Variational approach to Kirchhoff-type second-order impulsive differential systems

1 Introduction

In this study, we consider the impulsive nonlinear coupled differential system:

where 0 = t 0 < t 1 < t 2 < … < t n < t n + 1 = T , M 1 , M 2 : [ 0 , + ∞ ) → R are the continuous functions such that there are two positive constants m 0 and m 1 with m 0 ≤ M 1 ( x ) , M 2 ( x ) ≤ m 1 for all x ≥ 0 , F ( u ( t ) , v ( t ) ) : R 2 → R is a C 1 function, F u and F v denote the partial derivatives of F ( u ( t ) , v ( t ) ) with respect to u and v , respectively, norms ‖ u ‖ H 0 1 ( 0 , T ) and ‖ v ‖ H 0 1 ( 0 , T ) are specified later, I k , J k : R → R , k = 1 , 2 , … , n , are continuous, and

where

The equation of Kirchhoff type is

where L is the length of the string, h is the area of the cross-section, E is the Young’s modulus of the material, ρ is the mass density, and P 0 is the initial tension, which is proposed by Kirchhoff [1] as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. Some early classical studies dedicated to Kirchhoff equations can be seen in the studies of Bernstein [2] and Pohožaev [3]. However, after Lions [4] introduced an abstract framework to the following problem:

equation (2) began to arouse much attention. Some interesting results by variational methods can be found, for example, in [5–9].

On the other hand, impulsive differential equation is one of the main tools to study the dynamics of processes in which sudden, discontinuous jumps occur. In the last few years, variational methods are widely applied to study the existence and multiplicity of solutions for impulsive differential equations. The pioneering works of impulsive differential equations via variational methods were given in the studies of Tian and Ge [10] and Nieto and O’Regan [11]. Tian-Ge first studied the following impulsive differential equations by variational methods and obtained the existence of at least two positive solutions to the following problem:

where p > 1 , Φ p ( u ) = ∣ u ∣ p − 2 u , 0 = t 0 < t 1 < t 2 < … < t n < t n + 1 = T , Δ ( ρ ( t k ) Φ p ( u ′ ( t k ) ) ) = ρ ( t k + ) Φ p ( u ′ ( t k + ) ) − ρ ( t k − ) Φ p ( u ′ ( t k − ) ) , where u ′ ( t k + ) and u ′ ( t k − ) denote the right and left limits, respectively, of u ′ ( t ) at t = t k , I k ∈ C ( [ 0 , + ∞ ) , [ 0 , + ∞ ) ) , f ∈ C ( [ 0 , T ] × [ 0 , + ∞ ) , [ 0 , + ∞ ) ) , f ( t , 0 ) ≢ 0 , t ∈ [ 0 , T ] , ρ , s ∈ L ∞ [ 0 , T ] , ess inf [ 0 , T ] ρ > 0 , ess inf [ 0 , T ] s > 0 , 0 < ρ ( 0 ) , ρ ( T ) < ∞ , A ¯ ≤ 0 , B ¯ ≥ 0 , α , β , γ , σ > 0 , k = 1 , 2 , … , n .

Then, Nieto-O’Regan considered two classes of impulsive problems below, one of which is the linear problem:

the other is the nonlinear problem:

where 0 = t 0 < t 1 < t 2 < … < t n < t n + 1 = T , I k : R → R are continuous, f : [ 0 , T ] × R → R is continuous, d k are the fixed constants, k = 1 , 2 , … , n , σ ∈ L 2 ( 0 , T ) . They obtain the existence and multiplicity of solutions via variational methods.

Since then, there has been increasing research in this area. For some important and recent literature about such impulsive problems, we cite the works [12–18] for the interested readers.

The Kirchhoff-type second-order impulsive differential problems have been considered only by a few authors up to now. More precisely, Heidarkhani et al. [19] considered the following Kirchhoff-type second-order impulsive differential problem:

where K : [ 0 , + ∞ ) → R is a continuous function such that there are two positive constants m 0 and m 1 with m 0 ≤ K ( x ) ≤ m 1 for all x ≥ 0 , y ∈ L ∞ [ 0 , + ∞ ) , λ and μ are referred to as two control parameters, I k ∈ C ( R , R ) for 1 ≤ k ≤ p , 0 = t 0 < t 1 < t 2 < … < t p < + ∞ , Δ ( u ′ ( t k ) ) = u ′ ( t k + ) − u ′ ( t k − ) = lim t → t k + u ′ ( t ) − lim t → t k − u ′ ( t ) , f : [ 0 , + ∞ ) × R → R is an L 2 -Carathéodory function, and g : R → R is a Lipschitz continuous function. They obtained the existence of at least one weak solution and infinitely many weak solutions via variational methods when y ( t ) = q ( t ) and μ = λ , where q ∈ L ∞ [ 0 , + ∞ ) with 0 < q ̲ ≔ ess inf [ 0 , + ∞ ) q < q ¯ ≔ ess sup [ 0 , + ∞ ) q . When y ( t ) = q ( t ) , Caristi et al. [20] continued the study of problem (3) and obtained the existence of at least one weak solution, at least two weak solutions, and at least three weak solutions via variational approach. Based on previous works [19,20], Afrouzi et al. [21] also considered problem (3) when y ( t ) = q 2 , where q is a non-zero constant. The authors obtained the existence of at least three weak solutions using variational methods.

At present, to the best of our knowledge, there are few works on studying the existence and multiplicity of solutions for Kirchhoff-type second-order impulsive differential systems using variational methods. As a result, the goal of this study is to fill the gap in this area. Our main purpose is to study the existence and multiplicity of solutions of problem (1) under the nonlinearities F u and F v and impulsive functions I k , J k , k = 1 , 2 , … , n satisfying different growth conditions.

Throughout this study, we always assume that λ > − λ 1 and λ 1 is the first eigenvalue of the problem:

In this study, we make the following assumptions:

• (H1) F ( x , y ) is concave and there exist a , b > 0 and γ 1 , γ 2 ∈ [ 0 , 1 ) such that

  ∣ F x ( x , y ) ∣ ≤ a + b ∣ x ∣ γ 1 and ∣ F y ( x , y ) ∣ ≤ a + b ∣ y ∣ γ 2 , for ( x , y ) ∈ R 2 .

• (H2) There exist a k , b k > 0 and β k ∈ [ 0 , 1 ) ( k = 1 , 2 , … , n ) such that

  ∣ I k ( u ) ∣ ≤ a k + b k ∣ u ∣ β k and ∣ J k ( v ) ∣ ≤ a k + b k ∣ v ∣ β k , for u , v ∈ R .

• (H3) There exist c 1 > 0 and μ ∈ ( 1 , + ∞ ) such that

  ∣ F s ( s , t ) ∣ , ∣ F t ( s , t ) ∣ ≤ c 1 ( 1 + ∣ s ∣ μ + ∣ t ∣ μ ) .

• (H4) ∣ F s ( s , t ) ∣ , ∣ F t ( s , t ) ∣ = o ( ∣ s ∣ + ∣ t ∣ ) , as ∣ s ∣ + ∣ t ∣ → 0 .

• (H5) There exist c 2 > 0 , R 1 > 0 and 2 < θ < β such that

  F ( s , t ) ≥ c 2 ( ∣ s ∣ β + ∣ t ∣ β ) , ∣ s ∣ + ∣ t ∣ ≥ R 1 , and θ F ( s , t ) ≤ s F s ( s , t ) + t F t ( s , t ) ,

  where m 0 2 − m 1 θ > 0 .

• (H6)

  • (i) I k ( k = 1 , 2 , … , n ) satisfy I k ( x ) x ≥ 0 , for all x ∈ R ;

  • (ii) There exist θ > 2 , δ k > 0 , k = 1 , 2 , … , n , such that ∫ 0 x I k ( s ) d s ≤ δ k ∣ x ∣ θ , for x ∈ R \ { 0 } ;

  • (iii) I k ( x ) x ≤ θ ∫ 0 x I k ( s ) d s for x ∈ R \ { 0 } ;

    and

  • (i)′ J k ( k = 1 , 2 , … , n ) satisfy J k ( x ) x ≥ 0 , for all x ∈ R ;

  • (ii)′ There exist θ > 2 , δ k > 0 , k = 1 , 2 , … , n , such that ∫ 0 x J k ( s ) d s ≤ δ k ∣ x ∣ θ , for x ∈ R \ { 0 } ;

  • (iii)′ J k ( x ) x ≤ θ ∫ 0 x J k ( s ) d s , for x ∈ R \ { 0 } .

• (H7) There exist θ > 2 , δ k > 0 , k = 1 , 2 , … , n such that

  • (i) I k ( x ) x ≤ θ ∫ 0 x I k ( s ) d s < 0 , for x ∈ R \ { 0 } ;

  • (ii) ∫ 0 x I k ( s ) d s ≥ − δ k ∣ x ∣ θ , for x ∈ R \ { 0 } ;

    and

  • (i)′ J k ( x ) x ≤ θ ∫ 0 x J k ( s ) d s < 0 , for x ∈ R \ { 0 } ;

  • (ii)′ ∫ 0 x J k ( s ) d s ≥ − δ k ∣ x ∣ θ , for x ∈ R \ { 0 } .

The rest of this article is organized as follows. In Section 2, we present some preliminaries. In Section 3, we prove the main results via variational methods.

2 Variational structure

The following definition, lemma, and theorems will be needed in our argument.

Let C ( [ 0 , T ] ) be the space of all continuous functions on [ 0 , T ] with the norm

In the Sobolev space H 0 1 ( 0 , T ) , we consider the inner product

which induces the corresponding norm

It is a consequence of Poincaré’s inequality that

where λ 1 is the first eigenvalue of the Dirichlet problem (4). We can also define the inner product

which induces the corresponding norm

Then, the norm ‖ ⋅ ‖ is equivalent to the norm ‖ ⋅ ‖ H 0 1 ( 0 , T ) .

Set X = H 0 1 ( 0 , T ) × H 0 1 ( 0 , T ) , in the Sobolev space X , for any ( u , v ) ∈ X , we set the norm

and we also introduce the norm

Next, we will use variational methods to find the critical points of the functional Φ : X → R , defined by

for each ( u , v ) ∈ X , where M 1 ˜ ( t ) = ∫ 0 t M 1 ( ξ ) d ξ , M 2 ˜ ( t ) = ∫ 0 t M 2 ( ξ ) d ξ . It is clear that Φ is Fréchet differentiable at any ( u , v ) ∈ X and Φ ′ is continuous, with

Hence, the weak solutions of problem (1) are the corresponding critical points of Φ .

3 Proof of main results