%0 Journal Article
%T Algorithm for Solving a New System of Generalized Variational Inclusions in Hilbert Spaces
%A Shamshad Husain
%A Sanjeev Gupta
%J Journal of Calculus of Variations
%D 2013
%R 10.1155/2013/461371
%X We introduce and study a new system of generalized variational inclusions involving -cocoercive and relaxed -cocoercive operators, which contain the systems of variational inclusions and the systems of variational inequalities, variational inclusions, and variational inequalities as special cases. By using the resolvent technique for the -cocoercive operators, we prove the existence of solutions and the convergence of a new iterative algorithm for this system of variational inclusions in Hilbert spaces. An example is given to justify the main result. Our results can be viewed as a generalization of some known results in the literature. 1. Introduction Variational inclusions have been widely studied in recent years. The theory of variational inclusions includes variational, quasi-variational, variational-like inequalities as special cases. Various kinds of iterative methods have been studied to solve the variational inclusions. Among these methods, the resolvent operator technique to study the variational inclusions has been widely used by many authors. For details, we refer to [1¨C15]. For applications of variational inclusions, see [16]. Fang and Huang, Lan, Cho, and Verma, and kazmi investigated several resolvent operators for generalized operators such as -monotone, -monotone, -accretive, -accretive, -accretive, -monotone, -accretive, -proximal, and - -proximal mappings. For further details, we refer to [2¨C6, 8¨C10, 13] and the references therein. Very recently, Zou and Huang [15] introduced and studied -accretive operators, Xu and Wang [14] introduced and studied -monotone operators, and Ahmad et al. [1] introduced and studied -cocoercive operators. Inspired and motivated by researches going on in this area, we introduce and study a new system of generalized variational inclusions in Hilbert spaces. By using the resolvent operator technique for the -cocoercive operator, we develop a new class of iterative algorithms to solve a class of relaxed cocoercive variational inclusions associated with -cocoercive operators in Hilbert space. For illustration of Definitions 2, 5 and main result Theorem 19 Examples 4, 6, and 20 are given, respectively. Our results can be viewed as a refinement and improvement of Bai and Yang [2], Huang and Noor [17], and Noor et al. [11]. 2. Preliminaries Throughout this paper, we suppose that is a real Hilbert space endowed with a norm and an inner product , respectively. is the family of all the nonempty subsets of . In the sequel, let us recall some concepts. Definition 1 (see [18, 19]). A mapping is said to be(i) -Lipschitz
%U http://www.hindawi.com/journals/jcv/2013/461371/