, Volume 109, Issue 2, pp 495-505
Date: 05 Oct 2008

Unbounded Solutions for a Fractional Boundary Value Problems on the Infinite Interval

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Abstract

In this paper, we consider the fractional boundary value problem $$\left\{\begin{array}{l}\displaystyle D^{a}_{0+}u(t)+f(t,u(t))=0,\quad t\in(0,\infty),~\alpha\in (1,2),\\[2mm]\displaystyle u(0)=0,\quad\lim_{t\rightarrow\infty}D^{a-1}_{0+}u(t)=\beta u(\xi),\end{array}\right.$$ where D 0+ a is the standard Riemann-Liouville fractional derivative. By means of fixed point theorems, sufficient conditions are obtained that guarantee the existence of solutions to the above boundary value problem. The fractional modeling is a generalization of the classical integer-order differential equations and it is a very important tool for modeling the anomalous dynamics of numerous processes involving complex systems found in many diverse fields of science and engineering.

Supported by National Natural Sciences Foundation of China (10671012) and the Doctoral Program Foundation of Education Ministry of China (20050007011).