On the Existence of Solutions for Impulsive Duffing Dynamic Equations on Time Scales with Dirichlet Boundary Conditions

Abstract

By using critical point theory, some new sufficient conditions for the existence of solutions of impulsive Duffing dynamic equations on time scales with Dirichlet boundary conditions are obtained. Some examples are also given to illustrate our results.

1. Introduction

Consider the following Duffing dynamic equations on time scales with impulsive effects𝑢^ΔΔ(𝑡)+𝐶𝑢^Δ(𝜎(𝑡))−𝑟(𝑡)𝑢(𝜎(𝑡))+𝑓(𝜎(𝑡),𝑢(𝜎(𝑡)))=ℎ(𝑡),a.e.𝑡∈[0,𝜎(𝑇)]^𝜅2_𝕋,Δ𝑢^Δ(𝑡_𝑗)=𝑢^Δ(𝑡⁺_𝑗)−𝑢^Δ(𝑡⁻_𝑗)=𝐼_𝑗(𝑢(𝑡_𝑗)),𝑗=1,2,…,𝑝,𝑢(0)=0=𝑢(𝜎(𝑇)), (1.1) where 𝑇>0, 𝐶 is a regressive constant, 𝑡₀=0<𝑡₁<𝑡₂<⋯<𝑡_𝑝<𝑡_𝑝+1=𝜎(𝑇), 𝑟∈𝐿^∞[0,𝜎(𝑇)]_𝕋, ℎ∈𝐿²[0,𝜎(𝑇)]_𝕋, 𝑓∶[0,𝜎(𝜎(𝑇))]_𝕋×ℝ→ℝ is continuous, and 𝐼_𝑗∶×ℝ→ℝ, 𝑗=1,2,…,𝑝 are continuous.

Obviously, system (1.1) covers Duffing equations (when 𝕋=ℝ) 𝑢^′′𝑢(𝑡)+𝑔(𝑢(𝑡))=ℎ(𝑡),_′′(𝑡)+𝐶𝑢^′(𝑡)+𝑔(𝑡,𝑢(𝑡))=ℎ(𝑡). (1.2) The Duffing equation has been used to model the nonlinear dynamics of special types of mechanical and electrical systems. This differential equation has been named after the studies of Duffing in 1918 [1], has a cubic nonlinearity, and describes an oscillator. The main applications have been in electronics, but it can also have applications in mechanics and in biology. For example, the brain is full of oscillators at micro- and macrolevel [2]. There are applications in neurology, ecology, secure communications, chaotic synchronization, and so on. Due to the rich behaviour of these equations, the most general forced forms of the Duffing equation (1.2) have been studied by many researchers [3–12].

The study of dynamic equations on time scales goes back to its founder Stefan Hilger [13], and is a new area of still fairly theoretical exploration in mathematics. Motivating the subject is the notion that dynamic equations on time scales can build bridges between continuous and discrete equations. Further, the study of time scales has led to several important applications, for example, in the study of insect population models, neural networks, heat transfer, and epidemic models [14–16].

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 [17–24]). Applications of impulsive differential equations with or without delays occur in biology, medicine, mechanics, engineering, chaos theory, and so on (see [25–32]).

In addition, system (1.1) also includes 𝑢^ΔΔ(𝑡)+𝑓(𝜎(𝑡),𝑢(𝜎(𝑡)))=0,a.e.𝑡∈[0,𝜎(𝑇)]^𝜅2_𝕋,𝑢𝑢(0)=0=𝑢(𝜎(𝑇)),_′′(𝑡)+𝑟(𝑡)𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),a.e.𝑡∈[0,𝑇],Δ𝑢^′(𝑡_𝑗)=𝑢^′(𝑡⁺_𝑗)−𝑢^′(𝑡⁻_𝑗)=𝐼_𝑗(𝑡_𝑗),𝑗=1,2,…,𝑝,𝑢(0)=0=𝑢(𝑇), (1.3) which were studied by papers [33, 34], and some existence results were obtained by using some critical point theorems.

Our purpose in this paper is to study the variational structure of problem (1.1) in an appropriate space of functions and the existence of solutions for problem (1.1) by means of some critical point theorems. The organization of this paper is as follows. In Section 2, we make some preparations. In Section 3, we will study the variational structure of problem (1.1) and give some important lemmas which will be used in later section. In Section 4, by applying some critical point theorems, we establish sufficient conditions for the existence of solutions to problem (1.1). Some illustrative examples are also given in Section 4.

2. Preliminaries

In this section, we will first recall some basic definitions and lemmas which are used in what follows.

Definition 2.1 (see [13]). A time scale 𝕋 is an arbitrary nonempty closed subset of the real set ℝ with the topology and ordering inherited from ℝ. The forward and backward jump operators 𝜎,𝜌∶𝕋→𝕋, and the graininess 𝜇,𝜈∶𝕋→ℝ⁺ are defined, respectively, by 𝜎(𝑡)∶=inf{𝑠∈𝕋∶𝑠>𝑡},𝜌(𝑡)∶=sup{𝑠∈𝕋∶𝑠<𝑡},𝜇(𝑡)∶=𝜎(𝑡)−𝑡,𝜈(𝑡)∶=𝑡−𝜌(𝑡). (2.1) The point 𝑡∈𝕋 is called left dense, left scattered, right dense, or right scattered if 𝜌(𝑡)=𝑡, 𝜌(𝑡)<𝑡, 𝜎(𝑡)=𝑡, or 𝜎(𝑡)>𝑡, respectively. Points that are right dense and left dense at the same time are called dense. If 𝕋 has a left-scattered maximum 𝑚₁, defined 𝕋^𝜅=𝕋−{𝑚₁}; otherwise, set 𝕋^𝜅=𝕋. If 𝕋 has a right-scattered minimum 𝑚₂, defined 𝕋_𝜅=𝕋−{𝑚₂}; otherwise, set 𝕋_𝜅=𝕋.

Definition 2.2 (see [13]). For 𝑓∶𝕋→ℝ and 𝑡∈𝕋^𝜅, then the delta derivative of 𝑓 at the point 𝑡 is defined to be the number 𝑓^Δ(𝑡) (provided it exists) with the property that for each 𝜖>0, there is a neighborhood 𝑈 of 𝑡 such that |𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡)(𝜎(𝑡)−𝑠)|≤𝜖|𝜎(𝑡)−𝑠|,∀𝑠∈𝑈. (2.2)

Definition 2.3 (see [13]). A function 𝑓 is rd continuous provided it is continuous at each right-dense point in 𝕋 and has a left-sided limit at each left-dense point in 𝕋. The set of rd-continuous functions 𝑓 will be denoted by 𝐶_𝑟𝑑(𝕋).

Definition 2.4 (see [13]). Assume that 𝕋 is an arbitrary time scale. We say that a function 𝑝∶𝕋→ℝ is regressive provided 1+𝜇(𝑡)𝑝(𝑡)≠0,∀𝑡∈𝕋^𝜅. (2.3)

Definition 2.5 (see [13]). Assume that 𝕋 is an arbitrary time scale and a function 𝑝∶𝕋→ℝ is regressive, then we define the exponential function on 𝕋 by 𝑒_𝑝(𝑡,𝑠)=exp(^𝑡_𝑠𝜉_𝜇(𝜏)(𝑝(𝜏))Δ𝜏),for𝑠,𝑡∈𝕋, (2.4) in which 𝜉_ℎ1(𝑧)=ℎLog(1+𝑧ℎ)forℎ>0, (2.5)𝜉_ℎ(𝑧)=𝜉₀(𝑧)=𝑧 for ℎ=0, here Log is the principal logarithm function.

A function 𝑓∶𝕋→ℝ is called rd continuous provided it is continuous at each right-dense point and its left-sided limit exists (finite) at each left-dense point in 𝕋. We write 𝑓∈𝐶_𝑟𝑑(𝕋)=𝐶_𝑟𝑑(𝕋,ℝ). If 𝑓 is differentiable with 𝑓^Δ∈𝐶_𝑟𝑑(𝕋^𝜅), we write 𝑓∈𝐶¹_𝑟𝑑(𝕋). If 𝑓^Δ is differentiable on 𝕋^𝜅2=(𝕋^𝜅)^𝜅 with 𝑓^Δ2=(𝑓^Δ)^Δ∈𝐶_𝑟𝑑(𝕋^𝜅2), we write 𝑓∈𝐶²_𝑟𝑑(𝕋). Similar to the classical Riemann's definition of integrability, the concept of the Riemann delta integral on time scales is given in [35]. We know that many familiar functions, including monotone continuous, piecewise continuous, right-dense continuous functions, are Riemann delta integrable. In the following lemma, we present some properties of the integral that will be needed later.

Lemma 2.6 (see [35]). Let 𝑓,𝑔∶𝕋→ℝ be two functions and 𝑎,𝑏∈𝕋. Then one has the following. (1)Let 𝑓 and 𝑔 be Riemann delta integrable functions on [𝑎,𝑏], and 𝛼,𝛽∈ℝ. Then 𝛼𝑓+𝛽𝑔 is Riemann delta integrable and^𝑏_𝑎(𝛼𝑓(𝑠)+𝛽𝑔(𝑠))Δ𝑠=𝛼^𝑏_𝑎𝑓(𝑠)Δ𝑠+𝛽^𝑏_𝑎𝑔(𝑠)Δ𝑠. (2.6)(2)Fundamental Theorem of Calculus. Let 𝑓 be a continuous function on [𝑎,𝑏]_𝕋 such that 𝑓 is delta differentiable on [𝑎,𝑏). If 𝑓^Δ is Riemann delta integrable from 𝑎 to 𝑏, then^𝑏_𝑎𝑓^Δ(𝑠)Δ𝑠=𝑓(𝑏)−𝑓(𝑎). (2.7)(3)Integration by Parts. Let 𝑓 and 𝑔 be continuous functions on [𝑎,𝑏] that are differentiable on [𝑎,𝑏). If 𝑓^Δ and 𝑔^Δ are Riemann delta integrable from 𝑎 to 𝑏, then^𝑏_𝑎𝑓(𝑠)𝑔^Δ(𝑠)Δ𝑠=[𝑓(𝑠)𝑔(𝑠)]|^𝑏_𝑎−^𝑏_𝑎𝑓^Δ(𝑠)𝑔(𝜎(𝑠))Δ𝑠. (2.8)(4)If 𝑓 is Riemann delta integrable on [𝑎,𝑏], then so is |𝑓|, and|^𝑏_𝑎𝑓^Δ(𝑠)Δ𝑠|=^𝑏_𝑎|𝑓^Δ(𝑠)|Δ𝑠. (2.9)

The construction of the Δ-measure on 𝕋 and the following concepts can be found in [35]. (1)For each 𝑡₀∈𝕋⧵{max𝕋}, the single-point set 𝑡₀ is Δ-measurable, and its Δ-measure is given by𝜇_Δ({𝑡₀})=𝜎(𝑡₀)−𝑡₀=𝜇(𝑡₀). (2.10)(2)If 𝑎,𝑏∈𝕋 and 𝑎≤𝑏, then𝜇_Δ([𝑎,𝑏))=𝑏−𝑎,𝜇_Δ((𝑎,𝑏))=𝑏−𝜎(𝑎). (2.11)(3)If 𝑎,𝑏∈𝕋⧵{max𝕋} and 𝑎≤𝑏, then𝜇_Δ((𝑎,𝑏])=𝜎(𝑏)−𝜎(𝑎),𝜇_Δ([𝑎,𝑏])=𝜎(𝑏)−𝑎. (2.12)

The Lebesgue integral associated with the measure 𝜇_Δ on 𝕋 is called the Lebesgue delta integral. For a (measurable) set 𝐸⊂𝕋 and a function 𝑓∶𝐸→ℝ, the corresponding integral of 𝑓 on 𝐸 is denoted by ∫_𝐸𝑓(𝑠)Δ𝑠. All theorems of the general Lebesgue integration theory hold also for the Lebesgue delta integral on 𝕋. Moreover, comparing the Lebesgue delta integral with the Riemann delta integral, we have the following.

Lemma 2.7 (see [35]). Let [𝑎,𝑏] be a closed bounded interval in 𝕋 and let 𝑓 be a bounded real-valued function defined on [𝑎,𝑏]. If 𝑓 is Riemann delta integrable from 𝑎 to 𝑏, then 𝑓 is Lebesgue delta integrable on [𝑎,𝑏) and (𝑅)^𝑏_𝑎𝑓(𝑠)Δ𝑠=(𝐿)_[𝑎,𝑏)𝑓(𝑠)Δ𝑠, (2.13) where 𝑅 and 𝐿 indicate the Riemann delta integral and Lebesgue delta integral, respectively.

Assume that 𝑝∈ℝ and 𝑝>1. Let 𝐿^𝑝_𝐿𝑆[𝑎,𝑏] be the set of{𝑓∶[𝑎,𝑏]→ℝ∶^𝑏_𝑎|𝑓(𝑠)|^𝑝Δ𝑠<+∞}. (2.14) By Lemma 3.6 in [36], 𝐿^𝑝_𝐿𝑆[𝑎,𝑏] is a complete linear space with the norm ‖⋅‖_𝑝 defined by‖𝑓‖_𝑝=(^𝑏_𝑎|𝑓(𝑠)|^𝑝Δ𝑠)^1/𝑝. (2.15)

Let 𝐶_𝑇𝑆[𝑎,𝑏] denote the linear space of all continuous functions 𝑓∶[𝑎,𝑏]→ℝ with the maximum norm ‖𝑓‖_𝐶=max_{𝑡∈[𝑎,𝑏]}|𝑓(𝑡)|.

Lemma 2.8 (Hölder inequality [37]). Let 𝑓,𝑔∈𝐶_𝑟𝑑([𝑎,𝑏]),𝑝>1 and q the conjugate number of 𝑝. Then ^𝑏_𝑎|𝑓(𝑠)𝑔(𝑠)|Δ𝑠≤(^𝑏_𝑎|𝑓(𝑠)|^𝑝Δ𝑠)^1/𝑝⋅(^𝑏_𝑎|𝑔(𝑠)|^𝑞Δ𝑠)^1/𝑞. (2.16)

At the end of this section, we recall some notation and known results from critical point theory.

Let 𝕏 be a real normed space and let 𝜑 be a functional from 𝕏 to ℝ. (1)𝜑 is called weakly continuous if𝑥_𝑘⇀𝑥⇒𝜑(𝑥_𝑘)→𝜑(𝑥). (2.17)(2)𝜑 is called lower semicontinuous if𝑥_𝑘→𝑥⇒lim_𝑘→∞inf𝜑(𝑥_𝑘)≥𝜑(𝑥). (2.18)(3)𝜑 is called weakly lower semicontinuous if𝑥_𝑘⇀𝑥⇒lim_𝑘→∞inf𝜑(𝑥_𝑘)≥𝜑(𝑥). (2.19)

Let 𝕏 be a real Hilbert space, 𝜑∈𝐶¹(𝕏,ℝ), which means that 𝜑 is a continuously Fréchet-differentiable functional defined on 𝕏. 𝜑 is said to satisfy the Palais-Smale condition (P.S. condition) if any sequence {𝑥_𝑛}⊂𝕏 for which {𝜑(𝑥_𝑛)} is bounded and 𝜑^′(𝑥_𝑛)→0 as 𝑛→∞ possesses a convergent subsequence in 𝕏.

Let 𝐵_𝑟 be the open ball in 𝕏 with radius 𝑟 and centered at 0 and let 𝜕𝐵_𝑟 denote its boundary. The following lemma is taken from [38, 39] and will play an important role in the proof of our main results.

Lemma 2.9 (see [38]). If 𝕏 is a real normed space and 𝜑∶𝕏→ℝ is lower semicontinuous and convex, then 𝜑 is weakly lower semicontinuous.

Lemma 2.10 (see [39]). Let 𝕏 be a real Banach space, 𝜑∈𝐶¹(𝕏,ℝ) be weakly lower(upper) semicontinuous, and lim_{‖𝑥‖→∞}𝜑(𝑥)=+∞(lim_{‖𝑥‖→∞}𝜑(𝑥)=−∞). (2.20) Then, there exists 𝑥₀∈𝕏 such that 𝜑(𝑥₀)=inf_𝑥∈𝕏𝜑(𝑥)(𝜑(𝑥₀)=sup_𝑥∈𝕏𝜑(𝑥)). (2.21)

Lemma 2.11 (Mountain Pass Theorem [40]). Let 𝕏 be a real Hilbert space and let 𝜑∈𝐶¹(𝕏,ℝ) satisfy P.S. condition, and 𝜑 satisfies (J1)There are constants 𝜌,𝛼>0 such that 𝜑(𝑥)≥𝛼 for all 𝑥∈𝜕𝐵_𝜌, where 𝐵_𝜌={𝑥∈𝕏∶‖𝑥‖<𝜌}. (J2)𝜑(0)≤0 and there exists 𝑥₀∈𝐵_𝜌 such that 𝜑(𝑥₀)≤0. Then 𝜑 possesses a critical value 𝑐≥𝛼. Moreover, 𝑐 can be characterized as 𝑐=inf_𝑔∈Γmax_𝑠∈[0,1]𝜑(𝑔(𝑠)), (2.22) where Γ={𝑔∈𝐶([0,1],𝕏)∣𝑔(0)=0,𝑔(1)=𝑥₀}. (2.23)

3. Variational Structure

In this section, we will establish the corresponding variational framework for problem (1.1).

Let ℙ be the function space of the formℙ={𝑢∣𝑢∈𝐶¹_𝑟𝑑[0,𝜎(𝑇)]^𝜅_𝕋,𝑢∈𝐶_𝑇𝑆[0,𝜎(𝑇)]_𝕋,𝑢(0)=0=𝑢(𝜎(𝑇))}. (3.1)ℙ can be equipped with the inner product(𝑢,𝑣)_ℙ=₀^𝜎(𝑇)𝑢^Δ(𝑡)𝑣^Δ(𝑡)Δ𝑡,∀𝑢,𝑣∈ℙ, (3.2) inducing the norm‖𝑢‖_ℙ=(𝑢,𝑢)_ℙ=(₀^𝜎(𝑇)|𝑢^Δ(𝑡)|²Δ𝑡)^1/2,∀𝑢∈ℙ. (3.3)

Let {𝑢_𝑘}∈ℙ be a Cauchy sequence in ℙ, that is, ‖𝑢^Δ_𝑘−𝑢^Δ_𝑙‖₂=‖𝑢_𝑘−𝑢_𝑙‖_ℙ→0 as 𝑘,𝑙→∞. Since 𝐿²_𝐿𝑆[0,𝜎(𝑇)]_𝕋 is a complete space, there exists a 𝑣∈𝐿²_𝐿𝑆[0,𝜎(𝑇)]_𝕋 such that ‖𝑢^Δ_𝑘−𝑣‖₂→0 as 𝑘→∞. Define a function 𝑢 by𝑢(𝑡)=^𝑡₀𝑣(𝑠)Δ𝑠. (3.4) The function 𝑣 is called a weak derivative of 𝑢 and is also denoted by 𝑢^Δ.

Remark 3.1. If 𝑢 has a weak derivative, then this weak derivative is unique in 𝐿²_𝐿𝑆[0,𝜎(𝑇)]_𝕋, that is, if 𝑣₁,𝑣₂∈𝐿²_𝐿𝑆[0,𝜎(𝑇)]_𝕋 are both weak derivatives of 𝑢, then ‖𝑣₁−𝑣₂‖₂=0. If 𝑢 is (delta) differential on [0,𝜎(𝑇)]_𝕋, then its weak derivative is its (delta) derivative.

From (3.4), it is clear that 𝑢(0)=0 and|𝑢(𝜎(𝑇))|=|₀^𝜎(𝑇)𝑣(𝑠)Δ𝑠|=|₀^𝜎(𝑇)(𝑣(𝑠)−𝑢^Δ_𝑘≤(𝑠))Δ𝑠|_0𝜎(𝑇)|𝑣(𝑠)−𝑢^Δ_𝑘(𝑠)|Δ𝑠≤(₀^𝜎(𝑇)1²Δ𝑠)^1/2(₀^𝜎(𝑇)|𝑣(𝑠)−𝑢^Δ_𝑘(𝑠)|²Δ𝑠)^1/2=𝜎(𝑇)‖𝑣−𝑢^Δ_𝑘‖₂→0,as𝑘→∞. (3.5) Thus, 𝑢(𝜎(𝑇))=0.

Define the space ℍ asℍ={𝑢(𝑡)=^𝑡₀𝑣(𝑠)Δ𝑠∣∃{𝑢_𝑘}∈ℙsatisfying‖𝑢^Δ_𝑘−𝑣‖₂→0as𝑘→∞}. (3.6) It is clear that ℙ⊂ℍ, and we define the inner product (⋅,⋅)_ℍ in ℍ as(𝑢,𝑣)_ℍ=₀^𝜎(𝑇)𝑢^Δ(𝑡)𝑣^Δ(𝑡)Δ𝑡,∀𝑢,𝑣∈ℍ, (3.7) inducing the norm‖𝑢‖_ℍ=(𝑢,𝑢)_ℍ=(₀^𝜎(𝑇)|𝑢^Δ(𝑡)|²Δ𝑡)^1/2,∀𝑢∈ℍ, (3.8) where 𝑢^Δ represents the weak derivative of 𝑢. Moreover, we have the following result.

Lemma 3.2 (see [33]). ℍ is a complete space in the norm ‖⋅‖_ℍ and inequality ₀^𝜎(𝑇)|𝑢^𝜎(𝑡)|²Δ𝑡≤𝜎(𝑇)²₀^𝜎(𝑇)|𝑢^Δ(𝑡)|²Δ𝑡,∀𝑢∈ℍ (3.9) holds, that is, ‖𝑢^𝜎‖₂≤𝜎(𝑇)‖𝑢‖_ℍ.

Also, we consider the inner product(𝑢,𝑣)=₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑟(𝑡)𝑢^𝜎(𝑡)𝑣^𝜎(𝑡)Δ𝑡+₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑢^Δ(𝑡)𝑣^Δ(𝑡)Δ𝑡,∀𝑢,𝑣∈ℍ, (3.10) inducing the norm‖𝑢‖=(₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑟(𝑡)|𝑢^𝜎(𝑡)|²Δ𝑡+₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)|𝑢^Δ(𝑡)|²Δ𝑡)^1/2,∀𝑢∈ℍ. (3.11)

Throughout this paper, it will be assumed that ess inf_{𝑡∈[0,𝜎(𝑇)]𝕋}𝑟(𝑡)=𝑟₀>−𝜆₀, where 𝜆₀ is the smallest eigenvalue of the problem 𝑢^ΔΔ(𝑡)+𝜆𝑢^𝜎(𝑡)=0,𝑡∈[0,𝜎(𝑇)]^𝜅2_𝕋,𝑢(0)=0=𝑢(𝜎(𝑇)). (3.12) From [41], 𝜆₀ may be defined as𝜆₀=min_{𝑢≠0𝑢(0)=0=𝑢(𝜎(𝑇))}−∫₀^𝜎(𝑇)𝑢^ΔΔ(𝑡)𝑢^𝜎(𝑡)Δ𝑡‖𝑢^𝜎‖²₂. (3.13) Applying the integration by parts in Lemma 2.6, we have𝜆₀=min_{𝑢≠0𝑢(0)=0=𝑢(𝜎(𝑇))}‖𝑢‖²_ℍ‖𝑢^𝜎‖²₂>0, (3.14) which implies that‖𝑢^𝜎‖²₂≤1𝜆₀‖𝑢‖²_ℍ,∀𝑢∈ℍ. (3.15)

For convenience, we introduce the following notations: 𝑒^𝑙=min_{𝑡∈[0,𝜎(𝑇)]𝕋}𝑒_𝐶(𝑡,0),𝑒^𝐿=max_{𝑡∈[0,𝜎(𝑇)]𝕋}𝑒_𝐶(𝑡,0),ℎ₀=₀^𝜎(𝑇)|ℎ(𝑡)|Δ𝑡. (3.16)

Lemma 3.3. Suppose that (H1)𝑟₀𝑒^𝐿+𝜆₀𝑒^𝑙>0,forall𝑟₀∈(−𝜆₀,0). Then one has 𝜔‖𝑢‖²_ℍ≤‖𝑢‖²≤𝑒^𝐿(‖𝑟‖_∞𝜎²(𝑇)+1)‖𝑢‖²_ℍ, (3.17) which implies that the norm ‖⋅‖ and the norm ‖⋅‖_ℍ are equivalent, where 𝑟𝜔=⁰𝑒^𝐿𝜆₀+𝑒^𝑙,−𝜆₀<𝑟₀𝑒<0,_𝑙,𝑟₀≥0. (3.18)

Proof
Case 1 (𝑟₀≥0). For every 𝑢∈ℍ, we have ‖𝑢‖²=₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑟(𝑡)|𝑢^𝜎(𝑡)|²Δ𝑡+₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)|𝑢^Δ(𝑡)|²≥Δ𝑡_0𝜎(𝑇)𝑒_𝐶(𝑡,0)|𝑢^Δ(𝑡)|²Δ𝑡≥𝑒^𝑙₀^𝜎(𝑇)|𝑢^Δ(𝑡)|²Δ𝑡=𝑒^𝑙‖𝑢‖²_ℍ. (3.19) On the other hand, by Lemma 2.6, one has ‖𝑢‖²=₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑟(𝑡)|𝑢^𝜎(𝑡)|²Δ𝑡+₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)|𝑢^Δ(𝑡)|²Δ𝑡≤𝑒^𝐿(₀^𝜎(𝑇)|𝑟(𝑡)||𝑢^𝜎(𝑡)|²Δ𝑡+₀^𝜎(𝑇)|𝑢^Δ(𝑡)|²Δ𝑡)≤𝑒^𝐿(‖𝑟‖_∞𝜎²(𝑇)+1)₀^𝜎(𝑇)|𝑢^Δ(𝑡)|²Δ𝑡=𝑒^𝐿(‖𝑟‖_∞𝜎²(𝑇)+1)‖𝑢‖²_ℍ. (3.20) Therefore, the norm ‖⋅‖ and the norm ‖⋅‖_ℍ are equivalent.
Case 2 (−𝜆₀<𝑟₀<0). For every 𝑢∈ℍ, we have from (3.15) that ‖𝑢‖²=₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑟(𝑡)|𝑢^𝜎(𝑡)|²Δ𝑡+₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)|𝑢^Δ(𝑡)|²Δ𝑡≥𝑒^𝐿𝑟₀₀^𝜎(𝑇)|𝑢^𝜎(𝑡)|²Δ𝑡+𝑒^𝑙₀^𝜎(𝑇)|𝑢^Δ(𝑡)|²≥𝑟Δ𝑡₀𝑒^𝐿𝜆₀₀^𝜎(𝑇)|𝑢^Δ(𝑡)|²Δ𝑡+𝑒^𝑙₀^𝜎(𝑇)|𝑢^Δ(𝑡)|²𝑟Δ𝑡=(⁰𝑒^𝐿𝜆₀+𝑒^𝑙)₀^𝜎(𝑇)|𝑢^Δ(𝑡)|²𝑟Δ𝑡=(⁰𝑒^𝐿𝜆₀+𝑒^𝑙)‖𝑢‖²_ℍ. (3.21) Similar to Case 1, it is easy to obtain the desired results. This completes the proof.

Together with (3.15) and Lemma 3.3, it is not difficult to obtain the following.

Lemma 3.4 (Poincaré inequality). For all 𝑢∈ℍ, one has ₀^𝜎(𝑇)|𝑢^𝜎(𝑡)|²1Δ𝑡≤𝜆₀𝜔(₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑟(𝑡)|𝑢^𝜎(𝑡)|²Δ𝑡+₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)|𝑢^Δ(𝑡)|²Δ𝑡), (3.22) that is, ‖𝑢^𝜎‖₂≤(1/𝜆₀𝜔)‖𝑢‖.

Lemma 3.5. ‖𝑢‖_∞≤𝑎₀‖𝑢‖ for all 𝑢∈ℍ, where 𝑎₀=𝜎(𝑇)/𝜔.

Proof. For every 𝑢∈ℍ, we have |𝑢(𝑡)|=|𝑢(𝑡)−𝑢(0)|=|^𝑡₀𝑢^Δ≤(𝑠)Δ𝑠|_𝑡0|𝑢^Δ(𝑠)|Δ𝑠≤𝑡^1/2(^𝑡₀|𝑢^Δ(𝑠)|²Δ𝑠)^1/2≤𝑡^1/2‖𝑢‖_ℍ. (3.23) Therefore, ‖𝑢‖_∞≤𝜎(𝑇)‖𝑢‖_ℍ≤𝜎(𝑇)𝜔‖𝑢‖. (3.24) This completes the proof.

For 𝑢∈ℍ⋂𝐶²_𝑟𝑑[0,𝜎(𝑇)]^𝜅2_𝕋, we have that 𝑢 and 𝑢^Δ are both absolutely continuous. Hence, Δ𝑢^Δ(𝑡)=𝑢^Δ(𝑡⁺)−𝑢^Δ(𝑡⁻)=0 for any 𝑡∈[0,𝜎(𝑇)]^𝜅_𝕋.

If 𝑢∈ℍ, then 𝑢 is absolutely continuous and 𝑢^Δ∈𝐿²_𝑇𝑆[0,𝜎(𝑇)]_𝕋. In this case, Δ𝑢^Δ(𝑡)=𝑢^Δ(𝑡⁺)−𝑢^Δ(𝑡⁻)=0 may not hold for some 𝑡∈[0,𝜎(𝑇)]^𝜅_𝕋. It leads to the impulsive effects.

Take 𝑣∈ℍ and multiply the two sides of the equality𝑢^ΔΔ(𝑡)+𝐶𝑢^Δ(𝜎(𝑡))−𝑟(𝑡)𝑢(𝜎(𝑡))+𝑓(𝜎(𝑡),𝑢(𝜎(𝑡)))=ℎ(𝑡) (3.25) by 𝑒_𝐶(𝑡,0),𝑣^𝜎(𝑡) and integrate from 0 to 𝜎(𝑇); we have₀^𝜎(𝑇)(𝑒_𝐶(𝑡,0)𝑢^Δ(𝑡))^Δ𝑣^𝜎(𝑡)Δ𝑡=₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)ℎ(𝑡)𝑣^𝜎−(𝑡)Δ𝑡_0𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑟(𝑡)𝑢(𝜎(𝑡))𝑣^𝜎−(𝑡)Δ𝑡_0𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑓(𝜎(𝑡),𝑢(𝜎(𝑡)))𝑣^𝜎(𝑡)Δ𝑡. (3.26) Furthermore,₀^𝜎(𝑇)(𝑒_𝐶(𝑡,0)𝑢^Δ(𝑡))^Δ𝑣^𝜎=(𝑡)Δ𝑡_{𝑗=𝑝𝑗=0𝑡𝑗+1𝑡𝑗}(𝑒_𝐶(𝑡,0)𝑢^Δ(𝑡))^Δ𝑣^𝜎=(𝑡)Δ𝑡_{𝑗=𝑝𝑗=0}(𝑒_𝐶(𝑡⁻_𝑗+1,0)𝑢^Δ(𝑡⁻_𝑗+1)𝑣(𝑡⁻_𝑗+1)−𝑒_𝐶(𝑡⁺_𝑗,0)𝑢^Δ(𝑡⁺_𝑗)𝑣(𝑡⁺_𝑗)−^𝑡𝑗+1_𝑡𝑗𝑒_𝐶(𝑡,0)𝑢^Δ(𝑡)𝑣^Δ(𝑡)Δ𝑡)=−^𝑗=𝑝_𝑗=1𝑒_𝐶(𝑡_𝑗,0)Δ𝑢^Δ(𝑡_𝑗)𝑣(𝑡_𝑗)+𝑒_𝐶(𝜎(𝑇),0)𝑢^Δ(𝜎(𝑇))𝑣(𝜎(𝑇))−𝑒_𝐶(0,0)𝑢^Δ−(0)𝑣(0)_0𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑢^Δ(𝑡)𝑣^Δ(𝑡)Δ𝑡=−^𝑗=𝑝_𝑗=1𝑒_𝐶(𝑡_𝑗,0)𝐼_𝑗(𝑢(𝑡_𝑗))𝑣(𝑡_𝑗)−₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑢^Δ(𝑡)𝑣^Δ(𝑡)Δ𝑡. (3.27) Combining (3.26), we have₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑢^Δ(𝑡)𝑣^Δ(𝑡)Δ𝑡+₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑟(𝑡)𝑢(𝜎(𝑡))𝑣^𝜎(𝑡)Δ𝑡+^𝑗=𝑝_𝑗=1𝑒_𝐶(𝑡_𝑗,0)𝐼_𝑗(𝑢(𝑡_𝑗))𝑣(𝑡_𝑗)=₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑓(𝜎(𝑡),𝑢(𝜎(𝑡)))𝑣^𝜎(𝑡)Δ𝑡−₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)ℎ(𝑡)𝑣^𝜎(𝑡)Δ𝑡. (3.28) Considering the above, we introduce the following concept solution for problem (1.1).

Definition 3.6. We say that a function 𝑢∈ℍ is a weak solution of problem (1.1) if the identity ₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑢^Δ(𝑡)𝑣^Δ(𝑡)Δ𝑡+₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑟(𝑡)𝑢(𝜎(𝑡))𝑣^𝜎(𝑡)Δ𝑡+^𝑝_𝑗=1𝑒_𝐶(𝑡_𝑗,0)𝐼_𝑗(𝑢(𝑡_𝑗))𝑣(𝑡_𝑗)=₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑓(𝜎(𝑡),𝑢(𝜎(𝑡)))𝑣^𝜎(𝑡)Δ𝑡−₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)ℎ(𝑡)𝑣^𝜎(𝑡)Δ𝑡 (3.29) holds for any 𝑣∈ℍ.

Consider the functional 𝜑∶ℍ→ℝ defined by1𝜑(𝑢)=2‖𝑢‖²+₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)[ℎ(𝑡)𝑢^𝜎(𝑡)−𝐹(𝜎(𝑡),𝑢^𝜎(𝑡))]Δ𝑡+^𝑝_𝑗=1𝑒_𝐶(𝑡_𝑗,0)^{𝑢(𝑡𝑗)}₀𝐼_𝑗(𝑠)d𝑠, (3.30) where 𝐹(𝑡,𝑢)=∫^𝑢₀𝑓(𝑡,𝑠)d𝑠. Using the continuity of 𝑓 and 𝐼_𝑗, 𝑗=1,2,…,𝑝, one has that 𝜑∈𝐶¹(ℍ,ℝ). For any 𝑣∈ℍ, we have𝜑^′(𝑢)𝑣=₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑢^Δ(𝑡)𝑣^Δ(𝑡)Δ𝑡+₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑟(𝑡)𝑢^𝜎(𝑡)𝑣^𝜎−(𝑡)Δ𝑡_0𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑓(𝜎(𝑡),𝑢^𝜎(𝑡))𝑣^𝜎(𝑡)Δ𝑡+₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)ℎ(𝑡)𝑣^𝜎+(𝑡)Δ𝑡_𝑝𝑗=1𝑒_𝐶(𝑡_𝑗,0)𝐼_𝑗(𝑢(𝑡_𝑗))𝑣(𝑡_𝑗). (3.31) Therefore, the solutions of problem (1.1) are the corresponding critical points of 𝜑.

Lemma 3.7 (see [33]). The functionals 𝜓₁,𝜓₂,𝜓₃∶ℍ→ℝ defined by 𝜓₁(𝑢)=₀^𝜎(𝑇)𝐹(𝜎(𝑡),𝑢^𝜎(𝑡))Δ𝑡,𝜓₂(𝑢)=₀^𝜎(𝑇)ℎ(𝑡)𝑢^𝜎(𝑡)Δ𝑡,𝜓₃(𝑢)=^𝑝_𝑗=1^{𝑢(𝑡𝑗)}₀𝐼_𝑗(𝑠)d𝑠 (3.32) are weakly continuous on ℍ.

4. Main Results

Theorem 4.1. Assume that (H1) holds. Suppose further the following.(H2)There exist 𝑎,𝑏>0 and 𝜃∈[0,1) such that|𝑓(𝑡,𝑢)|≤𝑎+𝑏|𝑢|^𝜃,∀(𝑡,𝑢)∈[0,𝜎(𝑇)]_𝕋×ℝ. (4.1)(H3)There exist 𝑎_𝑗,𝑏_𝑗>0 and 𝜃_𝑗∈[0,1)(𝑗=1,2,…,𝑝) such that|𝐼_𝑗(𝑢)|≤𝑎_𝑗+𝑏_𝑗|𝑢|^𝜃𝑗,∀𝑢∈ℝ(𝑗=1,2,…,𝑝). (4.2) Then problem (1.1) has at least one weak solution.

Proof. According to Lemma 3.7, 𝜓₁(𝑢),𝜓₂(𝑢),𝜓₃(𝑢) are weakly continuous on ℍ. On the other hand, it is clear that ‖𝑢‖² is continuous and convex on ℍ. By Lemma 2.9,‖𝑢‖² is weakly lower semicontinuous. So 𝜑(𝑢) is weakly lower semicontinuous on ℍ.
From (H2)-(H3), we have1𝜑(𝑢)=2‖𝑢‖²+₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)[ℎ(𝑡)𝑢^𝜎(𝑡)−𝐹(𝜎(𝑡),𝑢^𝜎(𝑡))]Δ𝑡+^𝑝_𝑗=1𝑒_𝐶(𝑡_𝑗,0)^{𝑢(𝑡𝑗)}₀𝐼_𝑗≥1(𝑠)d𝑠2‖𝑢‖²−𝑒^𝐿₀^𝜎(𝑇)|ℎ(𝑡)𝑢^𝜎(𝑡)|Δ𝑡−𝑒^𝐿₀^𝜎(𝑇)(𝑎|𝑢^𝜎(𝑡)|+𝑏|𝑢^𝜎(𝑡)|^𝜃+1)Δ𝑡−𝑒^𝐿𝑝_𝑗=1^{𝑢(𝑡𝑗)}₀(𝑎_𝑗+𝑏_𝑗|𝑠|^𝜃𝑗≥1)d𝑠2‖𝑢‖²−𝑒^𝐿ℎ₀‖𝑢‖_∞−𝑒^𝐿𝜎(𝑇)(𝑎‖𝑢‖_∞+𝑏‖𝑢‖_∞^𝜃+1)−𝑒^𝐿𝑝_𝑗=1(𝑎_𝑗‖𝑢‖_∞+𝑏_𝑗‖𝑢‖^𝜃𝑗_∞⁺¹≥1)2‖𝑢‖²−𝑒^𝐿ℎ₀𝑎₀‖𝑢‖−𝑒^𝐿𝜎(𝑇)(𝑎𝑎₀‖𝑢‖+𝑏𝑎₀^𝜃+1‖𝑢‖^𝜃+1)−𝑒^𝐿𝑝_𝑗=1(𝑎_𝑗𝑎₀‖𝑢‖+𝑏_𝑗𝑎^𝜃𝑗₀⁺¹‖𝑢‖^𝜃𝑗+1) (4.3)(1)(2)(3)(4)(5) for all 𝑢∈ℍ. This implies that lim_{‖𝑢‖→∞}𝜑(𝑢)=∞, and 𝜑 is coercive. By Lemma 2.10, 𝜑 has a minimum point on ℍ, which is a critical point of 𝜑. Therefore, problem (1.1) has at least one weak solution. This completes the proof.

Example 4.2. Consider the following impulsive Duffing equations on time scales 𝑢^ΔΔ(𝑡)+2𝑢^Δ(𝑡)−3𝑢(𝑡)+1|𝑢(𝑡)|=10𝑒sin𝑡,a.e.𝑡∈[0,1]_𝕋,Δ𝑢^Δ(12)=𝑢^Δ(12⁺)−𝑢^Δ(12⁻)=³1𝑢(2),𝑢(0)=0=𝑢(𝜎(1)). (4.4) Then, problem (4.4) has at least one solution.

Proof. The result is easy to obtain from Theorem 4.1 and we should omit it. This completes the proof.

Theorem 4.3. Assume that (H1) holds. Suppose further the following. (H4)There exist 𝑎,𝑏,𝑐>0 and 𝜃∈[0,1) such that|𝑓(𝑡,𝑢)|≤𝑎|𝑢|+𝑏|𝑢|^𝜃+𝑐,∀(𝑡,𝑢)∈[0,𝜎(𝑇)]_𝕋×ℝ. (4.5)(H5)There exist 𝑎_𝑗,𝑏_𝑗,𝑐_𝑗>0 and 𝜃_𝑗∈[0,1)(𝑗=1,2,…,p) such that|𝐼_𝑗(𝑢)|≤𝑎_𝑗|𝑢|+𝑏_𝑗|𝑢|^𝜃𝑗+𝑐_𝑗,∀𝑢∈ℝ(𝑗=1,2,…,𝑝). (4.6)(H6)1/2−𝑎𝜎(𝑇)𝑒^𝐿𝑎²₀−∑^𝑝_𝑗=1𝑎_𝑗𝑒^𝐿𝑎²₀>0. Then problem (1.1) has at least one weak solution.

Proof. According to Theorem 4.1, we know that 𝜑(𝑢) is weakly lower semicontinuous on ℍ.
From (H4)–(H6), we have1𝜑(𝑢)=2‖𝑢‖²+₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)[ℎ(𝑡)𝑢^𝜎(𝑡)−𝐹(𝜎(𝑡),𝑢^𝜎(𝑡))]Δ𝑡+^𝑝_𝑗=1𝑒_𝐶(𝑡_𝑗,0)^{𝑢(𝑡𝑗)}₀𝐼_𝑗≥1(𝑠)d𝑠2‖𝑢‖²−𝑒^𝐿₀^𝜎(𝑇)|ℎ(𝑡)𝑢^𝜎(𝑡)|Δ𝑡−𝑒^𝐿₀^𝜎(𝑇)(𝑎|𝑢^𝜎(𝑡)|²+𝑏|𝑢^𝜎(𝑡)|^𝜃+1+𝑐|𝑢^𝜎(𝑡)|)Δ𝑡−𝑒^𝐿𝑝_𝑗=1^{𝑢(𝑡𝑗)}₀(𝑎_𝑗|𝑠|+𝑏_𝑗|𝑠|^𝜃𝑗+𝑐_𝑗≥1)d𝑠2‖𝑢‖²−𝑒^𝐿ℎ₀‖𝑢‖_∞−𝑒^𝐿𝜎(𝑇)(𝑎‖𝑢‖²_∞+𝑏‖𝑢‖_∞^𝜃+1+𝑐‖𝑢‖_∞)−𝑒^𝐿𝑝_𝑗=1(𝑎_𝑗‖𝑢‖²_∞+𝑏_𝑗‖𝑢‖^𝜃𝑗_∞⁺¹+𝑐_𝑗‖𝑢‖_∞≥1)2‖𝑢‖²−𝑒^𝐿ℎ₀𝑎₀‖𝑢‖−𝑒^𝐿𝜎(𝑇)(𝑎𝑎²₀‖𝑢‖²+𝑏𝑎₀^𝜃+1‖𝑢‖^𝜃+1+𝑐𝑎₀‖𝑢‖)−𝑒^𝐿𝑝_𝑗=1(𝑎_𝑗𝑎²₀‖𝑢‖²+𝑏_𝑗𝑎^𝜃𝑗₀⁺¹‖𝑢‖^𝜃𝑗+1+𝑐_𝑗𝑎₀1‖𝑢‖)=(2−𝑎𝜎(𝑇)𝑒^𝐿𝑎²₀−^𝑝_𝑗=1𝑎_𝑗𝑒^𝐿𝑎²₀)‖𝑢‖²−𝑒^𝐿ℎ₀𝑎₀‖𝑢‖−𝑒^𝐿𝜎(𝑇)(𝑏𝑎₀^𝜃+1‖𝑢‖^𝜃+1+𝑐𝑎₀‖𝑢‖)−𝑒^𝐿𝑝_𝑗=1(𝑏_𝑗𝑎^𝜃𝑗₀⁺¹‖𝑢‖^𝜃𝑗+1+𝑐_𝑗𝑎₀‖𝑢‖) (4.7)(6)(7)(8)(9)(10)(11)(12)(13) for all 𝑢∈ℍ. This implies that lim_{‖𝑢‖→∞}𝜑(𝑢)=∞, and 𝜑 is coercive. By Lemma 2.10, 𝜑 has a minimum point on ℍ, which is a critical point of 𝜑. Therefore, problem (1.1) has at least one weak solution. This completes the proof.

Example 4.4. Consider the following impulsive Duffing equations on time scales: 𝑢^ΔΔ(𝑡)+2𝑢^Δ1(𝑡)−3𝑢(𝑡)+5𝜎³(1)𝜈₀𝑢(𝑡)+|𝑢(𝑡)|=sin𝑡,a.e.𝑡∈[0,1]_𝕋,Δ𝑢^Δ(12)=𝑢^Δ(12⁺)−𝑢^Δ(12⁻1)=5𝜎²(1)𝜈₀1𝑢(2),𝑢(0)=0=𝑢(𝜎(1)), (4.8) where 𝜈₀=exp(₀^𝜎(1)log(1+3𝜇(𝑠))^1/𝜇(𝑠)Δ𝑠). (4.9) Then problem (4.8) has at least one solution.

Proof. By an easy calculation, we obtain 𝑒^𝐿=exp(₀^𝜎(1)log(1+3𝜇(𝑠))^1/𝜇(𝑠)Δ𝑠),𝑒^𝑙=1. (4.10) It is easy to verify that all the conditions of Theorem 4.3 are satisfied. From Theorem 4.3, problem (4.8) has at least one solution. This completes the proof.

Lemma 4.5. Assume that (H1) and (H3) hold. Suppose further the following. (H7)There exist constants 𝜇>2 and 𝑅>0, such that 0<𝜇𝐹(𝑡,𝑢)≤𝑢𝑓(𝑡,𝑢),∀|u|≥R,𝑡∈[0,𝜎(𝑇)]_𝕋, (4.11) where 𝐹(𝑡,𝑢)=∫^𝑢₀𝑓(𝑡,𝑠)d𝑠.Then 𝜑 satisfies the P.S. condition in ℍ.

Proof. Let {𝑢_𝑛} be a sequence in ℍ satisfying that {𝜑(𝑢_𝑛)} is bounded and 𝜑^′(𝑢_𝑛)→0 as 𝑛→+∞. Let Ω_𝑛={𝑡∈[0,𝜎(𝑇)]_𝕋∣|𝑢_𝑛(𝑡)|≥𝑅} for all 𝑛∈ℕ. By the continuity of 𝑓 and (H7), there exists a constant 𝑐₁>0 such that 𝛽𝜑(𝑢_𝑛)−(𝜑^′(𝑢_𝑛),𝑢_𝑛𝛽)=(2−1)‖𝑢_𝑛‖²+₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)[𝑢^𝜎_𝑛𝑓(𝜎(𝑡),𝑢^𝜎_𝑛)−𝛽𝐹(𝜎(𝑡),𝑢^𝜎_𝑛)]Δ𝑡+𝛽^𝑝_𝑗=1𝑒_𝐶(𝑡_𝑗,0)^{𝑢𝑛(𝑡𝑗)}₀𝐼_𝑗(𝑠)d𝑠−^𝑝_𝑗=1𝑒_𝐶(𝑡_𝑗,0)𝐼_𝑗(𝑢_𝑛(𝑡_𝑗))𝑢_𝑛(𝑡_𝑗)+₀^𝜎(𝑇)(𝛽−1)𝑒_𝐶(𝑡,0)ℎ(𝑡)𝑢^𝜎_𝑛𝛽Δ𝑡≥(2−1)‖𝑢_𝑛‖²+_Ω𝑛𝑒_𝐶(𝑡,0)[𝑢^𝜎_𝑛𝑓(𝜎(𝑡),𝑢^𝜎_𝑛)−𝛽𝐹(𝜎(𝑡),𝑢^𝜎_𝑛)]Δ𝑡−(𝛽+1)𝑒^𝐿𝑝_𝑗=1(𝑎_𝑗𝑎₀‖𝑢‖+𝑏_𝑗𝑎^𝜃𝑗₀⁺¹‖𝑢‖^𝜃𝑗+1)−(𝛽−1)𝑒^𝐿ℎ₀𝑎₀‖𝑢_𝑛‖−𝑐₁𝛽≥(2−1)‖𝑢_𝑛‖²−(𝛽+1)𝑒^𝐿𝑝_𝑗=1(𝑎_𝑗𝑎₀‖𝑢‖+𝑏_𝑗𝑎^𝜃𝑗₀⁺¹‖𝑢‖^𝜃𝑗+1)−(𝛽−1)𝑒^𝐿ℎ₀𝑎₀‖𝑢_𝑛‖−𝑐₁, (4.12) which implies that {𝑢_𝑛} is bounded in ℍ. Taking into account {𝑢_𝑛}⊂ℍ⊂𝐶_𝑇𝑆[0,𝜎(𝑇)]_𝕋 and employing Lemma 3.5, we know that {𝑢_𝑛} is relatively compact in 𝐶_𝑇𝑆[0,𝜎(𝑇)]_𝕋. Thus {𝑢_𝑛} possesses a convergent subsequence {𝑢_𝑛𝑘} in 𝐶_𝑇𝑆[0,𝜎(𝑇)]_𝕋, that is, ‖𝑢_𝑛𝑖−𝑢_𝑛𝑗‖_∞→0 as 𝑖,𝑗→+∞. since 𝑢_𝑛∈ℍ, we know that 𝑢^𝜎_𝑛(𝑇)=0 for all 𝑛∈ℕ. From the above inequality, it is easy to see that sup_{𝑡∈[0,𝜎(𝑇)]𝜅𝕋}|𝑢_𝑛𝑖(𝑡)−𝑢_𝑛𝑗(𝑡)|→0,as𝑖,𝑗→+∞. (4.13) From the continuity of 𝑓 and 𝐼_𝑗, 𝑗=1,2,…,𝑝, we have |𝑓(𝜎(𝑡),𝑢^𝜎_𝑛𝑖(𝑡))−𝑓(𝜎(𝑡),𝑢^𝜎_𝑛𝑗(𝑡))|→0,|𝐼_𝑗(𝑢_𝑛𝑖(𝑡))−𝐼_𝑗(𝑢_𝑛𝑗(𝑡))|→0 (4.14) hold uniformly for 𝑡∈[0,𝜎(𝑇)]^𝜅_𝕋 as 𝑖,𝑗→+∞. Set ‖𝑓(𝜎(𝑡),𝑢^𝜎_𝑛𝑖(𝑡))−𝑓(𝜎(𝑡),𝑢^𝜎_𝑛𝑗(𝑡))‖_∞=sup_{𝑡∈[0,𝜎(𝑇)]𝜅𝕋}|𝑓(𝜎(𝑡),𝑢^𝜎_𝑛𝑖(𝑡))−𝑓(𝜎(𝑡),𝑢^𝜎_𝑛𝑗(𝑡)),‖𝐼_𝑗(𝑢_𝑛𝑖(𝑡))−𝐼_𝑗(𝑢_𝑛𝑗(𝑡))‖_∞=sup_{𝑡∈[0,𝜎(𝑇)]𝜅𝕋}|𝐼_𝑗(𝑢_𝑛𝑖(𝑡))−𝐼_𝑗(𝑢_𝑛𝑗(𝑡))|. (4.15) Clearly, ‖𝑓(𝜎(𝑡),𝑢^𝜎_𝑛𝑖(𝑡))−𝑓(𝜎(𝑡),𝑢^𝜎_𝑛𝑗(𝑡))‖_∞→0,‖𝐼_𝑗(𝑢_𝑛𝑖(𝑡))−𝐼_𝑗(𝑢_𝑛𝑗(𝑡))‖_∞→0, (4.16) as 𝑖,𝑗→+∞. We rewrite (3.31) as (𝜑^′(𝑢),𝑣)=(𝑢,𝑣)−₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)𝑓(𝜎(𝑡),𝑢(𝜎(𝑡)))𝑣^𝜎+(𝑡)Δ𝑡_0𝜎(𝑇)𝑒_𝐶(𝑡,0)ℎ(𝑡)𝑣^𝜎(𝑡)Δ𝑡+^𝑝_𝑗=1𝑒_𝐶(𝑡_𝑗,0)𝐼_𝑗(𝑢(𝑡_𝑗))𝑣(𝑡_𝑗) (4.17) for any 𝑢,𝑣∈ℍ. Therefore, (𝑢_𝑛𝑖−𝑢_𝑛𝑗,𝑣)=(𝜑^′(𝑢_𝑛𝑖)−𝜑^′(𝑢_𝑛𝑗),𝑣)−^𝑝_𝑗=1𝑒_𝐶(𝑡_𝑗,0)[𝐼_𝑗(𝑢_𝑛𝑖(𝑡_𝑗))−𝐼_𝑗(𝑢_𝑛𝑗(𝑡_𝑗))]𝑣(𝑡_𝑗)+₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)[𝑓(𝜎(𝑡),𝑢_𝑛𝑖(𝜎(𝑡)))−𝑓(𝜎(𝑡),𝑢_𝑛𝑗(𝜎(𝑡)))]𝑣^𝜎(𝑡)Δ𝑡. (4.18) Thus, for any 𝑣∈ℍ, it follows that |(𝑢_𝑛𝑖−𝑢_𝑛𝑗,𝑣)|≤‖𝜑^′(𝑢_𝑛𝑖)−𝜑^′(𝑢_𝑛𝑗)‖‖𝑣‖+𝑒^𝐿𝑝_𝑗=1‖𝐼_𝑗(𝑢_𝑛𝑖(𝑡_𝑗))−𝐼_𝑗(𝑢_𝑛𝑗(𝑡_𝑗))‖_∞‖𝑣(𝑡_𝑗)‖_∞+𝑒^𝐿‖𝑓(𝜎(𝑡),𝑢_𝑛𝑖(𝜎(𝑡)))−𝑓(𝜎(𝑡),𝑢_𝑛𝑗(𝜎(𝑡)))‖_∞0^𝜎(𝑇)|𝑣^𝜎(𝑡)|Δ𝑡≤‖𝜑^′(𝑢_𝑛𝑖)−𝜑^′(𝑢_𝑛𝑗)‖‖𝑣‖+𝑒^𝐿𝑎₀^𝑝_𝑗=1‖𝐼_𝑗(𝑢_𝑛𝑖(𝑡_𝑗))−𝐼_𝑗(𝑢_𝑛𝑗(𝑡_𝑗))‖_∞‖𝑣‖+𝑒^𝐿𝜎(𝑇)‖𝑓(𝜎(𝑡),𝑢_𝑛𝑖(𝜎(𝑡)))−𝑓(𝜎(𝑡),𝑢_𝑛𝑗(𝜎(𝑡)))‖_∞‖𝑣^𝜎‖₂≤‖𝜑^′(𝑢_𝑛𝑖)−𝜑^′(𝑢_𝑛𝑗)‖‖𝑣‖+𝑒^𝐿𝑎₀^𝑝_𝑗=1‖𝐼_j(𝑢_𝑛𝑖(𝑡_𝑗))−𝐼_𝑗(𝑢_𝑛𝑗(𝑡_𝑗))‖_∞‖𝑣‖+𝑒^𝐿𝜎(𝑇)𝜆₀𝜔‖𝑓(𝜎(𝑡),𝑢_𝑛𝑖(𝜎(𝑡)))−𝑓(𝜎(𝑡),𝑢_𝑛𝑗(𝜎(𝑡)))‖_∞‖𝑣‖. (4.19) Hence, ‖𝑢_𝑛𝑖−𝑢_𝑛𝑗‖=sup_{‖𝑣‖≤1}|(𝑢_𝑛𝑖−𝑢_𝑛𝑗,𝑣)|≤‖𝜑^′(𝑢_𝑛𝑖)−𝜑^′(𝑢_𝑛𝑗)‖+𝑒^𝐿𝑎₀^𝑝_𝑗=1‖𝐼_𝑗(𝑢_𝑛𝑖(𝑡_𝑗))−𝐼_𝑗(𝑢_𝑛𝑗(𝑡_𝑗))‖_∞+𝑒^𝐿𝜎(𝑇)𝜆₀𝜔‖𝑓(𝜎(𝑡),𝑢_𝑛𝑖(𝜎(𝑡)))−𝑓(𝜎(𝑡),𝑢_𝑛𝑗(𝜎(𝑡)))‖_∞. (4.20) Observing that 𝜑^′(𝑢_𝑛)→0 as 𝑛→+∞, it is easy to see that ‖𝜑^′(𝑢_𝑛𝑖)−𝜑^′(𝑢_𝑛𝑗)‖→0,as𝑖,𝑗→+∞. (4.21) Together with (4.16)–(4.21), we get ‖𝑢_𝑛𝑖−𝑢_𝑛𝑗‖→0,as𝑖,𝑗→+∞. (4.22) In other words, {𝑢_𝑛} possesses a convergent subsequence {𝑢_𝑛𝑘} in ℍ. The P.S. condition is now satisfied. This completes the proof.

Theorem 4.6. Assume that (H1), (H3), and (H7) hold. Suppose further the following.(H8)There exists constant 𝜈≥2 such that0<𝜈𝐹(𝑡,𝑢)≤𝑢𝑓(𝑡,𝑢),∀𝑢∈[−1,0)(0,1],𝑡∈[0,𝜎(𝑇)]_𝕋, (4.23)(H9)𝑎²₀/2−𝑒^𝐿𝑀𝜎(𝑇)−𝑒^𝐿𝜎(𝑇)‖ℎ‖₂−𝑒^𝐿∑^𝑝_𝑗=1𝑎_𝑗−𝑒^𝐿∑^𝑝_𝑗=1𝑏_𝑗>0, where 𝑀∶=sup{𝐹(𝑡,𝑢)∶𝑡∈[0,𝜎(𝑇)]_𝕋,|𝑢|=1}. (4.24) Then problem (1.1) has at least one weak solution.

Proof. It suffices to show that 𝜑 possesses a nonzero critical point. Now, we need to verify that all assumptions of the Mountain Pass Theorem hold. The P.S. condition follows from Lemma 4.5. Next, we will check conditions (J1) and (J2) of Lemma 2.11. Assume that 0<‖𝑢‖_∞≤1. In view of (H8), for 𝜉∈(0,1], we get that dd𝜉[𝐹(𝑡,𝜉⁻¹𝑢)𝜉^𝜈d]=d𝜉[^𝜉−1𝑢₀𝑓(𝑡,𝑠)d𝑠𝜉^𝜈]=(𝜈𝐹(𝑡,𝜉⁻¹𝑢)−𝜉⁻¹𝑢𝑓(𝑡,𝜉⁻¹𝑢))𝜉^𝜈−1≤0. (4.25) Hence, 𝑢𝐹(𝑡,𝑢)≤𝐹(𝑡,|𝑢|)|𝑢|^𝜈,for0<|𝑢|≤1,𝑡∈[0,𝜎(𝑇)]_𝕋, (4.26) which implies that ₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)|𝐹(𝜎(𝑡),𝑢^𝜎(𝑡))|Δ𝑡≤₀^𝜎(𝑇)𝑒_𝐶𝑢(𝑡,0)|𝐹(𝜎(𝑡),^𝜎(𝑡)|𝑢^𝜎(𝑡)|)||𝑢^𝜎(𝑡)|^𝜈Δ𝑡≤𝑒^𝐿𝑀₀^𝜎(𝑇)|𝑢^𝜎(𝑡)|²Δ𝑡≤𝑒^𝐿𝑀𝜎²(𝑇)𝜔‖𝑢‖². (4.27) At the same time, we can obtain that ₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)|ℎ(𝑡)𝑢^𝜎(𝑡)|Δ𝑡≤𝑒^𝐿‖ℎ‖₂‖𝑢^𝜎‖₂≤𝑒^𝐿𝜎(𝑇)√𝜔‖ℎ‖₂‖𝑢‖,_𝑝𝑗=1𝑒_𝐶(𝑡_𝑗,0)^{𝑢(𝑡𝑗)}₀|𝐼_𝑗(𝑠)|d𝑠≤𝑒^𝐿𝑝_𝑗=1^{𝑢(𝑡𝑗)}₀(𝑎_𝑗+𝑏_𝑗|𝑠|^𝜃𝑗)d𝑠≤𝑒^𝐿𝑝_𝑗=1(𝑎_𝑗‖𝑢‖_∞+𝑏_𝑗‖𝑢‖^𝜃𝑗_∞⁺¹)≤𝑒^𝐿𝑝_𝑗=1(𝑎_𝑗𝑎₀‖𝑢‖+𝑏_𝑗𝑎^𝜃𝑗₀⁺¹‖𝑢‖^𝜃𝑗+1). (4.28) Together (4.27)-(4.28) and the expression of functional 𝜑, it follows that 1𝜑(𝑢)≥2‖𝑢‖²−𝑒^𝐿𝑀𝜎(𝑇)𝑎²₀‖𝑢‖²−𝑒^𝐿√𝜔𝑎²₀‖ℎ‖₂‖𝑢‖−𝑒^𝐿𝑝_𝑗=1(𝑎_𝑗𝑎₀‖𝑢‖+𝑏_𝑗𝑎^𝜃𝑗₀⁺¹‖𝑢‖^𝜃𝑗+1)1=(2−𝑒^𝐿𝑀𝜎(𝑇)𝑎²₀)‖𝑢‖²−𝑒^𝐿𝑝_𝑗=1𝑏_𝑗𝑎^𝜃𝑗₀⁺¹‖𝑢‖^𝜃𝑗+1−(𝑒^𝐿√𝜔𝑎²₀‖ℎ‖₂+𝑒^𝐿𝑝_𝑗=1𝑎_𝑗𝑎₀1)‖𝑢‖=(2−𝑒^𝐿𝑀𝜎(𝑇)𝑎²₀−𝑒^𝐿𝜎(𝑇)𝑎²₀‖ℎ‖₂−𝑒^𝐿𝑝_𝑗=1𝑎_𝑗𝑎²₀)‖𝑢‖²+(𝑒^𝐿𝜎(𝑇)𝑎²₀‖ℎ‖₂+𝑒^𝐿𝑝_𝑗=1𝑎_𝑗𝑎²₀)‖𝑢‖²−(𝑒^𝐿√𝜔𝑎²₀‖ℎ‖₂+𝑒^𝐿𝑝_𝑗=1𝑎_𝑗𝑎₀)‖𝑢‖−𝑒^𝐿𝑝_𝑗=1𝑏_𝑗𝑎^𝜃𝑗₀⁺¹‖𝑢‖^𝜃𝑗+1. (4.29) Setting 1‖𝑢‖=𝜌∶=𝑎₀=𝜔.𝜎(𝑇) (4.30) Then we have (𝑒^𝐿𝜎(𝑇)𝑎²₀‖ℎ‖₂+𝑒^𝐿𝑝_𝑗=1𝑎_𝑗𝑎²₀)𝜌²−(𝑒^𝐿√𝜔𝑎²₀‖ℎ‖₂+𝑒^𝐿𝑝_𝑗=1𝑎_𝑗𝑎₀)𝜌=0, (4.31) which implies from (H9) that 1𝜑(𝑢)≥(2−𝑒^𝐿𝑀𝜎(𝑇)𝑎²₀−𝑒^𝐿𝜎(𝑇)𝑎²₀‖ℎ‖₂−𝑒^𝐿𝑝_𝑗=1𝑎_𝑗𝑎²₀)𝜌²−𝑒^𝐿𝑝_𝑗=1𝑏_𝑗𝑎^𝜃𝑗₀⁺¹𝜌^𝜃𝑗+1=𝑎²₀2−𝑒^𝐿𝑀𝜎(𝑇)−𝑒^𝐿𝜎(𝑇)‖ℎ‖₂−𝑒^𝐿𝑝_𝑗=1𝑎_𝑗−𝑒^𝐿𝑝_𝑗=1𝑏_𝑗=𝛼>0. (4.32) By Lemma 3.5, if ‖𝑢‖=𝜌, then 0<‖𝑢‖_∞≤1. Set 𝐵_𝜌={𝑢∈ℍ∣‖𝑢‖<𝜌=1/𝑎₀}, then 𝜑(𝑢)≥𝛼>0 for all 𝑢∈𝜕𝐵_𝜌. Condition (J1) is satisfied.
Secondly, we verify the condition (J2) of Lemma 2.11. In view of assumption (H7), we have𝜇𝑢≤𝑓(𝑡,𝑢)𝜇𝐹(𝑡,𝑢),𝑢≥𝑅,𝑢≥𝑓(𝑡,𝑢)𝐹(𝑡,𝑢),𝑢≤−𝑅. (4.33)(4.34) Integrating (4.33) and (4.34) for 𝑢 on [𝑅,𝑢] and [𝑢,−𝑅], respectively, we get 𝑢𝜇ln𝑅≤ln𝐹(𝑡,𝑢)𝑅𝐹(𝑡,𝑢),𝑢≥𝑅,𝜇ln−𝑢≥ln𝐹(𝑡,−𝑅)𝐹(𝑡,𝑢),𝑢≤−𝑅, (4.35) that is, 𝑢𝐹(𝑡,𝑢)≥𝐹(𝑡,𝑅)(𝑅)^𝜇,𝑢≥𝑅,𝐹(𝑡,𝑢)≥𝐹(𝑡,−𝑅)(−𝑢𝑅)^𝜇,𝑢≤−𝑅. (4.36)(4.37) Combining (4.36) and (4.37), one has 𝐹(𝑡,𝑢)≥𝛼₁|𝑢|^𝜇,|𝑢|≥𝑅, (4.38) where 𝛼₁=𝑅^−𝜇min{min_{𝑡∈[0,𝜎(𝑇)]𝕋}𝐹(𝑡,𝑅),min_{𝑡∈[0,𝜎(𝑇)]𝕋}𝐹(𝑡,−𝑅)}>0. (4.39) On the other hand, by the continuity of 𝐹(𝑡,𝑢), 𝐹(𝑡,𝑢) is bounded on [0,𝜎(𝑇)]_𝕋×[−𝑅,𝑅], there exists 𝐾>0 such that 𝐹(𝑡,𝑢)≥−𝐾≥𝛼₁|𝑢|^𝜇−𝛼₁𝑅^𝜇−𝐾,|𝑢|≤𝑅. (4.40) Combining (4.38) and (4.40), we have 𝐹(𝑡,𝑢)≥𝛼₁|𝑢|^𝜇−𝛼₂,∀(𝑡,𝑢)∈[0,𝜎(𝑇)]_𝕋×ℝ, (4.41) where 𝛼₂=𝛼₁𝑅^𝜇+𝐾.
According to the Hölder inequality, we get₀^𝜎(𝑇)|𝑢^𝜎(𝑡)|²Δ𝑡≤𝜎(𝑇)^1−2/𝜇(₀^𝜎(𝑇)|𝑢^𝜎(𝑡)|^𝜇Δ𝑡)^2/𝜇. (4.42) Thus, ₀^𝜎(𝑇)|𝑢^𝜎(𝑡)|^𝜇Δ𝑡≥𝜎(𝑇)^(2−𝜇)/2(₀^𝜎(𝑇)|𝑢^𝜎(𝑡)|²Δ𝑡)^𝜇/2=𝜎(𝑇)^(2−𝜇)/2‖𝑢^𝜎‖^𝜇₂. (4.43) Together (4.41) and (4.43), we know that ₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)𝐹(𝜎(𝑡),𝑢^𝜎(𝑡))Δ𝑡≥₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)𝛼₁|𝑢^𝜎(𝑡)|^𝜇Δ𝑡−₀^𝜎(𝑇)𝛼₂𝑒_𝐶(𝑡,0)Δ𝑡≥𝛼₁𝑒^𝑙𝜎(𝑇)^(2−𝜇)/2‖𝑢^𝜎‖^𝜇₂−𝛼₂𝑒^𝐿𝜎(𝑇)≥𝛼₃‖𝑢^𝜎‖^𝜇₂−𝛼₄, (4.44) where 𝛼₃=𝛼₁𝑒^𝑙𝜎(𝑇)^(2−𝜇)/2, and 𝛼₄=𝛼₂𝑒^𝐿𝜎(𝑇).
Obviously, 𝜑(0)=0. Let 𝑣₀∈ℍ satisfy ‖𝑣₀‖=1. For any 𝜍>0, it follows from (4.44) and (4.28) that𝜑(𝜍𝑣₀𝜍)=²2‖𝑣₀‖²−₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)𝐹(𝜎(𝑡),𝜍𝑣^𝜎₀(𝑡))Δ𝑡+𝜍₀^𝜎(𝑇)𝑒_𝐶(𝑡,0)ℎ(𝑡)𝑣^𝜎₀(𝑡)Δ𝑡+^𝑝_𝑗=1𝑒_𝐶(𝑡_𝑗,0)^{𝜍𝑣0(𝑡𝑗)}₀𝐼_𝑗≤𝜍(𝑠)d𝑠₂2‖𝑣₀‖²−𝛼₃𝜍^𝜇‖𝑣^𝜎₀‖^𝜇₂+𝜍𝑒^𝐿𝜎(𝑇)√𝜔‖ℎ‖₂‖𝑣₀‖+𝑒^𝐿𝑝_𝑗=1(𝜍𝑎_𝑗𝑎₀‖𝑣₀‖+𝑏_𝑗𝜍^𝜃𝑗+1𝑎^𝜃𝑗₀⁺¹‖𝑣₀‖^𝜃𝑗+1)+𝛼₄. (4.45) Since 𝜇>2, we see that 𝜑(𝜍𝑣₀)→−∞ as 𝜍→+∞. we can choose sufficiently large 𝜍₀ such that 𝑢₀=𝜍₀𝑣₀∈ℍ satisfying 𝑢₀∈𝜕𝐵_𝜌 and 𝜑(𝑢₀)<0. Condition (J2) is satisfied.
According to Lemma 2.11, 𝜑 possesses a critical value c≥𝛼>0 given by𝑐=inf_𝑔∈Γmax_𝑠∈[0,1]𝜑(𝑔(𝑠)), (4.46) where Γ={𝑔∈𝐶([0,1],𝔼)∣𝑔(0)=0,𝑔(1)=𝑢₀}. (4.47) Therefore, problem (1.1) has at least one weak solution. This completes the proof.

Example 4.7. Let 𝕋=ℝ. Consider the following impulsive Duffing equations with Dirichlet boundary condition 𝑢^′′1(𝑡)+2𝑢^′1(𝑡)−𝑢(𝑡)+𝑓(𝑡,𝑢(𝑡))=10𝑒sin𝑡,a.e.𝑡∈[0,1],Δ𝑢^′(12)=𝑢^′(12⁺)−𝑢^′(12⁻1)=10𝑒⁵1𝑢(2),𝑢(0)=0=𝑢(1), (4.48) where 𝑢𝑓(𝑡,𝑢)=10𝑒^𝑡+1𝑢,∀(𝑡,𝑢)∈([0,1]×[−1,1]),₃10𝑒^𝑡+1,∀(𝑡,𝑢)∈([0,1]×(−∞,−1)(1,+∞)). (4.49) Then problem (4.48) has at least one solution.

Proof. By an easy calculation, 𝑒^𝐿=𝑒, 𝑒^𝑙=1, ‖ℎ‖₂≤1/10𝑒, 𝑎₀=1, 𝑎₁=0, 𝑏₁=1/10𝑒, and 𝑀=1/20𝑒. It is easy to verify that all the conditions of Theorem 4.6 are satisfied. From Theorem 4.6, problem (4.48) has at least one solution. This completes the proof.

Corollary 4.8. Assume that (H1), (H3), (H7), and (H8) hold. Suppose further the following. (H10)𝐼_𝑗(𝑡)≥0 for all 𝑡∈[0,𝜎(𝑇)]_𝕋, 𝑗=1,2,…,𝑝. (H11)𝑎²₀/2−𝑒^𝐿𝑀𝜎(𝑇)−𝑒^𝐿𝜎(𝑇)‖ℎ‖₂>0, where 𝑀∶=sup{𝐹(𝑡,𝑢)∶𝑡∈[0,𝜎(𝑇)]_𝕋,|𝑢|=1}. (4.50) Then problem (1.1) has at least one weak solution.

Corollary 4.9. Assume that (H1), (H3), (H7), (H8), and (H10) hold. Suppose further the following. (H12)ℎ(𝑡)≡0 for all 𝑡∈[0,𝜎(𝑇)]_𝕋.(H13)𝑎²₀/2>𝑒^𝐿𝑀𝜎(𝑇), where 𝑀∶=sup{𝐹(𝑡,𝑢)∶𝑡∈[0,𝜎(𝑇)]_𝕋,|𝑢|=1}. Then problem (1.1) has at least one weak solution.

Corollary 4.10. Assume that (H1), (H3), (H7), (H10), and (H12) hold. Suppose further the following. (H14)sup_{𝑡∈[0,𝜎(𝑇)]𝕋}lim_𝑠→0(𝑓(𝑡,𝑠)/𝑠)=0. Then problem (1.1) has at least one weak solution.

Proof. From the proof of Theorem 4.6, it is easy to see that 𝜑∈𝐶¹(𝔼,ℝ) and the condition (J2) of Lemma 2.11 is satisfied. We only need to verify the condition (J1) in Lemma 2.11.
In view of (H14), for 𝜖=𝜆₀𝜔/2𝑒^𝐿>0, there is a constant 𝛿>0 such that|𝑓(𝑡,𝑠)|≤𝜖|𝑠|,∀0≤|𝑠|≤𝛿,𝑡∈[0,𝜎(𝑇)]_𝕋. (4.51) As a result, we have |𝐹(𝜎(𝑡),𝑢^𝜎(𝑡))|=^𝑢𝜎₀^(𝑡)𝑓(𝜎(𝑡),𝑠)d𝑠≤^|𝑢𝜎₀^(𝑡)|𝜖𝜖|𝑠|d𝑠=2|𝑢^𝜎(𝑡)|² (4.52) which implies from Lemma 3.4 that ₀^𝜎(𝑇)|𝐹(𝜎(𝑡),𝑢^𝜎𝜖(𝑡))|Δ𝑡≤2‖𝑢^𝜎‖²₂≤𝜖2𝜆₀𝜔‖𝑢‖²,∀0≤|𝑠|≤𝛿. (4.53) From the above inequality, (H10) and (H12), it follows that 1𝜑(𝑢)≥2‖𝑢‖²−𝜖𝑒^𝐿2𝜆₀𝜔‖𝑢‖²=14‖𝑢‖². (4.54) Take 𝜌=𝛿/𝑎₀, 𝛼=𝛿²/4𝑎²₀. By Lemma 3.5, if ‖𝑢‖=𝜌, then 0<‖𝑢‖_∞≤𝛿. Set 𝐵_𝜌={𝑢∈ℍ∣‖𝑢‖<𝜌=𝛿/𝑎₀}, then 𝜑(𝑢)≥𝛼>0 for all 𝑢∈𝜕𝐵_𝜌. Condition (J1) is satisfied. Therefore, problem (1.1) has at least one weak solution. This completes the proof.

Remark 4.11. Let 𝐶=0, 𝑟(𝑡)≡0 and let 𝐼_𝑗(𝑡)≡0 for all 𝑡∈[0,𝜎(𝑇)]_𝕋, 𝑗=1,2,…,𝑝, then Corollary 4.10 reduces to Theorem 4.2 in [33].

Acknowledgment

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 10971183.