Positive periodic solutions for singular high-order neutral functional differential equations

Abstract

Abstract In this paper, we establish new results on the existence of positive periodic solutions for the following high-order neutral functional differential equation (NFDE) ( x ( t ) − c x ( t − σ ) ) ( 2 m ) + f ( x ( t ) ) x ′ ( t ) + g ( t , x ( t − δ ) ) = e ( t ) . $$\begin{array}{} (x(t)-cx(t-\sigma)) ^{(2m)}+f(x(t)) x'(t)+g(t,x(t-\delta))=e(t). \end{array}$$ The interesting thing is that g has a strong singularity at x = 0 and satisfies a small force condition at x = ∞, which is different from the corresponding ones known in the literature. Two examples are given to illustrate the effectiveness of our results.