Variational Approach to Impulsive Differential Equations with Singular Nonlinearities

Abstract

We discuss the existence of periodic solutions for nonautonomous second order differential equations with singular nonlinearities. Simple sufficient conditions that enable us to obtain many distinct periodic solutions are provided. Our approach is based on a variational method.

1. Introduction

Differential equations with impulsive effects appear naturally in the description of many evolution processes whose states experience sudden changes at certain times, called impulse moments. There is an extensive bibliography about the subject. For recent references, see [1].

Variational methods have been successfully employed to investigate regular second order differential equations with impulsive effects; See, for instance, [2–8]. In particular, the paper [8] considers the existence of 𝑛 distinct pairs of nontrivial solutions. However, very few papers have used variational methods to investigate the case of impulsive second order boundary value problems with singular nonlinearities. In fact, it seems that the work [9] is the first paper along this line. Singular boundary value problems without impulses have attracted the attention of many researchers; see [10] for details and references. This paper is devoted to the study of the existence and multiplicity of periodic solutions for impulsive second order differential equations with singular nonlinearities. More specifically, we consider the following impulsive problem: 𝑢^′′(𝑡)+𝜆𝑓(𝑡,𝑢(𝑡))=0,𝑡≠𝑡_𝑗,𝑡∈𝐼,−Δ𝑢^′(𝑡)=𝐼_𝑗𝑢𝑡_𝑗𝑢(0)(𝑇),𝑗=1,2,…,𝑝,=𝑢,𝑢^′(0)=𝑢^′(𝑇), (1) where 𝐼 denotes the real interval [0,𝑇], with 𝑇>0, 0=𝑡₀<𝑡₁<⋯<𝑡_𝑝<𝑡_𝑝+1=𝑇,𝜆 is a positive parameter, Δ𝑢^′(𝑡)=𝑢^′𝑡⁺_𝑗−𝑢^′𝑡⁻_𝑗, (2) and 𝑓∶𝐼×ℝ→ℝ presents a singularity with respect to its second argument at 𝑢=0.

Throughout this paper we will use the following notations. 𝐿^𝑝(𝐼) is the classical Lebesgue space of measurable functions 𝑢∶𝐼→ℝ such that |𝑢(⋅)|^𝑝 is integrable, and for 𝑢∈𝐿^𝑝(𝐼) we define its norm by ‖𝑢‖_𝐿^𝑝=^𝑇₀|𝑢(𝑡)|^𝑝𝑑𝑡^1/𝑝. (3) Let ‖𝑢‖_∞=sup{|𝑢(𝑡)|;𝑡∈𝐼} denote the norm of 𝑢∈𝐶(𝐼), the space of real-valued continuous functions. 𝑊^1;2(𝐼) is the classical Sobolev space of functions 𝑢∈𝐿²(𝐼) with their distributional derivatives 𝑢^′∈𝐿²(𝐼). We set 𝐻¹_𝑇={𝑢∈𝑊^1,2(𝐼);𝑢(0)=𝑢(𝑇)} and for 𝑢∈𝐻¹_𝑇 we define its norm by ‖𝑢‖_𝐻¹_𝑇=𝑢^′_𝐿2+‖𝑢‖_𝐿^21/2. (4) Wirtinger inequality ‖𝑢‖_𝐿²≤𝑇𝑢2𝜋^′_𝐿² (5) implies that we can consider on 𝐻¹_𝑇 the norm ‖𝑢‖=𝑢^′_𝐿2. (6)𝐻¹_𝑇 endowed with the norm ‖⋅‖ is a reflexive Banach space.

We introduce the following assumptions on the nonlinearity. (H1)(i)𝑓∶𝐼×(0;+∞)→ℝ is an 𝐿¹-Carathéodory function;(ii)lim_𝑢→0⁺𝑓(𝑡,𝑢)=−∞ for almost every 𝑡∈𝐼;(iii)there exist 𝑢₀≥1,𝑎,𝑏>0, and 𝛾∈(0,1) such that 𝑓(𝑡,𝑢₀)=0, and for almost all 𝑡∈𝐼 and all 𝑢∈(𝑢₀,+∞)𝑓(𝑡,𝑢)≤𝑎+𝑏𝑢^𝛾; (7)(iv)inf_𝑢>𝑢0+1∫^𝑇₀𝐹(𝑡,𝑢)𝑑𝑡>0, with 𝐹(𝑡;𝑢)=∫^𝑢_𝑢0𝑓(𝑡,𝑠)𝑑𝑠;(v)(𝜕𝐹/𝜕𝑡)(𝑡,𝑢) exists and is nonpositive for almost all 𝑡∈𝐼 and all 𝑢∈ℝ;(vi)lim_𝑢→+∞∫^𝑇₀𝐹(𝑡,𝑢)𝑑𝑡=+∞.

The jump functions 𝐼_𝑗, 𝑗=1,2,…,𝑝, satisfy the following: (H2)(i)𝐼_𝑗∶ℝ→ℝ is odd, continuous, and bounded;(ii)∫^𝑢₀𝐼_𝑗(𝑠)𝑑𝑠≤0.

Definition 1. A solution of (1) is a function 𝑢∈𝐶(𝐼) such that for every 𝑗=1,…,𝑝, 𝑢_𝑗=𝑢_{[𝑡𝑗,𝑡𝑗+1]} (8) is absolutely continuous with its derivatives 𝑢^′_𝑗 and 𝑢^′′_𝑗∈𝐿²(𝑡_𝑗,𝑡_𝑗+1) and satisfies the differential equation in (1) for 𝑡∈𝐼⧵{𝑡₁,𝑡₂,…,𝑡_𝑝}; the limits 𝑢^′(𝑡⁻_𝑗) and 𝑢^′(𝑡⁺_𝑗), 𝑗=1,2,…,𝑝, exist; the impulsive conditions and the boundary conditions in (1) hold.

2. The Main Result

In this section we state and prove our main result.

Theorem 2. Suppose that conditions (H1) and (H2) hold. Then for any 𝑛∈ℕ, there exists 𝜆_𝑛 such that for 𝜆>𝜆_𝑛 problem (1) has infinitely many distinct nontrivial solutions.

Proof. To give the proof of the main result, we first modify problem (1) to another one which is not singular.
For 𝜉∈(0,min(1,𝑢₀/4)), define a function 𝑓_𝜉 on 𝐼×[0,+∞) by 𝑓_𝜉()=𝑓()𝑡,𝑢𝑡,𝜉sin2𝜋𝜉/𝑢₀𝑢sin2𝜋𝑢₀,0≤𝑢≤𝑢₀,𝑓(𝑡,𝑢),𝑢>𝑢₀. (9)
Finally, we require 𝑓_𝜉(𝑡,−𝑢)=−𝑓_𝜉(𝑡,𝑢) for all 𝑢∈ℝ and almost 𝑡∈𝐼, to make 𝑓_𝜉 odd. Then 𝑓_𝜉 satisfies the following:(j)𝑓_𝜉 is an 𝐿¹-Carathéodory function and is odd in 𝑢;(jj)there exist 𝑎>0, 𝑏>0, 𝑏^′>0, and 𝛾∈(0,1) such that |𝑓_𝜉(𝑡,𝑢)|≤𝑎+𝑏𝑢^𝛾 for all 𝑢∈(−∞,−𝑢₀)∪(𝑢₀,+∞) and almost 𝑡∈𝐼 and |𝑓_𝜉(𝑡,𝑢)|≤𝑏^′𝑢 for all 𝑢∈(−𝑢₀,𝑢₀) and almost all 𝑡∈𝐼;(jjj)for almost all 𝑡∈𝐼 and all 𝑢∈ℝ, 𝐹_𝜉(𝑡,𝑢)∶=∫^𝑢_𝑢0𝑓_𝜉(𝑡,𝑠)𝑑𝑠 is such that 𝐹_𝜉(𝑡,0)=−^𝑢0₀𝑓_𝜉()𝑓()𝑡,𝑠𝑑𝑠=−𝑡,𝜉sin2𝜋𝜉/𝑢₀^𝑢0₀sin2𝜋𝑢₀=𝑓()𝑠𝑑𝑠𝑡,𝜉sin2𝜋𝜉/𝑢₀𝑢₀2𝜋₀^2𝜋𝐹−sin𝜁𝑑𝜁=0,foralmostall𝑡∈𝐼,_𝜉(𝑡,−𝑢)=_𝑢^−𝑢₀𝑓_𝜉(𝑡,𝑠)𝑑𝑠=−^𝑢0_−𝑢𝑓_𝜉(𝑡,𝑠)𝑑𝑠=−⁰_−𝑢𝑓_𝜉(𝑡,𝑠)𝑑𝑠+^𝑢0₀𝑓_𝜉(𝑡,𝑠)𝑑𝑠=−⁰_−𝑢𝑓_𝜉(𝑡,𝑠)𝑑𝑠=₀^−𝑢𝑓_𝜉(𝑡,𝑠)𝑑𝑠=−^𝑢₀𝑓_𝜉(𝑡,−𝜁)𝑑𝜁=^𝑢₀𝑓_𝜉()=𝑡,𝜁𝑑𝜁_𝑢00𝑓_𝜉(𝑡,𝑠)𝑑𝑠+^𝑢_𝑢0𝑓_𝜉(𝑡,𝑠)𝑑𝑠=𝐹_𝜉(𝑡,𝑢),foralmostall𝑡∈𝐼,all𝑢∈ℝ,inf_𝑢>𝑢0+1^𝑇₀𝐹_𝜉(𝑡,𝑢)𝑑𝑡=inf_𝑢>𝑢0+1^𝑇₀𝐹()𝐹𝑡,𝑢𝑑𝑡>0,_𝜉(𝑡,𝑢)<0,for0<𝑢<𝑢₀,foralmostall𝑡∈𝐼. (10)
We study the modified problem 𝑢^′′(𝑡)+𝜆𝑓_𝜉(𝑡,𝑢(𝑡))=0,𝑡≠𝑡_𝑗,𝑡∈𝐼,−Δ𝑢^′(𝑡)=𝐼_𝑗𝑢𝑡_𝑗𝑢(0)(𝑇),𝑗=1,2,…,𝑝,=𝑢,𝑢^′(0)=𝑢^′(𝑇). (11)
Consider the functional 𝜑_𝜉∶𝐻¹_𝑇→ℝ, defined by 𝜑_𝜉(𝑢)=12‖𝑢‖²−𝜆^𝑇₀𝐹_𝜉((𝑡−𝑡,𝑢))𝑑𝑡_{𝑝𝑗=1𝑢(𝑡𝑗)0}𝐼_𝑗(𝑠)𝑑𝑠. (12)
Clearly 𝜑_𝜉 is an even functional, 𝜑_𝜉(0)=0, and is Frechet differentiable, whose Frechet derivative at the point 𝑢∈𝐻¹_𝑇 is the functional 𝜑^′_𝜉(𝑢)∈(𝐻¹_𝑇)^∗ given by 𝜑^′_𝜉(𝑢)⋅𝑣=^𝑇₀𝑢^′(𝑡)⋅𝑣^′(𝑡)𝑑𝑡−𝜆^𝑇₀𝑓_𝜉((𝑡(𝑡)−𝑡,𝑢))⋅𝑣𝑑𝑡_𝑝𝑗=1𝐼_𝑗𝑢𝑡_𝑗𝑣𝑡_𝑗. (13)
Obviously, 𝜑^′_𝜉 is continuous. If 𝑢∈𝐻¹_𝑇 is a critical point of the functional 𝜑_𝜉, then 𝑢 is a solution of problem (11).
First, we show that 𝜑_𝜉 is bounded from below. Define a subset Ω of 𝐻¹_𝑇 as follows: Ω=𝑢∈𝐻¹_𝑇,min(𝑢)>1+𝑢₀∪𝑢∈𝐻¹_𝑇,max(𝑢)<−1+𝑢₀, (14) noticing that 𝜕Ω=𝑢∈𝐻¹_𝑇,𝑢(𝑡)≥1+𝑢₀∀𝑡∈𝐼,∃𝑡_𝑢∈𝐼with𝑢𝑡_𝑢=1+𝑢₀∪𝑢∈𝐻¹_𝑇,𝑢(𝑡)≤−1+𝑢₀∀𝑡∈𝐼,∃𝑡_𝑢∈𝐼with𝑢𝑡_𝑢=−1+𝑢₀. (15)
For 𝑢∈Ω, we have 𝜑_𝜉(𝑢)=12‖𝑢‖²−𝜆^𝑇₀𝐹_𝜉(𝑡,𝑢(𝑡))𝑑𝑡−^𝑝_𝑗=1^{𝑢(𝑡𝑗)}₀𝐼_𝑗(𝑠)≥1𝑑𝑠2‖𝑢‖²−𝜆^𝑇₀𝑎|𝑢(𝑡)|+𝑏|𝑢(𝑡)|^𝛾+1≥1𝑑𝑡2‖𝑢‖²−𝜆𝑎𝑇^3/2‖𝑢‖−𝑏𝜆𝑇^(3+𝛾)/2‖𝑢‖^𝛾+1>−∞. (16) If 𝑢∉Ω, we use the partition of the interval 𝐼=𝐼₁∪𝐼₂∪𝐼₃, where 𝐼₁=𝑡∈𝐼;𝑢(𝑡)∈−𝑢₀,𝑢₀,𝐼₂=𝑡∈𝐼;𝑢(𝑡)∈𝑢₀,1+𝑢₀∪−𝑢₀+1,−𝑢₀,𝐼₃=𝑡∈𝐼;𝑢(𝑡)∈(−∞,−1+𝑢₀∪1+𝑢₀,−∞. (17)
By (H2) and (jjj), we have 𝜑_𝜉(𝑢)=12‖𝑢‖²−𝜆^𝑇₀𝐹_𝜉((𝑡−𝑡,𝑢))𝑑𝑡_{𝑝𝑗=1𝑢(𝑡𝑗)0}𝐼_𝑗(𝑠)1𝑑𝑠≥2‖𝑢‖²−𝜆_𝐼3𝑎|𝑢(𝑡)|+𝑏|𝑢(𝑡)|^𝛾+1𝑑𝑡−𝜆_𝐼2𝐹_𝜉(𝑡,𝑢(𝑡))𝑑𝑡−𝜆_𝐼1𝐹_𝜉(𝑡,𝑢(𝑡))𝑑𝑡. (18) From (jjj), 𝐹_𝜉(𝑡,𝑢(𝑡))<0 on 𝐼₁, which implies 𝜑_𝜉(𝑢)≥12‖𝑢‖²−𝜆_𝐼3𝑎|𝑢(𝑡|𝑢(𝑡)|+𝑏)|^𝛾+1𝑑𝑡−𝜆_𝐼2𝐹_𝜉(𝑡,𝑢(𝑡))𝑑𝑡. (19) If 𝑡∈𝐼₂, then 𝑢(𝑡)∈[𝑢₀,𝑢₀+1]∪[−(𝑢₀+1),−𝑢₀]. This means that 𝑢(𝑡) is bounded uniformly in 𝑡∈𝐼₂. Since 𝑓_𝜉 is a Caratheodory function, it follows that 𝐹_𝜉 is bounded by some positive constant 𝐶. Hence, 𝜑_𝜉(𝑢)≥12‖𝑢‖²−𝜆𝑎𝑇^3/2‖𝑢‖−𝑏𝜆⋅𝑇^(3+𝛾)/2‖𝑢‖^𝛾+1−𝜆𝐶>−∞. (20) This shows that 𝜑_𝜉 is bounded from below.
Remark  3. There exists 𝑚>0 such that inf_{𝑢∈𝜕Ω}𝜑_𝜉(𝑢)≥−𝑚, whenever 𝜉∈(0,min(1,𝑢₀/4)).
Our next step is to show that 𝜑_𝜉 satisfies the Palais-Smale condition. For this purpose, let (𝑢_𝑘)_𝑘 be a sequence in 𝐻¹_𝑇 such that (𝜑(𝑢_𝑘))_𝑘 is bounded and lim_𝑘→+∞𝜑^′(𝑢_𝑘)=0. Then, there exists 𝑀>0 such that |𝜑_𝜉(𝑢_𝑘)|≤𝑀. In view of (16), if 𝑢_𝑘∈Ω, we have 1𝑀≥2𝑢_𝑘²−𝜆𝑎^′𝑇^3/2𝑢_𝑘−𝑏𝜆⋅𝑇^(3+𝛾)/2𝑢_𝑘^𝛾+1, (21) and if 𝑢_𝑘∉Ω, we can proceed as above to show that 12𝑢_𝑘²−𝜆𝑎𝑇^3/2𝑢_𝑘−𝑏𝜆⋅𝑇^(3+𝛾)/2𝑢_𝑘^𝛾+1−𝜆𝐶≤𝜑_𝜉𝑢_𝑘≤𝜑_𝜉𝑢_𝑘≤𝑀. (22) This shows that ‖𝑢_𝑘‖ is bounded. So that, (𝑢_𝑘)_𝑘∈ℕ is bounded in 𝐻¹_𝑇. From the reflexivity of 𝐻¹_𝑇, we may extract from (𝑢_𝑘)_𝑘∈ℕ a weakly convergent subsequence, which we label the same; that is, 𝑢_𝑘⇀𝑢 in 𝐻¹_𝑇. Since the injection of 𝐻¹_𝑇 into 𝐶(𝐼), with its natural norm, is continuous, it follows that 𝑢_𝑘⇀𝑢 in 𝐶(𝐼) and by Banach-Steinhaus theorem, the subsequence (𝑢_𝑘)_𝑘∈ℕ is bounded in 𝐻¹_𝑇 and hence, in 𝐶(𝐼). Moreover, the subsequence (𝑢_𝑘)_𝑘 is uniformly equicontinuous since, for 0≤𝑠≤𝑡≤𝑇, we have 𝑢_𝑘(𝑡)−𝑢_𝑘(𝑠)≤^𝑡_𝑠𝑢^′_𝑘(𝜏≤())𝑑𝜏𝑡−𝑠^1/2𝑡_𝑠𝑢^′_𝑘(𝜏)²𝑑𝜏^1/2≤(𝑡−𝑠)^1/2𝑢_𝑘≤𝐶(𝑡−𝑠)^1/2. (23) By Ascoli-Arzela theorem the subsequence (𝑢_𝑘)_𝑘∈ℕ is relatively compact in 𝐶(𝐼). By the uniqueness of the weak limit in 𝐶(𝐼), every uniformly convergent subsequence of (𝑢_𝑘)_𝑘∈ℕ converges to 𝑢. Thus, (𝑢_𝑘)_𝑘∈ℕ converges uniformly on 𝐼 to 𝑢.
Next, we will verify that (𝑢_𝑘)_𝑘∈ℕ strongly converges to 𝑢 in 𝐻¹_𝑇. By (13), we have 𝜑^′_𝜉𝑢_𝑘−𝜑^′_𝜉(𝑢)𝑢_𝑘=𝑢−𝑢_𝑘−𝑢²−𝜆^𝑇₀𝑓_𝜉𝑡,𝑢_𝑘(𝑡)−𝑓_𝜉(𝑡,𝑢(𝑡))𝑢_𝑘(𝑡)(𝑡)−−𝑢𝑑𝑡_𝑝𝑗=1𝐼_𝑗𝑢_𝑘𝑡_𝑗−𝐼_𝑗𝑢𝑡_𝑗𝑢_𝑘𝑡_𝑗−𝑢𝑡_𝑗. (24) The uniform convergence of (𝑢_𝑘)_𝑘∈ℕ to 𝑢 in 𝐶(𝐼) implies 𝜆^𝑇₀𝑓_𝜉𝑡,𝑢_𝑘(𝑡)−𝑓_𝜉(𝑡,𝑢(𝑡))𝑢_𝑘(𝑡)−𝑢(𝑡)𝑑𝑡⟶0,as𝑘⟶∞,_𝑝𝑗=1𝐼_𝑗𝑢_𝑘𝑡_𝑗−𝐼_𝑗𝑢𝑡_𝑗𝑢_𝑘𝑡_𝑗−𝑢𝑡_𝑗⟶0,as𝑘⟶∞. (25) Since 𝜑^′(𝑢_𝑘)→0 and 𝑢_𝑘⇀𝑢, as 𝑘→∞, lim_𝑘→∞𝜑^′_𝜉𝑢_𝑘−𝜑^′_𝜉(𝑢)𝑢_𝑘−𝑢=0. (26) In view of (24), (25), and (26), we obtain lim_𝑘→∞𝑢_𝑘−𝑢=0. (27) Thus, 𝜑 satisfies the Palais-Smale condition.
Let 𝑣₁,𝑣₂,…,𝑣_𝑛 denote the eigenfunctions of −𝑢^′′=𝜆𝑢, 𝑢(0)=𝑢(𝑇), 𝑢^′(0)=𝑢^′(𝑇), corresponding to the eigenvalues 𝜇₁,𝜇₂,…,𝜇_𝑛. We normalize 𝑣₁,𝑣₂,…,𝑣_𝑛, so that 𝑣_𝑘=1=𝜇_𝑘^𝑇₀𝑣_𝑘(𝑡)²𝑑𝑡,1≤𝑘≤𝑛. (28) For 𝑟>𝑢₀+1, we set 𝐾_𝑛(𝑟)=^𝑛_𝑚=1𝑐_𝑚𝑣_𝑚;^𝑛_𝑚=1𝑐²_𝑚=𝑟², (29) and 𝐸=𝐾_𝑛(𝑟)∩Ω. Then, for any 𝑢∈𝐸, we have for almost every 𝑡∈𝐼, 𝑢(𝑡)≥𝑢₀+1, or 𝑢(𝑡)≤−(𝑢₀+1), and 𝐹_𝜉(𝑡,−𝑢(𝑡))=𝐹_𝜉(𝑡,𝑢(𝑡))=_𝑢^𝑢(𝑡)₀𝑓_𝜉()=𝑡,𝑠𝑑𝑠_{𝑢𝑢(𝑡)0}𝑓(𝑡,𝑠)𝑑𝑠=𝐹(𝑡,𝑢(𝑡)). (30) It follows from (H1)(iv) that, for any 𝑢∈𝐸, ^𝑇₀𝐹_𝜉(𝑡,𝑢(𝑡))𝑑𝑡>0. (31) Let 𝛼_𝑛=inf_𝑢∈𝐸^𝑇₀𝐹_𝜉(𝑡,𝑢(𝑡))𝑑𝑡,𝛽_𝑛=inf^𝑝_{𝑢∈𝐸𝑗=1}^{𝑢(𝑡𝑗)}₀𝐼_𝑗(𝑠)𝑑𝑠. (32) It is clear that 𝛼_𝑛>0,𝛽_𝑛≤0. (33) Letting 𝜆_𝑛=12𝑟²−𝛽_𝑛⋅𝛼_𝑛⁻¹>0, (34) we see that when 𝜆>𝜆_𝑛 we have, for any 𝑢∈𝐸, 𝜑_𝜉(𝑢)≤12𝑟²−𝜆𝛼_𝑛−𝛽_𝑛<12𝑟²−𝜆_𝑛𝛼_𝑛−𝛽_𝑛=0. (35) On the other hand, there is a constant 𝜌∈(0,𝑢₀) such that 𝐵_𝜌={𝑢∈𝐻¹_𝑇;‖𝑢‖<𝜌} is not contained in Ω. So 𝐹_𝜉(𝑡,𝑢)<0 from (jjj). Hence, using (H2)(ii) we obtain, for 𝑢∈𝜕𝐵_𝜌∩𝑋, where 𝑋=[span{𝑣₁,𝑣₂,…,𝑣_𝑛}]^⟂,𝜑_𝜉(𝑢)=12‖𝑢‖²−𝜆^𝑇₀𝐹_𝜉((𝑡−𝑡,𝑢))𝑑𝑡_{𝑝𝑗=1𝑢𝑘(𝑡𝑗)0}𝐼_𝑗(𝑠)1𝑑𝑠>2𝜌²>0. (36)Remark  4. Sobolev inequality implies that 𝐸⊂𝐾_𝑛(𝑟)⧵𝐵_𝑢0.
Now, we apply Theorem 9.12 in [11] to conclude that 𝜑_𝜉 possesses infinitely many distinct pairs of nontrivial critical points. That is, problem (11) has infinitely many distinct pairs of distinct nontrivial solutions.
Finally, we must prove that there exists 𝜉₀∈(0,min(1,𝑢₀/4)) with the property that, for every 𝜉∈(𝜉₀,min(1,𝑢₀/4)), any positive solution 𝑢 of (11) satisfying 𝜑_𝜉(𝑢)≥−𝑚 is such that min𝑢≥𝑢₀ and hence 𝑢 is a solution of (1). We proceed by contradiction. Assume, on the contrary, that there are sequences (𝜉_𝑛)_𝑛∈ℕ and (𝑢_𝑛)_𝑛∈ℕ such that 𝜉_𝑛≤1𝑛;𝑢_𝑛isapositivesolutionof(11)with𝜉=𝜉_𝑛,𝜑_𝜉𝑛𝑢_𝑛≥−𝑚,min𝑢_𝑛<𝑢₀. (37)
First, we have 𝜆^𝑇₀𝑓_𝜉𝑛𝑡,𝑢_𝑛(𝑡)𝑑𝑡=−^𝑝_𝑗=0^{𝑡−𝑗+1}_𝑡+𝑗𝑢^′′_𝑛(𝑠)𝑑𝑠=−^𝑝_𝑗=0𝑢^′_𝑛𝑡⁻_𝑗+1−𝑢^′_𝑛𝑡⁺_𝑗=𝑢^′_𝑛(0)−𝑢^′_𝑛𝑡⁻₁+⋯+𝑢^′_𝑛𝑡_𝑝−𝑢^′_𝑛(𝑇)=^𝑝_𝑗=1𝐼_𝑗𝑢_𝑛𝑡_𝑗. (38)
Then, by (H1)(i), there exists 𝑐₁>0 such that 𝑓_𝜉𝑛⋅,𝑢_𝑛(⋅)_𝐿1≤𝑐₁, (39) which implies that 𝑢^′_𝑛𝐿∞≤𝑐₂, (40) for some constant 𝑐₂>0. Since 𝜑_𝜉𝑛(𝑢_𝑛)≥−𝑚, it follows that there must exist two constants 𝑅₁ and 𝑅₂, with 0<𝑅₁<𝑅₂, such that max_𝑡∈𝐼𝑢_𝑛(𝑡)∈[𝑅₁,𝑅₂]; otherwise, 𝑢_𝑛 would tend uniformly to +∞ and, in this case, 𝜑_𝜉𝑛(𝑢_𝑛) would go to −∞ (because of (H1)(vi) and (40)) and this contradicts 𝜑_𝜉𝑛(𝑢_𝑛)≥−𝑚. Also, min𝑢_𝑛<𝑢₀ implies that there exists an integer 𝑘_𝑛>1 such that min𝑢_𝑛≤𝑢₀−(𝑢₀/𝑘_𝑛). Since 𝑢_𝑛 is continuous, there exists 𝜏¹_𝑛 such that 𝑢_𝑛(𝜏¹_𝑛)=𝑢₀−(𝑢₀/𝑘_𝑛). Let 𝜏²_𝑛∈𝐼 be such that, for 𝑛 large enough, 𝑢_𝑛𝜏¹_𝑛=𝑢₀−𝑢₀𝑘_𝑛<𝑅₁=𝑢_𝑛𝜏²_𝑛. (41) Multiplying the differential equation in (11) by 𝑢^′_𝑛 and integrating the resulting equation on [𝜏¹_𝑛,𝜏²_𝑛], or on [𝜏²_𝑛,𝜏¹_𝑛], we get 𝐽∶=^𝜏2𝑛_𝜏1𝑛𝑢^′′_𝑛(𝑡)𝑢^′_𝑛(𝑡)𝑑𝑡+𝜆^𝜏2𝑛_𝜏1𝑛𝑓_𝜉𝑛𝑡,𝑢_𝑛(𝑡)𝑢^′_𝑛(𝑡)𝑑𝑡=0=_{𝑡𝑗∈[𝜏1𝑛,𝜏2𝑛]}^{𝑡−𝑗+1}_𝑡+𝑗𝑢^′′_𝑛(𝑡)⋅𝑢^′_𝑛(𝑡)𝑑𝑡+𝜆^{𝑡−𝑗+1}_𝑡+𝑗𝑓_𝜉𝑛𝑡,𝑢_𝑛(𝑡)𝑢^′_𝑛(𝑡)𝑑𝑡. (42) It is clear that 𝐽=𝐽₁+_{𝑡𝑗∈[𝜏1𝑛,𝜏2𝑛]}12𝑢^′_𝑛𝑡⁻_𝑗+1²−𝑢^′_𝑛𝑡⁺_𝑗², (43) where 𝐽₁=𝜆_{𝑡𝑗∈[𝜏1𝑛,𝜏2𝑛]}^{𝑡−𝑗+1}_𝑡+𝑗𝑓_𝜉𝑛𝑡,𝑢_𝑛(𝑡)𝑢^′_𝑛(𝑡)𝑑𝑡=𝜆_{𝑡𝑗∈[𝜏1𝑛,𝜏2𝑛]}^{𝑢𝑛(𝑡𝑘+1)}_{𝑢𝑛𝑡𝑘}𝑓_𝜉𝑛(𝑡,𝑠)𝑑𝑠. (44) Now, 𝐽=0 and ‖𝑢^′_𝑛‖_𝐿∞≤𝑐₂ imply that 𝐽₁ is bounded. Since 𝑓_𝜉𝑛𝑡,𝑢_𝑛(𝑡)𝑢^′_𝑛(𝑡)=𝑑𝐹𝑑𝑡^𝜉𝑛𝑡,𝑢_𝑛(𝑡)−𝐷₁𝐹_𝜉𝑛𝑡,𝑢_𝑛(𝑡), (45) it follows that 𝐽₁=𝜆_{𝑡𝑘∈(𝜏1𝑛,𝜏2𝑛)}^{𝑡−𝑘+1}_𝑡+𝑘𝑓_𝜉𝑛𝑡,𝑢_𝑛(𝑡)𝑢^′_𝑛(𝑡)𝑑𝑡=𝜆_{𝑡𝑘∈(𝜏1𝑛,𝜏2𝑛)}^{𝑢𝑛(𝑡𝑘+1)}_{𝑢𝑛𝑡𝑘}𝑓_𝜉𝑛(𝑡,𝑠)𝑑𝑠=𝜆𝐹_𝜉𝑛𝜏²_𝑛,𝑅₁−𝜆𝐹_𝜉𝑛𝜏¹_𝑛,𝑢₀−𝑢₀𝑘_𝑛−𝜆^𝜏2𝑛_𝜏1𝑛𝐷₁𝐹_𝜉𝑛𝑡,𝑢_𝑛(𝑡)𝑑𝑡≥𝜆𝐹_𝜉𝑛𝜏²_𝑛,𝑅₁−𝜆𝐹_𝜉𝑛𝜏¹_𝑛,𝑢₀−𝑢₀𝑘_𝑛=𝜆𝐹_𝜉𝑛𝜏²_𝑛,𝑅₁−^{𝑢0−(𝑢0/𝑘𝑛)}_𝑢0𝑓_𝜉𝑛𝜏¹_𝑛,𝑠𝑑𝑠. (46)
Now, we have that ^{𝑢0−(𝑢0/𝑘𝑛)}_𝑢0𝑓_𝜉𝑛𝜏¹_𝑛=𝑓𝜏,𝑠𝑑𝑠_1𝑛,𝜉_𝑛sin2𝜋𝜉_𝑛/𝑢₀^{𝑢0−(𝑢0/𝑘𝑛)}_𝑢0sin2𝜋𝑢₀=𝑓𝜏𝑠𝑑𝑠_1𝑛,𝜉_𝑛sin2𝜋𝜉_𝑛/𝑢₀𝑢₀12𝜋1−cos1−𝑘_𝑛. (47)
It follows from (H1)(ii) that 𝐹_𝜉𝑛𝜏¹_𝑛,𝑢₀−𝑢₀𝑘_𝑛=^{𝑢0−(𝑢0/𝑘𝑛)}_𝑢0𝑓_𝜉𝑛𝜏¹_𝑛,𝑠𝑑𝑠⟶−∞as𝑛⟶+∞. (48) This implies that 𝐽₁ is not bounded. We arrive at a contradiction. This completes the proof of our main result.

3. Example

Consider the boundary value problem 𝑢^′′(𝑡)+𝜆(𝑡+1+sin𝑡)³𝑢(𝑡)+3⋅ln³𝑢(𝑡)=0,𝑡≠𝑡_𝑗,𝑡∈𝐼,−Δ𝑢^′(𝑡)=−sin𝑢𝑡_𝑗𝑢(0)(𝑇),𝑗=1,2,3,=𝑢,𝑢^′(0)=𝑢^′(𝑇). (49)(i)𝑓∶[0,2𝜋]×(0;+∞)→ℝ, given by 𝑓(𝑡,𝑢)=(𝑡+1+sin𝑡)(√³𝑢+3)⋅ln√³𝑢, satisfies (H1) with 𝑇=2𝜋, 𝑢₀=1;𝑎=0;𝑏=8(𝜋+1);𝛾=2/3.(ii)For 𝑢>1, ln√³𝑢≤√³𝑢⇒(𝑡+1+sin𝑡)(√³𝑢+3)⋅ln√³𝑢≤8(𝜋+1)𝑢^2/3.(iii)𝐼_𝑗(𝑠)=−sin(𝑠) satisfies (H2).

We consider 𝑟≥𝑒 in the definition of 𝐾_𝑛(𝑟). We have ^𝑢₁3+√³𝑠⋅ln√³𝑠𝑑𝑠=^𝑢₁ln𝑠𝑑𝑠+3^𝑢1/3₁𝑥³⋅ln𝑥𝑑𝑥=𝑢ln𝑢−𝑢+3³√𝑢⁴164ln√³+𝑢−119.16 (50)

Then inf_𝑢>20^2𝜋𝐹(𝑡,𝑢)⋅𝑑𝑡>0,lim_𝑢→+∞0^2𝜋𝐹()𝛼𝑡,𝑢𝑑𝑡=+∞,_𝑛=inf_𝑢>20^2𝜋𝐹_𝜉(𝑡,𝑢(𝑡))𝑑𝑡=inf_𝑢>20^2𝜋(𝑡+1+sin𝑡)⋅₁^𝑢(𝑡)3+√³𝑠⋅ln√³𝑠𝑑𝑠𝑑𝑡≥inf_𝑢>20^2𝜋𝑢ln𝑢−𝑢+3³√𝑢⁴164ln√³𝑢−1+19×(𝑡)16𝑑𝑡≥2𝜋inf_𝑢>2𝑢ln𝑢−𝑢+1916+3³√𝑢⁴164ln√³𝑢−1≥𝜋. (51)

Also, 𝛼_𝑛=inf_𝑢>20^2𝜋𝐹_𝜉(𝑡,𝑢(𝑡))⋅𝑑𝑡=inf_𝑢>20^2𝜋(𝑡+1+sin𝑡)⋅^𝑢₁3+√³𝑠⋅ln√³≤()𝑠𝑑𝑠𝑑𝑡2𝜋+1inf_𝑢>2^𝑇₀𝑢ln𝑢−𝑢+1916+3³√𝑢⁴164ln√³≤()⋅𝑢−1𝑑𝑡2𝜋+1^𝑇₀34𝑢²(𝑡)≤()3𝑑𝑡2𝜋+14𝑟^2𝑛_𝑘=11𝜇_𝑘≤32𝑟^2𝑛_𝑘=11𝜇_𝑘, (52) where ∑^𝑛_𝑘=1(1/𝜇_𝑘)=∑^𝑛_𝑘=1(2𝜋/2𝜋𝑘)²<∑^𝑛_𝑘=1(1/𝑘²)<𝑛.

It is clear that 𝛽_𝑛=inf_𝑢>2−³_𝑗=1^{𝑢(𝑡𝑗)}₀sin𝑠𝑑𝑠=inf³_𝑢>2𝑗=1cos𝑢𝑡_𝑗𝛽−1≤0,_𝑛=inf_𝑢>2−³_𝑗=1^{𝑢(𝑡𝑗)}₀sin𝑠𝑑𝑠≥inf_𝑢>2−³_𝑗=1^{𝑢(𝑡𝑗)}₀𝑠𝑑𝑠≥inf_𝑢>2−³_𝑗=1𝑢𝑡_𝑗²23𝑑𝑠>−2𝑟². (53) Then 𝜆_𝑛=12𝑟²−𝛽_𝑛𝛼_𝑛⁻¹<2𝜋𝑟²,𝜆_𝑛=12𝑟²−𝛽_𝑛⋅𝛼_𝑛⁻¹>12𝑟²231𝑟²𝑛=1.3𝑛 (54) Applying our main result, we see that when 𝜆>𝜆_𝑛, for any 𝑛∈ℕ^∗, problem (49) has infinitely many distinct nontrivial solutions.

Acknowledgments

The authors wish to thank an anonymous referee for comments that led to the improvement of the presentation of the paper. Abdelkader Boucherif is grateful to KFUPM for its constant support.