Fixed Points by a New Iteration Method

Abstract

Abstract.The following result is shown. If T is a lipschitzian pseudo-contractive map of a compact convex subset E of a Hubert space into itself and x^ is any point in E, then a certain mean value sequence defined by xn+1 = anT[ßnTxn+ (1 -ßn)xn] + (1 -a")x" converges strongly to a fixed point of T, where {a"} and {/?"} are sequences of positive numbers that satisfy some conditions.It was recently shown in [1] that a mean value iteration method is available to find a fixed point of a strictly pseudo-contractive map. In this paper we shall prove that a certain sequence of points which is iteratively defined converges always to a fixed point of a lipschitzian pseudo-contractive map. For the definitions of a strictly pseudo-contractive map and a pseudo-contractive map in a Hubert space, see, for example, [3]. Theorem.If E is a convex compact subset of a Hubert space H, T is a lipschitzian pseudo-contractive map from E into itself and x, is any point in E, then the sequence {x"},?=1 converges strongly to a fixed point ofT, where xn is defined iteratively for each positive integer n by 71=1As a particular case, we may choose for instance an=ßn=n~'1/2.

Strong Convergence for Hybrid S-Iteration Scheme of Nonexpansive and Asymptotically Demicontractive Mappings

“…In the last few years or so, numerous papers have been published on the iterative approximation of fixed points of Lipschitz strongly pseudocontractive mappings using the Ishikawa iteration scheme (see, e.g., [8]). Results which had been known only in Hilbert spaces and only for Lipschitz mappings have been extended to more general Banach spaces (see, e.g., [3-6,14 and 15] and the references cited therein).…”

EQUILIBRIUM POINTS FOR A SYSTEM INVOLVING M-ACCRETIVE OPERATORS

“…Two well-known iteration schemes-the Mann iteration scheme (see, for example, [27]) and the Ishikawa iteration scheme (see, for example, [23])-have successfully been employed to approximate the kernels of accretive self-maps. However, if a map T is such that its range does not intersect its domain (and this occurs frequently in many applications), then neither the Mann nor the Ishikawa scheme may be well defined.…”

Approximating fixed points of nonexpansive type mappings

“…In this paper, we wish to use Conditions 1.1 and 1.2 to prove the convergence of Ishikawa iterates [3] of certain nonexpansive type mappings.…”