Strong Convergence for Hybrid ImplicitS-Iteration Scheme of Nonexpansive and Strongly Pseudocontractive Mappings

Abstract

Let be a nonempty closed convex subset of a real Banach space , let : → be nonexpansive, and let : → be Lipschitz strongly pseudocontractive mappings such that ∈ ( ) ∩ ( ) = { ∈ : = = } and − ≤ − and − ≤ − for all , ∈ . Let { } be a sequence in [0, 1] satisfying (i) ∑ ∞ =1 = ∞; (ii) lim → ∞ = 0. For arbitrary 0 ∈ , let { } be a sequence iteratively defined by = , = (1 − ) −1 + , ≥ 1. Then the sequence { } converges strongly to a common fixed point p of S and T.

Strong convergence of modified implicit hybrid S-iteration scheme for finite family of nonexpansive and asymptotically generalized $\Phi$-hemicontractive mappings

“…(i) Corollary 3.2 captures the result of Kang et al [17]. It follows that the result Kang et al [17] is a special case of our result.…”

“…(i) Corollary 3.2 captures the result of Kang et al [17]. It follows that the result Kang et al [17] is a special case of our result. Hence, our result extends and improves the results of Kang et al [17] and many others in the literature.…”

“…We remark that the iteration process of Kang et al [17] can be obtained as a special case of our iteration process. Our result mainly improves and extends the result of Kang et al [17] and several other results in the literation from the class of strongly psudocontractive mapping to the more general class asymptotically generalized Φ-hemicontractive mappings. A different approach is used to obtain our result and the necessity of applying condition (C3) for the two mappings is weaken to only one mapping.…”

“…(ii) In our result the necessity of applying condition (C3) for the two classes of mappings (nonexpansive mappings and strongly pseudocontractive mappings) to prove strong convergence as considered in [17] and [12] is weaken by applying it to only one family of mapping which is the nonexpansive mappings.…”

“…In 2014, Khan [17] proved the following result: Theorem 1.4 (see [17]). Let K be a nonempty closed convex subset of a real Banach space E, let S : K → K be a nonexpansive mapping, and let T : K → K be a Lipschitz strongly pseudocontractive mapping such that p ∈ F(S) F(T ) = {x ∈ K :…”