Some Strong Convergence Results of Random Iterative Algorithms With Errors in Banach Spaces

Abstract

Abstract. In this paper, we study the strong convergence and stability of a new two step random iterative scheme with errors for accretive Lipschitzian mapping in real Banach spaces. The new iterative scheme is more acceptable because of much better convergence rate and less restrictions on parameters as compared to random Ishikawa iterative scheme with errors. We support our analytic proofs by providing numerical examples. Applications of random iterative schemes with errors to variational inequality are also given. Our results improve and establish random generalization of results obtained by Chang [4], Zhang [31] and many others.

Stability and Strong Convergence Results for Random Jungck-Kirk-Noor Iterative Scheme

“…The theory of random fixed point theorems was initiated in 1950 by Prague school of probabilistic. After the classical results of Bharucha-Reid [3] in 1976, where he gave sufficient conditions for a stochastic analogue of Schouder's fixed point theorem for random operators, the theory of random fixed points received unprecedented attention by several researchers and many interesting results have appeared in the literature see [7,16,18,28].Špaček [29] and Hanš [8] established stochastic analogue of the Banach fixed point theorem in a separable metric space. Itoh [13] in 1979, generalized and extendedŠpaček and Hanš's theorem to a multivalued contraction random operators.…”

Exponential inequalities for Mann’s iterative scheme with functional random errors

“…Chugh et al [14] studied the strong convergence and stability of a new two-step random iterative scheme with errors for accretive Lipschitzian mapping in real Banach spaces. In [10], Cho et al has built a random Ishikawa's iterative sequence with errors for random strongly pseudo-contractive operator in separable Banach spaces and proved that under suitable conditions, this random iterative sequence with errors converges to a random fixed point of the operator.…”

“…In [4], a random fixed point theorem was obtained for the sum of a weaklystrongly continuous random operator and a nonexpansive random operator which contains as a special Krasnoselskii type of Edmund and O'Regan via the method of measurable selectors. We note some recent works on random fixed points in [1,3,5,13,14,16,27].…”

On convergence of random iterative schemes with errors for strongly pseudo-contractive Lipschitzian maps in real Banach spaces

“…With the developments in random fixed point theory, there has been a renewed interest in random iterative schemes [2,3,7,8,10]. In linear spaces, Mann and Ishikawa iterative schemes are two general iterative schemes which have been successfully applied to fixed point problems [1,5,6,13,14,16,19,26,28,37].…”

“…In linear spaces, Mann and Ishikawa iterative schemes are two general iterative schemes which have been successfully applied to fixed point problems [1,5,6,13,14,16,19,26,28,37]. Recently, many stability and convergence results of iterative schemes have been established, using Lipschitz accretive pseudo-contractive) and Lipschitz strongly accretive (or strongly pseudo-contractive) mappings in Banach spaces [9,10,12,13,22,23,24,32,37]. Since in deterministic case the consideration of error terms is an important part of an iterative scheme, therefore, we introduce a three step random iterative scheme with errors and prove that the iterative scheme is stable with respect to T with Lipschitz condition where T is a strongly accretive mapping in arbitrary real Banach space.…”