Positive solutions for integral nonlinear boundary value problem in fractional Sobolev spaces

Abstract

In the present paper, we consider an important problem from the point of view of application in sciences and engineering, namely, Riemann-Liouville nonlinear fractional boundary value problem. Under new minimal conditions on the parameters 0 ≤ s, 𝜏 ≤ 1, it is shown that, based on the upper and lower solutions method using Schauder fixed point theorem, the positive solutions in a Sobolev spaces exist. Moreover, our results are illustrated by a numerical example.

Existence and Uniqueness Results of Fractional Differential Inclusions and Equations in Sobolev Fractional Spaces

“…We replace the value of c 1 with the value obtained in (9). We obtain the integral equation (7). Conversely, if w satisfies ( 7) by Lemmas 1 and 2, we obtain D 0 + w(t) = y(t).…”

“…Fractional calculus is considered one of the most important areas of mathematics, which plays an important role in applications in many fields of science such as physics, biology, engineering, and others. Using different mathematical analysis techniques, many research papers were published on integral differential equations, as well as fractional differential equations (see [7][8][9][10][11][12][13][14]). For more explanations and notions related to the definitions and various issues of fractional integrals and derivatives, please see [15][16][17].…”

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

“…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 …”

“…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.…”

Recent advances in neural network methods for FDE and its application

“…The obtained fruits of this work can be effectively applied to control and design neural networks. Azzaoui et al 7 consider an important problem from the point of view of application in sciences and engineering, namely, Riemann-Liouville nonlinear fractional…”