Existence of solutions for a semipositone fractional boundary value pantograph problem

Hamid Boulares; Manar A. Alqudah; Thabet Abdeljawad; Hamid Boulares; Manar A. Alqudah; Thabet Abdeljawad

doi:10.3934/math.20221070

AIMS Mathematics

2022, Volume 7, Issue 10: 19510-19519. doi: 10.3934/math.20221070

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Existence of solutions for a semipositone fractional boundary value pantograph problem

1.
Laboratory of Analysis and Control of Differential Equations "ACED", Faculty MISM, Department of Mathematics, University 8 May 1945 Guelma, Algeria
2.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, Riyadh 11671, Saudi Arabia
3.
Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
4.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan

Received: 29 June 2022 Revised: 10 August 2022 Accepted: 16 August 2022 Published: 02 September 2022
MSC : 34B10, 34B15, 26A33

The boundary value problem (BVP) for a nonlinear non positone or semi-positone multi-point Caputo-Hadamard fractional differential pantograph problem is addressed in this study.

$ \begin{equation*} \mathfrak{D}_{1}^{\upsilon}x(\mathfrak{t})+\mathrm{f}(\mathfrak{t}, x( \mathfrak{t}), x(1+\lambda\mathfrak{t})) = 0, \ \mathfrak{t}\in(1, \mathfrak{b}) \end{equation*} $

$ \begin{equation*} x(1) = \delta_{1}, \ x(\mathfrak{b}) = \sum\limits_{i = 1}^{m-2}\zeta_{i}x(\mathfrak{\eta } _{i})+\delta_{2}, \ \delta_{i}\in\mathbb{R}, \ i = 1, 2, \end{equation*} $

where $ \lambda\in\left(0, \frac{\mathfrak{b-}1}{\mathfrak{b}}\right) $. The novelty in our approach is to show that there is only one solution to this problem using the Schauder fixed point theorem. Our results expand some recent research in the field. Finally, we include an example to demonstrate our findings.
- Caputo-Hadamard fractional derivative,
- BVP,
- changing sign nonlinearity,
- pantograph problem,
- Schauder fixed point theorem
Citation: Hamid Boulares, Manar A. Alqudah, Thabet Abdeljawad. Existence of solutions for a semipositone fractional boundary value pantograph problem[J]. AIMS Mathematics, 2022, 7(10): 19510-19519. doi: 10.3934/math.20221070

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Abstract

The boundary value problem (BVP) for a nonlinear non positone or semi-positone multi-point Caputo-Hadamard fractional differential pantograph problem is addressed in this study.

$ \begin{equation*} \mathfrak{D}_{1}^{\upsilon}x(\mathfrak{t})+\mathrm{f}(\mathfrak{t}, x( \mathfrak{t}), x(1+\lambda\mathfrak{t})) = 0, \ \mathfrak{t}\in(1, \mathfrak{b}) \end{equation*} $

$ \begin{equation*} x(1) = \delta_{1}, \ x(\mathfrak{b}) = \sum\limits_{i = 1}^{m-2}\zeta_{i}x(\mathfrak{\eta } _{i})+\delta_{2}, \ \delta_{i}\in\mathbb{R}, \ i = 1, 2, \end{equation*} $

where $ \lambda\in\left(0, \frac{\mathfrak{b-}1}{\mathfrak{b}}\right) $. The novelty in our approach is to show that there is only one solution to this problem using the Schauder fixed point theorem. Our results expand some recent research in the field. Finally, we include an example to demonstrate our findings.

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