Existence results for fractional differential equations in presence of upper and lower solutions

Rim Bourguiba; Rosana Rodríguez-López

doi:10.3934/dcdsb.2020180

Article Contents

2021, Volume 26, Issue 3: 1723-1747. Doi: 10.3934/dcdsb.2020180

This issue Previous Article A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems Next Article On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise

Existence results for fractional differential equations in presence of upper and lower solutions

Rim Bourguiba^1, and
Rosana Rodríguez-López^2, ,

1.
Université de Monastir, Faculté des Sciences de Monastir, , LR18ES17, 5000 Monastir, Tunisie
2.
Universidade de Santiago de Compostela, Instituto de Matemáticas, Santiago de Compostela, 15782, Spain

^* Corresponding author: Universidade de Santiago de Compostela, Instituto de Matemáticas, Santiago de Compostela, 15782, Spain
^* Corresponding author: Universidade de Santiago de Compostela, Instituto de Matemáticas, Santiago de Compostela, 15782, Spain

Received: August 2019

Revised: April 2020

Early access: June 2020

Published: March 2021

The second author is supported by grant numbers MTM2016-75140-P (AEI/FEDER, UE) and ED431C 2019/02 (GRC Xunta de Galicia)

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Abstract

In this paper, we study some existence results for fractional differential equations subject to some kind of initial conditions. First, we focus on the linear problem and we give an explicit form of solutions by reduction to an integral problem. We analyze some properties of the solutions to the linear problem in terms of its coefficients. Then we provide examples of application of the mentioned properties. Secondly, with the help of this theory, we study the nonlinear problem subject to initial value conditions. By using the upper and lower solutions method and the monotone iterative algorithm, we show the existence and localization of solutions to the nonlinear fractional differential equation.

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Mathematics Subject Classification: Primary: 34A08, 26A33; Secondary: 35G55.

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Figure 1. Graph of the solution to (17)

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Figure 2. Graph of the solution for a negative forcing term

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Table 1. Sufficient conditions for the nonnegativity of solutions

$ \begin{array}{c} f \geq 0 \mbox{ on } [0,b] \\ c\geq 0 \\ \lambda \geq 0 \\ R\geq 0 \end{array} $

$ \begin{array}{c} f \geq 0 \mbox{ on } [0,b] \\ c\geq 0 \\ \lambda < 0 \\ R\leq 0\end{array} $

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Table 2. Sufficient conditions for the nonpositivity of solutions

$ \begin{array}{c} f \leq 0 \mbox{ on } [0,b] \\ c\leq 0 \\ \lambda \geq 0 \\ R\geq 0 \end{array} $

$ \begin{array}{c} f \leq 0 \mbox{ on } [0,b] \\ c\leq 0 \\ \lambda\geq 0 \\ R\leq 0\end{array} $

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