Existence of a sequence of solutions for a mixed boundary value problem for a complete Sturm-Liouville equation

Heidarkhani, Shapour; Moradi, Shahin; El-Houari, Hamza

doi:10.22075/ijnaa.2025.39544.5567

Existence of a sequence of solutions for a mixed boundary value problem for a complete Sturm-Liouville equation

Articles in Press

Document Type : Research Paper

Authors

Shapour Heidarkhani ¹

Shahin Moradi ¹

Hamza El-Houari ²

¹ Department of Mathematics, Faculty of sciences, Razi University, 67149 Kermanshah, Iran

² AMNEA Group, Department of Mathematics, Faculty of Sciences and Techniques Errachidia, University Moulay Ismail of Meknès, Morocco

10.22075/ijnaa.2025.39544.5567

Abstract

This paper establishes the existence of infinitely many weak solutions for a mixed boundary value problem for a complete Sturm-Liouville equation. Our method relies on critical point theory, and we extend and enhance several recent findings. Additionally, we provide an example to illustrate the main results.

Keywords

Sturm-Liouville

Weak solutions

Infinitely many solutions

Variational methods

[1] G.A. Afrouzi, S. Moradi, and G. Caristi, Infinitely many solutions for impulsive nonlocal elastic beam equations, Differ. Equ. Dyn. Syst. 30 (2022), 287-300.

[2] E. Amoroso, G. Bonanno, G. Molica Bisci, G. D'Aguì, S. De Caro, S. Foti, D. O'Regan, and A. Testa, Second order differential equations for the power converters dynamical performance analysis, Math. Methods Appl. Sci. 45 (2022), 5573-5591.

[3] D. Averna, N. Giovannelli, and E. Tornatore, Existence of three solutions for a mixed boundary value problem with the Sturm‑Liouville equation, Bull. Korean Math. Soc. 49 (2012), 1213-1222.

[4] G. Bonanno, G. D'Aguì, and V. Morabito, Mixed boundary value problems involving Sturm‑Liouville differential equations with possibly negative coefficients, Bound. Value Probl. 2024 (2009), 1-16.

[5] G. Bonanno and E. Tornatore, Infinitely many solutions for a mixed boundary value problem, Ann. Pol. Math. 99 (2010), 285-293.

[6] L. Chen and J. Sun, Nonlinear boundary value problem for first order impulsive functional differential equations, J. Math. Anal. Appl. 318 (2006), 726-741.

[7] J. Chu and J.J. Nieto, Impulsive periodic solutions of first‑order singular differential equations, Bull. Lond. Math. Soc. 40 (2008), 143-150.

[8] G. D'Aguì, Existence results for a mixed boundary value problem with Sturm‑Liouville equation, Adv. Pure Appl. Math. 2 (2011), 237-248.

[9] G. D'Aguì, Multiplicity results for nonlinear mixed boundary value problem, Bound. Value Prob. 2012 (2012), 1-12.

[10] G. D'Aguì, A. Sciammetta, and E. Tornatore, Two non‑zero solutions for Sturm‑Liouville equations with mixed boundary conditions, Nonlinear Anal. RWA 47 (2019), 324-331.

[11] D. Ehsani, M.R. Mokhtarzadeh, and A. Razani, Expansions of Dirichlet to Neumann operators under perturbations, Appl. Anal. 89 (2010), no. 5, 789-799.

[12] H. El‑Houari, L.S. Chadli, and H. Moussa, Multiple solutions in fractional Orlicz‑Sobolev Spaces for a class of nonlocal Kirchhoff systems, Filomat 38 (2024), no. 8, 2857-2875.

[13] M. Makvand Chaharlang and A. Razani, Two weak solutions for some Kirchhoff‑type problem with Neumann boundary condition, Georgian Math. J. 28 (2021), no. 3, 429-438.

[14] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, in: CBMS Reg. Conf. Ser. Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986.

[15] M.A. Ragusa, A. Razani, and F. Safari, Existence of radial solutions for a $p (x)$ ‑Laplacian Dirichlet problem, Adv. Difference Equ. 2021 (2021), 1-14.

[16] F. Safari and A. Razani, Nonlinear nonhomogeneous Neumann problem on the Heisenberg group, Appl. Anal. 101 (2022), no. 7, 2387-2400.

[17] F. Safari and A. Razani, Positive weak solutions of a generalized supercritical Neumann problem, Iran. J. Sci. Technol. Trans. A: Sci. 44 (2020), no. 6, 1891-1898.

[18] R. Salvati, Multiple solutions for a mixed boundary value problem, Math. Sci. Res. J. 7 (2003), 275-283.

[19] J. Sun, H. Chen, J.J. Nieto, and M. Otero‑Novoa, Multiplicity of solutions for perturbed second‑order Hamiltonian systems with impulsive effects, Nonlinear Anal. TMA 72 (2010), 4575-4586.

[20] D. Zhang and B. Dai, Infinitely many solutions for a class of nonlinear impulsive differential equations with periodic boundary conditions, Compu. Math. Appl. 61 (2011), 3153-3160.