## Positive periodic solution for the generalized neutral differential equation with multiple delays and impulse.(English)Zbl 1406.34093

Summary: 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: $$x^\prime(t)=x(t)[a(t)-f(t,x(t),x(t-\tau_1(t,x(t))),\ldots,x(t-\tau_n(t,x(t))),x^\prime(t-\gamma_1(t,x(t))),\ldots,x^\prime(t-\gamma_m(t,x(t))))],t\neq t_k,k\in Z_+; x(t_k^+)=x(t_k^-)+\theta_k(x(t_k)),k\in Z_+$$. As applications of our results, we also give some applications to several Lotka-Volterra models and new results are obtained.

### MSC:

 34K40 Neutral functional-differential equations 34K13 Periodic solutions to functional-differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems
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