L-FUZZY FIXED POINTS FOR HYBRID L-FUZZY CONTRACTIONS IN b-METRIC SPACES WITH A MACHINE LEARNING APPROACH TO FRACTIONAL DIFFERENTIAL EQUATIONS

Abstract

Bilateral contraction, a recently introduced hybrid form of contraction, has opened new avenues in fixed point (FP) theory. In this paper, we build upon this concept and the structure of b-metric spaces (b-MS) to propose a novel contraction type, referred to as the Hybrid L-fuzzy contraction. This new approach unifies and extends key elements from the Jaggi and Dass–Gupta contraction principles within the framework of L-fuzzy set theory in b-metric spaces. Our aim is to broaden the theoretical foundation and enhance the applicability of these contractions. To illustrate the effectiveness of the proposed method, a detailed example is presented. Furthermore, we demonstrate its relevance through an application to a fractional differential equation (FDE). The existence of a solution is validated using a modern machine learning (ML) technique, specifically Physics-Informed Neural Networks (PINNs), which integrate the underlying differential equation directly into the learning process. This fusion of analytical theory and computational modeling highlights the potential of ML in supporting and extending traditional mathematical analysis.