Fixed point iteration for pseudocontractive maps

Abstract

Abstract. Let K be a compact convex subset of a real Hilbert space, H; T : K → K a continuous pseudocontractive map. Let {an}, {bn}, {cn}, {a n }, {b n } and {c n } be real sequences in [0,1] satisfying appropriate conditions. For arbitrary x 1 ∈ K, define the sequence {xn} ∞ n=1 iteratively by x n+1 = anxn + bnT yn + cnun; yn = a n xn + b n T xn + c n vn, n ≥ 1, where {un}, {vn} are arbitrary sequences in K. Then, {xn} ∞ n=1 converges strongly to a fixed point of T . A related result deals with the convergence of {xn} ∞ n=1 to a fixed point of T when T is Lipschitz and pseudocontractive. Our theorems also hold for the slightly more general class of continuous hemicontractive nonlinear maps.

Iterative approximation of solutions of nonlinear equations of Hammerstein type

“…For the iterative approximation of solutions of (1.4) and (1.5), the monotonicity/accretivity of A is crucial. The Mann iteration scheme (see, e.g., [26]) and the Ishikawa iteration scheme (see, e.g., [23]) have successfully been employed (see, e.g., [1,9,10,11,12,13,14,15,16,17,18,19,20,21,23,25,27,28,29,30,31,32,33,34]). Attempts to apply these methods to (1.8) have not provided satisfactory results.…”

Iterative Process for a Common Fixed Point of a Finite Family of Asymptotically k-Pseudocontractive Maps

“…The class of $k$-pseudocontractive maps and their likes have been studied by various authors (Browder & Petryshyn, 1967;Chidume & Moore, 1999;Chidume, 1984;Hicks & Kubicek, 1977;Moore & Nnoli, 2001;Moore, Nnubia, & Akabuike, 2015;Moore, Nnubia, & Nfor, 2010;Osilike, 1993;Weng, 1993;Wong, 1974).…”

Strong Convergence of Three Step Iteration Process for Nonexpansive and Strongly Pseudocontractive Mappings Apurva Kumar Das, Shailesh Dhar Diwan and Samir Dashputre

“…It is well known that the Mann iteration process does not converge always to fixed point of Lipschitz pseudocontractive mapping therefore the Ishikawa iteration process is significantly desirable for these type of mappings. A detail information regarding the convergence of Mann iteration process of such mapping is given in [5]. In a paper Rashwan [20] studied the convergence of Mann iterates to a common fixed point for a pair of mappings in normed space.…”