Qualitative results and Lyapunov-type inequality for a new \(\varphi\)-ABC fractional boundary value problem

Volume 41, Issue 1, pp 9--24 https://dx.doi.org/10.22436/jmcs.041.01.02

Publication Date: August 29, 2025 Submission Date: February 26, 2025 Revision Date: April 21, 2025 Accteptance Date: July 19, 2025

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Authors

S. T. M. Thabet - Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai 602 105‎, Tamil Nadu, India. - Department of Mathematics, Radfan University College, University of Lahej, Lahej, Yemen. - Department of Mathematics, College of Science, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul 02814, Republic of Korea. T. Abdeljawad - Department of Fundamental Sciences, Faculty of Engineering and Architecture, Istanbul Gelisim University, Avcılar- Istanbul, 34310, Turkey. - Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia. - Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Garankuwa, Medusa 0204, South Africa. I. Kedim - Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia. B. Abdalla - Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia.

Abstract

This paper investigates a new category of a fractional boundary value problem (BVP) involving the generalized Atangana-Baleanu-Caputo (ABC) derivatives of order belonging to \((2,3].\) First, the Green function with its properties for a proposed fractional \(\mathrm{BVP}\) are derived. Then, the theoretical results are established by various techniques. The existence and uniqueness of theorems of the solutions are proved by utilizing the Weissinger, the Aghajani fixed point theorems, the Meir-Keeler condensing operator, and the Kuratowski's measure of non-compactness techniques. The sufficient conditions of the existence and non-existence results of nontrivial solutions for the proposed problem are investigated by introducing the Lyapunov-type inequality (\(\mathbb{LTI}\)). Finally, our findings are compared with the existing results in the literature. The validity of the main outcomes is also tested by numerical examples with graphs and tables, and the strict minimum borders of eigenvalues for several fractional \(\mathrm{BVP}s\) are estimated. This work is the first to deal with \(\mathbb{LTI}\) for fractional \(\mathrm{BVP}\) in the sense of generalized ABC derivatives.

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Keywords

• Fractional Atangana-Baleanu-Caputo derivative
• fractional boundary value problem
• Weissinger fixed point theorems
• Meir-Keeler condensing operator
• Lyapunov-type inequality

MSC

•  26A33
•  34B27
•  26D10

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