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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 126, pp. 1--14.\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-2013/126\hfil Existence of solutions]
{Existence of solutions to second-order boundary-value problems
with small perturbations of impulses}

\author[G. Bonanno, B. Di Bella, J. Henderson \hfil EJDE-2013/126\hfilneg]
{Gabriele Bonanno, Beatrice Di Bella, Johnny Henderson}  % in alphabetical order

\address{Gabriele Bonanno \newline
Department of Civil, Information Technology, Construction,
Environmental Engineering  and Applied Mathematics,
 University of Messina, 98166 - Messina, Italy}
\email{bonanno@unime.it}

\address{Beatrice Di Bella \newline
Department of Mathematics and Computer Science,
University of Messina, 98166 - Messina, Italy}
\email{bdibella@unime.it}

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

\thanks{Submitted April 4, 2013. Published May 21, 2013.}
\subjclass[2000]{34B37, 34B15, 58E05}
\keywords{Dirichlet boundary condition;
impulsive effects; variational methods; \hfill\break\indent critical points}

\begin{abstract}
 In this article we study second-order impulsive differential
 equations with Dirichlet boundary conditions, depending on two 
 real parameters. We show that an appropriate growth condition of 
 the nonlinear term,  under small perturbations of impulsive terms, 
 ensures the existence of three solutions. The approach is based on 
 variational methods.
\end{abstract}

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

\section{Introduction}

Impulsive differential equations are recognized as adequate models to
study the evolution of processes that are subject to sudden changes in
their states. Processes with such a character arise naturally and often,
especially in engineering and physics. In fact, it is known that many
biological phenomena involving thresholds, optimal control models in economics,
pharmacokinetics and frequency modulated systems, do exhibit impulse effects.
For this reason, the theory of impulsive differential equations
has become an important area of investigation in recent years.
For an introduction of the basic theory of impulsive differential equations
in $\mathbb{R}^n$, see \cite{LAS} and \cite{BHN}.
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{Chen,Quian,Mawhin}).
Recently, some researchers have begun to study the existence of solutions
for impulsive boundary value problems by using variational methods
(see for instance \cite{Nieto}--\cite{Wang1}).

In this article we consider the  nonlinear Dirichlet boundary-value
problem
\begin{equation} \label{eq:impuls}
\begin{gathered}
-u''(t)+a(t)u'(t)+b(t)u(t)=\lambda g(t,u(t)) \quad t\in [0,T],\; t\not= t_j\\
u(0)=u(T)=0\\
\Delta u'(t_j)=u'(t^+_j)-u'(t^-_j)=\mu I_j(u(t_j)),\quad j=1,2,\dots,n
\end{gathered}
\end{equation}
where $\lambda \in ]0,+\infty[$, $\mu \in ]0,+\infty[$,
$g:[0,T]\times \mathbb{R} \to \mathbb{R}$, $a,b \in L^{\infty}([0,T])$ 
satisfy the conditions $\operatorname{ess\,inf}_{t \in[0,T]} a(t)\geq 0$,
$\operatorname{ess\,inf}_{t \in[0,T]}b(t) \geq 0$,
 $0=t_0<t_1<t_2<\dots <t_ n<t_{n+1}=T$,
$\Delta u'(t_j)=u'(t^+_j)-u'(t^-_j)= \lim_{t \to t^+_j}u'(t)
-\lim_{t \to t^-_j}u'(t)$, and $I_j: \mathbb{R} \to \mathbb{R}$
are continuous for every $j=1,2,\dots,n$.

By a classical solution of \eqref{eq:impuls}, we mean a function
$$
u \in \big\{w \in C([0,T]: w_{|{[t_j,t_{j+1}]}} \in H^2([t_j,t_{j+1}])\big\}
$$
that satisfies the equation in \eqref{eq:impuls} a.e. on
$[0,T]\setminus \{t_1,\dots,t_n\}$, the limits $u'(t^+_j)$, $u'(t^-_j)$,
$j=1,\dots,n$, exist, satisfy the impulsive conditions
$\Delta u'(t_j)=\mu I_j(u(t_j))$m and the boundary condition $u(0)=u(T)=0$.
Clearly, if $a, b, g$ are continuous, then the classical solution
$u \in C^2([t_j,t_{j+1}])$, $j=0,1,\dots,n$, and satisfies the equation
in \eqref{eq:impuls} for all $t \in [0,T]\setminus \{t_1,\dots,t_n\}$.

By using variational methods, we show the existence of three solutions
for this problem. More precisely, by choosing $\mu$ in a suitable way and
under a growth condition on the nonlinear term we prove that  \eqref{eq:impuls}
has at least three solutions for every $\lambda$ lying in a precise interval.
In particular, we obtain two main theorems.  In the first one
(Theorem \ref{thm3}) we require on the antiderivative of $g$ both
a growth more then quadratic in a suitable interval and a growth less
then quadratic at infinity, and at the same time, on the impulse $I_j$,
an asymptotic condition is required.  In the second one (Theorem \ref{thm4})
we establish the existence of at least three positive solutions uniformly
bounded without asymptotic conditions on $g$ and $I_j$.

As an example, we present a particular case of Theorem \ref{thm4}.

\begin{theorem} \label{thm0}
Let $g:\mathbb{R} \to \mathbb{R} $ be a nonnegative continuous and non-zero
function such that
\begin{equation}
\lim_{u \to 0^+}\frac{g(u)}{u}=\lim_{u \to+\infty}\frac{g(u)}{u}=0 \,.\label{eq:0}
\end{equation}
Then, for every
\[
\lambda > \frac{(12+T^2)e^{2T}}{2T(e^{T}-1)(e^{3T/4}-e^{T/4})}\inf_{d >0}
\frac{d^2}{\int_0^dg(x)\,dx}
\]
 and for every negative continuous function $I_j : \mathbb{R} \to \mathbb{R}$,
$j=1,\dots,n$, there exists $\delta^*>0$ such that, for each
$\mu \in ]0,\delta^*[$, the problem
\begin{equation}\label{eq:impuls0}
\begin{gathered}
-u''(t)+u'(t)+u(t)=\lambda g(u(t)) \quad t\in [0,T],\; t\not= t_j\\
u(0)=u(T)=0\\
\Delta u'(t_j)=u'(t^+_j)-u'(t^-_j)=\mu I_j(u(t_j)),\quad j=1,2,\dots,n
\end{gathered}
\end{equation}
has at least three non-zero solutions.
\end{theorem}

We wish to stress that in many papers, as for instance
in \cite{ZhangDai,ZouLi,LiuChen},
under assumptions similar to those of our results, the authors ensure
the existence of at least only one solution for  \eqref{eq:impuls} and,
 moreover, do not give an estimate of $\lambda$ and $\mu$ and an explicit
upper bound, uniformly with respect to parameters, of the solutions.

The remainder of the paper is organized as follows.
 In Section 2, some preliminary results will be given.
In Section 3, we will state and prove the main results of the paper,
as well as give some applications to \eqref{eq:impuls}.

\section{Preliminaries}

We consider the following problem,  which is slightly
different form \eqref{eq:impuls},
\begin{equation}\label{eq:impuls1}
\begin{gathered}
-(p(t)u'(t))'+q(t)u(t)=\lambda f(t,u(t)) \quad t\in [0,T],\; t\not= t_j\\
u(0)=u(T)=0\\
\Delta u'(t_j)=u'(t^+_j)-u'(t^-_j)=\mu I_j(u(t_j)),\quad j=1,2,\dots,n
\end{gathered}
\end{equation}
where $p \in C^1([0,T],[0,+\infty[)$, $q \in L^{\infty}([0,T])$ with
$\operatorname{ess\,inf}_{t \in[0,T]} q(t) \geq 0$.

It is easy to see that the solutions of \eqref{eq:impuls1} are solutions
of \eqref{eq:impuls} if
$$
p(t)=e^{-\int_0^t a(\tau)\,d\tau}, \quad
q(t)=b(t)e^{-\int_0^t a(\tau)\,d\tau}, \quad
 f(t,u)=g(t,u)e^{-\int_0^t a(\tau)\,d\tau} \,.
$$
Let us introduce some notation.
In the Sobolev space $H^1_0(0,T)$, consider the inner product
$$
(u,v)=\int_0^T p(t)u'(t)v'(t)\,dt+\int_0^T q(t)u(t)v(t)\,dt \,,
$$
which induces the norm
$$
\|u\|=\Big(\int_0^T p(t)(u'(t))^2\,dt +\int_0^T q(t)(u(t))^2\,dt\Big)^{1/2}.
$$
Let us recall the Poincar\`e type inequality
\begin{equation}
\Big[\int_0^Tu^2(t)\,dt \Big]^{1/2}\leq \frac{T}{\pi}
\Big[\int_0^T(u')^2(t)\,dt \Big]^{1/2}. \label{eq:poin}
\end{equation}

\begin{proposition} \label{propos:ineq}
 Let $u \in H^1_0(0,T)$. Then
\begin{equation}
\|u\|_{\infty }\leq \frac{1}{2}\sqrt{\frac{T}{p^*}}\|u\| \label{eq:poinc}
\end{equation}
where $p^*:= \min_{t \in [0,T]}p(t)$
\end{proposition}

\begin{proof}
In view of H\"older's inequality one has
$$
\|u\|_{\infty }\leq \frac{\sqrt T}{2}\|u'\|_{L^2([0,T])}
\leq \frac{1}{2}\sqrt{\frac{T}{p^*}}\|u\| \,.
$$
\end{proof}

Here and in the sequel $f : [0, T]\times \mathbb{R} \to \mathbb{R}$ is an
$L^1$-Carath\'eodory function, namely:
\begin{itemize}
\item[(F1)]
\begin{itemize}
\item[(a)] $t \to f(t,x)$ is measurable for every $x \in\mathbb{R}$;
\item[(b)] $x \to f(t,x)$ is continuous for almost every $t \in [0,T]$;
\item[(c)] for every $\rho >0$ there exists a function $l_\rho \in L^1([0,T])$
such that
$$
\sup_{|x|\leq \rho}|f(t,x)|\leq l_{\rho}(t)
$$
for almost every $t \in [0,T]$;
\end{itemize}
\end{itemize}

\begin{definition} \label{def2.1}\rm
A function $u \in H^1_0(0,T)$ is said to be a weak solution of
\eqref{eq:impuls1}, if $u$ satisfies
\begin{equation}\label{eq:debole}
\begin{aligned}
&\int_0^T p(t)u'(t)v'(t) \, dt +\int_0^Tq(t)u(t)v(t) dt \\
&-\lambda\int_0^Tf(t,u(t))v(t)\,dt+\mu\sum_{j=1}^n p(t_j)I_j(u(t_j))v(t_j)=0,
\end{aligned}
\end{equation}
for any $v\in H^1_0(0,T)$.
\end{definition}

\begin{lemma} \label{lem2.1}
$u \in H^1_0(0,T)$ is a weak solution of \eqref{eq:impuls1} if and only
if $u$ is a classical solution of \eqref{eq:impuls1}.
\end{lemma}

\begin{proof}
Let $u \in H^1_0(0,T)$ be a weak solution of \eqref{eq:impuls1}.
Then \eqref{eq:debole}
holds for any $v\in H^1_0(0,T)$. Fix $j \in \{0,1,2,\dots,n\}$ and let
$\bar{v} \in H^1_0(0,T)$ such that $\bar{v}(t)=0$ for all
$t \in [0,t_j]\cup[t_{j+1},T]$. Thus by
\eqref{eq:debole} we obtain
$$
\int_{t_j}^{t_{j+1}}[-(p(t)u'(t))'\bar{v}(t)+q(t)u(t)\bar{v}(t)]\,dt
-\lambda\int_{t_j}^{t_{j+1}}f(t,u(t))\bar{v}(t)\,dt=0 \,.
$$
This implies that
$$
-(p(t)u'(t))'+q(t)u(t)=\lambda f(t,u(t))
$$
for almost every $t \in [t_j,t_{j+1}]$. Hence, $u \in H^2(t_j,t_{j+1})$
and satisfies the equation
\begin{equation}
-(pu')'+qu=\lambda f(t,u) \quad \text{almost every }  t \in [0,T] \label{eq:sol}\,.
\end{equation}
Now multiplying by $v \in H^1_0(0,T)$ and integrating on $[0,T]$, we obtain that
$$
-\sum_{j=1}^n \Delta u'(t_j)p(t_j)v(t_j)+\int_0^T\left[-(p(t)u'(t))'+q(t)u(t)
 -\lambda f(t,u(t))\right]v(t)\,dt=0 \,.
$$
Taking again \eqref{eq:debole} into account,
we obtain
$$
\sum_{j=1}^n \Delta u'(t_j)p(t_j)v(t_j)=\mu\sum_{j=1}^nI_j(u(t_j))p(t_j)v(t_j) \,.
$$
Hence $\Delta u'(t_j)=\mu I_j(u(t_j)$, for every $j=1,2,\dots,n$,
and the impulsive condition in \eqref{eq:impuls1} is satisfied.
\end{proof}

Now, we define the functionals $\Phi, \Psi: H^1_0(0,T) \to \mathbb{R}$ by
\begin{equation}
\Phi(u)=\frac{1}{2}\|u\|^2\quad
\Psi(u)=\int_{0}^T F(t,u(t))\,dt-\frac{\mu}{\lambda}
\sum_{j=1}^n p(t_j)\int_0^{u(t_j)}I_j(x)\,dx, \label{eq:def}
\end{equation}
for each $u \in H^1_0(0,T)$, where $F(t,\xi)=\int_0^{\xi}f(t,x)\,dx$
for each $(t,\xi) \in [0,T]\times \mathbb{R}$.
Using the property of $f$ and the continuity of $I_j$, $j=1,2,\dots,n$,
we have that $\Phi, \Psi\,\in C^1(H^1_0(0,T),\mathbb{R})$ and for any
$v\in H^1_0(0,T)$, we have
$$
\Phi'(u)(v)=\int_0^T p(t)u'(t)v'(t)\,dt + \int_0^T q(t)u(t)v(t)\,dt
$$
and
$$
\Psi'(u)(v)=\int_0^Tf(t,u(t))v(t)\,dt
-\frac{\mu}{\lambda}\sum_{j=1}^n p(t_j)I_j(u(t_j))v(t_j).
$$
So, arguing in a standard way, it is possible to prove that the critical
points of the functional $E_{\lambda,\mu}(u):=\Phi(u)-\lambda  \Psi(u)$
are the weak solutions of problem \eqref{eq:impuls1} and so they are classical.

We now state two critical point theorems which are the main tools for the
proofs of our results. The following statement
comes easily by the results contained in \cite{BONMAR} and in \cite{BONCAND}.

\begin{theorem}[{\cite[Theorem 2.6]{BONMAR}}] \label{teo:bon}
 Let $X$ be a reflexive real Banach space; $\Phi:X \to \mathbb{R}$
be a sequentially weakly lower semicontinuous, coercive and continuously
G\^ateaux differentiable functional whose G\^ateaux derivative admits
a continuous inverse on $X^*$, $\Psi:X \to \mathbb{R}$ be a sequentially
weakly upper semicontinuous, continuously G\^ateaux differentiable functional
whose G\^ateaux derivative is compact, such that
$$
\Phi(0)= \Psi(0)=0 \,.
$$
Assume that there exist $r >0$ and $\bar{x}\in X$, with $r<\Phi(\bar{x})$
such that
\begin{itemize}
\item [(i)] $\sup_{\Phi(x)\leq r}\Psi(x)
<r \Psi(\bar{x})/ \Phi(\bar{x})$,
\item [(ii)] for each $\lambda$ in
\[
\Lambda_r:=\Bigl] \frac{\Phi(\bar{x})}{\Psi(\bar{x})},
\frac{r}{\sup_{\Phi(x)\leq r}\Psi(x)}\Bigr[\,, 
\]
the functional
$\Phi-\lambda \Psi$ is coercive.
\end{itemize}
Then, for each $\lambda \in \Lambda_r$ the functional $\Phi-\lambda \Psi$
has at least three distinct critical points in $X$.
\end{theorem}

\begin{theorem}[{\cite[Theorem 3.2]{BONCAND}}] \label{teo:can}
 Let $X$ be a reflexive real Banach space; $\Phi:X \to \mathbb{R}$
be a convex, coercive and continuously G\^ateaux differentiable functional
 whose G\^ateaux derivative admits a continuous inverse on $X^*$,
$\Psi:X \to \mathbb{R}$ be a continuously G\^ateaux differentiable
functional whose G\^ateaux derivative is compact, such that
$$
\inf_{X}\Phi=\Phi(0)= \Psi(0)=0 \,.
$$
Assume that there exist two positive constants  $r_1,r_2 >0$ and
$\bar{x}\in X$, with $2r_1<\Phi(\bar{x})<\frac{r_2}{2}$, such that
\begin{itemize}
\item[(j)] $\frac{\sup_{\Phi(x)\leq r_1}\Psi(x)}{r_1}<\frac{2}{3}\frac{\Psi(\bar{x})}{\Phi(\bar{x})}$,
\item[(jj)] $\frac{\sup_{\Phi(x)\leq r_2}\Psi(x)}{r_2}<\frac{1}{3}\frac{\Psi(\bar{x})}{\Phi(\bar{x})}$,
\item[(jjj)] for each $\lambda$ in
\[
\Lambda_{r_1,r_2}^*
:=\Bigl] \frac{3}{2}\frac{\Phi(\bar{x})}{\Psi(\bar{x})},
\min\big\{\frac{r_1}{\sup_{\Phi(x)\leq r_1}\Psi(x)},
\frac{r_2}{2\sup_{\Phi(x)\leq r_2}\Psi(x)}\big\}\Bigr[
\]
and for every
$x_1,x_2 \in X$, which are local minima for the functional
$\Phi-\lambda \Psi$, and such that
$\Psi(x_1)\geq 0$ and $\Psi(x_2)\geq 0$, one has
$\inf_{t \in [0,1]}\Psi(tx_1+(1-t)x_2) \geq 0$.
\end{itemize}
Then, for each $\lambda \in \Lambda_{r_1,r_2}^*$ the functional
$\Phi-\lambda \Psi$ has at least three distinct critical points which lie
in $\Phi^{-1}(]-\infty, r_2[)$.
\end{theorem}

\section{Main results}

First, we give the following lemma which we will use in the proof
of our main result.

\begin{lemma}\label{stime}
Assume that
\begin{itemize}
\item [(H1)] there exist constants $\alpha,\beta>0$ and $\sigma\in [0,1[$
such that
$$
|I_j(x)|\leq \alpha+\beta|x|^{\sigma} \quad \text{for all }
 x \in \mathbb{R}, \; j=1,2,\dots,n\,.
$$
\end{itemize}
Then, for any $u \in H^1_0(0,T)$, one has
\begin{equation} \label{dis7}
\big| \sum_{j=1}^n p(t_j)\int_0^{u(t_j)}I_j(x)\,dx \big|
\le \sum_{j=1}^n p(t_j) \Big( \alpha \|u\|_{\infty}
+ \frac{\beta}{\sigma+1} \|u\|_{\infty}^{\sigma+1}\Big).
\end{equation}
\end{lemma}

\begin{proof}
Thanks to condition (H1), one has
$$
\big|\int_0^{u(t_j)}I_j(x)\,dx \big|
\leq \alpha|u(t_j)|+\frac{\beta}{\sigma +1}|u(t_j)|^{\sigma +1}.
$$
Thus, \eqref{dis7} is obtained.
\end{proof}

\begin{remark} \label{rmk3.1} \rm
It is easy to verify that the condition
\begin{itemize}
\item[(H1')] There exist constants $\gamma_j,\beta_j>0$ and
$\sigma_j\in [0,1[$ , ($j=1,2,\dots,n$), such that
$$
|I_j(x)|\leq \gamma_j+\beta_j|x|^{\sigma_j} \quad \text{for all }
x \in \mathbb{R}, \; j=1,2,\dots,n\,.
$$
\end{itemize}
is equivalent to (H1). In fact, it is sufficient to put
$\beta:= \max_{1\leq j\leq n} \beta_j$,
$\gamma:= \max_{1\leq j\leq n} \gamma_j$,
$\alpha=\gamma+ \beta$ and $\sigma:= \max_{1\leq j\leq n} \sigma_j$.
\end{remark}

Now, put
\[
\tilde p:=\sum_{j=1}^n p(t_j),\quad
k:=\frac{6p^*}{12\|p\|_{\infty}+T^2\|q\|_{\infty}},\quad
\Gamma_c:=\frac{\alpha}{c}+ \big(\frac{\beta}{\sigma+1}\big)c^{\sigma-1},
\]
where $\alpha$, $\beta$, $\sigma$ are given by $\rm(h_1)$ and $c$ is
a positive constant.

\begin{theorem} \label{thm1}
Suppose that {\rm (F1), (H1)} are satisfied. Furthermore,
assume that there exist two positive constants $c,d$, with
$c<d$, such that
\begin{itemize}
\item [(A1)]  $F(t,\xi)\geq 0$ for all $(t, \xi) \, \in \left([0,\frac{T}{4}]\cup [\frac{3T}{4},T]\right) \times [0,d]$;
\item [(A2)] \[
 \frac{\int_0^T \max_{|\xi|\leq c}F(t,\xi)\,dt}{c^2}
<k\frac{\int_{T/4}^{3T/4}F(t,d)\,dt}{d^2};
\]
\item [(A3)]
\[
\limsup_{|\xi|\to +\infty} \frac{\sup_{t \in [0,T]}F(t,\xi)}{\xi^2}
\le \frac{\pi^2}{4T}
    \frac{\int_{0}^{T}\max_{|\xi|\leq c}F(t,\xi)\,dt}{c^2}.
\]
\end{itemize}
Then, for every $\lambda$ in
\[\Lambda:=\Big]\frac{2p^* }{kT}\frac{d^2}{\int_{T/4}^{3T/4}F(t,d)\,dt},
 \frac{2p^*}{T}\frac{c^2}{\int_0^T\max_{|\xi|\leq c}F(t,\xi)\,dt}\Big[ ,
\]
there exists
\[
\delta:=\frac{1}{T\tilde p}\min\Big\{\frac{2p^*c^2-\lambda T\int_{0}^{T}
\max_{|\xi|\leq c}F(t,\xi)\,dt}{c^2\Gamma_c},
\frac{k\lambda T \int_{T/4}^{3T/4}F(t,d)\,dt-2p^*d^2}{d^2
\Gamma_{(d/\sqrt k) }} \Big\}
\]
 such that, for each $\mu \in [0,\delta[$ the problem \eqref{eq:impuls1}
has at least three distinct classical solutions.
\end{theorem}

\begin{proof}
First, we observe that due to (A2) the interval $\Lambda$ is non-empty and,
consequently, one has $\delta>0$. Now,
fix $\lambda$ and $\mu$ as in the conclusion.
Our aim is to apply Theorem \ref{teo:bon}. For this end, take $X= H^1_0(0,T)$
and $\Phi,  \Psi$ as in \eqref{eq:def}.
Put
$r=2c^2p^*/T$. Taking \eqref{eq:poinc} into account, for every $u \in X$
such that $\Phi(u)\leq r$, one has
$\max_{t \in [0,T]}|u(t)| \leq c$.
Consequently, from Lemma \ref{stime} it follows that
$$
\sup_{\Phi(u)\leq r}\Psi(u)\leq\int_0^T\max_{|\xi|\leq c}F(t,\xi)\,dt
+\frac{\mu}{\lambda} \tilde p \Big(\alpha c
+\frac{\beta}{\sigma+1}c^{\sigma+1}\Big);
$$
that is,
$$
\frac{\sup_{\Phi(u)\leq r}\Psi(u)}{r}\leq \frac{T}{2p^*}
\Big[\frac{\int_0^T\max_{|\xi|\leq c}F(t,\xi)\,dt}{c^2}
+\frac{\mu}{\lambda} \tilde p \Gamma_c \Big]\,.
$$
Hence, bearing in mind that $\mu < \delta$, one has
\begin{equation}
\frac{\sup_{\Phi(u)\leq r}\Psi(u)}{r}<\frac{1}{\lambda}\,. \label{eq:1}
\end{equation}
Put
$$
\bar{v}(t)= \begin{cases}
 \frac{4d}{T}t, & t\in[0,T/4],\\
  d, & t\in ]T/4,3T/4], \\
  \frac{4d}{T}(T-t), & t\in ]3T/4,T].
  \end{cases}
$$
Clearly $\bar v \in X$. Moreover, one has
\begin{equation} \label{dis9}
\frac{8p^*}{T}d^2\le\|\bar v\|^2
\leq \frac{2d^2(12\|p\|_{\infty}+T^2\|q\|_{\infty})}{3T}
=\frac{4d^2p^*}{kT}.
\end{equation}
So, from $c<\sqrt{2} d$ we obtain
$r<\Phi(\bar{v})$. Moreover, again from the previous inequality, we have
$$
\Phi(\bar{v}) <\frac{2p^*d^2}{kT}.
$$
Now, due to Lemma \ref{stime}, (A1), \eqref{eq:poinc} and \eqref{dis9}
one has
\begin{align*}
\Psi(\bar{v})
&\geq \int_{T/4}^{3T/4}F(t,d)\,dt
 -\frac{\mu}{\lambda} \tilde p \Big( \alpha \|\bar v\|_{\infty}
 + \frac{\beta}{\sigma+1} \|\bar v\|_{\infty}^{\sigma+1}\Big)\\
&\ge\int_{T/4}^{3T/4}F(t,d)\,dt-\frac{\mu}{\lambda}
\frac{\tilde p \,d^2}{k}\Gamma_{(d /\sqrt k)}.
\end{align*}
So, we obtain
$$
\frac{\Psi(\bar v)}{\Phi(\bar v)}
\geq \frac{kT\int_{T/4}^{3T/4}F(t,d) \,dt
-\frac{\mu}{\lambda}\tilde p T d^2 \Gamma_{(d /\sqrt k)}}{2p^* d^2}.
$$
Since $\mu < \delta$, one has
\begin{equation}
\frac{\Psi(\bar v)}{\Phi(\bar v)}>\frac{1}{\lambda} \label{eq:2} \,.
\end{equation}
Therefore, from \eqref{eq:1} and \eqref{eq:2}, condition (i) of
 Theorem \ref{teo:bon} is fulfilled.

Now,  to prove the coercivity of the functional $\Phi -\lambda \Psi$,
due to (A3), we have
\[
\limsup_{|\xi|\to +\infty} \frac{\sup_{t \in [0,T]}F(t,\xi)}{\xi^2}
<\big(\frac{\pi^2 p^*}{2T^2}\big)\frac{1}{\lambda}.
\]
 So, we can fix $\varepsilon >0$ satisfying
$$
\limsup_{|\xi|\to +\infty} \frac{\sup_{t \in [0,T]}F(t,\xi)}{\xi^2}
<\varepsilon <\big(\frac{\pi^2 p^*}{2T^2}\big)\frac{1}{\lambda}\,.
$$
Then, there exists a positive constant $h$ such that
$$
F(t,\xi)\leq \varepsilon |\xi|^{2}+h \quad \forall t \in [0,T], \; \forall
\xi \in \mathbb{R}  \,.
$$
Taking into account Lemma \ref{stime}, Proposition \ref{propos:ineq}
and \eqref{eq:poin},
it follows that
\begin{align*}
&\Phi(u)-\lambda \Psi(u)\\
&\geq \frac{1}{2}\|u\|^2 -\lambda \varepsilon\|u\|^2_{L^2[0,T]}-\lambda hT
 -\mu \tilde p\Big[\alpha \frac{1}{2}\sqrt{\frac{T}{p^*}}\|u\|
 + \frac{\beta}{\sigma+1} \Big(\frac{1}{2}\sqrt{\frac{T}{p^*}}\Big)^{\sigma+1}
 \|u\|^{\sigma+1}\Big]\\
&\geq \Big(\frac{1}{2}-\lambda \varepsilon\frac{T^2}{\pi^2 p^*}\Big)\|u\|^2
 -\lambda hT
 -\mu \tilde p\Big[\alpha \frac{1}{2}\sqrt{\frac{T}{p^*}}\|u\|
 + \frac{\beta}{\sigma+1} \Big(\frac{1}{2}\sqrt{\frac{T}{p^*}}\Big)^{\sigma+1}
 \|u\|^{\sigma+1}\Big],
\end{align*}
for all $u \in H^1_0(0,T)$. So, the functional $\Phi -\lambda \Psi$ is coercive.
Now, the conclusion of Theorem \ref{teo:bon} can be used. It follows that,
for every
\[
\lambda \in \Big]\frac{2k }{T}\frac{d^2}{\int_{T/4}^{3T/4}F(t,d)\,dt},
 \frac{2}{T}\frac{c^2}{\int_0^1\max_{|\xi|\leq c}F(t,\xi)\,dt}\Big[ \,,
\]
the functional $\Phi-\lambda \Psi$ has at least three distinct critical
points in $X$, which are the weak solutions of the problem \eqref{eq:impuls1}.
This completes the proof.
\end{proof}

\begin{corollary} \label{cor1}
Suppose that {\rm (H1)} holds. Let $h \in L^1([0,T])$ be a nonnegative
and non-zero function and let $g:\mathbb{R} \to \mathbb{R}$ be a continuous
function. Put $h_0:=\int_{T/4}^{3T/4}h(t)dt$ and
$G(\xi)=\int_{0}^{\xi}g(x)dx$ for all $\xi \in \mathbb{R}$, and
assume that there exist two positive constants $c,d$, with
$c<d$, such that
\begin{itemize}
\item[(A1')] $G(\xi)\geq 0$ for all $\xi  \in  [0,d]$;
\item[(A2')] \[
\frac{\max_{|\xi|\leq c}G(\xi)}{c^2}<\frac{1}{2}\frac{h_0}{\|h\|_1}
\frac{ G(d)}{d^2};
\]
\item[(A3')] $ \limsup_{|\xi|\to +\infty}  G(\xi)/\xi^2\le 0$.
\end{itemize}
Then, for every
$\lambda$ in
$$
\Lambda:=\Big]\frac{4 }{Th_0}\frac{d^2}{ G(d)} ,
 \frac{2}{T\|h\|_1}\frac{c^2}{\max_{|\xi|\leq c} G(\xi)}\Big[\,,
$$
there exists
\[
\delta:=\frac{1}{T n}\min\Big\{\frac{2c^2-\lambda T\|h\|_1
\max_{|\xi|\leq c}G(\xi)}{c^2\Gamma_c},
\frac{\frac{\lambda T h_0}{2} G(d)-2d^2}{d^2
\Gamma_{(\sqrt 2 d) }} \Big\}
\]
 such that, for each $\mu \in [0,\delta[$ the problem
\begin{equation}\label{eq:impuls1par}
\begin{gathered}
-u''(t)=\lambda h(t)g(u(t)) \quad t\in [0,T],\; t\not= t_j\\
u(0)=u(T)=0\\
\Delta u'(t_j)=u'(t^+_j)-u'(t^-_j)=\mu I_j(u(t_j)),\quad j=1,2,\dots,n
\end{gathered}
\end{equation}
 has at least three classical solutions.
\end{corollary}

The proof of the above corollary follows from Theorem \ref{thm1}
by choosing $f(t,x)=h(t)g(x)$ for all $(t,x) \in [0,T]\times \mathbb{R}$
and taking into account that $k=1/2$.

\begin{remark} \label{rmk3.2} \rm
Clearly, if $g$ is nonnegative then assumption (A1') is verified
 and (A2') becomes
 $$
\frac{ G(c)}{c^2}<\frac{1}{2}\frac{h_0}{\|h\|_1}\,\frac{ G(d)}{d^2}.
$$
\end{remark}

Now, we state  a result without asymptotic conditions on $I_j$.
The following lemma  will be crucial in our arguments.

\begin{lemma}\label{teo:positive}
Suppose that {\rm (F1)} is satisfied. Moreover, assume that
 $f(t,x) \geq 0$ for all $(t,x) \in [0,T]\times \mathbb{R}$ and
$I_j(x) \leq 0$ for all $x \in \mathbb{R}$, $j=1,\dots,n$.
If $u$ is a classical solution of \eqref{eq:impuls1}, then $u(t) \geq 0$
for all $t \in [0,T]$.
\end{lemma}

\begin{proof} If $u$ is a classical solution of\eqref{eq:impuls1}, then
$$
\int_0^T(p(t)u'(t))'v(t)\,dt-\int_0^T q(t)u(t)v(t)\,dt
+\lambda \int_0^Tf(t,u(t))v(t)\,dt=0
$$
for all $v \in X$.
Let $v(t)=\max\{-u(t),0\}$ for all $t \in [0,T]$; clearly $v \in X$ and we have
\begin{align*}
0&=\sum_{j=0}^{n}\int _{t_j}^{t_{j+1}}(p(t)u'(t))'v(t)\,dt
 -\int_0^T q(t)u(t)v(t)\,dt+\lambda \int_0^T f(t,u(t))v(t)\,dt\\
&=\sum_{j=0}^{n}p(t)u'(t)v(t)\big|_{t_j}^{t_{j+1}}
 -\int _0^T p(t)u'(t)v'(t)\,dt-\int_0^T q(t)u(t)v(t)\,dt\\
&\quad +\lambda \int_0^T f(t,u(t))v(t)\,dt\\
&=-\sum_{j=1}^{n}\Delta u'(t_j)p(t_j)v(t_j)
 -\int _{0}^{T}p(t)u'(t)v'(t)\,dt-\int_0^T q(t)u(t)v(t)\,dt\\
&\quad +\lambda \int_0^T f(t,u(t))v(t)\,dt\\
&=-\mu \sum_{j=1}^{n}I_j(u(t_j))p(t_j)v(t_j)
 +\int _{0}^{T}p(t)(v'(t))^2\,dt+\int_0^T q(t)(v(t))^2\,dt\\
&\quad +\lambda \int_0^T f(t,u(t))v(t)\, dt \\
&\geq \|v\|^2 \,.
\end{align*}
So $v(t)=0$ for $t \in [0,T]$.
\end{proof}

Put
$$
\Im_c:=\sum_{j=1}^{n}\min_{|\xi| \leq c}\int_0^{\xi}I_j(x)\,dx,
\quad \text{for all} \;c>0 \,.
$$
Our other main result is as follows.

\begin{theorem} \label{thm2}
Suppose that  {\rm (F1)} is satisfied. Furthermore,
assume that there exist three positive constants $c_1,c_2,d$,
with $c_1<d<\sqrt{\frac{k}{2}}c_2$, such that
\begin{itemize}
\item[(B1)] $f(t,\xi)\geq 0$ for all $(t, \xi)  \in [0,\,T] \times [0,c_2]$;
\item[(B2)]
\[
 \frac{\int_0^T  F(t,c_1)\,dt}{c_1^2}<\frac{2}{3}k
\frac{\int_{T/4}^{3T/4}F(t,d)\,dt}{d^2};
\]

\item[(B3)]
\[ \frac{\int_0^T  F(t,c_2)\,dt}{c_2^2}<\frac{k}{3}
\frac{\int_{T/4}^{3T/4}F(t,d)\,dt}{d^2}.
\]
\end{itemize}
Let
\[
\Lambda':=\Big]\frac{3p^* }{kT}\frac{d^2}{\int_{T/4}^{3T/4}F(t,d)\,dt},
 \frac{p^*}{T}\min \Big\{\frac{2c_1^2}{\int_0^T F(t,c_1)\,dt},
\frac{c_2^2}{\int_0^T F(t,c_2)\,dt} \Big\}\Big[.
\]
Then, for every $\lambda \in \Lambda'$ and for every negative continuous
function $I_j$, $j=1,\dots,n$, there exists
 $$
\delta^*:=\frac{1}{T\|p\|_{\infty}}\min
\Big\{\frac{\lambda T\int_{0}^T F(t,c_1)\,dt-2p^*c_1^2}{\Im_{c_1}},
\frac{\lambda T \int_{0}^T F(t,c_2)\,dt-p^*c_2^2}{\Im_{c_2}} \Big\}
$$
such that, for each $\mu \in ]0,\delta^*[$ the problem \eqref{eq:impuls1}
has at least three classical solutions $u_i$, $i=1,2,3$, such that
$0<\|u_i\|_{\infty}\leq c_2$.
\end{theorem}

\begin{proof}
Without loss of generality, we can assume $f(t,x) \geq 0$ for all
$(t,x) \in [0,T] \times \mathbb{R}$.
Fix $\lambda, I_j$ and $\mu$ as in the conclusion and take $X, \Phi$ and
$\Psi$ as in the proof of Theorem \ref{thm1}.
Put $\bar{v}$ as in Theorem \ref{thm1}, $r_1=\frac{2p^*c_1^2}{T}$ and
$r_2=\frac{2p^*c_2^2}{T}$. Therefore, one has $2r_1<\Phi(\bar{v})<\frac{r_2}{2}$
and since $\mu<\delta^*$, one has
\begin{align*}
\frac{1}{r_1}\sup_{\Phi(u)<r_1}\Psi(u)
&\leq \frac{T}{2p^*c_1^2}\Big(\int_0^T F(t,c_1)\,dt
 -\frac{\mu}{\lambda}\|p\|_{\infty}\Im_{c_1}\Big)\\
&<\frac{1}{\lambda}
 <\frac{T}{3k}\frac{\int_{T/4}^{3T/4}F(t,d)\, dt}{d^2}\\
&\leq \frac{2}{3} \frac{\Psi(\bar{v})}{\Phi(\bar{v})} \,,
\end{align*}
and
\begin{align*}
\frac{2}{r_2}\sup_{\Phi(u)<r_2}\Psi(u)
&\leq \frac{T}{p^*c_2^2}\Big(\int_0^T F(t,c_2)\,dt
 -\frac{\mu}{\lambda}\|p\|_{\infty}\Im_{c_2}\Big)\\
&<\frac{1}{\lambda}
 <\frac{T}{3k}\frac{\int_{T/4}^{3T/4}F(t,d)\, dt}{d^2}\\
&\leq \frac{2}{3} \frac{\Psi(\bar{v})}{\Phi(\bar{v})} \,.
\end{align*}
Therefore, conditions (j) and (jj) of Theorem \ref{teo:can} are satisfied.
Finally, let $u_1$ and $u_2$ be two local minima for $\Phi-\lambda\Psi$.
 Then, $u_1$ and $u_2$ are critical points for $\Phi-\lambda\Psi$, and so,
they are weak solutions for the problem \eqref{eq:impuls1}.
Hence, owing to Lemma \ref{teo:positive}, we obtain $u_1(t)\geq 0$ and
$u_2(t)\geq 0$ for all $t \in [0,T]$. So, one has $\Psi(s u_1+(1-s)u_2)\geq 0$
for all $s \in [0,1]$.
From Theorem \ref{teo:can} the functional $\Phi-\lambda\Psi$ has at least
three distinct critical points which are weak solutions of \eqref{eq:impuls1}.
This complete the proof.
\end{proof}

Let $A(t)$ a primitive of $a(t)$, $g: [0,T]\times \mathbb{R} \to \mathbb{R}$
an $L^1$-Carath\'eodory function  and put
$$
G(t, \xi)=\int_0^\xi g(t,x)\, dx,\quad
\tilde{k}:=\frac{6e^{-A(T)}}{12+T^2\|be^{-A}\|_{\infty}} \,.
$$
By Theorems \ref{thm1} and \ref{thm2},
we obtain the following results for  problem \eqref{eq:impuls}.

\begin{theorem} \label{thm3}
Suppose that {\rm (H1)} holds. Furthermore, assume that there exist two
positive constants $c,d$, with $c<d$, such that
\begin{itemize}
\item[(I1)] $G(t,\xi)\geq 0$ for all $(t, \xi) \in \left([0,\frac{T}{4}]
 \cup [\frac{3T}{4},T]\right) \times [0,d]$;

\item[(I2)]
\[
 \frac{\int_0^T e^{-A(t)} \max_{|\xi|\leq c}G(t,\xi)\,dt}{c^2}<\tilde{k}
\frac{\int_{T/4}^{3T/4}e^{-A(t)}G(t,d)\,dt}{d^2};
\]
\item[(I3)] $ \limsup_{|\xi|\to +\infty} \frac{\sup_{t \in [0,T]}e^{-A(t)}G(t,\xi)}{\xi^2}\leq\frac{ \pi^2 }{4T}\frac{\int_{0}^{T}e^{-A(t)} \max_{|\xi|\leq c}G(t,\xi)\,dt}{c^2}$.
\end{itemize}
Let
$$
\Lambda:=\Big]\frac{2}{\tilde{k}Te^{\|a\|_{1}}}
\frac{d^2}{\int_{T/4}^{3T/4}e^{-A(t)}G(t,d)\,dt},
 \frac{2}{Te^{\|a\|_{1}}}\frac{c^2}{\int_0^Te^{-A(t)} 
\max_{|\xi|\leq c}G(t,\xi)\,dt}\Big[.
$$  
Then, for every $\lambda \in \Lambda$, there exists
\begin{align*}
\delta &:= \frac{1}{\tilde{e} T}\min\Big\{\frac{2c^2e^{-\|a\|_{1}}
 -\lambda T\int_{0}^{T}e^{-A(t)}
 \max_{|\xi|\leq c}G(t,\xi)\,dt}{c^2\Gamma_c}\,, \\
 &\quad \frac{\tilde{k}\lambda T \int_{T/4}^{3T/4}e^{-A(t)}G(t,d)\,dt
-2e^{-\|a\|_{1}}d^2}{d^2\Gamma_{(d/\sqrt k) }} \Big\}
 \end{align*}
such that, for each $\mu \in [0,\delta[$ the problem \eqref{eq:impuls} 
has at least three distinct classical solutions.
\end{theorem}

The above theorem follows immediately  from Theorem \ref{thm1} taking into 
account Section 2.


\begin{theorem} \label{thm4}
Assume that there exist three positive constants $c_1,c_2,d$, with 
$c_1<d<\sqrt{\frac{\tilde{k}}{2}}c_2$, such that
\begin{itemize}
\item[(J1)]  $g(t,\xi)\geq 0$ for all $(t, \xi)  \in [0,\,T] \times [0,c_2]$;

\item[(J2)] 
\[
\frac{\int_0^T e^{-A(t)}G(t,c_1)\,dt}{c_1^2}<\frac{2}{3}\tilde{k}
\frac{\int_{T/4}^{3T/4}e^{-A(t)}G(t,d)\,dt}{d^2};
\]
\item[(J3)] 
\[
\frac{\int_0^T e^{-A(t)}G(t,c_2)\,dt}{c_2^2}<\frac{\tilde{k}}{3}
\frac{\int_{T/4}^{3T/4}e^{-A(t)}G(t,d)\,dt}{d^2}.
\]
\end{itemize}
Then, for every $\lambda$  in
\begin{align*}
\Lambda'&:=\Big]\frac{3e^{-\|a\|_{1}} }{\tilde{k}T}
\frac{d^2}{\int_{T/4}^{3T/4}e^{-A(t)}G((t,d)\,dt},\\
&\quad \frac{e^{-\|a\|_{1}}}{T}\min 
\Big\{\frac{2c_1^2}{\int_0^T e^{-A(t)}G(t,c_1)\,dt},
\frac{c_2^2}{\int_0^T e^{-A(t)}G(t,c_2)\,dt} \Big\}\Big[ \,,
\end{align*}
 and for every negative continuous function $I_j$,
 $j=1,\dots,n$, there exists
\begin{align*}
\delta^*:=\frac{1}{T}\min\Big\{&\frac{\lambda T\int_{0}^T e^{-A(t)}G(t,c_1)\,dt
-2e^{-\|a\|_{1}}c_1^2}{\Im_{c_1}}\,,\\
&\frac{\lambda T \int_{0}^T e^{-A(t)}G(t,c_2)\,dt
-e^{-\|a\|_{1}}c_2^2}{\Im_{c_2}} \Big\}
\end{align*}
such that, for each $\mu \in ]0,\delta^*[$, problem \eqref{eq:impuls}
 has at least three classical solutions $u_i$, $i=1,2,3$, such that 
$0<\|u_i\|_{\infty}\leq c_2$.
\end{theorem}

The above theorem follows from Theorem \ref{thm2} when taking into 
account Section 2.

 We want to point out that the function $a(t)$ can be taken of any sign,
 provided changes are made to the constant $\tilde {k}$.
When $g:\mathbb{R} \to \mathbb{R}$ is a nonnegative continuous function, 
the assumptions in Theorem \ref{thm4} take a simpler form:

\begin{theorem} \label{thm5}
Put
$$
\theta:=\frac{\int_{T/4}^{3T/4}e^{-A(t)}\,dt}{\|e^{-A}\|_{1}}, \quad 
k^*:=\frac{2}{3}\tilde{k}\theta, \quad 
L:=\frac{e^{-\|a\|_{1}}}{T\|e^{-A}\|_{1}} \,.
$$
Assume that there exist three positive constants $c_1,c_2,d$, with 
$c_1<d<\frac{1}{2}\sqrt{\frac{3k^*}{\theta}}c_2$, such that
\begin{itemize}
\item[(J2'')] $ G(c_1)/c_1^2 < k^*G(d)/d^2$;
\item[(J3'')] $ G(c_2)/c_2^2 < \frac{k^*}{2} G(d)/d^2$.
\end{itemize}
Then, for every $\lambda$ in
$$
\Lambda'':=\Big]\frac{2L}{k^* }\frac{d^2}{G(d)}, L\min 
\big\{\frac{2c_1^2}{G(c_1)},\,\frac{c_2^2}{G(c_2)} \big\}\Big[ \,,
$$
 and for every negative continuous function $I_j$,
 $j=1,\dots,n$, there exists
$$
\delta^*:=\|e^{-A}\|_{1}\min\Big\{\frac{\lambda G(c_1)-2Lc_1^2}{\Im_{c_1}},
\frac{\lambda  G(c_2)-Lc_2^2}{\Im_{c_2}} \Big\}
$$ 
such that, for each $\mu \in ]0,\delta^*[$, problem \eqref{eq:impuls} 
has at least three classical solutions $u_i$, $i=1,2,3$, such 
that $0<\|u_i\|_{\infty}\leq c_2$.
\end{theorem}

Now using Theorem \ref{thm5}, we give the proof of Theorem \ref{thm0}. 


\begin{proof}[Proof of Theorem \ref{thm0}]
 Fix $\lambda >\lambda_1$, put $G(\xi)=\int_0^{\xi} g(x)\,dx$ for all 
$\xi \in \mathbb{R}$, and let $d>0$ such that $G(d)>0$ and 
$$
\lambda >\frac{(12+T^2)e^{2T}}{2T(e^{T}-1)(e^{3T/4}-e^{T/4})}\frac{d^2}{G(d)}.
$$ 
From \eqref{eq:0} there is $c_1>0$ such that $c_1<d$ and
 $G(c_1)/c_1^2< 2/(T(e^{T}-1)\lambda)$, and there is  $c_2>0$ such that 
\[
d<\sqrt{\frac{3e^{-T}}{12+T^2}}c_2,\quad 
\frac{G(c_2)}{c_2^2}<\frac{1}{T(e^{T}-1)\lambda}.
\]
 Therefore, Theorem \ref{thm5} ensures the conclusion.
\end{proof}

Finally, we give two applications of the results above.

\begin{example}\label{esempio1}\rm 
The  problem
\begin{equation}
\begin{gathered}
-u''(t)+(\frac{t}{\pi}-1)^2u(t)=\lambda u^2(3-4u)\sin t \quad
\text{a.e. in }  [0,\pi]\\
u(0)=u(\pi)=0\\
\Delta u'(t_1)=u'(t^+_1)-u'(t^-_1)=\mu (1-\sqrt[3]{u(t_1)})
\end{gathered}
\end{equation}
admits at least three non-trivial solutions for each $\lambda \in [7,20]$ 
and for each $0< \mu<\frac{1}{38 \pi}(1-\frac{63\lambda \pi}{4096})$.

Indeed, it is sufficient to apply Theorem \ref{thm1} by choosing, 
for instance, $c=1/64$ and $d=1/2$.
\end{example}

 We  remark that although \cite[Theorem 3.2]{ZouLi} can be applied,
it guarantees the existence of at least one solution, only. 
Our results go further than \cite[Theorem 1]{Dai},  
we have precise values of the parameter $\lambda$ for which the problem
admits solutions.

\begin{example}\label{esempio2}\rm 
Let $g:(t,x) \in (0,1]\times \mathbb{R} \to \mathbb{R}$, be defined as
$$
g(t,x)=\begin{cases}
 10^{-4}e^{t^2}/ \sqrt[4]{t} &\text{if }x \leq 10^{-2}\\
 x^2e^{t^2}/\sqrt[4]{t} &\text{if }10^{-2}<x \leq 1\\
 e^{t^2}/(x^2 \sqrt[4]{t}) &\text{if } x > 1.
  \end{cases}
$$
By Theorem \ref{thm4}, for each $\lambda \in [33,55]$ and each 
$\mu \in ]0,3.4\times 10^{-4}[$ the problem
\begin{gather*}
-u''(t)+2tu'(t)+(1-t)u(t)=\lambda g(t,u(t)) \quad\text{a.e. in }[0,1]\\
u(0)=u(1)=0\\
\Delta u'(t_1)=u'(t^+_1)-u'(t^-_1)=\mu (-1-|u(t_1)|^3)
\end{gather*}
admits at least three non-trivial solutions $u_i$, 
such that $0<|u_i(t)|<10^2$ for all $t\in [0,1]$, $i=1,2,3$.

It suffices to choose, for instance, $c_1=10^{-2}$, $c_2=10^2$, $d=1$.
\end{example}

 We observe that in Example \ref{esempio1} we do not have the negativity of 
the impulsive term, so we cannot apply Theorem \ref{thm4}.
On the other hand, in Example \ref{esempio2} the function is negative, 
but it does not have the sublinear growth.

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