 Research
 Open Access
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 methods
1 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
References
 Bainov, D, Simeonov, P: Systems with Impulse Effect. Ellis Horwood Series: Mathematics and Its Applications. Ellis Horwood, Chichester (1989) MATHGoogle Scholar
 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
 Carter, TE: Necessary and sufficient conditions for optimal impulsive rendezvous with linear equations of motion. Dyn. Control 10, 219227 (2000) MathSciNetView ArticleMATHGoogle Scholar
 George, PK, Nandakumaran, AK, Arapostathis, A: A note on controllability of impulsive systems. J. Math. Anal. Appl. 241, 276283 (2000) MathSciNetView ArticleMATHGoogle Scholar
 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
 Lakshmikantham, V, Bainov, DD, Simeonov, PS: Impulsive Differential Equations and Inclusions. World Scientific, Singapore (1989) View ArticleGoogle Scholar
 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
 Liu, X, Willms, AR: Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft. Math. Probl. Eng. 2, 277299 (1996) View ArticleMATHGoogle Scholar
 Nieto, JJ, RodríguezLópez, R: Boundary value problems for a class of impulsive functional equations. Comput. Math. Appl. 55, 27152731 (2008) MathSciNetView ArticleMATHGoogle Scholar
 Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations. World Scientific, Singapore (1995) MATHGoogle Scholar
 Zavalishchin, ST, Sesekin, AN: Dynamic Impulse System: Theory and Applications. Kluwer Academic, Dordrecht (1997) View ArticleMATHGoogle Scholar
 Hernandez, E, Henriquez, HR, McKibben, MA: Existence results for abstract impulsive secondorder neutral functional differential equations. Nonlinear Anal. TMA 70, 27362751 (2009) MathSciNetView ArticleMATHGoogle Scholar
 Li, J, Nieto, JJ, Shen, J: Impulsive periodic boundary value problems of firstorder differential equations. J. Math. Anal. Appl. 303, 288303 (2005) MathSciNetView ArticleGoogle Scholar
 Lin, XN, Jiang, DQ: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. J. Math. Anal. Appl. 321, 501514 (2006) MathSciNetView ArticleMATHGoogle Scholar
 Wang, H, Chen, H: Boundary value problem for secondorder impulsive functional differential equations. Appl. Math. Comput. 191, 582591 (2007) MathSciNetMATHGoogle Scholar
 Bai, L, Dai, B: Application of variational method to a class of Dirichlet boundary value problems with impulsive effects. J. Franklin Inst. 348, 26072624 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Bonanno, G, Di Bella, B, Henderson, J: Existence of solutions to secondorder boundaryvalue problems with small perturbations of impulses. Electron. J. Differ. Equ. 2013, 126 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Nieto, JJ, O’Regan, D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 10, 680690 (2009) MathSciNetView ArticleMATHGoogle Scholar
 Sun, J, Chen, H: Variational method to the impulsive equation with Neumann boundary conditions. Bound. Value Probl. 2009, Article ID 316812 (2009) MathSciNetMATHGoogle Scholar
 Sun, J, Chen, H, Nieto, JJ, OteroNovoa, M: The multiplicity of solutions for perturbed secondorder Hamiltonian systems with impulsive effects. Nonlinear Anal. TMA 72, 45754586 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Teng, K, Zhang, C: Existence of solution to boundary value problem for impulsive differential equations. Nonlinear Anal., Real World Appl. 11(5), 44314441 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Tian, Y, Ge, W: Applications of variational methods to boundary value problem for impulsive differential equations. Proc. Edinb. Math. Soc. 51, 509527 (2008) MathSciNetView ArticleMATHGoogle Scholar
 Zhang, D, Dai, B: Existence of solutions for nonlinear impulsive differential equations with Dirichlet boundary conditions. Math. Comput. Model. 53, 11541161 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Zhang, Z, Yuan, R: An application of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal., Real World Appl. 11, 155162 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Zhou, J, Li, Y: Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. Nonlinear Anal. TMA 71, 28562865 (2009) MathSciNetView ArticleMATHGoogle Scholar
 Zhang, H, Li, Z: Periodic and homoclinic solutions generated by impulses. Nonlinear Anal., Real World Appl. 12, 3951 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Yang, L, Chen, H: Existence and multiplicity of periodic solutions generated by impulses. Abstr. Appl. Anal. 2011, Article ID 310957 (2011) MathSciNetMATHGoogle Scholar
 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
 Ricceri, B: A general variational principle and some of its applications. J. Comput. Appl. Math. 113, 401410 (2000) MathSciNetView ArticleMATHGoogle Scholar
 Heidarkhani, S, Ferrara, M, Salari, A: Infinitely many periodic solutions for a class of perturbed secondorder differential equations with impulses. Acta Appl. Math. 139, 8194 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Bonanno, G, Candito, P: Nondifferentiable functionals and applications to elliptic problems with discontinuous nonlinearities. J. Differ. Equ. 244, 30313059 (2008) MathSciNetView ArticleMATHGoogle Scholar
 Bonanno, G, Marano, SA: On the structure of the critical set of nondifferentiable functions with a weak compactness condition. Appl. Anal. 89, 110 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Bonanno, G, Chinnì, A: Existence of three solutions for a perturbed twopoint boundary value problem. Appl. Math. Lett. 23, 807811 (2010) MathSciNetView ArticleMATHGoogle Scholar
 D’Aguì, G, Heidarkhani, S, Molica Bisci, G: Multiple solutions for a perturbed mixed boundary value problem involving the onedimensional pLaplacian. Electron. J. Qual. Theory Differ. Equ. 2013, 24 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Heidarkhani, S, Khademloo, S, Solimaninia, A: Multiple solutions for a perturbed fourthorder Kirchhoff type elliptic problem. Port. Math. 71(1), 3961 (2014) MathSciNetView ArticleMATHGoogle Scholar
 Bonanno, G, D’Aguì, G: Multiplicity results for a perturbed elliptic Neumann problem. Abstr. Appl. Anal. 2010, Article ID 564363 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Bonanno, G, Molica Bisci, G: Three weak solutions for elliptic Dirichlet problems. J. Math. Anal. Appl. 382, 18 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Bonanno, G, Molica Bisci, G, Rădulescu, V: Existence of three solutions for a nonhomogeneous Neumann problem through OrliczSobolev spaces. Nonlinear Anal. TMA 74(14), 47854795 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Bonanno, G, Molica Bisci, G, Rădulescu, V: Multiple solutions of generalized Yamabe equations on Riemannian manifolds and applications to EmdenFowler problems. Nonlinear Anal., Real World Appl. 12, 26562665 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Ferrara, M, Heidarkhani, S: Multiple solutions for perturbed pLaplacian boundary value problem with impulsive effects. Electron. J. Differ. Equ. 2014, 106 (2014) MathSciNetView ArticleMATHGoogle Scholar
 Ferrara, M, Khademloo, S, Heidarkhani, S: Multiplicity results for perturbed fourthorder Kirchhoff type elliptic problems. Appl. Math. Comput. 234, 316325 (2014) MathSciNetMATHGoogle Scholar
 Afrouzi, G, Hadjian, A, Rădulescu, V: Variational analysis for Dirichlet impulsive differential equations with oscillatory nonlinearity. Port. Math. 70(3), 225242 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Afrouzi, G, Hadjian, A, Rădulescu, V: Variational approach to fourthorder impulsive differential equations with two control parameters. Results Math. 65, 371384 (2014) MathSciNetView ArticleMATHGoogle Scholar
 Cencelj, M, Repovš, D, Virk, Ž: Multiple perturbations of a singular eigenvalue problem. Nonlinear Anal. TMA 119, 3745 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Ferrara, M, Molica Bisci, G, Repovš, D: Existence results for nonlinear elliptic problems on fractal domains. Adv. Nonlinear Anal. 5, 7584 (2016) MathSciNetMATHGoogle Scholar
 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
 Rădulescu, V, Repovš, D: Combined effects in nonlinear problems arising in the study of anisotropic continuous media. Nonlinear Anal. TMA 75, 15241530 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Repovš, D: Stationary waves of Schrödingertype equations with variable exponent. Anal. Appl. 13(6), 645661 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems. Springer, Berlin (1989) View ArticleMATHGoogle Scholar
 Zeidler, E: Nonlinear Functional Analysis and Its Applications, vol. II. Springer, Berlin (1985) View ArticleMATHGoogle Scholar
 Drábek, P, Milota, J: Methods of Nonlinear Analysis: Applications to Differential Equations. Birkhäuser, Basel (2007) MATHGoogle Scholar
 Candito, P, Molica Bisci, G: Existence of two solutions for a secondorder discrete boundary value problem. Adv. Nonlinear Stud. 11, 443453 (2011) MathSciNetView ArticleMATHGoogle Scholar