Existence of the positive solutions for boundary value problems of mixed differential equations involving the Caputo and Riemann-Liouville fractional derivatives (Q6157264)

In recent years, due to applications in mathematics, physics, biology, neural networks, and so on, the theory of fractional calculus has become the main focus of many scholars. At the same time, the theory of fractional differential equations is becoming more and more extensive and systematic. Some authors studied the existence of solutions for a class of mixed fractional differential equations. In this paper, a class of two point boundary value problems of mixed fractional differential equation systems was investigated, in which, the highest derivative terms are composite with the left-sided Riemann-Liouville and the right-sided Caputo fractional derivatives. The existence of the solutions for the boundary value problems was obtained by using the fixed-point theorem in cone expansion and compression of norm type. Finally, an example is given out to illustrate the main results.