On the browder-hartman-stampacchia variational inequality

Chang, S.S.,Ha, K.S.,Cho, Y.J.,Zhang, C.J. Korean Mathematical Society 1995 대한수학회지 Vol.32 No.3

The Hartman-Stampacchia variational inequality was first suggested and studied by Hartman and Stampacchia [8] in finite dimensional spaces during the time establishing the base of variational inequality theory in 1960s [4]. Then it was generalized by Lions et al. [6], [9], [10], Browder [3] and others to the case of infinite dimensional inequality [3], [9], [10], and the results concerning this variational inequality have been applied to many important problems, i.e., mechanics, control theory, game theory, differential equations, optimizations, mathematical economics [1], [2], [6], [9], [10]. Recently, the Browder-Hartman-Stampaccnia variational inequality was extended to the case of set-valued monotone mappings in reflexive Banach sapces by Shih-Tan [11] and Chang [5], and under different conditions, they proved some existence theorems of solutions of this variational inequality.

PERTURBED PROXIMAL POINT ALGORITHMS FOR GENERALIZED MIXED VARIATIONAL INEQUALITIES

Jeong, Jae-Ug The Youngnam Mathematical Society Korea 2002 East Asian mathematical journal Vol.18 No.1

In this paper, we study a class of variational inequalities, which is called the generalized set-valued mixed variational inequality. By using the properties of the resolvent operator associated with a maximal monotone mapping in Hilbert spaces, we have established an existence theorem of solutions for generalized set-valued mixed variational inequalities, suggesting a new iterative algorithm and a perturbed proximal point algorithm for finding approximate solutions which strongly converge to the exact solution of the generalized set-valued mixed variational inequalities.

GENERALIZED STRONGLY NONLINEAR QUASI-VARIATIONAL INEQUALITIES

Jeong, Jae-Ug 한국전산응용수학회 1999 Journal of applied mathematics & informatics Vol.6 No.1

In this paper we introduce and study a new class of vari-ational inequalities which are called the generalized strongly nonlin-ear quasi-variational inequalities. An algorithm for finding the ap-proximate solution of generalized strongly nonlinear quasi-variational inequalities is also given. These variational inequalities include the previously known classes of variational inequalities as special cases.

AN ITERATIVE METHOD FOR NONLINEAR MIXED IMPLICIT VARIATIONAL INEQUALITIES

JEONG, JAE UG The Honam Mathematical Society 2004 호남수학학술지 Vol.26 No.4

In this paper, we develop an iterative algorithm for solving a class of nonlinear mixed implicit variational inequalities in Hilbert spaces. The resolvent operator technique is used to establish the equivalence between variational inequalities and fixed point problems. This equivalence is used to study the existence of a solution of nonlinear mixed implicit variational inequalities and to suggest an iterative algorithm for solving variational inequalities. In our results, we do not assume that the mapping is strongly monotone.

Generalized variational-like inequalities with compositely monotone multifunctions

Lu-Chuan Ceng,이규명,Jen-Chih Yao 대한수학회 2008 대한수학회지 Vol.45 No.3

In this paper, we introduce two classes of generalized variational- like inequalities with compositely monotone multifunctions in Banach spaces. Using the KKM-Fan lemma and the Nadler’s result, we prove the existence of solutions for generalized variational-like inequalities with compositely relaxed η-α monotone multifunctions in reflexive Banach spaces. On the other hand we also derive the solvability of generalized variational-like inequalities with compositely relaxed η-α semimonotone multifunctions in arbitrary Banach spaces by virtue of the Kakutani-Fan- Glicksberg fixed-point theorem. The results presented in this paper extend and improve some earlier and recent results in the literature. In this paper, we introduce two classes of generalized variational- like inequalities with compositely monotone multifunctions in Banach spaces. Using the KKM-Fan lemma and the Nadler’s result, we prove the existence of solutions for generalized variational-like inequalities with compositely relaxed η-α monotone multifunctions in reflexive Banach spaces. On the other hand we also derive the solvability of generalized variational-like inequalities with compositely relaxed η-α semimonotone multifunctions in arbitrary Banach spaces by virtue of the Kakutani-Fan- Glicksberg fixed-point theorem. The results presented in this paper extend and improve some earlier and recent results in the literature.

EXISTENCE AND ITERATIVE APPROXIMATIONS OF SOLUTIONS FOR STRONGLY NONLINEAR VARIATIONAL-LIKE INEQUALITIES

Li, Jin-Song,Sun, Ju-He,Kang, Shin-Min The Youngnam Mathematical Society Korea 2011 East Asian mathematical journal Vol.27 No.5

In this paper, we introduce and study a new class of strongly nonlinear variational-like inequalities. Under suitable conditions, we prove the existence of solutions for the class of strongly nonlinear variational- like inequalities. By making use of the auxiliary principle technique, we suggest an iterative algorithm for the strongly nonlinear variational-like inequality and give the convergence criteria of the sequences generated by the iterative algorithm.

On Solvability of Generalized Nonlinear Variational-like Inequalities

Lili Zhang,Zeqing Liu,강신민 대한수학회 2008 대한수학회지 Vol.45 No.1

In this paper, we introduce and study a new class of generalized nonlinear variational-like inequalities. By employing the auxiliary principle technique we suggest an iterative algorithm to compute approximate solutions of the generalized nonlinear variational-like inequalities. We discuss the convergence of the iterative sequences generated by the algorithm in Banach spaces and prove the existence of solutions and convergence of the algorithm for the generalized nonlinear variational-like inequalities in Hilbert spaces, respectively. Our results extend, improve and unify several known results due to Ding, Liu et al, and Zeng, and others. In this paper, we introduce and study a new class of generalized nonlinear variational-like inequalities. By employing the auxiliary principle technique we suggest an iterative algorithm to compute approximate solutions of the generalized nonlinear variational-like inequalities. We discuss the convergence of the iterative sequences generated by the algorithm in Banach spaces and prove the existence of solutions and convergence of the algorithm for the generalized nonlinear variational-like inequalities in Hilbert spaces, respectively. Our results extend, improve and unify several known results due to Ding, Liu et al, and Zeng, and others.

Existence and iterative approximations of solutions for strongly nonlinear variational-like inequalities

Jinsong Li,Juhe Sun,강신민 영남수학회 2011 East Asian mathematical journal Vol.27 No.5

In this paper, we introduce and study a new class of strongly nonlinear variational-like inequalities. Under suitable conditions, we prove the existence of solutions for the class of strongly nonlinear variational-like inequalities. By making use of the auxiliary principle technique, we suggest an iterative algorithm for the strongly nonlinear variational-like inequality and give the convergence criteria of the sequences generated by the iterative algorithm.

EXISTENCE RESULTS FOR VECTOR NONLINEAR INEQUALITIES

Lee, Suk-Jin,Lee, Byung-Soo Korean Mathematical Society 2003 대한수학회논문집 Vol.18 No.4

The purpose of this paper is to consider some existence results for vector nonlinear inequalities without any monotonicity assumption. As consequences of our main result, we give some existence results for vector equilibrium problem, vector variational-like inequality problem and vector variational inequality problems as special cases.

ON SOLVABILITY OF GENERALIZED NONLINEAR VARIATIONAL-LIKE INEQUALITIES

Zhang, Lili,Liu, Zeqing,Kang, Shin-Min Korean Mathematical Society 2008 대한수학회지 Vol.45 No.1

In this paper, we introduce and study a new class of generalized nonlinear variational-like inequalities. By employing the auxiliary principle technique we suggest an iterative algorithm to compute approximate solutions of the generalized nonlinear variational-like inequalities. We discuss the convergence of the iterative sequences generated by the algorithm in Banach spaces and prove the existence of solutions and convergence of the algorithm for the generalized nonlinear variational-like inequalities in Hilbert spaces, respectively. Our results extend, improve and unify several known results due to Ding, Liu et al, and Zeng, and others.