## Journal of Applied Mathematics

### Positive Periodic Solution for the Generalized Neutral Differential Equation with Multiple Delays and Impulse

#### Abstract

By using a fixed point theorem of strict-set-contraction, which is different from Gaines and Mawhin's continuation theorem and abstract continuation theory for $k$-set contraction, we established some new criteria for the existence of positive periodic solution of the following generalized neutral delay functional differential equation with impulse: $x\mathrm{\text{'}}(t)=x(t)[a(t)-f(t,x(t),x(t-{\tau }_{\mathrm{1}}(t,x(t))),\dots ,x(t-{\tau }_{n}(t,x(t))),x\mathrm{\text{'}}(t-{\gamma }_{\mathrm{1}}(t,x(t))),\dots ,x\mathrm{\text{'}}(t-{\gamma }_{m}(t,x(t))))], t\ne {t}_{k}, k\in {Z}_{+}; x({t}_{k}^{+})=x({t}_{k}^{-})+{\theta }_{k}(x({t}_{k})), k\in {Z}_{+}$. As applications of our results, we also give some applications to several Lotka-Volterra models and new results are obtained.

#### Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 592513, 12 pages.

Dates
First available in Project Euclid: 2 March 2015

https://projecteuclid.org/euclid.jam/1425305566

Digital Object Identifier
doi:10.1155/2014/592513

Mathematical Reviews number (MathSciNet)
MR3176824

Zentralblatt MATH identifier
07010691

#### Citation

Luo, Zhenguo; Luo, Liping; Zeng, Yunhui. Positive Periodic Solution for the Generalized Neutral Differential Equation with Multiple Delays and Impulse. J. Appl. Math. 2014 (2014), Article ID 592513, 12 pages. doi:10.1155/2014/592513. https://projecteuclid.org/euclid.jam/1425305566

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