Existence theory for nonlocal boundary value problems involving mixed fractional derivatives

Abstract

In this paper, we develop the existence theory for a new kind of nonlocal three-point boundary value problems for differential equations and inclusions involving both left Caputo and right Riemann–Liouville fractional derivatives. The Banach and Krasnoselskii fixed point theorems and the Leray–Schauder nonlinear alternative are used to obtain the desired results for the singlevalued problem. The existence of solutions for the multivalued problem concerning the upper semicontinuous and Lipschitz cases is proved by applying nonlinear alternative for Kakutani maps and Covitz and Nadler fixed point theorem. Examples illustrating the main results are also presented.

Nonlinear Integro-Differential Equations Involving Mixed Right and Left Fractional Derivatives and Integrals with Nonlocal Boundary Data

“…A nonlinear fractional oscillator equation containing left Riemann-Liouville and right Caputo fractional derivatives was investigated in [17]. In a recent paper [18], the authors proved some existence results for nonlocal boundary value problems of differential equations and inclusions containing both left Caputo and right Riemann-Liouville fractional derivatives.…”

“…Motivated by aforementioned applications of integro-differential equations and [18], we introduce 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 and solve it subject to nonlocal boundary conditions. In precise terms, we prove existence and uniqueness of solutions for the problem given by…”

On resonant mixed Caputo fractional differential equations

“…The questions linked to the existence of solutions to BVPs for fractional differential equations have been studied by researchers using different methods, here we cite some such as fixed point theorems, the upper and lower solutions, Mawhin's coincidence degree theory, Laplace transform method, iteration methods, etc. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].…”

Existence Results for a Nonlocal Coupled System of Differential Equations Involving Mixed Right and Left Fractional Derivatives and Integrals

“…The left-sided and right-sided fractional derivatives were used to formulate the fractional diffusion-advection equation to study anomalous superdiffusive transport phenomena in [35]. For further details, we refer the reader to the articles [36][37][38][39]. 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.…”