A HYBRID PROXIMAL POINT ALGORITHM AND STABILITY FOR SET-VALUED MIXED VARIATIONAL INCLUSIONS INVOLVING (A, ${\eta}$)-ACCRETIVE MAPPINGS

Abstract

Abstract. A new class of nonlinear set-valued mixed variational inclu-sions involving (A;)-accretive mappings in Banach spaces is introducedand studied, which includes many kind of variational inclusion (inequal-ity) and complementarity problems as special cases. By using the resol-vent operator associated with (A;)-accretive operator due to Lan-Cho-Verma, the existence of solution for this kind of variational inclusion isproved, and a new hybrid proximal point algorithm is established and sug-gested, the convergence and stability theorems of iterative sequences gen-erated by new iterative algorithms are also given in q-uniformly smoothBanach spaces. 1. IntroductionThe variational inclusion, which was introduced and studied by Hassouniand Mouda [7], is a useful and important generalization of the variational in-equality. Various variational inclusions have been intensively studied in recentyears. Many authors (see, [1], [3], [4], [5], [6], [10], [11], [12], [14], [15], [19])introduced the concepts of -subdi erential operators, maximal -monotoneoperators, H-monotone operators, A-monotone operators, (H;)-monotone op-erators, (A;)-accretive mappings, (G;)-monotone operators, and de ned re-solvent operators associated with them, respectively.Moreover, by using the resolvent operator technique, many authors con-structed some approximation algorithms for some nonlinear variational inclu-sions in Hilbert spaces or Banach spaces. Recently, Verma [16] has developeda hybrid version of the Eckstein-Bertsekas [2] proximal point algorithm, intro-duced the algorithm based on the (A;)-maximal monotonicity framework andstudied convergence of the algorithm.