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

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

In this paper, we explored the existence and uniqueness of solutions for a boundary value problem involving (p, q)-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.

Keywords

(p, q)-difference equations / boundary value problems / integral conditions / existence results / fixed point theorems

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

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

[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

[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

[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

[5] R. J. Finkelstein, q-field theory , Letters Math. Phys., 34 (1995), 169-176. https://doi.org/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

[7] P. P. Kulish, E. V. Damaskinsky, On the q oscillator and the quantum algebra suq (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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[27] D. W. Boyd, J. S. Wong, On nonlinear contractions, Proc. Am. Math. Soc., 20 (1969), 458-464. https://doi.org/10.2307/2035677