Applications of the expanded Darbo's fixed point theorem to fractional differential equations with finite delay

Introduction/purpose: The aim of this paper is to study the existence of mild solutions to the initial value problem (IVP for short) of the Darboux problem for partial hyperbolic fractional differential equations via the Caputo derivative with finite delay in Fréchet spaces. Methods: In our analysis, Darbo's fixed point theorem is employed together with the notion of measure of noncompactness to establish the existence of solutions by reducing the research to proving the existence and uniqueness of fixed points of appropriate operators. Results: Under suitable conditions, existence of mild solutions to the considered fractional partial hyperbolic differential equations is proved. An illustrative example demonstrates the theoretical results applicability. Conclusion: Throughout this paper, sufficient conditions were considered to establish the existence and uniqueness of solutions to the Darboux problem by applying Darbo's fixed point theorem on unbounded interval. The study provides a rigorous methodological framework for analyzing similar classes of problems and contributes to the broader understanding of mild solutions in fractional calculus and functional analysis. The results offer potential applications for researchers investigating fractional differential equations in infinite-dimensional spaces.