Existence and Multiple of Positive Solution for Nonlinear Fractional Difference Equations with Parameter
Author(s)
Youji Xu
ABSTRACT
Let
,
. We study the existence and multiple positive solutions of n-th nonlinear discrete fractional boundary value problem of the form 
By using a fixed-point
theorem on cone, the parameter intervals of problem is established.
KEYWORDS
Fractional Difference Equations, Parameter Intervals, Positive Solution, Fixed-Point Theorem
Fractional Difference Equations, Parameter Intervals, Positive Solution, Fixed-Point Theorem
Cite this paper
Xu, Y. (2015) Existence and Multiple of Positive Solution for Nonlinear Fractional Difference Equations with Parameter. Journal of Applied Mathematics and Physics, 3, 757-760. doi: 10.4236/jamp.2015.37091.
Xu, Y. (2015) Existence and Multiple of Positive Solution for Nonlinear Fractional Difference Equations with Parameter. Journal of Applied Mathematics and Physics, 3, 757-760. doi: 10.4236/jamp.2015.37091.
References
[1] Miller, K.S. and Ross, B. (1988) Fractional Difference Calculus. Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, 139-152; Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 1989. [2] Agrawal, O.P. (2002) Formulation of Euler-Lagrange Equations for Fractional Variational Problems. Journal of Mathematical Analysis and Applications, 272, 368-379. http://dx.doi.org/10.1016/S0022-247X(02)00180-4 [3] Podlubny, I. (1999) Fractional Differential Equations. Academic Press, New York. [4] Lakshmikantham, V. and Vatsala, A.S. (2008) Basic Theory of Fractional Differential Equations. Nonlinear Analysis: Theory, Methods & Applications, 69, 2677-2682. http://dx.doi.org/10.1016/j.na.2007.08.042 [5] Bai, Z. and Lü, H. (2005) Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation. Journal of Mathematical Analysis and Applications, 311, 495-505. http://dx.doi.org/10.1016/j.jmaa.2005.02.052 [6] Salem, H.A.H. (2009) On the Fractional Order m-Point Boundary Value Problem in Reflexive Banach Spaces and Weak Topologies. Journal of Computational and Applied Mathematics, 224, 567-572. http://dx.doi.org/10.1016/j.cam.2008.05.033 [7] Atici, F.M. and Eloe, P.W. (2007) A Transform Method in Discrete Fractional Calculus. International Journal of Difference Equations, 2, 165-176. [8] Goodrich, C.S. (2010) Continuity of Solutions to Discrete Fractional Initial Value Problems. Computers and Mathematics with Applications, 59, 3489-3499. http://dx.doi.org/10.1016/j.camwa.2010.03.040 [9] Atici, F.M. and Eloe, P.W. (2009) Initial Value Prob-lems in Discrete Fractional Calculus. Proceedings of the American Mathematical Society, 137, 981-989. http://dx.doi.org/10.1090/S0002-9939-08-09626-3 [10] Atici, F.M. and Eloe, P.W. (2011) Two-Point Boundary Value Problems for Finite Fractional Difference Equations. Journal of Difference Equations and Applications, 17, 445-456. http://dx.doi.org/10.1080/10236190903029241 [11] Goodrich, C.S. (2010) Solutions to a Discrete Right-Focal Fractional Boundary Value Problem. International Journal of Difference Equations, 5, 195-216. [12] Guo, D. and Laksmikantham, V. (1988) Nonlinear Problems in Abstract Cones. Academic Press, London.
[1] Miller, K.S. and Ross, B. (1988) Fractional Difference Calculus. Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, 139-152; Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 1989. [2] Agrawal, O.P. (2002) Formulation of Euler-Lagrange Equations for Fractional Variational Problems. Journal of Mathematical Analysis and Applications, 272, 368-379. http://dx.doi.org/10.1016/S0022-247X(02)00180-4 [3] Podlubny, I. (1999) Fractional Differential Equations. Academic Press, New York. [4] Lakshmikantham, V. and Vatsala, A.S. (2008) Basic Theory of Fractional Differential Equations. Nonlinear Analysis: Theory, Methods & Applications, 69, 2677-2682. http://dx.doi.org/10.1016/j.na.2007.08.042 [5] Bai, Z. and Lü, H. (2005) Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation. Journal of Mathematical Analysis and Applications, 311, 495-505. http://dx.doi.org/10.1016/j.jmaa.2005.02.052 [6] Salem, H.A.H. (2009) On the Fractional Order m-Point Boundary Value Problem in Reflexive Banach Spaces and Weak Topologies. Journal of Computational and Applied Mathematics, 224, 567-572. http://dx.doi.org/10.1016/j.cam.2008.05.033 [7] Atici, F.M. and Eloe, P.W. (2007) A Transform Method in Discrete Fractional Calculus. International Journal of Difference Equations, 2, 165-176. [8] Goodrich, C.S. (2010) Continuity of Solutions to Discrete Fractional Initial Value Problems. Computers and Mathematics with Applications, 59, 3489-3499. http://dx.doi.org/10.1016/j.camwa.2010.03.040 [9] Atici, F.M. and Eloe, P.W. (2009) Initial Value Prob-lems in Discrete Fractional Calculus. Proceedings of the American Mathematical Society, 137, 981-989. http://dx.doi.org/10.1090/S0002-9939-08-09626-3 [10] Atici, F.M. and Eloe, P.W. (2011) Two-Point Boundary Value Problems for Finite Fractional Difference Equations. Journal of Difference Equations and Applications, 17, 445-456. http://dx.doi.org/10.1080/10236190903029241 [11] Goodrich, C.S. (2010) Solutions to a Discrete Right-Focal Fractional Boundary Value Problem. International Journal of Difference Equations, 5, 195-216. [12] Guo, D. and Laksmikantham, V. (1988) Nonlinear Problems in Abstract Cones. Academic Press, London.