Theory and Modern Applications

# Existence and uniqueness of a positive periodic solution for Rayleigh type ϕ-Laplacian equation

## Abstract

By using the Manásevich-Mawhin continuation theorem and some analysis skills, we establish some sufficient condition for the existence and uniqueness of positive T-periodic solutions for a generalized Rayleigh type ϕ-Laplacian operator equation. The results of this paper are new and they complement previous known results.

MSC:34K13, 34C25.

## 1 Introduction

During the past few years, many researchers have discussed the periodic solutions of a Rayleigh type differential equation (see [110]). For example, in 2009, Xiao and Liu [7] studied the Rayleigh type p-Laplacian equation with a deviating argument of the form

${\left({\varphi }_{p}\left({x}^{\prime }\left(t\right)\right)\right)}^{\prime }+f\left(t,{x}^{\prime }\left(t\right)\right)+g\left(t,x\left(t-\tau \left(t\right)\right)\right)=e\left(t\right).$

By using the coincidence degree theory, we establish new results on the existence of periodic solutions for the above equation. Afterward, Xiong and Shao [9] used the coincidence degree theory to establish new results on the existence and uniqueness of positive T-periodic solutions for the Rayleigh type p-Laplacian equation of the form

${\left({\varphi }_{p}\left({x}^{\prime }\left(t\right)\right)\right)}^{\prime }+f\left(t,{x}^{\prime }\left(t\right)\right)+g\left(t,x\left(t\right)\right)=e\left(t\right).$

In this paper, we consider the following Rayleigh type ϕ-Laplacian operator equation:

${\left(\varphi \left({x}^{\prime }\left(t\right)\right)\right)}^{\prime }+f\left(t,{x}^{\prime }\left(t\right)\right)+g\left(t,x\left(t\right)\right)=e\left(t\right),$
(1.1)

where the function $\varphi :\mathbb{R}\to \mathbb{R}$ is continuous and $\varphi \left(0\right)=0$. $f,g\in Car\left(\mathbb{R}×\mathbb{R},\mathbb{R}\right)$ is an ${L}^{p}$-Carathéodory function and $p=\frac{m}{m-1}$, $m\ge 2$, which means it is measurable in the first variable and continuous in the second variable. For every $0, there exists ${h}_{r,s}\in {L}^{p}\left[0,T\right]$ such that $|g\left(t,x\left(t\right)\right)|\le {h}_{r,s}$ for all $x\in \left[r,s\right]$ and a.e. $t\in \left[0,T\right]$; and f, g is a T-periodic function about t and $f\left(t,0\right)=0$. $e\in {L}^{p}\left(\left[0,T\right],\mathbb{R}\right)$ and is T-periodic.

Here $\varphi :\mathbb{R}\to \mathbb{R}$ is a continuous function and $\varphi \left(0\right)=0$, which satisfies

(A1) $\left(\varphi \left({x}_{1}\right)-\varphi \left({x}_{2}\right)\right)\left({x}_{1}-{x}_{2}\right)>0$ for $\mathrm{\forall }{x}_{1}\ne {x}_{2}$, ${x}_{1},{x}_{2}\in \mathbb{R}$;

(A2) there exists a function $\alpha :\left[0,+\mathrm{\infty }\right]\to \left[0,+\mathrm{\infty }\right]$, $\alpha \left(s\right)\to +\mathrm{\infty }$ as $s\to +\mathrm{\infty }$, such that $\varphi \left(x\right)\cdot x\ge \alpha \left(|x|\right)|x|$ for $\mathrm{\forall }x\in \mathbb{R}$.

It is easy to see that ϕ represents a large class of nonlinear operators, including ${\varphi }_{p}:\mathbb{R}\to \mathbb{R}$ is a p-Laplacian, i.e., ${\varphi }_{p}\left(x\right)={|x|}^{p-2}x$ for $x\in \mathbb{R}$.

We know that the study on ϕ-Laplacian is relatively infrequent, the main difficulty lies in the fact that the ϕ-Laplacian operator typically possesses more uncertainty than the p-Laplacian operator. For example, the key step for ${\varphi }_{p}$ to get a priori solutions, ${\int }_{0}^{T}{\left({\varphi }_{p}^{\prime }\left(x\left(t\right)\right)\right)}^{\prime }x\left(t\right)\phantom{\rule{0.2em}{0ex}}dt=-{\int }_{0}^{T}{|{x}^{\prime }\left(t\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dt$, is no longer available for general ϕ-Laplacian. So, we need to find a new method to get over it.

By using the Manásevich-Mawhin continuation theorem and some analysis skills, we establish some sufficient condition for the existence of positive T-periodic solutions of (1.1). The results of this paper are new and they complement previous known results.

## 2 Main results

For convenience, define

which is a Banach space endowed with the norm $\parallel \cdot \parallel$; define $\parallel x\parallel =max\left\{{|x|}_{0},{|{x}^{\prime }|}_{0}\right\}$ for all x, and

${|x|}_{0}=\underset{t\in \left[0,T\right]}{max}|x\left(t\right)|,\phantom{\rule{2em}{0ex}}|{x}^{\prime }{|}_{0}=\underset{t\in \left[0,T\right]}{max}|{x}^{\prime }\left(t\right)|.$

For the T-periodic boundary value problem

${\left(\varphi \left({x}^{\prime }\left(t\right)\right)\right)}^{\prime }=\stackrel{˜}{f}\left(t,x,{x}^{\prime }\right),$
(2.1)

here $\stackrel{˜}{f}:\left[0,T\right]×\mathbb{R}×\mathbb{R}\to \mathbb{R}$ is assumed to be Carathéodory.

Lemma 2.1 (Manásevich-Mawhin [11])

Let Ω be an open bounded set in ${C}_{T}^{1}$. If

1. (i)

for each $\lambda \in \left(0,1\right)$, the problem

${\left(\varphi \left({x}^{\prime }\right)\right)}^{\prime }=\lambda \stackrel{˜}{f}\left(t,x,{x}^{\prime }\right),\phantom{\rule{2em}{0ex}}x\left(0\right)=x\left(T\right),\phantom{\rule{2em}{0ex}}{x}^{\prime }\left(0\right)={x}^{\prime }\left(T\right)$

has no solution on Ω;

1. (ii)

the equation

$F\left(a\right):=\frac{1}{T}{\int }_{0}^{T}\stackrel{˜}{f}\left(t,x,{x}^{\prime }\right)\phantom{\rule{0.2em}{0ex}}dt=0$

has no solution on $\partial \mathrm{\Omega }\cap \mathbb{R}$;

1. (iii)

the Brouwer degree of F

$deg\left\{F,\mathrm{\Omega }\cap \mathbb{R},0\right\}\ne 0.$

Then the periodic boundary value problem (2.1) has at least one periodic solution on $\overline{\mathrm{\Omega }}$.

Lemma 2.2 If $\varphi \left(x\right)$ is bounded, then x is also bounded.

Proof Since $\varphi \left(x\right)$ is bounded, then there exists a positive constant N such that $|\varphi \left(x\right)|\le N$. From (A2), we have $\alpha \left(|x|\right)|x|\le \varphi \left(x\right)\cdot x\le |\varphi \left(x\right)|\cdot |x|\le N|x|$. Hence, we can get $\alpha \left(|x|\right)\le N$ for all $x\in \mathbb{R}$. If x is not bounded, then from the definition of α, we get $\alpha \left(|x|\right)>N$ for some $x\in \mathbb{R}$, which is a contradiction. So x is also bounded. □

Lemma 2.3 Suppose that the following condition holds:

(A3) $\left({x}_{1}-{x}_{2}\right)\left(g\left(t,{x}_{1}\right)-g\left(t,{x}_{2}\right)\right)<0$ for all t, ${x}_{1},{x}_{2}\in \mathbb{R}$, ${x}_{1}\ne {x}_{2}$.

Then (1.1) has at most one T-periodic solution in ${C}_{T}^{1}$.

Proof Assume that ${x}_{1}\left(t\right)$ and ${x}_{2}\left(t\right)$ are two T-periodic solutions of (1.1). Then we obtain

${\left(\varphi \left({x}_{1}^{\prime }\left(t\right)\right)-\varphi \left({x}_{2}^{\prime }\left(t\right)\right)\right)}^{\prime }+f\left(t,{x}_{1}^{\prime }\left(t\right)\right)-f\left(t,{x}_{2}^{\prime }\left(t\right)\right)+g\left(t,{x}_{1}\left(t\right)\right)-g\left(t,{x}_{2}\left(t\right)\right)=0.$
(2.2)

Set $u\left(t\right)={x}_{1}\left(t\right)-{x}_{2}\left(t\right)$. Now, we claim that

In contrast, in view of ${x}_{1},{x}_{2}\in {C}^{1}\left[0,T\right]$, for $t\in \mathbb{R}$, we obtain

$\underset{t\in \mathbb{R}}{max}u\left(t\right)>0.$

Then there must exist ${t}^{\ast }\in \mathbb{R}$ (for convenience, we can choose ${t}^{\ast }\in \left(0,T\right)$) such that

$u\left({t}^{\ast }\right)=\underset{t\in \left[0,T\right]}{max}u\left(t\right)=\underset{t\in \mathbb{R}}{max}u\left(t\right)>0,$

which implies that

${u}^{\prime }\left({t}^{\ast }\right)={x}_{1}^{\prime }\left({t}^{\ast }\right)-{x}_{2}^{\prime }\left({t}^{\ast }\right)=0$

and

${x}_{1}\left({t}^{\ast }\right)-{x}_{2}\left({t}^{\ast }\right)>0.$

By hypothesis (A3) and (2.2), we have

$\begin{array}{rcl}{\left(\varphi \left({x}_{1}^{\prime }\left({t}^{\ast }\right)\right)-\varphi \left({x}_{2}^{\prime }\left({t}^{\ast }\right)\right)\right)}^{\prime }& =& -\left[f\left({t}^{\ast },{x}_{1}^{\prime }\left({t}^{\ast }\right)\right)-f\left({t}^{\ast },{x}_{2}^{\prime }\left({t}^{\ast }\right)\right)\right]\\ -\left[g\left({t}^{\ast },{x}_{1}\left({t}^{\ast }\right)\right)-g\left({t}^{\ast },{x}_{2}\left({t}^{\ast }\right)\right)\right]\\ =& -\left[g\left({t}^{\ast },{x}_{1}\left({t}^{\ast }\right)\right)-g\left({t}^{\ast },{x}_{2}\left({t}^{\ast }\right)\right)\right]>0,\end{array}$

and there exists $\epsilon >0$ such that ${\left(\varphi \left({x}_{1}^{\prime }\left(t\right)\right)-\varphi \left({x}_{2}^{\prime }\left(t\right)\right)\right)}^{\prime }>0$ for all $t\in \left({t}^{\ast }-\epsilon ,{t}^{\ast }\right]$. Therefore, $\varphi \left({x}_{1}^{\prime }\left(t\right)\right)-\varphi \left({x}_{2}^{\prime }\left(t\right)\right)$ is strictly increasing for $t\in \left({t}^{\ast }-\epsilon ,{t}^{\ast }\right]$, which implies that

From (A1) we get

This contradicts the definition of ${t}^{\ast }$. Thus,

By using a similar argument, we can also show that

${x}_{2}\left(t\right)-{x}_{1}\left(t\right)\le 0.$

Therefore, we obtain

Hence, (1.1) has at most one T-periodic solution in ${C}_{T}^{1}$. The proof of Lemma 2.3 is now complete. □

For the sake of convenience, we list the following assumptions which will be used repeatedly in the sequel:

(H1) there exists a positive constant D such that $g\left(t,x\right)-e\left(t\right)<0$ for $x>D$ and $t\in \mathbb{R}$, $g\left(t,x\right)-e\left(t\right)>0$ for $x\le 0$ and $t\in \mathbb{R}$;

(H2) there exist constants $\sigma >0$ and $m\ge 2$ such that $f\left(t,u\right)u\ge \sigma {|u|}^{m}$ for $\left(t,u\right)\in \left[0,T\right]×\mathbb{R}$;

(H3) there exist positive constants ρ and γ such that $|f\left(t,u\right)|\le \rho {|u|}^{m-1}+\gamma$ for $\left(t,u\right)\in \left[0,T\right]×\mathbb{R}$;

(H4) there exist positive constants α, β, B such that

By using Lemmas 2.1-2.3, we obtain our main results.

Theorem 2.1 Assume that conditions (H1)-(H4) and (A3) hold. Then (1.1) has a unique positive T-periodic solution if $\sigma -\frac{\alpha {T}^{m-1}}{{2}^{m-1}}>0$.

Proof Consider the homotopic equation of (1.1) as follows:

${\left(\varphi \left({x}^{\prime }\left(t\right)\right)\right)}^{\prime }+\lambda f\left(t,{x}^{\prime }\left(t\right)\right)+\lambda g\left(t,x\left(t\right)\right)=\lambda e\left(t\right).$
(2.3)

By Lemma 2.3, it is easy to see that (1.1) has at most one T-periodic solution in ${C}_{T}^{1}$. Thus, to prove Theorem 2.1, it suffices to show that (1.1) has at least one T-periodic solution in ${C}_{T}^{1}$. To do this, we are going to apply Lemmas 2.1 and 2.2. Firstly, we will claim that the set of all possible T-periodic solutions of (2.3) is bounded. Let $x\left(t\right)\in {C}_{T}^{1}$ be an arbitrary solution of (2.3) with period T. As $x\left(0\right)=x\left(T\right)$, there exists ${t}_{0}\in \left[0,T\right]$ such that ${x}^{\prime }\left({t}_{0}\right)=0$, while $\varphi \left(0\right)=0$, we see

$\begin{array}{rcl}|\varphi \left({x}^{\prime }\left(t\right)\right)|& =& |{\int }_{{t}_{0}}^{t}{\left(\varphi \left({x}^{\prime }\left(s\right)\right)\right)}^{\prime }\phantom{\rule{0.2em}{0ex}}ds|\\ \le & \lambda {\int }_{0}^{T}|f\left(t,{x}^{\prime }\left(t\right)\right)|\phantom{\rule{0.2em}{0ex}}dt+\lambda {\int }_{0}^{T}|g\left(t,x\left(t\right)\right)|\phantom{\rule{0.2em}{0ex}}dt+\lambda {\int }_{0}^{T}|e\left(t\right)|\phantom{\rule{0.2em}{0ex}}dt,\end{array}$
(2.4)

where $t\in \left[{t}_{0},{t}_{0}+T\right]$.

We claim that there is a constant $\xi \in \mathbb{R}$ such that

$|x\left(\xi \right)|\le D.$
(2.5)

Let $\overline{t}$, $\underline{t}$ be, respectively, the global maximum point and the global minimum point of $x\left(t\right)$ on $\left[0,T\right]$; then ${x}^{\prime }\left(\overline{t}\right)=0$, and we claim that

${\left(\varphi \left({x}^{\prime }\left(\overline{t}\right)\right)\right)}^{\prime }\le 0.$
(2.6)

Assume, by way of contradiction, that (2.6) does not hold. Then ${\left(\varphi \left({x}^{\prime }\left(\overline{t}\right)\right)\right)}^{\prime }>0$ and there exists $\epsilon >0$ such that ${\left(\varphi \left({x}^{\prime }\left(t\right)\right)\right)}^{\prime }>0$ for $t\in \left(\overline{t}-\epsilon ,\overline{t}+\epsilon \right)$. Therefore $\varphi \left({x}^{\prime }\left(t\right)\right)$ is strictly increasing for $t\in \left(\overline{t}-\epsilon ,\overline{t}+\epsilon \right)$. From (A1) we know that ${x}^{\prime }\left(t\right)$ is strictly increasing for $t\in \left(\overline{t}-\epsilon ,\overline{t}+\epsilon \right)$. This contradicts the definition of $\overline{t}$. Thus, (2.6) is true. From $f\left(t,0\right)=0$, (2.3) and (2.6), we have

$g\left(\overline{t},x\left(\overline{t}\right)\right)-e\left(\overline{t}\right)\ge 0.$
(2.7)

Similarly, we get

$g\left(\underline{t},x\left(\underline{t}\right)\right)-e\left(\underline{t}\right)\le 0.$
(2.8)

In view of (H1), (2.7) and (2.8) imply that

$x\left(\overline{t}\right)\le D,\phantom{\rule{2em}{0ex}}x\left(\underline{t}\right)>0.$

Case (1): If $x\left(\underline{t}\right)\in \left(0,D\right)$, define $\xi =\overline{t}$, obviously, $|x\left(\xi \right)|\le D$.

Case (2): If $x\left(\underline{t}\right)\ge D$, from $x\left(\overline{t}\right)\le D$, we know $x\left(\overline{t}\right)=x\left(\underline{t}\right)$. Define $\xi =\overline{t}$, we have $|x\left(\xi \right)|=D$. This proves (2.5).

Then we have

$|x\left(t\right)|=|x\left(\xi \right)+{\int }_{\xi }^{t}{x}^{\prime }\left(s\right)\phantom{\rule{0.2em}{0ex}}ds|\le D+{\int }_{\xi }^{t}|{x}^{\prime }\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}t\in \left[\xi ,\xi +T\right]$

and

$|x\left(t\right)|=|x\left(t-T\right)|=|x\left(\xi \right)-{\int }_{t-T}^{\xi }{x}^{\prime }\left(s\right)\phantom{\rule{0.2em}{0ex}}ds|\le D+{\int }_{t-T}^{\xi }|{x}^{\prime }\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}t\in \left[\xi ,\xi +T\right].$

Combining the above two inequalities, we obtain

$\begin{array}{rl}{|x|}_{0}& =\underset{t\in \left[0,T\right]}{max}|x\left(t\right)|=\underset{t\in \left[\xi ,\xi +T\right]}{max}|x\left(t\right)|\\ \le \underset{t\in \left[\xi ,\xi +T\right]}{max}\left\{D+\frac{1}{2}\left({\int }_{\xi }^{t}|{x}^{\prime }\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds+{\int }_{t-T}^{\xi }|{x}^{\prime }\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds\right)\right\}\\ \le D+\frac{1}{2}{\int }_{0}^{T}|{x}^{\prime }\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds.\end{array}$
(2.9)

Since ${x}^{\prime }\left(t\right)$ is T-periodic, multiplying ${x}^{\prime }\left(t\right)$ and (2.3) and then integrating it from 0 to T, we have

$\begin{array}{rl}0& ={\int }_{0}^{T}{\left(\varphi \left({x}^{\prime }\left(t\right)\right)\right)}^{\prime }{x}^{\prime }\left(t\right)\phantom{\rule{0.2em}{0ex}}dt\\ =-\lambda {\int }_{0}^{T}f\left(t,{x}^{\prime }\left(t\right)\right){x}^{\prime }\left(t\right)\phantom{\rule{0.2em}{0ex}}dt-\lambda {\int }_{0}^{T}g\left(t,x\left(t\right)\right){x}^{\prime }\left(t\right)\phantom{\rule{0.2em}{0ex}}dt+\lambda {\int }_{0}^{T}e\left(t\right){x}^{\prime }\left(t\right)\phantom{\rule{0.2em}{0ex}}dt.\end{array}$
(2.10)

In view of (2.10), we have

$|{\int }_{0}^{T}f\left(t,{x}^{\prime }\left(t\right)\right){x}^{\prime }\left(t\right)\phantom{\rule{0.2em}{0ex}}dt|=|-{\int }_{0}^{T}g\left(t,x\left(t\right)\right){x}^{\prime }\left(t\right)\phantom{\rule{0.2em}{0ex}}dt+{\int }_{0}^{T}e\left(t\right){x}^{\prime }\left(t\right)\phantom{\rule{0.2em}{0ex}}dt|.$

From (H2), we know

$|{\int }_{0}^{T}f\left(t,{x}^{\prime }\left(t\right)\right){x}^{\prime }\left(t\right)\phantom{\rule{0.2em}{0ex}}dt|\ge \sigma {\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt.$

Set

${E}_{1}=\left\{t\in \left[0,T\right]\mid |x\left(t\right)|\le B\right\},\phantom{\rule{2em}{0ex}}{E}_{2}=\left\{t\in \left[0,T\right]\mid |x\left(t\right)|\ge B\right\}.$

From (H4), we have

$\begin{array}{c}\sigma {\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{\rule{1em}{0ex}}\le {\int }_{{E}_{1}+{E}_{2}}|g\left(t,x\left(t\right)\right)||{x}^{\prime }\left(t\right)|\phantom{\rule{0.2em}{0ex}}dt+{\int }_{0}^{T}|e\left(t\right)||{x}^{\prime }\left(t\right)|\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{\rule{1em}{0ex}}\le {\left({\int }_{{E}_{1}}|g\left(t,x\left(t\right)\right){|}^{\frac{m}{m-1}}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{m-1}{m}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{m}}+\alpha {\int }_{0}^{T}|x\left(t\right){|}^{m-1}|{x}^{\prime }\left(t\right)|\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{\rule{2em}{0ex}}+\beta {\int }_{0}^{T}|{x}^{\prime }\left(t\right)|\phantom{\rule{0.2em}{0ex}}dt+{\int }_{0}^{T}|e\left(t\right)||{x}^{\prime }\left(t\right)|\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{\rule{1em}{0ex}}\le {|{g}_{B}|}_{\frac{m}{m-1}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{m}}+\alpha {\left(D+\frac{1}{2}{\int }_{0}^{T}|{x}^{\prime }\left(t\right)|\phantom{\rule{0.2em}{0ex}}dt\right)}^{m-1}{\int }_{0}^{T}|{x}^{\prime }\left(t\right)|\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{\rule{2em}{0ex}}+\beta {T}^{\frac{m-1}{m}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{m}}+{\left({\int }_{0}^{T}|e\left(t\right){|}^{\frac{m}{m-1}}\right)}^{\frac{m-1}{m}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{m}}\hfill \\ \phantom{\rule{1em}{0ex}}={|{g}_{B}|}_{\frac{m}{m-1}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{m}}+\frac{\alpha }{{2}^{m-1}}{\left(\frac{2D}{{\int }_{0}^{T}|{x}^{\prime }\left(t\right)|\phantom{\rule{0.2em}{0ex}}dt}+1\right)}^{m-1}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right)|\phantom{\rule{0.2em}{0ex}}dt\right)}^{m}\hfill \\ \phantom{\rule{2em}{0ex}}+\beta {T}^{\frac{m-1}{m}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{m}}+{|e|}_{\frac{m}{m-1}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{m}},\hfill \end{array}$
(2.11)

where ${g}_{B}={max}_{|x|\le B}|g\left(t,x\left(t\right)\right)|$, ${|{g}_{B}|}_{\frac{m}{m-1}}={\left({\int }_{0}^{T}{|{g}_{B}|}^{\frac{m}{m-1}}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{m-1}{m}}$.

For the constant $\delta >0$, which is only dependent on $k>0$, we have

So, from (2.11), we have

$\begin{array}{c}\sigma {\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{\rule{1em}{0ex}}\le |{g}_{B}{|}_{\frac{m}{m-1}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{m}}+\frac{\alpha }{{2}^{m-1}}\left(1+\frac{2Dm}{{\int }_{0}^{T}|{x}^{\prime }\left(t\right)|\phantom{\rule{0.2em}{0ex}}dt}\right){\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right)|\phantom{\rule{0.2em}{0ex}}dt\right)}^{m}\hfill \\ \phantom{\rule{2em}{0ex}}+\beta {T}^{\frac{m-1}{m}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{m}}+|e{|}_{\frac{m}{m-1}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{m}}\hfill \\ \phantom{\rule{1em}{0ex}}=|{g}_{B}{|}_{\frac{m}{m-1}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{m}}+\frac{\alpha }{{2}^{m-1}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right)|\phantom{\rule{0.2em}{0ex}}dt\right)}^{m}+\frac{\alpha Dm}{{2}^{m-2}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right)|\phantom{\rule{0.2em}{0ex}}dt\right)}^{m-1}\hfill \\ \phantom{\rule{2em}{0ex}}+\beta {T}^{\frac{m-1}{m}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{m}}+|e{|}_{\frac{m}{m-1}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{m}}\hfill \\ \phantom{\rule{1em}{0ex}}\le |{g}_{B}{|}_{\frac{m}{m-1}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{m}}+\frac{\alpha {T}^{m-1}}{{2}^{m-1}}{\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{\alpha Dm{T}^{\frac{{\left(m-1\right)}^{2}}{m}}}{{2}^{m-2}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{m-1}{m}}\hfill \\ \phantom{\rule{2em}{0ex}}+\beta {T}^{\frac{m-1}{m}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{m}}+|e{|}_{\frac{m}{m-1}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{m}}\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{\alpha {T}^{m-1}}{{2}^{m-1}}{\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt+\frac{\alpha Dm{T}^{\frac{{\left(m-1\right)}^{2}}{m}}}{{2}^{m-2}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{m-1}{m}}\hfill \\ \phantom{\rule{2em}{0ex}}+\left(|{g}_{B}{|}_{\frac{m}{m-1}}+\beta {T}^{\frac{m-1}{m}}+|e{|}_{\frac{m}{m-1}}\right){\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{m}}.\hfill \end{array}$

Since $\sigma -\frac{\alpha {T}^{m-1}}{{2}^{m-1}}>0$, so it is easy to see that there is a constant ${M}_{1}^{\prime }>0$ (independent of λ) such that

${\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\le {M}_{1}^{\prime }.$

By applying Hölder’s inequality and (2.9), we have

${|x|}_{0}\le D+\frac{1}{2}{\int }_{0}^{T}|{x}^{\prime }\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds\le D+\frac{1}{2}{T}^{\frac{m-1}{m}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{m}}\le D+\frac{1}{2}{T}^{\frac{m-1}{m}}{\left({M}_{1}^{\prime }\right)}^{\frac{1}{m}}:={M}_{1}.$

In view of (2.4) and (H3), we have

$\begin{array}{rcl}{|\varphi \left({x}^{\prime }\right)|}_{0}& =& \underset{t\in \left[0,T\right]}{max}\left\{|\varphi \left({x}^{\prime }\left(t\right)\right)|\right\}\\ =& \underset{t\in \left[{t}_{0},{t}_{0}+T\right]}{max}\left\{|{\int }_{{t}_{0}}^{t}{\left(\varphi \left({x}^{\prime }\left(s\right)\right)\right)}^{\prime }\phantom{\rule{0.2em}{0ex}}ds|\right\}\\ \le & {\int }_{0}^{T}|f\left(t,{x}^{\prime }\left(t\right)\right)|\phantom{\rule{0.2em}{0ex}}dt+{\int }_{0}^{T}|g\left(t,x\left(t\right)\right)|\phantom{\rule{0.2em}{0ex}}dt+{\int }_{0}^{T}|e\left(t\right)|\phantom{\rule{0.2em}{0ex}}dt\\ \le & \rho {\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m-1}\phantom{\rule{0.2em}{0ex}}dt+\gamma T+{T}^{\frac{1}{m}}{\left({\int }_{0}^{T}|g\left(t,x\left(t\right)\right){|}^{\frac{m}{m-1}}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{m-1}{m}}\\ +{T}^{\frac{1}{m}}{\left({\int }_{0}^{T}|e\left(t\right){|}^{\frac{m}{m-1}}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{m-1}{m}}\\ \le & \rho {T}^{\frac{1}{m}}{\left({\int }_{0}^{T}|{x}^{\prime }\left(t\right){|}^{m}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{m-1}{m}}+\gamma T+{T}^{\frac{1}{m}}{\left({\int }_{0}^{T}|g\left(t,x\left(t\right)\right){|}^{\frac{m}{m-1}}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{m-1}{m}}\\ +{T}^{\frac{1}{m}}{\left({\int }_{0}^{T}{|e\left(t\right)|}^{\frac{m}{m-1}}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{m-1}{m}}\\ \le & \rho {T}^{\frac{1}{m}}{\left({M}_{1}^{\prime }\right)}^{\frac{m-1}{m}}+\gamma T+{T}^{\frac{1}{m}}{|{g}_{{M}_{1}}|}_{\frac{m-1}{m}}+{T}^{\frac{1}{m}}{|e|}_{\frac{m-1}{m}}:={M}_{2}^{\prime },\end{array}$

where $|{g}_{{M}_{1}}|={max}_{|x\left(t\right)|\le {M}_{1}}|g\left(t,x\left(t\right)\right)|$.

Thus, from Lemma 2.2, we know that there exists some positive constant ${M}_{2}>{M}_{2}^{\prime }+1$ such that, for all $t\in \mathbb{R}$,

$|{x}^{\prime }\left(t\right)|\le {M}_{2}.$

Set $M=\sqrt{{M}_{1}^{2}+{M}_{2}^{2}}+1$, we have

$\mathrm{\Omega }=\left\{x\in {C}_{T}^{1}\left(\mathbb{R},\mathbb{R}\right)\mid {|x|}_{0}\le M+1,|{x}^{\prime }{|}_{0}\le M+1\right\},$

we know that (2.4) has no solution on Ω as $\lambda \in \left(0,1\right)$ and when $x\left(t\right)\in \partial \mathrm{\Omega }\cap \mathbb{R}$, $x\left(t\right)=M+1$ or $x\left(t\right)=-M-1$, from (2.11) we know that $M+1>D$. So, from (H1) we see that

$\begin{array}{c}\frac{1}{T}{\int }_{0}^{T}\left\{g\left(t,M+1\right)-e\left(t\right)\right\}\phantom{\rule{0.2em}{0ex}}dt<0,\hfill \\ \frac{1}{T}{\int }_{0}^{T}\left\{g\left(t,-M-1\right)-e\left(t\right)\right\}\phantom{\rule{0.2em}{0ex}}dt>0.\hfill \end{array}$

So condition (ii) is also satisfied. Set

$H\left(x,\mu \right)=\mu x-\left(1-\mu \right)\frac{1}{T}{\int }_{0}^{T}\left\{g\left(t,x\right)-e\left(t\right)\right\}\phantom{\rule{0.2em}{0ex}}dt,$

where $x\in \partial \mathrm{\Omega }\cap \mathbb{R}$, $\mu \in \left[0,1\right]$, we have

$xH\left(x,\mu \right)=\mu {x}^{2}-\left(1-\mu \right)x\frac{1}{T}{\int }_{0}^{T}\left\{g\left(t,x\right)-e\left(t\right)\right\}\phantom{\rule{0.2em}{0ex}}dt>0,$

and thus $H\left(x,\mu \right)$ is a homotopic transformation and

$\begin{array}{rl}deg\left\{F,\mathrm{\Omega }\cap \mathbb{R},0\right\}& =deg\left\{-\frac{1}{T}{\int }_{0}^{T}\left\{g\left(t,x\right)-e\left(t\right)\right\}\phantom{\rule{0.2em}{0ex}}dt,\mathrm{\Omega }\cap \mathbb{R},0\right\}\\ =deg\left\{x,\mathrm{\Omega }\cap \mathbb{R},0\right\}\ne 0.\end{array}$

So condition (iii) is satisfied. In view of Lemma 2.1, there exists at least one solution with period T.

Suppose that $x\left(t\right)$ is the T-periodic solution of (1.1). We can easily show that (2.8) also holds. Thus,

which implies that (1.1) has a unique positive solution with period T. This completes the proof. □

We illustrate our results with some examples.

Example 2.1 Consider the following second-order p-Laplacian-like Rayleigh equation:

${\left({\varphi }_{p}\left({x}^{\prime }\left(t\right)\right)\right)}^{\prime }+\left(10+5{sin}^{2}t\right){x}^{\prime }\left(t\right)-\left(5x\left(t\right)+{sin}^{2}t-8\right)={e}^{{cos}^{2}t},$
(2.12)

where ${\varphi }_{p}\left(u\right)={|u|}^{p-2}u$.

Comparing (2.12) to (1.1), we see that $g\left(t,x\right)=-5x\left(t\right)-{sin}^{2}t+8$, $f\left(t,u\right)=\left(10+5{sin}^{2}t\right)u$, $e\left(t\right)={e}^{{cos}^{2}t}$, $T=\pi$. Obviously, we know that ${\varphi }_{p}$ is a homeomorphism from to satisfying (A1) and (A2). Consider $\left({x}_{1}-{x}_{2}\right)\left(g\left(t,{x}_{1}\right)-g\left(t,{x}_{2}\right)\right)=-5{\left({x}_{1}-{x}_{2}\right)}^{2}<0$ for ${x}_{1}\ne {x}_{2}$, then (A3) holds. Moreover, it is easily seen that there exists a constant $D=2$ such that (H1) holds. Consider $f\left(t,u\right)u=\left(10+5{sin}^{2}t\right){u}^{2}\ge 10{u}^{2}$, here $\sigma =10$, $m=2$, and $|f\left(t,u\right)|=|\left(10+5{sin}^{2}t\right)u|\le 15|u|+1$, here $\rho =15$, $\gamma =1$. So, we can get that conditions (H2) and (H3) hold. Choose $B>0$, we have $|g\left(t,x\right)|\le 5|x|+9$, here $\alpha =5$, $\beta =9$, then (H4) holds and $\sigma -\frac{\alpha T}{2}=10-\frac{5\pi }{2}>0$. So, by Theorem 2.1, we can get that (2.12) has a unique positive periodic solution.

Example 2.2 Consider the following second-order p-Laplacian-like Rayleigh equation:

${\left(\varphi \left({x}^{\prime }\left(t\right)\right)\right)}^{\prime }+\left(200+16{cos}^{2}t\right){\left({x}^{\prime }\left(t\right)\right)}^{3}-\left(20{x}^{3}\left(t\right)+10{cos}^{2}\left(t\right)-15\right)={e}^{{sin}^{2}t},$
(2.13)

where $\varphi \left(u\right)=u{e}^{{|u|}^{2}}$.

Comparing (2.13) to (1.1), we see that $g\left(t,x\right)=-20{x}^{3}-10{cos}^{2}t+15$, $f\left(t,v\right)=\left(200+16{cos}^{2}t\right){v}^{3}$, $e\left(t\right)={e}^{{sin}^{2}t}$, $T=\pi$. Obviously, we get

$\left(x{e}^{{|x|}^{2}}-y{e}^{{|y|}^{2}}\right)\left(x-y\right)\ge \left(|x|{e}^{{|x|}^{2}}-|y|{e}^{{|y|}^{2}}\right)\left(|x|-|y|\right)\ge 0$

and

$\varphi \left(x\right)\cdot x={|x|}^{2}{e}^{{|x|}^{2}}.$

So, we know that (A1) and (A2) hold. Consider $\left({x}_{1}-{x}_{2}\right)\left(g\left(t,{x}_{1}\right)-g\left(t,{x}_{2}\right)\right)=-20{\left({x}_{1}-{x}_{2}\right)}^{2}\left({x}_{1}^{2}+{x}_{1}{x}_{2}+{x}_{2}^{2}\right)<0$ for ${x}_{1}\ne {x}_{2}$, then (A3) holds. Moreover, it is easily seen that there exists a constant $D=1$ such that (H1) holds. Consider $f\left(t,v\right)v=\left(200+16{cos}^{2}t\right){v}^{4}\ge 200{v}^{4}$, here $\sigma =200$, $m=4$, and $|f\left(t,v\right)|=|\left(200+16{cos}^{2}t\right){v}^{3}|\le 216{|v|}^{3}+5$, here $\rho =216$, $\gamma =5$. So, we can get that conditions (H2) and (H3) hold. Choose $B>0$, we have $|g\left(t,x\right)|\le 20{|x|}^{3}+25$, here $\alpha =20$, $\beta =25$, then (H4) holds and $\sigma -\frac{\alpha {T}^{m-1}}{{2}^{m-1}}=200-\frac{20×{\pi }^{3}}{{2}^{3}}>0$. Therefore, by Theorem 2.1, we know that (2.13) has a unique positive periodic solution.

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## Acknowledgements

Research is supported by the National Natural Science Foundation of China (Nos. 11326124, 11271339).

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Correspondence to Yun Xin.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

YX and ZBC worked together in the derivation of the mathematical results. All authors read and approved the final manuscript.

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Xin, Y., Cheng, Z. Existence and uniqueness of a positive periodic solution for Rayleigh type ϕ-Laplacian equation. Adv Differ Equ 2014, 225 (2014). https://doi.org/10.1186/1687-1847-2014-225