A NEW GENERALIZED RESOLVENT AND APPLICATION IN BANACH MAPPINGS

Abstract. In this paper, we introduce a new generalized resolvent in aBanach space and discuss its some properties. Using these properties, weobtain an iterative scheme for nding a point which is a xed point ofrelatively weak nonexpansive mapping and a zero of monotone mapping.Furthermore, strong convergence of the scheme to a point which is a xedpoint of relatively weak nonexpansive mapping and a zero of monotonemapping is proved. 1. IntroductionLet Ebe a real Banach space with dual E . We denote by Jthe normalizedduality mapping from Einto 2 E . de ned byJx:= ff 2E : hx;fi= kxk 2 = kfk 2 kg;where h;idenotes the generalized duality pairing. It is well known that if Eis strictly convex then Jis single-valued and if Eis uniformly smooth then Jis uniformly continuous on bounded subsets of E. Moreover, if Eis a reexiveand strictly convex Banach space with a strictly convex dual, then J 1 is singlevalued, one-to-one, surjective, and it is the duality mapping from E into Eand thus JJ 1 = I