LAUSR

In this paper, we develop sufficient criteria for the existence of a nonoscillatory solution to the fractional neutral functional differential equation of the form: $$\begin{aligned} D^{\alpha }_t[x(t)+c x(t-\tau )]'+\sum ^m_{i=1}P_i(t)F_i(x(t-\sigma _i))=0,\quad t\ge t_0, \end{aligned}$$Dtα[x(t)+cx(t-τ)]′+∑i=1mPi(t)Fi(x(t-σi))=0,t≥t0,where $$D_t^{\alpha }$$Dtα is Liouville fractional derivatives of order $$\alpha \ge 0$$α≥0 on the half-axis, $$c\in \mathbb {R}$$c∈R, $$\tau $$τ, $$\sigma _i\in \mathbb {R}^+$$σi∈R+, $$P_i\in C([t_0, \infty ), \mathbb {R})$$Pi∈C([t0,∞),R), $$F_i\in C(\mathbb {R}, \mathbb {R}), ~ i=1,2,\ldots ,m$$Fi∈C(R,R),i=1,2,…,m, $$m \ge 1$$m≥1 is an integer. Our results are new and improve many known results on the integer-order functional differential equations.