\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 97, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/97\hfil 
Existence of infinitely many anti-periodic solutions]
{Existence of infinitely many anti-periodic solutions for second-order
impulsive differential inclusions}

\author[S. Heidarkhani, G. A. Afrouzi, A. Hadjian,  J. Henderson
 \hfil EJDE-2013/97\hfilneg]
{Shapour Heidarkhani, Ghasem A. Afrouzi,\\
  Armin Hadjian,  Johnny Henderson} 

\address{Shapour Heidarkhani \newline
Department of Mathematics, Faculty of Sciences, Razi
University,  67149 Kermanshah, Iran. \newline
School of Mathematics, Institute for Research in Fundamental
Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran}
\email{s.heidarkhani@razi.ac.ir}

\address{Ghasem A. Afrouzi \newline
Department of Mathematics, Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran}
\email{afrouzi@umz.ac.ir}

\address{Armin Hadjian\newline
Department of Mathematics, Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran}
\email{a.hadjian@umz.ac.ir}

\address{Johnny Henderson \newline
Department of Mathematics, Baylor University, Waco, TX 76798-7328, USA} 
\email{Johnny\_Henderson@baylor.edu}

\thanks{Submitted January 29, 2013. Published April 16, 2013.}
\subjclass[2000]{58E05, 49J52, 34A60}
\keywords{Differential inclusions; impulsive; anti-periodic solution; 
\hfill\break\indent non-smooth critical point theory}

\begin{abstract}
 In this article, we establish the existence of infinitely many
 anti-periodic solutions for a second-order impulsive differential
 inclusion with a perturbed nonlinearity and two parameters.
 The technical approach is mainly based on a critical point theorem
 for non-smooth functionals.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks



\section{Introduction}

The aim of this article is to show the existence
of infinitely many solutions for the following two parameter
second-order impulsive differential inclusion subject to
anti-periodic boundary conditions
\begin{equation}\label{e1.1}
\begin{gathered}
-(\phi_p(u'(x)))'+M\phi_p(u(x))\in\lambda F(u(x))+\mu G(x,u(x))\quad \text{in }
[0,T]\setminus Q,\\
-\Delta\phi_p(u'(x_k))=I_k(u(x_k)),\quad k=1,2,\dots,m,\\
u(0)=-u(T),\quad u'(0)=-u'(T),
\end{gathered}
\end{equation}
where $Q=\{x_1,x_2,\dots,x_m\}$, $p>1$, $T>0$,
$M\geq 0$, $\phi_p(x):=|x|^{p-2}x$,
$0=x_0<x_1<\dots<x_m<x_{m+1}=T$,
$\Delta\phi_p(u'(x_k)) :=\phi_p(u'(x_k^+))-\phi_p(u'(x_k^-))$,
with $u'(x_k^+)$ and $u'(x_k^-)$ denoting the right and left limits,
respectively, of $u'(x)$ at $x=x_k$,
 $I_k\in C(\mathbb{R},\mathbb{R})$, $k=1,2,\dots,m$, $\lambda$ is a positive
parameter, $\mu$ is a nonnegative parameter, and $F$ is a
multifunction defined on $\mathbb{R}$, satisfying
\begin{itemize}
\item[(F1)] $F:\mathbb{R}\to 2^{\mathbb{R}}$ is upper semicontinuous
with compact convex values;

\item[(F2)] $\min F,\,  \max F:\mathbb{R}\to\mathbb{R}$ are Borel
measurable;

\item[(F3)] $|\xi|\leq a(1+|s|^{r-1})$ for all $s\in\mathbb{R}$,
$\xi\in F(s)$, $r>1$ $(a>0)$.
\end{itemize}
Also, $G$ is a multifunction defined on $[0,T]\times\mathbb{R}$,
satisfying
\begin{itemize}
\item[(G1)] $G(x,\cdot):\mathbb{R}\to 2^{\mathbb{R}}$
is upper semicontinuous with compact convex values
for a.e. $x\in[0,T]\setminus Q$;

\item[(G2)] $\min G,\,  \max G:([0,T]\setminus Q)\times\mathbb{R}\to\mathbb{R}$
are Borel measurable;

\item[(G3)] $|\xi|\leq a(1+|s|^{r-1})$ for
a.e. $x\in[0,T]$, $s\in\mathbb{R}$, $\xi\in G(x,s)$, $r>1$ $(a>0)$.
\end{itemize}

 Impulsive differential equations are used to describe various models
of real-world processes that are subject to a sudden change.
These models are studied in physics, population dynamics,
ecology, industrial robotics, biotechnology, economics, optimal
control, and so forth. Associated with this development, a theory of
impulsive differential equations has been given extensive attention.
Differential inclusions arise in models for control systems,
mechanical systems, economical systems, game theory, and biological
systems to name a few. It is very important to study anti-periodic
boundary value problems because they can be applied to interpolation
problems \cite{DelKno}, antiperiodic wavelets \cite{Chen}, the Hill
differential operator \cite{DjaMity}, and so on. It is natural from
both a physical standpoint as well as a theoretical view to give
considerable attention to a synthesis involving problems for
impulsive differential inclusion with anti-periodic boundary
conditions.

 Recently, multiplicity of solutions for differential inclusions via
non-smooth variational methods and critical point theory has been
considered and here we cite the papers
\cite{Iann1,Iann2,Kris,KrisMarVar,TianHend}. For instance, in
\cite{Kris}, the author, employing a non-smooth Ricceri-type
variational principle \cite{Ricceri1}, developed by Marano and
Motreanu \cite{MarMot}, has established the existence of infinitely
many, radially symmetric solutions for a differential inclusion
problem in $\mathbb{R}^N$. Also, in \cite{KrisMarVar}, the authors
 extended a recent result of Ricceri concerning the existence of
three critical points of certain non-smooth functionals. Two
applications have been given, both in the theory of differential
inclusions; the first one concerns a non-homogeneous Neumann
boundary value problem, the second one treats a quasilinear elliptic
inclusion problem in the whole $\mathbb{R}^N$. In \cite{Iann1}, the
author, under convenient assumptions, has investigated the existence
of at least three positive solutions for a differential inclusion
involving the $p$-Laplacian operator on a bounded domain, with
homogeneous Dirichlet boundary conditions and a perturbed
nonlinearity depending on two positive parameters; his result also
ensured an estimate on the norms of the solutions independent of
both the perturbation and the parameters. Very recently, Tian and
Henderson in \cite{TianHend}, based on a non-smooth version of
critical point theory of Ricceri due to Iannizzotto \cite{Iann1},
have established the existence of at least three solutions for the
problem \eqref{e1.1} whenever $\lambda$ is large enough and $\mu$ is
small enough.

 In the present paper, motivated by \cite{TianHend}, employing an
 abstract critical point result (see Theorem \ref{the2.6}
below), we are interested in ensuring the existence of infinitely
many anti-periodic solutions for the problem \eqref{e1.1}; see
Theorem \ref{the3.1} below. We refer to \cite{BonaMoliRad}, in which
related variational methods are used for non-homogeneous problems.

 To the best of our knowledge, no investigation has been devoted to
establishing the existence of infinitely many solutions to such a
problem as \eqref{e1.1}. For a couple of references on impulsive
differential inclusions, we refer to \cite{ErbeKraw} and
\cite{FrigOreg}.

 A special case of our main result is the following theorem.

\begin{theorem}\label{t1.1}
Assume that {\rm(F1)--(F3)} hold, and
$I_i(0)=0$, $I_i(s)s<0$, $s\in\mathbb{R}$, $i=1,2,\dots,m$.
Furthermore, suppose that
\begin{gather*}
\liminf_{\xi\to+\infty} \frac{\sup_{|t|\leq\xi}\min\int_0^tF(s)ds}{\xi^p}=0,\\
\limsup_{\xi\to+\infty}\frac{\int_{0}^T\min\int_0^{\xi(\frac{T}{2}-x)}
F(s)\,ds\,dx}{\frac{1}{p}\xi^{p}\left(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\right)
-\sum_{i=1}^m\int_0^{\xi(\frac{T}{2}-x_{i})}I_i(s)ds}=+\infty.
\end{gather*}
Then, the problem \eqref{e1.1}, for $\lambda=1$ and $\mu=0$, admits
a sequence of pairwise distinct solutions.
\end{theorem}

\section{Basic definitions and preliminary results}

 Let $(X,\|\cdot\|_X)$ be a real Banach space. We
denote by $X^\ast$ the dual space of $X$, while
$\langle\cdot,\cdot\rangle$ stands for the duality pairing between
$X^\ast$ and $X$. A function $\varphi:X\to\mathbb{R}$ is
called locally Lipschitz if, for all $u\in X$, there exist a
neighborhood $U$ of $u$ and a real number $L>0$ such that
$$
|\varphi(v)-\varphi(w)|\leq L\|v-w\|_X\quad\text{for all } v,w\in U.
$$
If $\varphi$ is locally Lipschitz and $u\in X$, the generalized
directional derivative of $\varphi$ at $u$ along the direction
$v\in X$ is
$$
\varphi^\circ(u;v):=\limsup_{w\to u,\,\tau\to
0^+}\frac{\varphi(w+\tau v)-\varphi(w)}{\tau}.
$$
The generalized gradient of $\varphi$ at $u$ is the set
$$
\partial\varphi(u):=\{u^\ast\in X^\ast :
\langle u^\ast,v\rangle\leq\varphi^\circ(u;v)\;\text{for all}\;v\in
X\}.
$$
So $\partial\varphi:X\to 2^{X^\ast}$ is a multifunction. We
say that $\varphi$ has compact gradient if $\partial\varphi$ maps
bounded subsets of $X$ into relatively compact subsets of $X^\ast$.

\begin{lemma}[{\cite[Proposition 1.1]{MotPan}}] \label{lem2.1}
Let $\varphi\in C^1(X)$ be a functional. Then $\varphi$ is locally
Lipschitz and
\begin{gather*}
\varphi^\circ(u;v)=\langle\varphi'(u),v\rangle\quad
\text{for all } u,v\in X; \\
\partial\varphi(u)=\{\varphi'(u)\}\quad\text{for all }u\in X.
\end{gather*}
\end{lemma}

\begin{lemma}[{\cite[Proposition 1.3]{MotPan}}] \label{lem2.2}
Let $\varphi:X\to\mathbb{R}$ be a locally Lipschitz
functional. Then
$ \varphi^\circ(u;\cdot)$ is subadditive and
positively homogeneous for all $u\in X$, and
\[
\varphi^\circ(u;v)\leq L\|v\|\quad\text{for all }u,v\in X,
\]
with $L>0$ being a Lipschitz constant
for $\varphi$ around $u$.
\end{lemma}

\begin{lemma}[{\cite{Clarke}}] \label{lem2.3}
Let $\varphi:X\to\mathbb{R}$ be a locally Lipschitz
functional. Then $\varphi^\circ:X\times X\to\mathbb{R}$ is
upper semicontinuous and for all $\lambda\geq 0$, $u,v\in X$,
$$
(\lambda\varphi)^\circ(u;v)=\lambda\varphi^\circ(u;v).
$$
Moreover, if $\varphi,\psi:X\to\mathbb{R}$ are locally
Lipschitz functionals, then
$$
(\varphi+\psi)^\circ(u;v)\leq\varphi^\circ(u;v)+\psi^\circ(u;v)\quad
\text{for all } u,v\in X.
$$
\end{lemma}

\begin{lemma}[{\cite[Proposition 1.6]{MotPan}}] \label{lem2.4}
Let $\varphi,\psi:X\to\mathbb{R}$ be locally Lipschitz
functionals. Then
\begin{gather*}
\partial(\lambda\varphi)(u)=\lambda\partial\varphi(u)\quad
\text{for all } u\in X,\,\lambda\in\mathbb{R}, \\
\partial(\varphi+\psi)(u)\subseteq\partial\varphi(u)+\partial\psi(u)\quad
\text{for all } u\in X.
\end{gather*}
\end{lemma}

\begin{lemma}[{\cite[Proposition 1.6]{Iann1}}] \label{lem2.5}
Let $\varphi:X\to\mathbb{R}$ be a locally Lipschitz
functional with a compact gradient. Then $\varphi$ is sequentially
weakly continuous.
\end{lemma}


 We say that $u\in X$ is a (generalized) critical point of a locally
Lipschitz functional $\varphi$ if $0\in\partial\varphi(u)$; i.e.,
$$
\varphi^\circ(u;v)\geq 0\quad\text{for all}\;v\in X.
$$
When a non-smooth functional, $g:X\to(-\infty,+\infty)$, is
expressed as a sum of a locally Lipschitz function,
$\varphi:X\to\mathbb{R}$, and a convex, proper, and lower
semicontinuous function, $j:X\to(-\infty,+\infty)$; that is,
$g:=\varphi+j$, a (generalized) critical point of $g$ is every
 $u\in X$ such that
$$
\varphi^\circ(u;v-u)+j(v)-j(u)\geq 0
$$
for all $v\in X$ (see \cite[Chapter 3]{MotPan}).

 Hereafter, we assume that $X$ is a reflexive real Banach space,
$\mathcal{N}:X\to\mathbb{R}$ is a sequentially weakly lower
semicontinuous functional, $\Upsilon:X\to\mathbb{R}$ is a
sequentially weakly upper semicontinuous functional, $\lambda$ is a
positive parameter, $j:X\to(-\infty,+\infty)$ is a convex,
proper, and lower semicontinuous functional, and $D(j)$ is the
effective domain of $j$.
Write
$$
\mathcal{M}:=\Upsilon-j,\quad
I_\lambda:=\mathcal{N}-\lambda\mathcal{M}
=(\mathcal{N}-\lambda\Upsilon)+\lambda j.
$$
We also assume that $\mathcal{N}$ is coercive and
\begin{equation}\label{e2.1}
D(j)\cap\mathcal{N}^{-1}((-\infty,r))\neq\emptyset
\end{equation}
for all $r>\inf_X\mathcal{N}$. Moreover, owing to \eqref{e2.1} and
provided $r>\inf_X\mathcal{N}$, we can define
\begin{gather*}
\varphi(r):=\inf_{u\in\mathcal{N}^{-1}((-\infty,r))}
\frac{\big(\sup_{v\in\mathcal{N}^{-1}((-\infty,r))}\mathcal{M}(v)\big)
 -\mathcal{M}(u)}{r-\mathcal{N}(u)},
\\
\gamma:=\liminf_{r\to +\infty}\varphi(r),\quad
\delta:=\liminf_{r\to(\inf_X\mathcal{N})^+}\varphi(r).
\end{gather*}
If $\mathcal{N}$ and $\Upsilon$ are locally Lipschitz functionals,
in \cite[Theorem 2.1]{BonaMoli1} the following result is proved;
it is a more precise version of \cite[Theorem 1.1]{MarMot} (see
also \cite{Ricceri1}).

\begin{theorem}\label{the2.6}
Under the above assumption on $X,\,\mathcal{N}$ and $\mathcal{M}$,
one has
\begin{itemize}
\item[(a)] For every $r>\inf_X\mathcal{N}$ and every $\lambda\in
(0,1/\varphi(r))$, the restriction of the functional
$I_\lambda=\mathcal{N}-\lambda\mathcal{M}$ to
$\mathcal{N}^{-1}((-\infty,r))$ admits a global minimum, which is a
critical point (local minimum) of $I_\lambda$ in $X$.

\item[(b)] If $\gamma<+\infty$, then for each
$\lambda\in (0,1/\gamma)$, the following alternative holds: either
\begin{itemize}
\item[(b1)] $I_\lambda$ possesses a global
minimum, or
\item[(b2)] there is a sequence $\{u_n\}$ of critical points
(local minima) of $I_\lambda$ such that
$\lim_{n\to+\infty}\mathcal{N}(u_n)=+\infty$.
\end{itemize}

\item[(c)] If $\delta<+\infty$, then for each
$\lambda\in (0,1/\delta)$, the following alternative holds: either
\begin{itemize}
\item[(c1)] there is a global minimum of $\mathcal{N}$ which is a
 local minimum of $I_\lambda$, or
\item[(c2)] there is a sequence $\{u_n\}$ of pairwise distinct
critical points (local minima)
of $I_\lambda$, with $\lim_{n\to+\infty}\mathcal{N}(u_n)=\inf_X\mathcal{N}$,
which  converges weakly to a global minimum of $\mathcal{N}$.
\end{itemize}
\end{itemize}
\end{theorem}

Now we recall  some basic definitions and notation. On the
reflexive Banach space  $X:=\{u\in W^{1,p}([0,T]) :
u(0)=-u(T)\}$ we consider the norm
$$
\|u\|_X:=\Big(\int_0^T\big(|u'(x)|^p+M|u(x)|^p\big) dx\Big)^{1/p}
$$
for all $u\in X$, which is equivalent to the usual norm (note that
$M\geq 0$).
We recall that $X$ is compactly embedded into the space $C^0([0,T])$
endowed with the maximum norm $\|\cdot\|_{C^0}$.

\begin{lemma}[{\cite[Lemma 3.3]{TianHend}}]
Let $u\in X$. Then
\begin{equation}
\label{3}\|u\|_{C^0}\leq\frac{1}{2}T^{1/q}\|u\|_X,
\end{equation}
where $1/p+1/q=1$.
\end{lemma}

 Obviously, $X$ is compactly embedded into $L^\gamma([0,T])$ endowed
with the usual norm $\|\cdot\|_{L^\gamma}$, for all $\gamma\geq 1$.

\begin{definition}\label{def2.7} \rm
A function $u\in X$ is a weak solution of the problem \eqref{e1.1}
if there exists $u^\ast\in L^\gamma([0,T])$ (for some $\gamma>1$)
such that
$$
\int_0^T\Big[\phi_p(u'(x))v'(x)+M\phi_p(u(x))v(x)-u^\ast(x)v(x)\Big]dx
-\sum_{i=1}^m I_i(u(x_i))v(x_i)=0
$$
for all $v\in X$ and $u^\ast\in\lambda F(u(x))+\mu G(x,u(x))$ for
a.e. $x\in[0,T]$.
\end{definition}

\begin{definition}\label{def2.8}
By a solution of the impulsive differential inclusion \eqref{e1.1}
we will understand a function $u:[0,T]\setminus
Q\to\mathbb{R}$ is of class $C^1$ with $\phi_p(u')$
absolutely continuous, satisfying
\begin{gather*}
-(\phi_p(u'(x)))'+M\phi_p(u(x))=u^\ast\quad \text{in } [0,T]\setminus Q,\\
-\Delta\phi_p(u'(x_k))=I_k(u(x_k)),\quad k=1,2,\dots,m,\\
u(0)=-u(T),\quad u'(0)=-u'(T),
\end{gather*}
where $u^\ast\in\lambda F(u(x))+\mu G(x,u(x))$ and
$u^\ast\in L^\gamma([0,T])$ (for some $\gamma>1$).
\end{definition}

\begin{lemma}[{\cite[Lemma 3.5]{TianHend}}] \label{lem2.9}
If a function $u\in X$ is a weak solution of \eqref{e1.1}, then
$u$ is a classical solution of \eqref{e1.1}.
\end{lemma}

 We introduce for a.e. $x\in[0,T]$ and all $s\in\mathbb{R}$, the
Aumann-type set-valued integral
$$
\int_0^s F(t)dt=\Big\{\int_0^s
f(t)dt\,:\, f:\mathbb{R}\to\mathbb{R}\text{ is a measurable
selection of }F\Big\}
$$
and set $\mathcal{F}(u)=\int_0^T\min\int_0^u F(s)\,ds\,dx$ for all
$u\in L^p([0,T])$; the Aumann-type set-valued integral
$$
\int_0^s G(x,t)dt=\Big\{\int_0^s
g(x,t)dt\,:\, g:[0,T]\times\mathbb{R}\to\mathbb{R}\text{ is
a measurable selection of }G\Big\}
$$
and set $\mathcal{G}(u)=\int_0^T\min\int_0^u G(x,s)\,ds\,dx$ for all
$u\in L^p([0,T])$.

\begin{lemma}[{\cite[Lemma 3.1]{Iann2}}] \label{lem2.10}
The functionals $\mathcal{F,G}:L^p([0,T])\to\mathbb{R}$ are
well defined and Lipschitz on any bounded subset of $L^p([0,T])$.
Moreover, for all $u\in L^p([0,T])$ and all
$u^\ast\in\partial(\mathcal{F}(u)+\mathcal{G}(u))$,
$$
u^\ast(x)\in F(u(x))+G(x,u(x))\quad\text{for a.e. }x\in[0,T].
$$
\end{lemma}

 We define an energy functional for the problem \eqref{e1.1} by
setting
$$
I_\lambda(u)=\frac{1}{p}\|u\|_X^p-\lambda\mathcal{F}(u)-\mu\mathcal{G}(u)
-\sum_{i=1}^m\int_0^{u(x_i)}I_i(s)ds
$$
for all $u\in X$.

\begin{lemma}[{\cite[Lemma 4.4]{TianHend}}] \label{lem2.11}
The functional $I_\lambda:X\to\mathbb{R}$ is locally
Lipschitz. Moreover, for each critical point $u\in X$ of
$I_\lambda$, $u$ is a weak solution of \eqref{e1.1}.
\end{lemma}

\section{Main results}

We formulate our main result using the following assumptions:
\begin{itemize}
\item[(F4)]
\begin{align*}
&\liminf_{\xi\to+\infty}\frac{\sup_{|t|\leq\xi}\min\int_0^tF(s)ds}{\xi^p}\\
&<\frac{1}{p}(\frac{2}{T})^p\limsup_{\xi\to+\infty}
\frac{\int_{0}^T\min\int_0^{\xi(\frac{T}{2}-x)}
F(s)\,ds\,dx}{\frac{1}{p}\xi^{p}\big(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\big)
-\sum_{i=1}^m\int_0^{\xi(\frac{T}{2}-x_{i})}I_i(s)ds};
\end{align*}

\item[(I1)] $I_i(0)=0$, $I_i(s)s<0$, $s\in\mathbb{R}$, $i=1,2,\dots,m$.
\end{itemize}

\begin{theorem}\label{the3.1}
Assume that {\rm(F1)--(F4), (I1)} hold. Let
\begin{gather*}
\lambda_1:=1\Big/\limsup_{\xi\to+\infty}\frac{\int_{0}^T\min\int_0^{\xi(\frac{T}{2}-x)}
F(s)\,ds\,dx}{\frac{1}{p}\xi^{p}\left(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\right)
-\sum_{i=1}^m\int_0^{\xi(\frac{T}{2}-x_{i})}I_i(s)ds},
\\
\lambda_2:=1\Big/\liminf_{\xi\to+\infty}
\frac{\sup_{|t|\leq\xi}\min\int_0^tF(s)ds}{\frac{1}{p}
\left(\frac{2\xi}{T}\right)^p}.
\end{gather*}
Then, for every $\lambda\in (\lambda_1,\lambda_2)$, 
and every multifunction $G$ satisfying 
\begin{itemize}
\item[(G4)] $\int_0^{T}\min\int_0^t G(x,s)\,ds\,dx\geq 0$ for all
$t\in\mathbb{R}$, and
\item[(G5)] $G_\infty:=\lim_{\xi\to+\infty}\frac{\sup_{|t|\leq\xi}
\min\int_0^t G(x,s)ds}{\xi^p}<+\infty$,
\end{itemize}
if we put
\[
\mu_{G,\lambda}:=\frac{1}{pG_\infty}\frac{2^{p}}{T^p}\big(1-\lambda\frac{pT^p
}{2^p}\liminf_{\xi\to+\infty}\frac{\sup_{|t|\leq\xi}
\min\int_0^tF(s)ds}{\xi^p}\big),
\]
where $\mu_{G,\lambda}=+\infty$ when $G_\infty=0$,  problem
\eqref{e1.1} admits an unbounded sequence of solutions for every
$\mu\in [0,\mu_{G,\lambda})$ in $X$.
\end{theorem}

\begin{proof}
Our aim is to apply Theorem \ref{the2.6}(b) to \eqref{e1.1}. To this
end, we fix $\overline{\lambda}\in (\lambda_1,\lambda_2)$ and let $G$
be a multifunction satisfying (G1)--(G5). Since
$\overline{\lambda}<\lambda_2$, we have
$$
\mu_{G,\overline{\lambda}}=\frac{1}{pG_\infty}\frac{2^{p}}{T^p}
\big(1-\overline{\lambda}\frac{pT^p}{2^p}
\liminf_{\xi\to+\infty}\frac{\sup_{|t|\leq\xi}
\min\int_0^tF(s)ds}{\xi^p}\big)>0.
$$
Now fix $\overline{\mu}\in (0,\mu_{g,\overline{\lambda}})$, put
$\nu_1:=\lambda_1$, and
$$
\nu_2:=\frac{\lambda_2}{1+\frac{pT^p
}{2^p}\frac{\overline{\mu}}{\overline{\lambda}}\lambda_2\,
G_\infty}.
$$
If $G_\infty=0$, then $\nu_1 = \lambda_1$, $\nu_2=\lambda_2$ and
$\overline{\lambda}\in (\nu_1,\nu_2)$. If $G_\infty \neq 0$, since
$\overline{\mu}<\mu_{G,\overline{\lambda}}$, we have
$$
\frac{\overline{\lambda}}{\lambda_2}+\frac{pT^p
}{2^p}\overline{\mu}\,G_\infty<1,
$$
and so
$$
\frac{\lambda_2}{1+\frac{pT^p
}{2^p}\frac{\overline{\mu}}{\overline{\lambda}}\lambda_2\,
G_\infty} > \overline{\lambda},
$$
namely, $\overline{\lambda}<\nu_2$. Hence, taking into account
that $\overline{\lambda}>\lambda_1=\nu_1$, one has
$\overline{\lambda}\in(\nu_1,\nu_2)$. Now, set
$$
J(x,s):=F(s)+\frac{\overline{\mu}}{\overline{\lambda}}G(x,s)
$$
for all $(x,s)\in [0,T]\times\mathbb{R}$. Assume $j$ identically
zero in $X$ and for each $u\in X$ put
\begin{gather*}
\mathcal{N}(u):=\frac{1}{p}\|u\|_X^p-\sum_{i=1}^m\int_0^{u(x_i)}I_i(s)ds,\quad 
\Upsilon(u):=\int_0^T\min\int_0^u J(x,s)\,ds\,dx, \\
\mathcal{M}(u):=\Upsilon(u)-j(u)=\Upsilon(u), \\
I_{\overline{\lambda}}(u):=\mathcal{N}(u)-\overline{\lambda}\mathcal{M}(u)
=\mathcal{N}(u)-\overline{\lambda}\Upsilon(u).
\end{gather*}
It is a simple matter to verify that $\mathcal{N}$ is sequentially
weakly lower semicontinuous on $X$. Clearly, 
$\mathcal{N}\in C^1(X)$. By Lemma \ref{lem2.1}, $\mathcal{N}$ 
is locally Lipschitz on $X$. By Lemma \ref{lem2.10}, $\mathcal{F}$
 and $\mathcal{G}$ are locally Lipschitz on $L^p([0,T])$. So,
 $\Upsilon$ is locally Lipschitz on $L^p([0,T])$. Moreover, $X$ is 
compactly embedded into $L^p([0,T])$. So $\Upsilon$ is locally 
Lipschitz on $X$.
Furthermore, $\Upsilon$ is sequentially weakly
upper semicontinuous. For all $u\in X$, by $\rm(I_1)$,
$$
\int_0^{u(x_i)}I_i(s)ds<0,\quad i=1,2,\dots,m.
$$
So, we have
$$
\mathcal{N}(u)=\frac{1}{p}\|u\|_X^p-\sum_{i=1}^m
\int_0^{u(x_i)}I_i(s)ds>\frac{1}{p}\|u\|_X^p
$$
for all $u\in X$. Hence, $\mathcal{N}$ is coercive and
$\inf_X\mathcal{N}=\mathcal{N}(0)=0.$ We want to prove that, under
our hypotheses, there exists a sequence $\{\overline{u}_n\}\subset
X$ of critical points for the functional $I_{\overline{\lambda}}$,
that is, every element $\overline{u}_n$ satisfies
$$
I_{\overline{\lambda}}^\circ(\overline{u}_n,v-\overline{u}_n)\geq
0,\quad\text{for every }v\in X.
$$
Now, we claim that $\gamma<+\infty$. To see this, let $\{\xi_n\}$ 
be a sequence of positive numbers such that
$\lim_{n\to+\infty}\xi_n=+\infty$ and
\begin{equation}\label{e3.1}
\lim_{n\to+\infty}\frac{\sup_{|t|\leq
\xi_n}\min\int_0^t J(x,s)ds}{\xi_n^p}
=\liminf_{\xi\to+\infty}\frac{\sup_{|t|\leq\xi}\min\int_0^t
J(x,s)ds}{\xi^p}.
\end{equation}
Put
$$
r_n:=\frac{1}{p}\left(\frac{2\xi_n}{T^{1/q}}\right)^p,\quad 
\text{for all }n\in\mathbb{N}.
$$
Then, for all $v\in X$ with $\mathcal{N}(v)<r_n$, taking into
account that $\|v\|^p_X<p r_n$ and
$\|v\|_{C^0}\leq\frac{1}{2}T^{1/q}\|v\|_X$, one has
$|v(x)|\leq\xi_n$ for every $x\in[0,T]$. Therefore, for all
$n\in\mathbb{N}$,
\begin{align*}
\varphi(r_n)
&= \inf_{u\in\mathcal{N}^{-1}((-\infty,r))}
\frac{\big(\sup_{v\in\mathcal{N}^{-1}((-\infty,r))}\mathcal{M}(v)\big)
-\mathcal{M}(u)}{r-\mathcal{N}(u)}\\
&\leq  \frac{\sup_{\|v\|^p_X<p r_n}\big(\mathcal{F}(v)+\frac{\overline{\mu}}
{\overline{\lambda}}\mathcal{G}(v)\big)}{r_n}\\
&\leq  \frac{\sup_{|t|\leq\xi_n}\big(\int_0^T\min\int_0^t
F(s)\,ds\,dx+\frac{\overline{\mu}}{\overline{\lambda}}\int_0^T\min\int_0^t
G(x,s)\,ds\,dx\big)}{r_n}\\
&\leq
p\big(\frac{T}{2}\big)^p\Big[\frac{\sup_{|t|\leq\xi_n}\min\int_0^t
F(s)ds}{\xi_n^p}+
\frac{\overline{\mu}}{\overline{\lambda}}\frac{\sup_{|t|\leq\xi_n}\min\int_0^t
G(x,s)ds}{\xi_n^p}\Big].
\end{align*}
Moreover, from Assumptions (F4) and (G5), we have
$$
\lim_{n\to +\infty}\frac{\sup_{|t|\leq\xi_n}\min\int_0^t
F(s)ds}{\xi_n^p}+\lim_{n\to+\infty}
\frac{\overline{\mu}}{\overline{\lambda}}\frac{\sup_{|t|\leq\xi_n}\min\int_0^t
G(x,s)ds}{\xi_n^p}<+\infty,
$$
which follows
$$
\lim_{n\to +\infty}\frac{\sup_{|t|\leq\xi_n}\min\int_0^t
J(x,s)ds}{\xi_n^p}<+\infty.
$$
Therefore,
\begin{equation}\label{e3.2}
\gamma\leq\liminf_{n\to +\infty}\varphi(r_n) \leq
p\big(\frac{T}{2}\big)^p\liminf_{\xi\to+\infty}\frac{\sup_{|t|\leq\xi}\min\int_0^t
J(x,s)ds}{\xi^p} < +\infty.
\end{equation}
Since
$$
\frac{\sup_{|t|\leq\xi}\min\int_0^tJ(x,s)ds}{\xi^p}\leq
\frac{\sup_{|t|\leq\xi}\min\int_0^tF(s)ds}{\xi^p}+
\frac{\overline{\mu}}{\overline{\lambda}}\frac{\sup_{|t|\leq\xi}\min\int_0^tG(x,s)ds}{\xi^p},
$$
and taking (G5) into account, we get
\begin{equation}\label{e3.3}
\liminf_{\xi\to
+\infty}\frac{\sup_{|t|\leq\xi}\min\int_0^tJ(x,s)ds}{\xi^p}\leq
\liminf_{\xi\to+\infty}\frac{\sup_{|t|\leq\xi}\min\int_0^tF(s)ds}{\xi^p}
+\frac{\overline{\mu}}{\overline{\lambda}}\,G_\infty.
\end{equation}
Moreover, from Assumption (G4) we obtain
$$
\limsup_{\xi\to+\infty}\frac{\int_{0}^T\min\int_0^{\xi(\frac{T}{2}-x)}
J(x,s)\,ds\,dx}{\frac{1}{p}\xi^{p}\big(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\big)
-\sum_{i=1}^m\int_0^{\xi(\frac{T}{2}-x_{i})}I_i(s)ds}
$$
\begin{equation}\label{e3.4}
\geq
\limsup_{\xi\to+\infty}\frac{\int_{0}^T\min\int_0^{\xi(\frac{T}{2}-x)}
F(s)\,ds\,dx}{\frac{1}{p}\xi^{p}\left(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\right)
-\sum_{i=1}^m\int_0^{\xi(\frac{T}{2}-x_{i})}I_i(s)ds}.
\end{equation}
Therefore, from \eqref{e3.3} and \eqref{e3.4}, we observe that
\begin{align*}
\overline{\lambda}\in (\nu_1,\nu_2)
&\subseteq
\Big(\frac{1}{\limsup_{\xi\to+\infty}\frac{\int_{0}^T
\min\int_0^{\xi(\frac{T}{2}-x)}
J(x,s)\,ds\,dx}{\frac{1}{p}\xi^{p}
\big(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\big)
-\sum_{i=1}^m\int_0^{\xi(\frac{T}{2}-x_{i})}I_i(s)ds}},\\
&\quad \frac{1}{\liminf_{\xi\to+\infty}
\frac{\sup_{|t|\leq\xi}\min\int_0^tJ(x,s)ds}{\frac{1}{p}
(\frac{2\xi}{T})^p}}\Big)\\
&\subseteq (0,1/\gamma).
\end{align*}
For the fixed $\overline{\lambda}$, the inequality \eqref{e3.2}
ensures that the condition (b) of Theorem \ref{the2.6} can be
applied and either $I_{\overline{\lambda}}$ has a global minimum or
there exists a sequence $\{u_n\}$ of weak solutions of the problem
\eqref{e1.1} such that $\lim_{n\to\infty}\| u_{n} \|=+\infty$,
The other step is to show that for the fixed $\overline{\lambda}$
the functional $I_{\overline{\lambda}}$ has no global minimum. Let
us verify that the functional $I_{\overline{\lambda}}$ is
unbounded from below. Since
\begin{align*}
\frac{1}{\overline{\lambda}}
&<\limsup_{\xi\to+\infty}\frac{\int_{0}^T\min\int_0^{\xi(\frac{T}{2}-x)}
F(s)\,ds\,dx}{\frac{1}{p}\xi^{p}\big(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\big)
-\sum_{i=1}^m\int_0^{\xi(\frac{T}{2}-x_{i})}I_i(s)ds}
\\
&\leq \limsup_{\xi\to+\infty}\frac{\int_{0}^T\min\int_0^{\xi(\frac{T}{2}-x)}
J(x,s)\,ds\,dx}{\frac{1}{p}\xi^{p}\big(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\big)
-\sum_{i=1}^m\int_0^{\xi(\frac{T}{2}-x_{i})}I_i(s)ds},
\end{align*}
there exists a sequence $\{\eta_n\}$ of positive numbers and a
constant $\tau$ such that $\lim_{n\to +\infty}\eta_n=+\infty$ and
\begin{equation}\label{e3.5}
\frac{1}{\overline{\lambda}}<\tau<\frac{\int_{0}^T\min\int_0^{\eta_n(\frac{T}{2}-x)}
J(x,s)\,ds\,dx}{\frac{1}{p}{\eta_n}^{p}
\big(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\big)
-\sum_{i=1}^m\int_0^{{\eta_n}(\frac{T}{2}-x_{i})}I_i(s)ds}
\end{equation}
for each $n\in\mathbb{N}$ large enough. For all $n\in\mathbb{N}$,
set
\begin{equation*}
w_n(x):=\eta_n\big(\frac{T}{2}-x\big).
\end{equation*}
For any fixed $n\in\mathbb{N}$, it is easy to see that $w_n\in X$
and, in particular, one has
$$
\|w_n\|_X^p=\eta_n^{p}\Big(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\Big),
$$
and so
\begin{equation}\label{e3.6}
\mathcal{N}(w_n)=
\frac{1}{p}\eta_n^{p}\Big(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\Big)
-\sum_{i=1}^m\int_0^{\eta_n(\frac{T}{2}-x_{i})}I_i(s)ds.
\end{equation}
By \eqref{e3.5} and \eqref{e3.6}, we see that
\begin{align*}
I_{\overline{\lambda}}(w_n)
&= \mathcal{N}(w_n)-\overline{\lambda}\mathcal{M}(w_n)\\&=
\frac{1}{p}\eta_n^{p}\Big(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\Big)
-\sum_{i=1}^m\int_0^{\eta_n(\frac{T}{2}-x_{i})}I_i(s)ds
\\
&\quad - \overline{\lambda}\int_{0}^T
\min\int_0^{\eta_n(\frac{T}{2}-x)}J(x,s)\,ds\,dx\\
&< \Big(\frac{1}{p}\eta_n^{p}
\Big(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\Big)
-\sum_{i=1}^m\int_0^{\eta_n(\frac{T}{2}-x_{i})}I_i(s)ds\Big)
(1-\overline{\lambda}\tau)
\end{align*}
for every $n\in\mathbb{N}$ large enough. Since
$\overline{\lambda}\tau>1$ and $\lim_{n\to
+\infty}\eta_n=+\infty$, we have
$$
\lim_{n\to +\infty}I_{\overline{\lambda}}(w_n) =-\infty.
$$
Then, the functional $I_{\overline{\lambda}}$ is unbounded from
below, and it follows that $I_{\overline{\lambda}}$ has no global
minimum. Therefore, from part (b) of Theorem \ref{the2.6}, the
functional $I_{\overline{\lambda}}$ admits a sequence of critical
points $\{\overline{u}_n\}\subset X$ such that
$\lim_{n\to+\infty}\mathcal{N}(\overline{u}_n)=+\infty.$
Since $\mathcal{N}$ is bounded on bounded sets, and taking into
account that
$\lim_{n\to+\infty}\mathcal{N}(\overline{u}_n)=+\infty$,
then $\{\overline{u}_n\}$ has to be unbounded, i.e.,
$$
\lim_{n\to +\infty}\|\overline{u}_n\|_X=+\infty.
$$
Moreover, if $\overline{u}_n\in X$ is a critical point of
$I_{\overline{\lambda}}$, clearly, by definition, one has
$$
I_{\overline{\lambda}}^\circ(\overline{u}_n,v-\overline{u}_n)\geq
0,\quad\text{for every }v\in X.
$$
Finally, by Lemma \ref{lem2.11}, the critical points of
$I_{\overline{\lambda}}$ are weak solutions for the problem
\eqref{e1.1}, and by Lemma \ref{lem2.9}, every weak solution of
\eqref{e1.1} is a solution of \eqref{e1.1}. Hence, the assertion
follows.
\end{proof}

\begin{remark}\label{rem3.2} \rm
Under the conditions
\begin{gather*}
\liminf_{\xi\to+\infty}\frac{\sup_{|t|\leq\xi}\min\int_0^tF(s)ds}{\xi^p}=0,
\\
\limsup_{\xi\to+\infty}\frac{\int_{0}^T\min\int_0^{\xi(\frac{T}{2}-x)}
F(s)\,ds\,dx}{\frac{1}{p}\xi^{p}\big(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\big)
-\sum_{i=1}^m\int_0^{\xi(\frac{T}{2}-x_{i})}I_i(s)ds} =+\infty,
\end{gather*}
from Theorem \ref{the3.1}, we see that for every $\lambda > 0$ and
for each $\mu\in \big[0,\frac{2^p}{pT^p G_\infty}\big)$, 
problem \eqref{e1.1} admits a sequence of solutions which is
unbounded in $X$. Moreover, if $G_\infty=0$, the result holds for
every $\lambda>0$ and $\mu\geq 0$.
\end{remark}

 The following result is a special case of Theorem \ref{the3.1} with
$\mu = 0$.

\begin{theorem}\label{the3.3}
Assume that {\rm \rm(F1)--(F4), (I1)} hold. Then, for
each
\begin{align*}
\lambda&\in\Bigg(
\frac{1}{\limsup_{\xi\to+\infty}\frac{\int_{0}^T\min\int_0^{\xi(\frac{T}{2}-x)}
F(s)\,ds\,dx}{\frac{1}{p}\xi^{p}\big(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\big)
-\sum_{i=1}^m\int_0^{\xi(\frac{T}{2}-x_{i})}I_i(s)ds}},\\
&\quad \frac{1}{\liminf_{\xi\to+\infty}\frac{\sup_{|t|\leq\xi}\min\int_0^tF(s)ds}{\frac{1}{p}
(\frac{2\xi}{T})^p}}\Bigg),
\end{align*}
the problem
\begin{gather*}
 -(\phi_p(u'(x)))'+M\phi_p(u(x))\in\lambda
F(u(x))\quad \text{in } [0,T]\setminus Q,\\
-\Delta\phi_p(u'(x_k))=I_k(u(x_k)),\quad k=1,2,\dots,m,\\
u(0)=-u(T),\quad u'(0)=-u'(T)
\end{gather*}
has an unbounded sequence of solutions in $X$.
\end{theorem}

 Now, we present the following example to illustrate our results.

\begin{example} \label{examp3.4} \rm
Consider the problem
\begin{equation}\label{e3.7}
\begin{gathered} 
-(\phi_3(u'(x)))'+\phi_3(u(x))\in\lambda
F(u(x))\quad \text{in } [0,2]\setminus\{1\},\\
-\Delta\phi_3(u'(x_1))=I_1(u(x_1)),\quad x_1=1,\\
u(0)=-u(2),\quad u'(0)=-u'(2),
\end{gathered}
\end{equation}
where, for $s\in\mathbb{R}$,
\begin{equation*}
F(s)=\begin{cases}
\{0\}, & \text{if } |s|<2^{-1/3},\\
{[0,1]}, & \text{if } |s|=2^{-1/3},\\
\{s-2^{-1/3}+1\}, & \text{if } s>2^{-1/3},\\
\{s+2^{-1/3}+1\}, & \text{if } s<-2^{-1/3}.
\end{cases}
\end{equation*}
Simple calculations show that
$$
\sup_{|t|\leq 2^{-1/3}}\min\int_0^tF(s)ds=0
$$
and
\begin{align*}
&\frac{\int_{0}^2\min\int_0^{\xi(1-x)}F(s)\,ds\,dx}{\frac{5}{6}\xi^{3}
-\int_0^{\xi(1-x_1)}I_1(s)ds}\\
&= \frac{6}{5}\frac{1}{\xi^{3}}\int_{-1}^1\min\int_0^{\xi x}F(s)\,ds\,dx\\
&= \frac{6}{5}\frac{1}{\xi^{3}}\Big(\int_{-1}^{-2^{-1/3}}\int_0^{\xi
x}\max F(s)\,ds\,dx
+\int_{-2^{-1/3}}^{0}\int_0^{\xi x}\max F(s)\,ds\,dx\\
&\quad +\int_{0}^{2^{-1/3}}\int_0^{\xi x}\max F(s)\,ds\,dx
+\int_{2^{-1/3}}^{1}\int_0^{\xi x}\max F(s)\,ds\,dx\Big)>0
\end{align*}
for some $\xi\in\mathbb{R}$. So,
\begin{gather*}
\liminf_{\xi\to+\infty}\frac{\sup_{|t|\leq\xi}\min\int_0^tF(s)ds}{\frac{1}{3}
\xi^3}=0,\\
\limsup_{\xi\to+\infty}\frac{\int_{0}^2\min\int_0^{\xi(1-x)}
F(s)\,ds\,dx}{\frac{5}{6}\xi^{3}-\int_0^{\xi(1-x_1)}I_1(s)ds}>0.
\end{gather*}
Hence, using Theorem \ref{the3.3}, problem \eqref{e3.7}, for
$\lambda$ lying in a convenient interval, has an unbounded sequence
of solutions in $X:=\{u\in W^{1,3}([0,2]) : u(0)=-u(2)\}$.
 \end{example}

 Here we point out the following consequences of Theorem
\ref{the3.3}, using the assumptions
\begin{itemize}
\item[(F5)] $\liminf_{\xi\to+\infty}
\frac{\sup_{|t|\leq\xi}\min\int_0^tF(s)ds}{\xi^p}
<\frac{1}{p}\big(\frac{2}{T}\big)^p$;

\item[(F6)]
$\limsup_{\xi\to+\infty}\frac{\int_{0}^T\min\int_0^{\xi(\frac{T}{2}-x)}
F(s)\,ds\,dx}{\frac{1}{p}\xi^{p}\big(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\big)
-\sum_{i=1}^m\int_0^{\xi(\frac{T}{2}-x_{i})}I_i(s)ds}>1$.
\end{itemize}

\begin{corollary}\label{cor3.5}
Assume that {\rm(F1)--(F3), (F5)--(F6), (I1)} hold. 
Then, the problem
\begin{gather*}
 -(\phi_p(u'(x)))'+M\phi_p(u(x))\in
F(u(x))\quad \text{in }[0,T]\setminus Q,\\
-\Delta\phi_p(u'(x_k))=I_k(u(x_k)),\quad k=1,2,\dots,m,\\
u(0)=-u(T),\quad u'(0)=-u'(T)
\end{gather*}
has an unbounded sequence of solutions in $X$.
\end{corollary}

\begin{remark}
Theorem \ref{t1.1} in the Introduction is an immediate consequence
of Corollary \ref{cor3.5}.
\end{remark}

Now, we give the following consequence of the main result.

\begin{corollary}\label{cor3.7}
Let $F_1:\mathbb{R}\to 2^{\mathbb{R}}$ be an upper
semicontinuous multifunction with compact convex values, such that
$\min F_1, \max F_1:\mathbb{R}\to\mathbb{R}$ are Borel
measurable and $|\xi|\leq a(1+|s|^{r_1-1})$ for all
$s\in\mathbb{R},\,\xi\in F_1(s),\,r_1>1\,(a>0)$. Furthermore,
suppose that
\begin{itemize}
\item[(C1)] $\liminf_{\xi\to+\infty}\frac{\sup_{|t|\leq\xi}\min\int_0^tF_1(s)ds}
{\xi^p}<+\infty$;

\item[(C2)] $\limsup_{\xi\to+\infty}\frac{\int_{0}^T\min\int_0^{\xi(\frac{T}{2}-x)}
F_1(s)\,ds\,dx}{\frac{1}{p}\xi^{p}\big(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\big)
-\sum_{i=1}^m\int_0^{\xi(\frac{T}{2}-x_{i})}I_i(s)ds}=+\infty$.
\end{itemize}
Then, for every  multifunction $F_2:\mathbb{R}\to
2^{\mathbb{R}}$ which is upper semicontinuous with compact convex
values, $\min F_2, \max F_2:\mathbb{R}\to\mathbb{R}$
are Borel measurable and $|\xi|\leq b(1+|s|^{r_2-1})$ for all
$s\in\mathbb{R},\,\xi\in F_2(s),\,r_2>1\,(b>0)$, and satisfies
the conditions
$$
\sup_{t\in\mathbb{R}}\min\int_0^tF_2(s)ds\leq 0
$$
and
$$
\liminf_{\xi\to+\infty}\frac{\int_{0}^T\min\int_0^{\xi(\frac{T}{2}-x)}
F_2(s)\,ds\,dx}{\frac{1}{p}\xi^{p}\big(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\big)
-\sum_{i=1}^m\int_0^{\xi(\frac{T}{2}-x_{i})}I_i(s)ds}>-\infty,
$$
for each
$$
\lambda\in\Big(0,\
\frac{1}{\liminf_{\xi\to+\infty}\frac{\sup_{|t|\leq\xi}\min\int_0^tF_1(s)ds}{\frac{1}{p}
(\frac{2\xi}{T})^p}}\Big),
$$
and the problem
\begin{gather*}
-(\phi_p(u'(x)))'+M\phi_p(u(x))\in\lambda
(F_1(u(x))+F_2(u(x)))\quad \text{in } [0,T]\setminus Q,\\
-\Delta\phi_p(u'(x_k))=I_k(u(x_k)),\quad k=1,2,\dots,m,\\
u(0)=-u(T),\quad u'(0)=-u'(T)
\end{gather*}
has an unbounded sequence of solutions in $X$.
\end{corollary}

\begin{proof}
Set $F(t)=F_1(t)+F_2(t)$ for all $t\in\mathbb{R}$. Assumption
(C2) along with the condition
$$
\liminf_{\xi\to+\infty}\frac{\int_{0}^T\min\int_0^{\xi(\frac{T}{2}-x)}
F_2(s)\,ds\,dx}{\frac{1}{p}\xi^{p}\left(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\right)
-\sum_{i=1}^m\int_0^{\xi(\frac{T}{2}-x_{i})}I_i(s)ds}>-\infty
$$
yield
\begin{align*}
&\limsup_{\xi\to+\infty}\frac{\int_{0}^T\min\int_0^{\xi(\frac{T}{2}-x)}
F(s)\,ds\,dx}{\frac{1}{p}\xi^{p}\big(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\big)
-\sum_{i=1}^m\int_0^{\xi(\frac{T}{2}-x_{i})}I_i(s)ds}\\
&=\limsup_{\xi\to+\infty}\frac{\int_{0}^T\min\int_0^{\xi(\frac{T}{2}-x)}
F_1(s)\,ds\,dx+\int_{0}^T\min\int_0^{\xi(\frac{T}{2}-x)}
F_2(s)\,ds\,dx}{\frac{1}{p}\xi^{p}\big(T+\frac{2M}{p+1}(\frac{T}{2})^{p+1}\big)
-\sum_{i=1}^m\int_0^{\xi(\frac{T}{2}-x_{i})}I_i(s)ds}=+\infty.
\end{align*}
Moreover, Assumption (C1) and the condition
$$
\sup_{t\in\mathbb{R}}\min\int_0^tF_2(s)ds\leq 0
$$
ensure that
$$
\liminf_{\xi\to+\infty}\frac{\sup_{|t|\leq\xi}\min\int_0^tF(s)ds}{\xi^p}
\leq\liminf_{\xi\to+\infty}\frac{\sup_{|t|\leq\xi}\min\int_0^tF_1(s)ds}{\xi^p}
<+\infty.
$$
Since
$$
\frac{1}{\liminf_{\xi\to+\infty}\frac{\sup_{|t|\leq\xi}\min\int_0^tF(s)ds}{\frac{1}{p}
(\frac{2\xi}{T})^p}}\geq
\frac{1}{\liminf_{\xi\to+\infty}\frac{\sup_{|t|\leq\xi}\min\int_0^tF_1(s)ds}{\frac{1}{p}
(\frac{2\xi}{T})^p}},
$$
by applying Theorem \ref{the3.3}, we have the desired conclusion.
\end{proof}


\begin{remark}\label{r3} \rm
We observe that in Theorem \ref{the3.1}  we can replace
$\xi\to+\infty$ with $\xi\to 0^+$, and then by the same argument  as
in the proof of Theorem \ref{the3.1}, but using conclusion (c) of
Theorem \ref{the2.6} instead of (b),  problem \eqref{e1.1} has a
sequence of solutions, which strongly converges to 0 in $X$.
\end{remark}

\subsection*{Acknowledgments}
 Shapour Heidarkhani was supported by a grant 91470046 from IPM.

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\end{thebibliography}

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