On the nonlocal hybrid $ ({\mathsf{k}}, {\rm{\mathsf{φ}}}) $-Hilfer inverse problem with delay and anticipation

Abdelkrim Salim; Sabri T. M. Thabet; Imed Kedim; Miguel Vivas-Cortez; Abdelkrim Salim; Sabri T. M. Thabet; Imed Kedim; Miguel Vivas-Cortez

doi:10.3934/math.20241112

AIMS Mathematics

2024, Volume 9, Issue 8: 22859-22882. doi: 10.3934/math.20241112

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Research article

On the nonlocal hybrid $ ({\mathsf{k}}, {\rm{\mathsf{φ}}}) $-Hilfer inverse problem with delay and anticipation

1.
Faculty of Technology, Hassiba Benbouali University of Chlef, P. O. Box 151, Chlef 02000, Algeria; a.salim@univ-chlef.dz
2.
Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes, P. O. Box 89, Sidi Bel-Abbes 22000, Algeria; salim.abdelkrim@yahoo.com
3.
Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai 602105, Tamil Nadu, India
4.
Department of Mathematics, Radfan University College, University of Lahej, Lahej, Yemen
5.
Department of Mathematics, College of Science, Korea University, Seoul 02814, South Korea
6.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia; i.kedim@psau.edu.sa
7.
Faculty of Exact and Natural Sciences, School of Physical Sciences and Mathematics, Pontifical Catholic University of Ecuador, Sede Quito, Ecuador

Received: 18 May 2024 Revised: 02 July 2024 Accepted: 08 July 2024 Published: 24 July 2024
MSC : 26A33, 34A08, 34A12

This paper focused on establishing results regarding the existence of solutions for a class of nonlocal terminal value problems involving hybrid implicit nonlinear fractional differential equations with the $ ({\mathsf{k}}, {\rm{\mathsf{φ}}}) $-Hilfer fractional derivative, which includes both finite delay and anticipation arguments. Our analysis was based on the Banach fixed point technique, and the Schauder and Krasnoselskii fixed point theorems. Moreover, illustrative examples were considered to support our new results.
- $ ({\mathsf{k}}, {\rm{\mathsf{φ}}}) $-Hilfer fractional derivative,
- hybrid equations,
- terminal value problem,
- retarded arguments,
- advanced arguments,
- existence,
- uniqueness,
- nonlocal condition
Citation: Abdelkrim Salim, Sabri T. M. Thabet, Imed Kedim, Miguel Vivas-Cortez. On the nonlocal hybrid $ ({\mathsf{k}}, {\rm{\mathsf{φ}}}) $-Hilfer inverse problem with delay and anticipation[J]. AIMS Mathematics, 2024, 9(8): 22859-22882. doi: 10.3934/math.20241112

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Abstract

This paper focused on establishing results regarding the existence of solutions for a class of nonlocal terminal value problems involving hybrid implicit nonlinear fractional differential equations with the $ ({\mathsf{k}}, {\rm{\mathsf{φ}}}) $-Hilfer fractional derivative, which includes both finite delay and anticipation arguments. Our analysis was based on the Banach fixed point technique, and the Schauder and Krasnoselskii fixed point theorems. Moreover, illustrative examples were considered to support our new results.

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