Infinitely Many Periodic Solutions for a Class of Perturbed Second-Order Differential Equations with Impulses

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1 Acta Appl Math DOI.7/s Infinitely Many Periodic Solutions for a Class of Perturbed Second-Order Differential Equations with Impulses Shapour Heidarkhani Massimiliano Ferrara Amjad Salari Received: 8 March 4 / Accepted: September 4 Springer Science+Business Media Dordrecht 4 Abstract We investigate the existence of infinitely many periodic solutions generated by impulses for a class of perturbed second-order impulsive differential equations. Our approach is based on variational methods and critical point theory. Keywords Periodic solutions Perturbed impulsive differential equation Critical point theory Variational methods Introduction The aim of this paper is to investigate the existence of infinitely many periodic solutions generated by impulses for the following perturbed problem ü(t) + V u (t, u(t)) =, t (s k,s k ), u(s k ) = f k (u(s k )) + μg k (u(s k )), () u() u(t ) = u() u(t ) =, where s k, k =,,...,m, are instants in which the impulses occur and = s <s < <s m <s m+ = T, u(s k ) = u(s k + ) u(s k ) with u(s k ± ) = lim t sk ± u(t), f k (ξ) = grad ξ F k (ξ), g k (ξ) = grad ξ G k (ξ), F k,g k C (R N, R) such that F k () = G k () =, V C ([,T] R N, R), V ξ (t, ξ) = grad ξ V(t,ξ), >andμ are two parameters. The theory and applications of impulsive functional differential equations are emerging as an important area of investigation, since it is far richer than the corresponding theory of non-impulsive functional differential equations. Various population models, which S. Heidarkhani A. Salari Department of Mathematics, Faculty of Sciences, Razi University, 6749 Kermanshah, Iran S. Heidarkhani s.heidarkhani@razi.ac.ir M. Ferrara (B) Department of Law and Economics, University Mediterranea of Reggio Calabria, Via dei Bianchi,, 893 Reggio Calabria, Italy massimiliano.ferrara@unirc.it

2 S. Heidarkhani et al. are characterized by the fact that per sudden changing of their state and process under depends on their prehistory at each moment of time, can be expressed by impulsive differential equations with deviating argument, as population dynamics, ecology and epidemic, etc. We note that the difficulties dealing with such models are that such equations have deviating arguments and theirs states are discontinuous. In the last decades, impulsive differential equations have become more important in some mathematical models of real processes and phenomena studied in spacecraft control, impact mechanics, physics, chemistry, chemical engineering, population dynamics, biotechnology, economics and inspection process in operations research. It is now recognized that the theory of impulsive differential equations is a natural framework for a mathematical modelling of many natural phenomena. For the background, theory and applications of impulsive differential equations, we refer the interest readers to [4, 5,, 4, 7, 9,, 3, 6, 9, 3, 3, 39]. Impulsive differential equations have been studied extensively in the literature. There have been many approaches to study the existence of solutions of impulsive differential equations, such as fixed point theory, topological degree theory (including continuation method and coincidence degree theory) and comparison method (including upper and lower solution methods and monotone iterative method) and so on (see, for example, [,, 3,, 4, 5, 37] and references therein). Recently, in [3,, 8, 33 36, 4, 4, 43] usingvariational methods and critical point theory studied the existence and multiplicity of solutions of impulsive problems. All the results of [3,, 8, 33 36, 4, 4, 43] can be seen as generalizations of corresponding ones for second order ordinary differential equations. In other words, these results can be applied to impulsive systems when the impulses are absent and still give the existence of solutions in this situation. This, in some sense, means that the nonlinear term V u plays a more important role than the impulsive terms f k do in guaranteeing the existence of solutions in these results. In [4] the authors studied the existence of periodic and homoclinic solutions for a class of second order differential equations of the form () whenμ = via variational methods, and they showed that under appropriate conditions such a system possesses at least one non-zero periodic solution and at least one non-zero homoclinic solution and these solutions are generated by impulses when f =. In [38] the authors, based on variational methods and critical point theory studied the problem ()when μ =, and proved that such a problem admits at least one nonzero, two zero, or infinitely many periodic solutions generated by impulses under different assumptions, respectively. In the present paper, motivated by [38], employing a smooth version of [7, Theorem.] which is a more precise version of Ricceri s Variational Principle [3, Theorem.5] under some hypotheses on the behavior of the nonlinear terms at infinity, under conditions on the potentials of f and g we prove the existence of a definite interval about in which the problem () admits a sequence of solutions generated by impulsive which is unbounded in the space HT which will be introduced later (Theorem 3.). We also list some consequences of Theorem 3.. Replacing the conditions at infinity of the nonlinear terms, by a similar one at zero, the same results hold; see Theorem 3.4. Two examples of applications are pointed out (see Examples 3. and 3.). Definition. (see [4]) A solution of the problem () is called a solution generated by impulses if this solution is nontrivial when impulsive terms g k,f k forsome k m, but it is trivial when impulsive terms g k = f k forall k m. For example, if the problem () does not possess non-zero weak solution when g k = f k forall k m, then a non-zero weak solution for problem () with g k,f k for some k m is called a weak solution generated by impulses. For a through on the subject, we refer the reader to [6, 8, 5, 6, ].

3 Infinitely Many Periodic Solutions for a Class Preliminaries Our main tool to investigate the existence of infinitely many solutions generated by impulsive for the problem () is a smooth version of Theorem. of [7] which is a more precise version of Ricceri s Variational Principle [3, Theorem.5] that we now recall here. Theorem. Let X be a reflexive real Banach space, let Φ,Ψ : X R be two Gâteaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous, strongly continuous, and coercive and Ψ is sequentially weakly upper semicontinuous. For every r>inf X Φ, let us put and Then, one has ϕ(r) := inf u Φ (],r[) sup v Φ (],r]) Ψ(v) Ψ(u) r Φ(u) γ := lim inf ϕ(r), δ := lim inf ϕ(r). r + r (inf X Φ) + (a) for every r>inf X Φ and every ], ϕ(r) [, the restriction of the functional I = Φ Ψ to Φ (],r[) admits a global minimum, which is a critical point (local minimum) of I in X. (b) If γ<+ then, for each ], [, the following alternative holds: γ either (b ) I possesses a global minimum, or (b ) there is a sequence {u n } of critical points (local minima) of I such that lim Φ(u n) =+. n + (c) If δ<+ then, for each ], [, the following alternative holds: δ either (c ) there is a global minimum of Φ which is a local minimum of I, or (c ) there is a sequence of pairwise distinct critical points (local minima) of I which weakly converges to a global minimum of Φ. Let H T = { u :[,T] R N u is absolutely continuous, u() = u(t ), u L ( [,T], R N)}. Then H T is a Hilbert space with the inner product u, v = T [( u(t), v(t) ) + ( u(t), v(t) )] dt, u, v H T,

4 S. Heidarkhani et al. where (.,.) denotes the inner product in R N. The corresponding norm is defined by ( T ( u = u(t) + u(t) ) ) dt u HT. Obviously, HT is a separable and uniformly convex Banach space. Since (HT,. ) is compactly embedded in C([,T], RN ) (see [7]), there exists a positive constant C such that u C u, () where u = max t [,T ] u(t). We mean by a (weak) solution of the problem (), any u X such that T ( ( ) ) m ( u(t) v(t) Vu t,u(t) v(t) dt + f k u(sk ) ) m ( v(s k ) + μ g k u(sk ) ) v(s k ) = for every v X. To state our results concisely we introduce the following assumptions: (A) V is continuous differentiable and there exist positive constants a,a > such that a ξ V(t,ξ) a ξ for all (t, ξ) [,T] R N ; (A) V(t,ξ) V ξ (t, ξ)ξ V(t,ξ)for all (t, ξ) [.T ] R N. A special case of our main result is the following theorem. Theorem. Assume that Assumptions (A) and (A) hold. Let F k C (R N, R) and denote f k (ξ) = grad ξ F k (ξ) for every ξ R N for k =,,...,m. Furthermore, suppose that lim inf F k(t)] = and lim sup F k(ξ) =+. ξ + ξ + Then, the problem ü(x) + V u (t, u(t)) =, t (s k,s k ), u(s k ) = f k (u(s k )), u() u(t ) = u() u(t ) =, has an unbounded sequence of periodic solutions generated by impulses. 3 Main Results We formulate our main result as follows. Theorem 3. Assume that Assumptions (A) and (A) hold. Furthermore, suppose that (A3) lim inf ξ + F k (t)] < a 3 Ta lim sup ξ + Then, for each ], [ where := lim sup ξ + F k (ξ) Ta m F k (ξ).

5 Infinitely Many Periodic Solutions for a Class and := lim inf ξ + F k (t)] for every arbitrary negative function G k C (R N, R) such that G k () =, denoting g k (ξ) = grad ξ G k (ξ) for every ξ R N for k =,,...,m, satisfying the condition g := lim ξ G k(t)] < + (3) and for every μ [,μ g, [ where μ g, := max g ( lim inf t ξ [ m F k (t)] ξ + ), the C problem () has an unbounded sequence of periodic solutions generated by impulses. Proof Our goal is to apply Theorem.. Fix ], [ and let G k, k =,...,m be functions satisfying the condition (3). Since, <, one has μ g, >. Fix μ [,μ g, [ and put ν := and ν :=.Ifg + μ =, clearly, ν =, ν = and ]ν,ν [.Ifg, g since μ<μ g,, we obtain +μg <, and so >, namely, <ν + μ. Hence, since g > = ν, one has ]ν,ν [. Now, put Q k (ξ) = F k (ξ) + μ G k(ξ) for all ξ R N and k =,...,m.takex = HT and consider the functionals Φ, Ψ : X R for each u X, as follows T [ Φ(u) = u(t) V ( t,u(t) )] dt and ( m ( Ψ(u)= F k u(sk ) ) + μ m ( G k u(sk ) )). It is well known that Ψ is a Gâteaux differentiable functional and sequentially weakly upper semicontinuous whose Gâteaux derivative at the point u X is the functional Ψ (u) X, given by ( m ( Ψ (u)v = f k u(sk ) ) v(s k ) + μ m ( g k u(sk ) ) ) v(s k ), and Ψ : X X is a compact operator. Moreover, Φ is a Gâteaux differentiable functional whose Gâteaux derivative at the point u X is the functional Φ (u) X,givenby T ( ( ) ) Φ (u)v = u(t) v(t) Vu t,u(t) v(t) dt for every v X. Furthermore, Φ is sequentially weakly lower semicontinuous. Indeed, let u n X with u n u weakly in X, taking weakly lower semicontinuity of the norm into account, we have lim inf n + u n u and u n u uniformly on [,T]. Hence, since V is continuous, we have lim n + T [ u n (t) V ( t,u n (t) )] dt T, [ u(t) V ( t,u(t) )] dt,

6 S. Heidarkhani et al. that is lim inf n + Φ(u n ) Φ(u) which means Φ is sequentially weakly lower semicontinuous. By Assumption (A) we have a 3 u Φ(u) a 4 u, (4) where a 3 = min{,a } and a 4 = min{,a }. PutI := Φ Ψ. Similar to the proof of Lemma of [4], we observe that the weak solutions of the problem () are exactly the solutions of the equation I (u) =. Now, we want to verify that γ<+, whereγ is defined in Theorem.. Let{ξ n } be a real sequence such that ξ n + as n and max t ξn [ lim Q k(t)] = lim inf Q k(t)]. n ξn ξ + Put r n = ξ n a 3 for all n N.Taking() into account that, from (4)wehaveΦ (],r n ]) {u X; u ξ n }. Hence, taking Φ() = Ψ() = into account, for every n large enough, one has ϕ(r n ) = inf u Φ (],r n[) max t ξ n [ Q k(t)] = ξn a 3 max t ξ n [ F k(t)] + ξn a 3 Moreover, it follows from Assumption (A3) that (sup v Φ (],r n]) Ψ(v)) Ψ(u) sup v Φ (],r Ψ(v) n]) r n Φ(u) r n max t ξn m [ F k(t) μ G k(t)] μ n a 3 max t ξn [ G k(t)] n a 3. lim inf F k(t)] < +, ξ + so we obtain max t ξn [ lim F k(t)] < +. (5) n ξn Then, in view of (3) and(5), we have lim n max t ξn [ F k(t)] n a 3 + lim n μ max t ξn [ G k(t)] n a 3 < +, which follows Therefore, max m t ξn lim [ F k(t) μ G k(t)] < +. n ξn a 3 γ lim inf max m t ξn n) lim k(t) μ G k(t)] n + n ξn a 3 < +. (6)

7 Infinitely Many Periodic Solutions for a Class Since max t ξn [ Q k(t)] n a 3 taking (3) into account, one has lim inf ξ + max t ξn [ F k(t)] Q k(t)] lim inf ξ + n a 3 + Moreover, since G k for k =,,...,mis negative, we have μ max u ξn [ G k(t)] n a 3, F k(t)] μ + g. (7) lim sup Q k(ξ) lim sup F k(ξ). (8) ξ + Ta ξ + Ta Therefore, from (7)and(8), and from Assumption (A3) and (6) we observe ] ]ν,ν [, lim sup Q k (ξ) ξ + Ta lim inf ξ + Q k (t)] [ ], [. γ For the fixed, the inequality (6) concludes that the condition (b) of Theorem. can be applied and either I has a global minimum or there exists a sequence {u n } of weak solutions of the problem () such that lim n u =+. The other step is to show that for the fixed the functional I has no global minimum. Let us verify that the functional I is unbounded from below. Since < lim sup F k(ξ), ξ + Ta we can consider a real sequence {d n } and a positive constant τ such that d n + as n and <τ< F k(d n ) (9) Ta dn for each n N large enough. Let {w n } be a sequence in X defined by w n (t) = d n. () For any fixed n N, w n X and w n = Tdn, and in particular, taking (4) into account, we T [ Φ(w n ) = ẇn (t) V ( t,w n (t) )] T [ ( dt = V t,wn (t) )] dt a Tdn. () On the other hand, since G k for k =,,...,m is negative, from the definition of Ψ,we infer m Ψ(w n ) F k (d n ). ()

8 S. Heidarkhani et al. So, according to (9), ()and() we obtain I (w n ) = Φ(w n ) Ψ (w n ) a Td n ( ) m F k (d n ) <( τ)a Tdn for every n N large enough. Hence, the functional I is unbounded from below, and it follows that I has no global minimum. Therefore, applying Theorem. we deduce that there is a sequence {u n } X of critical points of I such that lim n Φ(u n ) =+,which from (4) it follows that lim n u n =+. From Assumptions (A) and (A), by the same arguing as given in [4, Theorem 4] we observe that the problem () does not admit any nonzero periodic solution when impulses are zero. Hence, by Definition., the problem () admits an unbounded sequence of periodic solutions generated by impulses. Remark 3. Theorem 3. gives back to Theorem.7 of [38] whenμ =. Remark 3. Under the conditions lim inf F k(t)] = and lim sup F k(ξ) =+, ξ + ξ + Theorem 3. concludes that for every > and for each μ [, g [ the problem () admits infinitely many periodic solutions generated by impulses. Moreover, if g =, the result holds for every >andμ. We now exhibit an example in which the hypotheses of Theorem 3. are satisfied. Example 3. Let m = T = N =, V(t,ξ) = ξ(ξ + arctan(ξ)) for every (t, ξ) [,T] R. By the same reasoning as given in Example of [4] we see that the function V satisfies Assumptions (A) and (A). Now, let {a n } and {b n } be sequences defined by b =, b n+ = bn 6 and a n = bn 4 for n N.Let b 3 ( t) + ift [,b ], (a f (ξ) = n bn 3) (a n t) + ift n= [a n,a n ], (bn+ 3 a n) (b n+ t) + ift n= [b n+,b n+ ], otherwise. Let F (ξ) = ξ f (x)dx for all ξ R. Then, F is a C function with F = f. From the computation of Example 3. of [8], we have F (a n ) = π a n + a n and F (b n ) = π b3 n + b F n, and so lim (a n) F n = and lim (b n) an n =. Therefore, bn F lim inf (ξ) F ξ = and lim sup (ξ) ξ =. Hence, using Theorem 3., the problem (), in this case, with g (ξ) = e ξ + (ξ + ) γ where ξ + = max{ξ,} and γ is a positive real number, putting G (ξ) = ξ g (x)dx for all ξ R,forevery(, μ) ], + [ [, + [ admits infinitely many periodic solutions generated by impulses. Remark 3.3 We explicitly observe that Assumption (A3) in Theorem 3. could be replaced by the following more general condition

9 Infinitely Many Periodic Solutions for a Class (A3 ) there exist two sequence {θ n } and {η n } with η n > a a 3 TCθ n for every n N and lim n + η n =+ such that lim n + max t ηn [ F k(t)] ( F k(θ n )) a 3 η n a Tθn < lim sup F k(ξ). ξ + a T Obviously, from (A3 ) we obtain (A3), by choosing θ n = foralln N. Moreover, if we assume (A3 ) instead of (A3) and set r n = a 3 η n for all n N, by the same arguing as inside in Theorem 3., we obtain ϕ(r n ) sup v Φ (],r Ψ(v) ( n]) F k(w n )) r n T [ ẇ n(t) V(t,w n (t))]dt max t η n [ F k(t)] ( F k(θ n )) a 3 η n a Tθn where w n (t) = θ n for every t [,T]. We have the same conclusion as in Theorem 3. with the interval ], [ replaced by the interval by ], lim sup F k (ξ) ξ + a T lim n + max t ηn [ F k (t)] ( F k (θ n)) a 3 η n a Tθ n Here, we point out a simple consequence of Theorem 3.. Corollary 3. Assume that Assumptions (A) and (A) hold. Furthermore, suppose that max (B) lim inf t ξ [ F k (t)] ξ + < a 3 ; (B) lim sup F k (ξ) ξ + >Ta. Then, for every arbitrary negative function G k C (R N, R) such that G k () =, denoting g k (ξ) = grad ξ G k (ξ) for every ξ R N for k =,,...,m, satisfying the condition (3) and for every μ [,μ g, [ where μ g, := ( lim inf g ξ + F k(t)] ), [. the problem ü(t) + V u (t, u(t)) =, t (s k,s k ), u(s k ) = f k (u(s k )) + μg k (u(s k )), u() u(t ) = u() u(t ) =, has an unbounded sequence of periodic solutions generated by impulses. Remark 3.4 Theorem. is an immediately consequence of Corollary 3. when μ =. We here give the following consequence of the main result.

10 Corollary 3.3 Let Fk C (R N, R) such that Fk all ξ R N for k =,,...,m. Assume that S. Heidarkhani et al. () = and let fk (ξ) = grad ξ F k (ξ) for max (D) lim inf t ξ [ Fk (t)] ξ + < + ; (D) lim sup Fk (ξ) ξ + =+. Then, for every Fk i C (R N, R) such that Fk i i () =, denoting fk (ξ) = grad ξ F k i (ξ) for all ξ R N for k =,,...,mfor i n, satisfying and for each { max sup ξ R { min lim inf ξ + ], ( F i k (ξ) ) ; i n} ( Fk i(ξ)) } ; i n >, a 3 lim inf ξ + ( F k (ξ)) for every arbitrary negative function G k C (R N, R) such that G k () =, denoting g k (ξ) = grad ξ G k (ξ) for every ξ R N for k =,,...,m, satisfying the condition (3) and for every μ [,μ g, [ where the problem μ g, := ( lim inf g ξ + [, F k (t)] ü(t) + V u (t, u(t)) =, t (s k,s k ), u(s k ) = n i= f k i(u(s k)) + μg k (u(s k )), u() u(t ) = u() u(t ) = has an unbounded sequence of periodic solutions generated by impulses. Proof Set F k (ξ) = n i= F k i(ξ) for all ξ RN. Assumption (D) along with the condition { min lim inf ξ + Fk i(ξ) } ; i n > ensures lim sup F k(ξ) n i= = lim sup F k i(ξ) =+. ξ + ξ + Moreover, Assumption (D) in conjunction with the condition { max sup ξ R F i k (ξ); i n } ),

11 Infinitely Many Periodic Solutions for a Class yields lim inf F k(t)] lim inf ξ + ξ + Hence, applying Theorem 3. we have the desired result. F k (t)] < +. Arguing as in the proof of Theorem 3., but using conclusion (c) of Theorem. instead of (b), the following result holds. Theorem 3.4 Assume that Assumptions (A) and (A) hold. Furthermore, suppose that (E) lim inf F k (t)] ξ + < a 3 lim sup F k (ξ) Ta ξ +. Then, for each ] 3, 4 [ where 3 := lim sup ξ + and 4 := F k (ξ) Ta lim inf ξ + F k (t)] for every arbitrary negative function G k C (R N, R) such that G k () =, denoting g k (ξ) = grad ξ G k (ξ) for every ξ R N for k =,,...,m, satisfying the condition max t ξ [ m g := lim G k(t)] < + (3) ξ + and for every μ [,μ g, [ where μ g, := g ( lim inf F k (t)] ξ + ), the problem () has an unbounded sequence of pairwise distinct periodic solutions generated by C impulses. Proof Fix ] 3, 4 [ and let G k for k =,,...,mbe functions satisfy the condition (3). Since, <, one has μ g, := max t ξ [ m ( lim inf F ) k(t)] >. g ξ + Fix μ ],μ g, [ and put ν 3 := 3 and ν 4 :=.Ifg + C μ =, clearly, ν 3 = 3, ν 4 = 4 a 3 4G and ]ν 3,ν 4 [.Ifg, since μ<μ g,, we obtain and so C a 3 μg <, 4 + C a 3 μ 4g >, namely, <ν 4. Hence, being in mind that > 3 = ν 3, one has ]ν 3,ν 4 [.,

12 S. Heidarkhani et al. Now, set Q k (ξ) = F k (ξ) + μ G k(ξ) for all ξ R N and k =,,...,m.since Q k(t)] F k(t)] + taking (3) into account, one has max t ξ [ m lim inf Q k(t)] lim inf ξ + μ max t ξ [ m F k(t)] ξ + G k(t)], + μ g. (4) Moreover, since G k is negative for k =,,...,m,fromassumption(e)weobtain m lim sup Q k(ξ) m lim sup F k(ξ). (5) ξ + ξ + Therefore, from (4) and(5), we observe ] ]ν 3,ν 4 [ lim sup ξ +, Q k (ξ) Ta lim inf ξ + Q k (t)] [ ] 3, 4 [. We take X, Φ, Ψ and I as in the proof of Theorem 3.. We observe that δ<+. Indeed, let {ξ n } be a sequence of positive numbers such that ξ n + as n + and max t ξn [ lim F k(t)] < +. n ξn Put r n = a 3ξn for all n N. By the same arguing as in the proof of Theorem 3., it follows C that δ<+. Let us verify that the functional I has not a local minimum at zero. For this, let {d n } be a sequence of positive numbers and τ>such that d n + as n and <τ< F k(d n ) Ta dn (6) for each n N large enough. Let {w n } be a sequence in X defined by w n (t) = d n.so, according to (), () and(6) we obtain ) m I (w n ) = Φ(w n ) Ψ (w n ) a Tdn ( F k (d n ) <( τ)a Tdn < for every n N large enough. Since I () =, this concludes that the functional I has not a local minimum at zero. Hence, the part (c) of Theorem. follows that there exists a sequence {u n } in X of critical points of I such that u n asn. From Assumptions (A) and (A), by the same arguing as given in [4, Theorem 4] we observe that the problem () does not admit any nonzero periodic solution when impulses are zero. Hence, by Definition., the problem () admits an unbounded sequence of pairwise distinct periodic solutions generated by impulses. Remark 3.5 Applying Theorem 3.4, results similar to Corollaries 3. and 3.3 can be obtained. We leave their formulation to the reader.

13 Infinitely Many Periodic Solutions for a Class We end this paper by presenting the following example in which the hypotheses of Theorem 3.4 are satisfied. Example 3. Let m = T = N = andv(t,ξ)= ξ for every (t, ξ) [,T] R. We have C =, a = a 3 = in this case. We easily observe that the function V satisfies Assumptions (A) and (A). Now, let { ξ(4 sin(ln( ξ )) cos(ln( ξ ))) if ξ R \{}, f (ξ) = if ξ =. Let F (ξ) = ξ f (x)dx for all ξ R. Then, F is a C function with F = f.moreover, one has lim inf max t ξ [ F (t)] ξ + = and lim sup F (ξ) ξ + = 3. Hence, using Theorem 3.4, the problem (), in this case, with g (ξ) = e ξ + (ξ + ) γ where ξ + = max{ξ,} and γ is a positive real number, putting G (ξ) = ξ g (x)dx for all ξ R, forevery (, μ) ] 6, [ [, + [ admits infinitely many periodic solutions generated by impulses. 4 References. Agarwal, R.P., O Regan, D.: A multiplicity result for second order impulsive differential equations via the Leffett Williams fixed point theorem. Appl. Math. Comput. 6, (5). Agarwal, R.P., O Regan, D.: Multiple nonnegative solutions for second order impulsive differential equations. Appl. Math. Comput. 4, 5 59 () 3. Bai, L., Dai, B.: Application of variational method to a class of Dirichlet boundary value problems with impulsive effects. J. Franklin Inst. 348, () 4. Bainov, D., Simeonov, P.: Systems with Impulse Effect. Ellis Horwood Series: Mathematics and Its Applications. Ellis Horwood, Chichester (989) 5. Benchohra, M., Henderson, J., Ntouyas, S.: Theory of Impulsive Differential Equations. Contemporary Mathematics and Its Applications, vol.. Hindawi Publishing Corporation, New York (6) 6. Bonanno, G., Molica Bisci, G.: A remark on perturbed elliptic Neumann problems. Studia Univ. Babeş- Bolyai, Mathematica, Volume LV, Number 4, December 7. Bonanno, G., Molica Bisci, G.: Infinitely many solutions for a boundary value problem with discontinuous nonlinearities. Bound. Value Probl. 9, (9) 8. Bonanno, G., Molica Bisci, G., Rădulescu, V.: Arbitrarily small weak solutions for a nonlinear eigenvalue problem in Orlicz Sobolev spaces. Monatshefte Math. 65(3-4), () 9. Bonanno, G., Molica Bisci, G., Rădulescu, V.: Infinitely many solutions for a class of nonlinear eigenvalue problems in Orlicz Sobolev spaces. C. R. Acad. Sci., Ser. Math. 349, (). Bonanno, G., Molica Bisci, G., Rădulescu, V.: Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz Sobolev spaces. Nonlinear Anal. TMA 75, (). 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. 3(6), 4 (3). Carter, T.E.: Necessary and sufficient conditions for optimal impulsive rendezvous with linear equations of motion. Dyn. Control, 9 7 () 3. Chen, J., Tisdel, C.C., Yuan, R.: On the solvability of periodic boundary value problems with impulse. J. Math. Anal. Appl. 33, 9 9 (7) 4. Chu, J., Nieto, J.J.: Impulsive periodic solutions of first-order singular differential equations. Bull. Lond. Math. Soc. 4, 43 5 (8) 5. Ferrara, M., Heidarkhani, S.: Multiple solutions for a perturbed p-laplacian boundary value problem with impulsive effects. Electron. J. Differ. Equ. 4(6), 4 (4) 6. Ferrara, M., Khademloo, S., Heidarkhani, S.: Multiplicity results for perturbed fourth-order Kirchhoff type elliptic problems. Appl. Math. Comput. 34, (4) 7. George, P.K., Nandakumaran, A.K., Arapostathis, A.: A note on controllability of impulsive systems. J. Math. Anal. Appl. 4, () 8. Graef, J.R., Heidarkhani, S., Kong, L.: Infinitely many solutions for systems of Sturm Liouville boundary value problems. Results Math. (4). doi:.7/s

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