Open Access

Multiple solutions for a class of perturbed second-order differential equations with impulses

  • Shapour Heidarkhani1Email author,
  • Massimiliano Ferrara2,
  • Amjad Salari1 and
  • Giuseppe Caristi3
Boundary Value Problems20162016:74

https://doi.org/10.1186/s13661-016-0581-z

Received: 1 January 2016

Accepted: 29 March 2016

Published: 5 April 2016

Abstract

The present paper is an attempt to investigate the existence of weak solutions for perturbed impulsive problems containing a Lipschitz nonlinear term. The study bases itself on the most recent variational approaches to the smooth functionals which are defined on reflexive Banach spaces. The findings of the study, finally, revealed that, under appropriate conditions, such problems possess at least three weak solutions. According to the results, these solutions are generated by impulses when the Lipschitz nonlinear term is zero.

Keywords

multiple solutions perturbed impulsive differential equation critical point theory variational methods

1 Introduction

This paper attempts to study the existence of three weak solutions for the perturbed impulsive problem
$$ \left \{ \textstyle\begin{array}{l} \ddot{u}(t)+V_{u}(t,u(t))=h(u(t)),\quad t\in(s_{k-1},s_{k}), \\ \Delta{\dot{u}(s_{k})}=\lambda f_{k}(u(s_{k}))+ \mu g_{k}(u(s_{k})), \\ u(0)- u(T)= \dot{u}(0)- \dot{u}(T)=0, \end{array}\displaystyle \right . $$
(1)
where \(s_{k}\), \(k=1,2,\ldots,m\), are instants in which the impulses occur and \(0=s_{0}< s_{1}< s_{2}\cdots< s_{m}< s_{m+1}=T\), \(\Delta{\dot{u}(s_{k})}= \dot{u}({s_{k}}^{+})-\dot{u}({s_{k}}^{-})\) with \(\dot{u}({s_{k}}^{\pm})= \lim_{t\to{{s_{k}}^{\pm}}}\dot{u}(t)\), \(f_{k}(\xi)= \operatorname{grad}_{\xi}F_{k}(\xi)\), \(g_{k}(\xi)=\operatorname{grad}_{\xi}G_{k}(\xi)\), \(h(\xi)=\operatorname{grad}_{\xi}H(\xi)\), \(F_{k},G_{k},H\in{\mathrm{C}}^{1}({\mathbb{R}}^{N},\mathbb{R})\), \(V\in{\mathrm{C}}^{1}([0,T]\times{\mathbb{R}}^{N}, \mathbb{R})\), \(V_{\xi}(t,\xi)= \operatorname{grad}_{\xi}V(t,\xi)\), \(h:{\mathbb{R}}^{N}\to {\mathbb{R}}^{N}\) is a Lipschitz continuous function with the Lipschitz constant \(L > 0\), i.e.,
$$\bigl\vert h(\xi_{1})-h(\xi_{2}) \bigr\vert \leq L \vert \xi_{1}-\xi_{2}\vert $$
for every \(\xi_{1},\xi_{2}\in\mathbb{R}^{N}\) and \(h(0)=0\), and \(\lambda>0\) and \(\mu\geq0\) are two parameters.

Impulsive differential equations emerge from the real world problems and are acclimated to be employed as handy means for the description of the processes which are endowed with abrupt discontinuous jumps. As for this, these processes are used in such a vast array of fields as control theory, biology, impact mechanics, physics, chemistry, chemical engineering, population dynamics, biotechnology, economics, optimization theory, and the inspection process in operations research. That is why the theory of impulsive differential equations is now highly appreciated as a natural theoretical basis for the mathematical modeling of the natural phenomena of various kinds. For a comprehensive background in the theory and the applications of the impulsive differential equations, we hereby refer the interested reader to [111].

There is already a large body of research on the notion of impulsive differential equations in the literature. The findings of most of these studies are mainly achieved through some such theories as fixed point theory, topological degree theory (including continuation method and coincidence degree theory) and comparison method (including upper and lower solutions method and monotone iterative method) (see, for example, [1215] and references therein). Recently, the existence and multiplicity of solutions for impulsive problems have been thoroughly investigated by [1625] using variational methods and the critical point theory, the whole findings of which can be considered as nothing but generalizations of the corresponding ones for the second-order ordinary differential equations. Put differently, the aforementioned achievements can be applied to impulsive systems in the absence of the impulses and still give the existence of solutions in this situation. This is, somehow, to say that the nonlinear term \(V_{u}\) functions more significantly as compared to the role played by the impulsive terms \(f_{k}\) in guaranteeing the existence of solutions in these results. In [26], which is a probe into the existence of periodic and homoclinic solutions for a class of second-order differential equations of the form (1) in the case \(\mu=0\), via variational methods, the results signify that such a system enjoys at least one non-zero periodic solution as well as one non-zero homoclinic solution under appropriate conditions, and these solutions are generated by impulses when \(f=0\). Based on the variational methods and the critical point theory, [27] has examined problem (1) in the case \(\mu=0\), by means of which the authors have proved that such a problem admits at least one non-zero, two non-zeros, or an infinite number of periodic solutions as yielded by the impulses under different assumptions, respectively. Most particularly, using a smooth version of Theorem 2.1 in [28] which is a more precise version of Ricceri’s variational principle ([29], Theorem 2.5) under some hypotheses on the behavior of the nonlinear terms at infinity, under conditions on the potentials of \(f_{k}\) and \(g_{k}\), [30] has proved that the existence of definite intervals about λ and μ, in which problem (1) in the case \(h\equiv0\) admits an unbounded sequence of solutions generated by impulses. Moreover, it has been proved that replacing the conditions at infinity of the nonlinear terms with a similar one at zero admits the same results.

In the present paper, employing two sorts of three critical points theorems obtained in [31, 32], which we will recall in the next section (Theorems 2.1 and 2.2), we establish the existence of at least three weak solutions for problem (1). We also verify that these solutions are generated by impulses when \(h\equiv0\); see Theorems 3.1 and 3.2. We say that a solution of the problem (1) is called a solution generated by impulses if this solution is nontrivial when impulsive terms \(f_{k},g_{k}\neq0\) for some \(1\leq k\leq m\), but it is trivial when impulsive terms are zero. For example, if the problem (1) does not possess non-zero weak solution when \(f_{k}=g_{k}\equiv0\) for all \(1\leq k\leq m\), then a non-zero weak solution for problem (1) with \(f_{k},g_{k}\neq0\) for some \(1\leq k\leq m\) is called a weak solution generated by impulses. Along the same lines of reasoning, these theorems (Theorems 2.1 and 2.2) have been successfully employed by [3335] to ensure the presence of at least three solutions for the perturbed boundary value problems.

The curious reader is also referred to [3641], which have verified the existence of multiple solutions for boundary value problems. For a thorough study of the subject, we also refer the reader to [4248].

The organization of the present paper is as follows. In Section 2 we recall some basic definitions and preliminary results, while Section 3 is devoted to the existence of multiple solutions for the impulsive differential problem (1).

2 Preliminaries

Our fundamental tool consists of three critical point theorems. In the first one, the coercivity of the functional \(\Phi -\lambda\Psi\) is essential. In the second one, a proper sign hypothesis has been assumed.

Theorem 2.1

([32], Theorem 2.6)

Let X be a reflexive real Banach space, \(\Phi:X \rightarrow \mathbb{R}\) be a coercive continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on \(X^{*}\), \(\Psi:X \rightarrow \mathbb{R}\) be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that \(\Phi(0)=\Psi(0)=0\).

Assume that there exist \(r>0\) and \(\bar {v}\in X\), with \(r< \Phi (\bar {v})\) such that
(a1): 

\(\frac{\sup_{ \Phi(u)\leq r}\Psi(u)}{r}< \frac{\Psi(\bar {v})}{\Phi(\bar {v})}\);

(a2): 

for each \(\lambda\in \Lambda_{r}:= \, ]\frac{\Phi(\bar {v})}{\Psi (\bar {v})}, \frac{r}{\sup_{\Phi(u)\leq r}\Psi(u)} [\) the functional \(\Phi-\lambda \Psi\) is coercive.

Then, for each \(\lambda\in\Lambda_{r}\) the functional \(\Phi -\lambda \Psi\) has at least three distinct critical points in X.

Theorem 2.2

([31], Theorem 3.3)

Let X be a reflexive real Banach space, \(\Phi:X \rightarrow \mathbb{R}\) be a convex, coercive and continuously Gâteaux differentiable functional whose derivative admits a continuous inverse on \(X^{\ast}\), \(\Psi:X \rightarrow \mathbb{R}\) be a continuously Gâteaux differentiable functional whose derivative is compact, such that
  1. 1.

    \(\inf_{X}\Phi=\Phi(0)=\Psi(0)=0\);

     
  2. 2.
    for every \(\lambda>0\) and for every \(u_{1},u_{2}\in X\) which are local minima for the functional \(\Phi-\lambda\Psi\) and such that \(\Psi (u_{1})\geq0\) and \(\Psi(u_{2})\geq 0\), one has
    $$\inf_{s\in[0,1]}\Psi \bigl(su_{1}+(1-s)u_{2} \bigr)\geq0. $$
     
Assume that there are two positive constants \(r_{1}\), \(r_{2}\) and \(\bar {v}\in X\), with \(2r_{1}<\Phi(\bar {v})<\frac{r_{2}}{2}\), such that
(b1): 

\(\frac{\sup_{u\in\Phi^{-1}(]-\infty ,r_{1}[)}\Psi(u)}{r_{1}}< \frac{2}{3}\frac{\Psi(\bar {v})}{\Phi(\bar {v})}\);

(b2): 

\(\frac{\sup_{u\in\Phi^{-1}(]-\infty ,r_{2}[)}\Psi(u)}{r_{2}}< \frac{1}{3}\frac{\Psi(\bar {v})}{\Phi(\bar {v})}\).

Then, for each \(\lambda\in\, ]\frac{3}{2}\frac{\Phi(\bar {v})}{\Psi(\bar {v})}, \min \{ \frac{r_{1}}{\sup_{u\in\Phi^{-1}(]-\infty,r_{1}[)}\Psi (u)}, \frac{\frac{r_{2}}{2}}{\sup_{u\in\Phi^{-1}(]-\infty,r_{2}[)}\Psi (u)} \} [\), the functional \(\Phi-\lambda\Psi\) has at least three distinct critical points which lie in \(\Phi^{-1}(]-\infty,r_{2}[)\).
In this paper we consider the Hilbert space
$$X= \bigl\{ u : [0,T]\rightarrow{\mathbb{R}}^{N} \mid u \text{ is absolutely continuous}, u(0)= u(T), \dot{u} \in {\mathrm{L}}^{2} \bigl([0,T],{\mathbb{R}}^{N} \bigr) \bigr\} $$
with the inner product
$$\langle u, v\rangle= \int_{0}^{T} \bigl[ \bigl( u(t),v(t) \bigr) + \bigl(\dot{u}(t), \dot{v}(t) \bigr) \bigr]\, {\mathrm{d}}t\quad \text{for all } u, v\in X, $$
where \((\cdot,\cdot)\) is the inner product in \({\mathbb{R}}^{N}\). Obviously, the corresponding norm into the above inner product is as follows:
$$\|u\|= \biggl( \int_{0}^{T} \bigl( \bigl\vert \dot{u}(t) \bigr\vert ^{2}+ \bigl\vert u(t) \bigr\vert ^{2} \bigr)\,{ \mathrm{d}}t \biggr)^{1\over 2}\quad \text{for all } u\in X, $$
and X with this norm is a separable and uniformly convex Banach space.
Since the embedding \(X\hookrightarrow{\mathrm{C}}([0,T],\mathbb{R}^{N})\) is compact (see [49]), one has
$$ C:=\sup_{u\in X\backslash\{0\}}\frac{\max_{t\in[0,T]}|u(t)|}{\|u\|}< \infty. $$
(2)
We say that \(u\in X\) is a weak solution of the problem (1) if
$$\begin{aligned}& \int_{0}^{T} \bigl[ \bigl(\dot{u}(t),\dot{v}(t) \bigr)- \bigl(V_{u} \bigl(t,u(t) \bigr),v(t) \bigr)+ \bigl(h \bigl(u(t) \bigr),v(t) \bigr) \bigr]\, {\mathrm{d}}t+\lambda \sum _{k=1}^{m} \bigl(f_{k} \bigl(u(s_{k}) \bigr),v(s_{k}) \bigr) \\& \quad {}+\mu \sum_{k=1}^{m} \bigl(g_{k} \bigl(u(s_{k}) \bigr),v(s_{k}) \bigr)=0 \end{aligned}$$
for every \(v\in X\).
Moreover, set
$$G^{\theta}:=\max_{|t|\leq\theta} \Biggl[-\sum _{k=1}^{m}G_{k}(t) \Biggr] $$
for every \(\theta>0\) and
$$G_{\eta}:=\inf_{[0,\eta]} \Biggl[-\sum _{k=1}^{m}G_{k}(t) \Biggr] $$
for every \(\eta>0\). It is obvious that \(G^{\theta}\geq0\) and \(G_{\eta}\leq0\).
We consider the following assumptions on V:
  1. (A1)

    V is continuously differentiable and there exist two positive constants \(a_{1},a_{2}> 0\) so that \(a_{1}|\xi|^{2}\leq -V(t,\xi)\leq a_{2}|\xi|^{2}\) for all \((t,\xi)\in[0,T]\times {\mathbb{R}}^{N}\);

     
  2. (A2)

    \(-V(t,\xi)\leq-(V_{\xi}(t,\xi),\xi)\leq-2V(t,\xi )\) for all \((t,\xi)\in[0.T]\times{\mathbb{R}}^{N}\);

     
  3. (A3)

    \(V_{\xi_{1}-\xi_{2}}(t,\xi_{1}-\xi_{2})=V_{\xi_{1}}(t,\xi_{1})-V_{\xi _{2}}(t,\xi_{2})\) for all \(t\in[0,T]\) and \(\xi_{1}, \xi_{2}\in\mathbb{R}^{N}\).

     
We assume throughout and without further mention that the Lipschitz constant \(L > 0\) of the function h meets the condition
$$\min \biggl\{ \frac{1}{2},a_{1} \biggr\} >{TLC}^{2}. $$

We require the proposition below in proving Theorem 3.1.

Proposition 2.3

Let the assumptions (A1), (A2), and (A3) be satisfied and \(K:X\to X^{*}\) be the operator defined by
$$K(u)v= \int_{0}^{T} \bigl[ \bigl(\dot{u}(t),\dot{v}(t) \bigr)- \bigl(V_{u} \bigl(t,u(t) \bigr),v(t) \bigr)+ \bigl(h \bigl(u(t) \bigr),v(t) \bigr) \bigr]\, {\mathrm{d}}t. $$
Then K admits a continuous inverse on \(X^{*}\).

Proof

Since \(|h(\xi_{1})-h(\xi_{2})| \leq L|\xi_{1}-\xi_{2}|\) for every \(\xi_{1},\xi_{2}\in\mathbb{R}^{N}\), using the Cauchy-Schwarz inequality one has \(-L|\xi_{1}-\xi_{2}|^{2}\leq(h(\xi_{1})-h(\xi_{2}),\xi_{1}-\xi_{2})\leq L|\xi_{1}-\xi_{2}|^{2}\) for every \(\xi_{1},\xi_{2}\in\mathbb{R}^{N}\). So, taking (2) into account, bearing in mind that \(h(0)=0\), we have
$$\begin{aligned} \bigl\langle K(u),u \bigr\rangle &= \int_{0}^{T} \bigl( \bigl\vert \dot{u}(t) \bigr\vert ^{2}- \bigl(V_{u} \bigl(t,u(t) \bigr),u(t) \bigr)+ \bigl(h \bigl(u(t) \bigr),u(t) \bigr) \bigr)\,{\mathrm{d}}t \\ &\geq \int_{0}^{T} \bigl( \bigl\vert \dot{u}(t) \bigr\vert ^{2}+ a_{1} \bigl\vert u(t) \bigr\vert ^{2} \bigr)\,{\mathrm{d}}t-L \int_{0}^{T} \bigl\vert u(t) \bigr\vert ^{2}\,{\mathrm{d}}t \\ &\geq \bigl(\min\{1,a_{1}\}-{TLC}^{2} \bigr) \|u \|^{2}, \end{aligned}$$
and because \(\min\{1,a_{1}\}\geq\min\{\frac{1}{2},a_{1}\}>{TLC}^{2}\), we have \(\lim_{u\rightarrow \infty}\frac{ \langle K(u),u\rangle}{\|u\|}= +\infty\), that is, K is coercive. For any \(u,v\in X\) one has
$$\begin{aligned} \bigl\langle K(u)-K(v), u- v \bigr\rangle =& \int_{0}^{T} \bigl(\dot{u}(t)- \dot{v}(t), \dot{u}(t)- \dot{v}(t) \bigr)\, {\mathrm{d}}t \\ &{}- \int_{0}^{T} \bigl(V_{u} \bigl(t,u(t) \bigr)- V_{v} \bigl(t,v(t) \bigr),u(t)- v(t) \bigr)\,{\mathrm{d}}t \\ &{}+ \int_{0}^{T} \bigl(h \bigl(u(t) \bigr)- h \bigl(v(t) \bigr),u(t)- v(t) \bigr)\,{\mathrm{d}}t \\ \geq& \int_{0}^{T} \bigl\vert \dot{u}(t)-\dot{v}(t) \bigr\vert ^{2}\,{\mathrm{d}}t + \int_{0}^{T}a_{1} \bigl\vert u(t)- v(t) \bigr\vert ^{2} \,{\mathrm{d}}t \\ &{}-L \int_{0}^{T} \bigl\vert u(t)-v(t) \bigr\vert ^{2}\,{\mathrm{d}}t \\ \geq& \bigl(\min\{1, a_{1}\}-{TLC}^{2} \bigr)\|u- v \|^{2}, \end{aligned}$$
so K is uniformly monotone. By Theorem 26.A(d) in [50], \(K^{-1}\) exists and is continuous on \(X^{*}\). □

3 Main results

In this section, we show our main results of the existence of at least three weak solutions for the problem (1).

To obtain our first result, we take the two positive constants θ and η in such a way that
$$\frac{(a_{2}+LTC^{2})T\eta^{2}}{ -\sum_{k=1}^{m}F_{k}(\eta )}< \frac{(a_{3}-LTC^{2})\theta^{2}}{ C^{2}\max_{|t|\leq \theta} [-\sum_{k=1}^{m}F_{k}(t) ]} $$
and taking
$$\lambda\in\Lambda_{1}\, := \biggl]\frac{(a_{2}+LTC^{2})T\eta ^{2}}{ -\sum_{k=1}^{m}F_{k}(\eta)}, \frac{(a_{3}-LTC^{2})\theta^{2}}{ C^{2}\max_{|t|\leq\theta} [-\sum_{k=1}^{m}F_{k}(t) ]} \biggr[, $$
set
$$ \delta_{\lambda}=\min \biggl\{ \frac{\theta ^{2}-\frac{C^{2}\lambda}{(a_{3}-LTC^{2})} \max_{|t|\leq \theta} [-\sum_{k=1}^{m}F_{k}(t) ]}{\frac {C^{2}}{(a_{3}-LTC^{2})}G^{\theta}}, \frac{\eta^{2}- \frac{\lambda}{(a_{2}+LTC^{2})T} [-\sum_{k=1}^{m}F_{k}(\eta) ]}{\frac {1}{(a_{2}+LTC^{2})T}G_{\eta}} \biggr\} $$
(3)
and
$$ \bar {\delta}_{\lambda}:=\min \biggl\{ \delta_{\lambda }, \frac{1}{ \max \{0,\frac{C^{2}}{(a_{3}-LTC^{2})} \limsup_{|t|\to\infty}\frac{\sup\sum_{k=1}^{m}[-G_{k}(t)]}{|t|^{2}} \}} \biggr\} , $$
(4)
where we say \({\rho}/{0}=+\infty\), so that, for example, \(\bar {\delta}_{\lambda}=+\infty\) when
$$\limsup_{|t|\to\infty}\frac{\sup\sum_{k=1}^{m}[-G_{k}(t)]}{|t|^{2}}\leq0 $$
and \(G_{\eta}=G^{\theta}=0\).

Theorem 3.1

Suppose that V satisfies the assumptions (A1), (A2), and (A3). Assume that there exist two positive constants θ and η such that \(\theta<\sqrt{T}C\eta\) and
  1. (A4)

    \(\frac{ \max_{|t|\leq \theta} [-\sum_{k=1}^{m}F_{k}(t) ]}{\theta^{2}}< \frac{a_{3}-LTC^{2}}{C^{2}(a_{2}+LTC^{2})T}\frac{ -\sum_{k=1}^{m}F_{k}(\eta)}{\eta^{2}}\), where \(a_{3}=\min\{\frac {1}{2},a_{1}\}\);

     
  2. (A5)

    \(\limsup_{|t|\to+\infty }\frac{\sum_{k=1}^{m}[-F_{k}(t)]}{|t|^{2}}\leq0\).

     
Then, for each \(\lambda\in\Lambda_{1}\) and for each arbitrary function \(G_{k}\in {\mathrm{C}}^{1}({\mathbb{R}}^{N},\mathbb{R})\) denoting \(g_{k}(\xi)=\operatorname{grad}_{\xi}G_{k}(\xi)\) for each \(\xi\in\mathbb{R}^{N}\) for \(k=1,2,\ldots,m\), fulfilling the condition
$$\limsup_{|t|\to\infty}\frac{\sum_{k=1}^{m}[-G_{k}(t)]}{|t|^{2}}< +\infty, $$
there exists \(\bar {\delta}_{\lambda}>0\) given by (4) such that, for each \(\mu\in[0,\bar {\delta}_{\lambda}[\), the problem (1) admits at least three distinct weak solutions in X.

Proof

Fix λ, \(G_{k}\) for \(k=1,2,\ldots,m\) and μ as in the conclusion. Our aim is applying Theorem 2.1 for the functionals \(\Phi, \Psi:X \to\mathbb{R}\), defined by
$$\Phi(u)= \int_{0}^{T} \biggl[\frac{1}{2} \bigl\vert \dot{u}(t) \bigr\vert ^{2}-V \bigl(t,u(t) \bigr) \biggr]\,{ \mathrm{d}}t+ \int_{0}^{T}H \bigl(u(t) \bigr)\,{\mathrm{d}}t $$
and
$$\Psi(u)=- \Biggl(\sum_{k=1}^{m}F_{k} \bigl(u(s_{k}) \bigr)+\frac{\mu }{\lambda} \sum _{k=1}^{m}G_{k} \bigl(u(s_{k}) \bigr) \Biggr). $$
It is easily observable that Ψ is a Gâteaux differentiable functional and sequentially weakly upper semicontinuous whose Gâteaux derivative at the point \(u\in X\) is the functional \(\Psi'(u)\in X^{*}\), given by
$$\Psi'(u)v= - \Biggl(\sum_{k=1}^{m} \bigl(f_{k} \bigl(u(s_{k}) \bigr),v(s_{k}) \bigr)+ \frac{\mu}{\lambda} \sum_{k=1}^{m} \bigl(g_{k} \bigl(u(s_{k}) \bigr),v(s_{k}) \bigr) \Biggr), $$
and \(\Psi':X \to X^{*}\) is a compact operator. Moreover, Φ is a Gâteaux differentiable functional whose Gâteaux derivative at the point \(u\in X\) is the functional \(\Phi'(u)\in X^{*}\), given by
$$\begin{aligned} \Phi'(u)v =& \int_{0}^{T} \bigl[ \bigl(\dot{u}(t),\dot{v}(t) \bigr)- \bigl(V_{u} \bigl(t,u(t) \bigr),v(t) \bigr) \bigr]\,{ \mathrm{d}}t \\ &{}+ \int_{0}^{T} \bigl(h \bigl(u(t) \bigr),v(t) \bigr) \,{ \mathrm{d}}t \end{aligned}$$
for every \(v\in X\), while Proposition 2.3 shows that \(\Phi'\) admits a continuous inverse on \(X^{*}\). Furthermore, Φ is sequentially weakly lower semicontinuous. Indeed, let \(u_{n}\in X\) with \(u_{n}\to u\) weakly in X, taking weakly lower semicontinuity of the norm, we have \(\liminf_{n\to+\infty}\|u_{n}\|\geq\|u\|\) and \(u_{n}\to u\) uniformly on \([0,T]\). Hence, since V and H are continuous, we have
$$\begin{aligned}& \lim_{n\to+\infty}\frac{1}{2} \int_{0}^{T} \bigl[ \bigl\vert \dot{u_{n}}(t) \bigr\vert ^{2}- V \bigl(t,u_{n}(t) \bigr) \bigr]\,{\mathrm{d}}t+ \int_{0}^{T}H \bigl(u_{n}(t) \bigr)\,{ \mathrm{d}}t \\& \quad \geq \frac{1}{2} \int_{0}^{T} \bigl[ \bigl\vert \dot{u}(t) \bigr\vert ^{2}- V \bigl(t,u(t) \bigr) \bigr]\,{\mathrm{d}}t+ \int_{0}^{T}H \bigl(u(t) \bigr)\,{\mathrm{d}}t. \end{aligned}$$
Thus \(\liminf_{n\to+\infty}\Phi(u_{n})\geq\Phi(u)\), that is, Φ is sequentially weakly lower semicontinuous. Like the proof of Lemma 1 of [26], we observe that the weak solutions of the problem (1) are concisely the solutions of the equation \(\Phi'(u)-\lambda\Psi'(u)=0\). Since \(-L|\xi|\leq|h(\xi)| \leq L|\xi|\) for every \(\xi\in\mathbb{R}^{N}\), we have \(|H(\xi)|\leq L|\xi|^{2}\) for all \(\xi\in\mathbb{R}^{N}\). In parallel lines with the assumption (A1),
$$ \bigl(a_{3} -LTC^{2} \bigr)\|u \|^{2} \leq\Phi(u)\leq \bigl(a_{4}+LTC^{2} \bigr) \|u\|^{2}, $$
(5)
where \(a_{4}=\min\{\frac{1}{2},a_{2}\}\). Put \(r:=\frac{\theta^{2}(a_{3}-LTC^{2})}{C^{2}}\) and \(w(t):=\eta\) for every \(t\in[0,T]\). Because \(\min\{\frac{1}{2},a_{1}\}>{TLC}^{2}\), we have \(\min \{1,a_{1}\}>{TLC}^{2}\), which means \(a_{3}-LTC^{2}>0\), and so \(r>0\). It is clear that \(w\in X\) and
$$\|w\|^{2}=T\eta^{2}. $$
Since \(\theta<\sqrt{T}C\eta\), using (5), we have \(0< r<\Phi(w)\). Taking (2) into account, from (5) we observe that
$$\begin{aligned} \Phi^{-1} \bigl( ]{-}\infty,r [ \bigr)&= \bigl\{ u\in X; \Phi(u)\leq r \bigr\} \\ &\subseteq \bigl\{ u\in X; \bigl(a_{3}-LTC^{2} \bigr)\|u \|^{2}\leq r \bigr\} \\ &\subseteq \bigl\{ u\in X; \bigl\vert u(t) \bigr\vert \leq\theta\text{ for each } t\in [0,T] \bigr\} , \end{aligned}$$
and it follows that
$$\begin{aligned} \sup_{u\in\Phi^{-1}(]-\infty,r])}\Psi(u) &=\sup_{u\in\Phi^{-1}(]-\infty,r])} \Biggl[ -\sum _{k=1}^{m}F_{k} \bigl(u(s_{k}) \bigr)- \frac{\mu}{\lambda} \sum _{k=1}^{m}G_{k} \bigl(u(s_{k}) \bigr) \Biggr] \\ &\leq \max_{|\xi|\leq \theta} \Biggl[-\sum_{k=1}^{m}F_{k}( \xi) \Biggr]+\frac{\mu}{\lambda }G^{\theta}. \end{aligned}$$
Moreover, we have
$$\begin{aligned} \Psi(w)&=-\sum_{k=1}^{m}F_{k} \bigl(w(t) \bigr) -\frac{\mu}{\lambda}\sum_{k=1}^{m}G_{k} \bigl(w(t) \bigr) \\ &\geq-\sum_{k=1}^{m}F_{k}(\eta) +\frac{\mu}{\lambda}G_{\eta}. \end{aligned}$$
So, we obtain
$$\begin{aligned} \frac{ \sup_{u\in\Phi^{-1}(]-\infty,r])}\Psi(u)}{r} &=\frac{ \sup_{u\in\Phi^{-1}(]-\infty,r])} [-\sum_{k=1}^{m} [F_{k}(u(s_{k}))+\frac{\mu}{\lambda}G_{k}(u(s_{k}))] ]}{r} \\ &\leq \frac{ \max_{|\xi|\leq \theta} [-\sum_{k=1}^{m}F_{k}(\xi) ]+\frac{\mu}{\lambda }G^{\theta}}{ \frac{\theta^{2}(a_{3}-LTC^{2})}{C^{2}}} \end{aligned}$$
(6)
and
$$\begin{aligned} \frac{\Psi(w)}{ \Phi(w)}&\geq\frac{ -\sum_{k=1}^{m}F_{k}(w(t))-\frac {\mu}{\lambda}\sum_{k=1}^{m}G_{k}(w(t))}{ \frac{(a_{2}+LTC^{2})\eta^{2}}{C^{2}}} \\ &\geq\frac{ -\sum_{k=1}^{m}F_{k}(\eta)+\frac{\mu }{\lambda}G_{\eta}}{ \frac{(a_{2}+LTC^{2})\eta^{2}}{C^{2}}}. \end{aligned}$$
(7)
Since \(\mu<\delta_{\lambda}\), one has
$$\mu< \frac{\theta^{2}-\frac{C^{2}}{a_{3}-LTC^{2}} \lambda \max_{|\xi|\leq \theta} [-\sum_{k=1}^{m}F_{k}(\xi) ]}{\frac {C^{2}}{(a_{3}-LTC^{2})}G^{\theta}}, $$
this means
$$\frac{ \max_{|\xi|\leq \theta} [-\sum_{k=1}^{m}F_{k}(\xi) ]+\frac{\mu}{\lambda} G^{\theta}}{ \frac{\theta^{2}(a_{3}-LTC^{2})}{C^{2}}} < \frac{1}{\lambda}. $$
Furthermore,
$$\mu< \frac{\eta^{2}-\frac{C^{2}}{a_{3}-LTC^{2}} \lambda [-\sum_{k=1}^{m}F_{k}(\eta) ]}{ \frac{C^{2}}{a_{3}-LTC^{2}}G_{\eta}}, $$
this means
$$\frac{ -\sum_{k=1}^{m}F_{k}(\eta)+\frac{\mu}{\lambda} G_{\eta}}{ \frac{\eta^{2}(a_{3}-LTC^{2})}{C^{2}}}>\frac {1}{\lambda}. $$
Then
$$ \frac{ \max_{|\xi|\leq \theta} [-\sum_{k=1}^{m}F_{k}(\xi) ]+\frac{\mu}{\lambda }G^{\theta}}{ \frac{\theta^{2}(a_{3}-LTC^{2})}{C^{2}}} < \frac{1}{\lambda}< \frac{ -\sum_{k=1}^{m}F_{k}(\eta )+\frac{\mu}{\lambda} G_{\eta}}{ \frac{\eta^{2}(a_{3}-LTC^{2})}{C^{2}}}. $$
(8)
Hereupon, from (6)-(8) we infer that the condition (a1) of Theorem 2.1 is achieved. Eventually, since \(\mu<\bar {\delta}_{\lambda}\), we can fix \(l>0\) in such a manner that
$$\limsup_{|\xi|\to\infty}\frac{\sum_{k=1}^{m} [-G_{k}(\xi) ]}{|\xi|^{2}}< l $$
and \(\mu l<\frac{a_{3}-LTC^{2}}{ C^{2}}\). Therefore, there exists a constant q such that
$$ \sum_{k=1}^{m} \bigl[-G_{k}(u) \bigr]\leq l |u|^{2}+q\quad \text{for all } u\in\mathbb{R}^{N} $$
(9)
for \(k=1,2,\ldots,m\). Now, fix \(0<\varepsilon< \frac{a_{3}-LTC^{2}}{C^{2} \lambda}-\frac {\mu l}{\lambda}\). Owing to the assumption (A4) there is a constant \(q_{\varepsilon}\) such that
$$ \sum_{k=1}^{m} \bigl[-F_{k}(u) \bigr]\leq\varepsilon |u|^{2}+q_{\varepsilon} \quad \text{for all } u\in\mathbb{R}^{N} $$
(10)
for \(k=1,2,\ldots,m\). Due to (5), (9), and (10) we have
$$\begin{aligned} \Phi(u)-\lambda \Psi(u) =& \int_{0}^{T} \biggl[\frac{1}{2} \bigl\vert \dot {u}(t) \bigr\vert ^{2}-V \bigl(t,u(t) \bigr)+H \bigl(u(t) \bigr) \biggr]\,{\mathrm{d}}t \\ &{}- \lambda \Biggl[-\sum_{k=1}^{m} \biggl[F_{k} \bigl(u(s_{k}) \bigr)+\frac{\mu}{\lambda }G \bigl(u(s_{k}) \bigr) \biggr] \Biggr] \\ \geq& \bigl(a_{3}-LTC^{2} \bigr)\|u\|^{2}-\lambda \varepsilon |u|^{2}-\lambda q_{\varepsilon}-\mu l |u|^{2}- \mu q \\ \geq& \bigl(a_{3}-LTC^{2}-\lambda C^{2} \varepsilon-\mu C^{2} l \bigr)\|u\|^{2} -\lambda q_{\varepsilon}-\mu q. \end{aligned}$$
This means that the functional \(\Phi-\lambda\Psi\) is coercive, and the assumption (a2) of Theorem 2.1 is verified. From (6) and (8),
$$\lambda\in \, \biggl]\frac{\Phi(w)}{\Psi(w)}, \frac{r}{\sup_{\Phi(u)\leq r}\Psi(u)} \biggr[ $$
and Theorem 2.1 (with \(\bar {v}=w\)) ensures that the problem (1) possesses at least three weak solutions in X. □

We now offer another version of Theorem 3.1 within which no asymptotic condition on the nonlinear term is necessary; contrarily, each constituent of \(f_{k}\) and \(g_{k}\) for \(k=1,2,\ldots,m\) is considered to be negative.

Fix positive constants \(\theta_{1}\), \(\theta_{2}\), and η in such a way that
$$\begin{aligned}& \frac{3}{2}\frac{(a_{2}+LTC^{2})T\eta^{2}}{ [-\sum_{k=1}^{m}F(\eta) ]} \\& \quad < \frac{a_{3}-LTC^{2}}{C^{2}}\min \biggl\{ \frac{\theta_{1}^{2}}{ \max_{|\xi|\leq \theta_{1}} [-\sum_{k=1}^{m}F_{k}(\xi) ]}, \frac{\theta_{2}^{2}}{2 \max_{|\xi|\leq \theta_{2}} [-\sum_{k=1}^{m}F_{k}(\xi) ]} \biggr\} \end{aligned}$$
and put
$$\begin{aligned} \Lambda_{2} :=& \, \biggl] \frac{3}{2}\frac{(a_{2}+LTC^{2}) T\eta^{2}}{ [-\sum_{k=1}^{m}F(\eta) ]}, \\ &\frac{a_{3}-LTC^{2}}{C^{2}}\min \biggl\{ \frac{\theta_{1}^{2}}{ \max_{|\xi|\leq \theta_{1}} [-\sum_{k=1}^{m}F_{k}(\xi) ]}, \frac{\theta_{2}^{2}}{2 \max_{|\xi|\leq \theta_{2}} [-\sum_{k=1}^{m}F_{k}(\xi) ]} \biggr\} \biggr[. \end{aligned}$$

By the above symbolization, we obtain the following multiplicity result.

Theorem 3.2

Order the Banach space X by the positive cone \(X^{+}\) (see Section 5.4 of [51]), and suppose that V satisfies in the assumptions (A1), (A2), and (A3), \(F_{k}\in C^{1}({\mathbb{R}}^{N},\mathbb{R})\), each component of \(f_{k}(\xi)= \operatorname{grad}_{\xi}F_{k}(\xi)\) for \(k=1,2,\ldots,m\) is negative and there exist three positive constants \(\theta_{1}\), \(\theta_{2}\), and η such that \(\theta_{1}< C\sqrt{\frac{T}{2}}\eta<\frac{\theta_{2}}{2}\sqrt{\frac {a_{3}-LTC^{2}}{a_{2}+LTC^{2}}}\) where \(a_{3}=\min\{\frac{1}{2},a_{1}\}\) and
$$\begin{aligned}& (\mathrm{B}1)\quad \max \biggl\{ \frac{ \max_{|\xi|\leq \theta_{1}} [-\sum_{k=1}^{m}F_{k}(\xi) ]}{\theta _{1}^{2}}, \frac{ 2\max_{|\xi|\leq \theta_{2}} [-\sum_{k=1}^{m}F_{k}(\xi) ]}{\theta _{2}^{2}} \biggr\} \\& \hphantom{(\mathrm{B}1)\quad}\quad < \frac{2}{3}\frac{a_{3}-LTC^{2}}{C^{2}(a_{2}+LTC^{2})T}\frac{ [-\sum_{k=1}^{m}F(\eta) ]}{\eta^{2}}. \end{aligned}$$
Then, for each \(\lambda\in\Lambda_{2}\) and for every arbitrary function \(G_{k}\in{\mathrm{C}}^{1}({\mathbb{R}}^{N},\mathbb{R})\) such that each component of \(g_{k}(\xi)=\operatorname{grad}_{\xi}G_{k}(\xi)\) for every \(\xi\in\mathbb{R}^{N}\) is negative for \(k=1,2,\ldots,m\), there exists \(\delta^{*}_{\lambda}>0\) defined by
$$\begin{aligned}& \min \biggl\{ \frac{(a_{3}-LTC^{2})\theta_{1}^{2}-C^{2}\lambda \max_{|\xi|\leq \theta_{1}} [-\sum_{k=1}^{m}F(\xi) ]}{C^{2}G^{\theta _{1}}}, \\& \quad \frac{ (a_{3}-LTC^{2})\theta_{2}^{2}-2C^{2}\lambda \max_{|\xi|\leq \theta_{2}} [-\sum_{k=1}^{m}F_{k}(\xi) ]}{2C^{2}G^{\theta _{2}}} \biggr\} \end{aligned}$$
such that, for each \(\mu\in[0,\delta^{*}_{\lambda}[\), the problem (1) possesses at least three weak solutions \(u_{1}\), \(u_{2}\), and \(u_{3}\) such that \(u_{i}(t)\in X^{+}\) (or \(u_{i}(t)\geq0\)) for all \(t\in[0,T]\) and \(i=1,2,3\).

Proof

Fix λ, \(G_{k}\) for \(k=1,2,\ldots,m\) and μ as in the conclusion and take X, Φ, and Ψ as in the proof of Theorem 3.1. Obviously, the regularity assumptions of Theorem 2.2 on Φ and Ψ are satisfied. Our goal is to check (b1) and (b2). For this purpose, put \(w(t)=\eta\) for every \(t\in[0,T]\),
$$r_{1}:= \frac{(a_{3}-LTC^{2})\theta_{1}^{2}}{C^{2}} $$
and
$$r_{2}:= \frac{(a_{3}-LTC^{2})\theta_{2}^{2}}{C^{2}}. $$
According to condition \(\theta_{1}< C\sqrt{\frac{T}{2}}\eta<\frac{\theta_{2}}{2} \sqrt{\frac{a_{3}-LTC^{2}}{a_{2}+LTC^{2}}}\), and from (5), we get
$$2r_{1}< \Phi(w)< \frac{r_{2}}{2}. $$
Since \(\mu<\delta^{*}_{\lambda}\) and \(G_{\eta}=0\), one has
$$\begin{aligned} \frac{ \sup_{u\in\Phi^{-1}(]-\infty,r_{1}])}\Psi(u)}{r_{1}} &=\frac{ \sup_{u\in\Phi^{-1}(]-\infty,r_{1}])} [-\sum_{k=1}^{m} [F_{k}(u(s_{k}))+\frac{\mu}{\lambda}G_{k}(u(s_{k}))] ]}{r_{1}} \\ &\leq\frac{ \max_{|\xi|\leq \theta_{1}} [-\sum_{k=1}^{m}F_{k}(\xi) ]+\frac{\mu }{\lambda}G^{\theta_{1}}}{ \frac{(a_{3}-LTC^{2})\theta_{1}^{2}}{C^{2}}} \\ &< \frac{1}{\lambda} < \frac{2}{3}\frac{ [-\sum_{k=1}^{m}F_{k}(\eta ) ]+\frac{\mu}{\lambda} G\mathcal{\eta}}{ \frac{(a_{2}+LTC^{2})T\eta^{2}}{C^{2}}} \\ &\leq\frac{2}{3}\frac{\Psi(w)}{\Phi(w)} \end{aligned}$$
and
$$\begin{aligned} \frac{2 \sup_{u\in\Phi^{-1}(]-\infty,r_{2}])}\Psi (u)}{r_{2}}&=\frac{2 \sup_{u\in\Phi^{-1}(]-\infty ,r_{2}])} [-\sum_{k=1}^{m} [F_{k}(u(s_{k}))+\frac{\mu}{\lambda}G_{k}(u(s_{k}))] ]}{r_{2}} \\ &\leq\frac{2 \sup_{|\xi|\leq\theta_{2}} [-\sum_{k=1}^{m}F_{k}(\xi) ]+2\frac{\mu}{\lambda}G^{\theta_{2}}}{ \frac{(a_{3}-LTC^{2})\theta_{2}^{2}}{C^{2}}} \\ &< \frac{1}{\lambda} < \frac{2}{3}\frac{ [-\sum_{k=1}^{m}F_{k}(\eta ) ]+\frac{\mu}{\lambda}G_{\eta}}{ \frac{(a_{2}+LTC^{2})T\eta^{2}}{C^{2}}} \\ &\leq\frac{2}{3}\frac{\Psi(w)}{\Phi(w)}. \end{aligned}$$
Therefore, (b1) and (b2) of Theorem 2.2 are fulfilled. In the following, we show that \(\Phi-\lambda\Psi\) satisfies the assumption 2 of Theorem 2.2. 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, thus, they are weak solutions for the problem (1). We want to show that they are nonnegative. Let \(u_{0}\) be a nontrivial weak solution of problem (1). Arguing by a contradiction, assume that the set \(\mathcal{A}= \{t \in[0,T] : u_{0}(t)<0 \}= \{t \in[0,T] : 0-u_{0}(t)\in X^{+}, u_{0}(t)\neq0 \}\) is non-empty and its measure is positive. Put
$$\bar{v}(t)=\left \{ \textstyle\begin{array}{l@{\quad}l} 0,& 0\leq u_{0}(t), \\ u_{0}(t), & u_{0}(t)< 0 \end{array}\displaystyle \right . $$
for all \(t \in[0,T]\). Clearly, \(\bar{v} \in X\). Since \(u_{0}\) is a weak solution of (1) we have
$$\begin{aligned}& \int_{0}^{T} \bigl[ \bigl(\dot{u}_{0}(t), \dot{\bar{v}}(t) \bigr)- \bigl(V_{u_{0}} \bigl(t,u_{0}(t) \bigr), \bar{v}(t) \bigr)+\bigl(h \bigl(u_{0}(t),\bar{v}(t) \bigr)\bigr) \bigr]\,{ \mathrm{d}}t \\& \quad =-\lambda\sum_{k=1}^{m} \bigl(f_{k} \bigl(u_{0}(s_{k}) \bigr), \bar{v}(s_{k}) \bigr)-\mu \sum_{k=1}^{m} \bigl(g_{k} \bigl(u_{0}(s_{k}) \bigr), \bar{v}(s_{k}) \bigr). \end{aligned}$$
Thus, from our sign assumptions on the data, since \(-L|\xi|^{2}\leq(h(\xi),\xi) \leq L|\xi|^{2}\) for every \(\xi\in\mathbb{R}^{N}\), we have
$$\begin{aligned} \begin{aligned} 0&\leq \bigl(\min \{1, a_{1}\}-{TLC}^{2} \bigr) \|u_{0}\|_{X(\mathcal{A})}^{2}\leq \int_{\mathcal{A}} \bigl( \bigl\vert \dot{u}_{0}(t) \bigr\vert ^{2}+a_{1} \bigl\vert u_{0}(t) \bigr\vert ^{2}-L \bigl\vert u_{0}(t) \bigr\vert ^{2} \bigr)\,{\mathrm{d}}t \\ &\leq \int_{0}^{T} \bigl[ \bigl(\dot{u}_{0}(t), \dot{u}_{0}(t) \bigr)- \bigl(V_{u_{0}} \bigl(t,u_{0}(t) \bigr),u_{0}(t) \bigr)+ \bigl(h \bigl(u_{0}(t) \bigr),u_{0}(t) \bigr) \bigr]\,{\mathrm{d}}t\leq0. \end{aligned} \end{aligned}$$
Hence, \(u_{0}=0\) in \(\mathcal{A}\) and this is absurd. Then we conclude \(u_{1}(t)\geq0\) and \(u_{2}(t)\geq0\) for every \(t\in[0,T]\). Thus, it follows that \(su_{1}+(1-s)u_{2}\geq0\) for all \(s\in[0,1]\), and that
$$-(\lambda f_{k}+\mu g_{k}) \bigl(su_{1}+(1-s)u_{2} \bigr)\geq0 \quad \text{for } k=1,2,\ldots,m, $$
and consequently, \(\Psi(su_{1}+(1-s)u_{2})\geq0\), for every \(s\in [0,1]\). Hence, since all the hypotheses of Theorem 2.2 are satisfied, it follows that, for every
$$\lambda\in \, \biggl]\frac{3}{2}\frac{\Phi(w)}{\Psi(w)}, \min \biggl\{ \frac{r_{1}}{ \sup_{u\in\Phi^{-1}(]-\infty,r_{1}[)}\Psi (u)}, \frac{{r_{2}}/{2}}{ \sup_{u\in\Phi^{-1} (]-\infty,r_{2}[)}\Psi(u)} \biggr\} \biggr[, $$
the functional \(\Phi-\lambda\Psi\) has at least three distinct critical points \(u_{i}\) for \(i=1,2,3\), such that \(0\leq u_{i}(t)<\theta_{2}\) for all \(t\in[0,T]\) and \(i=1,2,3\), which are the weak solutions of the problem (1), and the favorable result is achieved. □

In the following, we present a special case of Theorem 3.1.

Corollary 3.3

Suppose that V satisfies the assumptions (A1), (A2), and (A3), and
$$\liminf_{\xi\to0}\frac{ \max_{|t|\leq \xi} [-\sum_{k=1}^{m}F_{k}(t) ]}{\xi^{2}}= \limsup_{\xi \to +\infty} \frac{\sum_{k=1}^{m}[-F_{k}(\xi)]}{\xi^{2}}=0. $$
Then there is \(\lambda^{*}>0\) such that for each \(\lambda>\lambda^{*}\) and every arbitrary function \(G_{k}\in{\mathrm{C}}^{1}({\mathbb{R}}^{N},\mathbb{R})\), denoting \(g_{k}(\xi)=\operatorname{grad}_{\xi}G_{k}(\xi)\) for every \(\xi\in\mathbb{R}^{N}\) for \(k=1,2,\ldots,m\), satisfying the asymptotical condition
$$\limsup_{|t|\to\infty}\frac{\sum_{k=1}^{m}[-G_{k}(t)]}{ |t|^{2}}< +\infty, $$
there exists \(\delta^{*}_{\lambda}>0\) such that, for each \(\mu\in[0,\delta^{*}_{\lambda}[\), the problem (1) admits at least three distinct weak solutions in X.

Proof

Fix \(\lambda>\lambda^{*}:=\frac{(a_{2}+LTC^{2})T\eta^{2}}{ -\sum_{k=1}^{m}F_{k}(\eta)}\) for some \(\eta>0\). Recalling
$$\liminf_{\xi\to0}\frac{ \max_{|t|\leq \xi} [-\sum_{k=1}^{m}F_{k}(t) ]}{\xi^{2}}=0, $$
there exists a sequence \(\{\theta_{n}\}\subset\, ]0,+\infty[\) with this feature that \(\lim_{n\to\infty} \theta_{n}=0\) and
$$\lim_{n\to\infty}\frac{ \max_{|t|\leq \theta_{n}} [-\sum_{k=1}^{m}F_{k}(t) ]}{\theta_{n}^{2}}=0. $$
Hence, there exists \(\bar {\theta}>0\) such that
$$\frac{ \max_{|t|\leq \bar {\theta}} [-\sum_{k=1}^{m}F_{k}(t) ]}{\bar {\theta}^{2}}< \min \biggl\{ \frac{a_{3}-LTC^{2}}{C^{2}(a_{2}+LTC^{2})T}\frac{ -\sum_{k=1}^{m}F_{k}(\eta)}{\eta^{2}}; \frac{a_{3}-LTC^{2}}{ \lambda C^{2}} \biggr\} $$
and \(\bar {\theta}<\sqrt{T}C\eta\). The conclusion follows from Theorem 3.1. □

Now, as an example, we present the following consequence of Theorem 3.2 with \(m=T=N=1\).

Corollary 3.4

Suppose that V satisfies the assumptions (A1), (A2), and (A3), \(f_{1}:{\mathbb{R}}\to\mathbb{R}\) is a negative continuous function and \(h:{\mathbb{R}}\to\mathbb{R}\) is a Lipschitz continuous function with the Lipschitz constant \(L > 0\) such that \(h(0)=0\), \(\min\{1,a_{1}\}>2L\), and \(a_{2}+2L<16(a_{3}-2L)\) where \(a_{3}=\min\{\frac{1}{2},a_{1}\}\). Furthermore, assume that
$$\lim_{\xi\to0^{+}}\frac{f_{1}(\xi)}{\xi} =0 $$
and
$$\int_{0}^{1/2}f_{1}(x)\, \mathrm{d}x< \frac{3}{32}\frac{a_{2}+2L}{a_{3}-2L} \int _{0}^{4}f_{1}(x)\, \mathrm{d}x. $$
Then, for every \(\lambda\in\, ] \frac{3}{8}\frac{a_{2}+2L}{ -\int _{0}^{1/2}f_{1}(x)\,\mathrm{d}x}, \frac{4(a_{3}-2L) }{-\int_{0}^{4}f_{1}(x)\,\mathrm{d}x} [\), and for every arbitrary negative continuous function \(g_{1}:{\mathbb{R}}\to\mathbb{R}\), there exists \(\delta^{*}_{\lambda}>0\) such that, for each \(\mu\in[0,\delta^{*}_{\lambda}[\), the problem
$$ \left \{ \textstyle\begin{array}{l} u''(t)+V_{u}(t,u(t))=h(u(t)),\quad t\neq s_{1}, \\ \Delta{u'(s_{1})}=\lambda f_{1}(u(s_{1}))+ \mu g_{1}(u(s_{1})), \\ u(0)- u(1)= u'(0)- u'(1)=0, \end{array}\displaystyle \right . $$
(11)
possesses at least three weak solutions \(u_{1}\), \(u_{2}\), and \(u_{3}\) such that \(0\leq u_{i}(t)<4\) for all \(t\in[0,T]\) and \(i=1,2,3\).

Proof

Our goal is to use Theorem 3.2 by choosing \(m=T=N=1\), \(\theta_{2}=4\) and \(\eta=\frac{1}{2}\). Since \(c=\sqrt{2}\), we observe that
$$\frac{3}{2}\frac{(a_{2}+LTC^{2})T\eta^{2}}{ [-\sum_{k=1}^{m}F_{k}(\eta) ]}= \frac {3}{8}\frac{a_{2}+2L}{ -\int_{0}^{1/2}f_{1}(x)\,\mathrm{d}x} $$
and
$$\frac{a_{3}-LTC^{2}}{C^{2}} \frac{\theta_{2}^{2}}{2 \max_{|\xi|\leq \theta_{2}} [-\sum_{k=1}^{m}F_{k}(\xi) ]}=\frac{4(a_{3}-2L) }{-\int_{0}^{4}f_{1}(x)\,\mathrm{d}x}. $$
Moreover, since \(\lim_{\xi\to 0^{+}}\frac{f_{1}(\xi)}{\xi}=0\), one has
$$\lim_{\xi\to0^{+}} \frac{ \int_{0}^{\xi}f_{1}(x)\,\mathrm{d}x}{\xi^{2}}=0. $$
Then there exists a positive constant \(\theta_{1}<\frac{1}{2}\) such that
$$\frac{ \int_{0}^{\theta_{1}}f_{1}(x)\,\mathrm{d}x}{\theta_{1}^{2}}> \frac{4}{3}\frac{a_{3}-2L}{a_{2}+2L} \int_{0}^{\frac{1}{2}}f_{1}(x)\,\mathrm{d}x $$
and
$$\frac{\theta_{1}^{2}}{ \int_{0}^{\theta _{1}}f_{1}(x)\,\mathrm{d}x}< \frac{8}{ \int_{0}^{4}f_{1}(x)\,\mathrm{d}x}. $$
Finally, an easy calculation shows that all hypotheses of Theorem 3.2 are fulfilled, and the conclusion follows. □

Remark 3.1

From Assumptions (A1), (A2), and (A3), we can show, by the same reasoning as given in Theorem 4 of [26], that the problem (1) when \(h\equiv0\) does not possess any non-zero weak solution in the cases where impulsive terms are zero. Consequently, the ensured weak solutions for the problem (1) when \(h\equiv0\) in Theorems 3.1 and 3.2 and in Corollary 3.3 are generated by impulses when impulsive terms \(f_{k},g_{k}\neq0\) for some \(1\leq k\leq m\), as well as for the problem (11) when \(h\equiv0\) in Corollary 3.4 are generated by impulses when impulsive terms \(f_{1},g_{1}\neq0\).

Remark 3.2

The methods used here can be applied studying discrete boundary value problems as in [52].

4 Concluding remarks

The theory of impulsive dynamic equations is generally thought to provide a natural framework for mathematical modeling of many real world phenomena such as chemotherapy, population dynamics, optimal control, ecology, industrial robotics, physics phenomena, etc. The impulsive effects can be broadly found in numerous evolution processes where their states may undergo abrupt changes at specific moments of time. As far as the second-order dynamic equations are concerned, we often take into account the impulses in terms of position and velocity. In the motion of spacecraft, on the contrary, we are supposed to consider instantaneous impulses depending on the position leading to jump discontinuities in velocity, but with no changes in terms of position. Impulsive problems such as problem (1) are considered as highly important for the description of quite a large number of real world phenomena including biology (biological phenomena involving thresholds), medicine (bursting rhythm models), pharmacokinetics, mechanics, and engineering. To this end, we have established, in this paper, the existence criteria of at least three solutions for the perturbed impulsive problem (1) based on variational methods and the critical point theory, under suitable hypotheses. The results of the study, finally, illustrated that these solutions are generated by impulses while \(h\equiv0\).

Declarations

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Sciences, Razi University
(2)
Department of Law and Economics, University Mediterranea of Reggio Calabria
(3)
Department of Economics, University of Messina

References

  1. Bainov, D, Simeonov, P: Systems with Impulse Effect. Ellis Horwood Series: Mathematics and Its Applications. Ellis Horwood, Chichester (1989) MATHGoogle Scholar
  2. Benchohra, M, Henderson, J, Ntouyas, S: Theory of Impulsive Differential Equations. Contemporary Mathematics and Its Applications, vol. 2. Hindawi Publishing Corporation, New York (2006) View ArticleMATHGoogle Scholar
  3. Carter, TE: Necessary and sufficient conditions for optimal impulsive rendezvous with linear equations of motion. Dyn. Control 10, 219-227 (2000) MathSciNetView ArticleMATHGoogle Scholar
  4. George, PK, Nandakumaran, AK, Arapostathis, A: A note on controllability of impulsive systems. J. Math. Anal. Appl. 241, 276-283 (2000) MathSciNetView ArticleMATHGoogle Scholar
  5. Haddad, WM, Chellaboina, C, Nersesov, SG, Sergey, G: Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity and Control. Princeton University Press, Princeton (2006) View ArticleMATHGoogle Scholar
  6. Lakshmikantham, V, Bainov, DD, Simeonov, PS: Impulsive Differential Equations and Inclusions. World Scientific, Singapore (1989) View ArticleGoogle Scholar
  7. Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations. Series in Modern Applied Mathematics, vol. 6. World Scientific, Teaneck (1989) View ArticleMATHGoogle Scholar
  8. Liu, X, Willms, AR: Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft. Math. Probl. Eng. 2, 277-299 (1996) View ArticleMATHGoogle Scholar
  9. Nieto, JJ, Rodríguez-López, R: Boundary value problems for a class of impulsive functional equations. Comput. Math. Appl. 55, 2715-2731 (2008) MathSciNetView ArticleMATHGoogle Scholar
  10. Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations. World Scientific, Singapore (1995) MATHGoogle Scholar
  11. Zavalishchin, ST, Sesekin, AN: Dynamic Impulse System: Theory and Applications. Kluwer Academic, Dordrecht (1997) View ArticleMATHGoogle Scholar
  12. Hernandez, E, Henriquez, HR, McKibben, MA: Existence results for abstract impulsive second-order neutral functional differential equations. Nonlinear Anal. TMA 70, 2736-2751 (2009) MathSciNetView ArticleMATHGoogle Scholar
  13. Li, J, Nieto, JJ, Shen, J: Impulsive periodic boundary value problems of first-order differential equations. J. Math. Anal. Appl. 303, 288-303 (2005) MathSciNetView ArticleGoogle Scholar
  14. Lin, XN, Jiang, DQ: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. J. Math. Anal. Appl. 321, 501-514 (2006) MathSciNetView ArticleMATHGoogle Scholar
  15. Wang, H, Chen, H: Boundary value problem for second-order impulsive functional differential equations. Appl. Math. Comput. 191, 582-591 (2007) MathSciNetMATHGoogle Scholar
  16. Bai, L, Dai, B: Application of variational method to a class of Dirichlet boundary value problems with impulsive effects. J. Franklin Inst. 348, 2607-2624 (2011) MathSciNetView ArticleMATHGoogle Scholar
  17. Bonanno, G, Di Bella, B, Henderson, J: Existence of solutions to second-order boundary-value problems with small perturbations of impulses. Electron. J. Differ. Equ. 2013, 126 (2013) MathSciNetView ArticleMATHGoogle Scholar
  18. Nieto, JJ, O’Regan, D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 10, 680-690 (2009) MathSciNetView ArticleMATHGoogle Scholar
  19. Sun, J, Chen, H: Variational method to the impulsive equation with Neumann boundary conditions. Bound. Value Probl. 2009, Article ID 316812 (2009) MathSciNetMATHGoogle Scholar
  20. Sun, J, Chen, H, Nieto, JJ, Otero-Novoa, M: The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects. Nonlinear Anal. TMA 72, 4575-4586 (2010) MathSciNetView ArticleMATHGoogle Scholar
  21. Teng, K, Zhang, C: Existence of solution to boundary value problem for impulsive differential equations. Nonlinear Anal., Real World Appl. 11(5), 4431-4441 (2010) MathSciNetView ArticleMATHGoogle Scholar
  22. Tian, Y, Ge, W: Applications of variational methods to boundary value problem for impulsive differential equations. Proc. Edinb. Math. Soc. 51, 509-527 (2008) MathSciNetView ArticleMATHGoogle Scholar
  23. Zhang, D, Dai, B: Existence of solutions for nonlinear impulsive differential equations with Dirichlet boundary conditions. Math. Comput. Model. 53, 1154-1161 (2011) MathSciNetView ArticleMATHGoogle Scholar
  24. Zhang, Z, Yuan, R: An application of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal., Real World Appl. 11, 155-162 (2010) MathSciNetView ArticleMATHGoogle Scholar
  25. Zhou, J, Li, Y: Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. Nonlinear Anal. TMA 71, 2856-2865 (2009) MathSciNetView ArticleMATHGoogle Scholar
  26. Zhang, H, Li, Z: Periodic and homoclinic solutions generated by impulses. Nonlinear Anal., Real World Appl. 12, 39-51 (2011) MathSciNetView ArticleMATHGoogle Scholar
  27. Yang, L, Chen, H: Existence and multiplicity of periodic solutions generated by impulses. Abstr. Appl. Anal. 2011, Article ID 310957 (2011) MathSciNetMATHGoogle Scholar
  28. Bonanno, G, Molica Bisci, G: Infinitely many solutions for a boundary value problem with discontinuous nonlinearities. Bound. Value Probl. 2009, Article ID 670675 (2009) MathSciNetView ArticleMATHGoogle Scholar
  29. Ricceri, B: A general variational principle and some of its applications. J. Comput. Appl. Math. 113, 401-410 (2000) MathSciNetView ArticleMATHGoogle Scholar
  30. Heidarkhani, S, Ferrara, M, Salari, A: Infinitely many periodic solutions for a class of perturbed second-order differential equations with impulses. Acta Appl. Math. 139, 81-94 (2015) MathSciNetView ArticleMATHGoogle Scholar
  31. Bonanno, G, Candito, P: Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities. J. Differ. Equ. 244, 3031-3059 (2008) MathSciNetView ArticleMATHGoogle Scholar
  32. Bonanno, G, Marano, SA: On the structure of the critical set of non-differentiable functions with a weak compactness condition. Appl. Anal. 89, 1-10 (2010) MathSciNetView ArticleMATHGoogle Scholar
  33. Bonanno, G, Chinnì, A: Existence of three solutions for a perturbed two-point boundary value problem. Appl. Math. Lett. 23, 807-811 (2010) MathSciNetView ArticleMATHGoogle Scholar
  34. D’Aguì, G, Heidarkhani, S, Molica Bisci, G: Multiple solutions for a perturbed mixed boundary value problem involving the one-dimensional p-Laplacian. Electron. J. Qual. Theory Differ. Equ. 2013, 24 (2013) MathSciNetView ArticleMATHGoogle Scholar
  35. Heidarkhani, S, Khademloo, S, Solimaninia, A: Multiple solutions for a perturbed fourth-order Kirchhoff type elliptic problem. Port. Math. 71(1), 39-61 (2014) MathSciNetView ArticleMATHGoogle Scholar
  36. Bonanno, G, D’Aguì, G: Multiplicity results for a perturbed elliptic Neumann problem. Abstr. Appl. Anal. 2010, Article ID 564363 (2010) MathSciNetView ArticleMATHGoogle Scholar
  37. Bonanno, G, Molica Bisci, G: Three weak solutions for elliptic Dirichlet problems. J. Math. Anal. Appl. 382, 1-8 (2011) MathSciNetView ArticleMATHGoogle Scholar
  38. Bonanno, G, Molica Bisci, G, Rădulescu, V: Existence of three solutions for a non-homogeneous Neumann problem through Orlicz-Sobolev spaces. Nonlinear Anal. TMA 74(14), 4785-4795 (2011) MathSciNetView ArticleMATHGoogle Scholar
  39. Bonanno, G, Molica Bisci, G, Rădulescu, V: Multiple solutions of generalized Yamabe equations on Riemannian manifolds and applications to Emden-Fowler problems. Nonlinear Anal., Real World Appl. 12, 2656-2665 (2011) MathSciNetView ArticleMATHGoogle Scholar
  40. Ferrara, M, Heidarkhani, S: Multiple solutions for perturbed p-Laplacian boundary value problem with impulsive effects. Electron. J. Differ. Equ. 2014, 106 (2014) MathSciNetView ArticleMATHGoogle Scholar
  41. Ferrara, M, Khademloo, S, Heidarkhani, S: Multiplicity results for perturbed fourth-order Kirchhoff type elliptic problems. Appl. Math. Comput. 234, 316-325 (2014) MathSciNetMATHGoogle Scholar
  42. Afrouzi, G, Hadjian, A, Rădulescu, V: Variational analysis for Dirichlet impulsive differential equations with oscillatory nonlinearity. Port. Math. 70(3), 225-242 (2013) MathSciNetView ArticleMATHGoogle Scholar
  43. Afrouzi, G, Hadjian, A, Rădulescu, V: Variational approach to fourth-order impulsive differential equations with two control parameters. Results Math. 65, 371-384 (2014) MathSciNetView ArticleMATHGoogle Scholar
  44. Cencelj, M, Repovš, D, Virk, Ž: Multiple perturbations of a singular eigenvalue problem. Nonlinear Anal. TMA 119, 37-45 (2015) MathSciNetView ArticleMATHGoogle Scholar
  45. Ferrara, M, Molica Bisci, G, Repovš, D: Existence results for nonlinear elliptic problems on fractal domains. Adv. Nonlinear Anal. 5, 75-84 (2016) MathSciNetMATHGoogle Scholar
  46. Rădulescu, V: Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic, and Variational Methods. Contemporary Mathematics and Its Applications, vol. 6. Hindawi Publishing Corporation, New York (2008) View ArticleMATHGoogle Scholar
  47. Rădulescu, V, Repovš, D: Combined effects in nonlinear problems arising in the study of anisotropic continuous media. Nonlinear Anal. TMA 75, 1524-1530 (2012) MathSciNetView ArticleMATHGoogle Scholar
  48. Repovš, D: Stationary waves of Schrödinger-type equations with variable exponent. Anal. Appl. 13(6), 645-661 (2015) MathSciNetView ArticleMATHGoogle Scholar
  49. Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems. Springer, Berlin (1989) View ArticleMATHGoogle Scholar
  50. Zeidler, E: Nonlinear Functional Analysis and Its Applications, vol. II. Springer, Berlin (1985) View ArticleMATHGoogle Scholar
  51. Drábek, P, Milota, J: Methods of Nonlinear Analysis: Applications to Differential Equations. Birkhäuser, Basel (2007) MATHGoogle Scholar
  52. Candito, P, Molica Bisci, G: Existence of two solutions for a second-order discrete boundary value problem. Adv. Nonlinear Stud. 11, 443-453 (2011) MathSciNetView ArticleMATHGoogle Scholar

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