Existence results of solution for fractional Sturm–Liouville inclusion involving composition with multi-maps

ABSTRACT

This paper is introduced as complementary studies based on fractional Sturm–Liouville problems in a Banach space. We explore the existence results for new considered problems which can be considered as mixture of equations and inclusions. For the sake of that, we use jointly continuous composed functions with multi-valued maps and denote this form by eq-inclusion problems. The form of the solutions is calculated by the rules of Caputo derivative and the corresponding integral. The concept “continuous image of multi-valued maps” is useful to show that the strong results will be under inclusion hypothesis. The argument and fit technicals used here consider both Lipschitz and non-Lipschitz cases with using nonlinear alternative Leray Schauder type and Covitiz and Nadler theorems.

KEYWORDS:

• Sturm–Liouville operator
• fractional differential inclusion
• fixed point theorem
• existence and uniqueness
• finite separated conditions

MSC:

• 26A33
• 34A08
• 34A12
• 34A60

Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (G: 19-130-1441). The authors, therefore, gratefully acknowledge DSR technical and financial support.

Availability of data and materials

Not applicable.

Authors' contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, [grant number G: 19-130-1441]. The authors, therefore, gratefully acknowledge DSR technical and financial support.