Three Positive Periodic Solutions to Nonlinear Neutral Functional Differential Equations with Parameters on Variable Time Scales

Abstract

Using two successive reductions: B-equivalence of the system on a variable time scale to a system on a time scale and a reduction to an impulsive differential equation and by Leggett-Williams fixed point theorem, we investigate the existence of three positive periodic solutions to the nonlinear neutral functional differential equation on variable time scales with a transition condition between two consecutive parts of the scale $(d/dt)\left(x\right(t)+c(t\left)x\right(t-\alpha \left)\right)=a\left(t\right)g\left(x\right(t\left)\right)x\left(t\right)-{\sum }_{j=1}^n{\lambda }_j{f}_j(t,x(t-v_j\left(t\right)\left)\right)$, $(t,x)\in {\mathbb{T}}_0\left(x\right)$,$\Delta t{|}_{(t,x)\in {\mathcal{S}}_{2i}}={\Pi }_i^1(t,x)-t$, $\Delta x{|}_{(t,x)\in {\mathcal{S}}_{2i}}={\Pi }_i^2(t,x)-x$, where ${\Pi }_i^1(t,x)=t_{2i+1}+{\tau }_{2i+1}\left({\Pi }_i^2\right(t,x\left)\right)$ and ${\Pi }_i^2(t,x)=B_{i}x+J_i\left(x\right)+x,i=\mathrm{1,2},\dots .{\lambda }_j(j=\mathrm{1,2},\dots ,n)$ are parameters, ${\mathbb{T}}_0\left(x\right)$ is a variable time scale with $(\omega ,p)$-property, $c\left(t\right),a\left(t\right)$, $v_j(t),$ and $f_j(t,x)(j=\mathrm{1,2},\dots ,n)$ are $\omega$-periodic functions of $t$, $B_{i+p}=B_i,J_{i+p}\left(x\right)=J_i\left(x\right)$ uniformly with respect to $i\in \mathbb{Z}$.