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\begin{document}

\title[Variational approach to fourth-order impulsive differential
equations]
{Variational approach to fourth-order impulsive differential
equations with two control parameters}
%-----------------------------------------------------------------------------------------------------------
\author[G.A. Afrouzi]{Ghasem A. Afrouzi}
\address{Ghasem A. Afrouzi \newline
Department of Mathematics, Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran} \email{\href{mailto:
G.A. Afrouzi <afrouzi@umz.ac.ir>}{afrouzi@umz.ac.ir}}
%-----------------------------------------------------------------------------------------------------------
\author[A. Hadjian]{Armin Hadjian}
\address{Armin Hadjian\newline
Department of Mathematics, Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran} \email{\href{mailto: A.
Hadjian <a.hadjian@umz.ac.ir>}{a.hadjian@umz.ac.ir}}
%-----------------------------------------------------------------------------------------------------------
\author[V.D. R\u adulescu]{Vicen\c tiu D. R\u adulescu$^\ast$}
\address{Vicen\c tiu D. R\u adulescu \newline
Institute of Mathematics ``Simion Stoilow" of the Romanian Academy,
P.O. Box 1-764, 014700 Bucharest, Romania \& Department of
Mathematics, University of Craiova,  Street A.I. Cuza No. 13, 200585
Craiova, Romania} \email{\href{mailto: V.D. R\u adulescu
<vicentiu.radulescu@imar.ro>}{vicentiu.radulescu@imar.ro}}
%-----------------------------------------------------------------------------------------------------------
\keywords{Impulsive differential equations; Multiple solutions; Variational methods.\\
$^\ast$ Corresponding author\\
\hspace*{.3cm} {\it 2010 Mathematics Subject Classifications}:
34B15, 34B37, 58E30}

%-----------------------------------------------------------------------------------------------------------

\begin{abstract}
In this paper, we are concerned with the multiplicity of solutions for a fourth-order impulsive
differential equation with Dirichlet boundary conditions and two control parameters.
Using variational methods and a three critical points theorem, we give some new criteria
to guarantee that the impulsive problem has at least three classical solutions.
We also provide an example in order to illustrate the main abstract results of this paper.
\end{abstract}

\maketitle

\numberwithin{equation}{section}
%---------------------------------------------------------------------------------------------------------------

\section{Introduction}

In this paper, we consider the fourth-order boundary value problem
with two control parameters and impulsive effects
\begin{equation}\label{e1.1}
\left\{\begin{array}{ll}
u^{(iv)}(t)+A u''(t)+B u(t)=\lambda f(t,u(t))+\mu g(t,u(t)),\qquad  t\neq t_j,\;t\in[0,1],\\
\Delta(u''(t_j))=I_{1j}(u'(t_j)),\qquad j=1,2,\ldots, n,\\
-\Delta(u'''(t_j))=I_{2j}(u(t_j)),\qquad j=1,2,\ldots, n,\\
u(0)=u(1)=u''(0)=u''(1)=0,
\end{array}\right.
\end{equation}
where $A$ and $B$ are two real constants,
$f,g:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}$ are continuous,
$I_{1j},I_{2j}\in C(\mathbb{R};\mathbb{R})$ for $1\leq j\leq n$,
$0=t_0<t_1<t_2<\cdots<t_n<t_{n+1}=1$, the operator $\Delta$ is
defined as $\Delta(u(t_j)):=u(t_j^+)-u(t_j^-)$, where $u(t_j^+)$ and
$u(t_j^-)$ denote the right and the left limits, respectively, of
$u$ at $t_j$, and $\lambda>0$ and $\mu\geq 0$ are referred to as
control parameters.

Many dynamical systems describing models in applied sciences have an
{\it impulsive} dynamical behaviour due to abrupt changes at certain
instants during the evolution process. The rigorous mathematical
description of these phenomena leads to {\it impulsive differential
equations}; they describe various processes of the real world
described by models that are subject to {\it sudden changes} in
their states. Essentially, impulsive differential equations
correspond to a {\it smooth evolution} that may {\it change
instantaneously} or even {\it abruptly}, as happens in various
applications that describe mechanical or natural phenomena. These
changes correspond to impulses in the smooth system, such as for
example in the model of a mechanical clock. Impulsive differential
equations also study models in physics, population dynamics,
ecology, industrial robotics, biotechnology, economics, optimal
control, chaos theory. Associated with this development, a theory of
impulsive differential equations has been given extensive attention.
For an introduction of the basic theory of impulsive differential
equations in $\mathbb{R}^n$, see \cite{BenHenNto}, \cite{LakBaiSim},
and \cite{sape}. Some classical tools have been used to study such
problems in the literature, such as the coincidence degree theory of
Mawhin, the method of upper and lower solutions with the monotone
iterative technique, and some fixed point theorems in cones (see
\cite{ChenTisYuan,Mawhin,QuiLi}). Recently, the existence and
multiplicity of solutions for impulsive boundary value problems by
using variational methods and critical point theory has been
considered and here we cite the papers
\cite{NieOre,SuChYa,TianGe,WangYang,XiaoNieto,XieLuo}.

Motivated by the above facts, in this paper, our aim is to study the
existence of solutions for fourth-order impulsive boundary value
problem \eqref{e1.1}. By employing a three critical point theorem
which we recall in the next section (Theorem \ref{the2.1}), we
establish the exact collections of the parameters $\lambda$ and
$\mu$, for which problem \eqref{e1.1} admits at least three
solutions; see Theorem \ref{the3.1}.

%---------------------------------------------------------------------------------------------------------------

\section{Abstract Setting}
The original {\it three critical point} theorem is due to Pucci and Serrin \cite{puse1,puse2} and establishes that if
$X$ is a real Banach space and a function $f : X \rightarrow{\mathbb R}$ is of class $C^1$, satisfies the Palais-Smale condition, and
has two local minima, then $ f$ has at least three distinct critical points. This result has been extended in the framework of problems depending on a real parameter by Ricceri \cite{ricc}, who also
established a precise range of the parameter that guarantees the existence of at least three critical points.

Our main tool is a three critical point theorem that we recall here in a
convenient form. We also
refer the reader to the recent papers
\cite{AfHaHe,BonaMoBi,BoMoRa1,BoMoRa2,MoRa} where an
analogous variational approach has been developed on studying
different elliptic problems.

%----------------------------------------------------------------------------------------------------------

\begin{theorem}[{\cite[Theorem 3.6]{BonaMara}}]\label{the2.1}
Let $X$ be a reflexive real Banach space; $\Phi:X\rightarrow
\mathbb{R}$ be a coercive, continuously G\^{a}teaux differentiable
and sequentially weakly lower semicontinuous functional whose
G\^{a}teaux derivative admits a continuous inverse on $X^*;$
$\Psi:X\rightarrow \mathbb{R}$ be a continuously G\^{a}teaux
differentiable functional whose G\^{a}teaux derivative is compact
such that
$$
\Phi(0)=\Psi(0)=0.
$$
Assume that there exist $r>0$ and $\overline{x}\in X,$ with
$r<\Phi(\overline{x}),$ such that
\begin{enumerate}
\item [$\rm(a_1)$] $\frac{\sup_{\Phi(x)\leq r}\Psi(x)}{r}<\frac{\Psi(\overline{x})}{\Phi(\overline{x})};$
\item [$\rm(a_2)$] for each $\lambda\in
\Lambda_r:=]\frac{\Phi(\overline{x})}{\Psi(\overline{x})},\frac{r}{\sup_{\Phi(x)\leq
r}\Psi(x)}[$ the functional $I_\lambda:=\Phi-\lambda\Psi$ is
coercive.
\end{enumerate}
Then, for each $\lambda\in\Lambda_r$ the functional $I_\lambda$ has
at least three distinct critical points in $X$.
\end{theorem}

%----------------------------------------------------------------------------------------------------------

Here and in the sequel, we suppose that $A$ and $B$ satisfy the
following condition:
\begin{equation}\label{e2.1}
A\leq 0\leq B.
\end{equation}
Define
\begin{eqnarray*}
H_0^1([0,1])&:=&\bigg{\{}u\in L^2([0,1]) : u'\in L^2([0,1]),\;u(0)=u(1)=0\bigg{\}},\\
H^2([0,1])&:=&\bigg{\{}u\in L^2([0,1]) : u',u''\in
L^2([0,1])\bigg{\}}.
\end{eqnarray*}
Take $X:=H^2([0,1])\cap H_0^1([0,1])$ and define
\begin{equation}\label{e2.2}
\|u\|_X:=\left(\int_0^1\big(|u''(t)|^2-A|u'(t)|^2+B|u(t)|^2\big)\,dt\right)^{1/2},\qquad
u\in X.
\end{equation}
Since $A$ and $B$ satisfy \eqref{e2.1}, it is straightforward to
verify that \eqref{e2.2} defines a norm for the Sobolev space $X$
and this norm is equivalent to the usual norm defined as follows:
$$
\|u\|:=\left(\int_0^1|u''(t)|^2\,dt\right)^{1/2}
$$
It follows from \eqref{e2.1} that $\|u\|\leq\|u\|_X$. For the norm
in $C^1([0,1])$,
$$
\|u\|_\infty:=\max\Big\{\max_{t\in[0,1]}|u(t)|,\max_{t\in[0,1]}|u'(t)|\Big\},
$$
we have the following relation.

%----------------------------------------------------------------------------------------------------------

\begin{lemma}[{\cite[Lemma 2.1]{XieLuo}}]\label{lem2.2}
Let $M_1:=1+1/\pi$. Then $\|u\|_\infty\leq M_1\|u\|_X$ for all $u\in
X$.
\end{lemma}

%----------------------------------------------------------------------------------------------------------

Throughout the sequel,
$f,g:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}$ are continuous
functions, and $\lambda>0$ and $\mu\geq 0$ are real parameters. Put
$$
F(t,\xi):=\int_0^\xi f(t,x)\,dx\qquad \textrm{and} \qquad
G(t,\xi):=\int_0^\xi g(t,x)\,dx,
$$
for all $(t,\xi)\in[0,1]\times\mathbb{R}$.

Moreover, set $\displaystyle G^c:=\int_0^1\max_{|\xi|\leq
c}G(t,\xi)\,dt$ for all $c>0$ and $G_d:=\inf_{[0,1]\times[0,d]}G$
for all $d>0$. Clearly, $G^c\geq 0$ and $G_d\leq 0$.

%----------------------------------------------------------------------------------------------------------

We say that $u\in C([0,1])$ is a \textit{classical solution} of
problem \eqref{e1.1}, if it satisfies the equation in \eqref{e1.1}
a.e. on $[0,1]\setminus\{t_1,t_2,\ldots,t_n\}$, the limits
$u''(t_j^+), u''(t_j^-), u'''(t_j^+)$ and $u'''(t_j^-)$, $1\leq
j\leq n$, exist, satisfy two impulsive conditions in \eqref{e1.1}
and the boundary condition $u(0)=u(1)=u''(0)=u''(1)=0$.

A \textit{weak solution} of problem \eqref{e1.1} is a function $u\in
X$ such that the equality
\begin{eqnarray*}
&&\hspace{-0.7cm}\int_0^1\big(u''(t)v''(t)-Au'(t)v'(t)+Bu(t)v(t)\big)\,dt=-\sum_{j=1}^n
I_{2j}(u(t_j))v(t_j)-\sum_{j=1}^n I_{1j}(u'(t_j))v'(t_j)\\
&&\hspace{6.5cm}+\lambda\int_0^1 f(t,u(t))v(t)\,dt+\mu\int_0^1
g(t,u(t))v(t)\,dt
\end{eqnarray*}
holds for all $v\in X$.

We consider the functional
$I_\lambda:X\rightarrow\mathbb{R}$, defined by
\begin{equation}\label{e2.3}
I_\lambda(u):=\Phi(u)-\lambda\Psi(u),\qquad u\in X,
\end{equation}
where
\begin{equation}\label{e2.4}
\Phi(u):=\frac{1}{2}\|u\|_X+\sum_{j=1}^n\int_0^{u'(t_j)}I_{1j}(s)\,ds+\sum_{j=1}^n\int_0^{u(t_j)}I_{2j}(s)\,ds
\end{equation}
and
\begin{equation}\label{e2.5}
\Psi(u):=\int_0^1 F(t,u(t))\,dt+\frac{\mu}{\lambda}\int_0^1
G(t,u(t))\,dt.
\end{equation}
It is clear that $I_\lambda$ is differentiable at any $u\in X$ and
\begin{eqnarray*}
I'_\lambda(u)(v)&=&\int_0^1\big(u''(t)v''(t)-Au'(t)v'(t)+Bu(t)v(t)\big)\,dt+\sum_{j=1}^n
I_{2j}(u(t_j))v(t_j)\\
&&+\sum_{j=1}^n I_{1j}(u'(t_j))v'(t_j)-\lambda\int_0^1
f(t,u(t))v(t)\,dt-\mu\int_0^1 g(t,u(t))v(t)\,dt
\end{eqnarray*}
for any $v\in X$. Hence, a critical point of $I_\lambda$ give us a
weak solution of problem \eqref{e1.1}.

%---------------------------------------------------------------------------------------------------------------

\begin{lemma}[{\cite[Lemma 2.2]{XieLuo}}]\label{lem2.3}
If $u\in X$ is a weak solution of problem \eqref{e1.1}, then $u$ is
also a classical solution of problem \eqref{e1.1}.
\end{lemma}

%---------------------------------------------------------------------------------------------------------------

We assume throughout, and without further mention, that the
following conditions hold:

\begin{enumerate}
\item[$\rm(H1)$] Assume that $\{t_1,t_2,\ldots,t_n\}\subseteq[\frac{1}{4},\frac{3}{4}];$
\item[$\rm(H2)$] Assume that there exist two positive constants $k_1$ and $k_2$ such that for each $u\in
X$,
$$
0\leq\sum_{j=1}^n\int_0^{u'(t_j)}I_{1j}(s)\,ds\leq
k_1\max_{j\in\{1,2,\ldots,n\}}|u'(t_j)|^2
$$
and
$$
0\leq\sum_{j=1}^n\int_0^{u(t_j)}I_{2j}(s)\,ds\leq
k_2\max_{j\in\{1,2,\ldots,n\}}|u(t_j)|^2.
$$
\end{enumerate}
Also put $k_0:=2048\big(\frac{3}{8}-\frac{9}{10\cdot
4^4}A+\frac{79}{14\cdot 4^8}B\big)$ and $k_3:=k_0+k_2$. These
constants will be used in some of our hypotheses in the next
section.

%---------------------------------------------------------------------------------------------------------------

In conclusion, we cite a recent monograph by Krist\'aly, R\u
adulescu and Varga \cite{KrisRadVar} as a general reference on
variational methods adopted here.

%---------------------------------------------------------------------------------------------------------------

\section{Main Results and Proofs}

In this section, we present our main results on the existence of at
least three classical solutions for the problem \eqref{e1.1}.

In order to introduce our first result, we fix $c,d>0$ such that
$$
\frac{k_3d^2}{\displaystyle\int_{1/4}^{3/4}F(t,d)\,dt}<\frac{c^2}{\displaystyle
2M_1^2\int_0^1\max_{|\xi|\leq c}F(t,\xi)\,dt}
$$
and pick
\begin{equation}\label{e3.1}
\lambda\in\Lambda:=\left(\frac{k_3d^2}{\displaystyle\int_{1/4}^{3/4}F(t,d)\,dt},
\frac{c^2}{\displaystyle 2M_1^2\int_0^1\max_{|\xi|\leq
c}F(t,\xi)\,dt}\right)\,.
\end{equation}
Set
\begin{equation}\label{e3.2}
\delta:=\min\left\{\frac{\displaystyle c^2-2\lambda
M_1^2\int_0^1\max_{|\xi|\leq
c}F(t,\xi)\,dt}{2M_1^2G^c},\frac{\displaystyle
k_3d^2-\lambda\int_{1/4}^{3/4}F(t,d)\,dt}{G_d}\right\}
\end{equation}
and
\begin{equation}\label{e3.3}
\overline{\delta}:=\min\left\{\delta,\frac{1}
{\max\left\{0,4M_1^2\limsup\limits_{|\xi|\rightarrow+\infty}\frac{\sup_{t\in[0,1]}G(t,\xi)}{\xi^2}\right\}}\right\},
\end{equation}
where we read ${r}/{0}=+\infty$. For instance,
$\overline{\delta}=+\infty$ when
$\limsup_{|\xi|\rightarrow+\infty}\frac{\sup_{t\in[0,1]}G(t,\xi)}{\xi^2}\leq
0$ and $G_d=G^c=0$.

With the above notations we are able to prove the following multiplicity property.

%----------------------------------------------------------------------------------------------------------

\begin{theorem}\label{the3.1}
Assume that there exist two positive constants $c,d$, with
$c<\sqrt{2k_0}M_1d$, such that
\begin{enumerate}
\item[$\rm(A1)$] $F(t,\xi)\geq 0$, for each $(t,\xi)\in ([0,\frac{1}{4}]\cup[\frac{3}{4},1])\times [0,d];$
\item[$\rm(A2)$] $\displaystyle\frac{\displaystyle\int_0^1\max_{|\xi|\leq
c}F(t,\xi)\,dt}{c^2}<\displaystyle\frac{\displaystyle\int_{1/4}^{3/4}F(t,d)\,dt}{2k_3(M_1d)^2};$
\item[$\rm(A3)$] $\displaystyle\limsup_{|\xi|\rightarrow
+\infty}\frac{\sup_{t\in[0,1]}F(t,\xi)}{\xi^2}<\frac{\displaystyle\int_0^1\max_{|\xi|\leq
c}F(t,\xi)\,dt}{2c^2}$.
\end{enumerate}
Then, for every $\lambda\in\Lambda$, where $\Lambda$ is given by
\eqref{e3.1}, and for every continuous function
$g:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}$ such that
$$
\limsup_{|\xi|\rightarrow +\infty}
\frac{\sup_{t\in[0,1]}G(t,\xi)}{\xi^2}<+\infty,
$$
there exists $\overline{\delta}>0$ given by \eqref{e3.3} such that,
for each $\mu\in [0,\overline{\delta})$, problem \eqref{e1.1} admits
at least three classical solutions.
\end{theorem}

%----------------------------------------------------------------------------------------------------------

\begin{proof}
Fix $\lambda,g$ and $\mu$ as in the conclusion. By Lemma
\ref{lem2.3}, it suffices to show the functional $I_\lambda$ defined
in \eqref{e2.3} has at least three critical points in $X$. We prove
this by verifying the conditions given in Theorem \ref{the2.1}. Note
that $\Phi$ defined in \eqref{e2.4} is a nonnegative G\^{a}teaux
differentiable, coercive, and sequentially weakly lower
semicontinuous functional, and its G\^{a}teaux derivative admits a
continuous inverse on $X^*$. Moreover, $\Psi$ defined in
\eqref{e2.5} is a continuously G\^{a}teaux differentiable functional
whose G\^{a}teaux derivative is compact. We will verify $\rm(a_1)$
and $\rm(a_2)$ of Theorem \ref{the2.1}.

Let $w$ be the function defined by
\begin{equation*}
w(t):=\begin{cases}
64d(t^3-\frac{3}{4}t^2+\frac{3}{16}t), & t\in[0,1/4)\\
d, & t\in [1/4,3/4] \\
64d(-t^3+\frac{9}{4}t^2-\frac{27}{16}t+\frac{7}{16}), & t\in
(3/4,1],
\end{cases}
\end{equation*}
and put $r:=\frac{c^2}{2M_1^2}$. Clearly, $w\in X$ and from the
condition (H2) one has
$$
\Phi(w)\geq\frac{1}{2}\|w\|^2_X=2048\left(\frac{3}{8}-\frac{9}{10\cdot
4^4}A+\frac{79}{14\cdot 4^8}B\right)d^2=k_0d^2>r.
$$
Also, by using condition $\rm(A1)$, since $0\leq w(t)\leq d$ for
each $t\in[0,1]$, we infer
\begin{eqnarray*}
\Psi(w)&=&\int_0^1 F(t,w(t))\,dt+\frac{\mu}{\lambda}\int_0^1
G(t,w(t))\,dt\\
&\geq&\int_{1/4}^{3/4} F(t,d)\,dt+\frac{\mu}{\lambda}\int_0^1
G(t,w(t))\,dt\\
&\geq&\int_{1/4}^{3/4} F(t,d)\,dt+\frac{\mu}{\lambda}G_d.
\end{eqnarray*}
For all $u\in X$ satisfying $\Phi(u)\leq r$, by Lemma \ref{lem2.2},
definitively one has
$$
\|u\|^2_\infty\leq M_1^2\|u\|_X^2\leq 2M_1^2\Phi(u)\leq 2M_1^2r=c^2.
$$
Therefore
\begin{eqnarray*}
\hspace{-1.3cm}\frac{\sup_{\Phi(u)\leq
r}\Psi(u)}{r}&\leq&\frac{\displaystyle\int_0^1\max_{|\xi|\leq
c}F(t,\xi)\,dt +\frac{\mu}{\lambda}\int_0^1\max_{|\xi|\leq
c}G(t,\xi)\,dt}
{\frac{c^2}{2M_1^2}}\\
&=&2M_1^2\frac{\displaystyle\int_0^1\max_{|\xi|\leq
c}F(t,\xi)\,dt}{c^2}+2M_1^2\frac{\mu}{\lambda}\frac{G^c}{c^2}.
\end{eqnarray*}
From this, if $G^c=0$, it is clear that we get
\begin{equation}\label{e3.4}
\frac{\sup_{\Phi(u)\leq r}\Psi(u)}{r}<\frac{1}{\lambda},
\end{equation}
while, if $G^c>0$, it turns out to be true bearing in mind that
$$
\mu<\frac{\displaystyle c^2-2\lambda M_1^2\int_0^1\max_{|\xi|\leq
c}F(t,\xi)\,dt}{2M_1^2G^c}.
$$
On the other hand, taking into account (H2), we have
\begin{eqnarray*}
\Phi(w)&=&2048\left(\frac{3}{8}-\frac{9}{10\cdot
4^4}A+\frac{79}{14\cdot
4^8}B\right)d^2+\sum_{j=1}^m\int_0^d I_{2j}(s)\,ds\\
&\leq&k_0d^2+k_2d_2=k_3d^2,
\end{eqnarray*}
and so,
\begin{eqnarray*}
\frac{\Psi(w)}{\Phi(w)}&\geq&\frac{\displaystyle\int_{1/4}^{3/4}F(t,d)\,dt
+\frac{\mu}{\lambda}G_d}{k_3d^2}\\
&=&\frac{\displaystyle\int_{1/4}^{3/4}F(t,d)\,dt}{k_3d^2}+\frac{\mu}{\lambda}
\frac{G_d}{k_3d^2}.
\end{eqnarray*}
Hence, if $G_d=0$, we find
\begin{equation}\label{e3.5}
\frac{\Psi(w)}{\Phi(w)}>\frac{1}{\lambda},
\end{equation}
while, if $G_d<0$, the same relation holds since
$$
\mu<\frac{\displaystyle
k_3d^2-\lambda\int_{1/4}^{3/4}F(t,d)\,dt}{G_d}.
$$
Therefore, from \eqref{e3.4} and \eqref{e3.5}, condition $\rm(a_1)$
of Theorem \ref{the2.1} is verified.

Now, in order to prove the coercivity of the functional $I_\lambda$,
first we assume that
$$
\limsup_{|\xi|\rightarrow +\infty}\frac{\sup_{t\in[0,1]}F(t,\xi)}{\xi^2}>0.
$$
Therefore, fix
$$
\limsup_{|\xi|\rightarrow
+\infty}\frac{\sup_{t\in[0,1]}F(t,\xi)}{\xi^2}<\varepsilon<\frac{\displaystyle\int_0^1\max_{|\xi|\leq
c}F(t,\xi)\,dt}{c^2},
$$
from $\rm(A3)$ there is a function $h_\varepsilon\in L^1([0,1])$
such that
$$
F(t,\xi)\leq\varepsilon\xi^2+h_\varepsilon(t),
$$
for each $t\in[0,1]$ and $\xi\in\mathbb{R}$. Taking \eqref{e2.3}
into account and since $\lambda<\frac{\displaystyle
c^2}{\displaystyle 2M_1^2\int_0^1\max_{|\xi|\leq c}F(t,\xi)\,dt}$,
it follows that
\begin{eqnarray}\label{e3.6}
\lambda\int_0^1
F(t,u(t))\,dt&\leq&\lambda\left(\varepsilon\int_0^1(u(t))^2\,dt+\int_0^1 h_\varepsilon(t)\,dt\right)\nonumber\\
&<&\frac{c^2}{\displaystyle 2M_1^2\int_0^1\max_{|\xi|\leq
c}F(t,\xi)\,dt}\left(\varepsilon\int_0^1(u(t))^2\,dt+\int_0^1 h_\varepsilon(t)\,dt\right)\nonumber\\
&\leq&\frac{c^2}{\displaystyle 2M_1^2\int_0^1\max_{|\xi|\leq
c}F(t,\xi)\,dt}\Big(\varepsilon
M_1^2\|u\|_X^2+\|h_\varepsilon\|_{L^1([0,1])}\Big),
\end{eqnarray}
for each $u\in X$. Moreover, since $\mu<\overline{\delta}$, we obtain
$$
\limsup_{|\xi|\rightarrow+\infty}\frac{\sup_{t\in[0,1]}G(t,\xi)}{\xi^2}<\frac{1}{4\mu
M_1^2}\,.
$$
Thus, there is a function $h_\mu\in L^1([0,1])$ such that
$$
G(t,\xi)\leq\frac{1}{4\mu M_1^2}\xi^2+h_\mu(t),
$$
for each $t\in[0,1]$ and $\xi\in\mathbb{R}$. Thus, taking again
Lemma \ref{lem2.2} into account, it follows that
\begin{eqnarray}\label{e3.7}
\int_0^1G(t,u(t))\,dt&\leq&\frac{1}{4\mu
M_1^2}\int_0^1(u(t))^2\,dt+\int_0^1
h_\mu(t)\,dt\nonumber\\
&\leq&\frac{1}{4\mu}\|u\|_X^2+\|h_\mu\|_{L^1([0,1])},
\end{eqnarray}
for each $u\in X$. Finally, putting together \eqref{e3.6} and
\eqref{e3.7}, we have
\begin{eqnarray*}
I_\lambda(u)&=&\Phi(u)-\lambda\Psi(u)\\
&\geq&\frac{1}{2}\|u\|_X^2-\frac{c^2}{\displaystyle
2M_1^2\int_0^1\max_{|\xi|\leq c}F(t,\xi)\,dt}\Big(\varepsilon
M_1^2\|u\|_X^2+\|h_\varepsilon\|_{L^1([0,1])}\Big)\\
&&-\frac{1}{4}\|u\|_X^2-\mu\|h_\mu\|_{L^1([0,1])}\\
&=&\frac{1}{2}\left(\frac{1}{2}-\frac{c^2}{\displaystyle\int_0^1\max_{|\xi|\leq
c}F(t,\xi)\,dt}\varepsilon\right)\|u\|_X^2
-\frac{c^2\|h_\varepsilon\|_{L^1([0,1])}}{\displaystyle
2M_1^2\int_0^1\max_{|\xi|\leq
c}F(t,\xi)\,dt}-\mu\|h_\mu\|_{L^1([0,1])}.
\end{eqnarray*}
On the other hand, if
$$
\limsup_{|\xi|\rightarrow
+\infty}\frac{\sup_{t\in[0,1]}F(t,\xi)}{\xi^2}\leq 0,
$$
there exists a function $h_\varepsilon\in L^1([0,1])$ such that
$F(t,\xi)\leq h_\varepsilon(t)$ for each $t\in[0,1]$ and
$\xi\in\mathbb{R}$, and arguing as before we obtain
$$
I_\lambda(u)\geq\frac{1}{4}\|u\|_X^2-\frac{c^2\|h_\varepsilon\|_{L^1([0,1])}}
{\displaystyle 2M_1^2\int_0^1\max_{|\xi|\leq
c}F(t,\xi)\,dt}-\mu\|h_\mu\|_{L^1([0,1])}.
$$
Both cases lead to the coercivity of $I_\lambda$ and condition
$\rm(a_2)$ of Theorem \ref{the2.1} is verified.

Since, from \eqref{e3.4} and \eqref{e3.5},
$$
\lambda\in\Lambda\subseteq\left]\frac{\Phi(w)}{\Psi(w)},
\frac{r}{\sup_{\Phi(u)\leq r}\Psi(u)}\right[,
$$
Theorem \ref{the2.1} ensures the existence of at least three
critical points for the functional $I_\lambda$ and the proof is
complete.
\end{proof}

%----------------------------------------------------------------------------------------------------------

The technical approach used to prove the previous result uses some
ideas from \cite{BonaChi1}. In the cited work, the existence of at
least three classical solutions for a perturbed two-point boundary
value problem has been investigated under suitable conditions on the
potentials $F$ and $G$; see also \cite{DagHeiMolica}, where
analogous variational approaches have been developed to study
a perturbed mixed boundary value problems.

%----------------------------------------------------------------------------------------------------------
\section{Particular Cases and An Example}

A particular case of Theorem \ref{the3.1} is the following multiplicity property.

%----------------------------------------------------------------------------------------------------------

\begin{theorem}\label{the3.3}
Let $\alpha\in L^1([0,1])$ be a nonnegative and non-zero function.
Further, let $h:\mathbb{R}\rightarrow\mathbb{R}$ be a nonnegative
continuous function. Put
$\displaystyle\alpha_0:=\int_{1/4}^{3/4}\alpha(t)\,dt$ and
$\displaystyle H(\xi):=\int_0^\xi h(x)\,dx$ for all
$\xi\in\mathbb{R}$, and assume that there exist two positive
constants $c,d$, with $c<16\sqrt{6}M_1d$, such that
\begin{enumerate}
\item[$\rm(A4)$] $\displaystyle\frac{H(c)}{c^2}
<\frac{\alpha_0}{2(768+k_2)M_1^2\|\alpha\|_{L^1([0,1])}}\frac{H(d)}{d^2};$
\item[$\rm(A5)$] $\displaystyle\limsup_{|\xi|\rightarrow
+\infty}\frac{H(\xi)}{\xi^2}\leq 0$.
\end{enumerate}
Then, for each parameter $\lambda$ belonging to
$$
\Lambda:=\left(\frac{768+k_2}{\alpha_0}\frac{d^2}{H(d)},
\frac{1}{2M_1^2\|\alpha\|_{L^1([0,1])}}\frac{c^2}{H(c)}\right),
$$
and for every continuous function
$g:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}$ such that
$$
\limsup_{|\xi|\rightarrow +\infty}
\frac{\sup_{t\in[0,1]}G(t,\xi)}{\xi^2}<+\infty,
$$
there exists
$$
\delta:=\min\left\{\frac{c^2-2\lambda M_1^2\|\alpha\|_{L^1([0,1])}
H(c)}{2M_1^2G^c},\frac{(768+k_2)d^2-\lambda\alpha_0H(d)}{G_d}\right\}
$$
such that, for each
$$
\mu\in\left[0,\min\left\{\delta,\frac{1}{\max\left\{0,4M_1^2\limsup_{|\xi|\rightarrow
+\infty}
\frac{\sup_{t\in[0,1]}G(t,\xi)}{\xi^2}\right\}}\right\}\right),
$$
the problem
\begin{equation}\label{e3.8}
\left\{\begin{array}{ll}
u^{(iv)}(t)=\lambda \alpha(t)h(u(t))+\mu g(t,u(t)),\qquad  t\neq t_j,\;t\in[0,1],\\
\Delta(u''(t_j))=I_{1j}(u'(t_j)),\qquad j=1,2,\ldots, n,\\
-\Delta(u'''(t_j))=I_{2j}(u(t_j)),\qquad j=1,2,\ldots, n,\\
u(0)=u(1)=u''(0)=u''(1)=0,
\end{array}\right.
\end{equation}
admits at least three classical solutions.
\end{theorem}

%----------------------------------------------------------------------------------------------------------

A direct consequence of the previous multiplicity property reads as follows.

%----------------------------------------------------------------------------------------------------------

\begin{corollary}\label{cor3.4}
Let $\alpha\in L^1([0,1])$ be a nonnegative and non-zero function.
Moreover, let $h:\mathbb{R}\rightarrow\mathbb{R}$ be a nonnegative
(not identically zero) and continuous function such that
\begin{equation}\label{e3.9}
\liminf_{\xi\rightarrow
0^+}\frac{h(\xi)}{\xi}=\limsup_{|\xi|\rightarrow
+\infty}\frac{h(\xi)}{\xi}=0.
\end{equation}
Then, for each
$$
\lambda>\frac{768+k_2}{\alpha_0}\inf_{d\in S}\frac{d^2}{H(d)},
$$
where $S:=\{d>0 : H(d)>0\}$, the problem
\begin{equation*}
\left\{\begin{array}{ll}
u^{(iv)}(t)=\lambda \alpha(t)h(u(t)),\qquad  t\neq t_j,\;t\in[0,1],\\
\Delta(u''(t_j))=I_{1j}(u'(t_j)),\qquad j=1,2,\ldots, n,\\
-\Delta(u'''(t_j))=I_{2j}(u(t_j)),\qquad j=1,2,\ldots, n,\\
u(0)=u(1)=u''(0)=u''(1)=0,
\end{array}\right.
\end{equation*}
admits at least three classical solutions.
\end{corollary}

\begin{proof}
Fix $\lambda>\frac{768+k_2}{\alpha_0}\inf_{d\in S}\frac{d^2}{H(d)}$.
Then, there exists $\overline{d}>0$ such that $H(\overline{d})>0$
and
$\lambda>\frac{(768+k_2)\overline{d}^2}{\alpha_0H(\overline{d})}$.
By using condition \eqref{e3.9} one has
$$
\liminf_{\xi\to 0^+}\frac{H(\xi)}{\xi^2}=0.
$$
Therefore, we can find a positive constant $\overline{c}$ such that
$\overline{c}<16\sqrt{6}M_1d$ and
$$
\frac{H(\overline{c})}{\overline{c}^2}<
\min\left\{\frac{\alpha_0H(\overline{d})}{(768+k_2)(M_1\overline{d})^2},
\frac{1}{2\lambda M_1^2\|\alpha\|_{L^1([0,1])}}\right\}.
$$
Hence
$$
\lambda\in\left(\frac{768+k_2}{\alpha_0}\frac{\overline{d}^2}{H(\overline{d})},
\frac{1}{2M_1^2\|\alpha\|_{L^1([0,1])}}\frac{\overline{c}^2}{H(\overline{c})}\right).
$$
All the hypotheses of Theorem \ref{the3.3} are satisfied and problem
\eqref{e3.8} admits at least three distinct classical solutions. The
proof is complete.
\end{proof}

%----------------------------------------------------------------------------------------------------------

In conclusion we present a concrete example of application of
Corollary \ref{cor3.4}.

%----------------------------------------------------------------------------------------------------------

\begin{example}\label{ex3.5}\rm{
Consider the following problem:
\begin{equation}\label{e3.10}
\left\{\begin{array}{ll}
u^{(iv)}(t)=\lambda e^{16t}h(u(t)),\qquad  t\in[0,1]\setminus\{\frac{1}{2}\},\\
\Delta(u''(t_1))=2u'(t_1),\qquad t_1=\frac{1}{2},\\
-\Delta(u'''(t_1))=4u(t_1),\qquad t_1=\frac{1}{2},\\
u(0)=u(1)=u''(0)=u''(1)=0,
\end{array}\right.
\end{equation}
where $h:\mathbb{R}\rightarrow\mathbb{R}$ is defined by
\begin{equation*}
h(x):=\begin{cases}
x^2, & |x|\leq 1,\\
\frac{1}{x^2}, & |x|>1.
\end{cases}
\end{equation*}
Here, $I_{11}(s)=2s$ and $I_{21}(s)=4s$ for all $s\in\mathbb{R}$. It
is easy to verify that (H2) is satisfied with $k_1=1$ and $k_2=2$.
Direct calculations give
\begin{eqnarray*}
\liminf_{\xi\rightarrow 0^+}\frac{h(\xi)}{\xi}&=&\lim_{\xi\rightarrow 0^+}\xi=0,\\
\limsup_{|\xi|\rightarrow
+\infty}\frac{h(\xi)}{\xi}&=&\lim_{|\xi|\rightarrow
+\infty}\frac{1}{\xi^3}=0.
\end{eqnarray*}
Also we have
$$
\inf_{d\in S}\frac{d^2}{H(d)}=\inf_{0<d\leq 1}\frac{3}{d}=3.
$$
Put $\alpha(t)=e^{16t}$ for all $t\in\mathbb{R}$. Therefore,
$$
\alpha_0=\int_{1/4}^{3/4}\alpha(t)\,dt=\frac{1}{16}(e^{12}-e^4).
$$
From Corollary \ref{cor3.4}, for each parameter
$\lambda>\frac{36960}{e^{12}-e^4}$, problem \eqref{e3.10} admits at
least three classical solutions. In particular, the problem
\begin{equation*}
\left\{\begin{array}{ll}
u^{(iv)}(t)=e^{16t}h(u(t)),\qquad  t\in[0,1]\setminus\{\frac{1}{2}\},\\
\Delta(u''(t_1))=2u'(t_1),\qquad t_1=\frac{1}{2},\\
-\Delta(u'''(t_1))=4u(t_1),\qquad t_1=\frac{1}{2},\\
u(0)=u(1)=u''(0)=u''(1)=0,
\end{array}\right.
\end{equation*}
admits at least three classical solutions.}
\end{example}

\medskip
\textbf{Acknowledgements.} V. R\u{a}dulescu acknowledges the support through Grant CNCS PCE-47/2011.
%----------------------------------------------------------------------------------------------------------

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\end{document}