument with the following conditions: *(i)*;*(ii)*for fixed , the mapping is convex;*(iii)*;*(iv)* is convex in first argument and concave in the second argument.** Then is a solution of (4.1) if and only if
*

*Proof. *Assume that is a solution of (4.1) that is, . Therefore from the definition of (-) pseudomonotonicity of it follows that
Conversely, suppose satisfying (4.14), that is,
Choose any point and consider then .*Case I. *When .

Therefore from (4.16) we have
Now conditions (i) and (ii) imply that,
From (4.17) and (4.18) we have
Since is hemicontinuous in the first argument and taking , it implies that
Hence is a solution of (4.1). *Case II. *When , let .

From (4.16) we have
Now using (4.18), (4.21), and (iv) it follows that
Since is hemicontinuous in the first argument and letting , we get
*Case III. *When , let .

From (4.16), (4.18), and (iv) we have

Since is hemicontinuous in the first argument and taking , we get

Theorem 4.8. * Let be a nonempty bounded convex subset of a real reflexive Banach space . Suppose is (-)-pseudomonotone with respect to and is hemicontinuous in the first argument with the following conditions: *(i)*;*(ii)*for fixed , the mapping is convex and lower semicontinuous;*(iii)*;*(iv)* is convex in first argument and concave in the second argument, and lower semicontinuous in the first argument.**Then the problem (4.1) has a solution.*

*Proof. *Consider the two set valued mappings and such that

It is easy to see that solves the equilibrium problem (4.1) if and only if . First to show that is a KKM mapping. If possible let not be a KKM mapping. Then there exists such that , that means there exists a , where , , , but . Hence, ; for .

From (i) and (ii) it follows that
which is a contradiction. Hence is a KKM mapping.

From the (-)-pseudomonotonicity of it follows that . Therefore is also a KKM mapping.

Since is closed bounded and convex, it is weakly compact. From the assumptions, we know that is weakly closed for all . In fact, because and are lower semicontinuous. Therefore, is weakly compact in

Therefore from Lemma 2.11 and Theorem 4.7 it follows that .

So there exists such that , that is, (4.1) has a solution.

Theorem 4.9. * Let be a nonempty unbounded closed convex subset of a real reflexive Banach space . Suppose is (-)-pseudomonotone with respect to and is hemicontinuous in the first argument and satisfy the following assumptions: *(i)*;*(ii)*for fixed , the mapping is convex and lower semicontinuous;*(iii)*;*(iv)* is convex in first argument and concave in the second argument, and lower semicontinuous in the first argument.*(v)* is weakly coercive, that is, there exists such that**
Then (4.1) has a solution.*

*Proof. *For , assume .

Consider the problem: find such that
By Theorem 4.8 we know that the problem (4.29) has at least one solution .

Choose with as in condition (v). Then and
If for all , we may choose large enough so that by the assumption (v) imply that , which contradicts (4.30).

Therefore there exists such that . For any , we can choose small enough such that .

From (4.29) it follows that
Hence .

#### 5. Conclusions

The present work has been aimed to theoretically study the existence of solutions for variational-like inequality problems under a new concept relaxed (-)--invariant pseudomonotone maps in reflexive Banach spaces. We have also obtained existence of solutions of equilibrium problems with (-)-pseudomonotone mappings. More research and development activities is therefore needed on generalized monotonicity to demonstrate the equilibrium problem and variational inequality problem.

#### Acknowledgments

The authors are very much thankful to the editor and referees for their suggestions which helped to improve the presentation of this paper. The work of the authors was partially supported by CSIR, New Delhi, Grant 25 (0163)/08/EMR-II.