Existence Theory for an Arbitrary Order Fractional Differential Equation with Deviating Argument

Abstract

In this paper, we are concerned with the existence criteria for positive solutions of the following nonlinear arbitrary order fractional differential equations with deviating argument

$$\left \{\begin{array}{l@{\quad}l}D_{0^+}^{\alpha}u(t)+h(t)f(u(\theta(t)))=0, & t\in ( 0,1 ),\ n-1<\alpha\leq n,\\[3pt]u^{(i)}(0)=0, & i=0,1,2,\ldots,n-2,\\[3pt][D_{0^+}^{\beta} u(t)]_{t=1}=0, & 1\leq\beta\leq n-2,\end{array} \right .$$

where n>3 (n∈ℕ), $D_{0^{+}}^{\alpha}$ is the standard Riemann-Liouville fractional derivative of order α,f:[0,∞)→[0,∞), h(t):[0,1]→(0,∞) and θ:(0,1)→(0,1] are continuous functions. Some novel sufficient conditions are obtained for the existence of at least one or two positive solutions by using the Krasnosel’skii’s fixed point theorem, and some other new sufficient conditions are derived for the existence of at least triple positive solutions by using the fixed point theorems developed by Leggett and Williams etc. In particular, the existence of at least n or 2n−1 distinct positive solutions is established by using the solution intervals and local properties. From the viewpoint of applications, two examples are given to illustrate the effectiveness of our results.