Infinitely many solutions for a boundary value problem with impulsive effects

  • Gabriele Bonanno1Email author,

    Affiliated with

    • Beatrice Di Bella2 and

      Affiliated with

      • Johnny Henderson3

        Affiliated with

        Boundary Value Problems20132013:278

        DOI: 10.1186/1687-2770-2013-278

        Received: 24 October 2013

        Accepted: 1 December 2013

        Published: 20 December 2013

        Abstract

        In this paper we are interested in multiplicity results for a nonlinear Dirichlet boundary value problem subject to perturbations of impulsive terms. The study of the problem is based on the variational methods and critical point theory. Infinitely many solutions follow from a recent variational result.

        MSC:34B37, 34B15.

        Keywords

        impulsive differential equations critical points infinitely many solutions

        1 Introduction

        The theory of impulsive differential equations provides a general framework for the mathematical modeling of many real world phenomena; see, for instance, [13] and [4]. Indeed, many dynamical systems have an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. Impulsive differential equations are basic tools for studying these phenomena [5, 6].

        There are some common techniques to approach these problems: the fixed point theorems [7, 8], the method of upper and lower solutions [9], or the topological degree theory [1012]. On the other hand, in the last few years, some authors have studied the existence of solutions by variational methods; see [1319].

        Here, we use critical point theory to investigate the existence of infinitely many solutions for the following nonlinear impulsive differential problem:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-278/MediaObjects/13661_2013_Article_631_Equa_HTML.gif

        where λ ] 0 , + [ , μ ] 0 , + [ , g : [ 0 , T ] × R R , a , b L ( [ 0 , T ] ) with ess inf t [ 0 , T ] a ( t ) 0 and ess inf t [ 0 , T ] b ( t ) 0 , 0 = t 0 < t 1 < t 2 < < t n < t n + 1 = T , Δ u ( t j ) = u ( t j + ) u ( t j ) = lim t t j + u ( t ) lim t t j u ( t ) , and I j : R R are continuous for every j = 1 , 2 , , n .

        We establish some multiplicity results for problem ( D λ , μ ) under an appropriate oscillation behavior of the primitive of the nonlinearity g and a suitable growth of the primitive of I j at infinity, for all λ belonging to a precise interval and provided μ is small enough (Theorem 3.3, Theorem 3.4). It is worth noticing that, when the impulsive effects I j , j = 1 , , n , are sublinear at infinity, our results hold for all μ 0 (see Remark 3.1). Here, as an example of our results, we present the following special case of Theorem 3.3.

        Theorem 1.1 Let g : R [ 0 , ) be a continuous function and put G ( ξ ) = 0 ξ g ( t ) d t for every ξ R . Assume that
        lim inf ξ + G ( ξ ) ξ 2 = 0 and lim sup ξ + G ( ξ ) ξ 2 = + .
        Then there is δ ¯ > 0 , where δ ¯ = 4 n T e T , such that for each μ [ 0 , δ ¯ [ , the problem
        { u ( t ) + u ( t ) + u ( t ) = g ( t , u ( t ) ) , t [ 0 , T ] , t t j , u ( 0 ) = u ( T ) = 0 , Δ u ( t j ) = μ u ( t j ) , j = 1 , 2 , , n

        admits infinitely many pairwise distinct classical solutions.

        We explicitly observe that in Theorem 1.1 impulsive effects I j , j = 1 , , n , (that is, I j ( x ) = x for all x R ) are linear, contrary to the usual assumption of sublinearity of impulses; see [14, 16, 2022] and [23]. The rest of this paper is organized as follows. In Section 2, we introduce some notations and preliminary results. Moreover, the abstract critical point theorem (Theorem 2.1) is recalled. In Section 3, we obtain some existence results. In Section 4, we give some examples to illustrate our results.

        2 Preliminaries

        By a classical solution of ( D λ , μ ) we mean a function
        u { w C ( [ 0 , T ] ) : w | [ t j , t j + 1 ] H 2 ( [ t j , t j + 1 ] ) }

        that satisfies the equation in ( D λ , μ ) a.e. on [ 0 , T ] { t 1 , , t n } , the limits u ( t j + ) , u ( t j ) , j = 1 , , n , exist, that satisfies the impulsive conditions Δ u ( t j ) = μ I j ( u ( t j ) ) and the boundary conditions u ( 0 ) = u ( T ) = 0 . Clearly, if a, b and g are continuous, then a classical solution u C 2 ( [ t j , t j + 1 ] ) , j = 0 , 1 , , n , satisfies the equation in ( D λ , μ ) for all t [ 0 , T ] { t 1 , , t n } .

        We consider the following slightly different form of problem ( D λ , μ ):
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-278/MediaObjects/13661_2013_Article_631_Eque_HTML.gif

        where p C 1 ( [ 0 , T ] , ] 0 , + [ ) , and q L ( [ 0 , T ] ) with ess inf t [ 0 , T ] q ( t ) 0 .

        It is easy to see that, by choosing
        p ( t ) = e 0 t a ( τ ) d τ , q ( t ) = b ( t ) e 0 t a ( τ ) d τ , f ( t , u ) = g ( t , u ) e 0 t a ( τ ) d τ ,

        the solutions of ( S λ , μ ) are solutions of ( D λ , μ ).

        Let us introduce some notations. In the Sobolev space H 0 1 ( 0 , T ) , consider the inner product
        ( u , v ) = 0 T p ( t ) u ( t ) v ( t ) d t + 0 T q ( t ) u ( t ) v ( t ) d t ,
        which induces the norm
        u = ( 0 T p ( t ) ( u ( t ) ) 2 d t + 0 T q ( t ) ( u ( t ) ) 2 d t ) 1 / 2 .

        The following lemmas are useful for proving our main result. Their proofs can be found in [24].

        Lemma 1 ([[24], Proposition 2.1])

        Let u H 0 1 ( 0 , T ) . Then
        u 1 2 T p u ,
        (1)

        where p : = min t [ 0 , T ] p ( t ) .

        Here, and in the sequel, f : [ 0 , T ] × R R is an L1-Carathéodory function, namely:
        1. (a)

          t f ( t , x ) is measurable for every x R ;

           
        2. (b)

          x f ( t , x ) is continuous for almost every t [ 0 , T ] ;

           
        3. (c)
          for every ρ > 0 , there exists a function l ρ L 1 ( [ 0 , T ] ) such that
          sup | x | ρ | f ( t , x ) | l ρ ( t )
           

        for almost every t [ 0 , T ] .

        Definition 1 A function u H 0 1 ( 0 , T ) is said to be a weak solution of ( S λ , μ ) if u satisfies
        0 T p ( t ) u ( t ) v ( t ) d t + 0 T q ( t ) u ( t ) v ( t ) d t λ 0 T f ( t , u ( t ) ) v ( t ) d t + μ j = 1 n p ( t j ) I j ( u ( t j ) ) v ( t j ) = 0
        (2)

        for any v H 0 1 ( 0 , T ) .

        Lemma 2 ([[24], Lemma 2.1 ])

        u H 0 1 ( 0 , T ) is a weak solution of ( S λ , μ ) if and only if u is a classical solution of ( S λ , μ ).

        Now, we define the functionals Φ , Ψ : H 0 1 ( 0 , T ) R in the following way:
        Φ ( u ) = 1 2 u 2 and Ψ ( u ) = 0 T F ( t , u ( t ) ) d t μ λ j = 1 n p ( t j ) 0 u ( t j ) I j ( x ) d x ,
        (3)
        for each u H 0 1 ( 0 , T ) , where F ( t , ξ ) = 0 ξ f ( t , x ) d x for each ( t , ξ ) [ 0 , T ] × R . Using the property of f and the continuity of I j , j = 1 , 2 , , n , we have that Φ , Ψ C 1 ( H 0 1 ( 0 , T ) , R ) and for any v H 0 1 ( 0 , T ) , one has
        Φ ( u ) ( v ) = 0 T p ( t ) u ( t ) v ( t ) d t + 0 T q ( t ) u ( t ) v ( t ) d t
        and
        Ψ ( u ) ( v ) = 0 T f ( t , u ( t ) ) v ( t ) d t μ λ j = 1 n p ( t j ) I j ( u ( t j ) ) v ( t j ) .

        So, arguing in a standard way, it is possible to prove that the critical points of the functional E λ ( u ) : = Φ ( u ) λ Ψ ( u ) are the weak solutions of problem ( S λ , μ ) and so they are classical solutions.

        In the next section we shall prove our results applying the following infinitely many critical points theorem obtained in [25]. First, we recall the following definition.

        Definition 2 Let X be a real Banach space, Φ , Ψ : X R two Gâteaux differentiable functionals, r ] , + ] . We say that functional E : = Φ Ψ satisfies the Palais-Smale condition cut off upper at r (in short (PS) [ r ] -condition) if any sequence { u n } , such that

        (α) { E ( u n ) } is bounded,

        (β) lim n + E ( u n ) X = 0 ,

        (γ) Φ ( u n ) < r for all n N ,

        has a convergent subsequence.

        When r = + , the previous definition is the same as the classical definition of the Palais-Smale condition, while if r < + , such a condition is more general than the classical one. We refer to [25] for more details.

        Theorem 2.1 (see [25], Theorem 7.4)

        Let X be a real Banach space, and let Φ , Ψ : X R be two continuously Gâteaux differentiable functionals such that Φ is bounded from below. For every r > inf X Φ , let us put
        φ ( r ) : = inf x Φ 1 ( ] , r [ ) sup v Φ 1 ( ] , r [ ) Ψ ( v ) Ψ ( u ) r Φ ( u )
        and
        γ : = lim inf r + φ ( r ) , δ : = lim inf r ( inf X Φ ) + φ ( r ) .
        1. (a)

          If γ < + and for each λ ] 0 , 1 γ [ , the functional E λ = Φ λ Ψ satisfies the ( PS ) [ r ] -condition for all r R , then for each λ ] 0 , 1 γ [ , the following alternative holds: either

           

        (a1) E λ has a global minimum

        or

        (a2) there exists a sequence { u n } of critical points (local minima) of E λ such that lim n Φ ( u n ) = + .

        1. (b)

          If δ < + and for each λ ] 0 , 1 δ [ , the functional E λ = Φ λ Ψ satisfies the ( PS ) [ r ] -condition for all r R , then for each λ ] 0 , 1 δ [ , the following alternative holds: either

           

        (b1) there exists a global minimum of Φ which is a local minimum of E λ

        or

        (b2) there exists a sequence of pairwise distinct critical points (local minima) of E λ such that lim n + Φ ( u n ) = inf X Φ .

        We recall that Theorem 2.1 improves [[26], Theorem 2.5] since no assumptions with respect to weak topology of X are made. In particular, the set Φ 1 ( ] , r [ ) ¯ w is not involved in the definition of φ and the sequential weak lower semicontinuity of E λ is not required.

        3 Main results

        In this section, we present our main results. Put
        k : = 6 p 12 p + T 2 q .
        Moreover, let
        A : = lim inf ξ + 0 T max | x | ξ F ( t , x ) d t ξ 2 , B : = lim sup ξ + T / 4 3 T / 4 F ( t , ξ ) d t ξ 2 .

        Our first result is as follows.

        Theorem 3.1 Assume that

        (a1) F ( t , ξ ) 0 for all ( t , ξ ) ( [ 0 , T 4 ] [ 3 T 4 , T ] ) × R ;

        (a2) A < k B .

        Then, for every λ Λ : = ] 2 p k T B , 2 p T A [ and for every continuous function I j : R R , j = 1 , , n , whose potential I j ( ξ ) : = 0 ξ I j ( x ) d x , ξ R , satisfies

        (i1) sup ξ 0 I j ( ξ ) = 0 ;

        (i2) I : = lim sup ξ + j = 1 n max | t | ξ ( I j ( t ) ) ξ 2 < + ,

        there exists δ I , λ > 0 , where
        δ I , λ : = 1 p I ( 2 p T λ A ) ,

        such that for every μ [ 0 , δ I , λ [ , problem ( S λ , μ ) has an unbounded sequence of weak solutions.

        Proof First, we observe that owing to (a2) the interval Λ is non-empty. Moreover, for each λ Λ and taking into account that λ < 2 p T A , one has δ I , λ > 0 . Now, fix λ and μ as in the conclusion. Our aim is to apply Theorem 2.1. For this end, take X = H 0 1 ( 0 , T ) and Φ, Ψ as in (3).

        We divide our proof into three steps in order to show Theorem 3.1. First, we prove that E λ = Φ λ Ψ satisfies the (PS) [ r ] -condition for all r R . So, fix r R and let { u n } X be a sequence such that { E λ ( u n ) } is bounded, lim n + E λ ( u n ) X = 0 and Φ ( u n ) < r for all n N . From Φ ( u n ) < r , taking into account that Φ is coercive, { u n } is bounded in X. Since the embedding of X in C ( 0 , T ) is compact (see, for instance, [[27], Theorem 8.8]) and X is reflexive, up to a subsequence, { u n ( x ) } is uniformly convergent to u 0 ( x ) , and { u n } is weakly convergent to u 0 in X. The uniform convergence of { u n } , taking also into account Lebesgue’s theorem, ensures that
        lim n + [ 0 T f ( t , u n ( t ) ) ( u n ( t ) u 0 ( t ) ) d t μ λ j = 1 n p ( t j ) I j ( u n ( t j ) ) ( u n ( t j ) u 0 ( t j ) ) ] = 0 ,
        that is,
        lim n + Ψ ( u n ) ( u n u 0 ) = 0 .
        (4)
        Now, from lim n + E λ ( u n ) X = 0 , there is a sequence { ϵ n } , with ϵ n 0 + , such that
        | E λ ( u n ) ( v ) | ϵ n
        for all v X with v 1 and for all n N . Setting = u n u 0 u n u 0 , one has
        | E λ ( u n ) ( u n u 0 ) | ϵ n u n u 0
        (5)
        for all n N . Moreover, having in mind that a b 1 2 a 2 + 1 2 b 2 , one has
        Φ ( u n ) ( u n u 0 ) = 0 T p ( t ) u n ( t ) ( u n ( t ) u 0 ( t ) ) d t + 0 T q ( t ) u n ( t ) ( u n ( t ) u 0 ( t ) ) d t = u n 2 [ 0 T p ( t ) u n ( t ) u 0 ( t ) d t + 0 T q ( t ) u n ( t ) u 0 ( t ) d t ] u n 2 1 2 [ 0 T p ( t ) ( u n ( t ) ) 2 d t + 0 T p ( t ) ( u 0 ( t ) ) 2 d t + 0 T q ( t ) ( u n ( t ) ) 2 d t + 0 T q ( t ) ( u 0 ( t ) ) 2 d t ] = 1 2 u n 2 1 2 u 0 2 ,
        that is,
        Φ ( u n ) ( u n u 0 ) 1 2 u n 2 1 2 u 0 2 .
        (6)
        From (5) and (6) one has
        Φ ( u n ) ( u n u 0 ) λ Ψ ( u n ) ( u n u 0 ) ϵ n u n u 0 , 1 2 u n 2 1 2 u 0 2 λ Ψ ( u n ) ( u n u 0 ) ϵ n u n u 0 ,
        and owing to (4), one has
        lim sup n + 1 2 u n 2 1 2 u 0 2 .

        Hence, [[27], Proposition III.30] ensures that { u n } strongly converges to u 0 X and our claim is proved.

        Second, we wish to prove that
        γ < + .
        Let { ξ n } be a sequence of positive numbers such that ξ n + and
        lim n 0 1 max | x | ξ n F ( t , x ) d t ξ n 2 = A .
        Put r n = 2 p T ξ n 2 for all n N . By Lemma 1, for all u X , one has
        max t [ 0 , T ] | v ( t ) | ξ n
        for all v X such that v 2 < 2 r n . Hence, one has
        φ ( r n ) = inf u Φ 1 ( ] , r n [ ) sup v Φ 1 ( ] , r n [ ) Ψ ( v ) Ψ ( u ) r n Φ ( u ) sup v Φ 1 ( ] , r n [ ) Ψ ( v ) r n T 2 p 0 T max | x | ξ n F ( t , x ) d t ξ n 2 + μ λ T p 2 p j = 1 n max | x | ξ n ( I j ( x ) ) ξ n 2 .
        So, from assumptions (a2) and (i2),
        γ lim inf ξ + [ T 2 p 0 T max | x | ξ F ( t , x ) d t ξ 2 + μ λ T p 2 p j = 1 n max | x | ξ ( I j ( x ) ) ξ 2 ] < + .
        Assumption 0 < μ < δ I , λ immediately yields
        γ T 2 p A + μ λ T p 2 p I < T 2 p A + 1 λ T 2 p A λ = 1 λ ,

        that is, λ < 1 γ . The previous inequality assures that conclusion (a) of Theorem 2.1 can be used, for which either Φ λ Ψ has a global minimum or there exists a sequence { u n } of solutions of problem ( S λ , μ ) such that lim n u n = + .

        The final step is to verify that the functional Φ λ Ψ has no global minimum. From lim sup ξ + T / 4 3 T / 4 F ( t , ξ ) d t ξ 2 = B , and taking into account that λ > 2 p k T B , there is h R such that
        lim sup ξ + T / 4 3 T / 4 F ( t , ξ ) d t ξ 2 > h > 2 p λ k T .
        (7)
        So, there exists a sequence of positive numbers η n such that η n + and
        lim n + T / 4 3 T / 4 F ( t , η n ) d t η n 2 > h .
        It follows that there is ν N such that for all n > ν , one has
        T / 4 3 T / 4 F ( t , η n ) d t η n 2 > h .
        Now, consider a function v n X defined by setting
        v n ( x ) = { 4 η n x T , x [ 0 , T 4 ] , η n , x ] T 4 , 3 T 4 ] , 4 η n T ( T x ) , x ] 3 T 4 , T ] .
        Clearly, one has
        Φ ( v n ) ( 4 T p + T 3 q ) η n 2 = 2 p k T η n 2 .
        Moreover, bearing in mind (a1) and (i1),
        Φ ( v n ) λ Ψ ( v n ) 2 p k T η n 2 λ T / 4 3 T / 4 F ( t , η n ) d t < η n 2 ( 2 p k T λ h ) .
        (8)

        Putting together (7) and (8), we get that the functional Φ λ Ψ is unbounded from below and so it has no global minimum.

        Therefore, Theorem 2.1 assures that there is a sequence { u n } X of critical points of Φ λ Ψ such that lim n + u n = + and, taking into account the considerations made in Section 2, the theorem is completely proved. □

        Remark 3.1 Assume that f : [ 0 , 1 ] × R [ 0 , ) . Clearly, condition (a1) holds, and condition (a2) assumes the following simpler form:

        ( a 2 )
        lim inf ξ + 0 T F ( t , ξ ) d t ξ 2 < k lim sup ξ + T / 4 3 T / 4 F ( t , ξ ) d t ξ 2 .

        In particular, if lim inf ξ + 0 T F ( t , ξ ) d t ξ 2 = 0 and lim sup ξ + T / 4 3 T / 4 F ( t , ξ ) d t ξ 2 = + , then ( a 2 ) holds and problem ( S λ , μ ) has an unbounded sequence of weak solutions in X for every pair ( λ , μ ) ] 0 , + [ × [ 0 , 2 T I [ .

        Moreover, under the assumption I = 0 , Theorem 3.1 guarantees the existence of infinitely many solutions to problem ( S λ , μ ) for every μ 0 .

        As an example, we point out below a special case of Theorem 3.1.

        Corollary 3.1 Let f : R [ 0 , ) be a continuous function, put F ( ξ ) = 0 ξ f ( t ) d t for every ξ R , and let q C 0 ( [ 0 , T ] ) . Assume that
        lim inf ξ + F ( ξ ) ξ 2 < k 2 lim sup ξ + F ( ξ ) ξ 2 .
        Then, for each λ ] 4 p k T 2 1 lim sup ξ + F ( ξ ) ξ 2 , 2 p k T 2 1 lim inf ξ + F ( ξ ) ξ 2 [ , and for each continuous function I j : R [ 0 , ) such that lim ξ + I j ( x ) ξ = 0 , j = 1 , , n , the problem
        { ( p ( t ) u ( t ) ) + q ( t ) u ( t ) = λ f ( u ( t ) ) , t [ 0 , T ] , t t j , u ( 0 ) = u ( T ) = 0 , Δ u ( t j ) = I j ( u ( t j ) ) , j = 1 , 2 , , n ,

        admits infinitely many pairwise distinct classical solutions.

        Replacing the condition at infinity of the potential F by a similar one at zero, and arguing as in the proof of Theorem 3.1 but using conclusion (b) of Theorem 2.1 instead of (a), one establishes the following result. Put
        A : = lim inf ξ 0 + 0 T max | x | ξ F ( t , x ) d t ξ 2 , B : = lim sup ξ 0 + T / 4 3 T / 4 F ( t , ξ ) d t ξ 2 .

        Theorem 3.2 Assume that

        (a1) F ( t , ξ ) 0 for all ( t , ξ ) ( [ 0 , T 4 ] [ 3 T 4 , T ] ) × R ;

        (b2) A < k B .

        Then, for every λ Λ : = ] 2 p k T B , 2 p T A [ and for every continuous function I j : R R , j = 1 , 2 , , n , whose potential I j ( ξ ) : = 0 ξ I j ( x ) d x , ξ R , satisfies

        (i1) sup ξ 0 I j ( ξ ) = 0 ;

        (j2) I 0 : = lim sup ξ 0 + j = 1 n max | t | ξ ( I j ( t ) ) ξ 2 < + ,

        there exists δ I , λ > 0 , where
        δ I , λ : = 1 I 0 p ( 2 p T λ A )

        such that for every μ [ 0 , δ I , λ [ , problem ( S λ , μ ) has a sequence of non-zero weak solutions, which strongly converges to 0.

        Proof We take X, Φ and Ψ as in the proof of Theorem 3.1. Fix λ Λ , let I j be a function that satisfies assumptions (i1) and (j2) and take 0 μ < δ I , λ . Arguing as in the proof of Theorem 3.1, one has δ = lim inf r 0 + φ ( r ) < + . Now, arguing again as in the proof of Theorem 3.1, there is a sequence of positive numbers { η n } such that η n 0 + and T / 4 3 T / 4 F ( t , η n ) d t η n 2 > h for all n > ν and for some ν N . By choosing v n as in the proof of Theorem 3.1, the sequence { v n } strongly converges to 0 in X and Φ ( v n ) λ Ψ ( v n ) < 0 for each n > ν . Therefore, taking into account that ( Φ λ Ψ ) ( 0 ) = 0 , 0 is not a local minimum of Φ λ Ψ . The part (b) of Theorem 2.1 ensures that there exists a sequence { u n } in X of critical points of Φ λ Ψ such that lim n + u n = 0 and the proof is complete. □

        Let A ( t ) be a primitive of a ( t ) , g : [ 0 , T ] × R R an L 1 -Carathéodory function and put
        G ( t , ξ ) = 0 ξ g ( t , x ) d x , k ˜ : = 6 e a 1 ( 12 + T 2 b ) .
        Moreover, let
        α : = lim inf ξ + 0 T e A ( t ) max | x | ξ G ( t , x ) d t ξ 2 , β : = lim sup ξ + T / 4 3 T / 4 e A ( t ) G ( t , ξ ) d t ξ 2 .

        In virtue of Theorems 3.1 and 3.2, we obtain the following results for problem ( D λ , μ ).

        Theorem 3.3 Assume that

        (c1) G ( t , ξ ) 0 for all ( t , ξ ) ( [ 0 , T 4 ] [ 3 T 4 , T ] ) × R ;

        (c2) α < k ˜ β .

        Then, for every λ Λ : = ] 2 k ˜ T e a 1 β , 2 T e a 1 α [ and for every continuous function I j : R R , j = 1 , 2 , , n , whose potential I j ( ξ ) : = 0 ξ I j ( x ) d x , ξ R , satisfies

        (i1) sup ξ 0 I j ( ξ ) = 0 ,

        (i2) I : = lim sup ξ + j = 1 n max | t | ξ ( I j ( t ) ) ξ 2 < + ,

        there exists δ I , λ > 0 , where
        δ I , λ : = 1 I ( 2 T e a 1 λ α ) ,

        such that for each μ [ 0 , δ I , λ [ , problem ( D λ , μ ) has an unbounded sequence of weak solutions.

        Proof As seen in Section 2, we put p ( t ) = e A ( t ) , q ( t ) = b ( t ) e A ( t ) and f ( t , u ) = g ( t , u ) e A ( t ) , t [ 0 , T ] . Clearly, one has F ( t , u ) = e A ( t ) G ( t , u ) , A = α , B = β , p = 1 e a 1 , k k ˜ . Hence, from Theorem 3.1 the conclusion is achieved. □

        Remark 3.2 Theorem 1.1 in Introduction is an immediate consequence of Theorem 3.3. In fact, it is enough to observe that (c1) is verified and one has α = 0 and β = + , for which 1 Λ = ] 0 , + [ . Moreover, from I = lim ξ + j = 1 n ξ 2 / 2 ξ 2 = n 2 < + , one has δ ¯ = δ I , λ = 4 n T e T and the conclusion is achieved.

        Replacing the condition at infinity of the potential G by a similar one at zero, one establishes the following result. Put
        α : = lim inf ξ 0 + 0 T e A ( t ) max | x | ξ G ( t , x ) d t ξ 2 , β : = lim sup ξ 0 + T / 4 3 T / 4 e A ( t ) G ( t , ξ ) d t ξ 2 .

        Theorem 3.4 Assume

        (c1) G ( t , ξ ) 0 for all ( t , ξ ) ( [ 0 , T 4 ] [ 3 T 4 , T ] ) × R ;

        ( c 2 ) α < k ˜ β .

        Then, for every λ Λ : = ] 2 k ˜ T e a 1 β , 2 T e a 1 α [ and for every continuous function I j : R R , j = 1 , , n , whose potential I j ( ξ ) : = 0 ξ I j ( x ) d x , ξ R , satisfies

        (i1) sup ξ 0 I j ( ξ ) = 0 ,

        (j2) I 0 : = lim sup ξ 0 + j = 1 n max | t | ξ ( I j ( t ) ) ξ 2 < + ,

        there exists δ I , λ > 0 , where
        δ I , λ : = 1 I 0 ( 2 T e a 1 λ α ) ,

        such that for each μ [ 0 , δ I , λ [ , problem ( D λ , μ ) has a sequence of non-zero weak solutions, which strongly converges to 0.

        Proof The conclusion follows from Theorem 3.2 by arguing as in the proof of Theorem 3.3. □

        Remark 3.3 We point out that in Theorem 3.3 (as in Theorem 3.4) the assumption ess inf [ 0 , T ] a 0 can be deleted provided that we assume the constant k ˜ : = 6 min [ 0 , T ] e A ( t ) 12 + T 2 b e A and the interval Λ = ] 2 min [ 0 , T ] e A ( t ) k ˜ T β , 2 min [ 0 , T ] e A ( t ) T α [ .

        Finally, we observe that the existence of infinitely many solutions to problem ( D λ , μ ) can be obtained from Theorem 3.3 and Theorem 3.4 even under small perturbations of the nonlinearity. As an example, we point out the following consequence of Theorem 3.3.

        Corollary 3.2 Let g : [ 0 , T ] × R R be an L 1 -Carathéodory function satisfying (c1) and (c2) of Theorem 3.3.

        Then, for every λ Λ = ] 2 k ˜ T e a 1 β , 2 T e a 1 α [ , for every nonnegative L 1 -Carathéodory function h : [ 0 , T ] × R R , whose potential h ( t , ξ ) = 0 ξ h ( t , x ) d x satisfies
        H = lim sup ξ + 0 T H ( t , ξ ) d t ξ 2 < + ,
        and for every continuous function I j : R R , j = 1 , 2 , , n , whose potential I j ( ξ ) : = 0 ξ I j ( x ) d x , ξ R , satisfies (i1) and (i2) of Theorem 3.3, there exist γ H , λ > 0 and δ I , λ > 0 , where
        γ H , λ : = 1 H ( 2 p T λ α ) , δ I , λ : = 1 I ( 2 p T λ α )
        such that for all γ [ 0 , γ H , λ [ and for all μ [ 0 , δ I , λ [ , the problem
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-278/MediaObjects/13661_2013_Article_631_Equar_HTML.gif

        has an unbounded sequence of weak solutions.

        Proof It is enough to apply Theorem 3.3 to the following function:
        g ¯ ( t , x ) = g ( t , x ) + γ ¯ λ ¯ h ( t , x ) , ( t , x ) [ 0 , T ] × R ,
        where γ ¯ is fixed in [ 0 , γ H , λ [ and λ ¯ is fixed in Λ. In fact, one has
        α ¯ = lim inf ξ + 0 T e A ( t ) max | x | ξ G ¯ ( t , x ) d t ξ 2 α + γ ¯ λ ¯ H < α + γ H , λ λ ¯ H = α + 2 T e a 1 1 λ ¯ α = 2 T e a 1 1 λ ¯
        (9)
        and
        β ¯ = lim sup ξ + T / 4 3 T / 4 e A ( t ) G ¯ ( t , ξ ) d t ξ 2 lim sup ξ + T / 4 3 T / 4 e A ( t ) G ( t , ξ ) d t ξ 2 = β ,
        (10)

        for which α ¯ < 2 T e a 1 1 λ ¯ < 2 T e a 1 k ˜ T e a 1 β 2 = k ˜ β k ˜ β ¯ , that is, α ¯ < k ˜ β ¯ . Moreover, from (9) one has λ ¯ < 2 T e a 1 1 α ¯ and from (10) λ ¯ > 2 k ˜ T e a 1 1 β ¯ . Hence, λ ¯ ] 2 k ˜ T e a 1 1 β ¯ , 2 T e a 1 1 α ¯ [ and Theorem 3.3 ensures the conclusion. □

        4 Applications

        In many papers [13, 20, 22, 28] and [23], the authors obtain the existence of infinitely many solutions for problem ( D λ , μ ) while the impulsive term is supposed to be odd. The next examples provide problems that admit infinitely many solutions for which those other results cannot be applied.

        Example 4.1 Consider the following boundary value problem:
        { ( 4 t + 1 t + 1 u ( t ) ) + ( 1 + t ) u ( t ) = λ f ( t , u ( t ) ) , t [ 0 , 1 ] , t t j , u ( 0 ) = u ( 1 ) = 0 , Δ u ( t j ) = μ ( 2 u u 2 + 1 1 ) , j = 1 , 2 , , n ,
        (11)
        where f : [ 0 , 1 ] × R R is the function defined as follows:
        f ( t , u ) = { cos ( π 2 t ) u sin 2 ln ( u ) if  u > 0 , 0 if  u 0 .
        It is easy to see that conditions (a1), (a2), (i1) and (i2) of Theorem 3.1 hold. In particular, k = 3 4 3 + 1 and
        lim inf ξ + 0 1 max | x | ξ F ( t , x ) d t ξ 2 = 2 2 4 π , lim sup ξ + 1 / 4 3 / 4 F ( t , ξ ) d t ξ 2 = ( 2 + 2 ) 4 2 2 8 π .

        Then, for each λ [ 36 , 42 ] and for every μ 0 , problem (11) has an unbounded sequence of solutions in X.

        Now, we give an application of Theorem 3.4.

        Example 4.2 Consider the Dirichlet problem
        { u ( t ) + u ( t ) + u ( t ) = λ g ( t , u ( t ) ) , t [ 0 , 1 2 ] , t t 1 , u ( 0 ) = u ( 1 / 2 ) = 0 , Δ u ( t 1 ) = μ ( e u ( u 2 + 2 u ) ) ,
        (12)
        where g : [ 0 , 1 ] × R R is the function defined as follows:
        g ( t , u ) = { e t u ( 5 2 2 sin ( ln | u | ) cos ( ln | u | ) ) if  u 0 , 0 if  u = 0 .
        By a simple calculation, we get k = 24 49 e and
        lim inf ξ 0 + 0 T e A ( t ) max | x | ξ G ( t , x ) d t ξ 2 = 1 8 , lim sup ξ 0 + T / 4 3 T / 4 e A ( t ) G ( t , ξ ) d t ξ 2 = 9 16 .

        Then, from Theorem 3.4, for each λ [ 15 , 19 ] and for every μ [ 0 , 11 20 [ , problem (12) admits a sequence of pairwise distinct classical solutions strongly converging at 0. We observe that, in this case, as direct computations show, also zero is a solution of the problem.

        Declarations

        Authors’ Affiliations

        (1)
        Department of Civil, Information Technology, Construction, Environmental Engineering and Applied Mathematics, University of Messina
        (2)
        Department of Mathematics and Computer Science, University of Messina
        (3)
        Department of Mathematics, Baylor University

        References

        1. Benchohra M, Henderson J, Ntouyas S Contemporary Mathematics and Its Applications 2. In Theory of Impulsive Differential Equations. Hindawi Publishing Corporation, New York; 2006.View Article
        2. Chen L, Sun J: Nonlinear boundary value problem for first order impulsive functional differential equations. J. Math. Anal. Appl. 2006, 318: 726-741. 10.1016/j.jmaa.2005.08.012MathSciNetView Article
        3. Chu J, Nieto JJ: Impulsive periodic solutions of first-order singular differential equations. Bull. Lond. Math. Soc. 2008, 40(1):143-150. 10.1112/blms/bdm110MathSciNetView Article
        4. Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.View Article
        5. Agarwal RP, Franco D, O’Regan D: Singular boundary value problems for first and second order impulsive differential equations. Aequ. Math. 2005, 69: 83-96. 10.1007/s00010-004-2735-9MathSciNetView Article
        6. Baek H: Extinction and permanence of a three-species Lotka-Volterra system with impulsive control strategies. Discrete Dyn. Nat. Soc. 2008., 2008: Article ID 752403
        7. Lee EL, Lee YH: Multiple positive solutions of two point boundary value problems for second order impulsive differential equations. Appl. Math. Comput. 2004, 158: 745-759. 10.1016/j.amc.2003.10.013MathSciNetView Article
        8. Chen J, Tisdell CC, Yuan R: On the solvability of periodic boundary value problems with impulse. J. Math. Anal. Appl. 2007, 331: 902-912. 10.1016/j.jmaa.2006.09.021MathSciNetView Article
        9. Luo Z, Nieto JJ: New result for the periodic boundary value problem for impulsive integro-differential equations. Nonlinear Anal. TMA 2009, 70: 2248-2260. 10.1016/j.na.2008.03.004MathSciNetView Article
        10. Lee YH, Liu X: Study of singular boundary value problems for second order impulsive differential equation. J. Math. Anal. Appl. 2007, 331: 159-176. 10.1016/j.jmaa.2006.07.106MathSciNetView Article
        11. Mawhin J: Topological degree and boundary value problems for nonlinear differential equations. Lecture Notes in Math. 1537. In Topological Methods for Ordinary Differential Equations. Springer, Berlin; 1993:74-142.View Article
        12. Qian D, Li X: Periodic solutions for ordinary differential equations with sublinear impulsive effects. J. Math. Anal. Appl. 2005, 303: 288-303. 10.1016/j.jmaa.2004.08.034MathSciNetView Article
        13. Chen H, Li J: Variational approach to impulsive differential equations with Dirichlet boundary conditions. Bound. Value Probl. 2010., 2010: Article ID 3254152
        14. Liu Z, Chen H, Zhou T: Variational methods to the second-order impulsive differential equation with Dirichlet boundary value problem. Comput. Math. Appl. 2011, 61: 1687-1699. 10.1016/j.camwa.2011.01.042MathSciNetView Article
        15. Nieto JJ, O’Regan D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 2009, 70: 680-690.MathSciNetView Article
        16. Sun J, Chen H, Yang L: The existence and multiplicity of solutions for an impulsive differential equation with two parameters via a variational method. Nonlinear Anal. 2010, 73: 440-449. 10.1016/j.na.2010.03.035MathSciNetView Article
        17. Xiao J, Nieto JJ, Luo Z: Multiplicity of solutions for nonlinear second order impulsive differential equations with linear derivative dependence via variational methods. Commun. Nonlinear Sci. Numer. Simul. 2012, 17: 426-432. 10.1016/j.cnsns.2011.05.015MathSciNetView Article
        18. Zhang Z, Yuan R: An application of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal., Real World Appl. 2010, 11: 155-162. 10.1016/j.nonrwa.2008.10.044MathSciNetView Article
        19. Zhang D, Dai B: Infinitely many solutions for a class of nonlinear impulsive differential equations with periodic boundary conditions. Comput. Math. Appl. 2011, 61: 3153-3160. 10.1016/j.camwa.2011.04.003MathSciNetView Article
        20. Bai L, Dai B: An application of variational methods to a class of Dirichlet boundary value problems with impulsive effects. J. Franklin Inst. 2011, 348: 2607-2624. 10.1016/j.jfranklin.2011.08.003MathSciNetView Article
        21. Chen P, Tang XH: New existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. Math. Comput. Model. 2012, 55: 723-739. 10.1016/j.mcm.2011.08.046MathSciNetView Article
        22. Sun J, Chen H: Multiplicity of solutions for class of impulsive differential equations with Dirichlet boundary conditions via variant fountain theorems. Nonlinear Anal., Real World Appl. 2010, 11: 4062-4071. 10.1016/j.nonrwa.2010.03.012MathSciNetView Article
        23. Wang W, Yang X: Multiple solutions of boundary-value problems for impulsive differential equations. Math. Methods Appl. Sci. 2011, 34: 1649-1657. 10.1002/mma.1472MathSciNetView Article
        24. 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: 1-14.MathSciNet
        25. Bonanno G: A critical point theorem via the Ekeland variational principle. Nonlinear Anal. TMA 2012, 75(5):2992-3007. 10.1016/j.na.2011.12.003MathSciNetView Article
        26. Ricceri B: A general variational principle and some of its applications. J. Comput. Appl. Math. 2000, 113: 401-410. 10.1016/S0377-0427(99)00269-1MathSciNetView Article
        27. Brezis H: Analyse fonctionnelle; théorie et applications. Masson, Paris; 1983.
        28. Zhou J, Li Y: Existence and multiplicity of solutions for some Dirichlet problems with impulse effects. Nonlinear Anal. TMA 2009, 71: 2856-2865. 10.1016/j.na.2009.01.140View Article

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