Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of Lipschitz pseudocontractive mappings

Abstract

Let E be a real Banach space, {Ti} N=1 be a finite family of continuous pseudocon- tractive self mappings of E and G : E → E be a mapping which is both δ -strongly accretive and λ -strictly pseudocontractive of Browder-Petryshyn type such that δ +λ 1. We propose a new implicit iteration scheme with perturbed mapping G for the approximation of common fixed points of {Ti} N=1 . For an arbitrary initial point x0 ∈ E , the sequence {xn} ∞=1 is defined by xn = αn(xn−1 − λnG(xn−1)) + (1 − αn)Tnxn where Tn = Tn mod N , {αn} ∞=1 ⊂ (a,b) ⊂)0,1( and {λn} ∞=1 ⊂ (0,1(. We establish some weak convergence theorems for this implicit iteration scheme. Also, necessary and sufficient conditions for strong convergence of this implicit iteration scheme are obtained.