Abstract
Abstract This paper is devoted to the existence of a periodic solution for ϕ-Laplacian neutral differential equation as follows $$\begin{array}{} (\phi(x(t)-cx(t-\tau))')'=f(t,x(t),x'(t)). \end{array}$$ By applications of an extension of Mawhin’s continuous theorem due to Ge and Ren, we obtain that given equation has at least one periodic solution. Meanwhile, the approaches to estimate a priori bounds of periodic solutions are different from the corresponding ones of the known literature.
Highlights
We consider a kind of second order φ-Laplacian neutral di erential equation as follows (φ(x(t) − cx(t − τ)) ) = f (t, x(t), x (t)), (1.1)
In 2007, Zhu and Lu [1] discussed the existence of a periodic solution for a kind of p-Laplacian neutral di erential equation as follows (φp(x(t) − cx(t − τ)) ) + g(t, x(t − δ(t))) = p(t), where c is a constant and |c| ≠
In order to get around this di culty, Zhu and Lu translated the pLaplacian neutral di erential equation into a two-dimensional system (x (t) − cx (t − τ)) (t) = φq(x (t)) = |x (t)|q− x (t) x (t) = −g(t, x (t − δ(t))) + p(t), Shaowen Yao: School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China *Corresponding Author: Zhibo Cheng: School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China and Department of Mathematics, Sichuan University, Chengdu, 610064, China; E-mail: czb_1982@126.com
Summary
The topic has obvious intrinsic theoretical signi cance To answer this question, in this paper, we try to ll the gap and establish the existence of periodic solutions of (1.1) by employing the extension of Mawhin’s continuation theorem due to Ge and Ren. The obvious di culty lies in the following two aspects. Nλ is said to be M − compact in Ωif (c) there is a vector subspace Z of Z with dimZ =dimX and an operator R : Ω × [ , ] → X being continuous and compact such that for λ ∈ [ , ],
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