p-Laplasyenli kesirli q-fark denkleminin varlık ve teklik sonuçları

Öz

Bu çalışmanın amacı, Schaefer ve Banach'ın sabit nokta teoremlerini kullanarak p-Laplasyenli kesirli q-fark denkleminin sınır değer probleminin çözümlerinin varlığı ve tekliği üzerine bazı sonuçlar elde etmektir. Uygulama olarak, ana sonucu göstermek için bir örnek sunulmuştur.

Anahtar Kelimeler

• Kesirli q-fark denklem
• sabit nokta teoeremleri
• varlık
• teklik
• Schaefer sabit nokta teoremi
• Banach sabit nokta teoremi

Existence and uniqueness results for fractional q-difference equation with p-Laplacian

Öz

The aim of this paper is to obtain some results on the existence and uniqueness of solutions to the boundary value problem of the fractional q-difference equation with p-Laplacian using Schaefer's and Banach's fixed point theorems. As an application, an example is presented to illustrate the main result.

Anahtar Kelimeler

• Fractional q-difference equation
• fixed point theorems
• existence
• uniqueness
• Schaefer’s fixed point theorem
• Banach fixed point theorem

Kaynakça

1. Jackson, F., On q-functions and a certain difference operator, Trans. R. Soc. Edinb., 46, 253–281, (1908).
2. Jackson, F., On q-definite integrals, Pure Appl. Math. Q., 41,193–203, (1910).
3. Ahmad, B., Nietoa, J., Alsaedi, A. and Al-Hutami, H., Existence of solutions for nonlinear fractional q-difference integral equations with two fractional orders and nonlocal four-point boundary conditions, J. Franklin Inst., 351, 2890–2909, (2014).
4. Ahmad, B. and Nieto, J. J., On nonlocal boundary value problems of nonlinear q-difference equations, Adv. Difference Equ. (2012) 81.
5. Annaby, M. H. and Mansour, Z. S., Fractional q-difference equations. In: q-Fractional Calculus and Equations, Lecture Notes in Math., 2056, 223-270, (2012).
6. Bai, Z. and Lu, H., Positive solutions for boundary value problem of nonlinear fractional differential equation, J.Math. Anal. Appl., 311, 2, 495–505. (2005).
7. El-Shahed, M. and Al-Askar, F. M., Positive solutions for boundary value problem of nonlinear fractional q-difference equation, ISRN Mathematical Analysis, Article ID 385459, 12 pages, (2011).
8. Graef, J.R. and Kong, L., Positive solutions for a class of higher order boundary value problems with fractional q-derivatives, Appl. Math. Comput., 218, 9682–9689, (2012).

9. Thiramanus, P. and Tariboon, J., Nonlinear second-order q-difference equations with three-point boundary conditions, Comput. Appl. Math., 33, 385–397, (2014).
10. Zhou, W. X. and Liu, H. Z., Existence solutions for boundary value problem of nonlinear fractional q-difference equations, Adv. Difference Equ., 113, (2013).
11. Liang, S. and Zhang, J., Existence and uniqueness of positive solutions for three-point boundary value problem with fractional q-differences, J. Appl. Math. Comput., 40, 277-288, (2012).
12. El-Shahed, M. and Al-Askar, F., Positive solution for boundary value problem of nonlinear fractional q-difference equation, ISRN Math. Anal., 1-12, (2011).
13. Li, X., Han, Z. and Sun, S., Existence of positive solutions of nonlinear fractional q-difference equation with parameter, Adv. Difference Equ., 260, 1-13, (2013).
14. Ma, K., Sun, S. and Han, Z., Existence of solutions of boundary value problems for singular fractional q-difference equations, J. Appl. Math. Comput., 54, 23-40, (2017).
15. Miao, F. and Liang, S., Uniqueness of positive solutions for fractional q-difference boundary value problems with p-Laplacian operator, Electron. J. Differential Equations, 174, 1–11, (2013).
16. Rajkovi´c, P., Marinkovi´c, S. and Stankovi´c, M., Fractional integrals and derivatives in q-calculus, Appl. Anal. Discrete Math., 1,1, 311–323, (2007).
17. Ülke, Ö. and Topal, F.S., Existence and uniqueness of solutions for fractional q-difference equations, Miskolc Mathematical Notes, 24, 1, 473-487, (2023).
18. Ülke, Ö. and Topal, F.S., Existence of solutions for fractional q-difference equations, Studia Universitatis Babeș-Bolyai Mathematica, 68, 3, 573-591, (2023).
19. Jiang, M. and Zhong, S., Existence of solutions for nonlinear fractional q-difference equations with Riemann-Liouville type q-derivatives, J. Appl. Math. Comput., 47, 429–459, (2015).
20. Turan, N., Başarır, M. and Şahin, A., On the solutions of the second-order (𝑝,𝑞)-difference equation with an application to the fixed point theory, AIMS Mathematics, 9(5), 10679-10697, (2024).
21. Turan, N., Başarır, M. and Şahin, A., On the solutions of a nonlinear system of q-difference equations, Boundary Value Problems, 2024, 92, 1-19, (2024).
22. Yuan Q. and Yang W., Positive solution for q-fractional four-point boundary value problems with p-Laplacian operator, Journal of Inequalities and Applications, 481, 1-14, (2014).
23. Qin, Z., Sun, S. and Han, Z., Multiple positive solutions for nonlinear fractional q-difference equation with p-Laplacian operator, Turk J Math, 46, 2, 638 – 661, (2022).
24. Li, S., Zhang, Z. and Jiang, W., Multiple positive solutions for four-point boundary value problem of fractional delay differential equations with p-Laplacian operator, Applied Numerical Mathematics, 165, 348-356, (2021).
25. Liu, X., Jia, M. and Xiang, X., On the solvability of a fractional differential equation model involving the p-Laplacian operator, Computers and Mathematics with Applications, 64, 3267-3275, (2012).
26. Zhao, J., Positive solutions for a class of q-fractional boundary value problems with p- Laplacian, J. Nonlinear Sci. Appl., 8, 442-450, (2015).
27. Wang, J., Yu, C., Zhang, B. and Wang, S., Positive solutions for eigenvalue problems of fractional q-difference equation with φ-Laplacian, Advances in Difference Equations, 2021:499, (2021).
28. Kac, V. and Cheung, P., Quantum Calculus, Springer, New York, (2002).
29. Agarwal, R., Certain fractional q-integrals and q-derivatives, Proc. Camb. Philos. Soc., 66, 365-370, (1969).
30. Atici, F. and Eloe, P., Fractional q-calculus on a time scale, J. Nonlinear Math. Phys. 14, 3, 333-344, (2007).
31. Yu, C. L. and Wang, J. F., Existence of solutions for nonlinear second-order q-difference equations with first-order q-derivatives, Adv. Difference Equ., 124, (2013).
32. Ferreira, R.A.C., Positive solutions for a class of boundary value problems with fractional q-differences, Comput. Math. Appl., 61, 2, 367–373, (2011).
33. Schaefer, H., Über die methode der a priori-Schranken, Math. Ann., 129, 415-416, (1955).
34. Gao, H.,Li, Y. and Zhang, B., A fıxed point theorem of Krasnoselskii-Schaefer type and its applications in control and periodicity of integral equations, Fixed Point Theory, 12, 1, 91-112, (2011).