Initial-value problems for hybrid Hadamard fractional differential equations. (English) Zbl 1300.34012

Ahmad, Bashir; Ntouyas, Sotiris K.

Electron. J. Differ. Equ. 2014, Paper No. 161, 8 p. (2014).

From the introduction: We study the existence of solutions for an initial value problem of hybrid fractional differential equations of Hadamard type given by \[ \begin{gathered} _HD^\alpha\Biggl({x(t)\over f(t,x(t))}\Biggr)= g(t, x(t)),\quad 1\leq t\leq T,\;0<\alpha\leq 1,\\ _HJ^{1-\alpha} x(t)|_{t=1}= \eta,\end{gathered} \] where \(_HD^\alpha\) is the Hadamard fractional derivative, \(f\in C([1,T]\times \mathbb{R}\), \(\mathbb{R}\setminus\{0\})\) and \(g: C([1, T]\times \mathbb{R},\mathbb{R})\), \(_H J^{(\cdot)}\) is the Hadamard fractional integral and \(\eta\in\mathbb{R}\).
The main result is proved by means of a fixed point theorem due to Dhage. An example illustrating the existence result is also presented.

MSC:

34A08	Fractional ordinary differential equations
34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

Hadamard fractional derivative; initial value problem; fixed-point theorem; existence