Existence and uniqueness of solutions for (<i>p, q</i>)-difference equations with integral boundary conditions

Özlem Batıt Özen; Erbil Çetin; Öyküm Ülke; Aynur Şahin; Fatma Serap Topal; Özlem Batıt Özen; Erbil Çetin; Öyküm Ülke; Aynur Şahin; Fatma Serap Topal

doi:10.3934/era.2025142

Electronic Research Archive

2025, Volume 33, Issue 5: 3225-3245. doi: 10.3934/era.2025142

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Existence and uniqueness of solutions for (p, q)-difference equations with integral boundary conditions

1.
Department of Mathematics, Ege University, Bornova 35100, Izmir, Türkiye
2.
National Defence University, Air NCO Vocational HE School, Gaziemir 35410, Izmir, Türkiye
3.
Department of Mathematics, Faculty of Sciences, Sakarya University, Sakarya 54050, Türkiye

Received: 12 February 2025 Revised: 15 May 2025 Accepted: 16 May 2025 Published: 23 May 2025

In this paper, we explored the existence and uniqueness of solutions for a boundary value problem involving $ \left(p, q\right) $-difference equations with integral conditions. By employing well-established fixed-point theorems, we established new and significant results in this area. To further illustrate the applicability of our findings, we presented three concrete examples that demonstrate the validity of the theoretical results.
- $ \left(p, q\right) $-difference equations,
- boundary value problems,
- integral conditions,
- existence results,
- fixed point theorems
Citation: Özlem Batıt Özen, Erbil Çetin, Öyküm Ülke, Aynur Şahin, Fatma Serap Topal. Existence and uniqueness of solutions for (p, q)-difference equations with integral boundary conditions[J]. Electronic Research Archive, 2025, 33(5): 3225-3245. doi: 10.3934/era.2025142

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Abstract

In this paper, we explored the existence and uniqueness of solutions for a boundary value problem involving $ \left(p, q\right) $-difference equations with integral conditions. By employing well-established fixed-point theorems, we established new and significant results in this area. To further illustrate the applicability of our findings, we presented three concrete examples that demonstrate the validity of the theoretical results.

References

[1]	B. Daşbaşı, The fractional-order mathematical modeling of bacterial resistance against multiple antibiotics in case of local bacterial infection, Sakarya Univ. J. Sci., 21 (2017), 442–453. https://doi.org/10.16984/saufenbilder.298934 doi: 10.16984/saufenbilder.298934
[2]	S. Alkan, A numerical method for solution of integro-differential equations of fractional order, Sakarya Univ. J. Sci., 21 (2017), 82–89. https://doi.org/10.16984/saufenbilder.296796 doi: 10.16984/saufenbilder.296796
[3]	S. Çetinkaya, A. Demir, Time fractional equation with non-homogenous Dirichlet boundary conditions, Sakarya Univ. J. Sci., 24 (2020), 1185–1190. https://doi.org/10.16984/saufenbilder.749168 doi: 10.16984/saufenbilder.749168
[4]	İ. Devecioğlu, R. Mutlu, A conformal fractional derivative-based leaky integrate-and-fire neuron model, Sakarya Univ. J. Sci., 26 (2022), 568–578. https://doi.org/10.16984/saufenbilder.1041088 doi: 10.16984/saufenbilder.1041088
[5]	R. J. Finkelstein, q-field theory, Letters Math. Phys., 34 (1995), 169–176. https://doi.org/10.1007/BF00739095 doi: 10.1007/BF00739095
[6]	P. G. O. Freund, A. V. Zabrodin, The spectral problem for the q-Knizhnik-Zamolodchikov equation and continuous q-Jacobi polynomials, Commun. Math. Phys., 173 (1995), 17–42. https://doi.org/10.1007/BF02100180 doi: 10.1007/BF02100180
[7]	P. P. Kulish, E. V. Damaskinsky, On the q oscillator and the quantum algebra su_q (1, 1), J. Phys. A: Math. Gen., 23 (1990), L415. https://doi.org/10.1088/0305-4470/23/9/003
[8]	N. Allouch, J. R. Graef, S. Hamani, Boundary value problem for fractional q-difference equations with integral conditions in Banach spaces, Fractal Fract., 6 (2022), 2–11. https://doi.org/10.3390/fractalfract6050237 doi: 10.3390/fractalfract6050237
[9]	R. Chakrabarti, R. Jagannathan, A (p, q)-oscillator realization of two-parameter quantum algebras, J. Phys. A: Math. Gen., 24 (1991), L711. https://doi.org/10.1088/0305-4470/24/13/002 doi: 10.1088/0305-4470/24/13/002
[10]	M. N. Hounkonnou, J. D. B. Kyemba, R (p, q)-calculus: differentiation and integration, SUT J. Math., 49 (2013), 145–167. https://doi.org/10.55937/sut/1394548362
[11]	P. N. Sadjang, On the fundamental theorem of (p, q)-calculus and some (p, q)-Taylor formulas, Results Math., 73 (2018), 1–21. https://doi.org/10.48550/arXiv.1309.3934 doi: 10.48550/arXiv.1309.3934
[12]	V. Sahai, S. Yadav, Representations of two parameter quantum algebras and p, q-special functions, J. Math. Anal. Appl., 335 (2007), 268–279. https://doi.org/10.1016/j.jmaa.2007.01.072
[13]	N. Turan, M. Başarır, A. Şahin, On the solutions of a nonlinear system of q-difference equations, Boundary Value Probl., 92 (2024), 1–19. https://doi.org/10.1186/s13661-024-01896-6 doi: 10.1186/s13661-024-01896-6
[14]	N. Turan, M. Başarır, A. Şahin, On the solutions of the second-order (p, q)-difference equation with an application to the fixed-point theory, AIMS Math., 9 (2024), 10679–10697. https://doi.org/10.3934/math.2024521 doi: 10.3934/math.2024521
[15]	İ. Gençtürk, Boundary value problems for a second-order (p, q)-difference equation with integral conditions, Turkish J. Math., 46 (2022), 499–515. https://doi.org/10.3906/mat-2106-90 doi: 10.3906/mat-2106-90
[16]	Z. Qin, S. Sun, Positive solutions for fractional (p, q)-difference boundary value problems, J. Appl. Math. Comput., 68 (2022), 2571–2588. https://doi.org/10.1007/s12190-021-01630-w
[17]	P. Neang, K. Nonlaopon, J. Tariboon, S.K. Ntouyas, B. Ahmad, Existence and uniqueness results for fractional (p, q)-difference equations with separated boundary conditions, Mathematics, 10 (2022), 1–15. https://doi.org/10.3390/math10050767
[18]	N. Kamsrisuk, C. Promsakon, S. K. Ntouyas, J. Tariboon, Nonlocal boundary value problems for (p, q)-difference equations, Differ. Equations Appl., 10 (2018), 183–195. https://doi.org: 10.7153/dea-2018-10-11
[19]	J. Soontharanon, T. Sitthiwirattham, On fractional (p, q)-calculus, Adv. Differ. Equations, 35 (2020), 1–18. https://doi.org/10.1186/s13662-020-2512-7 doi: 10.1186/s13662-020-2512-7
[20]	A. Şahin, Z. Kalkan, The AA-iterative algorithm in hyperbolic spaces with applications to integral equations on time scales, AIMS Math., 9 (2024), 24480–24506. https://doi.org/10.3934/math.20241192 doi: 10.3934/math.20241192
[21]	Z. Kalkan, A. Şahin, A. Aloqaily, N. Mlaiki, Some fixed point and stability results in b-metric-like spaces with an application to integral equations on time scales, AIMS Math., 9 (2024), 11335–11351. https://doi.org/10.3934/math.2024556 doi: 10.3934/math.2024556
[22]	A. Şahin, E. Öztürk, G. Aggarwal, Some fixed-point results for the KF-iteration process in hyperbolic metric spaces, Symmetry, 15 (2023), 1–16. https://doi.org/10.3390/sym15071360 doi: 10.3390/sym15071360
[23]	A. Şahin, Z. Kalkan, H. Arısoy, On the solution of a nonlinear Volterra integral equation with delay, Sakarya Univ. J. Sci., 21 (2017), 1367–1376. https://doi.org/10.16984/saufenbilder.305632 doi: 10.16984/saufenbilder.305632
[24]	M. A. Krasnoselskii, Two remarks on the method of successive approximations, Usp. Mat. Nauk, 10 (1955), 123–127.
[25]	S. Banach, Sur les opérations dans les ensembles abstraites et leurs applications aux équations integrals, Fundam. Math., 3 (1922), 133–181. https://doi.org/10.4064/fm-3-1-133-181 doi: 10.4064/fm-3-1-133-181
[26]	T. Sitthiwirattham, J. Tariboon, S. K. Ntouyas, Three-point boundary value problems of nonlinear second-order q-difference equations involving different numbers of q, J. Appl. Math., 2013 (2013), 1–12. http://dx.doi.org/10.1155/2013/763786 doi: 10.1155/2013/763786
[27]	D. W. Boyd, J. S. Wong, On nonlinear contractions, Proc. Am. Math. Soc., 20 (1969), 458–464. https://doi.org/10.2307/2035677 doi: 10.2307/2035677

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