Existence results for a discrete fractional boundary value problem

David Barilla; Martin Bohner; Giuseppe Caristi; Shapour Heidarkhani; Shahin Moradi; David Barilla; Martin Bohner; Giuseppe Caristi; Shapour Heidarkhani; Shahin Moradi

doi:10.3934/era.2025073

Electronic Research Archive

2025, Volume 33, Issue 3: 1541-1565. doi: 10.3934/era.2025073

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Existence results for a discrete fractional boundary value problem

1.
Department of Economics, University of Messina, via dei Verdi, Messina, Italy
2.
Department of Mathematics and Statistics, Missouri S & T, Rolla, MO 65409, USA
3.
Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah 67149, Iran

Received: 14 June 2024 Revised: 28 February 2025 Accepted: 03 March 2025 Published: 19 March 2025

In this study, we investigate the existence of at least one solution and the existence of an infinite number of solutions for a discrete fractional boundary value problem. Requiring an algebraic condition on the nonlinear term for small values of the parameter, and requiring an additional asymptotical behavior of the potential at zero, we investigate the existence of at least one nontrivial solution for the problem. Moreover, under suitable assumptions on the oscillatory behavior of the nonlinearity at infinity, for exact collections of the parameter, we discuss the existence of a sequence of solutions for the problem. We also present some examples that illustrate the applicability of the main results.
- discrete fractional,
- existence of one solution,
- existence of an infinite number of solutions,
- variational methods
Citation: David Barilla, Martin Bohner, Giuseppe Caristi, Shapour Heidarkhani, Shahin Moradi. Existence results for a discrete fractional boundary value problem[J]. Electronic Research Archive, 2025, 33(3): 1541-1565. doi: 10.3934/era.2025073

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Abstract

In this study, we investigate the existence of at least one solution and the existence of an infinite number of solutions for a discrete fractional boundary value problem. Requiring an algebraic condition on the nonlinear term for small values of the parameter, and requiring an additional asymptotical behavior of the potential at zero, we investigate the existence of at least one nontrivial solution for the problem. Moreover, under suitable assumptions on the oscillatory behavior of the nonlinearity at infinity, for exact collections of the parameter, we discuss the existence of a sequence of solutions for the problem. We also present some examples that illustrate the applicability of the main results.

References

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