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.