Existence of Solutions for Sturm-Liouville Boundary Value Problem of Impulsive Differential Equations

Abstract

This paper is concerned with the existence of solutions for Sturm-Liouville boundary value problem of a class of second-order impulsive differential equations, under different assumptions on the nonlinearity and impulsive functions, existence criteria of single and multiple solutions are established. The main tools are variational method and critical point theorems. Some examples are also given to illustrate the main results.

1. Introduction

Impulsive differential equation is one of the main tools to study the dynamics of processes in which sudden changes occur. The theory of impulsive differential equation has recently received considerable attention, see [1–9]. Some classical tools such as fixed-point theorems in cones and the method of upper and lower solutions with monotone iterative technique have been widely used to study impulsive differential equations. In the last few years, variational method has been used to determine the existence of solutions for impulsive differential equation possessing a variational structure under certain boundary condition, one can refer to [10–26] and the references therein for detailed discussions.

Especially, in [12], Nieto and O’Regan studied the nonlinear Dirichlet impulsive problem: −𝑢^′′(𝑡)+𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡∈[0,𝑇]⧵𝑡₁,𝑡₂,…,𝑡_𝑝,Δ𝑢^′𝑡_𝑗=𝐼_𝑗𝑢𝑡⁻_𝑗𝑢(0)(𝑇),𝑗=1,2,…,𝑝,=𝑢=0, (1.1)(1.2)(1.3) where 𝑓∈𝐶([0,𝑇]×ℝ,ℝ),𝐼_𝑗∈𝐶(ℝ,ℝ),and 𝑗=1,2,…,𝑝. By the least action principle, the existence of a solution was obtained by assuming sublinear growth on the nonlinearity and impulses.

In [26], Zhou and Li discussed the problem: −𝑢^′′(𝑡)+𝑔(𝑡)𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡∈[0,𝑇]⧵𝑡₁,𝑡₂,…,𝑡_𝑝, (1.4) with impulse conditions (1.2), and Dirichlet boundary condition (1.3), where 𝑓∈𝐶([0,𝑇]×ℝ,ℝ) and 𝑔∈𝐿^∞([0,𝑇],ℝ). By using a symmetric Mountain Pass theorem, the existence result of an infinite number of solutions was obtained.

Tian and Ge in [20] studied the existence of multiple solutions for the following equation with impulsive effect: −𝑝(𝑡)𝑢^′′+𝑞(𝑡)𝑢=𝑓(𝑡,𝑢),𝑡∈[0,1]⧵𝑡₁,𝑡₂,…,𝑡_𝑝,−Δ𝑝𝑡_𝑗𝑢𝑡_𝑗=𝐼_𝑗𝑢𝑡_𝑗(0),𝑗=1,2,…,𝑝,𝛼𝑢−𝛽𝑢^′(0)=0,𝛾𝑢(1)+𝜎𝑢^′(1)=0. (1.5)(1.6)(1.7) By applying variational methods and upper and lower solutions methods, they obtained the existence of at least four solutions and gave some accurate characteristics of the solutions. In the special case of 𝑝(𝑡)≡1,𝑞(𝑡)≡𝜆 and 𝐼_𝑗(𝑢(𝑡_𝑗))=𝑑_𝑗, in [19] they studied the corresponding linear and nonlinear problem and established some existence results of positive solutions via critical point theory and variational methods. In [13], Sun and Chen considered the equation: −𝑝(𝑡)𝑢^′′+𝑟(𝑡)𝑢′+𝑞(𝑡)𝑢=𝑓(𝑡,𝑢),𝑡∈[0,1]⧵𝑡₁,𝑡₂,…,𝑡_𝑝, (1.8) with impulsive condition (1.6) and Neumann boundary condition 𝑢^′(0⁺)=𝑢^′(1⁻)=0, they got existence criteria of at least one solution, two solutions and infinitely many solutions under some different conditions.

In the above-cited articles, analogous results are also given when the impulses are absent, so they cannot reflect the impact of the impulses on the existence of the solutions. In [24], Zhang and Li studied the periodic solutions generated by impulses for a second-order impulsive differential equation. As defined in [24], a solution is called a solution generated by impulses if this solution is nontrivial only if impulsive terms are not zero.

Impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time. Many systems involve the impulsive condition which depends not only on the derivative of a function but also the function itself. A few papers discussed the solutions of impulsive differential equations involving impulses both on the function and on its derivative by critical point theory; see for example [21, 22].

Inspired by the above results, in this paper we consider the existence and multiplicity of solution to the following Sturm-Liouville boundary value problem of a class of second-order impulsive differential equations: −𝑢^′′(𝑡)+𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡∈[0,𝑇]⧵𝑡₁,𝑡₂,…,𝑡_𝑝,−Δ𝑢𝑡_𝑗=𝐴_𝑗𝑢𝑡⁻_𝑗,𝑗=1,2,…,𝑝,−Δ𝑢^′𝑡_𝑗=𝐼_𝑗𝑢𝑡⁻_𝑗−𝐴_𝑗𝑢^′𝑡⁺_𝑗(0),𝑗=1,2,…,𝑝,𝛼𝑢−𝛽𝑢^′(0)=0,𝛾𝑢(𝑇)+𝜎𝑢^′(𝑇)=0, (1.9)(1.10)(1.11)(1.12) where 𝑓∈𝐶([0,𝑇]×ℝ,ℝ), 𝛼,𝛽,𝛾,𝜎≥0, 𝛼+𝛽>0, 𝛾+𝜎>0. 𝑝∈ℕ, 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑝<𝑡_𝑝+1=𝑇, Δ𝑢(𝑡_𝑗)=𝑢(𝑡⁺_𝑗)−𝑢(𝑡⁻_𝑗), 𝑢(t⁺_𝑗) (or 𝑢(𝑡⁻_𝑗)) denotes the right limit (or left limit) of 𝑢(𝑡) at 𝑡=𝑡_𝑗,Δ𝑢^′(𝑡_𝑗)=𝑢^′(𝑡⁺_𝑗)−𝑢^′(𝑡⁻_𝑗),𝑢^′(𝑡⁺_𝑗) (or 𝑢^′(𝑡⁻_𝑗)) denotes the right limit (or left limit) of 𝑢^′(𝑡) at 𝑡=𝑡_𝑗, 𝐴_𝑗∈ℝ, 𝐼_𝑗∈𝐶(ℝ,ℝ), 𝑗=1,2,…,𝑝. For convenience, we denote (1.9)–(1.12) as problem (𝑃).

We begin by establishing the corresponding variational framework of problem (𝑃) in an appropriate space of functions. Then under the assumption that the nonlinearity and the impulsive functions are superlinear, we get the existence of at least one nontrivial solution. Furthermore, when 𝑓 is odd about the second variable and 𝐼_𝑗 are odd, the existence of an infinite number of solutions is obtained. Moreover, we study the existence of solutions generated by impulses. Under suitable hypothesis, existence criteria of at least one solution and infinitely many solutions generated by impulses of problem (𝑃) are established. The main tools are the Mountain Pass theorem, symmetric Mountain Pass theorem, and the least action principle.

Note that when 𝐴_𝑗=0, the impulsive conditions (1.10) and (1.11) of problem (𝑃) degenerate to the impulsive condition (1.2), and when 𝛽=𝜎=0, (1.12) is Dirichlet boundary condition; however, our assumptions on 𝑓 are different from the assumptions on 𝑓 in [12]. The construction of a suitable space of functions and corresponding energy functional become more complicated with the impulse effects of (1.10), (1.11), and Sturm-Liouville boundary conditions taken into consideration.

The rest of this paper is organized as follows. In Section 2, we list some preliminary results and several lemmas, which are important in proving the existence of solutions. In Section 3, we state and prove the main results and give some examples to illustrate them.

2. Preliminary

We define the space: 𝑋=𝑢∈𝐿²(0,𝑇)∣𝑢_{∣(𝑡𝑗,𝑡𝑗+1)}∈𝐻¹𝑡_𝑗,𝑡_𝑗+1𝑡,𝑗=0,1,…,𝑝,−Δ𝑢_𝑗=𝐴_𝑗𝑢𝑡⁻_𝑗,𝑗=1,…,𝑝,𝑢(0)=0if𝛽=0,𝑢(𝑇)=0if𝜎=0. (2.1) Its norm is induced by the inner product: (𝑢,𝑣)=^𝑇₀𝑢^′(𝑡)𝑣^′(𝑡)+𝑢(𝑡)𝑣(𝑡)𝑑𝑡,𝑢,𝑣∈𝑋, (2.2) that is, ‖𝑢‖=^𝑇₀𝑢^′(𝑡)²+𝑢²(𝑡)𝑑𝑡^1/2,𝑢∈𝑋. (2.3)

If 𝑢∈𝑋, then 𝑢_{∣(𝑡𝑗,𝑡𝑗+1)}∈𝐻¹(𝑡_𝑗,𝑡_𝑗+1), and thus the limits 𝑢(𝑡⁺_𝑗) and 𝑢(𝑡⁻_𝑗+1) exist, we define 𝑢^𝑗(𝑡)=𝑢(𝑡),𝑡∈𝑡_𝑗,𝑡_𝑗+1,𝑢𝑡⁺_𝑗,𝑡=𝑡_𝑗,𝑢𝑡⁻_𝑗+1,𝑡=𝑡_𝑗+1, (2.4) then 𝑢^𝑗∈𝐶[𝑡_𝑗,𝑡_𝑗+1].

Now we state some lemmas, which are needed in the proof of the main results.

Lemma 2.1. 𝑋 is a Hilbert space.

Proof. Clearly, 𝑋 is a linear space equipped with the inner product (2.2), so 𝑋 is an inner space. Next, we will show that 𝑋 is complete.
Suppose {𝑢_𝑛}⊂𝑋 is an arbitrary Cauchy sequence, from the definition of 𝑋, for 𝑗=0,1,…,𝑝, {𝑢^𝑗_𝑛} is also a Cauchy sequence in 𝐻¹(𝑡_𝑗,𝑡_𝑗+1). Then, there exists 𝑢^𝑗∈𝐻¹(𝑡_𝑗,𝑡_𝑗+1) with lim_𝑛→∞𝑢^𝑗_𝑛=𝑢^𝑗 in 𝐻¹(𝑡_𝑗,𝑡_𝑗+1); therefore, {𝑢^𝑗_𝑛} converges to 𝑢^𝑗 in 𝐶[𝑡_𝑗,𝑡_𝑗+1], that is, ‖𝑢^𝑗_𝑛−𝑢^𝑗‖_{𝐶[𝑡𝑗,𝑡𝑗+1]}→0 as 𝑛→∞. So we have 𝑢^𝑗𝑡_𝑗=lim_𝑛→∞𝑢^𝑗_𝑛𝑡_𝑗=lim_𝑛→∞𝑢_𝑛𝑡⁺_𝑗=lim_𝑛→∞1−𝐴_𝑗𝑢_𝑛𝑡⁻_𝑗=lim_𝑛→∞1−𝐴_𝑗𝑢_𝑛^𝑗−1𝑡_𝑗=1−𝐴_𝑗𝑢^𝑗−1𝑡_𝑗, (2.5) that is, 1−𝐴_𝑗𝑢^𝑗−1𝑡_𝑗=𝑢^𝑗𝑡_𝑗. (2.6) Set 𝑢(𝑡)=𝑢^𝑗(𝑡),𝑡∈𝑡_𝑗,𝑡_𝑗+1𝑢,𝑗=0,…,𝑝,₀0⁺𝑢,𝑡=0,_𝑝(𝑇⁻),𝑡=𝑇. (2.7) Then we have 1−𝐴_𝑗𝑢𝑡⁻_𝑗=1−𝐴_𝑗𝑢^𝑗−1𝑡_𝑗=𝑢^𝑗𝑡_𝑗=𝑢𝑡⁺_𝑗. (2.8) So −Δ𝑢(𝑡_𝑗)=𝐴_𝑗𝑢(𝑡⁻_𝑗) and 𝑢∈𝑋.
Observe that 𝑢_𝑛−𝑢²=^𝑝_𝑗=0^𝑡𝑗+1_𝑡𝑗𝑢^𝑗_𝑛^′(𝑡)−𝑢^𝑗′(𝑡)²+𝑢^𝑗_𝑛(𝑡)−𝑢^𝑗(𝑡)²=𝑑𝑡_𝑝𝑗=0𝑢^𝑗_𝑛−𝑢^𝑗2_{𝐻1(𝑡𝑗,𝑡𝑗+1)}, (2.9) since {𝑢^𝑗_𝑛} converge to 𝑢^𝑗 in 𝐻¹(𝑡_𝑗,𝑡_𝑗+1), then the Cauchy sequence {𝑢_𝑛} converge to 𝑢 in 𝑋, that is, 𝑋 is complete. So 𝑋 is a Hilbert space.

For 𝑢∈𝑋, we denote ‖𝑢‖_∞=max_{𝑗∈0,1,…,𝑝}max_{𝑡𝑡∈𝑗,𝑡𝑗+1}𝑢^𝑗(𝑡), (2.10) then we can get the following result.

Lemma 2.2. Let 𝑐=max_{𝑗∈{0,1,…,𝑝}}{(𝑡_𝑗+1−𝑡_𝑗)^−1/2,(𝑡_𝑗+1−𝑡_𝑗)^1/2}, then ‖𝑢‖_∞≤𝑐‖𝑢‖, (2.11) for every 𝑢∈𝑋.

Proof. Let 𝑢∈𝑋, by mean value theorem, there exists 𝜏∈[𝑡_𝑗,𝑡_𝑗+1], such that 𝑢^𝑗(𝜏)=1𝑡_𝑗+1−𝑡_𝑗^𝑡𝑗+1_𝑡𝑗𝑢^𝑗(𝜃)𝑑𝜃. (2.12) For 𝑡∈[𝑡_𝑗,𝑡_𝑗+1], from Hölder inequality, we have 𝑢^𝑗(𝑡)=𝑢^𝑗(𝜏)+^𝑡_𝜏𝑢^𝑗′(𝜃)≤1𝑑𝜃𝑡_𝑗+1−𝑡_𝑗^𝑡𝑗+1_𝑡𝑗𝑢^𝑗(𝜃)𝑑𝜃+^𝑡𝑗+1_𝑡𝑗𝑢^𝑗′(𝜃≤𝑡)𝑑𝜃_𝑗+1−𝑡_𝑗^{−1/2𝑡𝑗+1}_𝑡𝑗𝑢^𝑗(𝜃)²𝑑𝜃^1/2+𝑡_𝑗+1−𝑡_𝑗^{1/2𝑡𝑗+1}_𝑡𝑗𝑢^𝑗′(𝜃)²𝑑𝜃^1/2≤max_{𝑗∈0,1,…,𝑝}𝑡_𝑗+1−𝑡_𝑗^−1/2,𝑡_𝑗+1−𝑡_𝑗^{1/2𝑡𝑗+1}_𝑡𝑗𝑢^𝑗(𝜃)²+𝑢^𝑗′(𝜃)²𝑑𝜃^1/2=𝑐𝑢^𝑗_{𝐻1(𝑡𝑗,𝑡𝑗+1)}≤𝑐‖𝑢‖. (2.13)

Definition 2.3. A function 𝑢∈𝐿²(0,𝑇) is said to be a classical solution of problem (𝑃), if 𝑢_{|𝑗)(𝑡,𝑡𝑗+1}∈𝐶²(𝑡_j,𝑡_𝑗+1) for 𝑗=0,1,2,…,𝑝, and 𝑢 satisfies (1.9) on [0,𝑇]⧵{𝑡₁,𝑡₂,…,𝑡_𝑝}, the limits 𝑢^′(𝑡⁺_𝑗),𝑢^′(𝑡⁻_𝑗),𝑢(𝑡⁺_𝑗),𝑢(𝑡⁻_𝑗) exist, and (1.10), (1.11), and (1.12) hold.

For convenience, denote 𝛼₁=𝛼𝛽𝛼,𝛽≠0,0,𝛽=0,²=𝛾𝜎𝐹()=,𝜎≠0,0,𝜎=0,𝑡,𝑢^𝑢₀𝑓(𝑡,𝜉)𝑑𝜉. (2.14) For 𝑢∈𝑋, set 𝜑(𝑢)=12^𝑇₀𝑢^′(𝑡)²+𝑢²(𝑡1)𝑑𝑡+2𝛼₁𝑢²(0)+12𝛼₂𝑢²(𝑇)−^𝑇₀𝐹((𝑡−𝑡,𝑢))𝑑𝑡_{𝑝𝑗=1𝑢(𝑡−𝑗)0}𝐼_𝑗(𝑠)𝑑𝑠. (2.15) From the assumption of the continuity of 𝑓 and 𝐼_𝑗(𝑗=1,2,…,𝑝), we can see that 𝜑∈𝐶¹(𝑋,ℝ) and 𝜑^′(𝑢),𝑣=^𝑇₀𝑢^′(𝑡)𝑣^′(𝑡)+𝑢(𝑡)𝑣(𝑡)𝑑𝑡+𝛼₁𝑢(0)𝑣(0)+𝛼₂𝑢(𝑇)𝑣(𝑇)−^𝑇₀𝑓(𝑡,𝑢(𝑡))𝑣(𝑡)𝑑𝑡−^𝑝_𝑗=1𝐼_𝑗𝑢𝑡⁻_𝑗𝑣𝑡⁻_𝑗, (2.16) for 𝑢,𝑣∈𝑋.

Lemma 2.4. If the function 𝑢∈𝑋 is a critical point of 𝜑, then u is a classical solution of problem (𝑃).

Proof. Suppose that 𝑢∈𝑋 is a critical point of 𝜑, then (𝜑^′(𝑢),𝑣)=0 for every 𝑣∈𝑋. By (2.16), we have ^𝑇₀𝑢^′𝑣^′+𝑢𝑣𝑑𝑡+𝛼₁𝑢(0)𝑣(0)+𝛼₂𝑢(𝑇)𝑣(𝑇)−^𝑇₀𝑓(𝑡,𝑢)𝑣𝑑𝑡−^𝑝_𝑗=1𝐼_𝑗𝑢𝑡⁻_𝑗𝑣𝑡⁻_𝑗=0. (2.17) For 𝑗∈{0,1,2,…,𝑝}, choose 𝑣=0 on [0,𝑇]⧵(𝑡_𝑗,𝑡_𝑗+1) and 𝑣_{|(𝑡𝑗,𝑡𝑗+1)}∈𝐶^∞₀(𝑡_𝑗,𝑡_𝑗+1), then 𝑣∈𝑋 and ^𝑡𝑗+1_𝑡𝑗𝑢′𝑣′𝑑𝑡=^𝑡𝑗+1_𝑡𝑗𝑓(𝑡,𝑢)−𝑢𝑣𝑑𝑡. (2.18)
By the definition of 𝐻¹(𝑡_𝑗,𝑡_𝑗+1) and since 𝑢_{|𝑗)(𝑡,𝑡𝑗+1}∈𝐻¹(𝑡_𝑗,𝑡_𝑗+1)⊂𝐶[𝑡_𝑗,𝑡_𝑗+1],𝑓∈𝐶([0,𝑇]×ℝ,ℝ), (2.18) gives us ((𝑢_{|𝑗)(𝑡,𝑡𝑗+1})^′)^′∈𝐶[𝑡_𝑗,𝑡_𝑗+1], thus (𝑢_{|𝑗)(𝑡,𝑡𝑗+1})^′∈𝐶¹[𝑡_𝑗,𝑡_𝑗+1] and 𝑢_{|𝑗)(𝑡,𝑡𝑗+1}∈𝐶²[𝑡_𝑗,𝑡_𝑗+1], so the limits 𝑢^′(𝑡⁺_𝑗),𝑢^′(𝑡⁻_𝑗) exist, then integrating (2.18) by parts, we obtain ^𝑡𝑗+1_𝑡𝑗−𝑢^′′+𝑢−𝑓(𝑡,𝑢)𝑣𝑑𝑡=0,𝑣∈𝑋,𝑣_{|(𝑡𝑗,𝑡𝑗+1)}∈𝐶^∞₀𝑡_𝑗,𝑡_𝑗+1. (2.19) So −𝑢^′′+𝑢−𝑓(𝑡,𝑢)=0,𝑡∈𝑡_𝑗,𝑡_𝑗+1. (2.20) Thus, we can get that 𝑢 satisfies (1.9) for 𝑡∈(0,𝑇)⧵{𝑡₁,…,𝑡_𝑝}.
In view of (2.17), (2.19), and −Δ𝑢(𝑡_𝑗)=𝐴_𝑗𝑢(𝑡⁻_𝑗), we get that 0=^𝑝_𝑗=0^𝑡𝑗+1_𝑡𝑗𝑢′𝑣′𝑑𝑡+^𝑇₀𝑢−𝑓(𝑡,𝑢)𝑣𝑑𝑡+𝛼₁𝑢(0)𝑣(0)+𝛼₂𝑢(𝑇)𝑣(𝑇)−^𝑝_𝑗=1𝐼_𝑗𝑢𝑡⁻_𝑗𝑣𝑡⁻_𝑗=^𝑝_𝑗=0𝑢′𝑡⁻_𝑗+1𝑣𝑡⁻_𝑗+1−𝑢′𝑡⁺_𝑗𝑣𝑡⁺_𝑗+^𝑇₀−𝑢^′′+𝑢−𝑓(𝑡,𝑢)𝑣𝑑𝑡+𝛼₁𝑢(0)𝑣(0)+𝛼₂𝑢(𝑇)𝑣(𝑇)−^𝑝_𝑗=1𝐼_𝑗𝑢𝑡⁻_𝑗𝑣𝑡⁻_𝑗=^𝑝_𝑗=1𝑢^′𝑡⁻_𝑗𝑣𝑡⁻_𝑗−𝑢^′𝑡⁺_𝑗𝑣𝑡⁻_𝑗+𝑢^′𝑡⁺_𝑗𝑣𝑡⁻_𝑗−𝑢^′𝑡⁺_𝑗𝑣𝑡⁺_𝑗+𝑢^′(𝑇)𝑣(𝑇)−𝑢^′(0)𝑣(0)+𝛼₁𝑢(0)𝑣(0)+𝛼₂𝑢(𝑇)𝑣(𝑇)−^𝑝_𝑗=1𝐼_𝑗𝑢𝑡⁻_𝑗𝑣𝑡⁻_𝑗=−^𝑝_𝑗=1Δ𝑢^′𝑡_𝑗+𝐼_𝑗𝑢𝑡⁻_𝑗−𝐴_𝑗𝑢^′𝑡⁺_𝑗𝑣𝑡⁻_𝑗+𝛼₁𝑢(0)−𝑢^′(0)𝑣(0)+𝛼₂𝑢(𝑇)+𝑢^′(𝑇)𝑣(𝑇). (2.21) If we choose 𝑣∈𝑋, such that 𝑣(0)=𝑣(𝑇)=0, then it follows from the above equality that −Δ𝑢^′𝑡_𝑗=𝐼_𝑗𝑢𝑡⁻_𝑗−𝐴_𝑗𝑢^′𝑡⁺_𝑗, (2.22) and the impulsive condition (1.11) holds.
If 𝛽=𝜎=0, then from the definition of 𝑋, we know that 𝑢(0)=𝑢(𝑇)=0, and the boundary condition (1.12) holds. If 𝛽≠0, 𝜎≠0, in view of (2.21) and (2.22), we can conclude that 𝛼₁𝑢(0)−𝑢^′(0)𝑣(0)+𝛼₂𝑢(𝑇)+𝑢^′(𝑇)𝑣(𝑇)=0,𝑣∈𝑋. (2.23) Since 𝑣(0) and 𝑣(𝑇) are arbitrary, we deduce that 𝛼₁𝑢(0)−𝑢^′(0)=0,𝛼₂𝑢(𝑇)+𝑢^′(𝑇)=0, (2.24) and 𝑢 satisfies the boundary condition (1.12). If one of 𝛽 and 𝜎 is equal to 0, similarly we can obtain the conclusion.
Therefore, 𝑢 is a classical solution of problem (𝑃).

Lemma 2.5 (Mountain Pass theorem [10, Theorem 2.2]). Let 𝐸 be a real Banach space and 𝜑∈𝐶¹(𝐸,ℝ) satisfying the (PS) condition. Suppose 𝜑(0)=0 and(𝐶₁) there are constants 𝜌,𝛼>0 such that 𝜑|_𝜕𝐵𝜌≥𝛼, where 𝐵_𝜌={𝑥∈𝐸∶‖𝑥‖<𝜌};(𝐶₂) there is an 𝑒∈𝐸⧵𝐵_𝜌 such that 𝜑(𝑒)≤0.Then 𝜑 possesses a critical value 𝑐≥𝛼. Moreover c can be characterized as 𝑐=inf_𝑔∈Γmax_{𝑢∈𝑔([0,1])}𝜑(𝑢), (2.25) where Γ=𝑔∈𝐶([0,1],𝐸)∣𝑔(0)=0,𝑔(1)=𝑒. (2.26)

Lemma 2.6 (see [10, Theorem 9.12]). Let 𝐸 be an infinite dimensional Banach space, and let 𝜑∈𝐶¹(𝐸,ℝ) be even functional which satisfies the (PS) condition, and 𝜑(0)=0. If 𝐸=𝑉⊕𝑋, where 𝑉 is finite dimensional, and 𝜑 satisfies(𝐶^′₁) there are constants 𝜌,𝛼>0 such that 𝜑|_{𝜕𝐵𝜌⋂𝑋}≥𝛼, where 𝐵_𝜌={𝑥∈𝐸∶‖𝑥‖<𝜌};(𝐶^′₂) for each finite dimensional subspace 𝐸⊂𝐸, there is an 𝑅=𝑅(𝐸) such that 𝜑≤0 on 𝐸⧵𝐵_𝑅(𝐸₎.Then 𝜑 possesses an unbounded sequence of critical values.

3. Main Results

Lemma 3.1. If there exist 𝜇>2 and 𝛿>0 such that 0<𝜇𝐹(𝑡,𝑥)≤𝑥𝑓(𝑡,𝑥),for𝑡∈[0,𝑇],|𝑥|≥𝛿,0<𝜇^𝑥₀𝐼_𝑗(𝑠)𝑑𝑠≤𝑥𝐼_𝑗(𝑥),for|𝑥|≥𝛿,𝑗=1,2,…,𝑝, (3.1)(3.2) then the functional 𝜑 satisfies the (PS) condition.

Proof. Let {𝑢_𝑛} be a sequence in 𝑋 satisfying that {𝜑(𝑢_𝑛)} is bounded and 𝜑^′(𝑢_𝑛)→0 as 𝑛→∞. Now we shall prove that {𝑢_𝑛} is a bounded sequence in 𝑋. By (2.15), (2.16), (3.1), and (3.2), we have 𝜇𝜑𝑢_𝑛−𝜑^′𝑢_𝑛,𝑢_𝑛=𝜇2^𝑇₀𝑢^′_𝑛²+𝑢_𝑛²𝑑𝑡+𝜇𝛼₁2𝑢_𝑛(0)²+𝜇𝛼₂2𝑢_𝑛(𝑇)²−𝜇^𝑇₀𝐹𝑡,𝑢_𝑛𝑑𝑡−𝜇^𝑝_𝑗=1^{𝑢𝑛(𝑡−𝑗)}₀𝐼_𝑗(𝑠)−𝑑𝑠_𝑇0𝑢^′_𝑛²+𝑢_𝑛²𝑑𝑡−𝛼₁𝑢_𝑛(0)²−𝛼₂𝑢_𝑛(𝑇)²+^𝑇₀𝑓𝑡,𝑢_𝑛𝑢_𝑛𝑑𝑡+^𝑝_𝑗=1𝐼_𝑗𝑢_𝑛𝑡⁻_𝑗𝑢_𝑛𝑡⁻_𝑗≥𝜇2−1𝑢_𝑛²+𝛼₁𝑢_𝑛(0)²+𝛼₂𝑢_𝑛(𝑇)²+𝑇min_{[],|𝑥|𝑡∈0,𝑇≤𝛿}𝑓()()+𝑡,𝑥𝑥−𝜇𝐹𝑡,𝑥_𝑝𝑗=1min_{|𝑥|≤𝛿}𝐼_𝑗(𝑥)𝑥−𝜇^𝑥₀𝐼_𝑗(𝑠)≥𝜇𝑑𝑠2−1𝑢_𝑛²+𝑀, (3.3) where 𝑀=𝑇min_{𝑡∈[0,𝑇],|𝑥|≤𝛿}(𝑓(𝑡,𝑥)𝑥−𝜇𝐹(𝑡,𝑥))+∑^𝑝_𝑗=1min_{|𝑥|≤𝛿}(𝐼_𝑗(𝑥)𝑥−𝜇∫^𝑥₀𝐼_𝑗(𝑠)𝑑𝑠) is a constant. So {𝑢_𝑛} is bounded in 𝑋.
From the reflexivity of 𝑋, we may extract a weakly convergent subsequence that, for simplicity, we call {𝑢_𝑛}, 𝑢_𝑛⇀𝑢. In what follows, we will prove that {𝑢_𝑛} strongly converges to 𝑢. By 𝑢_𝑛⇀𝑢, we see that, for 𝑗=0,1,…,𝑝, 𝑢^𝑗_𝑛⇀𝑢^𝑗 in 𝐻¹(𝑡_𝑗,𝑡_𝑗+1), and {𝑢^𝑗_𝑛} converges to 𝑢^𝑗 in 𝐶[𝑡_𝑗,𝑡_𝑗+1]. From the continuity of 𝑓 and 𝐼_𝑗(𝑗=1,2,…,𝑝), we know that ^𝑇₀𝑓𝑡,𝑢_𝑛−𝑓(𝑡,𝑢)𝑢_𝑛−𝑢𝑑𝑡⟶0,as𝑛⟶∞,_𝑝𝑗=1𝐼_𝑗𝑢_𝑛𝑡⁻_𝑗−𝐼_𝑗𝑢𝑡⁻_𝑗𝑢_𝑛𝑡⁻_𝑗−𝑢𝑡⁻_𝑗⟶0,as𝑛⟶∞. (3.4) In view of (2.16), we get 𝜑^′𝑢_𝑛−𝜑^′(𝑢),𝑢_𝑛=𝑢−𝑢_𝑛−𝑢²+𝛼₁𝑢_𝑛(0)−𝑢(0)²+𝛼₂𝑢_𝑛(𝑇)−𝑢(𝑇)²+^𝑇₀𝑓𝑡,𝑢_𝑛−𝑓(𝑡,𝑢)𝑢_𝑛−𝑢𝑑𝑡+^𝑝_𝑗=1𝐼_𝑗𝑢_𝑛𝑡⁻_𝑗−𝐼_𝑗𝑢𝑡⁻_𝑗𝑢_𝑛𝑡⁻_𝑗−𝑢𝑡⁻_𝑗. (3.5) By 𝜑^′(𝑢_𝑛)→0 and 𝑢_𝑛⇀𝑢, we obtain that 𝜑^′𝑢_𝑛−𝜑^′(𝑢),𝑢_𝑛−𝑢⟶0,as𝑛⟶∞. (3.6) From the previous discussions, it is easy to see that ‖𝑢_𝑛−𝑢‖→0 as 𝑛→∞. Therefore, 𝜑 satisfies the (PS) condition.

Theorem 3.2. Suppose that (3.1), (3.2) hold and 𝑓()(𝑥)[],𝐼𝑡,𝑥=𝑜,𝑥⟶0,𝑡∈0,𝑇_𝑗(𝑥)=𝑜(𝑥),𝑥⟶0,𝑗=1,2,…,𝑝, (3.7)(3.8) then problem (𝑃) has at least one nontrivial solution.

Proof. In order to show that problem (𝑃) has at least one nontrivial solution, that is, 𝜑 has at least one nontrivial critical point, it is sufficient to check the conditions in Lemma 2.5. Clearly 𝜑∈𝐶¹(𝑋,𝑅), 𝜑(0)=0, and 𝜑 satisfies the (PS) condition from Lemma 3.1.
First, it follows from (3.7) and (3.8) that there exists a constant 𝜀>0, such that for |𝑥|≤𝜀: |𝐹(≤1𝑡,𝑥)|8𝑐²𝑇𝑥²,^𝑥₀𝐼_𝑗(𝑠)1𝑑𝑠≤8𝑐²𝑝𝑥²,𝑗=1,2,…,𝑝, (3.9) where 𝑐 is given in Lemma 2.2. Choose 𝑢∈𝑋 with ‖𝑢‖≤𝜀/𝑐, then ‖𝑢‖_∞≤𝜀, and by (3.9) and (2.11), we have 𝜑(𝑢)=12‖𝑢‖²+12𝛼₁𝑢²(0)+12𝛼₂𝑢²(𝑇)−^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡−^𝑝_𝑗=1^{𝑢(𝑡−𝑗)}₀𝐼_𝑗(𝑠)≥1𝑑𝑠2‖𝑢‖²−^𝑇₀18𝑐²𝑇𝑢²(𝑡)𝑑𝑡−^𝑝_𝑗=118𝑐²𝑝𝑢𝑡⁻_𝑗²≥12‖𝑢‖²−^𝑇₀18𝑐²𝑇‖𝑢‖²_∞𝑑𝑡−^𝑝_𝑗=118𝑐²𝑝‖𝑢‖²_∞=12‖𝑢‖²−14𝑐²‖𝑢‖²_∞≥12‖𝑢‖²−14‖𝑢‖²=14‖𝑢‖². (3.10) If we choose 𝛼=𝜀²/4𝑐², 𝜌=𝜀/𝑐, then 𝜑|_𝜕𝐵𝜌≥𝛼. Thus, condition (𝐶₁) of Lemma 2.5 holds.
By (3.1) and (3.2), we know that there are positive constants 𝑐₁,𝑐₂,𝑑₁,and𝑑₂ such that 𝐹(𝑡,𝑥)≥𝑐₁|𝑥|^𝜇−𝑐₂,(𝑡,𝑥)∈[0,𝑇]×ℝ,_𝑥0𝐼_𝑗(𝑠)𝑑𝑠≥𝑑₁|𝑥|^𝜇−𝑑₂,𝑥∈ℝ,𝑗=1,2,…,𝑝. (3.11)(3.12) If we choose 𝑢₀∈𝑋⧵{0}, then for any 𝑟>0, we have 𝜑𝑟𝑢₀=12𝑟𝑢₀²+12𝛼₁𝑟𝑢₀(0)²+12𝛼₂𝑟𝑢₀(𝑇)²−^𝑇₀𝐹𝑡,𝑟𝑢₀𝑑𝑡−^𝑝_𝑗=1^{𝑟𝑢0(𝑡−𝑗)}₀𝐼_𝑗(𝑠)≤1𝑑𝑠2𝑟𝑢₀²+𝑟²2𝛼₁𝑢₀(0)²+𝛼₂𝑢₀(𝑇)²−^𝑇₀𝑐₁𝑟𝑢₀^𝜇−𝑐₂𝑑𝑡−^𝑝_𝑗=1𝑑₁𝑟𝑢₀𝑡⁻_𝑗^𝜇+𝑝𝑑₂=𝑟²2𝑢₀²+𝛼₁𝑢₀(0)²+𝛼₂𝑢₀(𝑇)²−𝑐₁^𝑇₀𝑢₀^𝜇𝑑𝑡+𝑑₁^𝑝_𝑗=1𝑢₀𝑡⁻_𝑗^𝜇𝑟^𝜇+𝑐₂𝑇+𝑝𝑑₂. (3.13) Since 𝜇>2, we can find 𝑟>1 such that ‖𝑟𝑢₀‖>𝜌 and 𝜑(𝑟𝑢₀)<0. Hence the condition (𝐶₂) of Lemma 2.5 is satisfied.

Theorem 3.3. Under the same assumptions as Theorem 3.2, if 𝑓(𝑡,𝑥) is odd in 𝑥, and 𝐼_𝑗(𝑗=1,2,…,𝑝) are odd, then problem (𝑃) has infinitely many solutions.

Proof. The proof follows the analogous ideas as that we have developed for Theorem 3.2. By the assumption we know that 𝜑(𝑢) is even, 𝜑∈𝐶¹(𝑋,𝑅),𝜑(0)=0, and 𝜑 satisfies the (PS) condition from Lemma 3.1.
If we let 𝑉 be any finite dimensional subspace of 𝑋, and 𝑊=𝑉^⟂, then for any 𝑢∈𝑊 with ‖𝑢‖≤𝜀/𝑐, it follows from (3.10) that 𝜑(𝑢)≥𝛼,for𝑢∈𝜕𝐵_𝜌∩𝑊, (3.14) where 𝛼=𝜀²/4𝑐²,𝜌=𝜀/𝑐. So 𝜑 satisfies the condition (𝐶^′₁). On the other hand, fix 𝑢₀∈𝑉⧵{0}, the same as (3.13), we can find some 𝑟>1 such that 𝑟𝑢₀>𝜌,𝜑𝑟𝑢₀<0, (3.15) then (𝐶^′₂) of Lemma 2.6 is true. Thus, 𝜑 possesses an infinite number of critical points, that is, problem (𝑃) has infinitely many solutions.

When 𝐴_𝑗≡0 and 𝐼_𝑗≡0 for 𝑗=1,2,…,𝑝, problem (𝑃) degenerates to the following problem (𝑃₀): −𝑢^′′(𝑡)(𝑡)((𝑡[],(0)+𝑢=𝑓𝑡,𝑢)),𝑡∈0,𝑇𝛼𝑢−𝛽𝑢^′(0)=0,𝛾𝑢(𝑇)+𝜎𝑢^′(𝑇)=0. (3.16) Now, we show the nonexistence of solutions for problem (𝑃₀).

Lemma 3.4. Assume that 𝑓 satisfies 𝑥𝑓(𝑡,𝑥)≤0,(𝑡,𝑥)∈[0,𝑇]×ℝ. (3.17) Then, (𝑃₀) has no nontrivial solution.

Proof. If 𝑢 is a nontrivial solution of (𝑃₀), then we have 0=−^𝑇₀𝑢^′′𝑢𝑑𝑡+^𝑇₀𝑢²𝑑𝑡−^𝑇₀𝑓()=𝑡,𝑢𝑢𝑑𝑡_𝑇0𝑢^′2+𝑢²𝑑𝑡+𝛼₁𝑢²(0)+𝛼₂𝑢²(𝑇)−^𝑇₀𝑓()=‖𝑢‖𝑡,𝑢𝑢𝑑𝑡₂+𝛼₁𝑢²(0)+𝛼₂𝑢²(𝑇)−^𝑇₀𝑓(𝑡,𝑢)𝑢𝑑𝑡≥0. (3.18) Since 𝛼₁≥0,𝛼₂≥0 and (3.17), we obtain 𝑢≡0.

Remark 3.5. By Lemma 3.4, if we get a solution of problem (𝑃) under the assumption of (3.17), then the solution is generated by impulses.
The next theorem gives some sufficient conditions that problem (𝑃) has at least one solution and infinitely many solutions generated by impulses under the assumption of (3.17), compared with Theorem 3.2, we change the conditions on nonlinearity and make certain requirements about 𝐴_𝑗,𝑗=1,2,…,𝑝.

Theorem 3.6. Suppose 0≤𝐴_𝑗≤1 for 𝑗=1,2,…,𝑝, (3.2), (3.8), and (3.17) hold, and there exist 𝑎>0 and 𝜂∈(0,1) such that 𝑓(𝑡,𝑥)≥−𝑎|𝑥|^𝜂,(𝑡,𝑥)∈[0,𝑇]×ℝ. (3.19) Then problem (𝑃) has at least one nontrivial solution generated by impulses. Furthermore, if 𝑓(𝑡,𝑥) is odd in 𝑥, and 𝐼_𝑗(𝑗=1,2,…,𝑝) are odd, then problem (𝑃) has infinitely many solutions generated by impulses.

Proof. Firstly, we show that 𝜑 satisfies the (PS) condition. Let {𝑢_𝑛} be a sequence in 𝑋 satisfying that {𝜑(𝑢_𝑛)} is bounded and 𝜑^′(𝑢_𝑛)→0 as 𝑛→∞. Now, we shall prove that {𝑢_𝑛} is a bounded sequence in 𝑋.
Equation (3.17) implies that 𝐹(𝑡,𝑥)≤0,(𝑡,𝑥)∈[0,𝑇]×ℝ. (3.20) By (3.2), (3.19), and (3.20), we have 𝜇𝜑𝑢_𝑛−𝜑^′𝑢_𝑛,𝑢_𝑛=𝜇2−1^𝑇₀𝑢^′_𝑛²+𝑢_𝑛²𝜇𝑑𝑡+2−1𝛼₁𝑢_𝑛(0)²+𝜇2−1𝛼₂𝑢_𝑛(𝑇)²−𝜇^𝑇₀𝐹𝑡,𝑢_𝑛𝑑𝑡+^𝑇₀𝑓𝑡,𝑢_𝑛𝑢_𝑛𝑑𝑡−𝜇^𝑝_𝑗=1^{𝑢𝑛(𝑡−𝑗)}₀𝐼_𝑗(𝑠)𝑑𝑠+^𝑝_𝑗=1𝐼_𝑗𝑢_𝑛𝑡⁻_𝑗𝑢_𝑛𝑡⁻_𝑗≥𝜇2−1𝑢_𝑛²+^𝑇₀𝑓𝑡,𝑢_𝑛𝑢_𝑛𝑑𝑡−𝜇^𝑝_𝑗=1^{𝑢𝑛(𝑡−𝑗)}₀𝐼_𝑗(𝑠)𝑑𝑠+^𝑝_𝑗=1𝐼_𝑗𝑢_𝑛𝑡⁻_𝑗𝑢_𝑛𝑡⁻_𝑗≥𝜇2−1𝑢_𝑛²−𝑎𝑇𝑢_𝑛∞^𝜂+1+^𝑝_𝑗=1min_{|𝑥|≤𝛿}𝐼_𝑗(𝑥)𝑥−𝜇^𝑥₀𝐼_𝑗(𝑠)≥𝜇𝑑𝑠2−1𝑢_𝑛²−𝑎𝑇𝑐^𝜂+1𝑢_𝑛^𝜂+1+𝑀₁, (3.21) where 𝑀₁=∑^𝑝_𝑗=1min_{|𝑥|≤𝛿}(𝐼_𝑗(𝑥)𝑥−𝜇∫^𝑥₀𝐼_𝑗(𝑠)𝑑𝑠) is a constant. So {𝑢_𝑛} is bounded in 𝑋, since 𝜂<1,𝜇>2. Similar to the proof in Lemma 3.1, we can see that {𝑢_𝑛} has a convergent subsequence in 𝑋. So, 𝜑 satisfies the (PS) condition.
From (3.8), there exists a constant 𝜀₁>0 such that for 𝑗=1,2,…,𝑝, we have ^𝑥₀𝐼_𝑗(𝑠)1𝑑𝑠≤4𝑐²𝑝𝑥²,|𝑥|≤𝜀₁, (3.22) where 𝑐 is given in Lemma 2.2. Hence combining this with (3.20), for 𝑢∈𝑋 with ‖𝑢‖≤𝜀₁/𝑐, we have 𝜑(𝑢)=12‖𝑢‖²+12𝛼₁𝑢²(0)+12𝛼₂𝑢²(𝑇)−^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡−^𝑝_𝑗=1^{𝑢(𝑡−𝑗)}₀𝐼_𝑗(𝑠)≥1𝑑𝑠2‖𝑢‖²−^𝑝_𝑗=1^{𝑢(𝑡−𝑗)}₀𝐼_𝑗(𝑠)≥1𝑑𝑠2‖𝑢‖²−^𝑝_𝑗=114𝑐²𝑝𝑢²𝑡⁻_𝑗≥12‖𝑢‖²−14𝑐²‖𝑢‖²_∞≥12‖𝑢‖²−14‖𝑢‖²=14‖𝑢‖². (3.23) If we choose 𝛼=𝜀²₁/4𝑐²,𝜌=𝜀₁/𝑐, then 𝜑|_𝜕𝐵𝜌≥𝛼. Thus, condition (𝐶₁) in Lemma 2.5 holds.
For some constant 𝑘>0, let 𝑢(𝑡)=𝑘𝑡,𝑡∈0,𝑡₁,𝑘𝑡₁^𝑗_𝑖=11−𝐴_𝑖,𝑡∈𝑡_𝑗,𝑡_𝑗+1,𝑗=1,…,𝑝−1,𝑘𝑡₁∏^𝑝_𝑖=11−𝐴_𝑖𝑡_𝑝()𝑡−𝑇𝑡−𝑇,𝑡∈_𝑝,𝑇. (3.24) Clearly, 𝑢∈𝑋 and 𝑢(𝑡)≥0 for 𝑡∈[0,𝑇]. Then for any 𝑟>0, by (3.19), we have 𝐹(𝑡,𝑟𝑢(𝑡))=^𝑟₀^𝑢(𝑡)𝑓(𝑡,𝑠)𝑑𝑠≥−^𝑟₀^𝑢(𝑡)𝑎|𝑠|^𝜂𝑎𝑑𝑠=−|𝑟𝜂+1𝑢(𝑡)|^𝜂+1𝑎≥−‖𝑟𝜂+1𝑢‖_∞^𝜂+1. (3.25) In view of (3.25) and (3.12), one can get 𝜑(𝑟𝑢)=12‖𝑟𝑢‖²+12𝛼₁(𝑟𝑢(0))²+12𝛼₂(𝑟𝑢(𝑇))²−^𝑇₀𝐹(𝑡,𝑟𝑢)𝑑𝑡−^𝑝_𝑗=1^{𝑟𝑢(𝑡−𝑗)}₀𝐼_𝑗(𝑠)≤𝑟𝑑𝑠₂2‖𝑢‖²+𝛼₁(𝑢(0))²+𝛼₂(𝑢(𝑇))²+𝑎𝑇𝑐^𝜂+1‖𝜂+1𝑢‖^𝜂+1𝑟^𝜂+1−^𝑝_𝑗=1𝑑₁𝑢𝑡⁻_𝑗^𝜇𝑟^𝜇+𝑝𝑑₂. (3.26) Since 𝜇>2,𝜂<1, we can find 𝑟>1 such that ‖𝑟𝑢‖>𝜌 and 𝜑(𝑟𝑢)<0. Hence, the condition (𝐶₂) in Lemma 2.5 is satisfied.
Finally, by Remark 3.5 and Lemma 2.5, the problem (𝑃) has at least one nontrivial solution generated by impulses. Analogous to the proof of Theorem 3.3, we know that when 𝑓(𝑡,𝑥) is odd about 𝑥, and 𝐼_𝑗(𝑗=1,2,…,𝑝) are odd, problem (𝑃) has an infinite number of solutions which are generated by impulses.

Theorem 3.7. Suppose (3.17) holds, 0<𝐴_𝑗<1 for 𝑗=1,2,…,𝑝, and there exist positive constants 𝑎₀,𝑎_𝑗,𝑎_𝑗,𝑏_𝑗 and 𝜂₀,𝜂_𝑗,𝜈_𝑗 which satisfy 𝜂_𝑗∈(0,1) and 0<𝜈_𝑗<min{𝜂₀,1} such that(i)𝐼_j(𝑥)≤𝑎_𝑗|𝑥|^𝜂𝑗+𝑏_𝑗 for 𝑥∈ℝ,𝑗=1,2,…,𝑝;(ii)𝐼_𝑗(𝑥)≥𝑎_𝑗|𝑥|^𝜈𝑗 for |𝑥|≤1,𝑥≠0, and 𝑓(𝑡,𝑥)≥−𝑎₀|𝑥|^𝜂0 for |𝑥|≤1,𝑡∈[0,𝑇]. Then problem (𝑃) has at least one nontrivial solution generated by impulses.

Proof. Firstly, we show that 𝜑 is weakly lower semi continuous. Let {𝑢_𝑛} be a sequence which weakly converges to some 𝑢 in 𝑋, then ‖𝑢‖≤liminf_𝑛→∞𝑢_𝑛. (3.27) On the other hand, we have that {𝑢_𝑛|_{(𝑡𝑗,𝑡𝑗+1)}} weakly converges to 𝑢|_{(𝑡𝑗,𝑡𝑗+1)} in 𝐻¹(𝑡_𝑗,𝑡_𝑗+1), then {𝑢_𝑛|_{(𝑡𝑗,𝑡𝑗+1)}} uniformly converges to 𝑢|_{(𝑡𝑗,𝑡𝑗+1)} on [𝑡_𝑗,𝑡_𝑗+1], then lim_𝑛→∞12𝛼₁𝑢²_𝑛(0)+12𝛼₂𝑢²_𝑛(𝑇)−^𝑇₀𝐹𝑡,𝑢_𝑛(𝑡)𝑑𝑡−^𝑝_𝑗=1^{𝑢𝑛(𝑡−𝑗)}₀𝐼_𝑗(𝑠)𝑑𝑠=lim_𝑛→∞12𝛼₁𝑢²_𝑛(0)+12𝛼₂𝑢²_𝑛(𝑇)−^𝑝_𝑗=0^𝑡𝑗+1_𝑡𝑗𝐹𝑡,𝑢_𝑛(𝑡)𝑑𝑡−^𝑝_𝑗=1^{𝑢𝑛(𝑡−𝑗)}₀𝐼_𝑗(𝑠)=1𝑑𝑠2𝛼₁𝑢²(0)+12𝛼₂𝑢²(𝑇)−^𝑝_𝑗=0^𝑡𝑗+1_𝑡𝑗𝐹(𝑡,𝑢(𝑡))𝑑𝑡−^𝑝_𝑗=1^{𝑢(𝑡−𝑗)}₀𝐼_𝑗(𝑠)=1𝑑𝑠2𝛼₁𝑢²(0)+12𝛼₂𝑢²(𝑇)−^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡−^𝑝_𝑗=1^{𝑢(𝑡−𝑗)}₀𝐼_𝑗(𝑠)𝑑𝑠. (3.28) From (3.27) and (3.28), we conclude that 𝜑(𝑢)≤liminf_𝑛→∞𝜑(𝑢_𝑛).
Next, we verify that 𝜑 is coercive.
From (3.17), the hypothesis (i) and Lemma 2.2, we have 𝜑(𝑢)=12‖𝑢‖²+12𝛼₁𝑢²(0)+12𝛼₂𝑢²(𝑇)−^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡−^𝑝_𝑗=1^{𝑢(𝑡−𝑗)}₀𝐼_𝑗(𝑠)≥1𝑑𝑠2‖𝑢‖²−^𝑝_𝑗=1^{𝑢(𝑡−𝑗)}₀𝐼_𝑗(𝑠)≥1𝑑𝑠2‖𝑢‖²−^𝑝_𝑗=1^{‖𝑢‖∞}₀𝑎_𝑗|𝑥|^𝜂𝑗+𝑏_𝑗≥1𝑑𝑥2‖𝑢‖²−^𝑝_𝑗=1𝑎_𝑗𝜂_𝑗‖𝑢‖+1^{𝜂𝑗∞+1}+𝑏_𝑗‖𝑢‖_∞≥12‖𝑢‖²−^𝑝_𝑗=1𝑎_𝑗𝑐^𝜂𝑗+1𝜂_𝑗‖𝑢‖+1^𝜂𝑗+1+𝑏_𝑗𝑐‖𝑢‖, (3.29) which implies that 𝜑 is coercive since 𝜂_𝑗∈(0,1) for 𝑗=1,2,…,𝑝.
Therefore by the least action principle [27, Theorem 1.1], we obtain that 𝑐^′=inf_𝑋𝜑(𝑢) is a critical value of 𝜑, this means that there exists 𝑢∈𝑋 such that 𝜑(𝑢)=𝑐^′.
Finally, we show that 𝑢 is nontrivial.
In (3.24), if we choose positive constant 𝑘 such that 𝑘<1/𝑡₁, then 𝑢(𝑡)∈[0,1] for 𝑡∈[0,1], and for any 𝑟∈(0,1), by the assumption (ii) and with same calculation of (3.25), we have 𝐹(𝑡,𝑟𝑢)𝑎≥−⁰𝜂₀‖𝑟+1𝑢‖^𝜂0_∞⁺¹,^{𝑟𝑢(𝑡−𝑗)}₀𝐼_𝑗(𝑠)𝑑𝑠≥𝑎_𝑗𝜈_𝑗𝑟+1𝑢𝑡⁻_𝑗^𝜈𝑗+1. (3.30) Then 𝜑(𝑟𝑢)=12‖𝑟𝑢‖²+12𝛼₁(𝑟𝑢(0))²+12𝛼₂(𝑟𝑢(𝑇))²−^𝑇₀𝐹(𝑡,𝑟𝑢)𝑑𝑡−^𝑝_𝑗=1^{𝑟𝑢(𝑡−𝑗)}₀𝐼_𝑗(𝑠)≤𝑟𝑑𝑠₂2‖𝑢‖²+𝛼₁(𝑢(0))²+𝛼₂(𝑢(𝑇))²+𝑎₀𝑇𝑐^𝜂0+1𝜂₀‖+1𝑢‖^𝜂0+1𝑟^𝜂0+1−^𝑝_𝑗=1𝑎_𝑗𝜈_𝑗+1𝑢𝑡⁻_𝑗^𝜈𝑗+1𝑟^𝜈𝑗+1. (3.31) For 𝑟>0 small enough, we have that 𝜑(𝑟𝑢)<0 since 0<𝜈_𝑗<min{𝜂₀,1}. Hence, 𝜑(𝑢)=inf_𝑋𝜑(𝑢)<0, and by Remark 3.5, 𝑢 is a nontrivial solution generated by impulses.

Example 3.8. Let 𝑇=1,𝑡₁∈(0,1), consider the system: −𝑢^′′(𝑡)+𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡∈[0,1]⧵𝑡₁,𝑢𝑡⁻₁−𝑢𝑡⁺₁=13𝑢𝑡⁻₁,𝑢^′𝑡⁻₁−𝑢^′𝑡⁺₁=𝑢³𝑡⁻₁−13𝑢^′𝑡⁺₁,𝑢(0)−𝑢^′(0)=0,𝑢(1)+𝑢^′(1)=0. (3.32) Compare with problem (P), 𝐴₁=1/3,𝐼₁(𝑢)=𝑢³.
If we choose 𝑓(𝑡,𝑢)=𝑢^𝑝/𝑞, where 𝑝 and 𝑞 are odd numbers and 𝑝/𝑞>1, then the conditions of Theorem 3.3 are satisfied. So, the impulsive boundary value problem (3.32) has an infinite number of nontrivial solutions.

Example 3.9. Consider (3.32) with 𝑓(𝑡,𝑢) defined by 𝑓(𝑡,𝑢)=−𝑢𝑒^𝑢,𝑢≤1,−𝑒,𝑢>1. (3.33) If we choose 𝑎=𝑒,𝜂=1/2, the assumptions of Theorem 3.6 are satisfied, then the impulsive boundary value problem (3.32) has at least one nontrivial solution generated by impulses.

Example 3.10. Consider the following problem: −𝑢^′′(𝑡)(𝑡)((𝑡[],(0)+𝑢=𝑓𝑡,𝑢)),𝑡∈0,1𝛼𝑢−𝛽𝑢^′(0)=0,𝛾𝑢(1)+𝜎𝑢^′(1)=0. (3.34) If we choose 𝑓(𝑡,𝑢)=−𝑡𝑢^𝑚/𝑛−𝑢^𝑝/𝑞, where 𝑚,𝑛,𝑝,and𝑞 are odd numbers, then 𝑢𝑓(𝑡,𝑢)≤0, from Lemma 3.4, we know that (3.34) has no nontrivial solution.

Example 3.11. Let 𝑇=1,𝑡₁∈(0,1), we consider the following problem: −𝑢^′′(𝑡)+𝑢(𝑡)=−𝑢^1/9(𝑡),𝑡∈[0,1]⧵𝑡₁,𝑢𝑡⁻₁−𝑢𝑡⁺₁=13𝑢𝑡⁻₁,𝑢^′𝑡⁻₁−𝑢^′𝑡⁺₁=𝑢^1/3𝑡⁻₁−13𝑢^′𝑡⁺₁,𝑢(0)−𝑢^′(0)=0,𝑢(1)+𝑢^′(1)=0, (3.35) here 𝑓(𝑡,𝑢)=−𝑢^1/9,𝐴₁=1/3,𝐼₁(𝑢)=𝑢^1/3. In view of 𝑢𝑓(𝑡,𝑢)=−𝑢^10/9≤0 and Lemma 3.4, we get that the above problem without impulsive, that is, the problem: −𝑢^′′(𝑡)+𝑢(𝑡)=−𝑢^1/9(𝑡),𝑡∈[0,1],𝑢(0)−𝑢^′(0)=0,𝑢(1)+𝑢^′(1)=0 (3.36) has no nontrivial solution. In addition, it is easy to verify that the hypotheses of Theorem 3.7 are fulfilled. Thus, via Theorem 3.7, the impulsive boundary value problem (3.35) has at least one nontrivial solutions generated by impulses.

Acknowledgments

The research of H. R. Sun was supported by NSF of China (10801065), FRFCU (lzujbky-2011-43, lzujbky-2012-k25) and SRF for ROCS, and SEM. The research of J. J. Nieto was partially supported by Ministerio de Ciencia e lnnovación and FEDER, Project MTM2010-15314.