| Title:
|
Multiplicity and uniqueness for a class of discrete fractional boundary value problems (English) |
| Author:
|
Zhanmei, Lv |
| Author:
|
Yanping, Gong |
| Author:
|
Yi, Chen |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
59 |
| Issue:
|
6 |
| Year:
|
2014 |
| Pages:
|
673-695 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper deals with a class of discrete fractional boundary value problems. We construct the corresponding Green's function, analyse it in detail and establish several of its key properties. Then, by using the fixed point index theory, the existence of multiple positive solutions is obtained, and the uniqueness of the solution is proved by a new theorem on an ordered metric space established by M. Jleli, et al. (2012). (English) |
| Keyword:
|
fractional order |
| Keyword:
|
discrete fractional boundary value problem |
| Keyword:
|
fractional difference equation |
| Keyword:
|
positive solution |
| MSC:
|
26A33 |
| MSC:
|
39A05 |
| MSC:
|
39A12 |
| idZBL:
|
Zbl 06391456 |
| idMR:
|
MR3277733 |
| DOI:
|
10.1007/s10492-014-0079-x |
| . |
| Date available:
|
2014-11-10T09:19:19Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143994 |
| . |
| Reference:
|
[1] Alyousef, K.: Boundary Value Problems for Discrete Fractional Equations.Ph.D. dissertation, The University of Nebraska-Lincoln (2012). MR 3047184 |
| Reference:
|
[2] Anastassiou, G. A.: Nabla discrete fractional calculus and nabla inequalities.Math. Comput. Modelling 51 (2010), 562-571. Zbl 1190.26001, MR 2594707, 10.1016/j.mcm.2009.11.006 |
| Reference:
|
[3] Atici, F. M., Eloe, P. W.: A transform method in discrete fractional calculus.Int. J. Difference Equ. 2 (2007), 165-176. MR 2493595 |
| Reference:
|
[4] Atici, F. M., Eloe, P. W.: Initial value problems in discrete fractional calculus.Proc. Am. Math. Soc. 137 (2009), 981-989. Zbl 1166.39005, MR 2457438, 10.1090/S0002-9939-08-09626-3 |
| Reference:
|
[5] Atici, F. M., Eloe, P. W.: Linear systems of fractional nabla difference equations.Rocky Mt. J. Math. 41 (2011), 353-370. Zbl 1218.39003, MR 2794443, 10.1216/RMJ-2011-41-2-353 |
| Reference:
|
[6] cı, F. M. Atı, Eloe, P. W.: Two-point boundary value problems for finite fractional difference equations.J. Difference Equ. Appl. 17 (2011), 445-456. MR 2783359, 10.1080/10236190903029241 |
| Reference:
|
[7] cı, F. M. Atı, Eloe, P. W.: Gronwall's inequality on discrete fractional calculus.Comput. Math. Appl. 64 (2012), 3193-3200. MR 2989347, 10.1016/j.camwa.2011.11.029 |
| Reference:
|
[8] cı, F. M. Atı, $Ş$engül, S.: Modeling with fractional difference equations.J. Math. Anal. Appl. 369 (2010), 1-9. MR 2643839, 10.1016/j.jmaa.2010.02.009 |
| Reference:
|
[9] Bastos, N. R. O., Ferreira, R. A. C., Torres, D. F. M.: Discrete-time fractional variational problems.Signal Process. 91 (2011), 513-524. Zbl 1203.94022 |
| Reference:
|
[10] Deimling, K.: Nonlinear Functional Analysis.Springer, New York (1985). Zbl 0559.47040, MR 0787404 |
| Reference:
|
[11] Erbe, L. H., Hu, S. C., Wang, H. Y.: Multiple positive solutions of some boundary value problems.J. Math. Anal. Appl. 184 (1994), 640-648. Zbl 0805.34021, MR 1281534, 10.1006/jmaa.1994.1227 |
| Reference:
|
[12] Ferreira, R. A. C.: Positive solutions for a class of boundary value problems with fractional $q$-differences.Comput. Math. Appl. 61 (2011), 367-373. Zbl 1216.39013, MR 2754144, 10.1016/j.camwa.2010.11.012 |
| Reference:
|
[13] Ferreira, R. A. C.: A discrete fractional Gronwall inequality.Proc. Am. Math. Soc. 140 (2012), 1605-1612. Zbl 1243.26012, MR 2869144, 10.1090/S0002-9939-2012-11533-3 |
| Reference:
|
[14] Ferreira, R. A. C., Goodrich, C. S.: Positive solution for a discrete fractional periodic boundary value problem.Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 19 (2012), 545-557. Zbl 1268.26010, MR 3058228 |
| Reference:
|
[15] Goodrich, C. S.: Continuity of solutions to discrete fractional initial value problems.Comput. Math. Appl. 59 (2010), 3489-3499. Zbl 1197.39002, MR 2646320, 10.1016/j.camwa.2010.03.040 |
| Reference:
|
[16] Goodrich, C. S.: Solutions to a discrete right-focal fractional boundary value problem.Int. J. Difference Equ. 5 (2010), 195-216. MR 2771325 |
| Reference:
|
[17] Goodrich, C. S.: Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions.Comput. Math. Appl. 61 (2011), 191-202. Zbl 1211.39002, MR 2754129, 10.1016/j.camwa.2010.10.041 |
| Reference:
|
[18] Goodrich, C. S.: Existence of a positive solution to a system of discrete fractional boundary value problems.Appl. Math. Comput. 217 (2011), 4740-4753. Zbl 1215.39003, MR 2745153, 10.1016/j.amc.2010.11.029 |
| Reference:
|
[19] Goodrich, C. S.: Existence of a positive solution to systems of differential equations of fractional order.Comput. Math. Appl. 62 (2011), 1251-1268. Zbl 1253.34012, MR 2824712, 10.1016/j.camwa.2011.02.039 |
| Reference:
|
[20] Goodrich, C. S.: On positive solutions to nonlocal fractional and integer-order difference equations.Appl. Anal. Discrete Math. 5 (2011), 122-132. Zbl 1289.39008, MR 2809040, 10.2298/AADM110111001G |
| Reference:
|
[21] Goodrich, C. S.: On a discrete fractional three-point boundary value problem.J. Difference Equ. Appl. 18 (2012), 397-415. Zbl 1253.26010, MR 2901829, 10.1080/10236198.2010.503240 |
| Reference:
|
[22] Goodrich, C. S.: On a first-order semipositone discrete fractional boundary value problem.Arch. Math. 99 (2012), 509-518. Zbl 1263.26016, MR 3001554, 10.1007/s00013-012-0463-2 |
| Reference:
|
[23] Goodrich, C. S.: On a fractional boundary value problem with fractional boundary conditions.Appl. Math. Lett. 25 (2012), 1101-1105. Zbl 1266.39006, MR 2922519, 10.1016/j.aml.2011.11.028 |
| Reference:
|
[24] Goodrich, C. S.: On discrete sequential fractional boundary value problems.J. Math. Anal. Appl. 385 (2012), 111-124. Zbl 1236.39008, MR 2832079, 10.1016/j.jmaa.2011.06.022 |
| Reference:
|
[25] Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones.Notes and reports in mathematics in science and engineering 5 Academic Press, Boston (1988). Zbl 0661.47045, MR 0959889 |
| Reference:
|
[26] Holm, M.: The Theory of Discrete Fractional Calculus: Development and Application.Ph.D. dissertation, The University of Nebraska-Lincoln (2011). MR 2873503 |
| Reference:
|
[27] Jarad, F., Abdeljawad, T., Baleanu, D., Biçen, K.: On the stability of some discrete fractional nonautonomous systems.Abstr. Appl. Anal. 2012 (2012), Article ID 476581, 9 pages, doi:10.1155/2012/476581. Zbl 1235.93206, MR 2889092, 10.1155/2012/476581 |
| Reference:
|
[28] Jleli, M., Rajić, V. Č., Samet, B., Vetro, C.: Fixed point theorems on ordered metric spaces and applications to nonlinear elastic beam equations.J. Fixed Point Theory Appl. 12 (2012), 175-192. Zbl 1266.54089, MR 3034859, 10.1007/s11784-012-0081-4 |
| Reference:
|
[29] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and Applications of Fractional Differential Equations.North-Holland Mathematics Studies 204 Elsevier, Amsterdam (2006). Zbl 1092.45003, MR 2218073 |
| Reference:
|
[30] Lakshmikantham, V., Leela, S., Devi, J. Vasundhara: Theory of Fractional Dynamic Systems.Cambridge Academic Publishers, Cambridge (2009). |
| Reference:
|
[31] Lv, W. D.: Existence of solutions for discrete fractional boundary value problems with a $p$-Laplacian operator.Adv. Difference Equ. 2012 (2012), 10 pages, doi:10.1186/\allowbreak1687-1847-2012-163. MR 3016060, 10.1186/\allowbreak1687-1847-2012-163 |
| Reference:
|
[32] Miller, K. S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations.John Wiley & Sons New York (1993). Zbl 0789.26002, MR 1219954 |
| Reference:
|
[33] Pan, Y. Y., Han, Z. L., Sun, S., Zhao, Y.: The existence of solutions to a system of discrete fractional boundary value problems.Abstr. Appl. Anal. 2012 (2012), Article ID 707631, 15 pages, doi:10.1155/2012/707631. Zbl 1244.39006, MR 2898035, 10.1155/2012/707631 |
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