×

The existence of solutions to a system of discrete fractional boundary value problems. (English) Zbl 1244.39006

Summary: We study the existence of solutions for the boundary value problem \[ \begin{alignedat}{2}1 -\Delta^v y_1(t)& = f(y_1(t + v - 1), y_2 (t + \mu - 1)),\\ -\Delta^\mu y_2(t) &= g(y_1(t + v - 1), y_2(t + \mu - 1)),\\ y_1(v - 2)& = \Delta y_1(v + b) = 0,\\ y_2(\mu - 2)& = \Delta y_2(\mu + b) = 0,\end{alignedat} \] where \(1 < \mu, ~v \leq 2, ~f, g : \mathbb R \times \mathbb R \rightarrow \mathbb R\) are continuous functions, \(b \in \mathbb N_0\). The existence of solutions to this problem is established by the Guo-Krasnosel’kii theorem and the Schauder fixed-point theorem, and some examples are given to illustrate the main results.

MSC:

39A12 Discrete version of topics in analysis
34A08 Fractional ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495-505, 2005. · Zbl 1079.34048 · doi:10.1016/j.jmaa.2005.02.052
[2] M. Benchohra, S. Hamani, and S. K. Ntouyas, “Boundary value problems for differential equations with fractional order and nonlocal conditions,” Nonlinear Analysis, vol. 71, no. 7-8, pp. 2391-2396, 2009. · Zbl 1198.26007 · doi:10.1016/j.na.2009.01.073
[3] K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229-248, 2002. · Zbl 1014.34003 · doi:10.1006/jmaa.2001.7194
[4] X. Xu, D. Jiang, and C. Yuan, “Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation,” Nonlinear Analysis, vol. 71, no. 10, pp. 4676-4688, 2009. · Zbl 1178.34006 · doi:10.1016/j.na.2009.03.030
[5] Y. Zhao, S. Sun, Z. Han, and Q. Li, “Positive solutions to boundary value problems of nonlinear fractional differential equations,” Abstract and Applied Analysis, vol. 2011, Article ID 390543, 16 pages, 2011. · Zbl 1210.34009 · doi:10.1155/2011/390543
[6] Y. Zhao, S. Sun, Z. Han, and Q. Li, “The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 4, pp. 2086-2097, 2011. · Zbl 1221.34068 · doi:10.1016/j.cnsns.2010.08.017
[7] Y. Zhao, S. Sun, Z. Han, and M. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 217, no. 16, pp. 6950-6958, 2011. · Zbl 1227.34011 · doi:10.1016/j.amc.2011.01.103
[8] Y. Zhao, S. Sun, Z. Han, and Q. Li, “Theory of fractional hybrid differential equations,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1312-1324, 2011. · Zbl 1228.45017 · doi:10.1016/j.camwa.2011.03.041
[9] Y. Zhao, S. Sun, Z. Han, and W. Feng, “Positive solutions for a coupled system of nonlinear differential equations of mixed fractional orders,” Advances in Difference Equations, vol. 2011, article 10, 13 pages, 2011. · Zbl 1268.34035
[10] Y. Zhou, F. Jiao, and J. Li, “Existence and uniqueness for p-type fractional neutral differential equations,” Nonlinear Analysis, vol. 71, no. 7-8, pp. 2724-2733, 2009. · Zbl 1175.34082 · doi:10.1016/j.na.2009.01.105
[11] Y. Zhou, F. Jiao, and J. Li, “Existence and uniqueness for fractional neutral differential equations with infinite delay,” Nonlinear Analysis, vol. 71, no. 7-8, pp. 3249-3256, 2009. · Zbl 1177.34084 · doi:10.1016/j.na.2009.01.202
[12] Y. Zhou and F. Jiao, “Nonlocal Cauchy problem for fractional evolution equations,” Nonlinear Analysis, vol. 11, no. 5, pp. 4465-4475, 2010. · Zbl 1260.34017 · doi:10.1016/j.nonrwa.2010.05.029
[13] C. S. Goodrich, “Existence of a positive solution to a class of fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1050-1055, 2010. · Zbl 1204.34007 · doi:10.1016/j.aml.2010.04.035
[14] C. S. Goodrich, “Existence of a positive solution to systems of differential equations of fractional order,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1251-1268, 2011. · Zbl 1253.34012 · doi:10.1016/j.camwa.2011.02.039
[15] F. M. Atici and P. W. Eloe, “A transform method in discrete fractional calculus,” International Journal of Difference Equations, vol. 2, no. 2, pp. 165-176, 2007.
[16] F. M. Atici and P. W. Eloe, “Fractional q-calculus on a time scale,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 3, pp. 333-344, 2007. · Zbl 1157.81315 · doi:10.2991/jnmp.2007.14.3.4
[17] F. M. Atıcı and P. W. Eloe, “Two-point boundary value problems for finite fractional difference equations,” Journal of Difference Equations and Applications, vol. 17, no. 4, pp. 445-456, 2011. · Zbl 1215.39002 · doi:10.1080/10236190903029241
[18] F. M. Atici and P. W. Eloe, “Initial value problems in discrete fractional calculus,” Proceedings of the American Mathematical Society, vol. 137, no. 3, pp. 981-989, 2009. · Zbl 1166.39005 · doi:10.1090/S0002-9939-08-09626-3
[19] C. S. Goodrich, “Continuity of solutions to discrete fractional initial value problems,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3489-3499, 2010. · Zbl 1197.39002 · doi:10.1016/j.camwa.2010.03.040
[20] C. S. Goodrich, “Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions,” Computers & Mathematics with Applications, vol. 61, no. 2, pp. 191-202, 2011. · Zbl 1211.39002 · doi:10.1016/j.camwa.2010.10.041
[21] R. P. Agarwal, M. Meehan, and D. O’Regan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge, UK, 2001. · Zbl 0960.54027 · doi:10.1017/CBO9780511543005
[22] J. Schauder, “Der Fixpunktsatz in Funktionalräumen,” Studia Mathematica, vol. 2, pp. 171-180, 1930. · JFM 56.0355.01
[23] Y. Pan and Z. Han, “Existence of solutions for a coupled system of boundary value probelm of nonlinear fractional differential equations,” in Proceedings of the 5th International Congress on Mathematical Biology, vol. 1, pp. 109-114, 2011.
[24] C. S. Goodrich, “Solutions to a discrete right-focal fractional boundary value problem,” International Journal of Difference Equations, vol. 5, no. 2, pp. 195-216, 2010.
[25] C. S. Goodrich, “Existence of a positive solution to a system of discrete fractional boundary value problems,” Applied Mathematics and Computation, vol. 217, no. 9, pp. 4740-4753, 2011. · Zbl 1215.39003 · doi:10.1016/j.amc.2010.11.029
[26] J. Henderson, S. K. Ntouyas, and I. K. Purnaras, “Positive solutions for systems of nonlinear discrete boundary value problems,” Journal of Difference Equations and Applications, vol. 15, no. 10, pp. 895-912, 2009. · Zbl 1185.39003 · doi:10.1080/10236190802350649
[27] D. R. Dunninger and H. Wang, “Existence and multiplicity of positive solutions for elliptic systems,” Nonlinear Analysis, vol. 29, no. 9, pp. 1051-1060, 1997. · Zbl 0885.35028 · doi:10.1016/S0362-546X(96)00092-2
[28] C. S. Goodrich, “On positive solutions to nonlocal fractional and integer-order difference equations,” Applicable Analysis and Discrete Mathematics, vol. 5, no. 1, pp. 122-132, 2011. · Zbl 1289.39008 · doi:10.2298/AADM110111001G
[29] C. S. Goodrich, “On discrete sequential fractional boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 385, no. 1, pp. 111-124, 2012. · Zbl 1236.39008 · doi:10.1016/j.jmaa.2011.06.022
[30] R. A. C. Ferreira, “Positive solutions for a class of boundary value problems with fractional q-differences,” Computers & Mathematics with Applications, vol. 61, no. 2, pp. 367-373, 2011. · Zbl 1216.39013 · doi:10.1016/j.camwa.2010.11.012
[31] N. R. O. Bastos, R. A. C. Ferreira, and D. F. M. Torres, “Discrete-time fractional variational problems,” Signal Processing, vol. 91, no. 3, pp. 513-524, 2011. · Zbl 1203.94022 · doi:10.1016/j.sigpro.2010.05.001
[32] F. M. Atıcı and S. , “Modeling with fractional difference equations,” Journal of Mathematical Analysis and Applications, vol. 369, no. 1, pp. 1-9, 2010. · Zbl 1204.39004 · doi:10.1016/j.jmaa.2010.02.009
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.