Oscillation of differential equations in the frame of nonlocal fractional derivatives generated by conformable derivatives

Abstract

Abstract Recently, Jarad et al. in (Adv. Differ. Equ. 2017:247, 2017) defined a new class of nonlocal generalized fractional derivatives, called conformable fractional derivatives (CFDs), based on conformable derivatives. In this paper, sufficient conditions are established for the oscillation of solutions of generalized fractional differential equations of the form {Dα,ρax(t)+f1(t,x)=r(t)+f2(t,x),t>a,limt→a+aIj−α,ρx(t)=bj(j=1,2,…,m), $$ \textstyle\begin{cases} {}_{a}\mathfrak{D}^{\alpha,\rho}x(t)+f_{1}(t,x)=r(t)+f_{2}(t,x),\quad t>a, \\ \lim_{t \to a^{+}}{ {}_{a}\mathfrak{I}^{j-\alpha,\rho}x(t)}=b_{j} \quad (j=1,2,\ldots,m), \end{cases} $$ where m=⌈α⌉ $m=\lceil\alpha\rceil$, Dα,ρa ${}_{a}\mathfrak{D}^{\alpha,\rho}$ is the left-fractional conformable derivative of order α∈C $\alpha\in\mathbb{C}$, Re(α)≥0 $\operatorname{Re}(\alpha)\geq0$ in the Riemann–Liouville setting and Iα,ρa ${}_{a}\mathfrak {I}^{\alpha,\rho}$ is the left-fractional conformable integral operator. The results are also obtained for CFDs in the Caputo setting. Examples are provided to demonstrate the effectiveness of the main result.

Keywords

• Fractional conformable integrals
• Fractional conformable derivatives
• Fractional differential equations
• Oscillation theory