Positive periodic solution for p-Laplacian neutral Rayleigh equation with singularity of attractive type

In this paper, we consider a kind of p-Laplacian neutral Rayleigh equation with singularity of attractive type, (ϕp(u(t)−cu(t−δ))′)′+f(t,u′(t))+g(t,u(t))=e(t).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigl(\phi_{p} \bigl(u(t)-cu(t-\delta) \bigr)' \bigr)'+f \bigl(t,u'(t) \bigr)+g \bigl(t, u(t) \bigr)=e(t). $$\end{document} By applications of an extension of Mawhin’s continuation theorem, sufficient conditions for the existence of periodic solution are established.


Introduction
As is well known, the Rayleigh equation can be derived from many fields, such as physics, mechanics and engineering technique fields, and an important question is whether this equation can support periodic solutions. In 1977, Gaines and Mawhin [1] introduced some continuation theorems and applied this theorem to discussing the existence of solutions for the Rayleigh equation [1, p. 99] u + f u + g(t, u) = 0.
Recently there have been published some results on singular Rayleigh equations [12][13][14][15][16]. In 2015, Wang and Ma [15] investigated the following singular Rayleigh equation: where g had a repulsive singularity at the origin, i.e., (1.1) By applications of the limit properties of the time map, the authors obtained the result of the existence of periodic solution for this equation. Afterwards, by using topological degree theory, Chen and Lu [12] discussed that the existence of periodic solution for the following singular Rayleigh equations: (1. 2) The authors found new methods for estimating a lower priori bounds of periodic solutions to equation (1.2). Recently, Xin and Cheng [16] investigated a kind of neutral Rayleigh equation with singularity of repulsive type, where g(t, u) = g 1 (t, u) + g 0 (u) and g 0 had a strong singularity at u = 0, i.e., By applications of coincidence degree theory, the authors found the existence of positive periodic solution for equation (1.3).
All the aforementioned results are related to Rayleigh equation or neutral Rayleigh equation with singularity of repulsive type. Naturally, a new question arises: how p-Laplacian neutral Rayleigh equation works on singularity of attractive type? Besides practical interests, the topic has obvious intrinsic theoretical significance. To answer this question, in this paper, we consider a kind of p-Laplacian neutral Rayleigh equation with singularity of attractive type, where p > 1, ϕ p (u) = |u| p-2 u for u = 0 and ϕ p (0) = 0; |c| = 1 and δ is a constant with 0 ≤ δ < ω; e : R → R is continuous periodic functions with e(t + ω)e(t) ≡ 0 and T 0 e(t) dt = 0; f is for continuous functions defined on R 2 and periodic in t with f (t, ·) = f (t + ω, ·) and f (t, 0) = 0, g(t, u) = g 0 (u) + g 1 (t, u), here g 1 : R × (0, +∞) → R is an L 2 -Carathéodory function, g 1 (t, ·) = g 1 (t + ω, ·); g 0 ∈ C((0, ∞); R) has an attractive singularity at the origin, i.e., Obviously, the attractive condition (1.6) is in contradiction with the repulsive singularity of (1.1) and (1.4). Therefore, the above methods of [12,15,16] are no long applicable to the proof of existence of a periodic solution for (1.5) with singularity of attractive type. So we need to find a new method to get over it.
In this paper, by applications of an extension of Mawhin's continuation theorem in [17] and some analysis techniques, we see the existence of a positive periodic solution for (1.5). Our results improve and extend the results in [12,15,16].

Preliminary lemmas
For convenience, define which is a Banach space endowed with the norm · define by u = max{ u ∞ , u ∞ }, for all x, and Lemma 2.1 (see [18]) If |c| = 1, then the operator (Au)(t) := u(t)-cu(t -δ) has a continuous inverse A -1 on the space C ω , and satisfying Proof We first consider |c| < 1. From Lemma 2.1, we have Similarly, for |c| > 1, we can get Lemma 2.3 (see [19]) If u ∈ C 1 ω (R, R), and there exists a point t * ∈ [0, ω] such that |u(t * )| < d, then The following lemma involves the consequences of Theorem 3.1 of [17].
then Eq. (2.1) has at least one periodic solution on¯ .

Main results: positive periodic solution for (1.5)
In this section, we will consider the existence of a positive periodic solution for (1.5) with singularity.

Conclusions
In this article we introduce the existence of a periodic solution for a p-Laplacian neutral Rayleigh equation with singularity of attractive type. Due to the attractive condition being in contradiction with the repulsive condition, the methods of [12,15,16] are no long applicable to the proof of a periodic solution for equation (1.5) with singularity of attractive singularity. In this paper, we give attractive conditions (1.6) and (H 3 ), and we see the existence of a periodic solution for (1.5) by applications of the extension of Mawhin's continuation theorem [17]. Moreover, in view of the mathematical points, the results satisfying the conditions of an attractive singularity are valuable to understand the periodic solution for Rayleigh equations.