Abstract
In this paper, we prove the existence of solutions for a variational-like inequality and a generalized variational-like inequality in the relaxed η-α pseudomonotone and strictly η-quasimonotone cases in Banach spaces by using the KKM technique. The results presented in this paper improve and extend some corresponding results of several authors.
Similar content being viewed by others
1 Introduction
The variational inequality was first introduced and studied in the finite-dimensional Euclidean space by Giannessi [1]. Variational inequality problems play a critical role in many fields of science, engineering, and economics. In the last four decades, since the time of the celebrated Hartman-Stampacchia theorem (see [2, 3]), the existence of a solution of a variational inequality, a generalized variational inequality, and other related problems has become a basic research topic which continues to attract the attention of researchers in applied mathematics (see for instance [4–13], and the references therein).
In 1995, Chang et al. [14] introduced and studied the problem of the existence of solutions and the perturbation problem for some kind of variational inequalities with monotone and semimonotone mappings in nonreflexive Banach spaces. Recently, Verma [15] studied a class variational inequality relaxed monotone mapping. Moreover, Fang and Huang [16] obtained the existence of solutions for variational-like inequalities with relaxed η-α monotone mappings in reflexive Banach spaces. In 2003, Facchinei and Pang [17, 18] used the degree theory to obtain a necessary and sufficient condition of variational inequality problems for continuous pseudomonotone mappings in a finite-dimensional space. In 2008, Kien et al. [19] proposed some extensions of the results of Facchinei and Pang [17, 18] to the case of variational inequalities and generalized variational inequalities in infinite-dimensional reflexive Banach spaces.
On the other hand, Bai et al. [20] introduced the new concept of relaxed η-α pseudomonotone mappings. By using the KKM technique, they obtain some existence results for variational-like inequalities with relaxed η-α pseudomonotone mappings in reflexive Banach spaces. In 2007, Wu and Huang [21] introduced the two new concepts of relaxed η-α pseudomonotonicity and relaxed η-α demipseudomonotonicity in Banach spaces. In 2009, Pourbarat and Abbasi [22] tried to replace some conditions of the work of Wu and Huang [21] with some new conditions. Moreover, they present the solvability of variational-like inequalities with relaxed η-α monotone mappings in arbitrary Banach spaces (see also in [2, 15–20] and [23–28]).
Inspired and motivated by [19], we introduce a new definition of relaxed η-α pseudomonotone mappings and prove the existence of solutions for variational-like inequality and generalized variational-like inequality with relaxed η-α pseudomonotone mappings and strictly η-quasimonotone mappings in Banach spaces by using KKM technique. The results presented in this paper improve and extend some corresponding results of several authors.
2 Preliminaries
Let X be a real reflexive Banach space with dual space and denoted the pairing between and X. Let K be a nonempty subset of X, and denote the family of all the nonempty subset of X and and be mappings. The generalized variational-like inequality defined by K and Φ, denoted by , is the problem of finding a point such that
The set of all satisfying (2.1) is denoted by . If for all , where is a single-valued mapping, then the problem is called a variational-like inequality and the abbreviation is the problem of finding an such that
We introduce the definition of relaxed η-α pseudomonotone for α mapping which comes from a family of functions which contains all mappings α given in [20]. In fact, the new definition is an extension of Definition 2.1 in [20]. Then we recall some definitions and results which are needed in the sequel.
We introduce the family
We note that if , for all where k is a function from to with , then .
Definition 2.1 The mapping is said to be:
-
(i)
Relaxed η-α pseudomonotone if there exist and with , such that for every distinct points ,
(2.3)If for all distinct points x, y in K, then (2.3) becomes
and F is said to be relaxed α pseudomonotone.
-
(ii)
Strictly η-quasimonotone if there exist such that for every distinct points ,
(2.4)If for all distinct points x, y in K, then (2.4) becomes
and F is said to be strictly quasimonotone.
Definition 2.2 The mapping is said to be:
-
(i)
Relaxed η-α pseudomonotone if there exist and with ,
-
(ii)
Strictly η-quasimonotone if there exist such that
Example 2.3 If define by and
where , then the mapping F is a relaxed η-α pseudomonotone mapping with
But it is not a relaxed α-pseudomonotone mapping. In fact, if we let , , , but , which is a contradiction.
Example 2.4 If define by and
where . Then the mapping F is strictly η-quasimonotone but fails to be strictly quasimonotone since if and , then we have but .
Definition 2.5 ([20])
Let and be two mappings. F is said to be η-hemicontinuous if, for any fixed , the mapping defined by is continuous at 0+.
If , then F is said to be hemicontinuous.
Definition 2.6 ([29])
A mapping is said to be a KKM mapping if, for any , , where denotes the convex hull of .
Lemma 2.7 ([29])
Let K be a nonempty subset of a Hausdorff topological vector space X and let be a KKM mapping. If is closed in X for every x in K and compact for some , then
Lemma 2.8 (Michael selection theorem [30])
Let X be a paracompact space and Y be a Banach space. Then every lower semicontinuous multivalued mapping from X to the family of nonempty, closed, convex subsets of Y admits a continuous selection.
3 Generalized variational-like inequality with relaxed η-α pseudomonotone mappings
In this section, we will discuss the existence of solutions for the following variational-like inequality and generalized variational-like inequality with relaxed η-α pseudomonotone mappings.
Theorem 3.1 Let K be a nonempty closed convex subset of a real reflexive Banach space X. Let and be mappings. Assume that:
-
(i)
F is an η-hemicontinuous and relaxed η-α pseudomonotone;
-
(ii)
for all ;
-
(iii)
for all , .
Then is a solution of if and only if
Proof Suppose that is a solution of . Since F is relaxed η-α pseudomonotone, we have
and hence is a solution of (3.1). Conversely, suppose that is a solution of (3.1) and be any point. Letting , , we have . It follows from (3.1) that
By the conditions of η, we have
It follows from (3.2) and (3.3) that
So, we have
Letting , we get
□
Theorem 3.2 Let X be a real reflexive Banach space and be a closed convex set. Let and be are mappings. Assume that:
-
(i)
F is a relaxed η-α pseudomonotone mapping and η-hemicontinuous;
-
(ii)
for all ;
-
(iii)
for all , and η is lower semicontinuous;
-
(iv)
is lower semicontinuous.
Then the following statements are equivalent:
-
(a)
There exists a reference point such that the set
is bounded (possibly empty).
-
(b)
The variational-like inequality has a solution.
Moreover, if there exists a vector such that the set
is bounded and for all x, y in K, then the solution set is nonempty and bounded.
Proof Suppose that there exists a reference point , which satisfies (a). Then there exists an open ball, denoted by Ω such that
We combine this with the obvious property . Thus . Define the set-valued mappings , for any , by
and
We claim that T is a KKM mapping. Indeed, if T is not a KKM mapping, then there exists such that . That is, there exists a , , where , , , but . By the definition of T, we have
Since for (), it follows that
On the other hand, we note that
It is a contradiction and this implies that T is a KKM mapping. Now we show that for all . For any given , let . Thus, we have . Since F is a relaxed η-α pseudomonotone, we obtain . This implies that and so for all . It follows that S is also a KKM mapping.
From the assumptions, we know that is weakly closed. In fact, since η and α are lower semicontinuous, we see that is a weakly closed subset of . Since is a weakly compact and is a weakly closed subset of , we see that is weakly compact for each . Thus, the conditions of Lemma 2.7 are satisfied in the weak topology. By Lemma 2.7 and Theorem 3.1, we have
It follows that there exists such that
Hence .
Assume that (b) holds. We take any . That is,
By the relaxed η-α pseudomonotonicity of F, we have
Hence and (a) is valid.
Finally, suppose that there is some such that the set is bounded. Then is nonempty by virtue of the implication (a) ⇒ (b). To prove that is bounded, it suffices to show that . Assume that , but . Thus, we have
and
Substituting into the inequality in (3.4), we have
This implies that . From (3.5) and (3.6), we have . By (3.6) and F is relaxed η-α pseudomonotone, we obtain
It implies that . By Theorem 3.1 and (3.5), we get . Hence
It is a contradiction. Therefore . □
Theorem 3.3 Let X be a real reflexive Banach space and be a closed convex set. Let and be are mappings. Assume that:
-
(i)
Φ is a lower semicontinuous multifunction with nonempty closed convex values, where is endowed with the norm topology;
-
(ii)
Φ is a relaxed η-α pseudomonotone mapping;
-
(iii)
for all ;
-
(iv)
for all , and η is lower semicontinuous;
-
(v)
is lower semicontinuous.
Then the following statements are equivalent:
-
(a)
There exists a reference point such that the set
is bounded (possibly empty).
-
(b)
The generalized variational-like inequality has a solution.
Proof Since Φ is lower semicontinuous multifunction with nonempty closed convex values, by Michael’s selection theorem (see for instance [30]) it admits a continuous selection; that is, there exists a continuous mapping such that for every . If (a) holds, then there exists an open ball, denoted by Ω such that
We combine this with the obvious property . Thus, we have
Applying Theorem 3.2, we get . For any , if we choose then
It follows that .
We prove that (b) ⇒ (a). Assume that (b) holds. We take any . Thus there exists satisfying
Because Φ is a relaxed η-α pseudomonotone, we obtain
It follows that
Hence and (a) is valid. □
4 Generalized variational-like inequality with strictly η-quasimonotone mappings
In this section, we will discuss the existence of solutions for the following variational-like inequality and generalized variational-like inequality with strictly η-quasimonotone mappings.
Theorem 4.1 Let K be a nonempty closed convex subset of a real reflexive Banach space X. Let and be mappings. Assume that:
-
(i)
F is η-hemicontinuous and strictly η-quasimonotone;
-
(ii)
for all ;
-
(iii)
for all ;
-
(iv)
for any fixed , the mapping is convex.
Then is a solution of if and only if
Proof Suppose that is a solution of . That is . To show that . Assume that there exists such that . By the property of η, we have . Since F is strictly η-quasimonotone, we have . By the property of η again, we get . It is a contradiction. Hence .
Conversely, suppose that is a solution of (4.1) and is arbitrary. Letting , , we have . It follows from (4.1) that
By assumption, we have
It follows from (4.2) and (4.3) that
Since F is η-hemicontinuous and letting , we get
□
Theorem 4.2 Let X be a real reflexive Banach space and be a closed convex set. Let and be are mappings. Assume that:
-
(i)
F is a strictly η-quasimonotone mapping and η-hemicontinuous;
-
(ii)
for all ;
-
(iii)
for all ;
-
(iv)
for any fixed , the mapping is convex and η is lower semicontinuous.
Then the following statements are equivalent:
-
(a)
There exists a reference point such that the set
is bounded (possibly empty).
-
(b)
The variational-like inequality has a solution.
Moreover, if there exists a vector such that the set
is bounded, then the solution set is nonempty and bounded.
Proof Suppose that (a) holds. Then there exists a reference point and an open ball, denoted by Ω such that
We combine this with the obvious property . Thus . Defined the set-valued mappings , for any , by
and
Since η is lower semicontinuous, we find that and are weakly closed subsets of . We claim that T is a KKM mapping. Similar to the proof of Theorem 3.2 we show that T is a KKM mapping. Now we show that for all . For any given , we let . That is, . Since F is strictly η-quasimonotone, we have . This implies that and so for all . It follows that S is also a KKM mapping. Since is weakly compact and is a weakly closed subset of , we find that is weakly compact for each . Thus, the condition of Lemma 2.7 is satisfied in the weak topology. By Lemma 2.7 and Theorem 4.1, we have
It follows that there exists such that
Hence .
Assume that (b) holds. We take any , that is,
By the strict η-quasimonotonicity of F, we have
Hence and (a) is valid.
Finally, suppose that there is some such that the set is bounded. Then is nonempty by virtue of the implication (a) ⇒ (b). To prove that is bounded, it suffices to show that . Assume that . Thus, we have
Substituting into the inequality in (4.4), we have
This implies that . Therefore . □
Theorem 4.3 Let X be a real reflexive Banach space and be a closed convex set. Let and be are mappings. Assume that:
-
(i)
Φ is a lower semicontinuous multifunction with nonempty closed convex values, where is endowed with the norm topology;
-
(ii)
Φ is a strictly η-quasimonotone mapping;
-
(iii)
for all ;
-
(iv)
for all ;
-
(v)
for any fixed , the mapping is convex and η is lower semicontinuous.
Then the following statements are equivalent:
-
(a)
There exists a reference point such that the set
is bounded (possibly empty).
-
(b)
The generalized variational-like inequality has a solution.
Proof Since Φ is a lower semicontinuous multifunction with nonempty closed convex values, by Michael’s selection theorem (see for instance [30]) it admits a continuous selection; that is, there exists a continuous mapping such that for every . If (a) holds, then there exists an open ball, denoted by Ω, such that
We combine this with the obvious property . Then we have
Applying Theorem 4.2, we get . For any , if we choose then
It follows that .
We prove that (b) ⇒ (a). Assume that (b) holds. We take any . Thus there exists satisfying
Because Φ is strictly η-quasimonotone and Theorem 4.1, we obtain
It follows that
Hence and (a) is valid. □
References
Giannessi F: Theorems of alternative, quadratic programs and complementarity problems. In Variational Inequalities and Complementarity Problems. Edited by: Cottle RW, Giannessi F, Lions JL. Wiley, New York; 1980:151–186.
Hartmann P, Stampacchia G: On some nonlinear elliptic differential functional equations. Acta Math. 1966, 115: 153–188.
Kinderlehrer D, Stampacchia G: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York; 1980.
Aussel D, Hadjisavvas N: On quasimonotone variational inequalities. J. Optim. Theory Appl. 2004, 121: 445–450.
Bianchi M, Hadjisavvas N, Shaible S: Minimal coercivity conditions and exceptional families of elements in quasimonotone variational inequalities. J. Optim. Theory Appl. 2004, 122: 1–17.
Crouzeix JP: Pseudomonotone variational inequality problems: existence of solutions. Math. Program. 1997, 78: 305–314.
Daniilidis A, Hadjisavvas N: Coercivity conditions and variational inequalities. Math. Program. 1999, 86: 433–438. 10.1007/s101070050097
Konnov IV: Generalized monotone equilibrium problems and variational inequalities. In Handbook of Generalized Convexity and Generalized Monotonicity. Edited by: Hadjisavvas N, Komlósi S, Schaible S. Springer, Berlin; 2005:559–618.
Yao JC: Variational inequalities with generalized monotone operators. Math. Oper. Res. 1994, 19: 691–705. 10.1287/moor.19.3.691
Yao JC: Multi-valued variational inequalities with K -pseudomonotone operators. J. Optim. Theory Appl. 1994, 80: 63–74. 10.1007/BF02196593
Yao JC, Chadli O: Pseudomonotone complementarity problems and variational inequalities. In Handbook of Generalized Convexity and Generalized Monotonicity. Edited by: Hadjisavvas N, Komlósi S, Schaible S. Springer, Berlin; 2005:501–558.
Ricceri B: Basic existence theorems for generalized variational and quasi-variational inequalities. In Variational Inequalities and Network Equilibrium Problems. Edited by: Giannessi F, Maugeri A. Plenum, New York; 1995:251–255.
Yen ND: On a problem of B. Ricceri on variational inequalities. 5. In Fixed Point Theory and Applications. Edited by: Cho YJ, Kim JK, Kang SM. Nova Science Publishers, New York; 2004:163–173.
Chang SS, Lee BS, Chen YQ: Variational inequalities for monotone operators in nonreflexive Banach spaces. Appl. Math. Lett. 1995, 8: 29–34.
Verma RU: On generalized variational inequalities involving relaxed Lipschitz and relaxed monotone operators. J. Math. Anal. Appl. 1997, 213: 387–392. 10.1006/jmaa.1997.5556
Fang YP, Huang NJ: Existence results for systems of strong implicit vector variational inequalities. Acta Math. Hung. 2004, 103: 265–277.
Facchinei F, Pang JS I. In Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin; 2003.
Facchinei F, Pang JS II. In Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin; 2003.
Kien BT, Yao JC, Yen ND: On the solution existence of pseudomonotone variational inequalities. J. Glob. Optim. 2008, 41: 135–145. 10.1007/s10898-007-9251-0
Bai MR, Zhoua SZ, Ni GY: Variational-like inequalities with relaxed η - α pseudomonotone mappings in Banach spaces. Appl. Math. Lett. 2006, 19: 547–554. 10.1016/j.aml.2005.07.010
Wu KQ, Huang NJ: Vector variational-like inequalities with relaxed η - α pseudomonotone mappings in Banach spaces. J. Math. Inequal. 2007, 1: 281–290.
Pourbarat K, Abbasi M: On the vector variational-like inequalities with relaxed η - α pseudomonotone mappings. Iran. J. Math. Sci. Inform. 2009,4(1):37–42.
Goeleven D, Motreanu D: Eigenvalue and dynamic problems for variational and hemivariational inequalities. Commun. Appl. Nonlinear Anal. 1996, 3: 1–21.
Siddiqi AH, Ansari QH, Kazmi KR: On nonlinear variational inequalities. Indian J. Pure Appl. Math. 1994, 25: 969–973.
Verma RU: On monotone nonlinear variational inequalities problems. Comment. Math. Univ. Carol. 1998, 39: 91–98.
Cottle RW, Yao JC: Pseudomonotone complementarity problems in Hilbert spaces. J. Optim. Theory Appl. 1992, 78: 281–295.
Karamardian S: Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl. 1976, 18: 445–454. 10.1007/BF00932654
Ceng LC, Lee GM, Yao JC: Generalized variational-like inequalities with compositely monotone multifunctions. J. Korean Math. Soc. 2008,45(3):841–858. 10.4134/JKMS.2008.45.3.841
Yuan GXZ: KKM Theory and Applications in Nonlinear Analysis. Dekker, New York; 1999.
Michael EA: Continuous selections, I. Ann. Math. 1956, 63: 361–382. 10.2307/1969615
Acknowledgements
The first author would like to thank the Thailand Research Fund for financial support and the second author is also supported by the Royal Golden Jubilee Program under Grant PHD/0282/2550, Thailand.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The work presented here was carried out in collaboration between all authors. SP, C-FW and AA defined the research theme. SP and C-FW designed theorems and methods of proof and interpreted the results. AA proved the theorems, interpreted the results and wrote the paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Arunchai, A., Plubtieng, S. & Wen, CF. Existence theorems for relaxed η-α pseudomonotone and strictly η-quasimonotone generalized variational-like inequalities. J Inequal Appl 2014, 442 (2014). https://doi.org/10.1186/1029-242X-2014-442
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-442