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

## 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 $(d/dt)(x(t)+c(t)x(t-\alpha ))=a(t)g(x(t))x(t)-{\sum }_{j=1}^{n}{\lambda }_{j}{f}_{j}(t,x(t-{v}_{j}(t)))$, $(t,x)\in {\mathbb{T}}_{0}(x)$,$\Delta t{|}_{(t,x)\in {\mathcal{S}}_{2i}}={\Pi }_{i}^{1}(t,x)-t$, $\Delta x{|}_{(t,x)\in {\mathcal{S}}_{2i}}={\Pi }_{i}^{2}(t,x)-x$, where ${\Pi }_{i}^{1}(t,x)={t}_{2i+1}+{\tau }_{2i+1}({\Pi }_{i}^{2}(t,x))$ and ${\Pi }_{i}^{2}(t,x)={B}_{i}x+{J}_{i}(x)+x,\mathrm{ }i=1,2,\dots .\mathrm{ }{\lambda }_{j}\mathrm{ }(j=1,2,\dots ,n)$ are parameters, ${\mathbb{T}}_{0}(x)$ is a variable time scale with $(\omega ,p)$-property, $c(t),\mathrm{ }a(t)$, ${v}_{j}(t),$ and ${f}_{j}(t,x)\mathrm{ }(j=1,2,\dots ,n)$ are $\omega$-periodic functions of $t$, ${B}_{i+p}={B}_{i},\mathrm{ }{J}_{i+p}(x)={J}_{i}(x)$ 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

## Information

Published: 2012
First available in Project Euclid: 14 December 2012

zbMATH: 1235.34243
MathSciNet: MR2898062
Digital Object Identifier: 10.1155/2012/516476  