Multiple Positive Solutions for Singular Semipositone Periodic Boundary Value Problems with Derivative Dependence

Abstract

By constructing a special cone in 𝐶¹[0,2𝜋] and the fixed point theorem, this paper investigates second-order singular semipositone periodic boundary value problems with dependence on the first-order derivative and obtains the existence of multiple positive solutions. Further, an example is given to demonstrate the applications of our main results.

1. Introduction

In this paper, we are concerned with the existence of multiple positive solutions for the second-order singular semipositone periodic boundary value problems (PBVP, for short): 𝑢^′′(𝑡)+𝑎(𝑡)𝑢(𝑡)=𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡)(),𝑢(0)(),𝑡∈0,2𝜋=𝑢2𝜋,𝑢^′(0)=𝑢^′(2𝜋), (1.1) where 𝑎∈𝐶[0,2𝜋], the nonlinear term 𝑓(𝑡,𝑢,𝑣) may be singular at 𝑡=0, 𝑡=2𝜋, and 𝑢=0, also may be negative for some value of 𝑡, 𝑢, and 𝑣.

In recent years, second-order singular periodic boundary value problems have been studied extensively because they can be used to model many systems in celestial mechanics such as the N-body problem (see [1–11] and references therein). By applying the Krasnosel'skii's fixed point theorem, Jiang [5] proves the existence of one positive solution for the second-order PBVP 𝑢^′′(𝑡)+𝑚²()[],𝑢(0)()𝑢=𝑓𝑡,𝑢,𝑡∈0,2𝜋=𝑢2𝜋,𝑢^′(0)=𝑢^′(2𝜋), (1.2) where 𝑚∈(0,1/2) is a constant and 𝑓∈𝐶([0,2𝜋]×[0,+∞),[0,+∞)). Zhang and Wang [6] used the same fixed point theorem to prove the existence of multiple positive solutions for PBVP (1.2) when 𝑓(𝑡,𝑢) is nonnegative and singular at 𝑢=0, not singular at 𝑡=0, 𝑡=2𝜋. Lin et al. [7] only obtained the existence of one positive solution to PBVP (1.1) when 𝑓(𝑡,𝑢,𝑣)=𝑓(𝑡,𝑢),𝑓 is semipositone and singular only at 𝑢=0.All the above works were done under the assumption that the first-order derivative 𝑢′ is not involved explicitly in the nonlinear term 𝑓.

Motivated by the works of [5–7], the present paper investigates the existence of multiple positive solutions to PBVP (1.1). PBVP (1.1) has two special features. The first one is that the nonlinearity 𝑓 may depend on the first-order derivative of the unknown function 𝑢, and the second one is that the nonlinearity 𝑓(𝑡,𝑢,𝑣) is semipositone and singular at 𝑡=0, 𝑡=2𝜋, and 𝑢=0. We first construct a special cone different from that in [5–7] and then deduce the existence of multiple positive solutions by employing the fixed point theorem on a cone. Our results improve and generalize some related results obtained in [5–7].

A map 𝑢∈𝐶¹[0,2𝜋]∩𝐶²(0,2𝜋) is said to be a positive solution to PBVP(1.1) if and only if 𝑢 satisfies PBVP (1.1) and 𝑢(𝑡)>0 for 𝑡∈[0,2𝜋].

The contents of this paper are distributed as follows. In Section 2, we introduce some lemmas and construct a special cone, which will be used in Section 3. We state and prove the existence of at least two positive solutions to PBVP (1.1) in Section 3. Finally, an example is worked out to demonstrate our main results.

2. Some Preliminaries and Lemmas

Define the set functions Λ=𝑎∈𝐶[0,2𝜋]∶𝑎≻0,𝑡∈[0,2𝜋],₀^2𝜋𝑎^𝑝𝑑𝑡^1/𝑝≤𝐾2𝑞forsome𝑝≥1, (2.1) where 𝑞 is the conjugate exponent of 𝑝, 𝐾𝑞=1𝑞(2𝜋)^2/𝑞22+𝑞^1−2/𝑞Γ1/𝑞Γ1/2+1/𝑞²2,1≤𝑞<∞,𝜋,𝑞=∞, (2.2) where Γ is the Gamma function.

Given 𝑎∈Λ, let 𝐺(𝑡,𝑠) be the Green function for the equation 𝑢^′′(𝑡)𝑢(𝑡)(),𝑢(0)()+𝑎=0,𝑡∈0,2𝜋=𝑢2𝜋,𝑢^′(0)=𝑢^′(2𝜋). (2.3)

Now, the following Lemma follows immediately from the paper [7].

Lemma 2.1. 𝐺(𝑡,𝑠) has the following properties:(G¹)𝐺(𝑡,𝑠) is continuous in 𝑡 and 𝑠 for all 𝑡,𝑠∈[0,2𝜋];(G²)𝐺(𝑡,𝑠)>0 for all (𝑡,𝑠)∈[0,2𝜋]×[0,2𝜋],𝐺(0,𝑠)=𝐺(2𝜋,𝑠) and 𝜕𝐺/𝜕𝑡|_(0,𝑠)=𝜕𝐺/𝜕𝑡|_(2𝜋,𝑠);(G³) denote 𝑙₁=min_{0≤𝑡,𝑠≤2𝜋}𝐺(𝑡,𝑠) and 𝑙₂=max_{0≤𝑡,𝑠≤2𝜋}𝐺(𝑡,𝑠), then 𝑙₂>𝑙₁>0;(G⁴) there exist functions ℎ,𝐻∈𝐶²[0,2𝜋] such that 𝐺()=()𝐻(𝑡)ℎ(𝑠)+ℎ(𝑡)𝐻(𝑠)(𝑡)𝐻(𝑠)(𝑡)ℎ(𝑠)𝑡,𝑠(𝑡)ℎ(𝑠)(𝑡)𝐻(𝑠)(𝑡)𝐻(𝑠)(𝑡)ℎ(𝑠)𝛼+1𝛽−1+𝑐𝐻+𝑑ℎ,0≤𝑠≤𝑡≤2𝜋,𝛼𝐻+𝛽ℎ+𝑐𝐻+𝑑ℎ,0≤𝑡≤𝑠≤2𝜋, (2.4)where 𝛼,𝛽,𝑐,𝑑 are constants, 𝐻,ℎ are independent solutions of the linear differential equation 𝑢^′′+𝑎(𝑡)𝑢(𝑡)=0, and 𝐻′(𝑡)ℎ(𝑡)−ℎ′(𝑡)𝐻(𝑡)=1;(G⁵)𝐺^′_𝑡(𝑡,𝑠) is bounded on [0,2𝜋]×[0,2𝜋].
Denote 𝑙₃=max_{0≤𝑡,𝑠≤2𝜋}|𝐺^′_𝑡(𝑡,𝑠)|, then 𝑙₃>0.

Remark 2.2. Using paper [5], we can get 𝐺(𝑡,𝑠) when 𝑎(𝑡)≡𝑚² and 𝑚∈(0,1/2), obtaining 𝐺()=()()𝑡,𝑠sin𝑚𝑡−𝑠+sin𝑚2𝜋−𝑡+𝑠()()()2𝑚1−cos2𝑚𝜋,0≤𝑠≤𝑡≤2𝜋,sin𝑚𝑠−𝑡+sin𝑚2𝜋−𝑠+𝑡()𝐺2𝑚1−cos2𝑚𝜋,0≤𝑡≤𝑠≤2𝜋,_′𝑡()=()()𝑡,𝑠cos𝑚𝑡−𝑠−cos𝑚2𝜋−𝑡+𝑠2()()()1−cos2𝑚𝜋,0≤𝑠≤𝑡≤2𝜋,−cos𝑚𝑠−𝑡+cos𝑚2𝜋−𝑠+𝑡2()𝑙1−cos2𝑚𝜋,0≤𝑡<𝑠≤2𝜋,₁=sin2𝑚𝜋2𝑚(1−cos2𝑚𝜋),𝑙₂=sin𝑚𝜋𝑚(1−cos2𝑚𝜋),𝑙₃=12. (2.5)
Let 𝐸={𝑢∈𝐶¹[0,2𝜋]∶𝑢(0)=𝑢(2𝜋),𝑢^′(0)=𝑢^′(2𝜋)} with norm ‖𝑢‖=max{‖𝑢‖₀,‖𝑢^′‖₀}, where ‖𝑢‖₀=max_{𝑡∈[0,2𝜋]}|𝑢(𝑡)|. Then (𝐸,‖⋅‖) is a Banach space. Let 𝜎=∶min{𝑙₁/𝑙₂,𝑙₁/𝑙₃},𝐿=∶𝑙₃/𝑙₁, from Lemma 2.1, we know that 𝜎,𝐿 are both constants and 0<𝜎<1, 𝐿>0.
Define 𝐾=𝑢∈𝐸∶𝑢(𝑡)≥𝜎‖𝑢‖,𝑢^′(𝑡‖𝑢‖[,Ω)≤𝐿,∀𝑡∈0,2𝜋]_𝑟={𝑢∈𝐸∶‖𝑢‖<𝑟},∀𝑟>0. (2.6)
It is easy to conclude that 𝐾 is a cone of 𝐸 and Ω_𝑟 is an open set of 𝐸.

Lemma 2.3 (see [12]). Let 𝐸 be a Banach space and 𝑃 a cone in 𝐸. Suppose Ω₁ and Ω₂ are bounded open sets of 𝐸 such that 𝜃∈Ω₁⊂Ω₁⊂Ω₂ and suppose that 𝐴∶𝑃∩(Ω₂⧵Ω₁)→𝑃 is a completely continuous operator such that(1)inf_{𝑢∈𝑃∩𝜕Ω1}‖𝐴𝑢‖>0 and 𝑢≠𝜆𝐴𝑢 for 𝑢∈𝑃∩𝜕Ω₁,𝜆≥1; 𝑢≠𝜆𝐴𝑢 for 𝑢∈𝑃∩𝜕Ω₂,0<𝜆≤1,or(2)inf_{𝑢∈𝑃∩𝜕Ω2}‖𝐴𝑢‖>0 and 𝑢≠𝜆𝐴𝑢 for 𝑢∈𝑃∩𝜕Ω₂,𝜆≥1; 𝑢≠𝜆𝐴𝑢 for 𝑢∈𝑃∩𝜕Ω₁,0<𝜆≤1. Then 𝐴 has a fixed point in 𝑃∩(Ω₂⧵Ω₁).
For convenience, let us list some conditions for later use.(H₀)𝑎(𝑡)∈Λ,𝑓∶(0,2𝜋)×(0,+∞)×𝑅→𝑅 is continuous and there exists a constant 𝑀>0 such that 0≤𝑓(𝑡,𝑢,𝑣)+𝑀≤𝑔(𝑡)ℎ(𝑢,𝑣),∀(𝑡,𝑢,𝑣)∈(0,2𝜋)×(0,+∞)×𝑅, (2.7)where 𝑔∈𝐶((0,2𝜋),𝑅⁺),ℎ∈𝐶((0,+∞)×𝑅,𝑅⁺), and 0<∫₀^2𝜋𝑔(𝑡)𝑑𝑡<+∞;(H₁) there exist 𝑟₁>𝜎⁻¹2𝜋𝑀𝑙₂ and 𝑎(𝑡)∈𝐿[0,2𝜋] with ∫₀^2𝜋𝑎(𝑡)𝑑𝑡>(≥)𝑟₁𝑙₁⁻¹ such that 𝑀+𝑓(𝑡,𝑢,𝑣)≥(>)𝑎(𝑡),∀𝑡∈(0,2𝜋),𝑢∈0,𝑟₁,𝑣∈−𝐿𝑟₁+2𝜋𝑀𝑙₃,𝐿𝑟₁+2𝜋𝑀𝑙₃; (2.8)(H₂) there exists 𝑅₁>𝑟₁ such that max𝑙₂,𝑙₃₀^2𝜋𝑔(𝑡)𝑑𝑡<𝑅₁𝑀₀⁻¹, (2.9)where 𝑀₀=∶max{ℎ(𝑢,𝑣)∶𝑢∈[𝜎𝑅₁−2𝜋𝑀𝑙₂,𝑅₁],𝑣∈[−(𝐿𝑅₁+2𝜋𝑀𝑙₃),(𝐿𝑅₁+2𝜋𝑀𝑙₃)]};(H₃) there exists [𝛼^∗,𝛽^∗]⊂(0,2𝜋) such that lim_𝑢→+∞𝑓(𝑡,𝑢,𝑣)𝑢=+∞uniformlywithrespectto𝑡∈𝛼^∗,𝛽^∗,𝑣∈𝑅. (2.10)

3. Main Results

Theorem 3.1. Assume that conditions (H₀)–(H₃) are satisfied, then PBVP (1.1) has at least two positive solutions 𝑢₁,𝑢₂∈𝐶¹[0,2𝜋]∩𝐶²(0,2𝜋) such that 𝑟₁<‖𝑢₁+𝑀𝜔‖<𝑅₁<‖𝑢₂+𝑀𝜔‖, where 𝜔(𝑡)=∶∫₀^2𝜋𝐺(𝑡,𝑠)𝑑𝑠.

Proof. We consider the following PBVP: 𝑢^′′(𝑡)+𝑎(𝑡)𝑢(𝑡)=𝑓𝑡,𝑢(𝑡)−𝑀𝜔(𝑡),𝑢^′(𝑡)−𝑀𝜔^′(𝑡)(),𝑢(0)()+𝑀,𝑡∈0,2𝜋=𝑢2𝜋,𝑢^′(0)=𝑢^′(2𝜋). (3.1)
It is easy to see that if 𝑢∈𝐶¹[0,2𝜋]∩𝐶²(0,2𝜋) and 𝑟₁<‖𝑢‖<𝑅₁ is a positive solution of PBVP (3.1) with 𝑢(𝑡)>𝑀𝜔(𝑡) for 𝑡∈[0,2𝜋], then 𝑥(𝑡)=𝑢(𝑡)−𝑀𝜔(𝑡) is a positive solution of PBVP (1.1) and 𝑟₁<‖𝑥+𝑀𝜔‖<𝑅₁.
As a result, we will only concentrate our study on PBVP (3.1).
Define an operator 𝑇∶𝐾⧵{𝜃}→𝐸 by (𝑇𝑢)(𝑡)=∶₀^2𝜋𝐺(𝑡,𝑠)𝑓𝑠,𝑢(𝑠)−𝑀𝜔(𝑠),𝑢^′(𝑠)−𝑀𝜔^′(𝑠)+𝑀𝑑𝑠,∀𝑡∈[0,2𝜋], (3.2) where 𝐺(𝑡,𝑠) is the Green function to problem (2.3). (1) We first show that 𝑇∶𝐾∩(Ω_𝑅⧵Ω_𝑟1)→𝐾 is completely continuous for any 𝑅>𝑟₁.
For any 𝑢∈𝐾∩(Ω_𝑅⧵Ω_𝑟1), from (H₁), we have 𝑢(𝑡)−𝑀𝜔(𝑡)≥𝜎𝑟₁−2𝜋𝑀𝑙₂>0. So, by Lemma 2.1 and (3.2), (𝑇𝑢)(0)=(𝑇𝑢)(2𝜋),(𝑇𝑢)^′(0)=(𝑇𝑢)^′(),(𝑡)=2𝜋𝑇𝑢)(₀^2𝜋𝐺(𝑡,𝑠)𝑓𝑠,𝑢(𝑠)−𝑀𝜔(𝑠),𝑢^′(𝑠)−𝑀𝜔^′(𝑠)≥𝑙+𝑀𝑑𝑠₁𝑙₂𝑙₂₀^2𝜋𝐺(𝑡,𝑠)𝑓𝑠,𝑢(𝑠)−𝑀𝜔(𝑠),𝑢^′(𝑠)−𝑀𝜔^′(𝑠)≥𝑙+𝑀𝑑𝑠₁𝑙₂max_{[]𝜏∈0,2𝜋0}^2𝜋𝐺(𝜏,𝑠)𝑓𝑠,𝑢(𝑠)−𝑀𝜔(𝑠),𝑢^′(𝑠)−𝑀𝜔^′(𝑠)=𝑙+𝑀𝑑𝑠₁𝑙₂‖𝑇𝑢‖₀≥𝜎‖𝑇𝑢‖₀[],),∀𝑡∈0,2𝜋(𝑇𝑢^′(𝑡)=₀^2𝜋𝐺^′_𝑡(𝑡,𝑠)𝑓𝑠,𝑢(𝑠)−𝑀𝜔(𝑠),𝑢^′(𝑠)−𝑀𝜔^′(𝑠)≤+𝑀𝑑𝑠_02𝜋𝐺^′_𝑡(𝑡,𝑠)𝑓𝑠,𝑢(𝑠)−𝑀𝜔(𝑠),𝑢^′(𝑠)−𝑀𝜔^′(𝑠)≤𝑙+𝑀𝑑𝑠₃𝑙₁𝑙₁₀^2𝜋𝑓𝑠,𝑢(𝑠)−𝑀𝜔(𝑠),𝑢^′(𝑠)−𝑀𝜔^′(𝑠)≤𝑙+𝑀𝑑𝑠₃𝑙₁₀^2𝜋𝐺(𝜏,𝑠)𝑓𝑠,𝑢(𝑠)−𝑀𝜔(𝑠),𝑢^′(𝑠)−𝑀𝜔^′(𝑠)=𝑙+𝑀𝑑𝑠₃𝑙₁(𝑇𝑢)(𝜏),∀𝑡,𝜏∈[0,2𝜋]. (3.3)(3.4)(3.5)
From (3.5), we have (𝑇𝑢)(𝑡)≥(𝑙₁/𝑙₃)max_{𝜏∈[0,2𝜋]}|(𝑇𝑢)^′(𝜏)|≥𝜎‖(𝑇𝑢)^′‖₀. Therefore, (𝑇𝑢)(𝑡)≥𝜎‖𝑇𝑢‖, |(𝑇𝑢)^′(𝑡)|≤𝐿‖𝑇𝑢‖,forall𝑡∈[0,2𝜋],thatis,𝑇∶𝐾∩(Ω_𝑅⧵Ω_𝑟1)→𝐾.
Assume that 𝑢_𝑛,𝑢_∗∈𝐾∩(Ω_𝑅⧵Ω_𝑟1) with ‖𝑢_𝑛−𝑢_∗‖→0,𝑛→+∞. Thus, from (H₁), we have lim_𝑛→+∞𝑓𝑡,𝑢_𝑛(𝑡)−𝑀𝜔(𝑡),𝑢^′_𝑛(𝑡)−𝑀𝜔^′(𝑡)=𝑓𝑡,𝑢_∗(𝑡)−𝑀𝜔(𝑡),𝑢^′_∗(𝑡)−𝑀𝜔^′(𝑡)(),𝑓,𝑡∈0,2𝜋𝑡,𝑢_𝑛(𝑡)−𝑀𝜔(𝑡),𝑢^′_𝑛(𝑡)−𝑀𝜔^′(𝑡)≤𝑀+𝑀₁𝑔(𝑡),𝑡∈(0,2𝜋),𝑀+𝑀₁𝑔(𝑡)∈𝐿[0,2𝜋], (3.6) where 𝑀₁=∶max{ℎ(𝑢,𝑣)∶𝑢∈[𝜎𝑟₁−2𝜋𝑀𝑙₂,𝑅],𝑣∈[−(𝐿𝑅+2𝜋𝑀𝑙₃),(𝐿𝑅+2𝜋𝑀𝑙₃)]}.
Lemma 2.1 and Lebesgue-dominated convergence theorem guarantee that 𝑇𝑢_𝑛−𝑇𝑢_∗≤max𝑙₂,𝑙₃₀^2𝜋𝑓𝑡,𝑢_𝑛(𝑡)−𝑀𝜔(𝑡),𝑢^′_𝑛(𝑡)−𝑀𝜔^′(𝑡)−𝑓𝑡,𝑢_∗(𝑡)−𝑀𝜔(𝑡),𝑢^′_∗(𝑡)−𝑀𝜔^′(𝑡)𝑑𝑡⟶0,𝑛⟶+∞. (3.7) So, 𝑇∶𝐾∩(Ω_𝑅⧵Ω_𝑟1)→𝐾 is continuous.
For any bounded 𝐷⊂𝐾∩(Ω_𝑅⧵Ω_𝑟1), From Lemma 2.1 and (H₁), for any 𝑢∈𝐷, we have ‖𝑇𝑢‖≤max𝑙₂,𝑙₃₀^2𝜋𝑓𝑠,𝑢(𝑠)−𝑀𝜔(𝑠),𝑢^′(𝑠)−𝑀𝜔^′(𝑠)𝑙+𝑀𝑑𝑠≤max₂,𝑙₃₀^2𝜋𝑔(𝑠)ℎ𝑢(𝑠)−𝑀𝜔(𝑠),𝑢^′(𝑠)−𝑀𝜔^′(𝑠)𝑙𝑑𝑠≤max₂,𝑙₃𝑀₁₀^2𝜋𝑔(𝑠)𝑑𝑠, (3.8) which means the functions belonging to {(𝑇𝐷)(𝑡)} and the functions belonging to {(𝑇𝐷)^′(𝑡)} are uniformly bounded on [0,2𝜋]. Notice that (𝑇𝑢)′(𝑡)≤𝑙₃𝑀₁₀^2𝜋𝑔(𝑠)𝑑𝑠,𝑡∈[0,2𝜋],𝑢∈𝐷, (3.9) which implies that the functions belonging to {(𝑇𝐷)(𝑡)} are equicontinuous on [0,2𝜋]. From Lemma 2.1, we have 𝐺^′_𝑡()=()𝐻𝑡,𝑠𝛼+1^′(𝑡)ℎ(𝑠)+𝛽−1ℎ^′(𝑡)𝐻(𝑠)+𝑐𝐻^′(𝑡)𝐻(𝑠)+𝑑ℎ^′(𝑡)ℎ(𝑠),0≤𝑠≤𝑡≤2𝜋,𝛼𝐻^′(𝑡)ℎ(𝑠)+𝛽ℎ^′(𝑡)𝐻(𝑠)+𝑐𝐻^′(𝑡)𝐻(𝑠)+𝑑ℎ^′(𝑡)ℎ(𝑠),0≤𝑡<𝑠≤2𝜋, (3.10) where 𝛼,𝛽,𝑐,𝑑 are constants, ℎ,𝐻∈𝐶²[0,2𝜋] are independent solutions of the linear differential equation 𝑢^′′+𝑎(𝑡)𝑢(𝑡)=0, and 𝐻′(𝑡)ℎ(𝑡)−ℎ′(𝑡)𝐻(𝑡)=1.
It is easy to see that 𝐺^′_𝑡(𝑡,𝑠) is continuous in 𝑡 and 𝑠 for 0≤𝑠≤𝑡≤2𝜋 and 0≤𝑡<𝑠≤2𝜋. So, for any 𝑡₁,𝑡₂∈[0,2𝜋],𝑡₁<𝑡₂, we have ^𝑡1₀𝐺^′_𝑡𝑡₁,𝑠−𝐺^′_𝑡𝑡₂,𝑠𝑔(𝑠)𝑑𝑠⟶0,as𝑡₁⟶𝑡⁻₂,or𝑡₂⟶𝑡⁺₁,_𝑡2𝑡1𝐺^′_𝑡𝑡₁,𝑠−𝐺^′_𝑡𝑡₂,𝑠𝑔(𝑠)𝑑𝑠≤2𝑙₃^𝑡2_𝑡1𝑔(𝑠)𝑑𝑠⟶0,as𝑡₁⟶𝑡⁻₂,or𝑡₂⟶𝑡⁺₁,_𝑡2𝜋2𝐺^′_𝑡𝑡₁,𝑠−𝐺^′_𝑡𝑡₂,𝑠𝑔(𝑠)𝑑𝑠⟶0,as𝑡₁⟶𝑡⁻₂,or𝑡₂⟶𝑡⁺₁. (3.11)
Therefore, (𝑇𝑢)^′𝑡₁−(𝑇𝑢)^′𝑡₂=₀^2𝜋𝐺^′_𝑡𝑡₁,𝑠−𝐺^′_𝑡𝑡₂,𝑠𝑓𝑠,𝑢(𝑠)−𝑀𝜔(𝑠),𝑢^′(𝑠)−𝑀𝜔^′(𝑠)+𝑀𝑑𝑠≤𝑀₁₀^2𝜋𝐺^′_𝑡𝑡₁,𝑠−𝐺^′_𝑡𝑡₂,𝑠𝑔(𝑠)𝑑𝑠=𝑀₁^𝑡1₀𝐺^′_𝑡𝑡₁,𝑠−𝐺^′_𝑡𝑡₂,𝑠𝑔(𝑠)𝑑𝑠+^𝑡2_𝑡1𝐺^′_𝑡𝑡₁,𝑠−𝐺^′_𝑡𝑡₂𝑔(𝑠)+,𝑠𝑑𝑠_𝑡2𝜋2𝐺^′_𝑡𝑡₁,𝑠−𝐺^′_𝑡𝑡₂,𝑠𝑔(𝑠)𝑑𝑠⟶0,as𝑡₁⟶𝑡⁻₂or𝑡₂⟶𝑡⁺₁. (3.12) Thus, the functions belonging to {𝑇𝐷^′(𝑡)} are equicontinuous on [0,2𝜋]. By Arzela-Ascoli theorem, 𝑇𝐷 is relatively compact in 𝐶¹[0,2𝜋].
Hence, 𝑇∶𝐾∩(Ω_𝑅⧵Ω_𝑟1)→𝐾 is completely continuous for any 𝑅>𝑟₁. (2) We now show that inf_{𝑢∈𝐾∩𝜕Ω𝑟1}‖𝑇𝑢‖>0,𝑢≠𝜆𝑇𝑢,∀𝑢∈𝐾∩𝜕Ω_𝑟1,𝜆≥1. (3.13)
For any 𝑢∈𝐾∩𝜕Ω_𝑟1, we have 0<𝜎𝑟₁−2𝜋𝑀𝑙₂≤𝑢(𝑡)−𝑀𝜔(𝑡)≤𝑟₁,𝑢^′(𝑡)−𝑀𝜔^′(𝑡)≤𝑢^′(𝑡)+𝑀𝜔^′(𝑡)≤𝐿𝑟₁+2𝜋𝑀𝑙₃,∀𝑡∈[0,2𝜋]. (3.14) From (H₁) and (3.2), (𝑇𝑢)(𝑡)=₀^2𝜋𝐺(𝑡,𝑠)𝑓𝑠,𝑢(𝑠)−𝑀𝜔(𝑠),𝑢^′(𝑠)−𝑀𝜔^′(𝑠)+𝑀𝑑𝑠≥𝑙₁₀^2𝜋𝑎(𝑠)𝑑𝑠>𝑙₁𝑟₁𝑙₁⁻¹=𝑟₁>0. (3.15)
Suppose that there exist 𝜆₀≥1 and 𝑢₀∈𝐾∩𝜕Ω_𝑟1 such that 𝑢₀=𝜆₀𝑇𝑢₀, that is, for 𝑡∈[0,2𝜋], 𝑢₀(𝑡)≥𝑇𝑢₀(𝑡)=₀^2𝜋𝐺(𝑡,𝑠)𝑓𝑠,𝑢₀(𝑠)−𝑀𝜔(𝑠),𝑢^′₀(𝑠)−𝑀𝜔^′(𝑠)+𝑀𝑑𝑠≥𝑙₁₀^2𝜋𝑎(𝑠)𝑑𝑠>𝑙₁𝑟₁𝑙₁⁻¹=𝑟₁. (3.16) This is in contradiction with 𝑢₀∈𝐾∩𝜕Ω_𝑟1 and (3.13) holds. (3) Next, we show that 𝑢≠𝜆𝑇𝑢∀𝑢∈𝐾∩𝜕Ω_𝑅1,0<𝜆≤1. (3.17)
Suppose this is false, then there exist 𝜆₀∈(0,1] and 𝑢₀∈𝐾∩𝜕Ω_𝑅1 with 𝑢₀=𝜆₀𝑇𝑢₀, that is, for 𝑡∈[0,2𝜋], we have 𝑢₀(𝑡)≤𝑇𝑢₀(𝑡)=₀^2𝜋𝐺(𝑡,𝑠)𝑓𝑠,𝑢₀(𝑠)−𝑀𝜔(𝑠),𝑢^′₀(𝑠)−𝑀𝜔^′(𝑠)𝑢+𝑀𝑑𝑠,_′0(𝑡)=𝜆₀𝑇𝑢₀^′(𝑡)≤₀^2𝜋𝐺^′_𝑡(𝑡,𝑠)𝑓𝑠,𝑢(𝑠)−𝑀𝜔(𝑠),𝑢^′(𝑠)−𝑀𝜔^′(𝑠)+𝑀𝑑𝑠. (3.18)
From (H₂), we have 0<𝜎𝑅₁−2𝜋𝑀𝑙₂≤𝑢₀(𝑡)−𝑀𝜔(𝑡)≤𝑅₁,𝑢^′₀(𝑡)−𝑀𝜔^′(𝑡)≤𝑢^′₀(𝑡)+𝑀𝜔^′(𝑡)≤𝐿𝑅₁+2𝜋𝑀𝑙₃,∀𝑡∈[0,2𝜋]. (3.19) Therefore, by (3.18), (3.19), and (H₂), it follows that 𝑢₀(𝑡)≤𝑙₂₀^2𝜋𝑔(𝑠)ℎ𝑢₀(𝑠)−𝑀𝜔(𝑠),𝑢^′₀(𝑠)−𝑀𝜔^′(𝑠)𝑑𝑠≤𝑙₂𝑀₀₀^2𝜋𝑔(𝑠)𝑑𝑠<𝑅₁[],𝑢,∀𝑡∈0,2𝜋_′0(𝑡)≤𝑙₃₀^2𝜋𝑔(𝑠)ℎ𝑢₀(𝑠)−𝑀𝜔(𝑠),𝑢^′₀(𝑠)−𝑀𝜔^′(𝑠)𝑑𝑠≤𝑙₃𝑀₀₀^2𝜋𝑔(𝑠)𝑑𝑠<𝑅₁,∀𝑡∈[0,2𝜋]. (3.20) Thus, ‖𝑢‖<𝑅₁. This is in contradiction with 𝑢₀∈𝐾∩𝜕Ω_𝑅1 and (3.17) holds. (4) Choose 𝑁^∗=(1+2𝜋𝑀𝑙₂)[𝜎𝑙₁(𝛽^∗−𝛼^∗)]⁻¹+1. From (H₃), there exists 𝑅₂>max{𝑅₁,1} such that 𝑓(𝑡,𝑢,𝑣)≥𝑁^∗𝑢,∀𝑢≥𝑅₂,𝑣∈𝑅,𝑡∈𝛼^∗,𝛽^∗. (3.21)
Now, we show that inf_{𝑢∈𝐾∩𝜕Ω𝑅}‖𝑇𝑢‖>0,𝑢≠𝜆𝑇𝑢,∀𝑢∈𝐾∩𝜕Ω_𝑅,𝜆≥1, (3.22) where 𝑅=(𝑅₂+2𝜋𝑀𝑙₂)𝜎⁻¹.
For any 𝑢∈𝐾∩𝜕Ω_𝑅, we have 𝑢(𝑡)−𝑀𝜔(𝑡)≥𝜎𝑅−2𝜋𝑀𝑙₂=𝑅₂,∀𝑡∈[0,2𝜋]. (3.23) This and (3.21) together with (3.2) imply (𝑇𝑢)(𝑡)=₀^2𝜋𝐺(𝑡,𝑠)𝑓𝑠,𝑢(𝑠)−𝑀𝜔(𝑠),𝑢^′(𝑠)−𝑀𝜔^′(𝑠)+𝑀𝑑𝑠≥𝑙₁^𝛽∗_𝛼∗𝑓𝑠,𝑢(𝑠)−𝑀𝜔(𝑠),𝑢^′(𝑠)−𝑀𝜔^′(𝑠)𝑑𝑠≥𝑙₁𝑁^∗𝑅₂𝛽^∗−𝛼^∗>0. (3.24)
Suppose that there exist 𝜆₀≥1 and 𝑢₀∈𝐾∩𝜕Ω_𝑅 such that 𝑢₀=𝜆₀𝑇𝑢₀,then, for 𝑡∈[𝛼^∗,𝛽^∗], we have 𝑢₀(𝑡)≥𝑇𝑢₀(𝑡)=₀^2𝜋𝐺(𝑡,𝑠)𝑓𝑠,𝑢₀(𝑠)−𝑀𝜔(𝑠),𝑢^′₀(𝑠)−𝑀𝜔^′(𝑠)+𝑀𝑑𝑠≥𝑙₁^𝛽∗_𝛼∗𝑓𝑠,𝑢(𝑠)−𝑀𝜔(𝑠),𝑢^′(𝑠)−𝑀𝜔^′(𝑠)𝑑𝑠≥𝑙₁𝑁^∗𝑅₂𝛽^∗−𝛼^∗>𝑅₂+2𝜋𝑀𝑙₂𝜎⁻¹=𝑅. (3.25) This is in contradiction with 𝑢₀∈𝐾∩𝜕Ω_𝑅 and (3.22) holds.
Now, (3.13), (3.17), (3.22), and Lemma 2.3 guarantee that 𝑇 has two fixed points 𝑢₁∈𝐾∩(Ω_𝑅1⧵Ω_𝑟1), 𝑢₂∈𝐾∩(Ω_𝑅⧵Ω_𝑅1) with 𝑟₁<‖𝑢₁‖₁<𝑅₁<‖𝑢₂‖₁<𝑅. Clear, PBVP (3.1) has at least two positive solutions 𝑢₁,𝑢₂∈𝐶¹[0,2𝜋]∩𝐶²(0,2𝜋).

Remark 3.2. From the proof of Theorem 3.1, when 𝑓(𝑡,𝑢,𝑣) is nonnegative (i.e., 𝑀=0 in (H₀)), Theorem 3.1 still holds.

Corollary 3.3. Assume that (H₀)–(H₂) hold, then PBVP (1.1) has at least one positive solution 𝑢(𝑡) such that 𝑟₁<‖𝑢+𝑀𝜔‖<𝑅₁, where 𝜔(𝑡)=∶∫₀^2𝜋𝐺(𝑡,𝑠)𝑑𝑠.

Corollary 3.4. Assume that (H₀) and (H₃) hold, and
(H₄) there exist 𝑅₁>𝜎⁻¹2𝜋𝑀𝑙₂ such that max𝑙₂,𝑙₃₀^2𝜋𝑔(𝑡)𝑑𝑡<𝑅₁𝑀₀⁻¹, (3.26) where 𝑀₀=∶max{ℎ(𝑢,𝑣)∶𝑢∈[𝜎𝑅₁−2𝜋𝑀𝑙₂,𝑅₁]and𝑣∈[−(𝐿𝑅₁+2𝜋𝑀𝑙₃),(𝐿𝑅₁+2𝜋𝑀𝑙₃)]}. Then PBVP (1.1) has at least one positive solution 𝑢(𝑡) such that ‖𝑢+𝑀𝜔‖>𝑅₁, where 𝜔(𝑡)=∶∫₀^2𝜋𝐺(𝑡,𝑠)𝑑𝑠.

Example 3.5. Consider the following second-order singular semipositone PBVP: 𝑢^′′+1𝑢16𝑢=^9/4+𝑢^′2+18𝜋𝑢𝑡()−√2𝜋−𝑡3𝑡30𝜋cos(),𝑢(0)()12,𝑡∈0,2𝜋=𝑢2𝜋,𝑢^′(0)=𝑢^′(2𝜋). (3.27)

4. Conclusion

PBVP (3.27) has at least two positive solutions 𝑢₁,𝑢₂∈𝐶¹[0,2𝜋]∩𝐶²(0,2𝜋) and 𝑢₁(𝑡),𝑢₂(𝑡)>0 for 𝑡∈[0,2𝜋].

To see this, we will apply Theorem 3.1 with 𝑚=1/4, 𝑓(𝑡,𝑢,𝑣)=((𝑢^9/4+𝑣²+1)/8𝜋𝑢𝑡(2𝜋−𝑡))−(√3/30𝜋)cos(𝑡/12), 𝑔(𝑡)=1/𝑡(2𝜋−𝑡), ℎ(𝑢,𝑣)=(𝑢^9/4+𝑣²+1)/8𝜋𝑢, 𝑀=1/20𝜋.

From Remark 2.2, it is easy to see that 𝑙₁=2, 𝑙₂=2√2, 𝑙₃=1/2, 𝜎=√2/2, and 𝐿=1/4.

By simple computation, we easily get 0≤𝑓(𝑡,𝑢,𝑣)+𝑀≤𝑔(𝑡)ℎ(𝑢,𝑣) and ∫₀^2𝜋𝑔(𝑡)𝑑𝑡=𝜋. So (H₀) holds.

Taking 𝑟₁=1/2, 𝑎(𝑡)=1/4𝜋𝑡(2𝜋−𝑡), then 𝜎⁻¹2𝜋𝑀𝑙₂=√2⋅2𝜋⋅(1/20𝜋)⋅2√2=2/5<𝑟₁, ∫₀^2𝜋𝑎(𝑡)𝑑𝑡=1/4=𝑟₁𝑙₁⁻¹ and for any 𝑡∈(0,2𝜋), 𝑢∈(0,1/2], 𝑣∈[−7/40,7/40], 𝑢^9/4+𝑣²+18𝜋𝑢𝑡()−√2𝜋−𝑡3𝑡30𝜋cos+112≥20𝜋1/2^9/4+14𝜋𝑡()−√2𝜋−𝑡3+130𝜋≥20𝜋1/2^9/44(𝜋)²−√3+130𝜋+120𝜋4𝜋𝑡()>12𝜋−𝑡4𝜋𝑡().2𝜋−𝑡 (4.1) Thus, (H₁) holds.

Taking 𝑅₁=4, then for 𝑢∈[(9/5)√2,4],|𝑣|≤21/20, we have 𝑀₀≤572√22𝜋^9/2+2120²<5+172√()=2𝜋23+2+16536√.2𝜋 (4.2) So, 𝑀₀max{𝑙₂,𝑙₃}∫₀^2𝜋𝑔(𝑡)𝑑𝑡=2√2𝜋𝑀₀<65/18<4=𝑅₁. That is, (H₂) holds.

Let [𝛼^∗,𝛽^∗]=[𝜋/2,𝜋], then it is easy to check that (H₃) holds.

Thus all the conditions of Theorem 3.1 are satisfied, so PBVP (3.27) has at least two positive solutions 𝑢₁,𝑢₂∈𝐶¹[0,2𝜋]∩𝐶²(0,2𝜋) and 𝑢₁(𝑡),𝑢₂(𝑡)>0 for 𝑡∈[0,2𝜋].

Acknowledgments

Research supported by the Project of Shandong Province Higher Educational Science and Technology Program (J09LA08) and Reward Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (BS2011SF022), China.