Existence and Multiplicity of Solutions to a Boundary Value Problem for Impulsive Differential Equations

Abstract

We investigate the existence and multiplicity of solutions to a boundary value problem for impulsive differential equations. By using critical point theory, some criteria are obtained to guarantee that the impulsive problem has at least one solution, at least two solutions, and infinitely many solutions. Some examples are given to illustrate the effectiveness of our results.

1. Introduction

In this paper, we will investigate the existence and multiplicity of solutions to the boundary value problem for impulsive differential equations: −𝜌(𝑡)𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′+𝑠(𝑡)|𝑢(𝑡)|^𝑝−2𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡≠𝑡_𝑖[],Δ𝜌𝑡,a.e.𝑡∈0,𝑇_𝑖𝑢^′𝑡_𝑖^𝑝−2𝑢^′𝑡_𝑖=𝐼_𝑖𝑢𝑡_𝑖,𝑢(0)(𝑇)𝑖=1,2,…,𝑘,=𝑢=0, (1) where 𝑝≥2, 𝜌(𝑡), 𝑠(𝑡)∈𝐿^∞[0,𝑇], with ess inf_{𝑡∈[0,𝑇]}𝜌(𝑡)>0, ess inf_{𝑡∈[0,𝑇]}𝑠(𝑡)>0, and 1≤𝜌(𝑡)<+∞; 0<𝑠(𝑡)<+∞; 0=𝑡₀<𝑡₁⋯<𝑡_𝑘<𝑡_𝑘+1=𝑇, Δ(𝜌(𝑡_𝑖)|𝑢^′(𝑡_𝑖)|^𝑝−2𝑢^′(𝑡_𝑖))=𝜌(𝑡⁺_𝑖)|𝑢^′(𝑡⁺_𝑖)|^𝑝−2𝑢^′(𝑡⁺_𝑖)−𝜌(𝑡⁻_𝑖)|𝑢^′(𝑡⁻_𝑖)|^𝑝−2𝑢^′(𝑡⁻_𝑖), 𝑓∶[0,𝑇]×𝑅→𝑅 is continuous, 𝐼_𝑖∶𝑅→𝑅 are continuous.

Recently, there have been many papers concerned with boundary value problems for impulsive differential equations. Impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time. The theory of impulsive differential systems has been developed by numerous mathematicians (see [1–6]).

Impulsive and periodic boundary value problems have been studied extensively in the literature. There have been many approaches to study periodic solutions of differential equations, such as the method of lower and upper solutions, fixed point theory, and coincidence degree theory (see [7–10]). However, the study of solutions for impulsive differential equations using variational method has received considerably less attention (see, [11–18]). Variational method is, to the best of our knowledge, novel and it may open a new approach to deal with nonlinear problems with some type of discontinuities such as impulses.

Teng and Zhang in [15] studied the existence of solutions to the boundary value problem for impulsive differential equations −𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′=𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡),𝑡≠𝑡_𝑖,a.e.𝑡∈[0,𝑇],Δ𝑢^′𝑡_𝑖=𝐼_𝑖𝑢𝑡_𝑖𝑢(0)(𝑇),𝑖=1,2,…,𝑙,=𝑢=0. (2) By using variational methods and iterative methods they showed that there exists a solution for problem (2).

In this paper, we will need the following conditions.(A)𝐹(𝑡,𝑢) is measurable in 𝑡 for every 𝑢∈ℝ and continuously differentiable in 𝑢 for a.e. 𝑡∈[0,𝑇] and there exist 𝑎∈𝐶(ℝ⁺,ℝ⁺),𝑏∈𝐿¹(0,𝑇;ℝ⁺) such that |𝐹(𝑡,𝑢)|≤𝑎(|𝑢|)𝑏(𝑡),𝑓(𝑡,𝑢)≤𝑎(|𝑢|)𝑏(𝑡), (3) for all 𝑢∈ℝ and a.e. 𝑡∈[0,𝑇], where 𝐹(𝑡,𝑢)=∫^𝑢₀𝑓(𝑡,𝑠)d𝑠.(𝐻₀) There exist constants 𝑎,𝑏>0 and 𝑟∈[0,𝑝−1) such that 𝑓(𝑡,𝑢)≤𝑎+𝑏|𝑢|^𝑟,for(𝑡,𝑢)∈[0,𝑇]×𝑅. (4)(𝐻₁) There exist constants 𝑐>0 and 𝜏∈[𝑝,+∞) such that 𝑓(𝑡,𝑢)≤𝑐|𝑢|^𝜏−1+1,for(𝑡,𝑢)∈[0,𝑇]×𝑅. (5)(𝐻₂) There exist constants 𝑎_𝑖,𝑏_𝑖>0 and 𝑟_𝑖∈[0,𝑝−1)(𝑖=1,2,…,𝑘) such that 𝐼_𝑖(𝑢)≤𝑎_𝑖+𝑏_𝑖|𝑢|^𝑟𝑖,for𝑢∈𝑅. (6)(𝐻₃) There exist constants 𝜇>𝑝 and 𝑀>0 such that 0<𝜇𝐹(𝑡,𝑢)≤𝑢𝑓(𝑡,𝑢),for𝑡∈[0,𝑇],|𝑢|≥𝑀. (7)(𝐻₄)lim_𝑢→0𝑓(𝑡,𝑢)/|𝑢|^𝑝−1=0 uniformly, for 𝑡∈[0,𝑇] and 𝑢∈𝑅. (𝐻₅) there exist 𝜆>𝑝 and 𝛽>𝜆−𝑝 such that limsup_|𝑥|→∞𝐹(𝑡,𝑥)|𝑥|^𝜆<∞uniformly,fora.e.𝑡∈[0,𝑇],liminf_|𝑥|→∞𝑥𝑓(𝑡,𝑥)−2𝐹(𝑡,𝑥)|𝑥|^𝛽[].>0,uniformlyfora.e.𝑡∈0,𝑇 (8)(𝐻₆) there exists 𝜁_𝑖𝑗>0 such that 2^𝑡₀𝐼_𝑖(𝑠)d𝑠−𝐼_𝑖(𝑡)𝑡≥0∀𝑖∈{1,2,…,𝑘},|𝑡|≥𝜁_𝑖𝑗. (9)

We recall some facts which will be used in the proof of our main results. It has been shown, for instance, in [19] that the set of all eigenvalues of the following problem: −𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′+𝜆|𝑢(𝑡)|^𝑝−2𝑢(𝑡)=0,𝑡≠𝑡_𝑖[],𝑢(0)(𝑇),𝑡∈0,𝑇=𝑢=0 (10) is given by the sequence of positive numbers 𝜆_𝑘=𝑝−1𝑘𝜋_𝑝𝑇^𝑝,for𝑘=1,2,…, (11) where 𝜋_𝑝=2¹₀1(1−𝑠^𝑝)^1/𝑝d𝑠. (12) Each eigenvalue 𝜆_𝑘 is simple with the associated eigenfunction 𝜑_𝑘(𝑥)=sin_𝑝𝑘𝜋_𝑝𝑥𝑇,for0≤𝑥≤𝑇. (13)𝑋_𝑘 denotes the eigenspace associated to 𝜆_𝑘, then 𝑊₀^1,𝑝(0,𝑇)=⨁_𝑖∈𝑁𝑋_𝑖.

An outline of this paper is given as follows. In the next section, we present some preliminaries including some basic knowledge and critical point theory. In Section 3, by using the critical point theory, we will establish some sufficient conditions for the existence of solutions of system (1). In Section 4, some examples are given to verify and support the theoretical findings.

2. Preliminaries

In this section, we recall some basic facts which will be used in the proofs of our main results. In order to apply the critical point theory, we make a variational structure. From this variational structure, we can reduce the problem of finding solutions of (1) to the one of seeking the critical points of a corresponding functional.

In [20], the Sobolev space 𝑊₀^1,𝑝(0,𝑇) be the endowed with the norm ‖𝑢‖_𝑊0^1,𝑝=^𝑇₀𝑢^′(𝑡)^𝑝d𝑡^1/𝑝,𝑢∈𝑊₀^1,𝑝(0,𝑇). (14)

Throughout the paper, we also consider the norm ‖𝑢‖=^𝑇₀𝜌(𝑡)𝑢^′(𝑡)^𝑝+𝑠(𝑡)|𝑢(𝑡)|^𝑝d𝑡^1/𝑝,𝑢∈𝑊₀^1,𝑝(0,𝑇). (15) By Poincaré inequality: ^𝑇₀𝑢^′(𝑡)^𝑝d𝑡≥𝐶^𝑇₀|𝑢(𝑡)|^𝑝d𝑡, (16) where 𝐶=𝜆₁ is precisely the largest 𝐶>0 for which the above inequality holds true. Then ∫^𝑇₀|𝑢^′(𝑡)|^𝑝d𝑡−𝜆₁∫^𝑇₀|𝑢(𝑡)|^𝑝d𝑡≥0 while it minimizes and equals to zero exactly on the ray generated by the first eigenfunction sin_𝑝(𝜋_𝑝𝑡/𝑇).

Let us recall that |𝑢|_𝑝=^𝑇₀|𝑢(𝑡)|^𝑝d𝑡^1/𝑝,‖𝑢‖_∞=max_{[]𝑡∈0,𝑇}|𝑢(𝑡)|. (17) We denote by |⋅|_𝑝 the usual 𝐿^𝑝-norm. The 𝑛-dimensional Lebesgue measure of a set 𝐸⊆𝑅^𝑛 is denoted by |𝐸|. By the Sobolev embedding theorem, 𝑊₀^1,𝑝(0,𝑇)↪𝐿^𝑟[0,𝑇] continuously for 𝑟∈[1,+∞), and there exists 𝛾_𝑟>0 such that |𝑢|_𝑟≤𝛾_𝑟‖𝑢‖,∀𝑢∈𝑊₀^1,𝑝(0,𝑇). (18)

Lemma 1. There exist 𝐶₁, 𝐶₂>0 such that 𝐶₁‖𝑢‖_𝑊0^1,𝑝≤‖𝑢‖≤𝐶₂‖𝑢‖_𝑊0^1,𝑝,∀𝑢∈𝑊₀^1,𝑝(0,𝑇). (19)

Proof. Since ess inf_{𝑡∈[0,𝑇]}𝜌(𝑡)>0, ess inf_{𝑡∈[0,𝑇]}𝑠(𝑡)>0, 1≤𝜌(𝑡)<+∞, and 0<𝑠(𝑡)<+∞, we have that ess inf_{𝑡∈[0,𝑇]}𝜌(𝑡)∶=𝑚₁≥1, ess inf_{𝑡∈[0,𝑇]}𝑠(𝑡)∶=𝑚₂>−𝜆₁, where 𝜆₁ is a positive number and that there exists 𝑛₁∈(0,1) such that −𝑚₂≤𝜆₁(1−𝑛₁). Thus, by Poincaré inequality, we have 1−𝑛₁^𝑇₀𝑢^′(𝑡)^𝑝d𝑡≥1−𝑛₁𝜆₁^𝑇₀|𝑢(𝑡)|^𝑝d𝑡≥−𝑚₂^𝑇₀|𝑢(𝑡)|^𝑝d𝑡, (20) for all 𝑢∈𝑊₀^1,𝑝(0,𝑇). Thereby, for every 𝑢∈𝑊₀^1,𝑝(0,𝑇), ‖𝑢‖^𝑝=^𝑇₀𝜌(𝑡)𝑢^′(𝑡)^𝑝d𝑡+^𝑇₀𝑠(𝑡)|𝑢(𝑡)|^𝑝d𝑡≥𝑚₁^𝑇₀𝑢^′(𝑡)^𝑝d𝑡=𝑚₁‖𝑢‖^𝑝_𝑊01,𝑝=𝐶^𝑝₁‖𝑢‖^𝑝_𝑊01,𝑝. (21)
On the other hand, by Poincaré inequality, one has ‖𝑢‖^𝑝=^𝑇₀𝜌(𝑡)𝑢^′(𝑡)^𝑝d𝑡+^𝑇₀𝑠(𝑡)|𝑢(𝑡)|^𝑝≤𝜌d𝑡_∞𝑇0𝑢^′(𝑡)^𝑝d𝑡+‖𝑠‖_∞^𝑇₀|𝑢(𝑡)|^𝑝≤𝜌d𝑡_∞+‖𝑠‖_∞𝜆₁^𝑇₀𝑢^′(𝑡)^𝑝=𝜌d𝑡_∞+‖𝑠‖_∞𝜆₁‖𝑢‖^𝑝_𝑊01,𝑝=𝐶^𝑝₂‖𝑢‖^𝑝_𝑊01,𝑝. (22)
Take 𝐶₁=(𝑚₁+𝑛₁−1)^1/𝑝, 𝐶₂=(‖𝜌‖_∞+‖𝑠‖_∞/𝜆₁)^1/𝑝, then 𝐶₁‖𝑢‖_𝑊0^1,𝑝≤‖𝑢‖≤𝐶₂‖𝑢‖_𝑊0^1,𝑝. (23)
The proof is complete.

Lemma 2. There exists 𝐶₃>0 such that if 𝑢∈𝑊₀^1,𝑝(0,𝑇), then ‖𝑢‖_∞≤𝐶₃‖𝑢‖. (24)

Proof. If 𝑢∈𝑊₀^1,𝑝(0,𝑇), it follows from the mean value theorem that 1𝑇^𝑇₀𝑢(𝑠)d𝑠=𝑢(𝜉), (25) for some 𝜉∈(0,𝑇). Hence, for 𝑡∈[0,𝑇], by Hölder inequality and Poincaré inequality |𝑢(𝑡)|=𝑢(𝜉)+^𝑡_𝜉𝑢^′(𝑠)≤𝑢(𝜉+d𝑠)^𝑇₀𝑢^′(𝑠≤1)d𝑠𝑇^𝑇₀𝑢(𝑠)d𝑠+𝑇^1/𝑞𝑇₀𝑢^′(𝑡)^𝑝d𝑠^1/𝑝≤𝐶𝑇^1/𝑝+𝑇^1/𝑞𝑇₀𝑢^′(𝑡)^𝑝d𝑠^1/𝑝≤𝐶𝑇^1/𝑝+𝑇^1/𝑞1𝐶₁‖𝑢‖. (26) Hence, ‖𝑢‖_∞≤((𝐶/𝑇)^1/𝑝+𝑇^1/𝑞)(1/𝐶₁)‖𝑢‖=𝐶₃‖𝑢‖. The proof is complete.

Take 𝑣∈𝑊₀^1,𝑝(0,𝑇) and multiply the two sides of the equality −𝜌(𝑡)𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′+𝑠(𝑡)|𝑢(𝑡)|^𝑝−2𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)) (27) by 𝑣 and integrate it from 0 to 𝑇, we have −^𝑇₀𝜌(𝑡)𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′𝑣(𝑡)+d𝑡_𝑇0𝑠(𝑡)|𝑢(𝑡)|^𝑝−2𝑢(𝑡)𝑣(𝑡)=d𝑡_𝑇0𝑓(𝑡,𝑢(𝑡))𝑣(𝑡)d𝑡. (28) Moreover, −^𝑇₀𝜌(𝑡)𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′𝑣(𝑡)d𝑡=−^𝑘_𝑖=0^𝑡𝑖+1_𝑡𝑖𝜌(𝑡)𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′𝑣(𝑡)=d𝑡_𝑘𝑖=0𝜌𝑡⁻_𝑖+1𝑢^′𝑡⁻_𝑖+1^𝑝−2𝑢^′𝑡⁻_𝑖+1𝑣𝑡⁻_𝑖+1−𝜌𝑡⁺_𝑖𝑢^′𝑡⁺_𝑖^𝑝−2𝑢^′𝑡⁺_𝑖𝑣𝑡⁺_𝑖−^𝑡𝑖+1_𝑡𝑖𝜌(𝑡)𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)𝑣^′(𝑡)=d𝑡_𝑘𝑖=1𝜌𝑡⁺_𝑖𝑢^′𝑡⁺_𝑖^𝑝−2𝑢^′𝑡⁺_𝑖−𝜌𝑡⁻_𝑖×𝑢^′𝑡⁻_𝑖^𝑝−2𝑢^′𝑡⁻_𝑖−𝜌(𝑇)𝑢^′(𝑇)^𝑝−2𝑢^′(𝑇)𝑣(𝑇)+𝜌(0)𝑢^′(0)^𝑝−2𝑢^′(0)𝑣(0)+^𝑇₀𝜌(𝑡)𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)𝑣^′(𝑡)=d𝑡_𝑘𝑖=1𝐼_𝑖𝑢𝑡_𝑖𝑣𝑡_𝑖+^𝑇₀𝜌(𝑡)𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)𝑣^′(𝑡)d𝑡. (29) Combining (28), we have ^𝑇₀𝜌(𝑡)𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)𝑣^′(𝑡)+d𝑡_𝑇0𝑠(𝑡)|𝑢(𝑡)|^𝑝−2𝑢(𝑡)𝑣(𝑡)d𝑡+^𝑘_𝑖=1𝐼_𝑖𝑢𝑡_𝑖𝑣𝑡_𝑖=^𝑇₀𝑓(𝑡,𝑢(𝑡))𝑣(𝑡)d𝑡. (30) Considering the above, we introduce the following concept solution for problem (1).

Definition 3. We say that a function 𝑢∈𝑊₀^1,𝑝(0,𝑇) is a solution of problem (1) if the identity ^𝑇₀𝜌(𝑡)𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)𝑣^′(𝑡)+d𝑡_𝑇0𝑠(𝑡)|𝑢(𝑡)|^𝑝−2𝑢(𝑡)𝑣(𝑡)d𝑡+^𝑘_𝑖=1𝐼_𝑖𝑢𝑡_𝑖𝑣𝑡_𝑖=^𝑇₀𝑓(𝑡,𝑢(𝑡))𝑣(𝑡)d𝑡 (31) holds for any 𝑣∈𝑊₀^1,𝑝(0,𝑇).
Consider the functional 𝜑∶𝑊₀^1,𝑝(0,𝑇)→𝑅 defined by 𝜑(𝑢)=1𝑝‖𝑢‖^𝑝+^𝑘_𝑖=1^{𝑢(𝑡𝑖)}₀𝐼_𝑖(𝑡)d𝑡−^𝑇₀𝐹(𝑡,𝑢(𝑡))d𝑡. (32) Using the continuity of 𝑓 and 𝐼_𝑖, 𝑖=1,2,…,𝑘, one has that 𝜑∈𝐶¹(𝑊₀^1,𝑝(0,𝑇),𝑅). For any 𝑣∈𝑊₀^1,𝑝(0,𝑇), we have 𝜑^′(𝑢)𝑣=^𝑇₀𝜌(𝑡)𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)𝑣^′(𝑡)+d𝑡_𝑇0𝑠(𝑡)|𝑢(𝑡)|^𝑝−2𝑢(𝑡)𝑣(𝑡)+d𝑡_𝑘𝑖=1𝐼_𝑖𝑢𝑡_𝑖𝑣𝑡_𝑖−^𝑇₀𝑓(𝑡,𝑢(𝑡))𝑣(𝑡)d𝑡. (33) Thus, the solutions of problem (1) are the corresponding critical points of 𝜑.

Definition 4. Let 𝑋 be a normed space. A minimizing sequence for a function 𝜑∶𝑋→(−∞,+∞) is a sequence 𝑢_𝑘 such that 𝜑(𝑢_𝑘)→inf𝜑 whenever 𝑘→+∞.

Definition 5. Let 𝑋 be a Banach space and let 𝜑∶𝑋→(−∞,+∞). 𝜑 is said to be sequentially weakly lower semicontinuous if liminf_𝑘→+∞𝜑(𝑥_𝑘)≥𝜑(𝑥) as 𝑥_𝑘⇀𝑥 in 𝑋.

Definition 6. Let 𝐸 be a Banach space and let 𝑐∈𝑅. For any sequence {𝑢_𝑘} in 𝐸, if 𝜑(𝑢_𝑘) is bounded and 𝜑^′(𝑢_𝑘)→0 as 𝑘→+∞ possesses a convergent subsequence, then we say that 𝜑 satisfies the Palais-Smale condition (denoted by PS condition for short). We say that 𝜑 satisfies the Palais-Smale condition at level 𝑐 (denoted by (PS)_𝑐 condition for short) if there exists a sequence {𝑢_𝑘} in 𝐸 such that 𝜑(𝑢_𝑘)→𝑐 and 𝜑^′(𝑢_𝑘)→0 as 𝑘→+∞ implies that 𝑐 is a critical value of 𝜑.

Definition 7. Let 𝐸 be a Banach space and let 𝜑∶𝐸→(−∞,+∞). 𝜑 is said to be coercive if 𝜑(𝑢)→+∞ as ‖𝑢‖→+∞.

Lemma 8 (see [12]). If 𝜑 is sequentially weakly lower semicontinuous on a reflexive Banach space 𝑋 and has a bounded minimizing sequence, then 𝜑 has a minimum on 𝑋.

Definition 9. Let 𝑋 be a real Banach space with a direct sum decomposition 𝑋=𝑋¹⊕𝑋². The functional 𝜑∈𝐶¹(𝑋,𝑅) is said to have a local linking at 0, with respect to (𝑋¹,𝑋²), if, for some 𝑟>0,(i)𝜑(𝑢)≥0,𝑢∈𝑋¹,‖𝑢‖≤𝑟,(ii)𝜑(𝑢)≤0,𝑢∈𝑋²,‖𝑢‖≤𝑟.
If 𝜑 has a local linking at 0, then 0 is critical point (the trivial one). Suppose, furthermore, that there are two sequences of finite dimensional subspaces 𝑋¹₁⊂𝑋¹₂⊂⋯⊂𝑋¹ and 𝑋²₁⊂𝑋²₂⊂⋯⊂𝑋² such that 𝑋¹=_𝑛𝑋¹_𝑛,𝑋²=_𝑛𝑋²_𝑛. (34)

Definition 10 (see [13, Definition 2.2]). Let 𝐼∈𝐶¹(𝑋,ℝ). The functional 𝐼 satisfies the (𝐶)^∗ condition if every sequence (𝑢_𝛼𝑛) such that 𝛼_𝑛 is admissible and 𝑢_𝛼𝑛∈𝑋_𝛼𝑛,sup𝐼𝑢_𝛼𝑛𝑢<∞,1+_𝛼𝑛𝐼^′𝑢_𝛼𝑛→0 (35) contains a subsequence which converges to a critical point of 𝐼.

Lemma 11 (see [13]). Suppose that 𝜑∈𝐶¹(𝑋,𝑅) satisfies the following assumptions: (1)𝜑 satisfies the (𝐶)^∗ condition,(2)𝜑 has a local linking at 0,(3)𝜑 maps bounded sets into bounded sets,(4)for every 𝑚∈𝑁,  𝜑(𝑢)→−∞ as ‖𝑢‖→+∞, 𝑢∈𝑋¹_𝑚⊕𝑋². Then 𝜑 has at least two critical points.

Lemma 12 (see [14]). Let E be a Banach space. Let 𝜑∈𝐶¹(𝑋,𝑅) be an even functional which satisfies the 𝑃𝑆 condition and 𝜑(0)=0. If 𝐸=𝑉⊕𝑌, where 𝑉 is finite dimensional, and 𝜑 satisfies the following conditions:(1)there exist constants 𝜌,𝛼 such that 𝜑|_{𝜕𝐵𝜌∩𝑌}≥𝛼, where 𝐵_𝜌={𝑥∈𝐸∶‖𝑥‖<𝜌},(2)for each finite-dimensional subspace 𝑊⊂𝐸 there is 𝑅=𝑅(𝑊) such that 𝜑(𝑢)≤0, for all 𝑢∈𝑊 with ‖𝑢‖≥𝑅. Then 𝜑 has an unbounded sequence of critical values.

3. Existence of Periodic Solutions

Theorem 13. Assume that (𝐴),  (𝐻₀), and (𝐻₂) are satisfied, then problem (1) has at least one solution.

Proof. Let 𝑀₁=max{𝑎₁,𝑎₂,…,𝑎_𝑘}, 𝑀₂=max{𝑏₁,𝑏₂,…,𝑏_𝑘}. By Lemma 2, we have 𝜑(𝑢)=1𝑝‖𝑢‖^𝑝+^𝑘_𝑖=1^{𝑢(𝑡𝑖)}₀𝐼_𝑖(𝑡)d𝑡−^𝑇₀𝐹((𝑡≥1𝑡,𝑢))d𝑡𝑝‖𝑢‖^𝑝−^𝑘_𝑖=1^{𝑢(𝑡𝑖)}₀𝑎_𝑖+𝑏_𝑖|𝑢|^𝑟𝑖d𝑡−^𝑇₀𝑎|𝑢|+𝑏|𝑢|^𝑟+1≥1d𝑡𝑝‖𝑢‖^𝑝−𝑘𝑀₁‖𝑢‖_∞−𝑀₂^𝑘_𝑖=1‖𝑢‖^𝑟𝑖_∞⁺¹−𝑎𝑇‖𝑢‖_∞−𝑏𝑇‖𝑢‖_∞^𝑟+1≥1𝑝‖𝑢‖^𝑝−𝑘𝑀₁𝐶₃‖𝑢‖−𝑀₂^𝑘_𝑖=1𝐶^𝑟𝑖₃⁺¹‖𝑢‖^𝑟𝑖+1−𝑎𝑇𝐶₃‖𝑢‖−𝑏𝑇𝐶₃^𝑟+1‖𝑢‖^𝑟+1, (36) for all 𝑢∈𝑊₀^1,𝑝(0,𝑇). This implies that lim_{‖𝑢‖→∞}𝜑(𝑢)=∞, and 𝜑 is coercive.
On the other hand, we show that 𝜑 is weakly lower semicontinuous. If {𝑢_𝑘}_𝑘∈𝑁⊂𝑊₀^1,𝑝(0,𝑇), 𝑢_𝑘⇀𝑢, then we have that {𝑢_𝑘}_𝑘∈𝑁 converges uniformly to 𝑢 on [0,𝑇] and liminf_𝑘→∞‖𝑢_𝑘‖≥‖𝑢‖. Thus liminf_𝑘→∞𝜑𝑢_𝑘=liminf_𝑘→∞1𝑝𝑢_𝑘^𝑝+^𝑘_𝑖=1^{𝑢𝑘(𝑡𝑖)}₀𝐼_𝑖(𝑡)−d𝑡_𝑇0𝐹𝑡,𝑢_𝑘(𝑡)≥1d𝑡𝑝‖𝑢‖^𝑝+^𝑘_𝑖=1^{𝑢(𝑡𝑖)}₀𝐼_𝑖(𝑡)d𝑡−^𝑇₀𝐹((𝑡(𝑢).𝑡,𝑢))d𝑡=𝜑 (37)
By Lemma 8, 𝜑 has a minimum point on 𝑊₀^1,𝑝(0,𝑇), which is a critical point of 𝜑. Hence, problem (1) has at least one solution. The proof is complete.

We readily have the following corollary.

Corollary 14. Assume that (𝐴),  (𝐻₀), and (𝐻₂) are satisfied and 𝑓 and the impulsive functions 𝐼_𝑖(𝑖=1,2,…,𝑘) are bounded. Then problem (1) has at least one solution.

Lemma 15. Assume that (𝐻₁), (𝐻₂), and (𝐻₃) are satisfied, then 𝜑(𝑢) satisfies the 𝑃𝑆 condition.

Proof. Assume that {𝑢_𝑛}⊂𝑊₀^1,𝑝(0,𝑇) satisfies that 𝜑(𝑢_𝑛) is bounded and 𝜑^′(𝑢_𝑛)→0 as 𝑛→+∞. We will prove that the sequence {𝑢_𝑛} is bounded.
It follows from (𝐻₁), (𝐻₂), (𝐻₃), and Lemma 2, we have 𝜇𝜑𝑢_𝑛−𝜑^′𝑢_𝑛𝑢_𝑛=𝜇𝑝−1𝑢_𝑛^𝑝+^𝑘_𝑖=1𝜇^{𝑢𝑛(𝑡𝑖)}₀𝐼_𝑖(𝑡)𝑑𝑡−𝐼_𝑖𝑢_𝑛𝑡_𝑖𝑢_𝑛𝑡_𝑖−^𝑇₀𝜇𝐹𝑡,𝑢_𝑛(𝑡)−𝑓𝑡,𝑢_𝑛(𝑡)𝑢_𝑛(𝑡)≥𝜇d𝑡𝑝−1𝑢_𝑛^𝑝−×𝜇+1𝑘𝑀₁𝐶₃𝑢_𝑛+𝑀₂^𝑘_𝑖=1𝐶^𝑟𝑖₃⁺¹𝑢_𝑛^𝑟𝑖+1−_{{|𝑢𝑛|≤𝑢0}}𝜇𝐹𝑡,𝑢_𝑛(𝑡)−𝑓𝑡,𝑢_𝑛(𝑡)𝑢_𝑛(𝑡)≥𝜇d𝑡𝑝−1𝑢_𝑛^𝑝−×𝜇+1𝑘𝑀₁𝐶₃𝑢_𝑛+𝑀₂^𝑘_𝑖=1𝐶^𝑟𝑖₃⁺¹𝑢_𝑛^𝑟𝑖+1−𝐶₄. (38) Hence, {𝑢_𝑛} is bounded in 𝑊₀^1,𝑝(0,𝑇).
Since 𝑊₀^1,𝑝(0,𝑇) is a reflexive Banach space, passing to a subsequence if necessary, we may assume that there is a 𝑢∈𝑊₀^1,𝑝(0,𝑇) such that 𝑢_𝑛⇀𝑢in𝑊₀^1,𝑝(),𝑢0,𝑇_𝑛→𝑢in𝐿^𝑝[],𝑢0,𝑇_𝑛convergesuniformlyto𝑢on[0,𝑇]. (39) Notice that 𝜑^′𝑢_𝑛−𝜑^′(𝑢),𝑢_𝑛−𝑢=^𝑇₀𝜌(𝑡)𝑢^′_𝑛(𝑡)^𝑝−2𝑢^′_𝑛(𝑡)−𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)×𝑢^′_𝑛(𝑡)−𝑢^′(𝑡)+d𝑡_𝑇0𝑠(𝑡)𝑢_𝑛(𝑡)^𝑝−2𝑢_𝑛(𝑡)−|𝑢(𝑡)|^𝑝−2𝑢(𝑡)×𝑢_𝑛(𝑡)(𝑡)+−𝑢d𝑡_𝑘𝑖=1𝐼_𝑖𝑢_𝑛𝑡_𝑖−𝐼_𝑖𝑢𝑡_𝑖𝑢_𝑛𝑡_𝑖−𝑢𝑡_𝑖−^𝑇₀𝑓𝑡,𝑢_𝑛(𝑡)−𝑓(𝑡,𝑢(𝑡))𝑢_𝑛(𝑡)−𝑢(𝑡)d𝑡. (40) Recalling the following well-known inequality: for any 𝑥,𝑦∈𝑅^𝑁, |𝑥|^𝑝−2𝑥−𝑦^𝑝−2𝑦𝑥−𝑦≥𝑐_𝑝𝑥−𝑦^𝑝,𝑝≥2, (41) for some constant 𝑐_𝑝 (Lemma  4.2 in [21]) and using Schwarz inequality, we have 𝑐_𝑝^𝑇₀𝜌(𝑡)𝑢^′_𝑛(𝑡)−𝑢^′(𝑡)^𝑝d𝑡+𝑐_𝑝^𝑇₀𝑠(𝑡)𝑢_𝑛(𝑡)−𝑢(𝑡)^𝑝≤𝜑d𝑡_′𝑢_𝑛−𝜑^′(𝑢)𝑢_𝑛−−𝑢_𝑘𝑖=1𝐼_𝑖𝑢_𝑛𝑡_𝑖−𝐼_𝑖𝑢𝑡_𝑖𝑢_𝑛𝑡_𝑖−𝑢𝑡_𝑖+^𝑇₀𝑓𝑡,𝑢_𝑛(𝑡)−𝑓(𝑡,𝑢(𝑡))𝑢_𝑛(𝑡)−𝑢(𝑡)d𝑡. (42) Since 𝐼_𝑖𝑢_𝑛𝑡_𝑖−𝐼_𝑖𝑢𝑡_𝑖𝑢_𝑛𝑡_𝑖−𝑢𝑡_𝑖≤𝐼_𝑖𝑢_𝑛𝑡_𝑖+𝐼_𝑖𝑢𝑡_𝑖𝑢_𝑛−𝑢_∞. (43) By the assumption (𝐻₁), we have ^𝑇₀𝑓𝑡,𝑢_𝑛(𝑡)−𝑓(𝑡,𝑢(𝑡))𝑢_𝑛(𝑡)(𝑡)≤−𝑢d𝑡_𝑇0𝑓𝑡,𝑢_𝑛(𝑡)+𝑓(𝑡,𝑢(𝑡))𝑢_𝑛(𝑡)−𝑢(𝑡)d𝑡≤𝑐^𝑇₀2+𝑢_𝑛^𝜏−1+|𝑢|^𝜏−1𝑢_𝑛(𝑡)−𝑢(𝑡)d𝑡≤𝐶₅𝑢_𝑛−𝑢_∞1+𝑢_𝑛^𝜏−1_𝜏−1+|𝑢|^𝜏−1_𝜏−1. (44) From (39), it follows that 𝑢_𝑛→𝑢 in 𝑊₀^1,𝑝(0,𝑇). Thus, 𝜑(𝑢) satisfies the PS condition. The proof is complete.

Lemma 16. Assume that (𝐴), (𝐻₂), (𝐻₅), and (𝐻₆) are satisfied, then 𝜑 satisfies the (𝐶)^∗ condition.

Proof. Let {𝑢_𝛼𝑛} be a sequence in 𝑊₀^1,𝑝(0,𝑇) such that 𝛼_𝑛 is admissible and 𝑢_𝛼𝑛∈𝑋_𝛼𝑛,sup𝜑𝑢_𝛼𝑛𝑢<+∞,1+_𝛼𝑛𝜑^′𝑢_𝛼𝑛→0, (45) then there exist a constant 𝐶₄>0 such that 𝜑𝑢_𝛼𝑛≤𝐶₄,1+𝑢_𝛼𝑛𝜑^′𝑢_𝛼𝑛≤𝐶₄, (46) for all large 𝑛. On the other hand, by (𝐻₅), there are constants 𝐶₅>0 and 𝜌₁>0 such that 𝐹(𝑡,𝑥)≤𝐶₅|𝑥|^𝜆, (47) for all |𝑥|≥𝜌₁ and a.e. 𝑡∈[0,𝑇]. By (A) one has |𝐹(𝑡,𝑥)|≤max_{𝑠∈0,𝜌1}𝑎(𝑠)𝑏(𝑡), (48) for all |𝑥|≤𝜌₁ and a.e. 𝑡∈[0,𝑇]. It follows from (47) and (48) that |𝐹(𝑡,𝑥)|≤max_{𝑠∈0,𝜌1}𝑎(𝑠)𝑏(𝑡)+𝐶₅|𝑥|^𝜆. (49)
From (𝐻₂) and Lemma 2, we have that ^𝑘_𝑖=1^{|𝑢(𝑡𝑖}₀^)|𝐼_𝑖(𝑡)d𝑡≤^𝑘_𝑖=1^{|𝑢(𝑡𝑖}₀^)|𝑎_𝑖+𝑏_𝑖|𝑡|^𝑟𝑖d𝑡≤𝑀₁𝑘‖𝑢‖_∞+𝑀₂^𝑘_𝑖=1‖𝑢‖^𝑟𝑖_∞⁺¹≤𝑀₁𝑘𝐶₃‖𝑢‖+𝑀₂𝐶₃^𝑘_𝑖=1‖𝑢‖^𝑟𝑖+1 (50) for all 𝑢∈𝑊₀^1,𝑝(0,𝑇), where 𝑀₁={𝑎₁,𝑎₂,…,𝑎_𝑘}, 𝑀₂={𝑏₁,𝑏₂,…,𝑏_𝑘}. Combining (49), (50), and Hölder’s inequality, we have 1𝑝𝑢_𝛼𝑛^𝑝=𝜑𝑢_𝛼𝑛−^𝑘_𝑖=1^{𝑢𝛼𝑛(𝑡𝑖)}₀𝐼_𝑖(𝑡)d𝑡+^𝑇₀𝐹𝑡,𝑢_𝛼𝑛(𝑡)d𝑡≤𝐶₄+𝑀₁𝑘𝐶₃‖𝑢‖+𝑀₂𝐶₃^𝑘_𝑖=1‖𝑢‖^𝑟𝑖+1+𝐶₅^𝑇₀𝑢_𝛼𝑛(𝑡)^𝜆d𝑡+max_{𝑠∈0,𝜌1}𝑎(𝑠)^𝑇₀𝑏(𝑡)d𝑡≤𝐶₄+𝑀₁𝑘𝐶₃‖𝑢‖+𝑀₂𝐶₃^𝑘_𝑖=1‖𝑢‖^𝑟𝑖+1+𝐶₅^𝑇₀𝑢_𝛼𝑛(𝑡)^𝜆d𝑡+𝐶₆, (51) for all large 𝑛, where 𝐶₆=max_{𝑠∈[0,𝜌1]}𝑎(𝑠)∫^𝑇₀𝑏(𝑡)d𝑡. On the other hand, by (𝐻₅), there exist 𝐶₇>0 and 𝜌₂>0 such that 𝑥𝑓(𝑡,𝑥)−2𝐹(𝑡,𝑥)≥𝐶₇|𝑥|^𝛽, (52) for all |𝑥|≥𝜌₂ and a.e. 𝑡∈[0,𝑇]. By (A), 𝑥𝑓(𝑡,𝑥)−2𝐹(𝑡,𝑥)≤𝐶₈𝑏(𝑡), (53) for all |𝑥|≤𝜌₂ and a.e. 𝑡∈[0,𝑇], where 𝐶₈=(2+𝜌₂)max_{𝑠∈[0,𝜌2]}𝑎(𝑠). Combining (52) and (53), one has 𝑓(𝑡,𝑥),𝑥−2𝐹(𝑡,𝑥)≥𝐶₇|𝑥|^𝛽−𝐶₇𝜌^𝛽₂−𝐶₈𝑏(𝑡), (54) for all 𝑥∈ℝ^𝑁 and a.e. 𝑡∈[0,𝑇]. According to (𝐻₆), there exists 𝐶₉>0 such that 2^𝑡₀𝐼_𝑖𝑗(𝑠)d𝑠−𝐼_𝑖𝑗(𝑡)𝑡≥−𝐶₉∀𝑖=1,2,…,𝑘,𝑡∈ℝ. (55) Thus by (46), (54), and (55), we obtain 𝑝+1𝐶₄≥𝑝𝜑𝑢_𝛼𝑛−𝜑^′𝑢_𝛼𝑛,𝑢_𝛼𝑛=𝑝^𝑘_𝑖=1^{𝑢𝛼𝑛(𝑡𝑖)}₀𝐼_𝑖(𝑡)d𝑡−𝑢_𝛼𝑛𝐼_𝑖𝑢_𝛼𝑛+^𝑇₀𝑢_𝛼𝑛(𝑡)𝑓𝑡,𝑢_𝛼𝑛(𝑡)−𝑝𝐹𝑡,𝑢_𝛼𝑛(𝑡)=d𝑡_𝑘𝑖=1𝑝^{𝑢𝛼𝑛(𝑡𝑖)}₀𝐼_𝑖(𝑡)d𝑡−𝐼_𝑖𝑢_𝛼𝑛𝑡_𝑖𝑢_𝛼𝑛𝑡_𝑖+^𝑇₀∇𝐹𝑡,𝑢_𝛼𝑛(𝑡),𝑢_𝛼𝑛(𝑡)−𝑝𝐹𝑡,𝑢_𝛼𝑛(𝑡)d𝑡≥−𝑘𝐶₉+𝐶₇^𝑇₀𝑢_𝛼𝑛^𝛽d𝑡−𝐶₇𝜌^𝛽₂𝑇−𝐶₈×^𝑇₀𝑏(𝑡)d𝑡, (56) for all large 𝑛. From (56), ∫^𝑇₀|𝑢_𝛼𝑛|^𝛽d𝑡 is bounded. If 𝛽>𝜆, by Hölder’s inequality, we have ^𝑇₀𝑢_𝛼𝑛^𝜆d𝑡≤𝑇^{(𝛽−𝜆)/𝛽𝑇}₀𝑢_𝛼𝑛^𝛽d𝑡^𝜆/𝛽. (57) Since 𝜉_𝑖𝑗∈[0,1) for all 𝑖=1,2,…,𝑘, by (51) and (57), {𝑢_𝛼𝑛} is bounded in 𝑊₀^1,𝑝(0,𝑇). If 𝛽≤𝜆, by Lemma 2, we obtain ^𝑇₀𝑢_𝛼𝑛(𝑡)^𝜆d𝑡=^𝑇₀𝑢_𝛼𝑛(𝑡)^𝛽𝑢_𝛼𝑛(𝑡)^𝜆−𝛽≤𝑢d𝑡_{𝛼𝑛∞𝜆−𝛽𝑇0}𝑢_𝛼𝑛(𝑡)^𝛽d𝑡≤𝐶₁^𝜆−𝛽𝑢_𝛼𝑛^{𝜆−𝛽𝑇}₀𝑢_𝛼𝑛(𝑡)^𝛽d𝑡. (58) Since 𝜉_𝑖𝑗∈[0,1), 𝜆−𝛽<2, by (51) and (58), {𝑢_𝛼𝑛} is also bounded in 𝑊₀^1,𝑝(0,𝑇). Hence, {𝑢_𝛼𝑛} is also bounded in 𝑊₀^1,𝑝(0,𝑇). Going if necessary to a subsequence, we can assume that 𝑢_𝛼𝛼𝑛⇀𝑢 in 𝑊₀^1,𝑝(0,𝑇). As the same the proof of Lemma 15, Therefore, 𝑢_𝛼𝑛→𝑢 in 𝑊₀^1,𝑝(0,𝑇). Hence 𝜑 satisfies the (𝐶)^∗ condition.

Theorem 17. Assume that (𝐴), (𝐻₁), (𝐻₂), (𝐻₃), (𝐻₅), and (𝐻₆) are satisfied and the following conditions hold.(𝐻₇) There exist constants 𝛿>0, 𝐾>0, 𝛾>0 such that 𝛾𝑝|𝑢|^𝑝+2𝑘𝑀₁𝑇|𝑢|()≤1≤𝐹𝑡,𝑢2𝑝𝛾^𝑝_𝑝|𝑢|^𝑝+𝐾|𝑢|^𝑝+1,𝑓𝑜𝑟𝑡∈[0,𝑇],|𝑢|≤𝛿. (59)(𝐻₈)𝐼_𝑖(𝑢)(𝑖=1,2,…,𝑘) is nondecreasing. Then the problem (1) has at least two critical points.

Proof. Let 𝑋¹_𝑛=span{𝜆_𝑘+1,𝜆_𝑘+2,…,𝜆_𝑘+𝑛}, 𝑋²_𝑛=𝑋²=(𝑋¹)^⟂ for 𝑛∈𝑁. Then 𝑋^𝑗=⋃_𝑛∈𝑁𝑋^𝑗_𝑛,𝑗=1,2. If 𝑢∈𝑋¹ one has |𝑢|_𝑝≤𝛾_𝑝‖𝑢‖, and if 𝑢∈𝑋², we have ‖𝑢‖^𝑝≤𝛾|𝑢|^𝑝_𝑝.
Step 1.  𝜑 has a local linking at 0 with respect to (𝑋¹,𝑋²).
For 𝑢∈𝑋¹, using (𝐻₇) we have ^𝑘_𝑖=1^{𝑢(𝑡𝑖)}₀𝐼_𝑖(𝑡)d𝑡≥0. (60) It follows from (𝐻₈) that 𝜑(𝑢)=1𝑝‖𝑢‖^𝑝+^𝑘_𝑖=1^{𝑢(𝑡𝑖)}₀𝐼_𝑖(𝑡)d𝑡−^𝑇₀𝐹((𝑡≥1𝑡,𝑢))d𝑡𝑝‖𝑢‖^𝑝−12𝑝𝛾^𝑝_𝑝|𝑢|^𝑝_𝑝−𝐾|𝑢|^𝑝+1_𝑝+1≥1𝑝‖𝑢‖^𝑝−12𝑝𝛾^𝑝_𝑝𝛾^𝑝_𝑝‖𝑢‖^𝑝−𝐾𝛾^𝑝+1_𝑝+1‖𝑢‖^𝑝+1=1‖𝑢‖2𝑝^𝑝−𝐾𝛾^𝑝+1_𝑝+1‖𝑢‖^𝑝+1. (61) Thus, 𝜑(𝑢)≥0 for 𝑢∈𝑋¹ with ‖𝑢‖≤𝑟₁, where 𝑟₁>0 is small enough.
For 𝑢∈𝑋², with ‖𝑢‖≤𝑟₂∶=𝛿/𝐶₃, we have |𝑢|≤|𝑢‖_∞≤𝐶₃‖𝑢‖≤𝛿 since dim𝑋²=𝑘<+∞. Thus, From (𝐻₂) and (𝐻₈) it follows that 𝜑(𝑢)=1𝑝‖𝑢‖^𝑝+^𝑘_𝑖=1^{𝑢(𝑡𝑖)}₀𝐼_𝑖(𝑡)d𝑡−^𝑇₀𝐹((𝑡1𝑡,𝑢))d𝑡≤𝑝‖𝑢‖^𝑝+^𝑘_𝑖=1^{𝑢(𝑡𝑖)}₀𝑎_𝑖+𝑏_𝑖|𝑢|^𝑟𝑖−^𝑇₀𝛾𝑝|𝑢|^𝑝+2𝑘𝑀₁𝑇|𝑢|≤𝛾d𝑡𝑝|𝑢|^𝑝_𝑝+𝑘𝑀₁‖𝑢‖_∞+𝑀₂×^𝑘_𝑖=1‖𝑢‖^𝑟𝑖_∞⁺¹−𝛾𝑝|𝑢|^𝑝_𝑝−2𝑘𝑀₁‖𝑢‖_∞≤−𝑘𝑀₁‖𝑢‖_∞+𝑀₂^𝑘_𝑖=1‖𝑢‖^𝑟𝑖_∞⁺¹≤0. (62) Take 𝑟=min{𝑟₁,𝑟₂}. We know that 𝜑 has a local linking at 0 with respect to (𝑋¹,𝑋²).
Step  2.  𝜑 maps bounded sets into bounded sets.
Assume ‖𝑢‖≤𝑅₀, where 𝑅₀ is a constant. By (𝐻₈), one has 𝜑(𝑢)≤1𝑝‖𝑢‖^𝑝+^𝑘_𝑖=1^{𝑢(𝑡𝑖)}₀𝐼_𝑖(𝑡)d𝑡+^𝑇₀|𝐹((𝑡1𝑡,𝑢))|d𝑡≤𝑝‖𝑢‖^𝑝+𝑘𝑀₁‖𝑢‖_∞+𝑀₂^𝑘_𝑖=1‖𝑢‖^𝑟𝑖_∞⁺¹+1‖𝑢‖2𝑝^𝑝+𝐾𝛾^𝑝+1_𝑝+1‖𝑢‖^𝑝+1≤3‖𝑢‖2𝑝^𝑝+𝑘𝑀₁𝐶₃‖𝑢‖+𝑀₂×^𝑘_𝑖=1𝐶^𝑟𝑖₃⁺¹‖𝑢‖^𝑟𝑖+1+𝐾𝛾^𝑝+1_𝑝+1‖𝑢‖^𝑝+1≤3𝑅2𝑝^0𝑝+𝑘𝑀₁𝐶₃𝑅₀+𝑀₂×^𝑘_𝑖=1𝐶^𝑟𝑖₃⁺¹𝑅₀^𝑟𝑖+1+𝐾𝛾^𝑝+1_𝑝+1𝑅₀^𝑝+1<∞, (63) which implies that 𝜑 maps bounded sets into bounded sets.
Step 3.   For every 𝑚∈𝑁, 𝜑(𝑢)→−∞ as ‖𝑢‖→+∞,𝑢∈𝑋¹_𝑚⊕𝑋².
By (𝐻₅), for any 𝑀₀>0, there exists a constant ℎ(𝑀₀) such that 𝐹(𝑡,𝑢(𝑡))≥𝑀|𝑢|^𝑝−ℎ(𝑀₀) for all (𝑡,𝑢)∈[0,𝑇]×𝑅. Since dim(𝑋¹_𝑚⊕𝑋²) is of finite dimension, there exists 𝛾>0 such that ‖𝑢‖^𝑝≤𝛾|𝑢|^𝑝_𝑝 for all 𝑢∈𝑋¹_𝑚⊕𝑋², which implies 𝜑(𝑢)≤1𝑝‖𝑢‖^𝑝+^𝑘_𝑖=1^{𝑢(𝑡𝑖)}₀𝐼_𝑖(𝑡)|𝑢|d𝑡−𝑀^𝑝_𝑝(𝑀)𝑇≤1+ℎ𝑝‖𝑢‖^𝑝+𝑘𝑀₁‖𝑢‖_∞+𝑀₂×^𝑘_𝑖=1‖𝑢‖^𝑟𝑖_∞⁺¹−𝑀𝛾‖𝑢‖^𝑝(𝑀)𝑇≤1+ℎ𝑝−𝑀𝛾‖𝑢‖^𝑝+𝑘𝑀₁𝐶₃‖𝑢‖+𝑀₂^𝑘_𝑖=1𝐶^𝑟𝑖₃⁺¹‖𝑢‖^𝑟𝑖+1+ℎ(𝑀)𝑇. (64) Choosing 𝑀>𝛾/𝑝, we have 𝜑(𝑢)→−∞ as ‖𝑢‖→+∞,𝑢∈𝑋¹_𝑚⊕𝑋².
Summing up the above, and by Lemma 16, 𝜑 satisfies all the assumptions of Lemma 11. Hence by Lemma 11, problem (1) has at least one nontrivial solution. The proof of Theorem 17 is completed.

Theorem 18. Assume that (𝐴), (𝐻₁), (𝐻₂), (𝐻₃), and (𝐻₄) are satisfied and the following conditions hold.(𝐻₉) 𝑓(𝑡,𝑢)=−𝑓(𝑡,−𝑢)𝑓𝑜𝑟(𝑡,𝑢)∈[0,𝑇]×𝑅. (𝐻₁₀) 𝐼_𝑖(𝑢)=−𝐼_𝑖(−𝑢)(𝑖=1,2,…,𝑘) and nondecreasing. Then the problem (1) has an infinite number of nontrivial solutions.

Proof. 𝜑∈𝐶¹(𝑊₀^1,𝑝(0,𝑇),𝑅), by (𝐻₉) and (𝐻₁₀), 𝜑 is an even functional and 𝜑(0)=0.
First, we verify the condition (2) of Lemma 12. 𝜇𝑢≤𝑓(𝑡,𝑢)𝐹(𝑡,𝑢),𝑢≥𝑀₀,𝜇𝑢≥𝑓(𝑡,𝑢)𝐹(𝑡,𝑢),𝑢≤−𝑀₀. (65) Integrating (65) for 𝑢 from [𝑀₀,𝑢] and [𝑢,−𝑀₀], respectively, we have 𝑢𝜇ln𝑀₀𝐹()≤ln𝑡,𝑢𝐹𝑡,𝑢₀,𝑢≥𝑀₀,𝑀𝜇ln⁰𝐹()−𝑢≥ln𝑡,𝑢𝐹𝑡,−𝑢₀,𝑢≤−𝑀₀. (66) That is, 𝐹(𝑡,𝑢)≥𝐹𝑡,𝑀₀𝑢𝑀₀^𝜇,𝑢≥𝑀₀,𝐹(𝑡,𝑢)≥𝐹𝑡,−𝑀₀−𝑢𝑀₀^𝜇,𝑢≤−𝑀₀. (67)(68) Combining (67) and (68), we have 𝐹(𝑡,𝑢)≥𝛼₁|𝑢|^𝜇,|𝑢|≥𝑀₀, (69) where 𝛼₁=𝑢₀^−𝜇minmin_{[]𝑡∈0,𝑇}𝐹𝑡,𝑀₀,min_{[]𝑡∈0,𝑇}𝐹𝑡,−𝑀₀>0. (70) On the other hand, by the continuity of 𝐹(𝑡,𝑢), 𝐹(𝑡,𝑢) is bounded on [0,𝑇]×[−𝑀₀,𝑀₀], there exists 𝐾₁>0 such that 𝐹(𝑡,𝑢)≥−𝐾₁≥𝛼₁|𝑢|^𝜇−𝛼₁𝑀₀^𝜇−𝐾₁,|𝑢|≤𝑀₀. (71) Combining (69) and (71), we have 𝐹(𝑡,𝑢)≥𝛼₁|𝑢|^𝜇−𝛼₂,∀(𝑡,𝑢)∈[0,𝑇]×𝑅, (72) where 𝛼₂=𝛼₁+𝐾₁.
For arbitrary finite-dimensional subspace 𝑊⊂𝑊₀^1,𝑝(0,𝑇), and any 𝑢∈𝑊, there exists 𝐶₁₀=𝐶₁₀(𝑊)>0 such that |𝑢|_𝜇≥𝐶₁₀‖𝑢‖. (73) By (𝐻₂), (72), (73), and Lemma 2, we have 𝜑(𝑢)=1𝑝‖𝑢‖^𝑝+^𝑘_𝑖=1^{𝑢(𝑡𝑖)}₀𝐼_𝑖(𝑡)d𝑡−^𝑇₀𝐹((𝑡1𝑡,𝑢))d𝑡≤𝑝‖𝑢‖^𝑝+^𝑘_𝑖=1^{𝑢(𝑡𝑖)}₀𝑎_𝑖+𝑏_𝑖|𝑢|^𝑟𝑖−d𝑡_𝑇0𝛼₁𝑢^𝜇−𝛼₂1d𝑡≤𝑝‖𝑢‖^𝑝+𝑘𝑀₁‖𝑢‖_∞+𝑀₂^𝑘_𝑖=1‖𝑢‖^𝑟𝑖_∞⁺¹−𝛼₁|𝑢|^𝜇_𝜇+𝛼₂1𝑇≤𝑝‖𝑢‖^𝑝+𝑘𝑀₁𝐶₃‖𝑢‖+𝑀₂^𝑘_𝑖=1𝐶^𝑟𝑖₃⁺¹‖𝑢‖^𝑟𝑖+1−𝛼₁𝐶^𝜇₁₀‖𝑢‖^𝜇+𝛼₂𝑇, (74) for every 𝑢∈𝑊. This implies that 𝜑(𝑢)→−∞ as 𝑢∈𝑊 and ‖𝑢‖→∞. So there exists 𝑅(𝑊)>0 such that 𝜑≤0 on for all 𝑢∈𝑊 with ‖𝑢‖≥𝑅.
In the following, we verify the condition (1) of Lemma 12.
Let 𝑉=𝑋₁⨁𝑋₂, 𝑌=⨁^∞_𝑖=3𝑋_𝑖, then 𝑊₀^1,𝑝(0,𝑇)=𝑉⨁𝑌 and 𝑉 is finite dimensional. Using (𝐻₁₀) we have ^𝑘_𝑖=1^{𝑢(𝑡𝑖)}₀𝐼_𝑖(𝑡)d𝑡≥0. (75) By (𝐻₃) and (𝐻₄), we have lim_𝑢→0𝐹(𝑡,𝑢)𝑢^𝑝=0. (76) Hence, for 𝜖=1/2𝑝𝛾^𝑝_𝑝, there exists 𝛿>0 such that for every 𝑢 with |𝑢|≤𝛿, |𝐹(≤1𝑡,𝑢)|2𝑝𝛾^𝑝_𝑝|𝑢|^𝑝. (77) Hence, for any 𝑢∈𝑌 with ‖𝑢‖≤𝛿/𝐶₃, ‖𝑢‖_∞≤𝛿, by (18), (54) and (55), we have 𝜑(𝑢)=1𝑝‖𝑢‖^𝑝+^𝑘_𝑖=1^{𝑢(𝑡𝑖)}₀𝐼_𝑖(𝑡)d𝑡−^𝑇₀𝐹((𝑡1𝑡,𝑢))d𝑡≥𝑝‖𝑢‖^𝑝−^𝑇₀𝐹((𝑡1𝑡,𝑢))d𝑡≥𝑝‖𝑢‖^𝑝−12𝑝𝛾^𝑝_𝑝|𝑢|^𝑝_𝑝≥1𝑝‖𝑢‖^𝑝−12𝑝𝛾^𝑝_𝑝𝛾^𝑝_𝑝‖𝑢‖^𝑝=1‖𝑢‖2𝑝^𝑝. (78) Take 𝛼=(1/2𝑝)(𝛿^𝑝/𝐶^𝑝₃),𝜌=𝛿/𝐶₃, then 𝜑(𝑢)≥𝛼,∀𝑢∈𝜕𝐵_𝜌∩𝑌. (79) Hence, by Lemma 12 and Lemma 15, 𝜑 possesses infinite critical points, that is, problem (1) has infinite nontrivial solutions. The proof is complete.

4. Example

Example 19. Let 𝑝=2, 𝜌(𝑡)=1, 𝑠(𝑡)=𝑡, 𝑡₁=1/2. Consider the boundary value problem −𝑢^′′(𝑡)+𝑡𝑢(𝑡)=𝑡+𝑢(𝑡)[],𝑢(0)(1),a.e.𝑡∈0,1=𝑢=0,Δ𝑢^′𝑡₁=𝑢^′𝑡⁺₁−𝑢^′𝑡⁻₁=1+𝑢(𝑡). (80) It is easy to see that conditions (𝐻₀) and (𝐻₂) of Theorem 13 hold. According to Theorem 13, problem (80) has at least one solution.

Example 20. Let 𝑝=2, 𝜌(𝑡)=1, 𝑠(𝑡)=2−𝑡, 𝑡₁=1/3. Consider the boundary value problem −𝑢^′′(𝑡)+(2−𝑡)𝑢(𝑡)=(𝑢(𝑡))²[],𝑢(0)(1),a.e.𝑡∈0,1=𝑢=0,Δ𝑢^′𝑡₁=𝑢^′𝑡⁺₁−𝑢^′𝑡⁻₁=1+𝑢(𝑡). (81) It is easy to check that all the conditions of Theorem 17 are satisfied. Thus, according to Theorem 17, problem (81) has at least two critical points.

Example 21. Let 𝑝=3, 𝜌(𝑡)=1+2𝑡, 𝑠(𝑡)=2+𝑡, 𝑡₁=1/2. Consider the boundary value problem −(1+2𝑡)𝑢^′(𝑡)𝑢^′(𝑡)^′+(2+𝑡)𝑢(𝑡)=(𝑢(𝑡))³,a.e.𝑡∈[0,1],𝑢(0)=𝑢(1)=0,Δ𝑢^′𝑡₁=𝑢^′𝑡⁺₁−𝑢^′𝑡⁻₁=³𝑢(𝑡). (82) It is easy to check that all the conditions of Theorem 18 are satisfied. Therefore, according to Theorem 18, problem (82) has infinite nontrivial solutions.

Acknowledgment

This work is supported by the National Natural Sciences Foundation of China under Grant no. 10971183.