Generalized H-η-accretive operators in Banach spaces with application to variational inclusions

Abstract

In this paper, a new notion of a generalized H-η-accretive operator is introduced and studied, which provides a unifying framework for the generalized m-accretive operator and the H-η-monotone operator in Banach spaces. A resolvent operator associated with the generalized H-η-accretive operator is defined, and its Lipschitz continuity is shown. As an application, the solvability for a class of variational inclusions involving the generalized H-η-accretive operators in Banach spaces is considered. By using the technique of the resolvent mapping, an iterative algorithm for solving the variational inclusion in Banach spaces is constructed. Under some suitable conditions, it is proven that the solution for the variational inclusion and the convergence of the iterative sequence generated by the algorithm exist. 502Xue-ping LUO and Nan-jing HUANG of η-subdifferential operators, maximal η-monotone operators, H-monotone operators, (H, η)-monotone operators, A-monotone operators, (A, η)-monotone operators, G-η-monotone operators, and M -monotone operators in Hilbert spaces, respectively. Huang and Fang [21] were the first to introduced the generalized m-accretive operators which extended the notion of maximal η-monotone operators to Banach spaces. Fang and Huang [22][23] , Lan et al. [24] , Lan [25] , and Zou and Huang [26][27] investigated many accretive operators such as H-accretive operators, (H, η)-accretive operators, (A, η)-accretive operators, and H(·, ·)-accretive operators in Banach spaces, which generalized the notions of H-monotone operators, (H, η)-monotone operators, (A, η)-monotone operators, and M -monotone operators in Hilbert spaces. They also defined the associated resolvent operators and used the resolvent operator technique. They developed some iterative algorithms to approximate the solutions of variational inclusions. Further, Xia and Huang [28] , Ding and Feng [29] , Feng and Ding [30] , Lou et al. [31] , Ding and Wang [32] , and Luo and Huang [33] introduced the notions of general H-monotone operators, A-monotone operators, H-η-monotone operators, and B-monotone operators in Banach spaces, which generalized the corresponding notions of the monotone type operators mentioned above. In addition, the H-η-monotone operators are different from the generalized m-accretive operators in Banach spaces.Motivated and inspired by the research in this field, in this paper, we introduce a new notion of generalized H-η-accretive operators which can provide a unifying framework for the generalized m-accretive operators and the H-η-monotone operators in Banach spaces. Moreover, we give a definition of the resolvent operator for the generalized H-η-accretive operator and prove its Lipschitz continuity in Banach spaces. As an application, we consider the solvability for a class of variational inclusions involving the generalized H-η-accretive operators in Banach spaces. By using the technique of resolvent mapping, we construct an iterative algorithm for solving the variational inclusion in Banach spaces. Under some su...

Strict feasibility of variational inclusion problems in reflexive Banach spaces

“…(see, for example, [9,10,15,16,18,19,20,21,22,25,26,27,29]). In particular, the variational inclusion problem has been studied intensively by many authors in recent years (see, for example, [11,12,13,30]). Among the main research issues on the variational inclusion problem, one of the most interesting problems in the theoretical aspect is the development of an efficient and implementable algorithm.…”

“…Among the main research issues on the variational inclusion problem, one of the most interesting problems in the theoretical aspect is the development of an efficient and implementable algorithm. Various kinds of iterative algorithms have been designed and studied to find solutions for various kinds of variational inclusions (see, for example, [11,12,13]).…”