Abstract
In this paper, we initiate the study of existence of solutions for a fractional differential system which contains mixed Riemann–Liouville and Hadamard–Caputo fractional derivatives, complemented with nonlocal coupled fractional integral boundary conditions. We derive necessary conditions for the existence and uniqueness of solutions of the considered system, by using standard fixed point theorems, such as Banach contraction mapping principle and Leray–Schauder alternative. Numerical examples illustrating the obtained results are also presented.
Highlights
Fractional differential equations have played a very important role in almost all branches of applied sciences because they are considered a valuable tool to model many real world problems
To the best of the authors’ knowledge, there are some papers dealing with sequential mixed type fractional derivatives, but we not find in the literature papers dealing with coupled systems with sequential Riemann–Liouville and Hadamard–Caputo fractional differential equations. To fill this gap, in the present paper, we investigate the existence and uniqueness of solutions for the following coupled system of sequential Riemann–Liouville and Hadamard–Caputo fractional differential equations supplemented with nonlocal coupled fractional integral boundary conditions
(i) In (2), we studied a coupled system consisting by mixed Caputo and Hadamard fractional derivatives, while, in (4), we consider mixed Riemann–Liouville and Hadamard– Caputo fractional derivatives
Summary
Fractional differential equations have played a very important role in almost all branches of applied sciences because they are considered a valuable tool to model many real world problems. There are many papers studying existence and uniqueness results for boundary value problems and coupled systems of fractional differential equations and used mixed types of fractional derivatives, see [20,21,22,23,24,25,26,27,28,29]. (iii) In both problems (4) and (2), the same method is used to establish the existence and uniqueness results, and based on standard fixed point theorems, but their presentation in the framework of mixed coupled Caputo and Hadamard and Riemann–Liouville and Hadamard–Caputo fractional derivatives is new. We emphasize that our results are new and significantly enhance the existing literature on the topic, and, as far as we know, they are the first results concerning a coupled system with sequential mixed Riemann–Liouville and Hadamard–Caputo fractional derivatives
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