Fixed point theorems and applications to fractional problems with integral-type weak contractions in modular metric spaces

Abstract

Abstract Building upon [M. Paul, K. Sarkar, and K. Tiwary, “Fixed point theorems for integral-type weak-contraction mappings in modular metric spaces,” Gen. Math. , vol. 31, no. 1, pp. 21–37, 2023]’s foundational results on common fixed points in modular and convex metric spaces with integral-type contractions, we present a significant generalization of these theorems that fully leverages the unique advantages of modular spaces for analyzing fractional Caputo-type problems (1 < δ < 2). Our approach demonstrates how modular spaces provide superior handling of solution symmetries, memory effects, and asymptotic behaviors compared to classical metric spaces, yielding more precise and comprehensive results through enhanced flexibility in dealing with complex structural properties. In addition to improving theoretical knowledge of fixed points in generalized metric spaces, the developed framework provides a potent analytical tool for solving difficult fractional calculus problems, especially when modeling viscoelastic materials and anomalous diffusion processes, where traditional approaches are inadequate. These innovations open up new avenues for studying nonlinear operators and offer solid mathematical underpinnings for theoretical advancements as well as real-world applications in mathematical physics and engineering.