Abstract
In this paper, we study the existence and uniqueness of solutions for a new kind of nonlocal four-point fractional integro-differential system involving both left Caputo and right Riemann–Liouville fractional derivatives, and Riemann–Liouville type mixed integrals. The Banach and Schaefer fixed point theorems are used to obtain the desired results. An example illustrating the existence and uniqueness result is presented.
Highlights
Fractional-order boundary value problems involving different kinds of fractional derivatives and boundary conditions have been investigated by many researchers in recent years
In [11], the authors carried out dynamical analysis of time fractional order phytoplankton-toxic phytoplankton–zooplankton system
For more applications of fractional calculus on bioengineering, anomalous diffusion of contamination, earth system dynamics, open channel flow, transient flow, physical models, fluid mechanics, viscoelastic fluids, etc., we refer the reader to the articles [17,18,19,20,21,22,23,24,25,26]
Summary
Fractional-order boundary value problems involving different kinds of fractional derivatives and boundary conditions have been investigated by many researchers in recent years. There are fewer works on boundary value problems containing both left and right fractional derivatives. In a more recent work [40], the authors investigated the existence of solutions for a new kind of integro-differential equation involving right-Caputo and left-Riemann–Liouville fractional derivatives of different orders and right-left Riemann–Liouville fractional integrals equipped with nonlocal boundary conditions. In this paper, motivated by aforementioned work on mixed fractional differential equations, we introduce and study a new coupled system of nonlinear fractional differential equations, involving left Caputo and right Riemann–Liouville fractional derivatives of different orders and a pair of nonlinearities with one of them in terms of mixed fractional integrals in each equation of the system, equipped with four-point nonlocal coupled boundary conditions given by β p q β p q.
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