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We have suggested a new modified iterative scheme for approximating a common fixed point of two G-nonexpansive mappings. Our approach was based on an iterative scheme in the context of Banach spaces via directed graphs. First, we proved a weak convergence theorem using the Opial's property of the underlying space. A weak convergence result without the Opial's property was also given. After this, we established several strong convergence theorems using various mild conditions. We also carried out some numerical simulations to examine the main techniques. Eventually, we obtained an application of our result to solve split feasibility problems (SFP) in the context of G-nonexpansive mappings. [ABSTRACT FROM AUTHOR]

This article introduces a novel iterative process, denoted as F ★ , designed for the class of generalized α -Nonexpensive mappings. The study establishes strong and weak convergence theorems within the context of Banach spaces, supported by carefully chosen assumptions. The convergence results contribute to the theoretical foundation of iterative processes in functional analysis. The presented framework is applied to address nonlinear integral equations, showcasing the versatility and applicability of the proposed F * for the class of generalized iteration process. Additionally, the article includes numerical examples that not only validate the theoretical findings but also provide insights into the practical utility of the developed methodology. [ABSTRACT FROM AUTHOR]

We consider the Banach space C [ 0 , 1 ] , which is a symmetric Banach space, and prove the existence and approximation of numerical solutions for a broad class of third-order BVPs. Our approach is based on an integral operator that is constructed using Green's function. The Banach contraction principle (BCP) is applied to guarantee a unique solution to our problem. Moreover, in order to find the value of the numerical solution, this new operator is embedded within the three-step Noor iterative scheme; we named this new iterative scheme the Noor–Green iterative scheme. We provide a convergence theorem for the proposed scheme by employing suitable restrictions on the parameters involved in the problem and in the scheme. The results of the stability of our scheme are also reported. It is worth mentioning that unlike the concept of stability in the classical sense, our result for stability is based on the concept of weak w 2 stability. In order to support our findings, we carried out various numerical experiments using different third-order BVPs. Finally, we report on the application of our iterative scheme to solve a class of fractional BVPs in the same symmetric Banach space. Our results are essentially new in the present literature and extend several of the results found in the current literature. [ABSTRACT FROM AUTHOR]

The aim of this article is to design a new iteration process for solving certain fixed-point problems. In particular, we prove weak and strong convergence theorems for generalized nonexpansive mappings in the framework of uniformly convex Banach spaces. In addition, we discuss the stability of the solution under mild conditions. Further, we provide some numerical examples to indicate that the proposed method works properly. [ABSTRACT FROM AUTHOR]

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 w 2 -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. [ABSTRACT FROM AUTHOR]

We introduce a new iterative method in this article, called the D iterative approach for fixed point approximation. Analytically, and also numerically, we demonstrate that our established D I.P is faster than the well-known I.P of the prior art. Finally, in a uniformly convex Banach space environment, we present weak as well as strong convergence theorems for Suzuki's generalized nonexpansive maps. Our findings are an extension, refinement, and induction of several existing iterative literatures. [ABSTRACT FROM AUTHOR]

In this paper we discuss unique fixed point of mappings satisfying a locally contractive condition on a closed ball in a complete Gd-metric like space. Examples have been given to show the novelty of our work. Our results improve/generalize several well known recent and classical results. [ABSTRACT FROM AUTHOR]

This paper is a continuation of the investigations of F-contractions. The aim of this article is to extend the concept of Ciric type F-contraction on a closed ball. We presented the notion of F-contraction on a closed ball and introduced a new approach of fixed point theorems for Fcontraction on a closed ball in a complete metric space. Our results are very useful for the contraction of the mapping only on closed ball instead of the whole space. Some comparative examples are constructed which illustrate the superiority of our results. Our results provide extension as well as substantial generalizations and improvements of several well-known results in the existing comparable literature. [ABSTRACT FROM AUTHOR]