Multiple solutions for a class of perturbed secondorder differential equations with impulses
 Shapour Heidarkhani^{1}Email author,
 Massimiliano Ferrara^{2},
 Amjad Salari^{1} and
 Giuseppe Caristi^{3}
https://doi.org/10.1186/s136610160581z
© Heidarkhani et al. 2016
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 methods1 Introduction
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 [1–11].
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, [12–15] and references therein). Recently, the existence and multiplicity of solutions for impulsive problems have been thoroughly investigated by [16–25] 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 secondorder 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 secondorder 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 nonzero periodic solution as well as one nonzero 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 nonzero, two nonzeros, 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 nonzero weak solution when \(f_{k}=g_{k}\equiv0\) for all \(1\leq k\leq m\), then a nonzero 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 [33–35] to ensure the presence of at least three solutions for the perturbed boundary value problems.
The curious reader is also referred to [36–41], 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 [42–48].
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 \(\Phi(0)=\Psi(0)=0\).
 (a_{1}):

\(\frac{\sup_{ \Phi(u)\leq r}\Psi(u)}{r}< \frac{\Psi(\bar {v})}{\Phi(\bar {v})}\);
 (a_{2}):

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.
Theorem 2.2
([31], Theorem 3.3)
 1.
\(\inf_{X}\Phi=\Phi(0)=\Psi(0)=0\);
 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$$\inf_{s\in[0,1]}\Psi \bigl(su_{1}+(1s)u_{2} \bigr)\geq0. $$
 (b_{1}):

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

\(\frac{\sup_{u\in\Phi^{1}(]\infty ,r_{2}[)}\Psi(u)}{r_{2}}< \frac{1}{3}\frac{\Psi(\bar {v})}{\Phi(\bar {v})}\).
 (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 \((t,\xi)\in[0,T]\times {\mathbb{R}}^{N}\);
 (A2)
\(V(t,\xi)\leq(V_{\xi}(t,\xi),\xi)\leq2V(t,\xi )\) for all \((t,\xi)\in[0.T]\times{\mathbb{R}}^{N}\);
 (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 require the proposition below in proving Theorem 3.1.
Proposition 2.3
Proof
3 Main results
In this section, we show our main results of the existence of at least three weak solutions for the problem (1).
Theorem 3.1
 (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 \(a_{3}=\min\{\frac {1}{2},a_{1}\}\);
 (A5)
\(\limsup_{t\to+\infty }\frac{\sum_{k=1}^{m}[F_{k}(t)]}{t^{2}}\leq0\).
Proof
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.
By the above symbolization, we obtain the following multiplicity result.
Theorem 3.2
Proof
In the following, we present a special case of Theorem 3.1.
Corollary 3.3
Proof
Now, as an example, we present the following consequence of Theorem 3.2 with \(m=T=N=1\).
Corollary 3.4
Proof
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 nonzero 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 secondorder 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
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