# 一类无穷区间上的分数阶微分方程边值问题解的存在性The Existence of Solutions for a Class of Boundary Value Problem of Fractional Differential Equations on an Infinite Interval

Abstract: By using the Leray-Schauder nonlinear alternative theorem and the Leggett-Williams fixed point theorem, a class of boundary value problem for fractional differential equations with integral conditions on an infinite interval is investigated. Some sufficient conditions on the existence of at least one unbounded solution and three positive solutions are established. At last, two examples are given to illustrate the results.

[1] 白占兵. 分数阶微分方程边值问题理论及应用[M]. 北京: 中国科学技术出版社, 2013.

[2] Ji, Y.-D., Guo, Y.-P., et al. (2015) Existence of Positive Solutions of the Boundary Value Problem for Nonlinear Fractional Differential Equations. Journal of Advances in Difference Equations, 2015, 13.
https://doi.org/10.1186/s13662-014-0335-0

[3] Bai, Z.-B. (2000) The Method of Lower and Upper Solutions for a Bending of an Elastic Beam Equation. Journal of Mathematical Analysis and Applications, 248, 195-202.

[4] Zhang, S.-Q. (2009) Monotone Iterative Method for Initial Value Problem Involving Riemann-Liouville Fractional Derivatives. Journal of Nonlinear Analysis, 71, 2087-2093.

[5] Zhao, X.-K. and Ge, W.-G. (2010) Unbounded Solutions for a Fractional Boundary Value Problems on Infinite Interval. Acta Applicandae Mathematicae, 109, 495-505.
https://doi.org/10.1007/s10440-008-9329-9

[6] Wang, G.-T. (2015) Explicit Iteration and Unbounded Solutions for Fractional Integral Boundary Value Problem on an Infinite Interval. Journal of Applied Mathematics Letters, 47, 1-7.

[7] Liu, Y.S. (2003) Boundary Value Problem for Second Order Differential Equations on Unbounded Domain. Applied Mathematics and Computation, 135, 569-583.

[8] Liang, S.-H. and Zhang, J.-H. (2011) Existence of Three Positive Solutions of M-Point Boundary Value Problems for Some Nonlinear Fractional Differential Equations on an Infinite Interval. Journal of Computers and Mathematics with Applications, 61, 3343-3354.

[9] Liang, S.-H. and Zhang, J.-H. (2011) Existence of Multiple Positive Solutions for M-Point Fractional Boundary Value Problems on an Infinite Interval. Journal of Mathematical and Computer Modelling, 54, 1334-1346.

[10] Kou, C.-H., Zhou, H.-C. and Ye, Y. (2011) Existence of Solutions of Initial Value Problems for Nonlinear Fractional Differential Equations on the Half-Axis. Nonlinear Analysis, 74, 5975-5986.

[11] Su, X.-W. and Zhang, S.-Q. (2011) Unbounded Solutions to a Boundary Value Problem of Fractional Order on the Half-Line. Computers and Mathematics with Applications, 61, 1079-1087.

Top