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

Abstract

In this paper, we study the existence of solutions for a new nonlocal boundary value problem of integro-differential equations involving mixed left and right Caputo and Riemann–Liouville fractional derivatives and Riemann–Liouville fractional integrals of different orders. Our results rely on the standard tools of functional analysis. Examples are constructed to demonstrate the application of the derived results.

On resonant mixed Caputo fractional differential equations

“…In particular to have mixed fractional terms on both sides, that is, having fractional integrals or fractional derivatives on the left-and right-hand side respectively, is an important class that is not fully studied in the literature. There are some real difficulties to examine the existence and uniqueness of solutions for these types of equations, and further properties for these types of equations have been studied by few researchers using different techniques (see, for example, [4,9,14] for partial treatment).…”

“…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].…”

“…Recently much attention has been given to the solvability of such type of differential equations that have left and right fractional derivatives. Further, several works are also devoted to this type of study, for details, see [3,4,7,[12][13][14].…”

“…In [4], the authors investigated the existence and uniqueness of solution by the use of some fixed point theorems for the following type BVP:…”

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

“…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.…”