## Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions.(English)Zbl 1211.39002

Summary: We consider a discrete fractional boundary value problem of the form - $$\Delta^{\nu} y(t)=f(t+\nu - 1,y(t+\nu - 1)), y(\nu - 2)=g(y), y(\nu +b)=0$$, where $$f:[\nu-1,\dots,\nu+b-1]_{\mathbb{N}_{\nu-2}}$$ is continuous, $$g:\mathcal C([\nu-2,\nu+b]_{\mathbb{N}_{\nu-2}},\mathbb{R})$$ is a given functional, and $$1<\nu \leq 2$$. We give a representation for the solution to this problem. Finally, we prove the existence and uniqueness of solution to this problem by using a variety of tools from nonlinear functional analysis including the contraction mapping theorem, the Brouwer theorem, and the Krasnosel’skii theorem.

### MSC:

 39A10 Additive difference equations 26A33 Fractional derivatives and integrals 39A12 Discrete version of topics in analysis
Full Text:

### References:

 [1] Bai, Z.; Lü, H., Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311, 495-505 (2005) · Zbl 1079.34048 [2] Benchohra, M.; Hamani, S.; Ntouyas, S. K., Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal. TMA, 71, 2391-2396 (2009) · Zbl 1198.26007 [3] Diethelm, K.; Ford, N., Analysis of fractional differential equations, J. Math. Anal. Appl., 265, 229-248 (2002) · Zbl 1014.34003 [4] Xu, X.; Jiang, D.; Yuan, C., Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear Anal. TMA, 71, 4676-4688 (2009) · Zbl 1178.34006 [5] Atici, F. M.; Eloe, P. W., Two-point boundary value problems for finite fractional difference equations, J. Difference Equ. Appl. (2010) [6] C.S. Goodrich, Solutions to a discrete right-focal fractional boundary value problem, Int. J. Difference Equations 5 (2010) (in press).; C.S. Goodrich, Solutions to a discrete right-focal fractional boundary value problem, Int. J. Difference Equations 5 (2010) (in press). [7] Atici, F. M.; Eloe, P. W., Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc., 137, 3, 981-989 (2009) · Zbl 1166.39005 [8] Goodrich, C. S., Continuity of solutions to discrete fractional initial value problems, Comput. Math. Appl., 59, 3489-3499 (2010) · Zbl 1197.39002 [9] Atici, F. M.; Eloe, P. W., Fractional $$q$$-calculus on a time scale, J. Nonlinear Math. Phys., 14, 3, 333-344 (2007) [10] Atici, F. M.; Eloe, P. W., A transform method in discrete fractional calculus, Int. J. Difference Equ., 2, 2, 165-176 (2007) [11] Atici, F. M.; Şengül, S., Modeling with fractional difference equations, J. Math. Anal. Appl. (2010) · Zbl 1204.39004 [12] Avery, R. I.; Henderson, J., Twin solutions of boundary value problems for ordinary differential equations and finite difference equations, Comput. Math. Appl., 42, 695-704 (2001) · Zbl 1006.34022 [13] Cheung, W.; Ren, J.; Wong, P. J.Y.; Zhao, D., Multiple positive solutions for discrete nonlocal boundary value problems, J. Math. Anal. Appl., 330, 900-915 (2007) · Zbl 1120.39016 [14] Rodriguez, J.; Taylor, P., Weakly nonlinear discrete multipoint boundary value problems, J. Math. Anal. Appl., 329, 77-91 (2007) · Zbl 1159.39009 [15] Agarwal, R.; Meehan, M.; O’Regan, D., Fixed Point Theory and Applications (2001), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0960.54027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.