The purpose of this article is to study an implicit iteration process for a finite family of α-hemicontractive mappings in Hilbert spaces. Our results extend and generalize the recent results of Husain et al. [12] and Diwan et al. [8] from the classes of hemicontractive and α-demicontractive mappings respectively, to the more general class of α-hemicontractive mappings.
In this article, we develop a faster iteration method, called the A** iteration method, for approximating the fixed points of almost contraction mappings and generalized α-nonexpansive mappings. We establish some weak and strong convergence results of the A** iteration method for fixed points of generalized α-nonexpansive mappings in uniformly convex Banach spaces. We provide a numerical example to illustrate the efficiency of our new iteration method. The weak w2-stability result of the new iteration method is also studied. As an application of our main results, we approximate the solution of a fractional Volterra–Fredholm integro-differential equation. Our results improve and generalize several well-known results in the current literature.
This article presents a new three-step implicit iterative method. The proposed method is used to approximate the fixed points of a certain class of pseudocontractive-type operators. Additionally, the strong convergence results of the new iterative procedure are derived. Some examples are constructed to authenticate the assumptions in our main result. At the end, we use our new method to solve a fractional delay differential equation in the sense of Caputo. Our main results improve and generalize the results of many prominent authors in the existing literature.
In this paper, we propose a modified hybrid S-iteration scheme for finite family of nonexpansive and asymptotically generalized Φ-hemicontractive mappings in the frame work of real Banach spaces. We remark that the iteration process of Kang et al. [14] can be obtained as a special case of our iteration process. A different approach is used to obtain our result and the necessity of condition (C3) is not required to prove our strong convergence theorem. Our result mainly extends and complements the result of [14] and several other related results in the literature.
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