Existence and Uniqueness of Solutions for a Nonlinear Coupled System of Differential Equations Including Φk‐Caputo‐Type Fractional Derivative in Generalized Banach Spaces in the Sense of Perov

Source abstract

ABSTRACT The target of this paper is to address the existence and uniqueness of solutions for a system of nonlinear differential equations, including a newly defined generalized Caputo‐type fractional‐order derivative (called as ‐Caputo‐type fractional derivative) in generalized Banach spaces in the sense of Perov. Basically, the problem under consideration in this study can be turned into an equivalent fixed‐point problem. Afterward, under different assumptions on the data of the proposed problem, we apply some fixed‐point theorems on some appropriate spaces endowed with vector‐valued norms to establish the desired results. More precisely, we show that there is a unique solution for this type of problem according to Perov's fixed‐point theorem, which employs convergent‐to‐zero matrices and appropriate vector‐valued norms, namely the Chebyshev and Bielecki vector norms. Further, an analogue of Schauder's fixed‐point theorem in generalized Banach spaces in the sense of Perov has been successfully used to show that there exists at least one solution to the proposed system. Ultimately, some appropriate examples have been included to support the proven theorems. Importantly, our results extend and complement some previous results in the literature and contribute to the growing body of knowledge in fractional calculus and fixed‐point theory. Lastly, we end our study by suggesting further lines of research.