Mathematica Bohemica, online first, 14 pp.

# Positive periodic solutions of a neutral functional differential equation with multiple delays

## Yongxiang Li, Ailan Liu

#### Received April 11, 2016.   First published May 18, 2017.

Yongxiang Li, Ailan Liu, Department of Mathematics, Northwest Normal University, 967 Anning East Road, Lanzhou 730070, People's Republic of China, e-mail: liyxnwnu@163.com, 15339860773@163.com

Abstract: This paper deals with the existence of positive $\omega$-periodic solutions for the neutral functional differential equation with multiple delays $(u(t)-cu(t-\delta))'+a(t) u(t)=f(t, u(t-\tau_1), \cdots, u(t-\tau_n))$. The essential inequality conditions on the existence of positive periodic solutions are obtained. These inequality conditions concern with the relations of $c$ and the coefficient function $a(t)$, and the nonlinearity $f(t, x_1,\cdots, x_n)$. Our discussion is based on the perturbation method of positive operator and fixed point index theory in cones.

Keywords: neutral delay differential equation; positive periodic solution; cone; fixed point index

Classification (MSC 2010): 34K13, 34K40, 47H11

DOI: 10.21136/MB.2017.0050-16

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