Nonnegative Solutions of Initial Value Problems for Langevin Equations Involving Two Fractional Orders

Abstract

In this paper, we study the existence and uniqueness of nonnegative solutions of an initial value problem for Langevin equations involving two fractional orders:

$$\begin{aligned} \left\{ \begin{array}{lll}^c_0\!D^\beta _t({}^c_0\!D^\alpha _t-\gamma )x(t)=f(t,x(t)),&{}\quad \ 0< t<1,\\ x^{(k)}(0)=\mu _k,&{}\quad \ 0\le k<l,\\ x^{(\alpha +k)}(0)=\nu _k,&{}\quad \ 0\le k<n,\\ \end{array}\right. \end{aligned}$$

where $^c_0\!D^\alpha _t$ and $^c_0\!D^\beta _t$ are the Caputo fractional derivatives, $f:[0,1]\times {\mathbf {R}}\rightarrow R$ is a continuous function and $0<\gamma <\Gamma (\alpha +1)$, $m-1<\alpha \le m$, $n-1<\beta \le n$, $l=\max \{ m,n\}$, ${n,m}\in {\mathbf {N}}^+$, $\mu _j\ge 0,$ $\forall ~ j\in [0,m-1]$, $\nu _i-\gamma \mu _i\ge 0,$ $\forall ~i\in [0,n-1]$. The main tools are fixed point theorems in partially ordered metric spaces, which are different from methods used in literature. Moreover, two examples are given to illustrate the main results.