Fixed point iteration for pseudocontractive maps

Abstract

Let K K be a compact convex subset of a real Hilbert space, H H ; T : K → K T:K\rightarrow K a continuous pseudocontractive map. Let { a n } , { b n } , { c n } , { a n ′ } , { b n ′ } \{a_{n}\}, \{b_{n}\}, \{c_{n}\}, \{a_{n}^{’}\}, \{b_{n}^{’}\} and { c n ′ } \{c_{n}^{’}\} be real sequences in [0,1] satisfying appropriate conditions. For arbitrary x 1 ∈ K , x_{1}\in K, define the sequence { x n } n = 1 ∞ \{x_{n}\}_{n=1}^{\infty } iteratively by x n + 1 = a n x n + b n T y n + c n u n ; y n = a n ′ x n + b n ′ T x n + c n ′ v n , n ≥ 1 , x_{n+1} = a_{n}x_{n} + b_{n}Ty_{n} + c_{n}u_{n}; y_{n} = a_{n}^{’}x_{n} + b_{n}^{’}Tx_{n} + c_{n}^{’}v_{n}, n\geq 1, where { u n } , { v n } \{u_{n}\}, \{v_{n}\} are arbitrary sequences in K K . Then, { x n } n = 1 ∞ \{x_{n}\}_{n=1}^{\infty } converges strongly to a fixed point of T T . A related result deals with the convergence of { x n } n = 1 ∞ \{x_{n}\}_{n=1}^{\infty } to a fixed point of T T when T T is Lipschitz and pseudocontractive. Our theorems also hold for the slightly more general class of continuous hemicontractive nonlinear maps.