Acta Appl MathDOI 10.1007/s10440-014-9970-4

Infinitely Many Periodic Solutions for a Classof Perturbed Second-Order Differential Equationswith Impulses

Shapour Heidarkhani · Massimiliano Ferrara ·Amjad Salari

Received: 28 March 2014 / Accepted: 12 September 2014© Springer Science+Business Media Dordrecht 2014

Abstract We investigate the existence of infinitely many periodic solutions generated byimpulses for a class of perturbed second-order impulsive differential equations. Our ap-proach is based on variational methods and critical point theory.

Keywords Periodic solutions · Perturbed impulsive differential equation · Critical pointtheory · Variational methods

1 Introduction

The aim of this paper is to investigate the existence of infinitely many periodic solutionsgenerated by impulses for the following perturbed problem

⎧⎨

⎩

ü(t) + Vu(t, u(t)) = 0, t ∈ (sk−1, sk),�u̇(sk) = λfk(u(sk)) + μgk(u(sk)),u(0) − u(T ) = u̇(0) − u̇(T ) = 0,

(1)

where sk , k = 1,2, . . . ,m, are instants in which the impulses occur and 0 = s0 < s1 <· · · < sm < sm+1 = T , �u̇(sk) = u̇(sk+) − u̇(sk−) with u̇(sk±) = limt→sk± u̇(t), fk(ξ) =gradξ Fk(ξ), gk(ξ) = gradξ Gk(ξ), Fk,Gk ∈ C1(RN,R) such that Fk(0) = Gk(0) = 0,V ∈ C1([0, T ] ×RN,R), Vξ (t, ξ) = gradξ V (t, ξ), λ > 0 and μ ≥ 0 are two parameters.

The theory and applications of impulsive functional differential equations are emerg-ing as an important area of investigation, since it is far richer than the corresponding the-ory of non-impulsive functional differential equations. Various population models, which

S. Heidarkhani · A. SalariDepartment of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah, Iran

S. Heidarkhanie-mail: s.heidarkhani@razi.ac.ir

M. Ferrara (B)Department of Law and Economics, University Mediterranea of Reggio Calabria, Via dei Bianchi, 2,89131 Reggio Calabria, Italye-mail: massimiliano.ferrara@unirc.it

mailto:s.heidarkhani@razi.ac.irmailto:massimiliano.ferrara@unirc.it

S. Heidarkhani et al.

are characterized by the fact that per sudden changing of their state and process under de-pends on their prehistory at each moment of time, can be expressed by impulsive differentialequations with deviating argument, as population dynamics, ecology and epidemic, etc. Wenote that the difficulties dealing with such models are that such equations have deviatingarguments and theirs states are discontinuous. In the last decades, impulsive differentialequations have become more important in some mathematical models of real processes andphenomena studied in spacecraft control, impact mechanics, physics, chemistry, chemicalengineering, population dynamics, biotechnology, economics and inspection process in op-erations research. It is now recognized that the theory of impulsive differential equationsis a natural framework for a mathematical modelling of many natural phenomena. For thebackground, theory and applications of impulsive differential equations, we refer the interestreaders to [4, 5, 12, 14, 17, 19, 22, 23, 26, 29, 30, 32, 39].

Impulsive differential equations have been studied extensively in the literature. Therehave been many approaches to study the existence of solutions of impulsive differentialequations, such as fixed point theory, topological degree theory (including continuationmethod and coincidence degree theory) and comparison method (including upper and lowersolution methods and monotone iterative method) and so on (see, for example, [1, 2, 13, 21,24, 25, 37] and references therein). Recently, in [3, 11, 28, 33–36, 40, 42, 43] using varia-tional methods and critical point theory studied the existence and multiplicity of solutionsof impulsive problems. All the results of [3, 11, 28, 33–36, 40, 42, 43] can be seen as gen-eralizations of corresponding ones for second order ordinary differential equations. In otherwords, these results can be applied to impulsive systems when the impulses are absent andstill give the existence of solutions in this situation. This, in some sense, means that the non-linear term Vu plays a more important role than the impulsive terms fk do in guaranteeingthe existence of solutions in these results. In [41] the authors studied the existence of peri-odic and homoclinic solutions for a class of second order differential equations of the form(1) when μ = 0 via variational methods, and they showed that under appropriate conditionssuch a system possesses at least one non-zero periodic solution and at least one non-zerohomoclinic solution and these solutions are generated by impulses when f = 0. In [38] theauthors, based on variational methods and critical point theory studied the problem (1) whenμ = 0, and proved that such a problem admits at least one nonzero, two zero, or infinitelymany periodic solutions generated by impulses under different assumptions, respectively.

In the present paper, motivated by [38], employing a smooth version of [7, Theorem 2.1]which is a more precise version of Ricceri’s Variational Principle [31, Theorem 2.5] undersome hypotheses on the behavior of the nonlinear terms at infinity, under conditions on thepotentials of f and g we prove the existence of a definite interval about λ in which theproblem (1) admits a sequence of solutions generated by impulsive which is unbounded inthe space H 1T which will be introduced later (Theorem 3.1). We also list some consequencesof Theorem 3.1. Replacing the conditions at infinity of the nonlinear terms, by a similar oneat zero, the same results hold; see Theorem 3.4. Two examples of applications are pointedout (see Examples 3.1 and 3.2).

Definition 1.1 (see [41]) A solution of the problem (1) is called a solution generated byimpulses if this solution is nontrivial when impulsive terms gk, fk �= 0 for some 1 ≤ k ≤ m,but it is trivial when impulsive terms gk = fk ≡ 0 for all 1 ≤ k ≤ m.

For example, if the problem (1) does not possess non-zero weak solution whengk = fk ≡ 0 for all 1 ≤ k ≤ m, then a non-zero weak solution for problem (1) with gk, fk �= 0for some 1 ≤ k ≤ m is called a weak solution generated by impulses.

For a through on the subject, we refer the reader to [6, 8–10, 15, 16, 20].

Infinitely Many Periodic Solutions for a Class

2 Preliminaries

Our main tool to investigate the existence of infinitely many solutions generated by impul-sive for the problem (1) is a smooth version of Theorem 2.1 of [7] which is a more preciseversion of Ricceri’s Variational Principle [31, Theorem 2.5] that we now recall here.

Theorem 2.1 Let X be a reflexive real Banach space, let Φ,Ψ : X −→ R be two Gâteauxdifferentiable functionals such that Φ is sequentially weakly lower semicontinuous, stronglycontinuous, and coercive and Ψ is sequentially weakly upper semicontinuous. For everyr > infX Φ , let us put

ϕ(r) := infu∈Φ−1(]−∞,r[)

supv∈Φ−1(]−∞,r]) Ψ (v) − Ψ (u)r − Φ(u)

and

γ := lim infr→+∞ ϕ(r), δ := lim infr→(infX Φ)+ ϕ(r).

Then, one has

(a) for every r > infX Φ and every λ ∈ ]0, 1ϕ(r) [, the restriction of the functional Iλ =Φ − λΨ to Φ−1(]−∞, r[) admits a global minimum, which is a critical point (localminimum) of Iλ in X.

(b) If γ < +∞ then, for each λ ∈ ]0, 1γ[, the following alternative holds:

either

(b1) Iλ possesses a global minimum,

or

(b2) there is a sequence {un} of critical points (local minima) of Iλ such that

limn→+∞Φ(un) = +∞.

(c) If δ < +∞ then, for each λ ∈ ]0, 1δ[, the following alternative holds:

either

(c1) there is a global minimum of Φ which is a local minimum of Iλ,

or

(c2) there is a sequence of pairwise distinct critical points (local minima) of Iλ whichweakly converges to a global minimum of Φ .

Let

H 1T ={u : [0, T ] →RN ∣∣ u is absolutely continuous, u(0) = u(T ), u̇ ∈ L2([0, T ],RN)}.

Then H 1T is a Hilbert space with the inner product

〈u,v〉 =∫ T

0

[(u(t), v(t)

) + (u̇(t), v̇(t))]dt, ∀u,v ∈ H 1T ,

S. Heidarkhani et al.

where (.,.) denotes the inner product in RN . The corresponding norm is defined by

‖u‖ =(∫ T

0

(∣∣u̇(t)

∣∣2 + ∣∣u(t)∣∣2)dt

) 12 ∀u ∈ H 1T .

Obviously, H 1T is a separable and uniformly convex Banach space.Since (H 1T ,‖.‖) is compactly embedded in C([0, T ],RN) (see [27]), there exists a posi-

tive constant C such that

‖u‖∞ ≤ C‖u‖, (2)where ‖u‖∞ = maxt∈[0,T ] ‖u(t)‖. We mean by a (weak) solution of the problem (1), anyu ∈ X such that

∫ T

0

(u̇(t)v̇(t) − Vu

(t, u(t)

)v(t)

)dt + λ

m∑

k=1fk

(u(sk)

)v(sk) + μ

m∑

k=1gk

(u(sk)

)v(sk) = 0

for every v ∈ X.To state our results concisely we introduce the following assumptions:

(A1) V is continuous differentiable and there exist positive constants a1, a2 > 0 such thata1|ξ |2 ≤ −V (t, ξ) ≤ a2|ξ |2 for all (t, ξ) ∈ [0, T ] ×RN ;

(A2) −V (t, ξ) ≤ −Vξ (t, ξ)ξ ≤ −2V (t, ξ) for all (t, ξ) ∈ [0.T ] ×RN .A special case of our main result is the following theorem.

Theorem 2.2 Assume that Assumptions (A1) and (A2) hold. Let Fk ∈ C1(RN,R) and de-note fk(ξ) = gradξ Fk(ξ) for every ξ ∈RN for k = 1,2, . . . ,m. Furthermore, suppose that

lim infξ−→+∞

max|t |≤ξ [−∑mk=1 Fk(t)]ξ 2

= 0 and lim supξ−→+∞

−∑mk=1 Fk(ξ)ξ 2

= +∞.

Then, the problem⎧⎨

⎩

ü(x) + Vu(t, u(t)) = 0, t ∈ (sk−1, sk),�u̇(sk) = fk(u(sk)),u(0) − u(T ) = u̇(0) − u̇(T ) = 0,

has an unbounded sequence of periodic solutions generated by impulses.

3 Main Results

We formulate our main result as follows.

Theorem 3.1 Assume that Assumptions (A1) and (A2) hold. Furthermore, suppose that

(A3) lim infξ−→+∞max|t |≤ξ [−

∑mk=1 Fk(t)]

ξ2<

a3C2T a2

lim supξ−→+∞−∑mk=1 Fk(ξ)

ξ2.

Then, for each λ ∈ ]λ1, λ2[ where

λ1 := 1lim supξ−→+∞

−∑mk=1 Fk(ξ)T a2ξ

2

Infinitely Many Periodic Solutions for a Class

and

λ2 := 1lim infξ−→+∞

max|t |≤ξ [−∑m

k=1 Fk(t)]ξ2a3C2

,

for every arbitrary negative function Gk ∈ C1(RN,R) such that Gk(0) = 0, denotinggk(ξ) = gradξ Gk(ξ) for every ξ ∈RN for k = 1,2, . . . ,m, satisfying the condition

g∞ := limξ−→∞

max|t |≤ξ [−∑mk=1 Gk(t)]ξ2a3C2

< +∞ (3)

and for every μ ∈ [0,μg,λ[ where μg,λ := 1g∞ (1 − λ lim infξ→+∞max|t |≤ξ [−

∑mk=1 Fk(t)]

ξ2a3C2

), the

problem (1) has an unbounded sequence of periodic solutions generated by impulses.

Proof Our goal is to apply Theorem 2.1. Fix λ ∈ ]λ1, λ2[ and let Gk , k = 1, . . . ,m be func-tions satisfying the condition (3). Since, λ < λ2, one has μg,λ > 0. Fix μ ∈ [0,μg,λ[ and putν1 := λ1 and ν2 := λ2

1+ μλ

λ2g∞. If g∞ = 0, clearly, ν1 = λ1, ν2 = λ2 and λ ∈ ]ν1, ν2[. If g∞ �= 0,

since μ < μg,λ, we obtainλλ2

+μg∞ < 1, and so λ21+ μ

λλ2g∞

> λ, namely, λ < ν2. Hence, since

λ > λ1 = ν1, one has λ ∈ ]ν1, ν2[. Now, put Qk(ξ) = Fk(ξ) + μλ Gk(ξ) for all ξ ∈ RN andk = 1, . . . ,m. Take X = H 1T and consider the functionals Φ , Ψ : X → R for each u ∈ X, asfollows

Φ(u) =∫ T

0

[1

2

∣∣u̇(t)

∣∣2 − V (t, u(t))

]

dt

and

Ψ (u) = −(

m∑

k=1Fk

(u(sk)

) + μλ

m∑

k=1Gk

(u(sk)

))

.

It is well known that Ψ is a Gâteaux differentiable functional and sequentially weakly uppersemicontinuous whose Gâteaux derivative at the point u ∈ X is the functional Ψ ′(u) ∈ X∗,given by

Ψ ′(u)v = −(

m∑

k=1fk

(u(sk)

)v(sk) + μ

λ

m∑

k=1gk

(u(sk)

)v(sk)

)

,

and Ψ ′ : X → X∗ is a compact operator. Moreover, Φ is a Gâteaux differentiable functionalwhose Gâteaux derivative at the point u ∈ X is the functional Φ ′(u) ∈ X∗, given by

Φ ′(u)v =∫ T

0

(u̇(t)v̇(t) − Vu

(t, u(t)

)v(t)

)dt

for every v ∈ X. Furthermore, Φ is sequentially weakly lower semicontinuous. Indeed, letun ∈ X with un → u weakly in X, taking weakly lower semicontinuity of the norm intoaccount, we have lim infn→+∞ ‖un‖ ≥ ‖u‖ and un → u uniformly on [0, T ]. Hence, since Vis continuous, we have

limn→+∞

1

2

∫ T

0

[∣∣u̇n(t)

∣∣2 − V (t, un(t)

)]dt ≥ 1

2

∫ T

0

[∣∣u̇(t)

∣∣2 − V (t, u(t))]dt,

S. Heidarkhani et al.

that is lim infn→+∞ Φ(un) ≥ Φ(u) which means Φ is sequentially weakly lower semicon-tinuous. By Assumption (A1) we have

a3‖u‖2 ≤ Φ(u) ≤ a4‖u‖2, (4)

where a3 = min{ 12 , a1} and a4 = min{ 12 , a2}. Put Iλ := Φ − λΨ . Similar to the proof ofLemma 1 of [41], we observe that the weak solutions of the problem (1) are exactly thesolutions of the equation I ′

λ(u) = 0. Now, we want to verify that γ < +∞, where γ is

defined in Theorem 2.1. Let {ξn} be a real sequence such that ξn → +∞ as n → ∞ and

limn→∞

max|t |≤ξn [−∑m

k=1 Qk(t)]ξ 2n

= lim infξ→+∞

max|t |≤ξ [−∑mk=1 Qk(t)]ξ 2

.

Put rn = ξ2n a3C2 for all n ∈N. Taking (2) into account that, from (4) we have Φ−1(]−∞, rn]) ⊆{u ∈ X; ‖u‖∞ ≤ ξn}. Hence, taking Φ(0) = Ψ (0) = 0 into account, for every n large enough,one has

ϕ(rn) = infu∈Φ−1(]−∞,rn[)

(supv∈Φ−1(]−∞,rn]) Ψ (v)) − Ψ (u)rn − Φ(u) ≤

supv∈Φ−1(]−∞,rn]) Ψ (v)rn

≤ max|t |≤ξn [−∑m

k=1 Qk(t)]ξ2n a3C2

= max|t |≤ξn∑m

k=1 [ − Fk(t) − μλ Gk(t)]ξ2n a3C2

≤ max|t |≤ξn [−∑m

k=1 Fk(t)]ξ2n a3C2

+ μλ

max|t |≤ξn [−∑m

k=1 Gk(t)]ξ2n a3C2

.

Moreover, it follows from Assumption (A3) that

lim infξ−→+∞

max|t |≤ξ [−∑mk=1 Fk(t)]ξ 2

< +∞,

so we obtain

limn→∞

max|t |≤ξn [−∑m

k=1 Fk(t)]ξ 2n

< +∞. (5)

Then, in view of (3) and (5), we have

limn→∞

max|t |≤ξn [−∑m

k=1 Fk(t)]ξ2n a3C2

+ limn→∞

μ

λ

max|t |≤ξn [−∑m

k=1 Gk(t)]ξ2n a3C2

< +∞,

which follows

limn→∞

max|t |≤ξn∑m

k=1 [ − Fk(t) − μλ Gk(t)]ξ2n a3C2

< +∞.

Therefore,

γ ≤ lim infn→+∞ ϕ(rn) ≤ limn→∞

max|t |≤ξn∑m

k=1 [ − Fk(t) − μλ Gk(t)]ξ2n a3C2

< +∞. (6)

Infinitely Many Periodic Solutions for a Class

Since

max|t |≤ξn [−∑m

k=1 Qk(t)]ξ2n a3C2

≤ max|t |≤ξn [−∑m

k=1 Fk(t)]ξ2n a3C2

+ μλ

max|u|≤ξn [−∑m

k=1 Gk(t)]ξ2n a3C2

,

taking (3) into account, one has

lim infξ→+∞

max|t |≤ξ [−∑mk=1 Qk(t)]ξ2a3C2

≤ lim infξ→+∞

max|t |≤ξ [−∑mk=1 Fk(t)]ξ2a3C2

+ μλ

g∞. (7)

Moreover, since Gk for k = 1,2, . . . ,m is negative, we have

lim sup|ξ |→+∞

−∑mk=1 Qk(ξ)T a2ξ 2

≥ lim sup|ξ |→+∞

−∑mk=1 Fk(ξ)T a2ξ 2

. (8)

Therefore, from (7) and (8), and from Assumption (A3) and (6) we observe

λ ∈ ]ν1, ν2[ ⊆]

1

lim sup|ξ |−→+∞−∑mk=1 Qk(ξ)

T a2ξ2

,1

lim infξ−→+∞max|t |≤ξ [−

∑mk=1 Qk(t)]

ξ2a3C2

[

⊆]

0,1

γ

[

.

For the fixed λ, the inequality (6) concludes that the condition (b) of Theorem 2.1 can beapplied and either Iλ has a global minimum or there exists a sequence {un} of weak solutionsof the problem (1) such that limn→∞ ‖u‖ = +∞.

The other step is to show that for the fixed λ the functional Iλ has no global minimum.Let us verify that the functional Iλ is unbounded from below. Since

1

λ< lim sup

|ξ |→+∞−∑mk=1 Fk(ξ)

T a2ξ 2,

we can consider a real sequence {dn} and a positive constant τ such that dn → +∞ asn → ∞ and

1

λ< τ <

−∑mk=1 Fk(dn)T a2d2n

(9)

for each n ∈ N large enough. Let {wn} be a sequence in X defined by

wn(t) = dn. (10)

For any fixed n ∈ N, wn ∈ X and ‖wn‖2 = T d2n , and in particular, taking (4) into account,we

Φ(wn) =∫ T

0

[1

2

∣∣ẇn(t)

∣∣2 − V (t,wn(t)

)]

dt =∫ T

0

[−V (t,wn(t))]

dt ≤ a2T d2n. (11)

On the other hand, since Gk for k = 1,2, . . . ,m is negative, from the definition of Ψ , weinfer

Ψ (wn) ≥ −m∑

k=1Fk(dn). (12)

S. Heidarkhani et al.

So, according to (9), (11) and (12) we obtain

Iλ(wn) = Φ(wn) − λΨ (wn) ≤ a2T d2n − λ(

−m∑

k=1Fk(dn)

)

< (1 − λτ)a2T d2n

for every n ∈ N large enough. Hence, the functional Iλ is unbounded from below, and itfollows that Iλ has no global minimum. Therefore, applying Theorem 2.1 we deduce thatthere is a sequence {un} ⊂ X of critical points of Iλ such that limn→∞ Φ(un) = +∞, whichfrom (4) it follows that limn→∞ ‖un‖ = +∞. From Assumptions (A1) and (A2), by thesame arguing as given in [41, Theorem 4] we observe that the problem (1) does not admit anynonzero periodic solution when impulses are zero. Hence, by Definition 1.1, the problem (1)admits an unbounded sequence of periodic solutions generated by impulses. �

Remark 3.1 Theorem 3.1 gives back to Theorem 1.7 of [38] when μ = 0.

Remark 3.2 Under the conditions

lim infξ−→+∞

max|t |≤ξ [−∑mk=1 Fk(t)]ξ 2

= 0 and lim supξ−→+∞

−∑mk=1 Fk(ξ)ξ 2

= +∞,

Theorem 3.1 concludes that for every λ > 0 and for each μ ∈ [0, 1g∞ [ the problem (1) admits

infinitely many periodic solutions generated by impulses. Moreover, if g∞ = 0, the resultholds for every λ > 0 and μ ≥ 0.

We now exhibit an example in which the hypotheses of Theorem 3.1 are satisfied.

Example 3.1 Let m = T = N = 1, V (t, ξ) = −ξ(ξ + arctan(ξ)) for every (t, ξ) ∈[0, T ] × R. By the same reasoning as given in Example 1 of [41] we see that the func-tion V satisfies Assumptions (A1) and (A2). Now, let {an} and {bn} be sequences defined byb1 = 2, bn+1 = b6n and an = b4n for n ∈N. Let

f1(ξ) =

⎧⎪⎪⎨

⎪⎪⎩

b31

√1 − (1 − t)2 + 1 if t ∈ [0, b1],

(an − b3n)√

1 − (an − 1 − t)2 + 1 if t ∈ ⋃∞n=1[an − 2, an],(b3n+1 − an)

√1 − (bn+1 − 1 − t)2 + 1 if t ∈ ⋃∞n=1[bn+1 − 2, bn+1],

1 otherwise.

Let −F1(ξ) =∫ ξ

0 f1(x)dx for all ξ ∈ R. Then, −F1 is a C1 function with −F ′1 = f1.From the computation of Example 3.1 of [18], we have −F1(an) = π2 an + an and−F1(bn) = π2 b3n + bn, and so limn→∞ −F1(an)a2n = 0 and limn→∞

−F1(bn)b2n

= ∞. Therefore,lim infξ→∞ −F1(ξ)ξ2 = 0 and lim supξ→∞ −F1(ξ)ξ2 = ∞. Hence, using Theorem 3.1, the prob-lem (1), in this case, with g1(ξ) = e−ξ+(ξ+)γ where ξ+ = max{ξ,0} and γ is a positive realnumber, putting −G1(ξ) =

∫ ξ0 g1(x)dx for all ξ ∈R, for every (λ,μ) ∈ ]0,+∞[×[0,+∞[

admits infinitely many periodic solutions generated by impulses.

Remark 3.3 We explicitly observe that Assumption (A3) in Theorem 3.1 could be replacedby the following more general condition

Infinitely Many Periodic Solutions for a Class

(A3′) there exist two sequence {θn} and {ηn} with ηn >√

a2a3

T Cθn for every n ∈ N andlimn→+∞ ηn = +∞ such that

limn→+∞

max|t |≤ηn [−∑m

k=1 Fk(t)] − (−∑m

k=1 Fk(θn))a3C2

η2n − a2T θ2n< lim sup

|ξ |−→+∞−∑mk=1 Fk(ξ)

a2T ξ 2.

Obviously, from (A3′) we obtain (A3), by choosing θn = 0 for all n ∈ N. Moreover, if weassume (A3′) instead of (A3) and set rn = a3C2 η2n for all n ∈ N, by the same arguing as insidein Theorem 3.1, we obtain

ϕ(rn) ≤supv∈Φ−1(]−∞,rn]) Ψ (v) − (−

∑mk=1 Fk(wn))

rn −∫ T

0 [ 12 |ẇn(t)|2 − V (t,wn(t))]dt

≤ max|t |≤ηn [−∑m

k=1 Fk(t)] − (−∑m

k=1 Fk(θn))a3C2

η2n − a2T θ2nwhere wn(t) = θn for every t ∈ [0, T ]. We have the same conclusion as in Theorem 3.1 withthe interval ]λ1, λ2[ replaced by the interval by

]1

lim sup|ξ |−→+∞−∑mk=1 Fk(ξ)

a2T ξ2

,1

limn→+∞max|t |≤ηn [−

∑mk=1 Fk(t)]−(−

∑mk=1 Fk(θn))

a3C2

η2n−a2T θ2n

[

.

Here, we point out a simple consequence of Theorem 3.1.

Corollary 3.2 Assume that Assumptions (A1) and (A2) hold. Furthermore, suppose that

(B1) lim infξ−→+∞max|t |≤ξ [−

∑mk=1 Fk(t)]

ξ2<

a3C2

;

(B2) lim supξ−→+∞−∑mk=1 Fk(ξ)

ξ2> T a2.

Then, for every arbitrary negative function Gk ∈ C1(RN,R) such that Gk(0) = 0, denotinggk(ξ) = gradξ Gk(ξ) for every ξ ∈ RN for k = 1,2, . . . ,m, satisfying the condition (3) andfor every μ ∈ [0,μg,1[ where

μg,1 := 1g∞

(

1 − lim infξ→+∞

max|t |≤ξ [−∑mk=1 Fk(t)]ξ2a3C2

)

,

the problem⎧⎨

⎩

ü(t) + Vu(t, u(t)) = 0, t ∈ (sk−1, sk),�u̇(sk) = fk(u(sk)) + μgk(u(sk)),u(0) − u(T ) = u̇(0) − u̇(T ) = 0,

has an unbounded sequence of periodic solutions generated by impulses.

Remark 3.4 Theorem 2.2 is an immediately consequence of Corollary 3.2 when μ = 0.

We here give the following consequence of the main result.

S. Heidarkhani et al.

Corollary 3.3 Let F 1k ∈ C1(RN,R) such that F 1k (0) = 0 and let f 1k (ξ) = gradξ F 1k (ξ) forall ξ ∈RN for k = 1,2, . . . ,m. Assume that(D1) lim infξ−→+∞

max|t |≤ξ [−∑m

k=1 F 1k (t)]ξ2

< +∞;(D2) lim supξ−→+∞

−∑mk=1 F 1k (ξ)ξ2

= +∞.Then, for every F ik ∈ C1(RN,R) such that F ik (0) = 0, denoting f ik (ξ) = gradξ F ik (ξ) for allξ ∈RN for k = 1,2, . . . ,m for 2 ≤ i ≤ n, satisfying

max{

supξ∈R

(−F ik (ξ)); 2 ≤ i ≤ n

}≤ 0

and

min

{

lim infξ→+∞

(−F ik (ξ))ξ 2

; 2 ≤ i ≤ n}

> −∞,

for each

λ ∈]

0,a3

C2 lim infξ→+∞(−F 1

k(ξ))

ξ2

[

,

for every arbitrary negative function Gk ∈ C1(RN,R) such that Gk(0) = 0, denotinggk(ξ) = gradξ Gk(ξ) for every ξ ∈ RN for k = 1,2, . . . ,m, satisfying the condition (3) andfor every μ ∈ [0,μg,λ[ where

μg,λ := 1g∞

(

1 − λ lim infξ→+∞

max|t |≤ξ [−∑mk=1 F 1k (t)]ξ2a3C2

)

,

the problem⎧⎨

⎩

ü(t) + Vu(t, u(t)) = 0, t ∈ (sk−1, sk),�u̇(sk) = λ∑ni=1 f ik (u(sk)) + μgk(u(sk)),u(0) − u(T ) = u̇(0) − u̇(T ) = 0

has an unbounded sequence of periodic solutions generated by impulses.

Proof Set Fk(ξ) = ∑ni=1 F ik (ξ) for all ξ ∈ RN . Assumption (D2) along with the condi-tion

min

{

lim infξ→+∞

−F ik (ξ)ξ 2

; 2 ≤ i ≤ n}

> −∞ensures

lim supξ−→+∞

−∑mk=1 Fk(ξ)ξ 2

= lim supξ→+∞

−∑mk=1∑n

i=1 Fik (ξ)

ξ 2= +∞.

Moreover, Assumption (D1) in conjunction with the condition

max{

supξ∈R

−F ik (ξ); 2 ≤ i ≤ n}

≤ 0

Infinitely Many Periodic Solutions for a Class

yields

lim infξ−→+∞

max|t |≤ξ [−∑mk=1 Fk(t)]ξ 2

≤ lim infξ−→+∞

max|t |≤ξ [−∑mk=1 F 1k (t)]ξ 2

< +∞.

Hence, applying Theorem 3.1 we have the desired result. �

Arguing as in the proof of Theorem 3.1, but using conclusion (c) of Theorem 2.1 insteadof (b), the following result holds.

Theorem 3.4 Assume that Assumptions (A1) and (A2) hold. Furthermore, suppose that

(E1) lim infξ−→0+max|t |≤ξ [−

∑mk=1 Fk(t)]

ξ2<

a3C2T a2

lim supξ−→0+−∑mk=1 Fk(ξ)

ξ2.

Then, for each λ ∈ ]λ3, λ4[ where

λ3 := 1lim supξ−→0+

−∑mk=1 Fk(ξ)T a2ξ

2

and

λ4 := 1lim infξ−→0+

max|t |≤ξ [−∑m

k=1 Fk(t)]ξ2a3C2

,

for every arbitrary negative function Gk ∈ C1(RN,R) such that Gk(0) = 0, denotinggk(ξ) = gradξ Gk(ξ) for every ξ ∈RN for k = 1,2, . . . ,m, satisfying the condition

g0 := limξ−→0+

max|t |≤ξ [−∑mk=1 Gk(t)]ξ2a3C2

< +∞ (13)

and for every μ ∈ [0,μg,λ[ where μg,λ := 1g0 (1−λ lim infξ→0+max|t |≤ξ [−

∑mk=1 Fk(t)]

ξ2a3C2

), the prob-

lem (1) has an unbounded sequence of pairwise distinct periodic solutions generated byimpulses.

Proof Fix λ ∈ ]λ3, λ4[ and let Gk for k = 1,2, . . . ,m be functions satisfy the condition (13).Since, λ < λ2, one has

μg,λ :=1

g0

(

1 − λ lim infξ→0+

max|t |≤ξ [−∑mk=1 Fk(t)]ξ2a3C2

)

> 0.

Fix μ ∈ ]0,μg,λ[ and put ν3 := λ3 and ν4 := λ41+ C2a3

μ

λλ4G0

. If g0 = 0, clearly, ν3 = λ3, ν4 = λ4and λ ∈ ]ν3, ν4[. If g0 �= 0, since μ < μg,λ, we obtain

λ

λ4+ C

2

a3μg0 < 1,

and soλ4

1 + C2a3

μ

λλ4g0

> λ,

namely, λ < ν4. Hence, being in mind that λ > λ3 = ν3, one has λ ∈ ]ν3, ν4[.

S. Heidarkhani et al.

Now, set Qk(ξ) = Fk(ξ) + μλ Gk(ξ) for all ξ ∈RN and k = 1,2, . . . ,m. Since

max|t |≤ξ [−∑mk=1 Qk(t)]ξ2a3C2

≤ max|t |≤ξ [−∑m

k=1 Fk(t)]ξ2a3C2

+ μλ

max|t |≤ξ [−∑mk=1 Gk(t)]ξ2a3C2

,

taking (13) into account, one has

lim infξ→0+

max|t |≤ξ [−∑mk=1 Qk(t)]ξ2a3C2

≤ lim infξ→0+

max|t |≤ξ [−∑mk=1 Fk(t)]ξ2a3C2

+ μλ

g0. (14)

Moreover, since Gk is negative for k = 1,2, . . . ,m, from Assumption (E1) we obtain

lim supξ−→0+

−∑mk=1 Qk(ξ)ξ 2

≥ lim supξ−→0+

−∑mk=1 Fk(ξ)ξ 2

. (15)

Therefore, from (14) and (15), we observe

λ ∈ ]ν3, ν4[ ⊆]

1

lim sup|ξ |−→0+−∑mk=1 Qk(ξ)

T a2ξ2

,1

lim infξ−→0+max|t |≤ξ [−

∑mk=1 Qk(t)]

ξ2a3C2

[

⊆ ]λ3, λ4[.

We take X, Φ , Ψ and Iλ as in the proof of Theorem 3.1. We observe that δ < +∞. Indeed,let {ξn} be a sequence of positive numbers such that ξn → 0+ as n → +∞ and

limn→∞

max|t |≤ξn [−∑m

k=1 Fk(t)]ξ 2n

< +∞.

Put rn = a3ξ2nC2 for all n ∈ N. By the same arguing as in the proof of Theorem 3.1, it followsthat δ < +∞. Let us verify that the functional Iλ has not a local minimum at zero. For this,let {dn} be a sequence of positive numbers and τ > 0 such that dn → 0+ as n → ∞ and

1

λ< τ <

−∑mk=1 Fk(dn)T a2d2n

(16)

for each n ∈ N large enough. Let {wn} be a sequence in X defined by wn(t) = dn. So,according to (11), (12) and (16) we obtain

Iλ(wn) = Φ(wn) − λΨ (wn) ≤ a2T d2n − λ(

−m∑

k=1Fk(dn)

)

< (1 − λτ)a2T d2n < 0

for every n ∈ N large enough. Since Iλ(0) = 0, this concludes that the functional Iλ hasnot a local minimum at zero. Hence, the part (c) of Theorem 2.1 follows that there exists asequence {un} in X of critical points of Iλ such that ‖un‖ → 0 as n → ∞. From Assump-tions (A1) and (A2), by the same arguing as given in [41, Theorem 4] we observe that theproblem (1) does not admit any nonzero periodic solution when impulses are zero. Hence, byDefinition 1.1, the problem (1) admits an unbounded sequence of pairwise distinct periodicsolutions generated by impulses. �

Remark 3.5 Applying Theorem 3.4, results similar to Corollaries 3.2 and 3.3 can be ob-tained. We leave their formulation to the reader.

Infinitely Many Periodic Solutions for a Class

We end this paper by presenting the following example in which the hypotheses of The-orem 3.4 are satisfied.

Example 3.2 Let m = T = N = 1 and V (t, ξ) = − 12 |ξ |2 for every (t, ξ) ∈ [0, T ] × R. Wehave C = √2, a2 = a3 = 12 in this case. We easily observe that the function V satisfiesAssumptions (A1) and (A2). Now, let

f1(ξ) ={

ξ(4 − 2 sin(ln(|ξ |)) − cos(ln(|ξ |))) if ξ ∈R \ {0},0 if ξ = 0.

Let −F1(ξ) =∫ ξ

0 f1(x)dx for all ξ ∈ R. Then, −F1 is a C1 function with −F ′1 = f1. More-over, one has lim infξ−→0+

max|t |≤ξ [−F1(t)]ξ2

= 1 and lim supξ−→0+ −F1(ξ)ξ2 = 3. Hence, usingTheorem 3.4, the problem (1), in this case, with g1(ξ) = e−ξ+(ξ+)γ where ξ+ = max{ξ,0}and γ is a positive real number, putting −G1(ξ) =

∫ ξ0 g1(x)dx for all ξ ∈ R, for every

(λ,μ) ∈ ] 16 , 14 [ × [0,+∞[ admits infinitely many periodic solutions generated by impulses.

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