Abstract
In this paper, we establish a Lazer-Leach-type condition depending on the delay for the existence of positive periodic solutions for singular differential equations with a deviating argument at resonance. The proof of the main result is based on the phase-plane analysis and topological degree methods.
Highlights
1 Introduction In this paper, we deal with the following delay differential equations at resonance: x + n x + g x(t), x(t – τ ) = p(t), ( . )
Where g : (, ∞) × R → R is continuous, τ ≥ is a constant, p(t) is continuous and π periodic, and the function g has a singularity of repulsive type at the origin for its first variable, that is, lims →∞ g(s, s ) = –∞
The periodic problems of singular differential equations had attracted the attentions of many researchers during more than the last two decades because of their background in applied science [ – ]
Summary
We deal with the following delay differential equations at resonance:. Wang and Ma [ ] first studied the resonant singular equation x + n x + g(x) = p(t), Li Advances in Difference Equations (2017) 2017:203 where g has a singularity and satisfies lim g(x) = g(+∞) They obtained the existence of π -periodic solutions of Wang [ ] discussed the periodic problem of the resonant Liénard equation with constant delay but without singularity and established some Lazer-Leach-type conditions depending on the delay. In Section , we state some lemmas to prove the main theorem. We take the initial point (xk(t m,k), yk(t m,k)) of the mth rotation that satisfies xk t m,k = , yk t m,k = xk t m,k > , and θk t m,k π = – (m – )π, where m = , ,.
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