A SURVEY ON MULTIPLICITY RESULTS FOR FRACTIONAL DIFFERENCE EQUATIONS AND VARIATIONAL METHOD

Document Type : Original Article

Author
Mohsen Khaleghi Moghadam
Department of Basic Sciences, Sari Agricultural Sciences and Natural Resources University, Sari, Iran.
10.22034/maco.3.2.5
Abstract
In this paper, we deal with the existence and multiplicity solutions, for the following fractional discrete boundary-value problem
{
T +1∇α
k (k∇α
0 (u(k))) + k∇α
0
(T +1∇α
k (u(k))) = λf(k, u(k)), k ∈ [1, T]N0
,
u(0) = u(T + 1) = 0,
where 0 ≤ α ≤ 1 and 0∇α
k is the left nabla discrete fractional difference and k∇α
T +1 is the right nabla discrete fractional difference and f : [1, T]N0 × R → R is a continuous function and λ > 0 is a parameter. The technical approach is based on the critical point theory and some local minimum theorems for differentiable functionals. Several examples are included to illustrate the main results.

Keywords

[1] T. Abdeljawad, On delta and nabla Gerasimov-Caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc. 2013 (2013).
[2] FM. Atici and PW. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ. Spec. Ed. I 2009, 3 (2009).
[3] T. Abdeljawad and F. Atici, On the definitions of nabla fractional operators, Abstr. Appl. Anal., 2012 (2012), Article ID 406757, 13 pages.
[4] V. Ambrosio and G. Molica Bisci, Periodic solutions for nonlocal fractional equations, Comm. Pure Appl. Math. 16(1) (2017) 331–344.
[5] A. Anastassiou, Discrete fractional calculus and inequalities, arXiv:0911.3370v1, (2009).
[6] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl. 1(2) (2015) 73–85.
[7] W. Dong, J. Xu and D. O’Regan, Solutions for a fractional difference boundary value problem, Adv. Difference Equ. 2013, 2013:319
[8] H. L. Gray and N. F. Zhang, On a new definition of the fractional difference, Math. Comp. 50, no. 182 (1988) 513–529.
[9] ZS. Xie, YF. Jin and CM. Hou, Multiple solutions for a fractional difference boundary value problem via variational approach, Abstr. Appl. Anal. 2012, 143914 (2012).
[10] FM. Atici and PW. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc. 137(3) (2009) 981–989.
[11] FM. Atici and PW. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ. Spec. Ed. I (3) (2009) 1–12.
[12] FM. Atici and PW. Eloe, Two-point boundary value problems for finite fractional difference equations, J. Difference Equ. Appl. 17(04) (2011) 445–456.
[13] FM. Atici and S. Şengül, Modeling with fractional difference equations, J. Math. Anal. Appl. 369 (2010) 1–9.
[14] RP. Agarwal, K. Perera, and D. O’Regan, Multiple positive solutions of singular discrete p-Laplacian problems via variational methods, Adv. Diff. Equ. 2 (2005) 93–99.
[15] RP. Agarwal, K. Perera, and D. O’Regan, Multiple positive solutions of singular and nonsingular discrete problems via variational methods, Nonl. Anal. TMA 58(2004) 69–73.
[16] G. Bonanno, Relations between the mountain pass theorem and local minima, Adv. Nonlinear Anal. 1 (2012) 205–220.
[17] G. Bonanno, P. Candito and G. D’Aguì, Variational methods on finite dimensional Banach spaces and discrete problems, Adv. Nonlinear Stud. 14 (2014) 915–939.
[18] G. Bonanno, P. Candito and G. D’Aguì, Positive solutions for a nonlinear parameter-depending algebraic system, Electron. J. Differential Equations, Vol. 2015 (2015), No. 17, pp. 1–14.
[19] G. Bonanno and G. D’Aguì, Two non-zero solutions for elliptic Dirichlet problems, Z. Anal. Anwend. 35 (2016) 449–464.
[20] G. Bonanno, B. Di Bella and D. O’Regan, Non-trivial solutions for nonlinear fourth-order elastic beam equations, Compu. Math. Appl. 62 (2011) 1862–1869.
[21] G. Bonanno and SA. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal. 89 (2010) 1–10.
[22] P. Candito, G. D’Aguì, and R. Livrea, Two positive solutions for a nonlinear parameter-depending algebraic system, Dolomites Res. Notes Approx. 14.2 (2021) 10–17.
[23] J. Chu and D. Jiang, Eigenvalues and discrete boundary value problems for the one-dimensional pLaplacian, J. Math. Anal. Appl. 305 (2005) 452–465.
[24] G. D’Aguì, J. Mawhin and A. Sciammetta, Positive solutions for a discrete two point nonlinear boundary value problem with p-Laplacian, Math. Anal. Appl. 447 (2017) 383–397.
[25] S. Dhar and L. Kong, A critical point approach to multiplicity results for a fractional boundary value problem, Bull. Malays. Math. Sci. Soc., 43.5 (2020) 3617–3633.
[26] C. Goodrich and AC. Peterson, Discrete fractional calculus, Springer international publishing Switzerland, New York, 2015.
[27] K. Ghanbari and Y. Gholami, New classes of Lyapunov type inequalities of fractional ∆-Difference SturmLiouville Problems With Applications, Bull. Iranian Math. Soc. Vol. 43 No. 2 (2017) 385 408.
[28] J. Henderson and HB. Thompson, Existence of multiple solutions for second order discrete boundary value problems, Comput. Math. Appl. 43 (2002) 1239–1248.
[29] D. Jiang, J. Chu, D. O’Regan and RP. Agarwal, Positive solutions for continuous and discrete boundary value problems to the one-dimensional p-Laplacian, Math. Inequal. Appl. 7 (2004) 523 -534.
[30] F. Jiao, Y. Zhou, Existence results for fractional boundary value problem via critical point theory, Int. J. Bifurc. Chaos Appl. Sci. Eng. 22 (2012) Article ID 1250086.
[31] M. Khaleghi Moghadam and M. Avci, Existence results to a nonlinear p(k)-Laplacian difference equation, J. Difference Equ. Appl. 23(10) (2017) 1652–1669.
[32] F. Kamache, R. Guefaifia and S. Boulaaras, Multiplicity of solutions for a class of nonlinear fractional boundary value systems via variational approach, J. Appl. Nonlinear Dyn 11.4 (2022) 789–803.
[33] M. Khaleghi Moghadam, S. Heidarkhani and J. Henderson, Infinitely many solutions for perturbed difference equations, J. Difference Equ. Appl. 207 (2014) 1055–1068.
[34] M. Khaleghi Moghadam and J. Henderson, Triple solutions for a dirichlet boundary value problem involving a perturbed discrete p(k)- laplacian operator, Open Math. J. 15 (2017) 1075–1089.
[35] M. Khaleghi Moghadam, L. Li and S. Tersian, Existence of three solutions for a discrete anisotropic boundary value problem, Bull. Iranian Math. Sco. 44.4 (2018) 1091–1107.
[36] P. Li, L. Xu, H. Wang and Y. Wang, The existence of solutions for perturbed fractional differential equations with impulses via Morse theory, Bound. Value Probl., 2020(1) (2020) 1–13.
[37] J. Liu, and W. Yu, Two solutions to Kirchhoff-type fourth-order implusive elastic beam equations, Bound. Value Probl. 2021.1 (2021): 1–10.
[38] M. Ousbika, and Z. Allali, A discrete problem involving the p(k)− Laplacian operator with three variable exponents, Int. J. Nonlinear Anal. Appl. 12.1 (2021) 521–532.
[39] SG. Samko, AA. Kilbas and OI. Marichev, Fractional integrals and derivatives: Theory and Applications, Breach Science Publishers: London, UK, 1993.
[40] A. R. A. Sanchez, and L. J. C. Morales, Existence of weak solution for a non-linear parabolic problem with fractional derivates, J. Math Computer Sci., Volume 30, Issue 3, (2023) 226–254.
Mathematical Analysis and Convex Optimization
Volume 3, Issue 2
December 2022
Pages 35-58