Opuscula Math. 44, no. 2 (2024), 167-195
https://doi.org/10.7494/OpMath.2024.44.2.167

Opuscula Mathematica

# Green's functions and existence of solutions of nonlinear fractional implicit difference equations with Dirichlet boundary conditions

Nikolay D. Dimitrov

Abstract. This article is devoted to deduce the expression of the Green's function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In this case, due to the points where some of the fractional operators are applied, we are in presence of an implicit fractional difference equation. So, due to such a property, it is more complicated to calculate and manage the expression of the Green's function than in the explicit case studied in a previous work of the authors. Contrary to the explicit case, where it is shown that the Green's function is constructed as finite sums, the Green's function constructed here is an infinite series. This fact makes necessary to impose more restrictive assumptions on the parameters that appear in the equation. The expression of the Green's function will be deduced from the Laplace transform on the time scales of the integers. We point out that, despite the implicit character of the considered equation, we can have an explicit expression of the solution by means of the expression of the Green's function. These two facts are not incompatible. Even more, this method allows us to have an explicit expression of the solution of an implicit problem. Finally, we prove two existence results for nonlinear problems, via suitable fixed point theorems.

Keywords: fractional difference, Dirichlet conditions, Green's function, existence of solutions.

Mathematics Subject Classification: 26A33, 39A12, 39A27.

Full text (pdf)

1. R.P. Agarwal, M. Meehan, D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge, 2001. https://doi.org/10.1017/CBO9780511543005
2. F.M. Atici, P.W. Eloe, A transform method in discrete fractional calculus, Int. J. Differ. Equ. 2 (2007), no. 2, 165-176.
3. F.M. Atici, P.W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc. 137 (2009), no. 3, 981-989. https://doi.org/10.1090/S0002-9939-08-09626-3
4. F.M. Atici, P.W. Eloe, Two-point boundary value problems for finite fractional difference equations, J. Difference Equ. Appl. 17 (2011), no. 4, 445-456. https://doi.org/10.1080/10236190903029241
5. F.M. Atici, P.W. Eloe, Linear forward fractional difference equations, Commun. Appl. Anal. 19 (2015), 31-42.
6. M. Bohner, A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications, Birkhäuser Boston Inc., Boston, MA, 2001.
7. R. Bourguiba, A. Cabada, O.K. Wanassi, Existence of solutions of discrete fractional problem coupled to mixed fractional boundary conditions, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 116 (2022), Article no. 175. https://doi.org/10.1007/s13398-022-01321-6
8. A. Cabada, N. Dimitrov, Nontrivial solutions of non-autonomous Dirichlet fractional discrete problems, Fract. Calc. Appl. Anal. 23 (2020), no. 4, 980-995. https://doi.org/10.1515/fca-2020-0051
9. A. Cabada, N. Dimitrov, J. Jagan Mohan, Green's functions for fractional difference equations with Dirichlet boundary conditions, Chaos Solitons Fractals 153 (2021), Article no. 111455. https://doi.org/10.1016/j.chaos.2021.111455
10. R. Donahue, The Development of a Transform Method for Use in Solving Difference Equations, Master’s Thesis, University of Dayton, 1987.
11. C.S. Goodrich, On positive solutions to nonlocal fractional and integer-order difference equations, Appl. Anal. Discrete Math. 5 (2011), 122-132. https://doi.org/10.2298/AADM110111001G
12. C.S. Goodrich, An Analysis of Nonlocal Boundary Value Problems of Fractional and Integer Order, DigitalCommons@University of Nebraska - Lincoln, 2012.
13. C. Goodrich, A.C. Peterson, Discrete Fractional Calculus, Springer Cham, 2015.
14. J. Henderson, Nontrivial solutions for a nonlinear ν-th order Atici-Eloe fractional difference equation satisfying Dirichlet boundary conditions, Differ. Equ. Appl. 14 (2022), no. 2, 137-143. https://doi.org/10.7153/dea-2022-14-08
15. N. Heymans, I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheol. Acta. 45 (2006), 765-771. https://doi.org/10.1007/s00397-005-0043-5
16. A.A. Kilbas, H.H. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
17. M.A. Krasnosel'skii, P.P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer-Verlag New York, 1984.
18. R.L. Magin, Fractional Calculus in Bioengineering, Begell House Publisher, Connecticut, 2006.
19. K. Mehrez, S.M. Sitnik, Functional inequalities for the Mittag-Leffler functions, Results Math. 72 (2017), no. 1-2, 703-714. https://doi.org/10.1007/s00025-017-0664-x
20. K.S. Miller, B. Ross, Fractional difference calculus, Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, 139-152; Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 1989.
21. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
22. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives - Theory and Applications, Gordon & Breach, Linghorne, 1993.
23. N. Shobanadevi, J. Jagan Mohan, Analysis of discrete Mittag-Leffler functions, Int. J. Anal. Appl. 7 (2015), 2, 129-144.
24. B.J. West, M. Bologna, P. Grigolini, Physics of Fractal Operators, Springer, New York, 2003.
• https://orcid.org/0000-0003-1488-935X
• CITMAga, 15782, Santiago de Compostela, Galicia, Spain
• Departamento de Estatística, Análise Matemática e Optimización, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, Galicia, Spain
• Communicated by P.A. Cojuhari.
• Revised: 2023-09-04.
• Accepted: 2023-11-13.
• Published online: 2024-01-15.