Existence of Solution for Impulsive Differential Equations with Nonlinear Derivative Dependence via Variational Methods

Abstract

We use variational methods and iterative methods to investigate the solutions of impulsive differential equations with nonlinear derivative dependence. The conditions for the existence of solutions are established. The main results are also demonstrated with examples.

1. Introduction

Many dynamical systems have an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. The mathematical description of these phenomena leads to impulsive differential equations. Recent development in this field has been motivated by many applied problems, such as control theory, population dynamics, and medicine [1–9].

We consider the following nonlinear Dirichlet boundary value problems for impulsive differential equations: −𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′+𝑔(𝑡)|𝑢(𝑡)|^𝑝−2𝑢(𝑡)=𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡),𝑡≠𝑡_𝑗,a.e.𝑡∈[0,𝑇],Δ𝑢^′𝑡_𝑗=𝑢^′𝑡⁺_𝑗^𝑝−2𝑢^′𝑡⁺_𝑗−𝑢^′𝑡⁻_𝑗^𝑝−2𝑢^′𝑡⁻_𝑗=𝐼_𝑗𝑢𝑡_𝑗𝑢(0)(𝑇),𝑗=1,2,…,𝑙,=𝑢=0,(1) where 𝑝≥2, 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑙<𝑡_𝑙+1=𝑇, 𝑔∈𝐿^∞[0,𝑇], 𝑓∶[0,𝑇]×𝑅×𝑅→𝑅 is continuous, and 𝐼_𝑗∶𝑅→𝑅, 𝑗=1,2,…,𝑙, are continuous.

The characteristic of (1) is the presence of the first order derivative in the nonlinearity term. Most of the results concerning the existence of solutions of these equations are obtained using upper and lower solutions methods, coincidence degree theory, and fixed point theorems [10–14]. However, to the best of our knowledge, there are few papers concerned with the existence of solutions for impulsive boundary value problems like problem (1) by using variational methods. Motivated by [15, 16], in this paper we will fill the gap in this area.

When there is no derivative in the nonlinearity term, problem (1) has been extensively studied by [17–23], using variational methods. We know, contrary to these equations, (1) is not variational and the well-developed critical point theory is of no avail for, at least, a direct attack to problem (1). The technique used in this paper consists of, associating with problem (1), a family of the following Dirichlet boundary value problems with no dependence on the derivative of the solution. Namely, for each 𝑤∈𝑊₀^1,𝑝, we consider the problem −𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′+𝑔(𝑡)|𝑢(𝑡)|^𝑝−2𝑢(𝑡)=𝑓𝑡,𝑢(𝑡),𝑤^′(𝑡),𝑡≠𝑡_𝑗,a.e.𝑡∈[0,𝑇],Δ𝑢^′𝑡_𝑗=𝑢^′𝑡⁺_𝑗^𝑝−2𝑢^′𝑡⁺_𝑗−𝑢^′𝑡⁻_𝑗^𝑝−2𝑢^′𝑡⁻_𝑗=𝐼_𝑗𝑢𝑡_𝑗𝑢(0)(𝑇),𝑗=1,2,…,𝑙,=𝑢=0.(2) Now problem (2) is variational and we can treat it by variational methods.

In this paper, we need the following conditions.(𝑓₁)𝑓∶[0,𝑇]×𝑅×𝑅→𝑅 is measurable in 𝑡∈[0,𝑇] for every (𝑥,𝜁)∈𝑅×𝑅 and continuous in (𝑥,𝜁)∈𝑅×𝑅 for a.e. 𝑡∈[0,𝑇].(𝑓₂)𝑓(𝑡,𝑥,𝜁)=𝑜(|𝑥|^𝑝−1) as 𝑥→0 uniformly for 𝑡∈[0,𝑇] and 𝜁∈𝑅 and 𝑓(𝑡,𝑥,0)≠0 for 𝑡∈[0,𝑇] and 𝑥∈𝑅.(𝑓₃) There exist constants 𝐶>0 and 𝑟∈(𝑝,+∞) such that &𝑓(𝑡,𝑥,𝜁)≤𝐶1+|𝑥|^𝑟−1,∀𝑡∈[0,𝑇],𝑥∈𝑅,𝜁∈𝑅.(3)(𝑓₄) There exist constants 𝜇>𝑝 and 𝑥₀>0 such that ()(),[],|𝑥|0<𝜇𝐹𝑡,𝑥,𝜁≤𝑥𝑓𝑡,𝑥,𝜁∀𝑡∈0,𝑇≥𝑥₀,𝜁∈𝑅,(4) where 𝐹(𝑡,𝑥,𝜁)=∫^𝑥₀𝑓(𝑡,𝑠,𝜁)𝑑𝑠.(𝑓₅) There exist constants 𝑎,𝑏>0 such that 𝐹(𝑡,𝑥,𝜁)≥𝑎|𝑥|^𝜇−𝑏,∀𝑡∈[0,𝑇],𝑥∈𝑅,𝜁∈𝑅.(5)(𝑓₆) The function 𝑓 satisfies the following conditions: 𝑓(𝑡,𝑥,𝜁)−𝑓𝑡,𝑥^′,𝜁≤𝐿₁𝑥−𝑥^′𝑝−1,∀𝑡∈[0,𝑇],𝑥,𝑥^′∈−𝜌₁,𝜌₁𝑓(),𝜁∈𝑅,𝑡,𝑥,𝜁−𝑓𝑡,𝑥,𝜁^′≤𝐿₂𝜁−𝜁^′𝑝−1,∀𝑡∈[0,𝑇],𝑥∈−𝜌₁,𝜌₁,𝜁,𝜁^′∈𝑅,(6) where 𝜌₁ is a constant.(𝐼₁)𝐼_𝑗(𝑗=1,2,…,𝑙) are odd and nondecreasing, and there exist constants 𝑎_𝑗>0, 𝑏_𝑗>0, and 𝑟_𝑗∈[0,𝑝−1),𝑗=1,2,…,𝑙 such that 𝐼_𝑗(𝑢)≤𝑎_𝑗+𝑏_𝑗|𝑢|^𝑟𝑗,for𝑗=1,2,…,𝑙.(7)(𝐼₂) There exists 0<𝜃≤𝑝 such that 𝜃^𝑢₀𝐼_𝑗(𝑠)𝑑𝑠≥𝑢𝐼_𝑗(𝑢)>0,∀𝑢>0.(8)(𝐼₃)|𝐼_𝑗(𝑢)−𝐼_𝑗(𝑣)|≤𝛼_𝑗|𝑢−𝑣|^𝑝−1, for all 𝑢,𝑣∈[−𝜌₁,𝜌₁], 𝑗=1,2,…,𝑙.

The paper is organized as follows: Section 2 is the preliminaries of this paper, Section 3 is devoted to show the solvability of problem (2), and Section 4 will show the solvability of problem (1).

2. Preliminaries

Firstly, we recall some facts which will be used in the proof of our main result. It has been shown, for instance, in [16] that the set of all eigenvalues of the following problem 𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′+𝜆|𝑢(𝑡)|^𝑝−2𝑢(𝑡)[],𝑢(0)(𝑇)=0,𝑡∈0,𝑇=𝑢=0(9) is given by the sequence of positive numbers 𝜆_𝑘=𝑝−1𝑘𝜋_𝑝𝑇^𝑝,𝑘=1,2,…,(10) where 𝜋_𝑝=2¹₀𝑑𝑠(1−𝑠^𝑝)^1/𝑝=2𝜋.𝑝sin𝜋/𝑝(11) Each eigenvalue 𝜆_𝑘 is simple with the associated eigenfunction 𝜑_𝑘(𝑥)=sin_𝑝𝑘𝜋_𝑝𝑥𝑇,for0≤𝑥≤𝑇.(12) We recall the well-known characterization of 𝜆₁ as the best constant in the Poincaré inequality ^𝑇₀𝑢^′(𝑡)^𝑝𝑑𝑡≥𝜆₁^𝑇₀|𝑢(𝑡)|^𝑝𝑑𝑡,∀𝑢∈𝑊₀^1,𝑝(0,𝑇).(13)

Let 𝑊₀^1,𝑝(0,𝑇) be the Sobolev space endowed with the norm ‖𝑢‖=^𝑇₀𝑢^′(𝑡)^𝑝𝑑𝑡+^𝑇₀𝑔(𝑡)|𝑢(𝑡)|^𝑝𝑑𝑡^1/𝑝.(14) Throughout the paper, it will be assumed that essinf_{𝑡∈[0,𝑇]}𝑔(𝑡)=𝑚>−𝜆₁. We also consider the norm ‖𝑢‖_𝑊0^1,𝑝=^𝑇₀𝑢^′(𝑡)^𝑝𝑑𝑡^1/𝑝,‖𝑢‖_𝑝=^𝑇₀|𝑢(𝑡)|^𝑝𝑑𝑡^1/𝑝.(15)

We need the following Lemmas.

Lemma 1 (see [24, Lemma 2.4]). If essinf_{𝑡∈[0,𝑇]}𝑔(𝑡)=𝑚>−𝜆₁, then the norms ‖⋅‖ and ‖⋅‖_𝑊0^1,𝑝 are equivalent.

Lemma 2 (see [24, Lemma 2.5]). There exists 𝐶₂>0 such that if 𝑢∈𝑊₀^1,𝑝, then ‖𝑢‖_∞≤𝑇𝑞+1^1/𝑞𝑇₀𝑢^′(𝑠)^𝑝𝑑𝑠^1/𝑝,‖𝑢‖(16)_∞≤𝐶₂‖𝑢‖.(17)

Lemma 3 (see [24, Lemma 2.6]). Let 𝑢∈𝑊₀^1,𝑝(0,𝑇); then there exists 𝐶₁∈(0,1) such that ‖𝑢‖_𝑝≤_𝑝1𝜆₁𝐶₁‖𝑢‖.(18)

For 𝑢∈𝑊^1,𝑝(0,𝑇), we have that 𝑢 and 𝑢^′ are both absolutely continuous. Hence Δ𝑢^′(𝑡)=𝑢^′(𝑡⁺)−𝑢^′(𝑡⁻)=0 for any 𝑡∈[0,𝑇]. If 𝑢∈𝑊₀^1,𝑝(0,𝑇), then 𝑢 is absolutely continuous. In this case, Δ𝑢^′(𝑡)=𝑢^′(𝑡⁺)−𝑢^′(𝑡⁻)=0 may not hold for some 𝑡∈[0,𝑇]. It leads to the impulsive effects. As a consequence, we need to introduce a different concept of solution.

Definition 4. A function 𝑢∈𝑢∈𝑊₀^1,𝑝()∶𝑢0,𝑇_′𝑝−2𝑢^′∈𝑊^1,∞(0,𝑇)⧵𝑡₁,𝑡₂,…,𝑡_𝑙(19) is said to be a classical solution of problem (1) if 𝑢 satisfies the equation a.e. on [0,𝑇]⧵{𝑡₁,𝑡₂,…,𝑡_𝑙} and the limits 𝑢^′(𝑡⁺_𝑗), 𝑢^′(𝑡⁻_𝑗) and 𝑗=1,2,…,𝑙, exist and satisfy the impulsive condition Δ𝑢^′𝑡_𝑗=𝑢^′𝑡⁺_𝑗^𝑝−2𝑢^′𝑡⁺_𝑗−𝑢^′𝑡⁻_𝑗^𝑝−2𝑢^′𝑡⁻_𝑗=𝐼_𝑗𝑢𝑡_𝑗(20) and the boundary condition 𝑢(0)=𝑢(𝑇)=0.

We have the following fact. Take 𝑣∈𝑊₀^1,𝑝(0,𝑇) and multiply the equation in problem (1) by 𝑣 and integrate from 0 to 𝑇: −^𝑇₀𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′𝑣(𝑡)+𝑑𝑡_𝑇0𝑔(𝑡)|𝑢(𝑡)|^𝑝−2𝑢(𝑡)𝑣(𝑡)=𝑑𝑡_𝑇0𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡)𝑣(𝑡)𝑑𝑡.(21) The first term is now −^𝑇₀𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′𝑣(𝑡)𝑑𝑡=−^𝑙_𝑗=0^𝑡𝑗+1_𝑡𝑗𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′𝑣(𝑡)𝑑𝑡,_{𝑡𝑗+1𝑡𝑗}𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′𝑣(𝑡)=𝑢𝑑𝑡_′𝑡⁻_𝑗+1^𝑝−2𝑢^′𝑡⁻_𝑗+1𝑣𝑡_𝑗+1−𝑢^′𝑡⁺_𝑗^𝑝−2𝑢^′𝑡⁺_𝑗𝑣𝑡_𝑗−^𝑡𝑗+1_𝑡𝑗𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)𝑣^′(𝑡)𝑑𝑡.(22) Hence −^𝑇₀𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′𝑣(𝑡)=𝑑𝑡_𝑙𝑗=1Δ𝑢^′𝑡_𝑗𝑣𝑡_𝑗−𝑢^′(𝑇)^𝑝−2𝑢^′(𝑇)𝑣(𝑇)+𝑢^′(0)^𝑝−2𝑢^′(0)𝑣(0)+^𝑇₀𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)𝑣^′(𝑡)=𝑑𝑡_𝑙𝑗=1𝐼_𝑗𝑢𝑡_𝑗𝑣𝑡_𝑗+^𝑇₀𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)𝑣^′(𝑡)𝑑𝑡.(23)

Definition 5. We say that a function 𝑢∈𝑊₀^1,𝑝(0,𝑇) is a weak solution of problem (1) if the identity ^𝑇₀𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)𝑣^′(𝑡)𝑑𝑡+^𝑇₀𝑔(𝑡)|𝑢(𝑡)|^𝑝−2𝑢(𝑡)𝑣(𝑡)+𝑑𝑡_𝑙𝑗=1𝐼_𝑗𝑢𝑡_𝑗𝑣𝑡_𝑗=^𝑇₀𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡)𝑣(𝑡)𝑑𝑡(24) holds for any 𝑣∈𝑊₀^1,𝑝(0,𝑇).

Proposition 6. Under the hypotheses (𝑓₁) and (𝑓₂), the functional 𝜑_𝑤∶𝑊₀^1,𝑝(0,𝑇)→𝑅 defined by 𝜑_𝑤(𝑢)=1𝑝‖𝑢‖^𝑝−^𝑇₀𝐹𝑡,𝑢(𝑡),𝑤^′(𝑡)+𝑑𝑡_{𝑙𝑗=1𝑢(𝑡𝑗)0}𝐼_𝑗(𝑠)𝑑𝑠(25) is continuous and differentiable and 𝜑^′_𝑤(𝑢)𝑣=^𝑇₀𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)𝑣^′(𝑡)+𝑑𝑡_𝑇0𝑔(𝑡)|𝑢(𝑡)|^𝑝−2𝑢(𝑡)𝑣(𝑡)+𝑑𝑡_𝑙𝑗=1𝐼_𝑗𝑢𝑡_𝑗𝑣𝑡_𝑗−^𝑇₀𝑓𝑡,𝑢(𝑡),𝑤^′(𝑡)𝑣(𝑡)𝑑𝑡(26) for any 𝑣∈𝑊₀^1,𝑝(0,𝑇). Moreover, the critical point of 𝜑_𝑤 is a classical solutions of problem (2).

Proof. Using the assumptions (𝑓₁), we can obtain the continuity and differentiability of 𝜑_𝑤 and that 𝜑^′_𝑤∶𝑊₀^1,𝑝(0,𝑇)→(𝑊₀^1,𝑝(0,𝑇))^∗ is defined by 𝜑^′_𝑤(𝑢)𝑣=^𝑇₀𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)𝑣^′(𝑡)+𝑑𝑡_𝑇0𝑔(𝑡)|𝑢(𝑡)|^𝑝−2𝑢(𝑡)𝑣(𝑡)+𝑑𝑡_𝑙𝑗=1𝐼_𝑗𝑢𝑡_𝑗𝑣𝑡_𝑗−^𝑇₀𝑓𝑡,𝑢(𝑡),𝑤^′(𝑡)𝑣(𝑡)𝑑𝑡(27) for any 𝑣∈𝑊₀^1,𝑝(0,𝑇). It follows that the critical point of 𝜑_𝑤 is the weak solution of (2). Moreover, it is a classical solution of problem (2).
Evidently, 𝑢(0)=𝑢(𝑇)=0 since 𝑢∈𝑊₀^1,𝑝(0,𝑇). By the definition of weak solution, we have ^𝑇₀𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)𝑣^′(𝑡)𝑑𝑡+^𝑇₀𝑔(𝑡)|𝑢(𝑡)|^𝑝−2𝑢(𝑡)𝑣(𝑡)+𝑑𝑡_𝑙𝑗=1𝐼_𝑗𝑢𝑡_𝑗𝑣𝑡_𝑗−^𝑇₀𝑓𝑡,𝑢(𝑡),𝑤^′(𝑡)𝑣(𝑡)𝑑𝑡=0.(28) Choose 𝑣∈𝑊₀^1,𝑝(0,𝑇) with 𝑣(𝑡)=0 for every 𝑡∈[0,𝑡_𝑗]⋃[𝑡_𝑗+1,𝑇]; then ^𝑡𝑗+1_𝑡𝑗𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)𝑣^′(𝑡)+𝑑𝑡_{𝑡𝑗+1𝑡𝑗}𝑔(𝑡)|𝑢(𝑡)|^𝑝−2𝑢(𝑡)𝑣(𝑡)=𝑑𝑡_{𝑡𝑗+1𝑡𝑗}𝑓𝑡,𝑢(𝑡),𝑤^′(𝑡)𝑣(𝑡)𝑑𝑡.(29) This implies that −𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′+𝑔(𝑡)|𝑢(𝑡)|^𝑝−2𝑢(𝑡)=𝑓𝑡,𝑢(𝑡),𝑤^′(𝑡),a.e.𝑡∈𝑡_𝑗,𝑡_𝑗+1.(30) Hence, |𝑢|^𝑝−2𝑢^′∈𝑊^1,∞(𝑡_𝑗,𝑡_𝑗+1) for every 𝑗=1,2,…,𝑙. The impulsive condition in (2) is satisfied. This completes the proof.

We will obtain the critical points of 𝜑_𝑤 by using the Mountain Pass Theorem. Therefore, we state this theorem precisely.

Lemma 7 (see [25]). Let 𝑋 be a real Banach space and 𝐼∈𝐶^′(𝑋,𝑅) satisfy (PS)-condition. Suppose that 𝐼 satisfies the following conditions:
(i)   𝐼(0)=0;
(ii) there exists constants 𝜌,𝛼>0 such that 𝐼|_{𝜕𝐵𝜌(0)}≥𝛼;
(iii) there exists 𝑒∈𝑋⧵𝐵_𝜌(0) such that 𝐼(𝑒)≤0.
Then 𝐼 possesses a critical value 𝑐≥𝛼 given by 𝑐=inf_𝑔∈Γmax_[]𝑠∈0,1𝐼𝑔(𝑠),(31) where 𝐵_𝜌(0) is an open ball in 𝑋 of radius 𝜌 centered at 0 and Γ=𝑔∈𝐶([0,1],𝑋)∶𝑔(0)=0,𝑔(1)=𝑒.(32)

3. The Solvability of (2)

Theorem 8. Suppose that (𝑓₁)--(𝑓₅) and (𝐼₁) hold; then there exist positive constants 𝑑₁ and 𝑑₂ such that, for each 𝑤∈𝑊₀^1,𝑝(0,𝑇), problem (2) has one solution 𝑢_𝑤 such that 𝑑₁≤‖𝑢_𝑤‖≤𝑑₂.

Proof. (I) We show that 𝜑_𝑤 satisfies the (PS)-condition.
Assume that {𝑢_𝑛}_𝑛∈𝐍⊂𝑊₀^1,𝑝(0,𝑇) is a sequence such that {𝜑_𝑤(𝑢_𝑛)}_𝑛∈𝐍 is bounded and 𝜑^′_𝑤(𝑢_𝑛)→0 as 𝑛→+∞. We will prove that the sequence {𝑢_𝑛}_𝑛∈𝐍 is bounded. Obviously, there exists a constant 𝐶₃>0 such that 𝜑_𝑤𝑢_𝑛≤𝐶₃,𝜑^′_𝑤𝑢_𝑛≤𝐶₃for𝑛∈𝐍.(33) We set 𝑚₀=max𝑎₁,𝑎₂,…,𝑎_𝑙,𝑀₀=max𝑏₁,𝑏₂,…,𝑏_𝑙.(34) From (17), (25), (26), and (33), (𝑓₄), and (𝐼₁), we have 𝜇𝜑_𝑤𝑢_𝑛−𝜑^′_𝑤𝑢_𝑛𝑢_𝑛=𝜇𝑝−1𝑢_𝑛^𝑝−^𝑇₀𝜇𝐹𝑡,𝑢_𝑛,𝑤^′−𝑓𝑡,𝑢_𝑛,𝑤^′𝑢_𝑛+𝑑𝑡_𝑙𝑗=1𝜇^{𝑢𝑛(𝑡𝑗)}₀𝐼_𝑗(𝑠)𝑑𝑠−𝐼_𝑗𝑢_𝑛𝑡_𝑗𝑢_𝑛𝑡_𝑗≥𝜇𝑝−1𝑢_𝑛^𝑝−^𝑙_𝑗=1𝑎_𝑗+𝑏_𝑗𝑢_𝑛𝑡_𝑗^𝑟𝑗𝑢_𝑛𝑡_𝑗−^𝑙_𝑗=1𝑎_𝑗+𝑏_𝑗|𝑠|^𝑟𝑗𝑑𝑠−𝐶₄≥𝜇𝑝−1𝑢_𝑛^𝑝−𝜇+1𝐶₂𝑙𝑚₀𝑢_𝑛+𝑀₀^𝑙_𝑗=1𝐶^𝑟𝑗₂⁺¹𝑢_𝑛^𝑟𝑗+1−𝐶₄.(35) By 𝑝>𝑟_𝑗+1, we obtain that {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 can assume that 𝑢_𝑛⇀𝑢in𝑊₀^1,𝑝(),𝑢0,𝑇_𝑛⟶𝑢in𝐿^𝑝(),𝑢0,𝑇_𝑛⟶𝑢uniformlyin𝐶[0,𝑇].(36) Hence ^𝑇₀𝑓𝑡,𝑢_𝑛,𝑤^′−𝑓𝑡,𝑢,𝑤^′𝑢_𝑛𝑢_𝑛(𝑡)−𝑢(𝑡)𝑑𝑡⟶0,(37)_𝑙𝑗=1𝐼_𝑗𝑢_𝑛𝑡_𝑗−𝐼_𝑗𝑢𝑡_𝑗𝑢_𝑛𝑡_𝑗−𝑢𝑡_𝑗⟶0.(38) Notice that 𝜑^′_𝑤𝑢_𝑛−𝜑^′_𝑤(𝑢),𝑢_𝑛=−𝑢_𝑇0𝑢^′_𝑛(𝑡)^𝑝−2𝑢^′_𝑛(𝑡)−𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)×𝑢^′_𝑛(𝑡)−𝑢^′(𝑡+)𝑑𝑡_𝑇0𝑔(𝑡)𝑢_𝑛(𝑡)^𝑝−2𝑢_𝑛(𝑡)−|𝑢(𝑡)|^𝑝−2𝑢(𝑡)×𝑢_𝑛(𝑡)(𝑡+−𝑢)𝑑𝑡_𝑙𝑗=1𝐼_𝑗𝑢_𝑛𝑡_𝑗−𝐼_𝑗𝑢𝑡_𝑗𝑢_𝑛𝑡_𝑗−𝑢𝑡_𝑗+^𝑇₀𝑓𝑡,𝑢_𝑛,𝑤^′−𝑓𝑡,𝑢,𝑤^′𝑢_𝑛(𝑡)−𝑢(𝑡)𝑑𝑡.(39) From [26, Lemma 4.2], we see that there exists 𝐶_𝑝>0, for any 𝑥,𝑦∈𝑅: |𝑥|^𝑝−2𝑥−𝑦^𝑝−2𝑦𝑥−𝑦≥𝐶_𝑝𝑥−𝑦^𝑝,𝑝≥2.(40) Combining this inequality with (39), we have 𝐶_𝑝^𝑇₀𝑢^′_𝑛(𝑡)−𝑢^′(𝑡)^𝑝+𝑔(𝑡)𝑢_𝑛(𝑡)−𝑢(𝑡)^𝑝≤𝜑𝑑𝑡_′𝑤𝑢_𝑛−𝜑^′_𝑤(𝑢)𝑢_𝑛+−𝑢_𝑙𝑗=1𝐼_𝑗𝑢_𝑛𝑡_𝑗−𝐼_𝑗𝑢𝑡_𝑗𝑢_𝑛𝑡_𝑗−𝑢𝑡_𝑗+^𝑇₀𝑓𝑡,𝑢_𝑛,𝑤^′−𝑓𝑡,𝑢,𝑤^′𝑢_𝑛(𝑡)−𝑢(𝑡)𝑑𝑡.(41) It follows from (37)–(39) that 𝑢_𝑛→𝑢 in 𝑊₀^1,𝑝(0,𝑇). Hence, 𝜑_𝑤 satisfies (PS)-condition.
(II) We verify assumption (ii) of Lemma 7.
By (𝑓₂), there exists 𝛿∈(0,1) such that 𝑓(≤𝜆𝑡,𝑥,𝜁)¹𝐶₁2|𝑥|^𝑝−1for𝑡∈[0,𝑇],|𝑥|≤𝛿,𝜁∈𝑅.(42) Since 𝐹(𝑡,0,𝜁)=0, it follows that 𝐹(≤𝜆𝑡,𝑥,𝜁)¹𝐶₁|𝑥|2𝑝^𝑝for𝑡∈[0,𝑇],|𝑥|≤𝛿,𝜁∈𝑅.(43) Using (𝐼₁), we have ^𝑙_𝑗=1^{𝑢(𝑡𝑗)}₀𝐼_𝑗(𝑡)𝑑𝑡≥0.(44) Hence, from (18), (25), (35), (𝑓₂), and (𝐼₁), for ‖𝑢‖≤𝛿/𝐶₂, we have 𝜑_𝑤(𝑢)=1𝑝‖𝑢‖^𝑝−^𝑇₀𝐹𝑡,𝑢(𝑡),𝑤^′(𝑡)+𝑑𝑡_{𝑙𝑗=1𝑢(𝑡𝑗)0}𝐼_𝑗(𝑡)≥1𝑑𝑡𝑝‖𝑢‖^𝑝−𝜆₁𝐶₁2𝑝^𝑇₀|𝑢(𝑡)|^𝑝≥1𝑑𝑡𝑝‖𝑢‖^𝑝−1‖𝑢‖2𝑝^𝑝=1‖𝑢‖2𝑝^𝑝.(45) Set 𝛼=(1/2𝑝)(𝛿/𝐶₂)^𝑝, 𝜌=𝛿/𝐶₂. Equation (45) shows that ‖𝑢‖=𝜌 implies that 𝜑_𝑤(𝑢)≥𝛼; that is, 𝜑_𝑤 satisfies assumption (ii) of Lemma 7.
(III) We verify assumption (iii) of Lemma 7.
By (𝐼₁) and (𝑓₅), we know that for 𝑠>1𝜑_𝑤()=1𝑠𝑢𝑝‖𝑠𝑢‖^𝑝−^𝑇₀𝐹𝑡,𝑠𝑢(𝑡),𝑤^′(𝑡)+𝑑𝑡_{𝑙𝑗=1𝑠𝑢(𝑡𝑗)0}𝐼_𝑗(𝑡)≤1𝑑𝑡𝑝‖𝑢‖^𝑝𝑠^𝑝+𝑙𝐶₂‖𝑢‖^𝑙_𝑗=1𝑎_𝑗+𝑏_𝑗|𝑠|^𝑟𝑗𝐶^𝑟𝑗₂‖𝑢‖^𝑟𝑗−𝑎|𝑠|^𝜇𝑇₀|𝑢(𝑡)|^𝜇𝑑𝑡+𝑏𝑇.(46) Take 𝑣₀∈𝑊₀^1,𝑝(0,𝑇) such that ‖𝑣₀‖=1. Since 𝜇>𝑝, 𝑎>0, and 0≤𝑟_𝑗<𝑝−1, (46) implies that there exists 𝜉₁>1 such that ‖𝑒‖>𝜌, and 𝜑_𝑤(𝑒)<0 if we set 𝑒=𝜉₁𝑣₀. By Lemma 7, 𝜑_𝑤 possesses a critical value 𝑐^′≥𝛼>0 given by 𝑐^′=inf_𝑔∈Γmax_[]𝑠∈0,1𝜑_𝑤𝑔(𝑠),(47) where Γ={𝑔∈𝐶([0,1],𝑊₀^1,𝑝(0,𝑇))∶𝑔(0)=0,𝑔(1)=𝑒}. Obviously, 𝜑_𝑤(0)=0, so according to Lemma 7, there exists 𝑢_𝑤≠0 and 𝑢_𝑤∈𝑋, such that 𝜑_𝑤𝑢_𝑤=𝑐^′,𝜑_𝑤𝑢_𝑤=0.(48)
(IV) We prove that 𝑑₁≤‖𝑢_𝑤‖≤𝑑₂.
Since 𝑢_𝑤 is the solution of problem (2), we have ^𝑇₀𝑢^′_𝑤^𝑝𝑑𝑡+^𝑇₀𝑔(𝑡)𝑢_𝑤^𝑝𝑑𝑡+^𝑙_𝑗=1𝐼_𝑗𝑢_𝑤𝑡_𝑗𝑢_𝑤𝑡_𝑗=^𝑇₀𝑓𝑡,𝑢_𝑤,𝑤^′𝑢_𝑤𝑑𝑡.(49) So 𝑢_𝑤^𝑝+^𝑙_𝑗=1𝐼_𝑗𝑢_𝑤𝑡_𝑗𝑢_𝑤𝑡_𝑗=^𝑇₀𝑓𝑡,𝑢_𝑤,𝑤^′𝑢_𝑤𝑑𝑡.(50) It follows from (𝑓₂) and (𝑓₃) that, given 𝜖>0, there exists a positive constant 𝐶_𝜖, independent of 𝑤, such that 𝑓(𝑡,𝑥,𝜁)≤𝜖|𝑥|^𝑝−1+𝐶_𝜖|𝑥|^𝑟−1.(51) Hence, using (𝐼₁), we have 𝑢_𝑤^𝑝≤𝑢_𝑤^𝑝+^𝑙_𝑗=1𝐼_𝑗𝑢_𝑤𝑡_𝑗𝑢_𝑤𝑡_𝑗=^𝑇₀𝑓𝑡,𝑢_𝑤,𝑤^′𝑢_𝑤𝑑𝑡≤𝜖^𝑇₀𝑢_𝑤^𝑝𝑑𝑡+𝐶_𝜖^𝑇₀𝑢_𝑤^𝑟𝑑𝑡.(52) By Sobolev embedding theorem, we obtain 𝑢_𝑤^𝑝1≤𝜖⋅𝜆₁𝐶₁𝑢_𝑤^𝑝+𝐶_𝜖⋅1𝜆₁𝐶₁^𝑟/𝑝𝑢_𝑤^𝑟.(53) Set 𝜖/𝜆₁𝐶₁=𝜖^′ and 𝐶_𝜖/(𝜆₁𝐶₁)^𝑟/𝑝=𝐶^′_𝜖; then 1−𝜖^′𝑢_𝑤^𝑝≤𝐶^′_𝜖𝑢_𝑤^𝑟,(54) which implies 𝑢_𝑤≥𝑑₁.(55) From the inf max characterization of 𝑢_𝑤 in (III), we obtain 𝜑_𝑤𝑢_𝑤≤max𝜑_𝑤𝑠𝑣₀(56) with 𝑣₀ chosen in (III). We estimate 𝜑_𝑤(𝑠𝑣₀) using (𝑓₅) and (𝐼₁): 𝜑_𝑤𝑠𝑣₀≤1𝑝𝑠^𝑝−𝑎|𝑠|^𝜇𝑇₀𝑣₀^𝜇𝑑𝑡+𝑏𝑇−𝑎_𝑗𝑇−𝑏_𝑗|𝑠|^𝑟𝑗𝑇₀𝑣₀^𝑟𝑗𝑑𝑡=∶ℎ(𝑠),(57) whose maximum is achieved at some 𝑠₀>0 and the value ℎ(𝑠₀) can be taken as 𝑑₂. Clearly it is independent of 𝑤. Which implies 𝑢_𝑤≤𝑑₂.(58) This completes the proof.

Lemma 9. Suppose that (𝑓₁)--(𝑓₃) hold and (𝑓₄), (𝑓₅), and (𝐼₁) hold only for positive 𝑥; then there exist positive constants 𝑑₃, and 𝑑₄ such that, for each 𝑤∈𝑊₀^1,𝑝(0,𝑇), problem (2) has a positive solution 𝑢_𝑤 such that 𝑑₃≤‖𝑢_𝑤‖≤𝑑₄.

Proof. Set 𝑓()=𝑓()𝑡,𝑢,𝜁𝑡,𝑢,𝜁,if𝑢≥0,0,if𝑢<0,𝐼_𝑗(𝑢)=𝐼_𝑗(𝑢),if𝑢≥0,0,if𝑢<0.(59) Consider the function −𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′+𝑔(𝑡)|𝑢(𝑡)|^𝑝−2𝑢(𝑡)=𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡),𝑡≠𝑡_𝑗,a.e.𝑡∈[0,𝑇],Δ𝑢^′𝑡_𝑗=𝐼_𝑗𝑢𝑡_𝑗𝑢(0)(𝑇),𝑗=1,2,…,𝑙,=𝑢=0.(60) Obviously, 𝑓 satisfies (𝑓₁)--(𝑓₅), and 𝐼_𝑗 satisfies (𝐼₁), so using Theorem 8, we obtained a solution 𝑢_𝑤 of (60). Multiplying the equation by 𝑢⁻_𝑤 and integrating by parks, we conclude that 𝑢⁻_𝑤≡0. So 𝑢_𝑤 is positive.

Theorem 10. Suppose that (𝑓₁)--(𝑓₅) and (𝐼₂) hold; then there exist positive constants 𝑑₅ and 𝑑₆ such that, for each 𝑤∈𝑊₀^1,𝑝(0,𝑇), problem (2) has one solution 𝑢_𝑤 such that 𝑑₅≤‖𝑢_𝑤‖≤𝑑₆.

Proof. (I) We show that 𝜑_𝑤 satisfies the (PS)-condition.
Assume that {𝑢_𝑛}_𝑛∈𝐍⊂𝑊₀^1,𝑝(0,𝑇) is a sequence such that {𝜑_𝑤(𝑢_𝑛)}_𝑛∈𝐍 is bounded and 𝜑^′_𝑤(𝑢_𝑛)→0 as 𝑛→+∞. We will prove that the sequence {𝑢_𝑛}_𝑛∈𝐍 is bounded.
It follows from (𝑓₃), (𝑓₄), and (𝐼₂) that 𝜇𝜑_𝑤𝑢_𝑛−𝜑^′_𝑤𝑢_𝑛𝑢_𝑛=𝜇𝑝−1𝑢_𝑛^𝑝−^𝑇₀𝜇𝐹𝑡,𝑢_𝑛,𝑤^′−𝑓𝑡,𝑢_𝑛,𝑤^′𝑢_𝑛+𝑑𝑡_𝑙𝑗=1𝜇^{𝑢𝑛(𝑡𝑗)}₀𝐼_𝑗(𝑠)𝑑𝑠−𝐼_𝑗𝑢_𝑛𝑡_𝑗𝑢_𝑛𝑡_𝑗≥𝜇𝑝−1𝑢_𝑛^𝑝−_{|𝑢𝑛|≥𝑥0}𝜇𝐹𝑡,𝑢_𝑛,𝑤^′−𝑓𝑡,𝑢_𝑛,𝑤^′𝑢_𝑛−𝑑𝑡_{|𝑢𝑛|≤𝑥0}𝜇𝐹𝑡,𝑢_𝑛,𝑤^′−𝑓𝑡,𝑢_𝑛,𝑤^′𝑢_𝑛≥𝜇𝑑𝑡𝑝−1𝑢_𝑛^𝑝−𝐶₆.(61) Hence, {𝑢_𝑛} is bounded in 𝑊₀^1,𝑝(0,𝑇). The following proof of (PS)-condition is similar to that in (I) of Theorem 8.
(II) We verify assumption (ii) of Lemma 7.
Using (𝐼₂), we have ^𝑙_𝑗=1^{𝑢(𝑡𝑗)}₀𝐼_𝑗(𝑡)𝑑𝑡≥0.(62) The proof is similar to that in (II) of Theorem 8.
(III) We verify assumption (iii) of Lemma 7.
Take 𝑣₀∈𝑊₀^1,𝑝(0,𝑇) such that ‖𝑣₀‖=1. By (𝐼₂) and (𝑓₅), using Sobolev embedding theorem, we have 𝜑_𝑤𝑠𝑣₀=1𝑝𝑠^𝑝−^𝑇₀𝐹𝑡,𝑠𝑣₀(𝑡),𝑤^′(𝑡)+^𝑙_𝑗=1^{𝑠𝑣0(𝑡𝑗)}₀𝐼_𝑗(𝑡)≤1𝑑𝑡𝑝𝑠^𝑝+𝑙𝑠‖𝑢‖^𝜃_∞−𝐶₇|𝑠|^𝜇𝑇₀𝑣₀(𝑡)^𝜇𝑑𝑡−𝐶₈𝑇≤1𝑝𝑠^𝑝+𝐶₉𝑠^𝜃−𝐶₁₀|𝑠|^𝜇−𝐶₈𝑇.(63) Since 𝜇>𝑝≥𝜃, (63) implies that there exists 𝜉₂∈𝑅⧵{0} such that ‖𝑒‖>𝜌 and 𝜑_𝑤(𝑒)<0 if we set 𝑒=𝜉₂𝑣₀. By Lemma 7, 𝜑_𝑤 possess a critical value 𝑐^′′≥𝛼>0, given by 𝑐^′′=inf_𝑔∈Γmax_[]𝑠∈0,1𝜑_𝑤𝑔(𝑠),(64) where Γ={𝑔∈𝐶([0,1],𝑊₀^1,𝑝(0,𝑇))∶𝑔(0)=0,𝑔(1)=𝑒}. Obviously, 𝜑_𝑤(0)=0, so according to Lemma 7, there exists 𝑢_𝑤≠0 and 𝑢_𝑤∈𝑋, such that 𝜑_𝑤𝑢_𝑤=𝑐^′′,𝜑_𝑤𝑢_𝑤=0.(65)
(IV) We prove that 𝑑₅≤‖𝑢_𝑤‖≤𝑑₆.
Like (IV) of Theorem 8, we can obtain that there exist 𝑑₅>0 such that 𝑢_𝑤≥𝑑₅.(66) It follows from 𝑢_𝑤 satisfying (65) that 𝜇𝑝−1𝑢_𝑤^𝑝≤𝜇𝜑_𝑤𝑢_𝑤−𝜑^′_𝑤𝑢_𝑤𝑢_𝑤+𝐶₁₁=𝜇𝜑_𝑤𝑢_𝑤+𝐶₁₁=𝜇inf_𝑔∈Γmax_[]𝑠∈0,1𝜑_𝑤𝑔(𝑠)+𝐶₁₁≤𝜇max_𝑠≥0𝜑_𝑤𝑠𝑣₀+𝐶₁₁1≤𝜇𝑝|𝑠|^𝑝+^𝑙_𝑗=1^{𝑠𝑣0(𝑡𝑗)}₀𝐼_𝑗(𝑡)|𝑠|𝑑𝑡−𝑎^𝜇𝑇₀𝑣₀^𝜇𝑑𝑡−𝑏𝑇+𝐶₁₁1≤𝜇𝑝|𝑠|^𝑝+|𝑠|^𝜃𝑙𝑣₀^𝜃_∞−𝑎|𝑠|^𝜇𝑇₀𝑣₀^𝜇𝑑𝑡+𝑏𝑇+𝐶₁₁.(67) Let 𝑟(𝑡)=1𝑝|𝑡|^𝑝+𝑡^𝜃𝑙𝑣₀^𝜃_∞−𝑎𝑡^𝜇𝑇₀𝑣₀^𝜇𝑑𝑡,𝑡≥0.(68) Since 𝜇>𝑝≥𝜃, then 𝑟(𝑡) can achieve its maximum at some 𝑡, which implies there exist 𝑑₆>0 such that 𝑢_𝑤≤𝑑₆.(69) This completes the proof.

Lemma 11. Suppose that (𝑓₁)--(𝑓₃) hold and (𝑓₄), (𝑓₅), and (𝐼₂) hold only for positive 𝑥, then there exist positive constants 𝑑₇, and 𝑑₈ such that, for each 𝑤∈𝑊₀^1,𝑝(0,𝑇), problem (2) has a positive solution 𝑢_𝑤 such that 𝑑₇≤‖𝑢_𝑤‖≤𝑑₈.

Proof. The proof is similar to that of Lemma 9, so we omit it.

Remark 12. By the same method, we can discuss the negative solution of (2).

4. The Solvability of (1)

Theorem 13. Assume that (𝑓₁)--(𝑓₆) and (𝐼₁) and (𝐼₃) hold; then problem (1) has one nontrivial solution provided 𝐿𝑘=²𝜆₁^−1/𝑝𝐶₁^{−1−(1/𝑝)}𝐶_𝑝−𝐿₁𝜆₁⁻¹𝐶₁⁻¹−𝐶^𝑝₂∑^𝑙_𝑗=1𝛼_𝑗(70) satisfying 0<𝑘<1.

Proof. We construct a sequence {𝑢_𝑛}∈𝑊₀^1,𝑝(0,𝑇) as solutions of the following problem: −𝑢^′_𝑛(𝑡)^𝑝−2𝑢^′_𝑛(𝑡)^′+𝑔(𝑡)𝑢_𝑛(𝑡)^𝑝−2𝑢_𝑛(𝑡)=𝑓𝑡,𝑢_𝑛(𝑡),𝑢^′_𝑛−1(𝑡),𝑡≠𝑡_𝑗,a.e.𝑡∈[0,𝑇],Δ𝑢^′_𝑛𝑡_𝑗=𝐼_𝑗𝑢_𝑛𝑡_𝑗𝑢,𝑗=1,2,…,𝑙,_𝑛(0)=𝑢_𝑛(𝑇)=01.1_𝑛 obtained in Theorem 8, starting with an arbitrary 𝑢₀∈𝑊₀^1,𝑝(0,𝑇). It follows from the Sobolev embedding theorem that ‖𝑢_𝑛‖_∞≤𝜌₁. Using 1.1_𝑛 and (1.1_𝑛+1), we obtain ^𝑇₀𝑢^′_𝑛(𝑡)^𝑝−2𝑢^′_𝑛(𝑡)𝑢^′_𝑛+1−𝑢^′_𝑛+𝑑𝑡_𝑇0𝑔(𝑡)𝑢_𝑛(𝑡)^𝑝−2𝑢_𝑛(𝑡)𝑢_𝑛+1−𝑢_𝑛+𝑑𝑡_𝑙𝑗=1𝐼_𝑗𝑢_𝑛𝑡_𝑗𝑢_𝑛+1𝑡_𝑗−𝑢_𝑛𝑡_𝑗=^𝑇₀𝑓𝑡,𝑢_𝑛,𝑢^′_𝑛−1𝑢_𝑛+1−𝑢_𝑛𝑑𝑡,_𝑇0𝑢^′_𝑛+1(𝑡)^𝑝−2𝑢^′_𝑛+1(𝑡)𝑢^′_𝑛+1−𝑢^′_𝑛+𝑑𝑡_𝑇0𝑔(𝑡)𝑢_𝑛+1(𝑡)^𝑝−2𝑢_𝑛+1(𝑡)𝑢_𝑛+1−𝑢_𝑛+𝑑𝑡_𝑙𝑗=1𝐼_𝑗𝑢_𝑛+1𝑡_𝑗𝑢_𝑛+1𝑡_𝑗−𝑢_𝑛𝑡_𝑗=^𝑇₀𝑓𝑡,𝑢_𝑛+1,𝑢^′_𝑛𝑢_𝑛+1−𝑢_𝑛𝑑𝑡.(71) Hence, 𝐶_𝑝^𝑇₀𝑢^′_𝑛+1−𝑢^′_𝑛^𝑝+𝑔(𝑡)𝑢_𝑛+1−𝑢_𝑛^𝑝≤𝑑𝑡_𝑇0𝑓𝑡,𝑢_𝑛+1,𝑢^′_𝑛−𝑓𝑡,𝑢_𝑛,𝑢^′_𝑛−1𝑢_𝑛+1−𝑢_𝑛+𝑑𝑡_𝑙𝑗=1𝐼_𝑗𝑢_𝑛𝑡_𝑗−𝐼_𝑗𝑢_𝑛+1𝑡_𝑗×𝑢_𝑛+1𝑡_𝑗−𝑢_𝑛𝑡_𝑗=^𝑇₀𝑓𝑡,𝑢_𝑛+1,𝑢^′_𝑛−𝑓𝑡,𝑢_𝑛,𝑢^′_𝑛×𝑢_𝑛+1−𝑢_𝑛+𝑑𝑡_𝑇0𝑓𝑡,𝑢_𝑛,𝑢^′_𝑛−𝑓𝑡,𝑢_𝑛,𝑢^′_𝑛−1𝑢_𝑛+1−𝑢_𝑛+𝑑𝑡_𝑙𝑗=1𝐼_𝑗𝑢_𝑛𝑡_𝑗−𝐼_𝑗𝑢_𝑛+1𝑡_𝑗×𝑢_𝑛+1𝑡_𝑗−𝑢_𝑛𝑡_𝑗≤𝐿₁^𝑇₀𝑢_𝑛+1−𝑢_𝑛^𝑝𝑑𝑡+𝐿₂^𝑇₀𝑢^′_𝑛−𝑢^′_𝑛−1^𝑝−1×𝑢_𝑛+1−𝑢_𝑛+𝑑𝑡_𝑙𝑗=1𝛼_𝑗𝑢_𝑛+1𝑡_𝑗−𝑢_𝑛𝑡_𝑗^𝑝.(72) Using (17) and (18) and by Hölder inequality, we have 𝑢_𝑛+1−𝑢_𝑛^𝑝−1≤𝐿₂𝜆₁^−1/𝑝𝐶₁^−1/𝑝𝐶_𝑝−𝐿₁𝜆₁⁻¹𝐶₁⁻¹−𝐶^𝑝₂∑^𝑙_𝑗=1𝛼_𝑗^𝑇0𝑢^′_𝑛−𝑢^′_𝑛−1^𝑝𝑑𝑡^{(𝑝−1)/𝑝}=𝐿₂𝜆₁^−1/𝑝𝐶₁^−1/𝑝𝐶_𝑝−𝐿₁𝜆₁⁻¹𝐶₁⁻¹−𝐶^𝑝₂∑^𝑙_𝑗=1𝛼_𝑗𝑢_𝑛−𝑢_𝑛−1𝑊^𝑝−1_01,𝑝≤𝐿₂𝜆₁^−1/𝑝𝐶₁^{−1−(1/𝑝)}𝐶_𝑝−𝐿₁𝜆₁⁻¹𝐶₁⁻¹−𝐶^𝑝₂∑^𝑙_𝑗=1𝛼_𝑗𝑢_𝑛−𝑢_𝑛−1^𝑝−1;(73) that is, 𝑢_𝑛+1−𝑢_𝑛≤𝑘^1/(𝑝−1)𝑢_𝑛−𝑢_𝑛−1.(74) Since 𝑘 is less than 1, then it follows that {𝑢_𝑛} strongly converges in 𝑊₀^1,𝑝(0,𝑇), as it easily follows proving that {𝑢_𝑛} is a Cauchy sequence in 𝑊₀^1,𝑝(0,𝑇). By Theorem 8, we know that 𝑢≠0. In this way we obtain a nontrivial solution of (1).

Example 14. Let 𝑡₁∈(0,𝑇) and 𝑝=2; consider the following nonlinear Dirichlet impulsive problem: −𝑢^′′(𝑡)+(−1−𝑡)𝑢(𝑡)=𝑢³(𝑡)1+sin𝑢^′(𝑡),a.e.𝑡∈[0,𝑇],𝑡≠𝑡₁,Δ𝑢^′𝑡₁=³𝑢𝑡₁,𝑢(0)=𝑢(𝑇)=0.(75)
Compared with (1), 𝑓(𝑡,𝑢,𝑢^′)=𝑢³(1+sin𝑢^′) and 𝐼₁(𝑢)=𝑢^1/3. We can take 𝜇=3 and 𝜃=5/3. Then by simple computation, it is easy to verify that all conditions of Theorem 13 are satisfied. Hence, by Theorem 13, problem (75) has one nontrivial solution.

Theorem 15. Assume that (𝑓₁)--(𝑓₆) and (𝐼₂) and (𝐼₃) hold; then problem (1) has one nontrivial solution provided 𝐿𝑘=²𝜆₁^−1/𝑝𝐶₁^{−1−(1/𝑝)}𝐶_𝑝−𝐿₁𝜆₁⁻¹𝐶₁⁻¹−𝐶^𝑝₂∑^𝑙_𝑗=1𝛼_𝑗(76) satisfying 0<𝑘<1.

Proof. We construct a sequence {𝑢_𝑛}∈𝑊₀^1,𝑝(0,𝑇) as solutions of the problem 1.1_𝑛 obtained in Theorem 10, starting with an arbitrary 𝑢₀∈𝑊₀^1,𝑝(0,𝑇). The rest of proof is similar to that of Theorem 13, so we omit it.

Example 16. Let 𝑡₁∈(0,𝑇) and 𝑝=4; consider the following nonlinear Dirichlet impulsive problem: −𝑢^′(𝑡)²𝑢^′(𝑡)^′−𝑡²|𝑢(𝑡)|²𝑢(𝑡)=𝑢⁷(𝑡)2+cos𝑢^′(𝑡),a.e.𝑡∈[0,𝑇],𝑡≠𝑡₁,Δ𝑢^′𝑡₁=𝑢²𝑡₁,𝑢(0)=𝑢(𝑇)=0.(77)
Compared with (1), 𝑓(𝑡,𝑢,𝑢^′)=𝑢⁷(2+cos𝑢^′) and 𝐼₁(𝑢)=𝑢². We can take 𝜇=6 and 𝑟₁=2. Then by simple computation, it is easy to verify that all conditions of Theorem 15 are satisfied. Hence, by Theorem 15, problem (77) has one nontrivial solution.

Remark 17. In order to discuss the positive solutions of (1), we can argue as in Theorem 8, by replacing 𝑓 by 𝑓. By the same method, we can discuss the negative solutions of (1).

Acknowledgments

The authors are grateful to the referees for their useful suggestions. This work is partially supported by the National Natural Science Foundation of China (no. 71201013), the Humanities and Social Sciences Project of the Ministry of Education of China (no. 12YJC630118), the Innovation Platform Open Funds for Universities in Hunan Province (no. 13K059), and the Provincial Natural Science Foundation of Hunan (no. 11JJ3012).