Positive solutions for a fractional configuration of the Riemann‐Liouville semilinear differential equation

Abstract

The objective of this work is to analyze some criteria of the existence of positive solutions for a fractional configuration of the semilinear differential equation equipped with the Riemann‐Liouville operator. To achieve our aim, we first introduce an operator and transform our main problems into equivalent fixed point problems. After that, based on the fixed point theorem attributed to Krasnoselskii and the nonlinear alternative of Leray‐Schauder in a cone, we prove our main results of existence of solutions to our problems in a well‐defined Banach spaces. With the help of illustrative examples at the end of the study, we validate our theoretical outcomes according to the method implemented in theorems.

Existence study of semilinear fractional differential equations and inclusions for multi–term problem under Riemann–Liouville operators

“…In previous works, [3][4][5][6] the authors obtained approximate solutions for some multi-order and multi-term fractional boundary value problems by implementing numerical algorithms and some stability results for a system of coupled fractional differential equations have been presented in Etemad et al 7 and Rezapour et al 8 Also, many papers on the positive solution in a cone have been published recently. [9][10][11][12] For more details and to know all about the various definitions of fractional derivatives and integrals as well as its properties, readers are advised to visit the following books. [13][14][15] Fractional differential equations and inclusions generalize ordinary differential equations and inclusions to arbitrary noninteger orders.…”

“…Many works have been published in the field of fractional differential and integro‐differential equations using different techniques of mathematical analysis; for example, Boucenna et al 1 studied some existence and uniqueness results of solutions for initial value problem in a Sobolev space; also Boulfoul et al 2 established some existence and uniqueness results for integro‐differential on an unbounded domain in a weighted Banach space. In previous works, 3–6 the authors obtained approximate solutions for some multi–order and multi–term fractional boundary value problems by implementing numerical algorithms and some stability results for a system of coupled fractional differential equations have been presented in Etemad et al 7 and Rezapour et al 8 Also, many papers on the positive solution in a cone have been published recently 9–12 . For more details and to know all about the various definitions of fractional derivatives and integrals as well as its properties, readers are advised to visit the following books 13–15 …”