The goal of the current study is to introduce a new class of proximal-point mappings that are associated with a new class of ${\mathcal{B}}^l$-co-monotone mappings that are being defined. The ${\mathcal{B}}^l$-co-monotone mapping is the sum of co-coercive and symmetric monotone mappings and an extension of the $C_n$-monotone mapping. The investigation is further discussed, along with its application, which involves a variational inclusion problem (VIP) in Banach spaces. Moreover, the study proposes an iterative algorithm and systematically investigates the convergence characteristics of its generated sequences. For the purpose of illustrating our findings, a simplified numerical example is created to show the convergence graph by using the MATLAB 2015a.

This study explores certain discoveries regarding fixed points under a functional contraction incorporating two test functions within the setup of a metric space equipped with a locally transitive relation. The outcomes investigated herewith refined, expanded, and strengthened several previously established fixed‐point findings. We provide numerous illustrative instances that demonstrated the credibility of our findings. We investigate the existence and uniqueness of solutions to specific nonlinear integral equations in order to ensure the practical use of our findings.

This study focuses on a three-step iterative scheme, referred to as the NIP iteration, for the approximation of fixed points associated with nonexpansive mappings in uniformly convex Banach spaces. Weak convergence is established using Fejér monotonicity, asymptotic regularity and the demiclosedness principle. Strong convergence is proved under uniform convexity, compactness, and Condition (I) of Senter and Dotson. A numerical convergence and computational-cost comparison is developed numerically, showing that the NIP iteration performs better than the Ishikawa, S, Noor, Abbas–Nazir and SP schemes. Numerical experiments for nonlinear nonexpansive mappings validate the theoretical findings. An application to convex optimization via fixed point reformulation is also presented, illustrating the effectiveness of the method.

The aim of this article is to derive fixed point findings under a suitable class of Suzuki nonlinear contractions besides utilizing a specific binary relation (not necessarily transitive) on a metric space. Established results are sharpened forms over several corresponding results of the existing literature, such as relation‐theoretic nonlinear contractive and nonlinear contractive (without use of relation) types of results, which will be derived from our newly established results. Furthermore, an example is constructed to demonstrate the realized improvement of established results. Finally, an application of our findings is incorporated, i.e., a solution of a nonlinear integral problem is discussed.

This manuscript seeks to study couple fixed-point findings over a relational extended contraction through a locally, finitely transitive binary relation and a pair of generalized altering distance functions. Various prior discoveries are expanded, refined, modified, compiled, and enhanced by our findings. We craft some instances for demonstrating the practical relevance of our findings. We embark on our research in an attempt to recognize a unique solution to a boundary value problem connected with a second-order ordinary differential equation.