Existence and Uniqueness of Positive Solutions for a Fractional Switched System

Abstract

We discuss the existence and uniqueness of positive solutions for the following fractional switched system: (${Dc}_{0+}^{\alpha }u(t)+f_{\sigma (t)}(t,u(t))+{\mathrm{g}}_{\sigma (t)}(t,u(t))=\mathrm{0},\mathrm{}\mathrm{t}\in J=[\mathrm{0,1}]$); $\mathrm{(u}(\mathrm{0})=u^{\prime \prime }(\mathrm{0})=\mathrm{0},u(\mathrm{1})={\int }_0^{1}u(s)\hspace{0.17em}d\mathrm{s)}$, where ${Dc}_{0+}^{\alpha }$ is the Caputo fractional derivative with , $\sigma (t):J\to \{\mathrm{1,2},\dots ,N\}$ is a piecewise constant function depending on $t$, and ${ℝ}^{+}=[0,+\mathrm{\infty })$,$f_i,g_i\in C[J\times {ℝ}^{+},{ℝ}^{+}$], $i=\mathrm{1,2},\dots ,N$. Our results are based on a fixed point theorem of a sum operator and contraction mapping principle. Furthermore, two examples are also given to illustrate the results.