Muhammad Nazam

In this paper, we introduce a new class of generalized contractions, called enriched (Ψ, φ )‐contractions. Within this framework, we establish several fixed‐point theorems together with corresponding convergence results, which extend and unify a range of existing results in the literature. To demonstrate the applicability of the proposed approach, we provide illustrative examples and show that the obtained fixed‐point and convergence results ensure the existence of solutions for certain classes of integral equations. These findings broaden the theoretical scope of contraction mappings and highlight their usefulness in the analysis of integral equations.

This paper introduces the generalized fuzzy contraction in k-fuzzy metric spaces, a novel extension of the classical Meir-Keeler contraction principle. The main discovery is that several fixed point theorems, previously limited to classical or fuzzy metric spaces, remain valid under more general k-fuzzy conditions. This breakthrough broadens the applicability of fixed point theory to systems with multi-parameter uncertainty and non-uniform convergence, thereby offering a new mathematical tool for analyzing complex real-world problems in control theory, optimization, and data science.

This paper investigates optimal solutions for best proximity points through the framework of generalized interpolative proximal contractions. We introduce a new method that uses interpolation techniques to handle a wider class of mappings by expanding the concepts of classical proximal contraction. In the absence of a precise solution, best proximity point theorems investigate the existence of such best proximity points for approximate solutions to the fixed point problem. This article aims to develop the best proximity point theorems for contractive non-self mappings via interpolation to generate global optimal approximate solutions to particular fixed point equations. In addition to demonstrating the existence of the optimal proximity points, iterative techniques are also offered to locate such optimal approximative solutions. We illustrate the utility of our findings with a few instances. The value of our research is illustrated with a few examples and applications.

We introduce an ordered implicit relation and obtain related fixed point theorems in rectangular cone $b$-metric space, as an extension of the results on cone metric, rectangular cone metric and cone $b$-metric space. We provide some examples as an explanation of established outcomes. Our theorems universalize many fixed point outcomes in literature(\cite{23}, \cite{k6}). An homotopy result as an application of main theorem is given, which further applied to the human aging process. The obtained fixed point results do not follow from the results proved by Ercan \cite{er}, as the present article deals with nonlinear contractions. The convergence of the sequence generated by Urysohn integral operator is also shown by using fixed point technique.

In this paper, we investigate fixed points of mappings that contract the perimeters of polygons. Our study is motivated by the idea that the perimeter, as a global geometric measure, provides a more natural and flexible framework than the individual edge lengths when analyzing contraction properties in metric spaces. We extend Petrov’s fixed point theorem from triangles topolygons with an arbitrary number of vertices and establish conditions under which such mappings admit unique fixed points. The methodology relies on generalizations of contraction mappings and properties of metric spaces. Our main contributions include a new perimeter-based contraction principle, a demonstration of its application in proving Banach’s contraction theorem, and examples that validate the theoretical results. This generalization enriches the existing literature on fixed point theory and opens avenues for further applications in geometric analysis and related areas.

We introduce some new generalized proximal interpolative contraction principles that produce corresponding proximal interpolative contraction principles and proximal contraction principles as special cases. We prove various best proximity point theorems using introduced generalized proximal interpolative contraction principles. The obtained results improve and generalize many best proximity point theorems published earlier in the literature. Some examples and applications are given to demonstrate the usefulness of our results.

This paper explores the significance and implications of common γ ‐fuzzy fixed point results related to γ ‐fuzzy contraction as a novel form of contraction in complete b ‐metric space. Theoretical developments and theorems provide a solid foundation for understanding utilization of the properties of γ ‐fuzzy contraction, showcasing its efficiency through numerous examples and establishing stability and convergence properties. Moreover, some conventional fixed point results are incorporated as the direct consequences of our main findings.