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The existence of a positive solution of \(D^{\alpha}[x(t) - x(0)] = x(t) f (t,x_t)\). (English) Zbl 1121.34064

Summary: We prove the existence of a positive solution for a delay differential equation with fractional order \[ \begin{cases} D^\alpha[x(t)-x(0)]=x(t)f(t,x_t), \quad & t\in I,\\ x(t)=\varphi(t)\geq 0,\quad & t\in[-\tau,0],\end{cases} \] where \(I=[0,T]\), \(0<\alpha<1\), \(D^\alpha\) is the standard Riemann-Liouville fractional derivative, \(\varphi\in C\) and \(f:I\times C\to \mathbb R^+\) is continuous.

MSC:

34K05 General theory of functional-differential equations
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