## 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

### Keywords:

Delay; Fractional differential equation
Full Text:

### References:

 [1] S. Zhang, The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl. 252 (2000), 804–812. · Zbl 0972.34004 [2] A. Babakhani, V. Daftardar-Gejji, Existence of positive solutions of nonlinear fractional differential equations, J. Math. Anal. Appl. 278 (2003), 434–442. · Zbl 1027.34003 [3] C. Bai, J. Fang, The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations, Applied Mathematics and Computation 150 (2004), 611–621. · Zbl 1061.34001 [4] V. Daftardar-Gejji, Positive solutions of a system of non-autonomous fractional differential equations, J. Math. Anal. Appl. 302 (2005), 56–64. · Zbl 1064.34004 [5] C. Bai, Positive solutions for nonlinear fractional differential equations with coefficient that changes sign, Nonlinear Analysis, 64 (2006), 677–685. · Zbl 1152.34304 [6] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. · Zbl 0924.34008 [7] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, New Jersey, 2000. · Zbl 0998.26002 [8] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. · Zbl 0789.26002 [9] Chengkui Zhong, Xianlin Fan, Wenyuan Chen, Nonlinear Functional Analysis and Its Application, Lanzhou Univ. Press, 1998.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.