H-Accretive Operators Resolvent Operator Technique Solving Variational Inclusions in Banach Spaces

K e y w o r d s H a c c r e t i v e operator, Resolvent operator technique, Variational inclusion, Iterative algorithm. 1. I N T R O D U C T I O N A N D P R E L I M I N A R I E S Variational inequalities and variational inclusions are among the most interesting and important mathematical problems and have been studied intensively in the past years since they have wide applications in mechanics, physics, optimization and control, nonlinear programming, economics and transportation equilibrium, and engineering sciences, etc. (see, for example, [1-20]). In the theory of variational inequalities and vm'iational inclusions, the development of an efficient and implementable iterative algorithm is interesting and important. Various kinds of iterative algorithms to solve the variational inequalities and inclusions have been developed by many authors. For details, we can refer to [1-6,8-13,15-20] and the references therein. Among these methods, the resolvent operator techniques for solving variational inequalities and variational inclusions are interesting and important. Recently, Huang and Fang [12] introduced a new class of maximal r]-monotone mapping in Hilbert spaces, which is a generalization of the classical maximal monotone mapping, and studied This work was supported by the National Science Foundation of China. *Author to whom all correspondence shouid be addressed. 0893-9659/04/$ see front matter (~ 2004 Elsevier Ltd. All rights reserved. Typeset by A~tS-TEX doi: I 0.1016/j .aml. 2003.09.003 648 Y.-P. FANG AND N.-J. HUANG the properties of the resolvent operator associated with the maximal u-monotone mapping. They also introduced and studied a new class of nonlinear variational inclusions involving maximal u-monotone mapping in Hilbert spaces. For some related works, we refer to [8] and the references therein. In this paper, we further generalize the resolvent operator technique by introducing a new class of H-accretive operators in Banach spaces. We extend the concept of resolvent operators associated with the classical m-accretive operators to the new/./-accretive operators. By using the new resolvent operator technique, we study the approximate solution of a class of variational inclusions with/-/-accretive operators in Banach spaces. In what follows, we always let X be a real Banach space with dual space X*, (., .} be the dual pair between X and X*, and 2 x denote the family of all the nonempty subsets of X. The generalized duality mapping Jq : X ~ 2 X* is defined by Jq(x) = { f* e X* : @, f*} = IIxllq and Nf*lI = HxHq-1} , V x E X , where q > 1 is a constant. In particular, J2 is the usual normalized duality mapping. It is known that, in general, Jq(x) = IIx]lq-2J2(x), for all x ¢ 0, and Jq is single-valued if X* is strictly convex. In the sequel, unless otherwise specified, we always suppose that X is a real Banach space such that Jq is single-valued and H is a Hilbert space. If X = T{, then J2 becomes the identity mapping of T{. The modulus of smoothness of X is the function Px : [0, oc) --~ [0, cx~) defined by p x ( t ) = sup ~ (llx + Yll + I1~ YlI) 1= Ilxll --1, IIYLI -< t , A Banach space X is called uniformly smooth if lira Px (t) = O. t--~0 t X is called q-uniformly smooth if there exists a constant c > O, such that px( t ) <_ ct q, q > l. Note that Jq is single-valued if X is uniformly smooth. In the study of characteristic inequalities in q-uniformly smooth Banach spaces, Xu [21] proved the following theorem. THEOIkEM X. Let X be a real uniformly smooth Banach space. Then, X is q-uniformly smooth i f and only i f there exists a constant Cq > O, such that for all x, y ~ X, II~ + yll q < II~ll q + q (y, J~(~) ) + c~llyll ". DEFINITION 1.1. be Let T, H : X --+ X be two single-valued operators. The operator T is said to