On positive solutions of fractional pantograph equations within function-dependent kernel Caputo derivatives

Ridha Dida; Hamid Boulares; Bahaaeldin Abdalla; Manar A. Alqudah; Thabet Abdeljawad; Ridha Dida; Hamid Boulares; Bahaaeldin Abdalla; Manar A. Alqudah; Thabet Abdeljawad

doi:10.3934/math.20231172

AIMS Mathematics

2023, Volume 8, Issue 10: 23032-23045. doi: 10.3934/math.20231172

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On positive solutions of fractional pantograph equations within function-dependent kernel Caputo derivatives

1.
Depatement of Mathematics, Faculty of Sciences, University Badji Mokhtar Annaba, P.O. Box 12, Annaba, 23000, Algeria
2.
Laboratory of Analysis and Control of Differential Equations "ACED'', University of 8 May 1945 Guelma, P.O. Box 401, 24000 Guelma, Algeria
3.
Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
4.
Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
5.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
6.
Department of Mathematics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Korea

Received: 24 May 2023 Revised: 02 July 2023 Accepted: 06 July 2023 Published: 19 July 2023
MSC : 26A33, 34B10, 34B15

Our main interest in this manuscript is to explore the main positive solutions (PS) and the first implications of their existence and uniqueness for a type of fractional pantograph differential equation using Caputo fractional derivatives with a kernel depending on a strictly increasing function $ \Psi $ (shortly $ \Psi $-Caputo). Such function-dependent kernel fractional operators unify and generalize several types of fractional operators such as Riemann-Liouvile, Caputo and Hadamard etc. Hence, our investigated qualitative concepts in this work generalise and unify several existing results in literature. Using Schauder's fixed point theorem (SFPT), we prove the existence of PS to this equation with the addition of the upper and lower solution method (ULS). Furthermore using the Banach fixed point theorem (BFPT), we are able to prove the existence of a unique PS. Finally, we conclude our work and give a numerical example to explain our theoretical results.
- fractional differential equations,
- fixed point theorem,
- positive solution,
- $ \Psi $-Caputo fractional derivative
Citation: Ridha Dida, Hamid Boulares, Bahaaeldin Abdalla, Manar A. Alqudah, Thabet Abdeljawad. On positive solutions of fractional pantograph equations within function-dependent kernel Caputo derivatives[J]. AIMS Mathematics, 2023, 8(10): 23032-23045. doi: 10.3934/math.20231172

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Abstract

Our main interest in this manuscript is to explore the main positive solutions (PS) and the first implications of their existence and uniqueness for a type of fractional pantograph differential equation using Caputo fractional derivatives with a kernel depending on a strictly increasing function $ \Psi $ (shortly $ \Psi $-Caputo). Such function-dependent kernel fractional operators unify and generalize several types of fractional operators such as Riemann-Liouvile, Caputo and Hadamard etc. Hence, our investigated qualitative concepts in this work generalise and unify several existing results in literature. Using Schauder's fixed point theorem (SFPT), we prove the existence of PS to this equation with the addition of the upper and lower solution method (ULS). Furthermore using the Banach fixed point theorem (BFPT), we are able to prove the existence of a unique PS. Finally, we conclude our work and give a numerical example to explain our theoretical results.

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