Existence results for a nonlinear coupled system involving both Caputo and Riemann-Liouville generalized fractional derivatives and coupled integral boundary conditions. (English) Zbl 1462.34009
Fractional Calculus has been recently an interesting topic of research for many researchers due to its wide applications in modeling various phenomena in physics and engineering. Different definitions of fractional calculus have been introduced by mathematicians. The main reason behind this special interest about this topic is that many systems can be modeled with fractional derivatives better than usual derivatives where the fractional ones can provide a good explanation to the behavior of certain physical systems. This paper studies some interesting results concerning the existence and uniqueness of solutions for nonlinear coupled system, defined in the context of generalized Caputo and Riemann-Liouville fractional derivatives, with nonlocal coupled Riemann-Stieltjes integral (generalized form of Riemann integral) boundary conditions. For more fundamental information about fractional definitions and mixed-type fractional derivatives, we refer the reader for this interesting book: [R. Almeida et al., “The variable-order fractional calculus of variations”, Preprint, arXiv:1805.0720l]. With the help of Schaefer’s fixed-point theorem and Banach contraction mapping principle, the authors prove in this research paper the existence and uniqueness of solutions (see lemma 11, theorem 12, and theorem 15 in this research paper) for their proposed boundary value problem of nonlinear coupled differential equations, defined in the context of generalized forms of Caputo and Riemann-Liouville fractional derivatives, with coupled integral boundary conditions. At the end of the results, the authors provide an example to apply their results numerically. In conclusion, investigating new proposed problems defined with the help of fractional definitions is always interesting and can provide new insights to discover the beauty of this topic of research and its applications in various field of natural sciences and engineering.
Reviewer: Mohammed Kaabar (Gelugor)
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 47N20 | Applications of operator theory to differential and integral equations |
Keywords:
Caputo and Riemann-Liouville generalized fractional derivatives; coupled system; integral boundary conditions; existence; fixed pointCitations:
Zbl 1423.00007; Zbl 1436.35323; Zbl 1437.35697; Zbl 1427.35316; Zbl 1432.34012; Zbl 1433.35185; Zbl 1428.35662; Zbl 1400.26012; Zbl 1452.35248; Zbl 1452.35237References:
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