Periodic solutions of p-Laplacian differential equations with jumping nonlinearity across half-eigenvalues

Abstract

This paper aims to investigate the existence of periodic solutions for $p$-Laplacian differential equations with jumping nonlinearity under the frame of half-eigenvalue. Based on the continuity theorem, some new results are obtained, which enrich and generalize the previous results.

1. Introduction

In this paper, we consider the existence of periodic solutions of $p$-Laplacian differential equations with jumping nonlinearity of variable coefficients as follows.

(1.1)\begin{align} & ( \phi_{p}(u'(t)))'+a_{+}(t)\phi_{p}(u^{+}(t))-a_{-}(t)\phi_{p}(u^{-}(t))+W_{0}(t,u(t))=e(t),\quad\text{and} \end{align}

(1.2)\begin{align} & \quad( \phi_{p}(u'(t)))'+a_{+}(t)\phi_{p}(u^{+}(t))-a_{-}(t)\phi_{p}(u^{-}(t))+W_{1}(t,u(t),u'(t))=e(t), \end{align}

where $W_{0}(t,u)=g_{0}(u)+g_{1}(t,u)$, $W_{1}(t,u,u')=\overline {g}_{0}(u)+\overline {g}_{1}(t,u')$; $g_{0},\overline {g}_{0}\in C(\mathbb {R},\mathbb {R})$; $g_{1},\overline {g}_{1}\in C([0,T]\times \mathbb {R},\mathbb {R})$ and are $T$-periodic in $t$; $a_{\pm }\in C([0,T],(0,\infty ))$ and are $T$-periodic in $t$; $e\in C([0,T],\mathbb {R})$ and is $T$-periodic in $t$; $\phi _{p}(s)=|s|^{p-2}s \ (s\neq 0)$, $\phi _{p}(0)=0$, $1< p<\infty$, $1/p+1/q=1;$ $u^{+}(t) = \max \{ u(t),0\}$, $u^{-}(t) = \max \{- u(t),0\}.$

In several decades ago, Dancer [6] and Fučik [11] presented the definition of the Dancer–Fučik spectrum for second-order linear differential equation with periodic boundary condition, which is a generalization of the common spectrum. Since then, many scholars devoted to investigating the existence of periodic solutions for second-order non-dissipative differential equation under the frame of Dancer–Fučik spectrum when asymptotic behaviours of potential function are given (see [8–10, 13, 16, 20, 21, 27] and references therein). After that, Del Pino, Manásevich, Murúa [7] gave the concept of Dancer–Fučik spectrum for second-order $p$-Laplacian differential equation with periodic boundary condition, i.e., if there exists a pair of positive constants $(\alpha,\beta )$ such that

(1.3)\begin{equation} \left\{ \begin{array}{@{}l@{}} (\phi _p (u'))' + \alpha\phi _p (u^ + ) - \beta\phi _p (u^ - ) = 0, \\ u(0) = u(T),u'(0) = u'(T) \end{array} \right. \end{equation}

has a nontrivial solutions, then we say that $(\alpha,\beta )$ lies in the Dancer–Fučik spectrum $S_{p}$ of the problem (). In the case $p=2$, the set $S_{2}$ is the classical Dancer–Fučik spectrum. It is known that the set $S_{p}$ has the structure (see [7]) $S_{p}=\bigcup _{k=0}^{+\infty }C_{k}$, where

\begin{align*} C_{0}& =\{(\alpha,\beta)|\alpha\geq 0, \beta\geq 0, \alpha=0\ \mbox{or}\ \beta=0\},\\ C_{k}& =\left\{(\alpha,\beta)|\alpha> 0, \beta>0, C_{n}=\left\{(a,b)|a> 0, b>0, \frac{1}{{\sqrt[p]{a}}} + \frac{1}{{\sqrt[p]{b}}} = \frac{T}{{n\pi_p }} \right\}\right. \end{align*}

in which $n\geq 1$ and $\pi _p$ is given by

\[ \pi _p = 2(p - 1)^{\frac{1}{p}} \int_0^{1} {(1 - t^{p} )}\,{\rm d}t = \frac{{2\pi (p - 1)^{\frac{1}{p}} }}{{p\sin \frac{\pi }{p}}}. \]

The problem (1.3) is non-resonant if and only if $(\alpha,\beta )\notin S_{p}$, otherwise it is resonant. The Dancer–Fučik spectrum $S_{p}$ has been used to investigate the existence of solutions of the following $p$-Laplacian problem.

(1.4)\begin{equation} \left\{ \begin{array}{@{}l} ( \phi_{p}(u'))'+g(t,u)=0,\\ u(0) = u(T),u'(0) = u'(T), \end{array} \right. \end{equation}

where $g(t,u)\in C([0,T]\times \mathbb {R},\mathbb {R})$. Assuming that the potential function $g$ satisfies the following asymptotic estimates.

(1.5)\begin{align} & p_1 \mathop { \le \lim \inf }\limits_{u\to +\infty } \frac{{g(t,u)}}{{|u|^{p - 2} u}} \le \mathop {\lim \sup }\limits_{u \to +\infty } \frac{{g(t,u)}}{{|u|^{p - 2} u}} \le p_2\ \text{uniformly for}\ \mbox{a.e.}\ t\in [0,T], \end{align}

(1.6)\begin{align} & q_1 \mathop { \le \lim \inf }\limits_{u \to - \infty } \frac{{g(t,u)}}{{|u|^{p - 2}u}} \le \mathop {\lim \sup }\limits_{u \to - \infty } \frac{{g(t,u)}}{{|u|^{p - 2} u}} \le q_2, \ \text{uniformly for}\ \mbox{a.e.}\ t\in [0,T], \end{align}

where $p_i,q_i, i=1.2$ are positive constant satisfying

(1.7)\begin{equation} \frac{{T }}{{(n + 1)\pi _p }} < \frac{1}{{\sqrt[p]{{p_2 }}}} + \frac{1}{{\sqrt[p]{{q_2 }}}} \le \frac{1}{{\sqrt[p]{{p_1 }}}} + \frac{1}{{\sqrt[p]{{q_1 }}}} < \frac{{T }}{{n\pi _p }},\end{equation}

then existence results for periodic solutions have been obtained using Leray–Schauder degree theory and the comparison argument, see [7]. The condition (1.7) extends the preceding idea of non-resonance ($(\alpha,\beta )\notin S_{p}$), in the sense that the asymptotes of the ratio $\frac {g(u)}{|u|^{p-2}u}$ are now forced to lie in the rectangle $[p_1,p_2]\times [q_1,q_2]\subset \mathbb {R}^{2}$, and this rectangle lies between two consecutive curves of the spectrum $S_{p}$, so does not intersect $S_{p}$.

A further extension was introduced by Nkashama and Robinson [21] in the case $p = 2$. This extension uses an averaging method to allow the asymptotes of the above ratio to cross the spectrum $S_{p}$, so long as suitable averages of this ratio do not. This averaging method was extended to more general $p$ by Chang and Qiao in [5]. The non-resonance condition in [5] assumes that $g$ satisfies

(1.8)\begin{align} & p_1 \mathop { \le \lim \inf }\limits_{u\to +\infty } \frac{{pG(t,u)}}{{|u|^{p } }} \le \mathop {\lim \sup }\limits_{u \to +\infty } \frac{{pG(t,u)}}{{|u|^{p} }} \le p_2\ \text{uniformly for}\ \mbox{a.e.}\ t\in [0,T], \end{align}

(1.9)\begin{align} & q_1 \mathop { \le \lim \inf }\limits_{u \to - \infty } \frac{{pG(t,u)}}{{|u|^{p }}} \le \mathop {\lim \sup }\limits_{u \to - \infty } \frac{{pG(t,u)}}{{|u|^{p } }} \le q_2, \ \text{uniformly for}\ \mbox{a.e.}\ t\in [0,T] \end{align}

with $([p_1,p_2]\times [q_1,q_2])\cap S_{p}=\emptyset,$ where $G(t,u)=\int _{0}^{u}g(t,s)\,{\rm d}s.$ Since

\[ \mathop {\lim \inf }\limits_{u \to \pm \infty } \frac{{g(t,u)}}{{|u|^{p - 2} u}} \le \mathop {\lim \inf }\limits_{u \to \pm \infty } \frac{{pG(t,u)}}{{|u|^{p} }}\mathop { \le \lim \sup }\limits_{u \to \pm \infty } \frac{{pG(t,u)}}{{|u|^{p} }} \le \mathop {\lim \sup }\limits_{u \to \pm \infty } \frac{{g(t,u)}}{{|u|^{p - 2} u}} , \]

we see that the former non-resonance condition (1.7), in terms of $\frac {{g(t,u)}}{{|u|^{p - 2} u}}$, implies the latter condition (1.8) and (1.9), that is, the latter condition is more general. In [17], Liu and Li considered the classical oscillation equation with $p$-Laplacian operator as follows.

\[ ( \phi_{p}(u'))'+g(u)=e(t), \]

where $g$ satisfies

(1.10)\begin{equation} \mathop {\lim }\limits_{u \to +\infty } \frac{{pG(u)}}{{|u|^{p} }} = \omega,\mathop {\lim }\limits_{u \to - \infty } \frac{{pG(u)}}{{|u|^{p} }} = \nu\end{equation}

and $(\omega,\nu )\notin S_{p}.$ Clearly, the (1.10) implies the (1.8) and (1.9) when $g$ is autonomous. Under the condition that there exist positive constants $\ b, \ d_{1}, \ d_{2}$ such that $\ d_{1}\leq \frac {{g(u)}}{{|u|^{p - 2} u}}\leq d_{2} \ \mbox {for all} \ |u|\geq b$, which is different from the assumptions given in [5], the existence of periodic solutions was obtained. For more articles concerning this topic, the readers can refer to [1, 14, 15, 26] and its references.

Recently, Binding and Rynne [3] presented the concept of half-eigenvalue for the following $p$-Laplacian differential equations with periodic boundary condition, i.e., if $\lambda$ is constant such that the following problem

(1.11)\begin{equation} \begin{cases} (\phi _p (u'))' + a_{+}(t)\phi _p (u^ + ) -a_{-}(t)\phi _p (u^ - ) + \lambda \phi _p (u) = 0, \\ u(0) = u(T),u'(0) = u'(T) \end{cases} \end{equation}

possesses a nontrivial solution, then $\lambda$ is termed half-eigenvalue. Let $\sigma =\bigcup _{m=0}^{+\infty }\sigma _{m}$, where

(1.12)\begin{equation} \sigma_{m}=\{\lambda\in\mathbb{R}| \ (1.11)\ \text{has a solution with 2m zeroes in }[0,T)\}.\end{equation}

By Prüfer approach, the characteristics of the set $\sigma$ were also given (see [18, theorem 3.1]). It contains two sequences of half-eigenvalues $\{\lambda _{2m}^{min}\}_{m\in \mathbb {Z}^{+}}$ and $\{\lambda _{2m}^{max}\}_{m\in \mathbb {Z}^{+}}$ that satisfy

\[ \lambda_{0}^{min}\leq \lambda_{0}^{max}<\lambda_{2}^{min}\leq \lambda_{2}^{max}<{\cdot}{\cdot}\cdot{<}\lambda_{2m}^{min}\leq \lambda_{2m}^{max}<{\cdot}{\cdot}\cdot\rightarrow +\infty, \]

$\sigma _{0}=\{\lambda _{0}^{min},\lambda _{0}^{max}\}$. Moreover, if we regard the half eigenvalues $\lambda _{n}^{{\rm max,min}}$ as functions of the coefficient pairs $(a^{+}, a^{-})$, i.e., $\lambda _{n}^{{\rm max,min}}(a^{+}, a^{-})$, and we suppose that ${a_{\pm }\in L^{\gamma }}$, it was shown in Li and Yan [18] that $\lambda _{n}^{{\rm max,min}}(a^{+}, a^{-})$ depends continuously on $(a^{+}, a^{-})$, with respect to the weak topology on $(L^{\gamma })^{2}$. Furthermore, it should be mentioned that if $a_{\pm }$ are constants, (1.11) becomes

(1.13)\begin{equation} \left\{ \begin{array}{@{}l} (\phi _p (u'))' + \widetilde{a}_{+}\phi _p (u^ + ) - \widetilde{a}_{-}\phi _p (u^ - ) = 0, \\ u(0) = u(T),u'(0) = u'(T), \end{array} \right. \end{equation}

where $\widetilde {a}_{\pm }=a_{\pm }+\lambda$. Clearly, it reduces to the problem of Dancer–Fučik spectrum.

For the problems of half-eigenvalues with variable coefficients, it can be traced back to [2, 4]. After that, more and more scholars devoted to investigating half-eigenvalues with different boundary value problems. For the aspects of existence of solutions, recently, based on the Landesman–Lazer conditions, Genoud and Rynne [12] made use of the shooting method to investigate the existence of solutions for $p$-Laplacian differential equations with Dirichlet boundary condition under the frame of half-eigenvalue as follows.

(1.14)\begin{equation} \left\{ \begin{array}{@{}l} -(\phi _p (u'))' -a_{+}\phi _p (u^ + ) +a_{-}\phi _p (u^ - ) -\lambda \phi _p (u) = f(t,u),\ t\in (0,1),\\ u(0) =u(1)=0. \end{array} \right. \end{equation}

Rynne [22] made a further study on this type of problem with the dissipation term. For more papers concerning this topic, we refer to Rynne [23–25] and references therein. However, it should be pointed out that there are few works on discussing the existence of periodic solutions for $p$-Laplacian problems with jumping nonlinearities under the frame of half-eigenvalue.

Motivated by the works mentioned above, our paper aims to investigate the existence of periodic solutions of $p$-Laplacian differential equations with jump nonlinearity of variable coefficients under the frame of half-eigenvalue. The variable coefficients and $p$-Laplacian operator bring many difficulties such as the prior estimation and convergence analysis. We make use of some analytical skills and the continuity theorem to overcome these difficulties and obtain some new results, which generalize the existence of periodic solutions under the framework of Dancer–Fučik spectral with constant coefficients to the case of half-eigenvalues with variable coefficients. Moreover, the model is more general and complex.

2. Main results

To begin with, we show some necessary basic knowledge and signs. Let $C([0,T],\mathbb {R})$ with norm $\| u \|_{\infty }={\max }_{t\in [0,T]}|u(t)|$ and $L^{p}([0,T],\mathbb {R})$ with norm $\| u \|_{p}=\left (\int _{0}^{T}|u(t)|^{p} {\rm d}t\right )^{\frac {1}{p}}.$ Set $a_{\pm,M}={\max }_{t\in [0,T]}a_{\pm }(t)$, $a_{\pm,L}={\min }_{t\in [0,T]}a_{\pm }(t)$. And define the following Banach space

\[ C_{T}^{1} = \{ u \in C^{1}([0,T],\mathbb{R})|\ u(0)=u(T),\ u'(0)=u'(T)\} \]

endowed the norm $\|u\|=\max \{\| u \|_{\infty },\| u' \|_{\infty }\}.$

Next, let us introduce the famous continuity theorem for $p$-Laplacian equation.

Lemma 2.1 [19]

Assume that $\Omega$ is an open bounded set in $C_{T}^{1}$ such that the following conditions hold:

${ (i)}$ For each $\lambda \in (0,1)$, the problem

(2.1)\begin{equation} \left\{ \begin{array}{@{}l} ( \phi_{p}(u'))'=\lambda f(t,u,u'),\\ u(0) = u(T),u'(0) = u'(T) \end{array} \right. \end{equation}

has no solution on $\partial \Omega$.

${(ii)}$ The equation

(2.2)\begin{equation} F(a):=\frac{1}{T}\int_{0}^{T}f(t,a,0)\,{\rm d}t=0 \end{equation}

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

${(iii)}$ The Brouwer degree

(2.3)\begin{equation} \mbox{deg}_{B}(F,\mathbb{R}\cap\partial\Omega,0)\neq 0.\end{equation}

Then, when $\lambda =1$, the problem (2.1) has a solution in $\overline {\Omega }$.

Remark 2.2 In lemma 2.1, it should be mentioned that for given $f$, a is being regarded as a constant function for $\partial \Omega \cap \mathbb {R}$ to make sense.

In order to get our main results of problems (1.1) and (1.2), we first consider the existence of periodic solutions for the following auxiliary dissipative problems.

(2.4)\begin{equation} \left\{ \begin{array}{@{}l} ( \phi_{p}(u'))'+\hbar\varepsilon\phi_{p}(u')+a_{+}(t)\phi_{p}(u^{+})-a_{-}(t)\phi_{p}(u^{-})+W_{0}(t,u)=e(t),\ t\in [0,T],\\ u(0) = u(T),u'(0) = u'(T) \end{array} \right.\end{equation}

and

(2.5)\begin{equation} \left\{ \begin{array}{@{}l} ( \phi_{p}(u'))'+\hbar\varepsilon\phi_{p}(u')+a_{+}(t)\phi_{p}(u^{+})-a_{-}(t)\phi_{p}(u^{-})\\ \quad +W_{1}(t,u,u')=e(t),\ t\in [0,T],\\ u(0) = u(T),u'(0) = u'(T), \end{array} \right.\end{equation}

where $\varepsilon >0$ is a constant and $\hbar$ represents a symbol defined by

\[ \hbar=\left\{ \begin{array}{@{}ll} 1, & \int_{0}^{T} a_{+}(s)\phi_{p}(u^{+}(s))u'(s)\,{\rm d}s\geq\int_{0}^{T} a_{-}(s)\phi_{p}(u^{-}(s))u'(s)\,{\rm d}s,\\ - 1, & \int_{0}^{T} a_{+}(s)\phi_{p}(u^{+}(s))u'(s)\,{\rm d}s<\int_{0}^{T} a_{-}(s)\phi_{p}(u^{-}(s))u'(s)\,{\rm d}s. \end{array} \right. \]

Now, we introduce some auxiliary lemmas.

Lemma 2.3 For $2\leq p<\infty$, assume that the following conditions hold.

$(H1)$ There exist nonnegative functions $a_{i}>0,i=1.2,$ and constant $\theta \in [0,p-1)$ such that for $(t,u)\in [0,T]\times \mathbb {R}$,

\[ |g_{1}(t,u)|\leq a_{1}(t)+a_{2}(t)|u|^{\theta}. \]

$(H2)$ There exists constant $a_{3}>0$ such that if $|u|>a_{3}$, for $t\in [0,T],u\in \mathbb {R}$,

\[ \text{sgn}\{u\}(a_{+}(t)\phi_{p}(u^{+})+W_{0}(t,u))>\|e\|_{\infty} \]

or

\[ \mbox{sgn}\{u\}({-}a_{-}(t)\phi_{p}(u^{-})+W_{0}(t,u))>\|e\|_{\infty}. \]

Then the problem (2.4) has at least one solution.

Proof. Consider the following homotopy equation with periodic boundary condition

(2.6)\begin{equation} ( \phi_{p}(u'))'+\lambda \hbar\varepsilon\phi_{p}(u')+\lambda a_{+}(t)\phi_{p}(u^{+})-\lambda a_{-}(t)\phi_{p}(u^{-})+\lambda W_{0}(t,u)=\lambda e(t),\end{equation}

where $\lambda \in (0,1)$.

First, if $u\in C_{T}^{1}$ is a solution of (2.6), we claim that there exist positive constants $b_{i},i=1,2$ such that $\|u\|_{\infty }\leq b_{1}+b_{2}\|u'\|_{{1}}$. Let $t_{1}$ and $t_{2}$ represent minimum and maximum points respectively. Thus, we know that

\[ \phi_{p}(u'(t_{1}))=0,(\phi_{p}(u'(t_{1})))'\geq 0\ \mbox{and}\ \phi_{p}(u'(t_{2}))=0,(\phi_{p}(u'(t_{2})))'\leq 0. \]

Thus, we need to consider the following three cases.

\begin{align*} & (A1):u(t_1 ) \le 0,u(t_2 ) \ge 0, \\ & (A2):u(t_1 ) \ge 0,u(t_2 ) \ge 0, \\ & (A3):u(t_1 ) \le 0,u(t_2 ) \le 0. \end{align*}

For the case (A1), it is clear that $\|u\|_{\infty }\leq \|u'\|_{{1}}$. For the case (A2), from (2.6), we have

\[ a_{+}(t_{1})\phi_{p}(u^{+}(t_{1}))+W_{0}(t_{1},u(t_{1})))\leq e(t_{1})\leq \|e\|_{\infty} \]

and

\[ a_{+}(t_{2})\phi_{p}(u^{+}(t_{2}))+W_{0}(t_{2},u(t_{2})))\geq e(t_{2})\geq{-}\|e\|_{\infty}. \]

Thus, we know that

\[ \{ a_{+}(t)\phi_{p}(u^{+}(t))+W_{0}(t,u(t)):t\in [0,T]\}\bigcap [-\|e\|_{\infty},\|e\|_{\infty}]\neq \emptyset, \]

i.e., there exists $t_{3}\in [0,T]$ such that

\[ |a_{+}(t_{3})\phi_{p}(u^{+}(t_{3}))+W_{0}(t_{3},u(t_{3}))|\leq \|e\|_{\infty}. \]

From $(H2)$, we can obtain that $|u(t_{3})|\leq a_{3}.$ Thus, the claim is proved. By the similar way to the case $(A2)$, the case $(A3)$ is also true. Here, we omit the detail.

Multiplying the both sides of (2.6) by $u'$ and integrating from $0$ to $T$, we have

(2.7)\begin{align} & \hbar\varepsilon\int_{0}^{T}|u'(s)|^{p}\,{\rm d}s+\int_{0}^{T} a(s)\phi_{p}(u^{+}(s))u'(s)\,{\rm d}s-\int_{0}^{T} b(s)\phi_{p}(u^{-}(s))u'(s)\,{\rm d}s\nonumber\\ & \quad+\int_{0}^{T}g_{1}(s,u(s))u'(s)\,{\rm d}s=\int_{0}^{T}e(s)u'(s)\,{\rm d}s. \end{align}

By the definition of $\hbar$, we get two cases as follows.

\begin{align*} & (i):\ \varepsilon\int_{0}^{T}|u'(s)|^{p}\,{\rm d}s\leq{-}\int_{0}^{T}g_{1}(s,u(s))u'(s)\,{\rm d}s+\int_{0}^{T}e(s)u'(s)\,{\rm d}s.\\ & (ii):\ \varepsilon\int_{0}^{T}|u'(s)|^{p}\,{\rm d}s\leq\int_{0}^{T}g_{1}(s,u(s))u'(s)\,{\rm d}s-\int_{0}^{T}e(s)u'(s)\,{\rm d}s. \end{align*}

Moreover, by $(H1)$ and the basic inequality $(x+y)^{r}\leq 2^{r}(x^{r}+y^{r}), x,y,r>0$, we have

\begin{align*} & \int_{0}^{T}|g_{1}(s,u(s))u'(s)|\,{\rm d}s\\ & \quad \leq \left(\int_{0}^{T}|g_{1}(s,u(s))|^{q}\,{\rm d}s\right)^{\frac{1}{q}} \left(\int_{0}^{T}|u'(s)|^{p}\,{\rm d}s\right)^{\frac{1}{p}}\\ & \quad \leq \left(\int_{0}^{T}(a_{1}(t)+a_{2}(t)|u(s)|^{\theta})^{q}\,{\rm d}s\right)^{\frac{1}{q}} \left(\int_{0}^{T}|u'(s)|^{p}\,{\rm d}s\right)^{\frac{1}{p}}\\ & \quad \leq \left[2^{\frac{1}{q}+1}\|a_{1}\|_{\infty}T^{\frac{1}{q}}+2^{\frac{1}{q}+1}\|a_{2}\|_{\infty}\left(\int_{0}^{T}|u(s)|^{\theta q}\,{\rm d}s\right)^{\frac{1}{q}}\right]\left(\int_{0}^{T}|u'(s)|^{p}\,{\rm d}s\right)^{\frac{1}{p}}\\ & \quad \leq \left(2^{\frac{1}{q}+1}\|a_{1}\|_{\infty}T^{\frac{1}{q}}+2^{\frac{2}{q}+1+\theta}\|a_{2}\|_{\infty}b_{1}^{\theta}T^{\frac{1}{q}}\right)\left(\int_{0}^{T}|u'(s)|^{p}\,{\rm d}s\right)^{\frac{1}{p}}\\ & \qquad+2^{\frac{2}{q}+1+\theta}\|a_{2}\|_{\infty}b_{2}^{\theta}T^{\frac{1}{q}}\left(\int_{0}^{T}|u'(s)|\,{\rm d}s\right)^{\theta}\left(\int_{0}^{T}|u'(s)|^{p}\,{\rm d}s\right)^{\frac{1}{p}}\\ & \quad \leq \left(2^{\frac{1}{q}+1}\|a_{1}\|_{\infty}T^{\frac{1}{q}}+2^{\frac{2}{q}+1+\theta}\|a_{2}\|_{\infty}b_{1}^{\theta}T^{\frac{1}{q}}\right)\left(\int_{0}^{T}|u'(s)|^{p}\,{\rm d}s\right)^{\frac{1}{p}}\\ & \qquad+2^{\frac{2}{q}+1+\theta}\|a_{2}\|_{\infty}b_{2}^{\theta}T^{\frac{1+\theta}{q}}\left(\int_{0}^{T}|u'(s)|^{p}\,{\rm d}s\right)^{\frac{1+\theta}{p}}. \end{align*}

Thus, from (i) or (ii), we have

\begin{align*} & \varepsilon\int_{0}^{T}|u'(s)|^{p}\,{\rm d}s\nonumber\\ & \leq \left.\left(2^{\frac{1}{q}+1}\|a_{1}\|_{\infty}T^{\frac{1}{q}}+2^{\frac{2}{q}+1+\theta}\|a_{2}\|_{\infty}b_{1}^{\theta}T^{\frac{1}{q}}+\int_{0}^{T}|e(s)|^{q}\right)^{\frac{1}{q}}\right)\left(\int_{0}^{T}|u'(s)|^{p}\,{\rm d}s\right)^{\frac{1}{p}}\\ & \quad+ 2^{\frac{2}{q}+1+\theta}\|a_{2}\|_{\infty}b_{2}^{\theta}T^{\frac{1+\theta}{q}}\left(\int_{0}^{T}|u'(s)|^{p}\,{\rm d}s\right)^{\frac{1+\theta}{p}}. \end{align*}

So, we can get

(2.8)\begin{align} \varepsilon \left(\int_{0}^{T}|u'(s)|^{p}\,{\rm d}s\right)^{\frac{1}{q}}& \leq 2^{\frac{1}{q}+1}\|a_{1}\|_{\infty}T^{\frac{1}{q}}+2^{\frac{2}{q}+1+\theta}\|a_{2}\|_{\infty}b_{1}^{\theta}T^{\frac{1}{q}}+\left(\int_{0}^{T}|e(s)|^{q}\right)^{\frac{1}{q}} \nonumber\\ & \quad+2^{\frac{2}{q}+1+\theta}\|a_{2}\|_{\infty}b_{2}^{\theta}T^{\frac{1+\theta}{q}}\left(\int_{0}^{T}|u'(s)|^{p}\,{\rm d}s\right)^{\frac{\theta}{p}}. \end{align}

Noting that $\theta \in [0,p-1)$, we claim that $\|u'\|_{{p}}$ is bounded, i.e, there exists a positive constant $D$ such that $\|u'\|_{{p}}\leq D$. Otherwise, we can find the sequence $\{u'_{n}(t)\}_{n\in \mathbb {Z^{+}}}$ which satisfies (2.8) such that

\[ \|u'_{n}\|_{{p}}\to \infty,\text{as } n\to \infty. \]

Moreover, multiplying (2.8) by $\frac {1}{\|u'_{n}\|_{{p}}^{\theta }}$, since $\theta \in [0,p-1)$, we have

\begin{align*} & \varepsilon \|u'_{n}\|_{{p}}^{p-1-\theta}\nonumber\\ & \quad\leq \frac{2^{\frac{1}{q}+1}\|a_{1}\|_{\infty}T^{\frac{1}{q}}+2^{\frac{2}{q}+1+\theta}\|a_{2}\|_{\infty}b_{1}^{\theta}T^{\frac{1}{q}}+\left(\int_{0}^{T}|e(s)|^{q}\right)^{\frac{1}{q}}}{\|u'_{n}\|_{{p}}^{\theta}}\\ & \qquad +2^{\frac{2}{q}+1+\theta}\|a_{2}\|_{\infty}b_{2}^{\theta}T^{\frac{1+\theta}{q}}, \end{align*}

which implies a contradiction as $n\to \infty$. Thus, the claim is true. Therefore, we have

\[ \|u\|_{\infty}\leq b_{1}+b_{2}\|u'\|_{{1}}\leq b_{1}+b_{2}T^{\frac{1}{q}}\|u'\|_{{p}}\leq b_{1}+b_{2}T^{\frac{1}{q}}D:=\overline{D}. \]

On the other hand, (2.6) is equivalent to

(2.9)\begin{equation} ( \phi_{p}(u'(t))e^{\lambda \hbar\varepsilon t})'=\lambda \left[{-}a_{+}(t)\phi_{p}(u^{+})+ a_{-}(t)\phi_{p}(u^{-})-W_{0}(t,u)+ e(t)\right]e^{\lambda \hbar\varepsilon t}.\end{equation}

Since $u(0)=u(T)$, there exists $t^{*}\in (0,T)$ such that $u'(t^{*})=0$. Integrating from $t^{*}$ to $t$ on (2.9), we have

\begin{align*} phi_{p}(u')& =\lambda e^{-\lambda \hbar\varepsilon t}\int_{t^{*}}^{t}({-}a_{+}(s)\phi_{p}(u^{+}(s))+ a_{-}(s)\phi_{p}(u^{-}(s))\\ & \quad -W_{0}(s,u(s))+ e(s))e^{\lambda \hbar\varepsilon s}\,{\rm d}s. \end{align*}

Thus, we can obtain

(2.10)\begin{equation} |u'(t)|^{p-1}\leq e^{2\varepsilon T}(Ta_{+,M}\overline{D}^{p-1}+Ta_{-,M}\overline{D}^{p-1}+lT+T\|e\|_{\infty}):=K^{p-1},\end{equation}

where $l={\max }_{|u|\leq \overline {D},0\leq t\leq T}|W_{0}(t,u(t))|$. It implies that $\|u'\|_{\infty }\leq K.$

Let

\[ \Omega=\{u(t)\in C^{1}_{T}:\|u\|\leq M+1\}, \]

where $M>K+\overline {D}$ such that $a_{+}(t)\phi _{p}(M+1)+W_{0}(t,M+1)-e(t)>0$ and $a_{-}(t)\phi _{p}(-M-1)+W_{0}(t,-M-1)-e(t)<0$. Clearly, the problem (2.4) has no solution on $\mathbb {R}\cap \partial \Omega$. Moreover, we can get

\begin{align*} & F(M+1)=\frac{1}{T}\int_{0}^{T}-(a_{+}(s)\phi_{p}(M+1)+W_{0}(s,M+1)-e(s))\,{\rm d}s<0,\\ & F({-}M-1)=\frac{1}{T}\int_{0}^{T}-(a_{-}(s)\phi_{p}({-}M-1)+W_{0}(s,-M-1)-e(s))\,{\rm d}s>0. \end{align*}

Thus, $F(a)$ has no solution on $\mathbb {R}\cap \partial \Omega$. Making a homotopy

\begin{align*} H(u,\lambda)& =\lambda u+(1-\lambda)\frac{1}{T}\int_{0}^{T}({-}a_{+}(s)\phi_{p}(u^{+}(s))+ a_{-}(s)\phi_{p}(u^{-}(s))\\ & \quad -W_{0}(s,u(s))+ e(s))\,{\rm d}s, \end{align*}

where $u\in \mathbb {R}\cap \partial \Omega$, we have

\begin{align*} uH(u,\lambda)& =\lambda u^{2}+(1-\lambda)\frac{1}{T}u\int_{0}^{T}({-}a_{+}(s)\phi_{p}(u^{+}(s))+ a_{-}(s)\phi_{p}(u^{-}(s))\\ & \quad -W_{0}(s,u(s))+ e(s))\,{\rm d}s<0. \end{align*}

Based on the homotopic transformation, we have

\[ \mbox{deg}_{B}(F,\mathbb{R}\cap\partial\Omega,0)=\mbox{deg}_{B}(u,\mathbb{R}\cap\partial\Omega,0)\neq 0. \]

Thus, the problem (2.4) has at least one solution.

Lemma 2.4 For $1< p<\infty,$ assume that the following conditions hold.

$(H3)$ There exist functions $a_{i}>0,i=1.2,$ and constant $\theta \in [0,p-1)$ such that for $(t,v)\in [0,T]\times \mathbb {R},$

\[ |\overline{g}_{1}(t,v)|\leq a_{1}(t)+a_{2}(t)|v|^{\theta}. \]

$(H4)$ There exist positive constants $\varrho,$ $d$ and $\gamma \in (0,p-1]$ such that for $t\in [0,T]$,

\[ {\mbox{sgn}\{u\}\overline{g}_{0}(u)}\geq \varrho |u|^{\gamma},\ |u|\geq d. \]

Then the problem (2.5) has at least one solution.

Proof. First, we can obtain the priori estimate that if $u\in C_{T}^{1}$ is a solution of the problem (2.5), then we have $|u(t)|\leq L,$ where $L$ is a positive constant and independent of $\lambda$. Otherwise, we can find solutions $\{u_{n}(t)\}$ of the problem (2.5) corresponding with $\lambda =\lambda _{n}$, namely, $u_{n}(t)$ satisfies the following equation

(2.11)\begin{align} & ( \phi_{p}(u_{n}'))'+\lambda_{n}\hbar \varepsilon\phi_{p}(u_{n}')+\lambda_{n} a_{+}(t)\phi_{p}(u_{n}^{+})\notag\\ & \quad -\lambda_{n} a_{-}(t)\phi_{p}(u_{n}^{-})+\lambda_{n} W_{1}(t,u_{n},u'_{n})=\lambda_{n} e(t)\end{align}

and periodic boundary condition such that

\[ \|u_{n}\|_{\infty}\to \infty,\ \mbox{as}\ n\to \infty. \]

For any $u \in C_{T}^{1}\setminus \{0\}$, let $z_{n}(t)=\frac {u_{n}(t)}{\left \| u_{n} \right \|_{\infty }}$, so $\|z_{n}\|_{\infty }=1$. Moreover, multiplying (2.11) by $\frac {u_{n}'}{\left \| u_{n} \right \|^{p}_{\infty }}$ and integrating from 0 to $T$, we have

(2.12)\begin{align} & \hbar\varepsilon\int_{0}^{T}|z_{n}'(s)|^{p}\,{\rm d}s+\int_{0}^{T} a(s)\phi_{p}(z_{n}^{+}(s))z_{n}'(s)\,{\rm d}s-\int_{0}^{T} b(s)\phi_{p}(z_{n}^{-}(s))z_{n}'(s)\,{\rm d}s\nonumber\\ & \quad+\frac{1}{{\|u_{n}\|}_{\infty}^{p-1}}\int_{0}^{T}\overline{g}_{1}(s,u'_{n}(s))z_{n}'(s)\,{\rm d}s=\frac{1}{{\|u_{n}\|}_{\infty}^{p-1}}\int_{0}^{T}e(s)z_{n}'(s)\,{\rm d}s. \end{align}

By definition of $\hbar$, we get two cases as follows.

\begin{align*} (i):\ \varepsilon\int_{0}^{T}|z_{n}'(s)|^{p}\,{\rm d}s& \leq{-}\frac{1}{{\|u_{n}\|}_{\infty}^{p-1}}\int_{0}^{T}\overline{g}_{1}(s,u'_{n}(s))z_{n}'(s)\,{\rm d}s\notag\\ & \quad +\frac{1}{{\|u_{n}\|}_{\infty}^{p-1}}\int_{0}^{T}e(s)z_{n}'(s)\,{\rm d}s.\\ (ii):\ \varepsilon\int_{0}^{T}|z_{n}'(s)|^{p}\,{\rm d}s& \leq \frac{1}{{\|u_{n}\|}_{\infty}^{p-1}}\int_{0}^{T}\overline{g}_{1}(s,u'_{n}(s))z_{n}'(s)\,{\rm d}s\notag\\ & \quad -\frac{1}{{\|u_{n}\|}_{\infty}^{p-1}}\int_{0}^{T}e(s)z_{n}'(s)\,{\rm d}s. \end{align*}

By $(H3)$, we can get

\begin{align*} & \frac{1}{{\|u_{n}\|}_{\infty}^{p-1}}\int_{0}^{T}\overline{g}_{1}(s,u'_{n}(s))z_{n}'(s)\,{\rm d}s\\ & \quad\leq\frac{1}{{\|u_{n}\|}_{\infty}^{p-1}}\left(\int_{0}^{T}|\overline{g}_{1}(s,u'_{n}(s))|^{q}\,{\rm d}s\right)^{\frac{1}{q}} \left(\int_{0}^{T}|z'_{n}(s)|^{p}\,{\rm d}s\right)^{\frac{1}{p}}\\ & \quad\leq\frac{1}{{\|u_{n}\|}_{\infty}^{p-1}}\left(\int_{0}^{T}(a_{1}(t)+a_{2}(t)|u'_{n}(s)|^{\theta})^{q}\,{\rm d}s\right)^{\frac{1}{q}} \left(\int_{0}^{T}|z'_{n}(s)|^{p}\,{\rm d}s\right)^{\frac{1}{p}} \\ & \quad\leq\frac{2^{\frac{1}{q}+1}\|a_{1}\|_{\infty}T^{\frac{1}{q}}}{{\|u_{n}\|}_{\infty}^{p-1}}\left(\int_{0}^{T}|z'_{n}(s)|^{p}\,{\rm d}s\right)^{\frac{1}{p}}\\ & \qquad +\frac{2^{\frac{1}{q}+1}\|a_{2}\|_{\infty}}{{\|u_{n}\|}_{\infty}^{p-\theta-1}}\left(\int_{0}^{T}|z'_{n}(s)|^{\theta q}\,{\rm d}s\right)^{\frac{1}{q}}\left(\int_{0}^{T}|z'_{n}(s)|^{p}\,{\rm d}s\right)^{\frac{1}{p}}\\ & \quad\leq\frac{2^{\frac{1}{q}+1}\|a_{1}\|_{\infty}T^{\frac{1}{q}}}{{\|u_{n}\|}_{\infty}^{p-1}}\left(\int_{0}^{T}|z'_{n}(s)|^{p}\,{\rm d}s\right)^{\frac{1}{p}}\\ & \qquad +\frac{2^{\frac{1}{q}+1}\|a_{2}\|_{\infty}T^{\frac{p-\theta-1}{p}}}{{\|u_{n}\|}_{\infty}^{p-\theta-1}}\left(\int_{0}^{T}|z'_{n}(s)|^{p}\,{\rm d}s\right)^{\frac{1+\theta}{p}} . \end{align*}

Thus, from (i) or (ii), one has

\begin{align*} \varepsilon\int_{0}^{T}|z_{n}'(s)|^{p}\,{\rm d}s& \leq \frac{2^{\frac{1}{q}+1}\|a_{1}\|_{\infty}T^{\frac{1}{q}}}{{\|u_{n}\|}_{\infty}^{p-1}}\left(\int_{0}^{T}|z'_{n}(s)|^{p}\,{\rm d}s\right)^{\frac{1}{p}}\\ & \quad +\frac{2^{\frac{1}{q}+1}\|a_{2}\|_{\infty}T^{\frac{p-\theta-1}{p}}}{{\|u_{n}\|}_{\infty}^{p-\theta-1}}\left(\int_{0}^{T}|z'_{n}(s)|^{p}\,{\rm d}s\right)^{\frac{1+\theta}{p}}\\ & \quad+ \frac{1}{{\|u_{n}\|}_{\infty}^{p-1}} \left(\int_{0}^{T}|e(s)|^{q}\right)^{\frac{1}{q}}\left(\int_{0}^{T}|z_{n}'(s)|^{p}\,{\rm d}s\right)^{\frac{1}{p}}, \end{align*}

which yields that

(2.13)\begin{align} \varepsilon\left(\int_{0}^{T}|z_{n}'(s)|^{p}\,{\rm d}s\right)^{\frac{1}{q}}& \leq \frac{2^{\frac{1}{q}+1}\|a_{1}\|_{\infty}T^{\frac{1}{q}}}{{\|u_{n}\|}_{\infty}^{p-1}}+\frac{2^{\frac{1}{q}+1}\|a_{2}\|_{\infty}T^{\frac{p-\theta-1}{p}}}{{\|u_{n}\|}_{\infty}^{p-\theta-1}}\left(\int_{0}^{T}|z'_{n}(s)|^{p}\,{\rm d}s\right)^{\frac{\theta}{p}}\nonumber\\ & \quad+ \frac{1}{{\|u_{n}\|}_{\infty}^{p-1}} \left(\int_{0}^{T}|e(s)|^{q}\right)^{\frac{1}{q}}. \end{align}

We claim that $\|z_{n}'\|_{{p}}\to 0$ as $n\to \infty$. If not, we have the following two cases.

\begin{align*} & (B1)\ \|z_{n}'\|_{{p}}\to c\ \mbox{as}\ n\to \infty,\ \mbox{where}\ c \ \mbox{is a positive constant};\\ & (B2)\ \|z_{n}'\|_{{p}}\to \infty\ \mbox{as}\ n\to \infty. \end{align*}

For the case $(B1)$, from (2.13), we have

\begin{align*} & \left(\int_{0}^{T}|z_{n}'(s)|^{p}\,{\rm d}s\right)^{\frac{\theta}{p}}\left[\varepsilon\left(\int_{0}^{T}|z_{n}'(s)|^{p}\,{\rm d}s\right)^{\frac{p-\theta-1}{p}}-\frac{2^{\frac{1}{q}+1}\|a_{2}\|_{\infty}T^{\frac{p-\theta-1}{p}}}{{\|u_{n}\|}_{\infty}^{p-\theta-1}}\right]\\ & \quad\leq \frac{2^{\frac{1}{q}+1}\|a_{1}\|_{\infty}T^{\frac{1}{q}}}{{\|u_{n}\|}_{\infty}^{p-1}}+ \frac{1}{{\|u_{n}\|}_{\infty}^{p-1}} \left(\int_{0}^{T}|e(s)|^{q}\right)^{\frac{1}{q}}, \end{align*}

which implies that $\varepsilon c^{p-1}\leq 0$ as $n\to \infty$. This is impossible. Clearly, for the case $(B2)$, we can also get a contradiction. Thus, $\|z_{n}'\|_{{p}}\to 0$ as $n\to \infty$. Moreover, for any $t\in [0,T]$, by Hölder inequality, we have

\[ |z_{n}(t)-z_{n}(0)|\leq T^{\frac{1}{q}}\left(\int_{0}^{T}|z_{n}'(s)|^{p}\,{\rm d}s\right)^{\frac{1}{p}}\to 0,\ \mbox{as}\ n \to \infty. \]

Since $\|z_{n}\|=1$, we can obtain

\[ \lim_{n\to \infty} z_{n}(t)={\pm} 1,\quad \forall t\in [0,T]\Rightarrow \lim_{n\to \infty} u_{n}(t)=\lim_{n\to \infty} \|u_{n}\|_{\infty}z_{n}(t)={\pm} \infty. \]

Thus, we have

(2.14)\begin{equation} \lim_{n\to \infty} a_{{\pm}}(t)\phi_{p}(u_{n}^{{\pm}}(t))={\pm} \infty,\ \forall t\in [0,T].\end{equation}

Similarly, we have

(2.15)\begin{align} \varepsilon\left(\int_{0}^{T}|u_{n}'(s)|^{p}\,{\rm d}s\right)^{\frac{1}{q}}& \leq 2^{\frac{1}{q}+1}\|a_{1}\|_{\infty}T^{\frac{1}{q}}+2^{\frac{1}{q}+1}\|a_{2}\|_{\infty}T^{\frac{p-\theta-1}{p}}\left(\int_{0}^{T}|u'_{n}(s)|^{p}\,{\rm d}s\right)^{\frac{\theta}{p}}\nonumber\\ & \quad+ \left(\int_{0}^{T}|e(s)|^{q}\right)^{\frac{1}{q}}. \end{align}

Clearly, $\|u'_{n}\|_{{p}}$ is bounded, i.e, there exists a positive constant $\overline {L}$ such that ${\|u'_{n}\|_{{p}}\leq \overline {L}}$. Otherwise,

\[ \|u'_{n}\|_{{p}}\to \infty,\ \mbox{as}\ n\to \infty. \]

From (2.15), we have

(2.16)\begin{equation} \varepsilon\|u_{n}\|_{{p}}^{\frac{p-\theta-1}{p}}\leq \frac{2^{\frac{1}{q}+1}\|a_{1}\|_{\infty}T^{\frac{1}{q}}+ \left(\int_{0}^{T}|e(s)|^{q}\right)^{\frac{1}{q}}}{\|u_{n}\|_{{p}}^{\frac{\theta}{p}}}+2^{\frac{1}{q}+1}\|a_{2}\|_{\infty}T^{\frac{p-\theta-1}{p}},\end{equation}

which implies a contradiction. Thus, $\|u'_{n}\|_{{p}}$ is bounded. Moreover, we can also get

(2.17)\begin{align} \left|\int_{0}^{T}\phi_{p}(u_{n}'(s))\,{\rm d}s\right|& \leq \int_{0}^{T}|u_{n}'(s)|^{p-1}\,{\rm d}s\leq T^{\frac{1}{p}}\left(\int_{0}^{T}|u_{n}'(s)|^{p}\,{\rm d}s\right)^{\frac{1}{q}}\nonumber\\ & \leq T^{\frac{1}{p}} \overline{L}^{p-1}, \end{align}

and

(2.18)\begin{align} \left|\int_{0}^{T}\overline{g}_{1}(s,u_{n}'(s))\,{\rm d}s\right|& \leq \|a_{1}\|_{\infty}T+\|a_{2}\|_{\infty}\int_{0}^{T}|u'_{n}|\,{\rm d}s\nonumber\\ & \leq\|a_{1}\|_{\infty}T+\|a_{2}\|_{\infty}T^{\frac{1}{q}}\left(\int_{0}^{T}|u'_{n}|^{p}\,{\rm d}s\right)^{\frac{1}{p}}\nonumber\\ & \leq\|a_{1}\|_{\infty}T+\|a_{2}\|_{\infty}T^{\frac{1}{q}}\overline{L}. \end{align}

By $(H4)$, we have

(2.19)\begin{align} & \text{if }\lim_{n\to \infty} u_{n}(t)=\infty,\ \overline{g}_{0}(u_{n})\geq \varrho{|u_{n}|}^{\gamma}\Rightarrow \overline{g}_{0}(u_{n})\to \infty \ \mbox{as}\ n \to \infty \ \mbox{or}\nonumber\\ & \text{if }\lim_{n\to \infty} u_{n}(t)={-}\infty,\ \overline{g}_{0}(u_{n})\leq{-} \varrho{|u_{n}|}^{\gamma}\Rightarrow \overline{g}_{0}(u_{n})\to -\infty \ \mbox{as}\ n \to \infty, \end{align}

which implies that

(2.20)\begin{equation} \lim_{n\to\infty}\left|\int_{0}^{T}(e(s)- a_{+}(s)\phi_{p}(u_{n}^{+}(s))+a_{-}(s)\phi_{p}(u_{n}^{-}(s))\,{\rm d}s\right|=\infty.\end{equation}

However, integrating from 0 to $T$ on (2.11), we have

\begin{align*} & \hbar\varepsilon\int_{0}^{T}\phi_{p}(u_{n}'(s))\,{\rm d}s+\int_{0}^{T} a_{+}(s)\phi_{p}(u_{n}^{+}(s))\,{\rm d}s-\int_{0}^{T} a_{-}(s)\phi_{p}(u_{n}^{-}(s))\,{\rm d}s\nonumber\\ & \quad+ \int_{0}^{T}W_{1}(s,u_{n}(s),u'_{n}(s))\,{\rm d}s =\int_{0}^{T}e(s)\,{\rm d}s. \end{align*}

This implies a contradiction for $n$ large enough. Therefore, $|u(t)|\leq L$. By the similar way to lemma 2.2, we can obtain that there exists a positive constant $\Theta$ such that $|u'(t)|\leq \Theta$. Moreover, $\mbox {deg}_{B}(F,\mathbb {R}\cap \partial \Omega,0)\neq 0$. Thus, the problem (2.5) has at least one solution.

Remark 2.5 In view of the continuity theorem, the key step is to obtain a priori bound estimation of solutions. In fact, if $W_{1}(t,u,u')$ is written as $\overline {g}_{0}(u)+\overline {g}_{1}(t,u,u')$, by the method used in lemma 2.3, we can not get (2.13) that make us fail to obtain the prior estimation when $1< p<\infty$.

Now, we state and prove our main results for problems (1.1) and (1.2).

Theorem 2.6 For $2\leq p<\infty$, assume that $(H1)$ and the following conditions hold.

$(H5)$ There exist positive constants $\delta,\ \zeta,\ d$ such that for $t\in [0,T],$

\[ \delta\leq\frac{g_{0}(u)}{\phi_{p}(u)}\leq \zeta,\ |u|\geq d. \]

$(H6)$ $\mathop {\lim }\limits _{|u| \to \infty } \frac {{pG_{0}(u)}}{{|u|^{p} }} = \eta,$ where $\eta \in (0,\infty )\setminus \bigcup _{k=0}^{\infty }[\lambda _{2k}^{min},\lambda _{2k}^{max}]$.

Then the problem (1.1) has at least one solution.

Proof. In view of lemma 2.3, we claim that if $\{u_n(t)\}$ are solutions of the problem (2.4) corresponding to $\varepsilon = \{\varepsilon _n\}$, where $\varepsilon _n\to 0$ as $n\to \infty$, then there exist constants $\widetilde {D},\ \widetilde {K}>0$, independent of $\varepsilon _n$, such that for $t\in [0,T]$,

\[ |u_{n}(t)|\leq \widetilde{D},\ |u_{n}'(t)|\leq \widetilde{K}. \]

Otherwise,

\[ \|u_{n}\|_{\infty}\to \infty\ \mbox{as}\ n\to \infty. \]

For any $u \in C_{T}^{1}\setminus \{0\}$, let $z_{n}(t)=\frac {u_{n}(t)}{\left \| u_{n} \right \|_{\infty }}$, so $\|z_{n}\|_{\infty }=1$. Moreover, dividing the problem (2.4) by $\left \| u_{n} \right \|_{\infty }^{p-1}$, we have

(2.21)\begin{align} & ( \phi_{p}(z_{n}'))'+\hbar \varepsilon_{n}\phi_{p}(z_{n}')+ a_{+}(t)\phi_{p}(z_{n}^{+})- a_{-}(t)\phi_{p}(z_{n}^{-})+ \frac{g_{0}(u_{n})}{\left\| u_{n} \right\|_{\infty}^{p-1}}\nonumber\\ & \quad- \frac{g_{1}(t,u_{n}(t))}{\left\| u_{n} \right\|_{\infty}^{p-1}}= \frac{e(t)}{\left\| u_{n} \right\|_{\infty}^{p-1}}, \end{align}

which is equivalent to

(2.22)\begin{align} & ( \phi_{p}(z_{n}')e^{\varepsilon_{n}\hbar t})'+ e^{\varepsilon_{n} \hbar t}a_{+}(t)\phi_{p}(z_{n}^{+})- e^{\varepsilon_{n}\hbar t}a_{-}(t)\phi_{p}(z_{n}^{-})+ e^{\varepsilon_{n}\hbar t} \frac{g_{0}(u_{n})}{\left\| u_{n} \right\|_{\infty}^{p-1}}\nonumber\\ & \quad- e^{\varepsilon_{n}\hbar t}\frac{g_{1}(t,u_{n}(t))}{\left\| u_{n} \right\|_{\infty}^{p-1}}= e^{\varepsilon_{n}\hbar t} \frac{e(t)}{\left\| u_{n} \right\|_{\infty}^{p-1}}. \end{align}

Since $z_{n}(0)=z_{n}(T)$, there exists $t_{*}\in (0,T)$ such that $z_{n}'(t_{*})=0$. Thus, we have

\begin{align*} | \phi_{p}(z_{n}')|e^{\varepsilon_{n}\hbar t}& \leq\int_{t_{*}}^{t}\left(\left|a_{+}(s)\phi_{p}(z_{n}^{+})\right|+ \left|a_{-}(t)\phi_{p}(z_{n}^{-})\right|+\left|\frac{g_{0}(u_{n})}{\left\| u_{n} \right\|_{\infty}^{p-1}}\right|\right.\\ & \left. \quad +\left| \frac{g_{1}(t,u_{n}(t))}{\left\| u_{n} \right\|_{\infty}^{p-1}}\right|+\left|\frac{e(s)}{\left\| u_{n} \right\|_{\infty}^{p-1}}\right|\right)e^{\varepsilon_{n}\hbar s}\,{\rm d}s. \end{align*}

From $\varepsilon _{n} \to 0$ as $n\to \infty$, there exists a positive constant $d_{1}$ such that $e^{\varepsilon _{n}\hbar t}\leq d_{1}$. Moreover, we have

(2.23)\begin{align} \int_{t_{*}}^{t} \left| \frac{g_{1}(t,u_{n}(t))}{\left\| u_{n} \right\|_{\infty}^{p-1}}e^{\varepsilon_{n}\hbar s}\,{\rm d}s\right|& \leq \frac{d_{1}}{\left\| u_{n} \right\|_{\infty}^{p-1}}\int_{0}^{T} | g_{1}(s,u_{n}(s))|\,{\rm d}s\nonumber\\ & \leq \frac{d_{1}}{\left\| u_{n} \right\|_{\infty}^{p-1}}\left(\|a_{1}\|_{\infty}T+\|a_{2}\|_{\infty}\int_{0}^{T} |u_{n}(s)|^{\theta}\,{\rm d}s\right)\nonumber\\ & \leq \frac{d_{1}T\|a_{1}\|_{\infty}}{\left\| u_{n} \right\|_{\infty}^{p-1}}+ \frac{d_{1} T \|a_{2}\|_{\infty}}{\left\| u_{n} \right\|_{\infty}^{p-\theta-1}}, \end{align}

which implies that there exists a constant $M_{1}>0$ which is independent of $n$ such that

(2.24)\begin{equation} \int_{t_{*}}^{t} \left| \frac{g_{1}(t,u_{n}(t))}{\left\| u_{n} \right\|_{\infty}^{p-1}}e^{\varepsilon_{n}\hbar s}\,{\rm d}s\right|\leq M_{1}.\end{equation}

Based on $\|z_{n}\|=1$ and $(H5)$, for any $t \in [0,T]$, there exists a constant $M_{2}>0$ which is independent of $n$ such that

\[ \int_{t_{*}}^{t}(\left|a_{+}(s)\phi_{p}(z_{n}^{+})\right|+ \left|a_{-}(s)\phi_{p}(z_{n}^{-})\right|+\left|\frac{g_{0}(u_{n})}{\left\| u_{n} \right\|_{\infty}^{p-1}}\right|)e^{\varepsilon_{n}\hbar s}\,{\rm d}s\leq M_{2}, \]

which together with $e\in C([0,T])$ and the monotonicity of $\phi _{p}(\cdot )$ yield that

(2.25)\begin{equation} |\phi_{p}(z_{n}'(t))|\leq M_{3},\ \mbox{i.e.},\ |z_{n}'(t)|\leq M_{3}^{q-1},\end{equation}

where $M_{3}>0$ is a constant and independent of $n$. Thus, from the definition of $z_{n}(t)$, it follows that $z_{n}(t)$ is uniformly bounded. By the standard argument, we can get that $z_{n}(t)$ is also equicontinuous. Moreover, from (2.22), we can also find a constant $M_{4}>0$ which is independent of $n$ such that

\[ |( \phi_{p}(z_{n}'(t)))'|\leq M_{4}. \]

Furthermore, $\forall t_{1},\ t_{2}\in [0,1]$, assuming that $t_{1}\leq t_{2},$ for any $n\in \mathbb {N}$, we have

\begin{align*} | \phi_{p}(z_{n}'(t_{2}))-\phi_{p}(z_{n}'(t_{1}))|& \leq \int_{t_{1}}^{t_{2}}|( \phi_{p}(z_{n}'(t)))'\,{\rm d}s|\\ & \leq M_{4}|t_{2}-t_{1}| \to 0\ \mbox{uniformly as}\ t_{1}\to t_{2}, \end{align*}

which together with the monotonicity and uniformly continuity of $\phi _{p}(\cdot )$ on $[-M_{3}^{q-1},M_{3}^{q-1}]$ yield that $z_{n}'(t)$ is also uniformly bounded and equicontinuous. By Arzelá–Ascoli theorem, there exists a subsequence of $\{z_{n}\}$ (without loss of generality, take its subsequence) and $z(t)\in C_{T}^{1}$ such that for any $t\in [0,T]$,

(2.26)\begin{equation} \mathop {\lim }\limits_{n \to \infty } z_n (t) = z(t),\mathop {\lim }\limits_{n \to \infty } z_n' (t) = z'(t).\end{equation}

Thus, there exists a $\widetilde {t}\in [0,T]$ such that $z(\widetilde {t})\neq 0.$ Multiplying both sides of (2.4) by $\frac {u_{n}'(t)}{\|u_{n}\|^{p}}$ and integrating from $\widetilde {t}$ to $t$, one has

(2.27)\begin{align} & \int_{\widetilde{t}}^{t}( \phi_{p}(z_{n}'))'z_{n}'\,{\rm d}s\,{+}\,\varepsilon_{n}\hbar\int_{\widetilde{t}}^{t}|z_{n}|^{p}\,{\rm d}s\,{+}\int_{\widetilde{t}}^{t} a_{+}(s)\phi_{p}(z_{n}^{+})z_{n}'\,{\rm d}s\,{-}\int_{\widetilde{t}}^{t} a_{-}(s)\phi_{p}(z_{n}^{-})z_{n}'\,{\rm d}s\nonumber\\ & \quad+ \frac{G_{0}(u_{n}(t))-G_{0}(u_{n}(\widetilde{t}))}{\left\| u_{n} \right\|_{\infty}^{p}}- \int_{\widetilde{t}}^{t}\frac{ g_{1}(s,u_{n}(s))z_{n}'}{\left\| u_{n} \right\|_{\infty}^{p-1}}\,{\rm d}s=\int_{\widetilde{t}}^{t}\frac{e(s)z_{n}'}{\left\| u_{n} \right\|_{\infty}^{p-1}}\,{\rm d}s, \end{align}

Following $(H4)$, we know that

\[ \mathop {\lim }\limits_{|u| \to \infty } \frac{{pG_{0}(u)}}{{|u|^{p} }} = \eta. \]

For any $\epsilon >0$, there exists a constant $l_{1}>0$ such that

\[ \left|\frac{{pG_{0}(u)}}{{|u|^{p} }}-\eta\right|\leq \epsilon,\ |u|\geq l_{1},\ \mbox{i.e.}, \left|G_{0}(u)-\frac{\eta|u|^{p}}{p}\right|\leq \epsilon\frac{|u|^{p}}{p},\ |u|\geq l_{1}. \]

By the continuity of $G_{0}$, define

\[ l_{2}:=\max_{|u|\leq l_{l}}\left|G_{0}(u)-\frac{\eta|u|^{p}}{p}\right|. \]

Thus, we have

\[ \left|G_{0}(u_{n})-\frac{\eta|u_{n}|^{p}}{p}\right|\leq \epsilon\frac{|u_{n}|^{p}}{p}+l_{2}\Rightarrow \left|\frac{G_{0}(u_{n})}{\left\| u_{n} \right\|_{\infty}^{p}}-\frac{\eta|z_{n}|^{p}}{p}\right|\leq \epsilon+\frac{l_{2}}{\|u_{n}\|^{p}}. \]

Taking limit in the above equality, by arbitrariness of $\epsilon$, we can obtain

(2.28)\begin{equation} \lim_{n\to\infty}\frac{G_{0}(u_{n})}{\left\| u_{n} \right\|_{\infty}^{p}}=\frac{\eta|z|^{p}}{p}.\end{equation}

Moreover, by (2.24) and (2.25), we have

(2.29)\begin{equation} \int_{\widetilde{t}}^{t}\frac{ g_{1}(s,u_{n}(s))z_{n}'(s)}{\left\| u_{n} \right\|_{\infty}^{p-1}}\,{\rm d}s, \int_{\widetilde{t}}^{t}\frac{e(s)z_{n}'}{\left\| u_{n} \right\|_{\infty}^{p-1}}\,{\rm d}s\to 0,\ \mbox{as}\ n\to \infty.\end{equation}

Furthermore, we know that

(2.30)\begin{equation} \int_{\widetilde{t}}^{t}( \phi_{p}(z_{n}'(s)))'z_{n}'(s)\,{\rm d}s=\frac{1}{q}|\phi_{p}(z_{n}'(t))|^{q}-\frac{1}{q}|\phi_{p}(z_{n}'(\widetilde{t}))|^{q}.\end{equation}

By (2.28)–(2.30), taking limit $n\to \infty$ in (2.27), we have

(2.31)\begin{align} & \frac{1}{q}|\phi_{p}(z'(t))|^{q}+\int_{\widetilde{t}}^{t} a_{+}(s)\phi_{p}(z^{+}(s))z'(s)\,{\rm d}s\notag\\ & \quad -\int_{\widetilde{t}}^{t} a_{-}(s)\phi_{p}(z^{-}(s))z'(s)\,{\rm d}s+\frac{\eta|z(t)|^{p}}{p}=\sigma,\end{align}

where

\[ \sigma=\frac{1}{q}|\phi_{p}(z'(\widetilde{t}))|^{q}+\frac{\eta|z(\widetilde{t})|^{p}}{p}>0. \]

Now, we claim that $z'(t)$ has only limited zero points on $[0, T].$ If not, assume that $z'(t)$ has unlimited zero points $\{\mu _{i}\}_{i\in \mathbb {N}}$ on $[0, T].$ Without loss of generality, there exists a $\mu _{*}$ such that

\[ \lim_{i\to \infty}\mu_{i}=\mu_{*}. \]

Following the continuity of $z'(t)$, we have $z'(\mu _{*})=0$ and

(2.32)\begin{equation} \int_{\widetilde{t}}^{\mu_{*}} a_{+}(s)\phi_{p}(z^{+}(s))z'(s)\,{\rm d}s-\int_{\widetilde{t}}^{\mu_{*}} a_{-}(s)\phi_{p}(z^{-}(s))z'(s)\,{\rm d}s+\frac{\eta|z(\mu_{*})|^{p}}{p}=\sigma.\end{equation}

By the above equality, we can get $z(\mu _{*})\neq 0$. In fact, if $z(\mu _{*})=0$, we have

(2.33)\begin{equation} \int_{\widetilde{t}}^{\mu_{*}} a_{+}(s)\phi_{p}(z^{+}(s))z'(s)\,{\rm d}s-\int_{\widetilde{t}}^{\mu_{*}} a_{-}(s)\phi_{p}(z^{-}(s))z'(s)\,{\rm d}s=\sigma.\end{equation}

By choosing a suitable radius $r_{1}$ and $r_{2}$ such that $\widetilde {t}$ belonging to the left neighbourhood of $(\mu _{*}-r_{1},\mu _{*})$ or the right neighbourhood of $(\mu _{*},\mu _{*}+r_{2})$, we just need to consider the following four cases.

\begin{align*} & (C1)\ z(t)\leq 0\ z'(t)\geq 0,\ t\in [\widetilde{t}, \mu_{*}],\\ & (C2)\ z(t)\geq 0\ z'(t)\leq 0,\ t\in [\widetilde{t}, \mu_{*}],\\ & (C3)\ z(t)\geq 0\ z'(t)\geq 0,\ t\in [\mu_{*},\widetilde{t}],\\ & (C4)\ z(t)\leq 0\ z'(t)\leq 0,\ t\in [\mu_{*},\widetilde{t}]. \end{align*}

For the case $(C1)$, we can obtain

\begin{align*} 0<\frac{1}{q}|\phi_{p}(z'(\widetilde{t}))|^{q}+\frac{\eta|z(\widetilde{t})|^{p}}{p}=\sigma& \leq a_{-,L}\int_{\widetilde{t}}^{\mu_{*}} \phi_{p}(z(s))z'(s)\,{\rm d}s\\ & ={-}\frac{a_{-,L}}{p}|z(\widetilde{t})|^{p}<0, \end{align*}

which is impossible. For the case $(C2)$, it follows

\begin{align*} 0<\frac{1}{q}|\phi_{p}(z'(\widetilde{t}))|^{q}+\frac{\eta|z(\widetilde{t})|^{p}}{p}=\sigma& \leq a_{+,L}\int_{\widetilde{t}}^{\mu_{*}} \phi_{p}(z(s))z'(s)\,{\rm d}s\\ & ={-}\frac{a_{+,L}}{p}|z(\widetilde{t})|^{p}<0, \end{align*}

which implies a contradiction. Moreover, by the same way, we can also obtain a contradiction under the case of $(C3)$ or $(C4)$. Thus, $z(\mu _{*})\neq 0$. Without loss of generality, assume that $z(\mu _{*})>0$. According to its continuity, there exist constants $\xi,\nu >0$ such that

\[ z(t)>\xi,\quad t\in [\mu_{*}-\nu,\mu_{*}+\nu]. \]

Choosing $n$ large enough, we can also get

\[ z_{n}(t)>\xi,\quad t\in [\mu_{*}-\nu,\mu_{*}+\nu]. \]

Since $\mu _{*}$ is a limiting point of $\{\mu _{i}\}$, so we can find two zero points called $\mu ^{*},\ \mu ^{**}$ of $z'(t)$ in $[\mu _{*}-\nu,\mu _{*}+\nu ]$. Integrating from $\mu ^{*}$ to $\mu ^{**}$ on (2.27), taking limit $n\to \infty$, we can obtain

(2.34)\begin{equation} \lim_{n\to \infty} \left(\int_{\mu^{*}}^{\mu^{**}} \frac{g_{0}(u_{n}(s))}{\left\| u_{n} \right\|_{\infty}^{p-1}}e^{\varepsilon_{n}\hbar s}\,{\rm d}s+\int_{\mu^{*}}^{\mu^{**}} e^{\varepsilon_{n}\hbar s} a_{+}(s)\phi_{p}(z_{n}(s))\,{\rm d}s\right)=0.\end{equation}

On the other hand, for $t\in [\mu ^{*}, \mu ^{**}]$, choose $n$ large enough such that

\[ u_{n}(t)\geq z_{n}(t)\|u\|_{\infty}\geq \xi\|u\|_{\infty}\geq d. \]

Then, we have

\[ \frac{g_{0}(u_{n}(t))}{\left\| u_{n} \right\|^{p-1}}=\frac{g_{0}(u_{n}(t))}{\phi_{p}(u_{n}(t))}\phi_{p}(z_{n}(t))\geq\delta \phi_{p}(\xi). \]

Thus, we can obtain

(2.35)\begin{align} & \lim_{n\to \infty}\left(\int_{\mu^{*}}^{\mu^{**}} \frac{g_{0}(u_{n}(s))}{\left\| u_{n} \right\|^{p-1}}e^{\varepsilon_{n} \hbar s}\,{\rm d}s+\int_{\mu^{*}}^{\mu^{**}} e^{\varepsilon_{n}\hbar s}a_{+}(s)\phi_{p}(z(s))\,{\rm d}s\right)\nonumber\\ & \quad\geq (a_{{+},L}+\delta)\phi_{p}(\xi)(\mu^{**}-\mu^{*})>0, \end{align}

which contradicts to (2.34). Thus, $z'(t)$ has only limited zero points on $[0, T].$ Differentiating (2.31), one has

\[ z'(t)\left(( \phi_{p}(z'(t)))'+a(t)\phi_{p}(z^{+}(t))-b(t)\phi_{p}(z^{-}(t))+\eta\phi_{p}(z(t))\right)=0. \]

Since $z'(t)$ has only limited zero points on $[0, T]$, we have

(2.36)\begin{equation} (\phi_{p}(z'(t)))'+a(t)\phi_{p}(z^{+}(t))-b(t)\phi_{p}(z^{-}(t))+\eta\phi_{p}(z(t))=0, \end{equation}

which together with $\eta \in (0,\infty )\setminus \bigcup _{k=0}^{\infty }[\lambda _{2k}^{min},\lambda _{2k}^{max}]$ yield $z(t)\equiv 0$ for $t\in [0,T]$. In fact, if there exists a $z\in C_{T}^{1}$ such that (2.36) is satisfied on a subset $\Delta \subset [0,T]$ and $z\equiv 0$ on $[0,T]\setminus \Delta,$ it must be an eigenvector corresponding to a half-eigenvalue, which implies a contradiction. Therefore, $z(t)\equiv 0$. However, it contradicts to $\|z\|_{\infty }=1$. Thus, there exists a constant $\widetilde {D} >0$ which is independent of $\varepsilon _n$ such that $|u_{n}(t)|\leq \widetilde {D}$.

Since $u_{n}(0)=u_{n}(T)$, there exists $t_{n}\in (0,T)$ such that $u_{n}'(t_{n})=0$. By the same way to lemma 2.2, we can also get that there exists a constant $\widetilde {K} >0$ that is independent of $\varepsilon _n$ such that $|u'_{n}(t)|\leq \widetilde {K}$. By Arzelá–Ascoli theorem (without loss of generality, take its subsequence), there exist $u\in C_{T}^{1}$ and $t_{0}\in [0,T]$ such that $\mathop {\lim }\limits _{n \to \infty } t_n = t_{0}$,

(2.37)\begin{equation} \mathop {\lim }\limits_{n \to \infty } u_n (t) = u(t),\quad \mathop {\lim }\limits_{n \to \infty } u_n' (t) = u'(t),\ \forall t\in [0,T].\end{equation}

Taking limit $n\to \infty$ in the following equation

\[ \phi_{p}(u_{n}')e^{\varepsilon_{n} \hbar t}=\int_{t_{n}}^{t}\left({-}a_{+}(s)\phi_{p}(u_{n}^{+})+ a_{-}\phi_{p}(u_{n}^{-}) -W_{0}(s,u(s))+e(s)\right)e^{\varepsilon_{n} \hbar s}\,{\rm d}s, \]

we have

\[ \phi_{p}(u')=\int_{t_{0}}^{t}\left({-}a_{+}(s)\phi_{p}(u^{+})+ a_{-}\phi_{p}(u^{-}) -W_{0}(s,u(s))+e(s)\right)\,{\rm d}s, \]

which implies that for all $t\in [0,T]$

\[ (\phi_{p}(u'(t)))'+a_{+}(t)\phi_{p}(u^{+}(t))-a_{-}(t)\phi_{p}(u^{-}(t))+W_{0}(t,u(t))=e(t) \]

and $u(0) = u(T),u'(0) = u'(T).$ Thus, the problem (1.1) has at least one periodic solution.

Remark 2.7 It should be mentioned that in the separated boundary conditions, typical existence results in this area usually assume that $\eta$ lies in the gap between half-eigenvalues with consecutive nodal counts, not between half-eigenvalues with the same nodal counts that may make the problem not have a solution (see [23, 24]).

Theorem 2.8 For $1< p<\infty,$ if $(H3),$ $(H5)$ and $(H6)$ hold. Then the problem (1.2) has at least one periodic solutions.

Proof. Based on lemma 2.4, by similar way to theorem 2.6, the conclusion of theorem 2.8 is also true. Here, we omit the detail.

Remark 2.9 If $\overline {g}_{1}=0$, $a_{\pm }$ are positive constants and $\widetilde {{g}}_{0}(u)$ satisfies $(H5)$ and $\mathop {\lim }\limits _{|u| \to \infty } \frac {{pG_{0}(u)}}{{|u|^{p} }} = \eta$, letting

\[ \widetilde{{g}}_{0}(u)=\overline{g}_{0}(u)+a_{+}\phi_{p}(u^{+})-a_{-}\phi_{p}(u^{-}), \]

by theorem 2.8, the problem (1.2) has at least one periodic solutions, provided that $(a_{+}+\eta,a_{-}+\eta )\notin S_{p},$ which is corresponding to theorem 2.3 of [17]. Thus, our main results generalize their conclusions.