References
- E. Zeidler, Nonlinear Functional Analysis and Its Applications, I: Fixed-Point Theorems, Springer-Verlag, New York, 1993.
- R. I. Avery and J. Henderson, Three symmetric positive solutions for a second-order boundary value problem, Appl. Math. Lett. 13 (2000), no. 3, 1-7. https://doi.org/10.1016/S0893-9659(99)00177-9
- R. I. Avery and A. C. Peterson, Three Positive Fixed Points of Nonlinear Operators on Ordered Banach Spaces, Computers and Mathematics with Applications. 42 (2001), 313-322. https://doi.org/10.1016/S0898-1221(01)00156-0
- J. Henderson and H. B. Thompson, Multiple symmetric positive solutions for a second order boundary value problem, Proceedings American Mathematical Society. 128 (2000), 2373-2379. https://doi.org/10.1090/S0002-9939-00-05644-6
- E. Kaufmann, Multiple positive solutions for higher order boundary value problems, Rocky Mountain J. Math. 28 (1998), 1017-1028. https://doi.org/10.1216/rmjm/1181071751
- R. P. Agarwal, D. O'Regan and P. J. Y. Vong, Positive Solutions of Differential. Difference and Integral Equations, Kluwer Academic, Dordrecht, 1999.
- R. W. Leggett and L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J. 28 (1979), 673-688. https://doi.org/10.1512/iumj.1979.28.28046
- R. I. Avery, A generalization of the Leggett-Williams fixed point theorem, MSR Hotline. 2 (1998), 9-14.
- R. P. Agarwal, D. ORegan, and P. J. Y. Wong, Positive Solutions of Differential, Difference, Intergral Equations, Kluwer Academic, Dordrecht, 1999.
- F. M. Atici and P. W. Eloe, A transform method in discrete fractional calculus, Int. J. Difference Equ. 2 (2007), no. 2, 165-176.
- C. S. Goodrich, Solutions to a discrete right-focal fractional boundary value problem, Int. J. Difference Equ. 5 (2010), 195-216.
- F. M. Atici and P. W. Eloe, Two-point boundary value problems for finite fractional difference equations, J. Difference Equ. Appl. 17 (2011), 445-456. https://doi.org/10.1080/10236190903029241
- F. M. Atici and P. W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ. Spec. Ed. I (3) (2009), 1-12.
- M. Holm, Sum and difference compositions and applications in discrete fractional calculus, Cubo 13 (2011), 153-184. https://doi.org/10.4067/S0719-06462011000300009
- F. M. Atici and P. W. Eloe, Two-point boundary value problems for finite fractional difference equations, Journal of Difference Equations and Applications 17 (2011), no. 4, 445-456. https://doi.org/10.1080/10236190903029241
- C. S. Goodrich, On discrete sequential fractional boundary value problems, Journal of Mathematical Analysis and Applications 385 (2012), no. 1, 111-124. https://doi.org/10.1016/j.jmaa.2011.06.022
- C. S. Goodrich, Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions, Computers and Mathematics with Applications 61 (2011), no. 2, 191-202. https://doi.org/10.1016/j.camwa.2010.10.041
- Zh. Huang and C. Hou, Solvability of Nonlocal Fractional Boundary Value Problems, Discrete Dynamics in Nature and Society Volume 2013, Article ID 943961, 9 pages.
- Z. Xie, Y. Jin, and C. Hou, Multiple solutions for a fractional difference boundary value problem via variational approach, Abstract and Applied Analysis, vol. 2012, Article ID 143914, 16 pages.
- F. M. Atic and S. Sengul, Modeling with fractional difference equations, Journal of Mathematical Analysis and Applications 369 (2010), 1-9. https://doi.org/10.1016/j.jmaa.2010.02.009
- M. Holm, The Theory of Discrete Fractional Calculus: Development and Application, DigitalCommons@University of Nebraska-Lincoln, 2011.
- F. M. Atici and P. W. Eloe, Linear systems of fractional nabla difference equations, Rocky Mountain J. Math. 41 (2011), 353-370. https://doi.org/10.1216/RMJ-2011-41-2-353
- D. Dahal, D. Duncan, and C. S. Goodrich, Systems of semipositone dis-crete fractional boundary value problems, J. Difference Equ. Appl., doi: 10.1080/10236198.2013.856073.
- R. A. C. Ferreira, Nontrivial solutions for fractional q-difference boundary value problems, Electron. J. Qual. Theory Differ. Equ. (2010), 10.
- R. A. C. Ferreira, Positive solutions for a class of boundary value problems with fractional q-differences, Comput. Math. Appl. 61 (2011), 367-373. https://doi.org/10.1016/j.camwa.2010.11.012
- R. A. C. Ferreira, A discrete fractional Gronwall inequality, Proc. Amer. Math. Soc. 140 (2012), 1605-1612. https://doi.org/10.1090/S0002-9939-2012-11533-3
- R. A. C. Ferreira, Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one, J. Difference Equ. Appl. 19 (2013), 712-718. https://doi.org/10.1080/10236198.2012.682577
- R. A. C. Ferreira and D. F. M. Torres, Fractional h-difference equations arising from the calculus of variations, Appl. Anal. Discrete Math. 5 (2011), 110-121. https://doi.org/10.2298/AADM110131002F
- R. A. C. Ferreira and C. S. Goodrich, Positive solution for a discrete fractional periodic boundary value problem, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 19 (2012), 545-557.
- C. S. Goodrich, Continuity of solutions to discrete fractional initial value problems, Comput. Math. Appl. 59 (2010), 3489-3499. https://doi.org/10.1016/j.camwa.2010.03.040
- C. S. Goodrich, Existence of a positive solution to a system of discrete fractional boundary value problems, Appl. Math. Comput. 217 (2011), 4740-4753. https://doi.org/10.1016/j.amc.2010.11.029
- C. S. Goodrich, On a discrete fractional three-point boundary value problem, J. Difference Equ. Appl. 18 (2012), 397-415. https://doi.org/10.1080/10236198.2010.503240
- C. S. Goodrich, On a fractional boundary value problem with fractional boundary conditions, Appl. Math. Lett. 25 (2012), 1101-1105. https://doi.org/10.1016/j.aml.2011.11.028
- C. S. Goodrich, On a first-order semipositone discrete fractional boundary value problem, Arch. Math. (Basel) 99 (2012), 509-518. https://doi.org/10.1007/s00013-012-0463-2
- C. S. Goodrich, On semipositone discrete fractional boundary value problems with nonlocal boundary conditions, J. Difference Equ. Appl. 19 (2013), 1758-1780. https://doi.org/10.1080/10236198.2013.775259