On the solutions of the second-order $ (p, q) $-difference equation with an application to the fixed-point theory

Nihan Turan; Metin Başarır; Aynur Şahin; Nihan Turan; Metin Başarır; Aynur Şahin

doi:10.3934/math.2024521

AIMS Mathematics

2024, Volume 9, Issue 5: 10679-10697. doi: 10.3934/math.2024521

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On the solutions of the second-order $ (p, q) $-difference equation with an application to the fixed-point theory

1.
Department of Mathematics, Faculty of Sciences, Sakarya University, Sakarya 54050, Turkey
2.
Department of Mathematics, Faculty of Arts and Sciences, İstanbul Beykent University, İstanbul 34500, Turkey

Received: 23 January 2024 Revised: 02 March 2024 Accepted: 08 March 2024 Published: 18 March 2024
MSC : 39A13, 39A21, 47H10

In this paper, we examined the existence and uniqueness of solutions to the second-order $ (p, q) $-difference equation with non-local boundary conditions by using the Banach fixed-point theorem. Moreover, we introduced a special case of this equation called the Euler-Cauchy-like $ (p, q) $-difference equation and provide its solution. We also studied the oscillation of solutions for this equation in $ (p, q) $-calculus and proved the $ (p, q) $-Sturm-type separation theorem and $ (p, q) $-Kneser theorem about the oscillation of solutions.
- $ (p, q) $-derivative,
- second-order difference equation,
- time scale,
- oscillations,
- fixed-points
Citation: Nihan Turan, Metin Başarır, Aynur Şahin. On the solutions of the second-order $ (p, q) $-difference equation with an application to the fixed-point theory[J]. AIMS Mathematics, 2024, 9(5): 10679-10697. doi: 10.3934/math.2024521

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Abstract

In this paper, we examined the existence and uniqueness of solutions to the second-order $ (p, q) $-difference equation with non-local boundary conditions by using the Banach fixed-point theorem. Moreover, we introduced a special case of this equation called the Euler-Cauchy-like $ (p, q) $-difference equation and provide its solution. We also studied the oscillation of solutions for this equation in $ (p, q) $-calculus and proved the $ (p, q) $-Sturm-type separation theorem and $ (p, q) $-Kneser theorem about the oscillation of solutions.

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