## 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 $i\in \mathbb{Z}$.

## Citation

Yongkun Li. Chao Wang. "Three Positive Periodic Solutions to Nonlinear Neutral Functional Differential Equations with Parameters on Variable Time Scales." J. Appl. Math. 2012 1 - 28, 2012. https://doi.org/10.1155/2012/516476

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