In this article, we establish new fixed point theorems for self-mappings defined on complete parametric metric spaces. By introducing a generalized integral-type contractive condition involving an auxiliary control function. The approach relies on the concept of convergence and completeness in parametric settings, which naturally generalizes classical metric frameworks. As an application, we demonstrate the existence of solutions to nonlinear integral equations, highlighting the applicability of the proposed framework. Several examples are presented to illustrate the effectiveness and novelty of the obtained results.

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The International Scientific Research Organization for Science, Engineering and Technology (ISROSET) is a Non-Profit Organization; The ISROSET is dedicated to improvement in academic sectors of Science (Chemistry, Bio-chemistry, Zoology, Botany, Biotechnology, Pharmaceutical Science, Bioscience, Bioinformatics, Biometrics, Biostatistics, Microbiology, Environmental Management, Medical Science, Forensic Science, Home Science, Library Science, Material Science Military Science, Physical Science, Physical Education Science, Educational Science, Fisheries, seed technology, Agriculture, Forestry Science, Mathematics, Physics, Statistics and Geology/Earth Science), Computer Science, Engineering and Information Technology, Commerce, Management, Economics Sociology and Social Science.

In this paper, we establish several new fixed point (FP) theorems for fuzzy mappings in the framework of complete $b$-fuzzy metric spaces (FMS). We introduce generalized contractive conditions that extend and unify a wide class of existing FP principles in fuzzy and non-fuzzy settings. Our results cover and generalize many classical theorems, and their strength is demonstrated by an application to the existence of fuzzy solutions of nonlinear integral equations. The findings highlight the relevance of $b$-FMSs in handling uncertainty and imprecision arising in real-world models

This article's goal is to use the Φp operator to prove a few fixed point theorems in complete metric space. Furthermore, we investigate whether fixed points for self mappings that meet the requirements of rational expression exist and are unique in a complete metric space. Our findings expand upon and generalize a great deal of previously published research.