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THE EXISTENCE OF THE EXTREMAL SOLUTION FOR THE BOUNDARY VALUE PROBLEMS OF VARIABLE FRACTIONAL ORDER DIFFERENTIAL EQUATION WITH CAUSAL OPERATOR

Division of Dynamics and Control, School of Mathematics and Statistics, Shandong University of Technology, 255000 Zibo, China

E-mail Address: jjfrun@sdut.edu.cn

Corresponding author.

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JUAN L. G. GUIRAO

Departamento de Matemática Aplicaday Estadística, Universidad Politécnica de Cartagena, 30203 Cartagena, Spain

E-mail Address: juan.garcia@upct.es

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, and

TAREQ SAEED

Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, 80203, Jeddah 21589, Saudi Arabia

E-mail Address: tsalmalki@kau.edu.sa

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https://doi.org/10.1142/S0218348X20400253Cited by:12 (Source: Crossref)

This article is part of the issue:

Special Issue on Fractal and Fractional with Applications to Nature
Guest Editors: Abdon Atangana, Emile Franc Goufo Ndongmo, José Francisco Gómez Aguilar, Muhammad Altaf Khan and Ilknur Koca

Abstract

In this study, the two-point boundary value problem is considered for the variable fractional order differential equation with causal operator. Under the definition of the Caputo-type variable fractional order operators, the necessary inequality and the existence results of the solution are obtained for the variable order fractional linear differential equations according to Arzela–Ascoli theorem. Then, based on the proposed existence results and the monotone iterative technique, the existence of the extremal solution is studied, and the relative results are obtained based on the lower and upper solution. Finally, an example is provided to illustrate the validity of the theoretical results.

Keywords:

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Vol. 28, No. 08

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History

Received 23 December 2019

Accepted 26 February 2020

Published: September 1, 2020

Information

This is an Open Access article in the “Special Issue on Fractal and Fractional with Applications to Nature” published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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THE EXISTENCE OF THE EXTREMAL SOLUTION FOR THE BOUNDARY VALUE PROBLEMS OF VARIABLE FRACTIONAL ORDER DIFFERENTIAL EQUATION WITH CAUSAL OPERATOR

Abstract

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