## Journal of Applied Mathematics

### 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 $(d/dt)(x(t)+c(t)x(t-\alpha ))=a(t)g(x(t))x(t)-{\sum }_{j=1}^{n}{\lambda }_{j}{f}_{j}(t,x(t-{v}_{j}(t)))$, $(t,x)\in {\mathbb{T}}_{0}(x)$,$\Delta t{|}_{(t,x)\in {\mathcal{S}}_{2i}}={\Pi }_{i}^{1}(t,x)-t$, $\Delta x{|}_{(t,x)\in {\mathcal{S}}_{2i}}={\Pi }_{i}^{2}(t,x)-x$, 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 $i\in \mathbb{Z}$.

#### Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 516476, 28 pages.

Dates
First available in Project Euclid: 14 December 2012

https://projecteuclid.org/euclid.jam/1355495069

Digital Object Identifier
doi:10.1155/2012/516476

Mathematical Reviews number (MathSciNet)
MR2898062

Zentralblatt MATH identifier
1235.34243

#### Citation

Li, Yongkun; Wang, Chao. Three Positive Periodic Solutions to Nonlinear Neutral Functional Differential Equations with Parameters on Variable Time Scales. J. Appl. Math. 2012 (2012), Article ID 516476, 28 pages. doi:10.1155/2012/516476. https://projecteuclid.org/euclid.jam/1355495069