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Relaxed ηα quasimonotone and application to the generalized variationallike inequality problem
Journal of Inequalities and Applications volume 2013, Article number: 488 (2013)
Abstract
In this paper, some new mappings called relaxed ηα quasimonotone and a relaxed ηα properly quasimonotone operator are first introduced. The relationships between them are obtained. After this, the variationallike inequality problem and the relaxed Minty variationallike inequality problem are discussed by use of the proposed generalized monotone operators. Furthermore, we give the gap function of the two variationallike inequalities and two kinds of optimization problems. Finally, we point out that the two optimization problems are equivalent under some conditions.
MSC:90C26, 90C30.
1 Introduction
As we know, variational inequality theory plays an important role in many fields, such as optimal control, mechanics, economics, transportation equilibrium, engineering sciences. It is well known that the role of generalized monotonicity of the operator in the variational inequality problem corresponds to the role of generalized convexity of the objective function in the mathematical programming problem. From this, the importance of the study of generalized monotonicity is evident.
In recent years, a number of authors have proposed many important generalizations of monotonicity. In [1], Karamardian and Schaible gave seven kinds of generalized monotone mappings which, in the case of gradient mappings, were related to a generalized convex function. In [2], Fang and Huang introduced a new concept of relaxed ηα monotonicity and obtained the existence for variationallike inequalities with relaxed ηα monotone mappings in a reflexive Banach space. Bai et al. [3] introduced relaxed ηα pseudomonotone and established the existence for variationallike inequalities with relaxed ηα pseudomonotone mappings in a reflexive Banach space. In [4], a series of sufficient and necessary conditions were given that related the generalized invexity of the function θ with the generalized invex monotonicity of its gradient function ∇θ. In [5], Yang introduced a gap function for many generalized variational inequalities. The relationships between the generalized convexity of functions and generalized monotone operators also have been investigated by many authors (see [6–9]). Ansari et al. [10] considered different kinds of generalized vector variationallike inequality problems.
Based on the results in [2, 3] and [5], relaxed ηα quasimonotone, relaxed ηα properly quasimonotone are proposed in this paper. With a more weakly monotone assumption, the existence for variationallike inequalities with a relaxed ηα quasimonotone mapping in a reflexive Banach space is discussed. After this, by the gap function, the equivalence between two kinds of optimization problems is obtained.
The paper is organized as follows. In Section 2, some concepts, basic assumptions and preliminary results are presented. In Section 3, the existence for variationallike inequalities with relaxed ηα quasimonotone mappings in a reflexive Banach space is established. In Section 4, the gap function of the relaxed Minty variationallike inequality is given, and the optimization problem is studied with it.
2 Definitions and preliminary
In this paper, let X be a reflexive Banach space, and dual space {X}^{\ast}, K is the nonempty subset of X. Now we recall some basic definitions as follows.
Definition 2.1 [2]
A mapping T:K\to {X}^{\ast} is said to be relaxed ηα monotone if there exist a mapping \eta :K\times K\to X and a function \alpha :X\to R with \alpha (tz)={t}^{p}\alpha (z) for all t>0 and z\in X such that
where p>1 is a constant.
Definition 2.2 [3]
A mapping T:K\to {X}^{\ast} is said to be relaxed ηα pseudomonotone if there exist a mapping \eta :K\times K\to X and a function \alpha :X\to R with \alpha (tz)={t}^{p}\alpha (z) for all t>0 and z\in X such that, for any x,y\in K, we have
where p>1 is a constant.
Based on this, we give the definition of relaxed ηα quasimonotone operator.
Definition 2.3 A mapping T:K\to {X}^{\ast} is said to be relaxed ηα quasimonotone if there exist a mapping \eta :K\times K\to X and a function \alpha :X\to R with \alpha (tz)={t}^{p}\alpha (z) for all t>0 and z\in X such that, for any x,y\in K, we have
where p>1 is a constant.
Special cases:

(1)
If \eta (x,y)=xy, \mathrm{\forall}x,y\in K, and \alpha (xy)\equiv 0, then (1) implies
\u3008Ty,xy\u3009>0\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}\u3008Tx,xy\u3009\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in K
and T is quasimonotone.

(2)
If \alpha (xy)\equiv 0, then (1) implies
\u3008Ty,\eta (x,y)\u3009>0\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}\u3008Tx,\eta (x,y)\u3009\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in K
and T is invex quasimonotone.
Remark 2.1 From the above definition, we can see that the following relationships hold. Relaxed ηα monotone ⇒ relaxed ηα pseudomonotone ⇒ relaxed ηα quasimonotone.
Example 2.1 Let K\in (\mathrm{\infty},+\mathrm{\infty}), T(x)={x}^{2}, x\in R,
It is clear that the mapping T is a relaxed ηα quasimonotone operator. But it is not a relaxed ηα pseudomonotone operator. If we let y=0, x=1, \u3008Ty,\eta (x,y)\u3009=0, but \u3008Tx,\eta (x,y)\u3009<\alpha (xy), which is a contradiction.
Bai, in reference [3], proved that T was relaxed ηα pseudomonotone, but not a relaxed ηα monotone mapping.
Definition 2.4 A mapping T:K\to {X}^{\ast} is said to be relaxed ηα properly quasimonotone if there exist a mapping \eta :K\times K\to X and a function \alpha :X\to R with \alpha (tz)={t}^{p}\alpha (z) for all t>0 and z\in X such that, for any {x}_{1},\dots ,{x}_{n}\in K and y\in cov\{{x}_{1},\dots ,{x}_{n}\}, there exists i\in \{1,2,\dots ,n\} such that
Remark 2.2 When \eta ({x}_{i},y)={x}_{i}y, \alpha ({x}_{i}y)\equiv 0, T is a properly quasimonotone mapping.
Condition A [4]

(1)
The function \eta :X\times X\to X is skew, i.e., \eta (x,y)+\eta (y,x)=0, \mathrm{\forall}x,y\in X.

(2)
The η is an affine function in the first argument.
Remark 2.3

(i)
When η is a skew function, we have \eta (x,x)=0;

(ii)
When η is a skew function and an affine function in the first argument, η is an affine function in the second argument;

(iii)
There exists a function η, which satisfies Condition A, for example, \eta (x,y)=3(xy), \mathrm{\forall}x,y\in X.
Theorem 2.1 Suppose that the mapping T:K\to {X}^{\ast} is relaxed ηα properly quasimonotone, and η satisfies Condition A, then T is relaxed ηα quasimonotone.
Proof Let x,y\in K, and
Let {y}_{t}=y+t(xy), t\in (0,1). T is relaxed ηα properly quasimonotone, we have
or
From Condition A and (5), we obtain
Let t\to 0, and we imply that
With Condition A and (3),
It is a contradiction with (7), so (5) is not true, and (4) is correct. In (4), let t\to 0, and we have
So, T is relaxed ηα quasimonotone. The proof is completed. □
3 Variationallike inequality problem (VLIP)
In this section, we discuss the following variational inequality problem.

(i)
Variationallike inequality problem (VLIP):
Find x\in K such that \u3008Tx,\eta (x,y)\u3009\ge 0, \mathrm{\forall}y\in K, we denote by S(T,K) the set of solutions.

(ii)
Minty variationallike inequality problem (MVLIP):
Find x\in K such that \u3008Ty,\eta (x,y)\u3009\ge 0, \mathrm{\forall}y\in K, we denote by SM(T,K) the set of solutions.

(iii)
Local Minty variationallike inequality problem (LMXLIP):
Find x\in K such that \u3008Ty,\eta (x,y)\u3009\ge 0, there exists a neighborhood U of x, \mathrm{\forall}y\in K\cap U, we denote by \mathit{LM}(T,K) the set of solutions.
Now we give a new generalized variationallike inequality.

(iv)
Relaxed Minty variationallike inequality problem (RMXLIP):
Find x\in K such that \u3008Ty,\eta (x,y)\u3009\ge \alpha (yx), \mathrm{\forall}y\in K, we denote by \mathit{RM}(T,K) the set of solutions.
Definition 3.1 A function F:K\to {2}^{X} is called a KKMfunction if for every finite subset \{{x}_{1},{x}_{2},\dots ,{x}_{n}\} of X, the convex hull conv(\{{x}_{1},{x}_{2},\dots ,{x}_{n}\})\subset {\bigcap}_{i=1}^{n}F({x}_{i}).
Lemma 3.1 [3]
Let a nonempty subset K\subset X and a KKMfunction F:K\to {2}^{X}, if F(x) is a compact set, \mathrm{\forall}x\in K, then {\bigcap}_{x\in K}F(x)\ne \mathrm{\varnothing}.
Theorem 3.1 Let K be a nonempty convex subset of a real reflexive Banach space X, if T:K\to {X}_{\ast} is hemicontinuous and η is a weak^{∗} compactvalued, which satisfies Condition A, then \mathit{RM}(T,K)\subset S(T,K).
Proof Suppose that x\in \mathit{RM}(T,K), but x\notin S(T,K), so there exists a point y\in K such that \u3008Tx,\eta (x,y)\u3009<0. Since Tx is weak^{∗} compactvalued, there exists \epsilon >0 such that
Let set V=\{{x}^{\ast}\in {X}^{\ast}:\u3008{x}^{\ast},\eta (y,x)\u3009<\epsilon \}, v is a weak^{∗} compact set, and Tx\in V.
Let {x}_{t}=ty+(1t)x, t\in (0,1], T is hemicontinuous, there exists \delta >0 such that T({x}_{t})\in V, \mathrm{\forall}t\in (0,\delta ).
That is,
On the other hand, x\in \mathit{RM}(T,K), so for all t\in (0,\delta ), we have
That is,
When t\to 0, (8) and (9) contradict, so \mathit{RM}(T,K\subset S(T,K)). □
Theorem 3.2 Let K be a nonempty convex subset of the real reflexive Banach space X. Let T:K\to {X}^{\ast} be relaxed ηα quasimonotone, η satisfy Condition A, and suppose that \u3008Tx,\eta (x,y)\u3009, \alpha (xy) is the continuity of y. Then one of the following assertions holds:

(i)
T is relaxed ηα properly quasimonotone;

(ii)
\mathit{LM}(T,K)\ne \mathrm{\varnothing}.
Proof Suppose that T is not relaxed ηα properly quasimonotone, then there exist {x}_{1},{x}_{2},\dots ,{x}_{n}\in K and y\in conv\{{x}_{1},{x}_{2},\dots ,{x}_{n}\} such that
By the continuity of y, there exists a neighborhood U of y. For any z\in U\cap K, one has
By relaxed ηα quasimonotone,
From y\in conv\{{x}_{1},{x}_{2},\dots ,{x}_{n}\} and Condition A, it follows easily that
By Condition A, we have
That is, y\in \mathit{LM}(T,K), \mathit{LM}(T,K)\ne \mathrm{\varnothing}.
When (ii) is not true, we could get (i) similarly. □
Theorem 3.3 Let K be a nonempty, compact, and convex subset of a real reflexive Banach space X. Let T:K\to {X}^{\ast} be relaxed ηα quasimonotone and upper hemicontinuous with weakly* compact values, and let η satisfy Condition A. Then S(T,K)\ne \mathrm{\varnothing}.
Proof According to Theorem 3.2, we have either \mathit{LM}(T,K)\ne \mathrm{\varnothing} or T is relaxed ηα properly quasimonotone. If \mathit{LM}(T,K)\ne \mathrm{\varnothing}, similarly to reference [6], we can get S(T,K)\ne \mathrm{\varnothing}.
On the other hand, if T is relaxed ηα properly quasimonotone, define the setvalued mapping G:K\to {2}^{{X}^{\ast}} by
For every {x}_{1},\dots ,{x}_{n}\in K, and y\in conv\{{x}_{1},\dots ,{x}_{n}\}, relaxed ηα properly quasimonotone implies that y\in {\bigcup}_{i=1}^{n}G({x}_{i}). In addition, for each x\in K, G(x) is closed, K is compact, so for each x\in K, G(x) is compact. By Lemma 3.1, we obtain {\bigcap}_{x\in K}G(x)\ne \mathrm{\varnothing}, which implies that \mathit{RM}(T,K)\ne \mathrm{\varnothing}. Finally, by Theorem 3.1, we get the result S(T,K)\ne \mathrm{\varnothing}. □
4 Gap function and application to the mathematical programming problem
In this section, we discuss the gap function of the relaxed Minty variationallike inequality, and use it to study the optimization problem.
Definition 4.1 [5]
A function \varphi :K\to R is said to be a gap function for the variational inequality (VI) if it satisfies the following properties:

(i)
\varphi (x)\le 0, \mathrm{\forall}x\in K;

(ii)
\varphi ({x}_{0})=0 if and only if {x}_{0} solves (VI).
For the variationallike inequality \u3008Tx,\eta (y,x)\u3009\ge 0, \mathrm{\forall}y\in K, Yang in reference [5] gives its gap function:
He pointed out that the solution of the variationallike inequality was the solution of the following optimization problem:
Now we give the gap function of the relaxed Minty variationallike inequality as follows:
Theorem 4.1 Let \eta (x,x)=0, \alpha (0)=0. Then {\varphi}_{2}(x) is the gap function of the relaxed Minty variationallike inequality.
Proof (i) For x\in K, then
(ii) If {\varphi}_{2}(x)=0, then
That is, {x}_{0}\in \mathit{RM}(T,K).
Conversely, if {x}_{0}\in \mathit{RM}(T,K), then \u3008Ty,\eta (y,{x}_{0})\u3009\alpha (y{x}_{0})\ge 0, \mathrm{\forall}y\in K. So, {\varphi}_{2}({x}_{0})\ge 0, by {\varphi}_{2}({x}_{0})\le 0, we can get {\varphi}_{2}({x}_{0})=0. The proof is completed. □
For the relaxed Minty variationallike inequality, we discuss the following optimization problem:
It is clear that the solution of the relaxed Minty variationallike inequality is also the solution of (P_{2}).
Finally, we discuss the relationship between the problems of (P_{1}) and (P_{2}).
Theorem 4.2 Let K be a nonempty closed convex subset of the real reflexive Banach space X, and let T:K\to {X}^{\ast} be hemicontinuous and relaxed ηα pseudomonotone. Assume that:

(i)
\eta (x,x)=0 for all x in K;

(ii)
for any fixed y, z in K, the mapping x\mapsto \u3008Tz,\eta (x,y)\u3009 is convex.
Then the problems of (P_{1}) and (P_{2}) are equivalent.
Proof From Theorem 3.1 in reference [3], we know that the solution of the variationallike inequality and the solution of the relaxed Minty variationallike inequality are equivalent. So, we can get that the problems of (P_{1}) and (P_{2}) are equivalent. □
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Acknowledgements
The authors are thankful to Professor Xin Min Yang for his teaching and valuable comments on the original version of this paper. This research is partially supported by the National Natural Science Foundation of China (11201240).
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Chen, Q., Luo, J. Relaxed ηα quasimonotone and application to the generalized variationallike inequality problem. J Inequal Appl 2013, 488 (2013). https://doi.org/10.1186/1029242X2013488
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DOI: https://doi.org/10.1186/1029242X2013488