Study on a kind of fourth-order p-Laplacian Rayleigh equation with linear autonomous difference operator

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• 作者：Yun Xin ; Xuefeng Han ; Zhibo Cheng
• 关键词：periodic solution ; p ; Laplacian ; fourth ; order ; linear autonomous difference operator ; Rayleigh type
• 刊名：Boundary Value Problems
• 出版年：2017
• 出版时间：December 2017
• 年：2017
• 卷：2017
• 期：1
• 全文大小：1586KB
• 刊物主题：Difference and Functional Equations; Ordinary Differential Equations; Partial Differential Equations; Analysis; Approximations and Expansions; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1687-2770
• 卷排序：2017

文摘

In this paper, we consider the following fourth-order Rayleigh type p-Laplacian generalized neutral differential equation with linear autonomous difference operator: $$ \bigl(\varphi_{p} \bigl(x(t)-c(t)x \bigl(t-\delta(t) \bigr) \bigr)'' \bigr)''+f \bigl(t,x'(t) \bigr)+g \bigl(t,x \bigl(t-\tau(t) \bigr) \bigr)=e(t). $$ By applications of coincidence degree theory and some analysis skills, sufficient conditions for the existence of periodic solutions are established.Keywordsperiodic solutionp-Laplacianfourth-orderlinear autonomous difference operatorRayleigh typeMSC34C2534K1334K401 IntroductionIn this paper, we consider the following fourth-order Rayleigh type p-Laplacian neutral differential equation with linear autonomous difference operator: $$ { } \bigl(\varphi_{p} \bigl(x(t)-c(t)x \bigl(t-\delta(t) \bigr) \bigr)'' \bigr)''+f \bigl(t,x'(t) \bigr)+g \bigl(t,x \bigl(t-\tau(t) \bigr) \bigr)=e(t), $$ (1.1) where \(p\geq2\), \(\varphi_{p}(x)=|x|^{p-2}x\) for \(x\neq0\) and \(\varphi_{p}(0)=0\); \(|c(t)|\neq1\), \(c,\delta\in C^{2}(\mathbb{R},\mathbb {R})\) and c, δ are T-periodic functions for some \(T > 0\); f and g are continuous functions defined on \(\mathbb{R}^{2}\) and periodic in t with \(f(t,\cdot)=f(t+T,\cdot)\), \(g(t,\cdot)=g(t+T,\cdot)\) and \(f(t,0)=0\), \(e, \tau:\mathbb{R}\rightarrow\mathbb{R}\) are continuous periodic functions with \(e(t+T)\equiv e(t)\) and \(\tau(t+T)\equiv\tau(t)\).In recent years, there has been a good amount of work on periodic solutions for fourth-order differential equations (see [1–17] and the references cited therein). For example, in [12], applying Mawhin’s continuation theorem, Shan and Lu studied the existence of periodic solution for a kind of fourth-order p-Laplacian functional differential equation with a deviating argument as follows: $$ { } \bigl[\varphi_{p} \bigl(u''(t) \bigr) \bigr]''+f \bigl(u(t) \bigr)u'(t)+g \bigl(t,u(t),u \bigl(t-\tau(t) \bigr) \bigr)=e(t). $$ (1.2) Afterwards, Lu and Shan [8] observed a fourth-order p-Laplacian differential equation $$ { } \bigl[\varphi_{p} \bigl(u''(t) \bigr) \bigr]''+f \bigl(u''(t) \bigr)+g \bigl(u \bigl(t-\tau(t) \bigr) \bigr)=e(t) $$ (1.3) and presented sufficient conditions for the existence of periodic solutions for (1.3). Recently, by means of Mawhin’s continuation theorem, Wang and Zhu [14] studied a kind of fourth-order p-Laplacian neutral functional differential equation $$ { } \bigl[\varphi_{p} \bigl(x(t)-cx(t-\delta) \bigr)'' \bigr]''+f \bigl(x(t) \bigr)x'(t)+g \bigl(t,x \bigl(t-\tau \bigl(t, \vert x\vert _{\infty}\bigr) \bigr) \bigr)=e(t). $$ (1.4) Some sufficient criteria to guarantee the existence of periodic solutions were obtained.However, the fourth-order p-Laplacian neutral differential equation (1.1), which includes the p-Laplacian neutral differential equation, has not attracted much attention in the literature. In this paper, we try to fill the gap and establish the existence of periodic solution of (1.1) using Mawhin’s continuation theory. Our new results generalize some recent results contained in [2, 8, 12, 14] in several aspects.