From α-fuzzy Fixed Points to Nonlinear Cauchy Differential Inclusions in Intuitionistic Fuzzy Metric Spaces

Abstract

This study delves into the concept of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>α</mml:mi> </mml:math> -fuzzy mappings and their associated <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>α</mml:mi> </mml:math> -fuzzy fixed points within the framework of Hausdorff intuitionistic fuzzy metric-like spaces. A general fixed point theorem for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>α</mml:mi> </mml:math> -fuzzy mappings is established in complete HIFMS and its subclasses. The results unify and generalize several classical and fuzzy fixed point theorems, extending them to more complex fuzzy structures. Additionally, recognizing the significant applications of differential inclusions as set-valued mappings, the research explores first-order nonlinear Cauchy differential inclusions within Hausdorff intuitionistic fuzzy metric spaces by leveraging the derived theoretical results. The findings demonstrate the robustness of fuzzy fixed point theory in modeling systems with uncertainty and imprecision, with potential applications in control theory and optimization. The gap between fuzzy metric theory and differential inclusions.