Asymptotic behavior of positive solutions of nonlinear fractional differential equations with Caputo-type Hadamard derivative
OMS-Vol. 3 (2019), Issue 1, pp. 40–48 Open Access Full-Text PDF
Said R. Grace, Shurong Sun, Zhenlai Han
Abstract:In this paper we are concerned with the problem of asymptotic integration of positive solutions of higher order fractional differential equations with Caputo-type Hadamard derivative of the form \(^{C,H}D_{a}^r x(t)=e(t)+f(t,x(t)), \; a>1,\) where \(r = n +\alpha -1, \alpha\in (0, 1), n \in \mathbb{Z}^+\). We shall apply our technique to investigate the oscillatory and asymptotic behavior of all solutions of the integral equation \(x(t)=e(t)+\int_a ^t (\ln\frac{t}{s} )^{r-1} k(t,s)f(s,x(s))\frac{ds}{s}, \; a>1,\) \(r\) is as above.
