Convergence Theorems for Strongly Pseudo-contractive and Strongly Accretive Maps

Abstract

SupposeEis an arbitrary real Banach space andKis a nonempty closed convex and bounded subset ofE. SupposeT:K→Kis a uniformly continuous strong pseudo-contraction with constantk∈(0,1). Suppose {an}, {bn}, {cn}, {a′n}, {b′n}, and {c′n} are sequences in (0,1) satisfying the following conditions: (i)an+bn+cn=1=a′n+b′n+c′n∀integersn≥0; (ii) limbn=limb′n=limc′n=0; (iii) Σbn=∞; (iv) Σcn<∞. For arbitraryx0,u0,v0∈K, define the sequence {xn}∞n=0iteratively byxn+1=anxn+bnTyn+cnun;yn=a′nxn+b′nTxn+c′nvn,n≥0, where {un}, {vn} are arbitrary sequences inK. Then {xn} converges strongly to the unique fixed point ofT. Related results deal with the iterative solutions of nonlinear equations involving set-valued, strongly accretive operators.