Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of Lipschitz pseudocontractive mappings
Lu-Chuan Ceng, Adrian Petruşel, Jen‐Chih Yao · Journal of Mathematical Inequalities · 2007
(communicated by R. Verma) Abstract. Let E be a real Banach space, {Ti}Ni=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}Ni=1. For an arbitrary initial point x0 ∈ E, the sequence {xn}∞n=1 is defined by xn = αn(xn−1 − λnG(xn−1)) + (1 − αn)Tnxn where Tn = Tn mod N, {αn}∞n=1 ⊂ [a, b] ⊂]0, 1 [ and {λn}∞n=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. 1.