Multiple solutions to fractional difference boundary value problems. (English) Zbl 1469.39002

Summary: The following fractional difference boundary value problems \(\Delta^v y(t)=-f(t+v-1,y(t+v-1))\), \(y(v-2)=y(v+b+1)=0\) are considered, where \(1<v\leq 2\) is a real number and \(\Delta^v y(t)\) is the standard Riemann-Liouville fractional difference. Based on the Krasnosel’skiĭ theorem and the Schauder fixed point theorem, we establish some conditions on \(f\) which are able to guarantee that this FBVP has at least two positive solutions and one solution, respectively. Our results significantly improve and generalize those in the literature. A number of examples are given to illustrate our main results.


39A12 Discrete version of topics in analysis
34A08 Fractional ordinary differential equations
Full Text: DOI


[1] Díaz, J. B.; Osler, T. J., Differences of fractional order, Mathematics of Computation, 28, 185-202 (1974) · Zbl 0282.26007
[2] Gray, H. L.; Zhang, N. F., On a new definition of the fractional difference, Mathematics of Computation, 50, 182, 513-529 (1988) · Zbl 0648.39002 · doi:10.2307/2008620
[3] Atici, F. M.; Eloe, P. W., A transform method in discrete fractional calculus, International Journal of Difference Equations, 2, 2, 165-176 (2007)
[4] Atici, F. M.; Eloe, P. W., Initial value problems in discrete fractional calculus, Proceedings of the American Mathematical Society, 137, 3, 981-989 (2009) · Zbl 1166.39005 · doi:10.1090/S0002-9939-08-09626-3
[5] Atıcı, F. M.; Şengül, S., Modeling with fractional difference equations, Journal of Mathematical Analysis and Applications, 369, 1, 1-9 (2010) · Zbl 1204.39004 · doi:10.1016/j.jmaa.2010.02.009
[6] Atıcı, F. M.; Eloe, P. W., Two-point boundary value problems for finite fractional difference equations, Journal of Difference Equations and Applications, 17, 4, 445-456 (2011) · Zbl 1215.39002 · doi:10.1080/10236190903029241
[7] Goodrich, C. S., On nonlinear boundary conditions satisfying certain asymptotic behavior, Nonlinear Analysis: Theory, Methods & Applications, 76, 58-67 (2013) · Zbl 1264.34030 · doi:10.1016/j.na.2012.07.023
[8] Goodrich, C. S., Some new existence results for fractional difference equations, International Journal of Dynamical Systems and Differential Equations, 3, 1-2, 145-162 (2011) · Zbl 1215.39004 · doi:10.1504/IJDSDE.2011.038499
[9] Goodrich, C. S., On a fractional boundary value problem with fractional boundary conditions, Applied Mathematics Letters, 25, 8, 1101-1105 (2012) · Zbl 1266.39006 · doi:10.1016/j.aml.2011.11.028
[10] Goodrich, C. S., On a discrete fractional three-point boundary value problem, Journal of Difference Equations and Applications, 18, 3, 397-415 (2012) · Zbl 1253.26010 · doi:10.1080/10236198.2010.503240
[11] Goodrich, C. S., Continuity of solutions to discrete fractional initial value problems, Computers & Mathematics with Applications, 59, 11, 3489-3499 (2010) · Zbl 1197.39002 · doi:10.1016/j.camwa.2010.03.040
[12] Goodrich, C. S., Nonlocal systems of BVPs with asymptotically sublinear boundary conditions, Applicable Analysis and Discrete Mathematics, 6, 2, 174-193 (2012) · Zbl 1299.34065 · doi:10.2298/AADM120329010G
[13] Holm, M., Sum and difference compositions in discrete fractional calculus, Cubo, 13, 3, 153-184 (2011) · Zbl 1248.39003 · doi:10.4067/S0719-06462011000300009
[14] Wu, G.-C.; Baleanu, D., New applications of the variational iteration method-from differential equations to \(q\)-fractional difference equations, Advances in Difference Equations, 2013, article 21 (2013) · Zbl 1365.39006 · doi:10.1186/1687-1847-2013-21
[15] Kang, S. G.; Zhao, X. H.; Chen, H. Q., Positive solutions for boundary value problems of fractional difference equations depending on parameters, Advances in Difference Equations, 2013, article 376 (2013) · Zbl 1347.26022
[16] Kang, S. G.; Li, Y.; Chen, H. Q., Positive solutions to boundary value problem of fractional difference equation with nonlocal conditions, Advances in Difference Equations, 2014, article 7 (2014) · Zbl 1419.39001
[17] Agarwal, R. P.; Meehan, M.; O’Regan, D., Fixed Point Theory and Applications (2001), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0960.54027 · doi:10.1017/CBO9780511543005
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