Multiple Positive Solutions of a Second Order Nonlinear Semipositone 𝑚-Point Boundary Value Problem on Time Scales

Abstract

In this paper, we study a general second-order 𝑚-point boundary value problem for nonlinear singular dynamic equation on time scales 𝑢^Δ∇(𝑡)+𝑎(𝑡)𝑢^Δ(𝑡)+𝑏(𝑡)𝑢(𝑡)+𝜆𝑞(𝑡)𝑓(𝑡,𝑢(𝑡))=0, 𝑡∈(0,1)_𝕋, 𝑢(𝜌(0))=0, 𝑢(𝜎(1))=∑^𝑚−2_𝑖=1𝛼_𝑖𝑢(𝜂_𝑖). This paper shows the existence of multiple positive solutions if 𝑓 is semipositone and superlinear. The arguments are based upon fixed-point theorems in a cone.

1. Introduction

In this paper, we consider the following dynamic equation on time scales: 𝑢^Δ∇(𝑡)+𝑎(𝑡)𝑢^Δ(𝑡)+𝑏(𝑡)𝑢(𝑡)+𝜆𝑞(𝑡)𝑓(𝑡,𝑢(𝑡))=0,𝑡∈(0,1)_𝕋,𝑢(𝜌(0))=0,𝑢(𝜎(1))=^𝑚−2_𝑖=1𝛼_𝑖𝑢(𝜂_𝑖), (1.1) where 𝛼_𝑖≥0,0<𝜂_𝑖<𝜂_𝑖+1<1; for all 𝑖=1,2,…,𝑚−2; 𝑓,𝑞,𝑎 and 𝑏 satisfy(C1)𝑞∈𝐿 is continuously and nonnegative function and there exists 𝑡₀∈(𝜌(0),𝜎(1)) s.t. 𝑞(𝑡₀)>0, 𝑞(𝑡) may be singular at 𝑡=𝜌(0),𝜎(1);(C2)𝑎∈𝐶([0,1],[0,+∞)), 𝑏∈𝐶([0,1],(−∞,0]).

In the past few years, the boundary value problems of dynamic equations on time scales have been studied by many authors (see [1–15] and references therein). Recently, multiple-point boundary value problems on time scale have been studied, for instance, see [1–9].

In 2008, Lin and Du [2] studied the 𝑚-point boundary value problem for second-order dynamic equations on time scales:𝑢^Δ∇(𝑡)+𝑓(𝑡,𝑢)=0,𝑡∈(0,𝑇)∈𝕋,𝑢(0)=0,𝑢(𝑇)=^𝑚−2_𝑖=1𝑘_𝑖𝑢(𝜉_𝑖), (1.2) where 𝕋 is a time scale. This paper deals with the existence of multiple positive solutions for second-order dynamic equations on time scales. By using Green's function and the Leggett-Williams fixed point theorem in an appropriate cone, the existence of at least three positive solutions of the problem is obtained.

In 2009, Topal and Yantir [1] studied the general second-order nonlinear 𝑚-point boundary value problems (1.1) with no singularities and the case. The authors deal with the determining the value of 𝜆; the existences of multiple positive solutions of (1.1) are obtained by using the Krasnosel'skii and Legget-William fixed point theorems.

Motivated by the abovementioned results, we continue to study the general second-order nonlinear 𝑚-point boundary value problem (1.1), but the nonlinear term may be singularity and semipositone.

In this paper, the nonlinear term 𝑓 of (1.1) is suit to and semipositone and the superlinear case, we will prove our two existence results for problem (1.1) by using Krasnosel'skii fixed point theorem. This paper is organized as follows. In Section 2, starting with some preliminary lemmas, we state the Krasnosel'skii fixed point theorem. In Section 3, we give the main result which state the sufficient conditions for the 𝑚-point boundary value problem (1.1) to have existence of positive solutions.

2. Preliminaries

In this section, we state the preliminary information that we need to prove the main results. From Lemmas 2.1 and 2.3 in [1], we have the following lemma.

Lemma 2.1 (see [1]). Assuming that (C2) holds. Then the equations 𝜙₁^Δ∇(𝑡)+𝑎(𝑡)𝜙^Δ₁(𝑡)+𝑏(𝑡)𝜙₁(𝑡)=0,𝑡∈(0,1)_𝕋,𝜙₁(𝜌(0))=0,𝜙₁(𝜎(1))=1,𝜙₂^Δ∇(𝑡)+𝑎(𝑡)𝜙^Δ₂(𝑡)+𝑏(𝑡)𝜙₂(𝑡)=0,𝑡∈(0,1)_𝕋,𝜙₂(𝜌(0))=1,𝜙₂(𝜎(1))=0 (2.1)(2.2) have unique solutions 𝜙₁ and 𝜙₂, respectively, and(a)𝜙₁ is strictly increasing on [𝜌(0),𝜎(1)],(b)𝜙₂ is strictly decreasing on [𝜌(0),𝜎(1)].

For the rest of the paper we need the following assumption:(C3)0<∑^𝑚−2_𝑖=1𝛼_𝑖𝜙₁(𝜂_𝑖)<1.

Lemma 2.2 (see [1]). Assuming that (C2) and (C3) hold. Let 𝑦∈𝐶[𝜌(0),𝜎(1)]. Then boundary value problem 𝑥^Δ∇(𝑡)+𝑎(𝑡)𝑥^Δ(𝑡)+𝑏(𝑡)𝑥(𝑡)+𝑦(𝑡)=0,𝑡∈(0,1)_𝕋,𝑥(𝜌(0))=0,𝑥(𝜎(1))=^𝑚−2_𝑖=1𝛼_𝑖𝑥(𝜂_𝑖) (2.3) is equivalent to integral equation 𝑥(𝑡)=^𝜎(1)_𝜌(0)𝐻(𝑡,𝑠)𝑝(𝑠)𝑦(𝑠)∇𝑠+𝐴𝜙₁(𝑡), (2.4) where 𝑝(𝑡)=𝑒_𝑎1(𝜌(𝑡),𝜌(0)),𝐴=1−∑^𝑚−2_𝑖=1𝛼_𝑖𝜙₁(𝜂_𝑖)^𝑚−2_𝑖=1𝛼_𝑖^𝜎(1)_𝜌(0)𝐻(𝜂_𝑖,𝑠)𝑝(𝑠)𝑦(𝑠)∇𝑠,1𝐻(𝑡,𝑠)=𝜙^Δ₁𝜙(𝜌(0))¹(𝑠)𝜙₂𝜙(𝑡),𝑠≤𝑡,₁(𝑡)𝜙₂(𝑠),𝑡≤𝑠. (2.5)(2.6)

Proof. First we show that the unique solution of (2.3) can be represented by (2.4). From Lemma 2.1, we know that the homogenous part of (2.3) has two linearly independent solution 𝜙₁ and 𝜙₂ since 𝜙₁(𝜌(0))𝜙^Δ₁𝜙(𝜌(0))₂(𝜌(0))𝜙^Δ₂(𝜌(0))=−𝜙^Δ₁(𝜌(0))≠0. (2.7)
Now by the method of variations of constants, we can obtain the unique solution of (2.3) which can be represented by (2.4) where 𝐴 and 𝐻 are as in (2.5) and (2.6), respectively. Next we check the function defined in (2.4) is the solution of the boundary value problem (2.3). For this purpose we first show that (2.4) satisfies (2.3). From the definition of Green's function (2.6), we get 1𝑥(𝑡)=𝜙^Δ₁((𝜌(0))^𝑡𝜌(0)𝜙₁(𝑠)𝜙₂(𝑡)𝑝(𝑠)𝑦(𝑠)∇𝑠+_𝑡^𝜎(1)𝜙₁(𝑡)𝜙₂(𝑠)𝑝(𝑠)𝑦(𝑠)∇𝑠)+𝐴𝜙₁(𝑡). (2.8) Hence, the derivatives 𝑥^Δ and 𝑥^Δ∇ are as follows: 𝑥^Δ1(𝑡)=𝜙^Δ₁(𝜌(0))(𝜙^Δ₂(𝑡)^𝑡_𝜌(0)𝜙₁(𝑠)𝑝(𝑠)𝑦(𝑠)∇𝑠+𝜙^Δ₁(𝑡)_𝑡^𝜎(1)𝜙₂(𝑠)𝑝(𝑠)𝑦(𝑠)∇𝑠)+𝐴𝜙^Δ₁𝑥(𝑡),_Δ∇1(𝑡)=𝜙^Δ₁(𝜌(0))(𝜙₂^Δ∇(𝑡)^𝜌(𝑡)_𝜌(0)𝜙₁(𝑠)𝑝(𝑠)𝑦(𝑠)∇𝑠+𝜙^Δ₂(𝑡)𝜙₁(𝑡)𝑝(𝑡)𝑦(𝑡)+𝜙₁^Δ∇(𝑡)^𝜎(1)_𝜌(𝑡)𝜙₂(𝑠)𝑝(𝑠)𝑦(𝑠)∇𝑠+𝜙^Δ₁(𝑡)𝜙₂(𝑡)𝑝(𝑡)𝑦(𝑡))+𝐴𝜙₁^Δ∇(𝑡). (2.9) Replacing the derivatives in (2.3), we deduce that 𝑥^Δ∇(𝑡)+𝑎(𝑡)𝑥^Δ(𝑡)+𝑏(𝑡)𝑥(𝑡)=𝐴(𝜙₁^Δ∇(𝑡)+𝑎(𝑡)𝜙^Δ₁(𝑡)+𝑏(𝑡)𝜙₁(𝑡))1+(𝜙^Δ₁(𝜌(0))^𝑡𝜌(0)𝜙₁(𝑠)𝑝(𝑠)𝑦(𝑠)∇𝑠)(𝜙₂^Δ∇(𝑡)+𝑎(𝑡)𝜙^Δ₂(𝑡)+𝑏(𝑡)𝜙₂(𝑡))1+(𝜙^Δ₁(𝜌(0))^𝑡𝜎(1)𝜙₂(𝑠)𝑝(𝑠)𝑦(𝑠)∇𝑠)(𝜙₁^Δ∇(𝑡)+𝑎(𝑡)𝜙^Δ₁(𝑡)+𝑏(𝑡)𝜙₁(𝑡))+1𝜙^Δ₁(𝜌(0))(𝜙₂^Δ∇(𝑡)_𝑡^𝜌(𝑡)𝜙₁(𝑠)𝑝(𝑠)𝑦(𝑠)∇𝑠+𝜙₁^Δ∇(𝑡)^𝑡_𝜌(𝑡)𝜙₂(𝑠)𝑝(𝑠)𝑦(𝑠)∇𝑠)+1𝜙^Δ₁(𝜌(0))(𝜙^Δ₂(𝑡)𝜙₁(𝑡)−𝜙^Δ₁(𝑡)𝜙₂(𝑡))𝑝(𝑡)𝑦(𝑡)=1𝜙^Δ₁(𝜌(0))(𝜙₂^Δ∇(𝑡)(𝜌(𝑡)−𝑡)𝜙₁(𝑡)𝑝(𝑡)𝑦(𝑡)−𝜙₁^Δ∇(𝑡)𝜙₂(𝑡)𝑝(𝑡)𝑦(𝑡)+𝜙^Δ₂(𝑡)𝜙₁(𝑡)𝑝(𝑡)𝑦(𝑡)−𝜙^Δ₁(𝑡)𝜙₂(𝑡)𝑝(𝑡)𝑦(𝑡))=1𝜙^Δ₁(𝜌(0))𝑝(𝑡)𝑦(𝑡)(𝜙^Δ₂(𝑡)𝜙₁(𝑡)−𝜙^Δ₁(𝑡)𝜙₂(𝑡))+1𝜙^Δ₁(𝜌(0))𝑝(𝑡)𝑦(𝑡)(𝜌(𝑡)−𝑡)(𝜙₂^Δ∇(𝑡)𝜙₁(𝑡)−𝜙₁^Δ∇(𝑡)𝜙₂(𝑡))=1𝜙^Δ₁(𝜌(0))𝑝(𝑡)𝑦(𝑡){(𝜙^Δ₂(𝑡)𝜙₁(𝑡)−𝜙^Δ₁(𝑡)𝜙₂(𝑡))+(𝜌(𝑡)−𝑡)(𝜙^Δ₂(𝑡)𝜙₁(𝑡)−𝜙^Δ₁(𝑡)𝜙₂(𝑡))^∇}=1𝜙^Δ₁(𝜌(0))𝑝(𝑡)𝑦(𝑡)(𝜙^Δ₂(𝜌(𝑡))𝜙₁(𝜌(𝑡))−𝜙^Δ₁(𝜌(𝑡))𝜙₂(𝜌(𝑡)))=1𝜙^Δ₁(𝜌(0))𝑝(𝑡)𝑦(𝑡)𝑒_Θ𝑎(𝜌(𝑡),𝜌(0))(−𝜙^Δ₁(𝜌(0)))=−𝑦(𝑡). (2.10)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) Therefore the function defined in (2.4) satisfies (2.3). Further we obtain that the boundary value conditions are satisfied by (2.4). The first condition follows from (2.5) and (2.6) and Lemma 2.1. Now we verify the second boundary condition. Since 1𝐻(𝜎(1),𝑠)=𝜙^Δ₁𝜙(𝜌(0))¹(𝑠)𝜙₂(𝜎(1))=0, (2.11) we obtain that 𝑥(𝜎(1))=^𝜎(1)_𝜌(0)𝐻(𝜎(1),𝑠)𝑝(𝑠)𝑦(𝑠)∇𝑠+𝐴𝜙₁(𝜎(1))=𝐴. (2.12) On the other hand, by using (2.5), we find that ^𝑚−2_𝑖=1𝛼_𝑖𝑥(𝜂_𝑖)=^𝑚−2_𝑖=1𝛼_𝑖(^𝜎(1)_𝜌(0)𝐻(𝜂_𝑖,𝑠)𝑝(𝑠)𝑦(𝑠)∇𝑠+𝐴𝜙₁(𝜂_𝑖=))_{𝑚−2𝑖=1}𝛼_𝑖(^𝜎(1)_𝜌(0)𝐻(𝜂_𝑖+1,𝑠)𝑝(𝑠)𝑦(𝑠)∇𝑠1−∑^𝑚−2_𝑖=1𝛼_𝑖𝜙₁(𝜂_𝑖)^𝑚−2_𝑖=1𝛼_𝑖𝜙₁(𝜂_𝑖)^𝜎(1)_𝜌(0)𝐻(𝜂_𝑖=1,𝑠)𝑝(𝑠)𝑦(𝑠)∇𝑠)1−∑^𝑚−2_𝑖=1𝛼_𝑖𝜙₁(𝜂_𝑖)^𝑚−2_𝑖=1𝛼_𝑖^𝜎(1)_𝜌(0)𝐻(𝜂_𝑖,𝑠)𝑝(𝑠)𝑦(𝑠)∇𝑠=𝐴. (2.13) Combining the two equations above finishes the proof.

Lemma 2.3. Green's function 𝐻(𝑡,𝑠) has the following properties: 𝜙𝐻(𝑡,𝑠)≤𝐻(𝑡,𝑡),^Δ1(𝜌(0))‖𝜙₁‖‖𝜙₂‖𝐻(𝑡,𝑡)𝐻(𝑠,𝑠)≤𝐻(𝑡,𝑠)≤𝐻(𝑠,𝑠),𝐻(𝑡,𝑡)≤𝜙₁(𝑡)‖𝜙₂‖𝜙^Δ₁.(𝜌(0)) (2.14)

Lemma 2.4. Assume that (C2) and (C3) hold. Let 𝑢 be a solution of boundary value problem (1.1) if and only if 𝑢 is a solution of the following integral equation: 𝑢(𝑡)=^𝜎(1)_𝜌(0)𝐺(𝑡,𝑠)𝑝(𝑠)𝑞(𝑠)𝑓(𝑠,𝑢(𝑠))∇𝑠, (2.15) where 1𝐺(𝑡,𝑠)=𝐻(𝑡,𝑠)+1−∑^𝑚−2_𝑖=1𝛼_𝑖𝜙₁(𝜂_𝑖)^𝑚−2_𝑖=1𝛼_𝑖𝐻(𝜂_𝑖,𝑠)𝜙₁(𝑡). (2.16)

The proofs of the Lemmas 2.3 and 2.4 can be obtained easily by Lemmas 2.1 and 2.2.

Lemma 2.5. Green's function 𝐺(𝑡,𝑠) defined by (2.16) has the following properties: 𝐶₂𝜙₁(𝑡)𝐻(𝑠,𝑠)≤𝐺(𝑡,𝑠)≤𝐶₁𝐻(𝑠,𝑠),𝐺(𝑡,𝑠)≤𝐶₃𝜙₁(𝑡), (2.17) where 𝐶₁=1+‖𝜙₁‖1−∑^𝑚−2_𝑖=1𝛼_𝑖𝜙₁(𝜂_𝑖)^𝑚−2_𝑖=1𝛼_𝑖,𝐶₂=𝜙^Δ₁(𝜌(0))‖𝜙₁‖‖𝜙₂‖11−∑^𝑚−2_𝑖=1𝛼_𝑖𝜙₁(𝜂_𝑖)^𝑚−2_𝑖=1𝛼_𝑖𝐻(𝜂_𝑖,𝜂_𝑖𝐶),₃=‖𝜙₂‖𝜙^Δ₁+1(𝜌(0))1−∑^𝑚−2_𝑖=1𝛼_𝑖𝜙₁(𝜂_𝑖)^𝑚−2_𝑖=1𝛼_𝑖𝐻(𝜂_𝑖,𝜂_𝑖). (2.18)

Proof. From Lemma 2.3, we have 1𝐺(𝑡,𝑠)≤𝐻(𝑠,𝑠)+1−∑^𝑚−2_𝑖=1𝛼_𝑖𝜙₁(𝜂_𝑖)^𝑚−2_𝑖=1𝛼_𝑖𝐻(𝑠,𝑠)‖𝜙₁‖≤𝐶₁𝐻(𝑠,𝑠),𝐺(𝑡,𝑠)≤‖𝜙₂‖𝜙^Δ₁𝜙(𝜌(0))¹1(𝑡)+1−∑^𝑚−2_𝑖=1𝛼_𝑖𝜙₁(𝜂_𝑖)^𝑚−2_𝑖=1𝛼_𝑖𝐻(𝜂_𝑖,𝜂_𝑖)𝜙₁(𝑡)≤𝐶₃𝜙₁1(𝑡),𝐺(𝑡,𝑠)≥1−∑^𝑚−2_𝑖=1𝛼_𝑖𝜙₁(𝜂_𝑖)^𝑚−2_𝑖=1𝛼_𝑖𝐻(𝜂_𝑖,𝑠)𝜙₁≥1(𝑡)1−∑^𝑚−2_𝑖=1𝛼_𝑖𝜙₁(𝜂_𝑖)^𝑚−2_𝑖=1𝛼_𝑖𝜙^Δ₁(𝜌(0))‖𝜙₁‖‖𝜙₂‖𝐻(𝜂_𝑖,𝜂_𝑖)𝐻(𝑠,𝑠)𝜙₁(𝑡)≥𝐶₂𝜙₁(𝑡)𝐻(𝑠,𝑠). (2.19) The proof is complete.

The following theorems will play major role in our next analysis.

Theorem 2.6 (see [16]). Let 𝑋 be a Banach space, and let 𝑃⊂𝑋 be a cone in 𝑋. Let Ω₁,Ω₂ be open subsets of 𝑋 with 0∈Ω₁⊂Ω₁⊂Ω₂, and let 𝑆∶𝑃→𝑃 be a completely continuous operator, such that, either (1)‖𝑆𝑤‖≤‖𝑤‖, 𝑤∈𝑃∩𝜕Ω₁, ‖𝑆𝑤‖≥‖𝑤‖, 𝑤∈𝑃∩𝜕Ω₂, or (2)‖𝑆𝑤‖≥‖𝑤‖, 𝑤∈𝑃∩𝜕Ω₁, ‖𝑆𝑤‖≤‖𝑤‖𝑤∈𝑃∩𝜕Ω₂. Then 𝑆 has a fixed point in 𝑃∩Ω₂⧵Ω₁.

3. Main Results

We make the following assumptions:(H1)𝑓(𝑡,𝑢)∈𝐶([𝜌(0),𝜎(1)]×[0,+∞),(−∞,+∞)), moreover there exists a function 𝑔(𝑡)∈𝐿¹([𝜌(0),𝜎(1)],(0,+∞)) such that 𝑓(𝑡,𝑢)≥−𝑔(𝑡), for any 𝑡∈(𝜌(0),𝜎(1)), 𝑢∈[0,+∞).(H∗1)𝑓(𝑡,𝑢)∈𝐶((𝜌(0),𝜎(1))×[0,+∞),(−∞,+∞)) may be singular at 𝑡=𝜌(0),𝜎(1), moreover there exists a function 𝑔(𝑡)∈𝐿¹((𝜌(0),𝜎(1)),(0,+∞)) such that 𝑓(𝑡,𝑢)≥−𝑔(𝑡), for any 𝑡∈(𝜌(0),𝜎(1)), 𝑢∈[0,+∞).(H2)𝑓(𝑡,0)>0, for 𝑡∈[𝜌(0),𝜎(1)].(H3) There exists [𝜃₁,𝜃₂]∈(𝜌(0),𝜎(1)) such that lim_𝑢↑+∞min_{𝑡∈[𝜃1,𝜃2]}(𝑓(𝑡,𝑢)/𝑢)=+∞.(H4)∫^𝜎(1)_𝜌(0)𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)𝑓(𝑠,𝑧)∇𝑠<+∞ for any 𝑧∈[0,𝑚], 𝑚>0 is any constant.

In fact, we only consider the boundary value problem𝑥^Δ∇(𝑡)+𝑎(𝑡)𝑥^Δ(𝑡)+𝑏(𝑡)𝑥(𝑡)+𝜆𝑞(𝑡)[𝑓(𝑡,[𝑥(𝑡)−𝑣(𝑡)]^∗)+𝑔(𝑡)]=0,𝜆>0,𝑥(𝜌(0))=0,𝑥(𝜎(1))=^𝑚−2_𝑖=1𝛼_𝑖𝑥(𝜂_𝑖), (3.1) where 𝑦(𝑡)^∗=𝑦(𝑡),𝑦(𝑡)≥0,0,𝑦(𝑡)<0, (3.2) and 𝑣(𝑡)=𝜆∫^𝜎(1)_𝜌(0)𝐺(𝑡,𝑠)𝑝(𝑠)𝑞(𝑠)𝑔(𝑠)∇𝑠, which is the solution of the boundary value problem 𝑣^Δ∇(𝑡)+𝑎(𝑡)𝑣^Δ(𝑡)+𝑏(𝑡)𝑣(𝑡)+𝜆𝑞(𝑡)𝑔(𝑡)=0,𝑣(𝜌(0))=0,𝑣(𝜎(1))=^𝑚−2_𝑖=1𝛼_𝑖𝑣(𝜂_𝑖). (3.3) From Lemma 2.1, it is easy to verify that 𝑣(𝑡)≤𝜆𝐶₀𝜙₁(𝑡) and 𝐶₀=𝐶₃∫^𝜎(1)_𝜌(0)𝑞(𝑠)𝑔(𝑠)∇𝑠.

We will show that there exists a solution 𝑥 for boundary value problem (3.1) with 𝑥(𝑡)≥𝑣(𝑡),𝑡∈[𝜌(0),𝜎(1)]. If this is true, then 𝑢(𝑡)=𝑥(𝑡)−𝑣(𝑡) is a nonnegative solution (positive on (𝜌(0),𝜎(1))) of boundary value problem (3.1). Since for any 𝑡∈(𝜌(0),𝜎(1)), from (𝑢(𝑡)+𝑣(𝑡))^Δ∇+𝑎(𝑡)(𝑢(𝑡)+𝑣(𝑡))^Δ+𝑏(𝑡)(𝑢(𝑡)+𝑣(𝑡))=−𝜆𝑞(𝑡)[𝑓(𝑡,𝑢)+𝑔(𝑡)], (3.4) we have 𝑢^Δ∇(𝑡)+𝑎(𝑡)𝑢^Δ(𝑡)+𝑏(𝑡)𝑢(𝑡)=−𝜆𝑞(𝑡)𝑓(𝑡,𝑢). (3.5) As a result, we will concentrate our study on boundary value problem (3.1).

We note that 𝑥(𝑡) is a solution of (3.1) if and only if𝑥(𝑡)=𝜆^𝜎(1)_𝜌(0)𝐺(𝑡,𝑠)𝑝(𝑠)𝑞(𝑠)(𝑓(𝑡,[𝑥(𝑡)−𝑣(𝑡)]^∗)+𝑔(𝑡))∇𝑠. (3.6)

For our constructions, we will consider the Banach space 𝑋=𝐶[𝜌(0),𝜎(1)] equipped with standard norm ‖𝑥‖=max_0≤𝑡≤1|𝑥(𝑡)|,𝑥∈𝑋. We define a cone 𝑃 by𝐶𝑃={𝑥∈𝑋∣𝑥(𝑡)≥²𝐶₁𝜙₁(𝑡)‖𝑥‖,𝑡∈[𝜌(0),𝜎(1)]}, (3.7) where 𝜙₁ is defined by Lemma 2.1 (namely, 𝜙₁ is solution (2.1)). Define an integral operator 𝑇∶𝑃→𝑋 by𝑇𝑥(𝑡)=𝜆^𝜎(1)_𝜌(0)𝐺(𝑡,𝑠)𝑝(𝑠)𝑞(𝑠)(𝑓(𝑡,[𝑥(𝑡)−𝑣(𝑡)]^∗)+𝑔(𝑡))∇𝑠. (3.8) Notice, from (3.8) and Lemma 2.5, we have 𝑇𝑥(𝑡)≥0 on [0,1] for 𝑥∈𝑃 and 𝑇𝑥(𝑡)=𝜆^𝜎(1)_𝜌(0)𝐺(𝑡,𝑠)𝑝(𝑠)𝑞(𝑠)(𝑓(𝑡,[𝑥(𝑡)−𝑣(𝑡)]^∗)+𝑔(𝑡))∇𝑠≤𝐶₁𝜆^𝜎(1)_𝜌(0)𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)(𝑓(𝑡,[𝑥(𝑡)−𝑣(𝑡)]^∗)+𝑔(𝑡))∇𝑠, (3.9) then ‖𝑇𝑥‖≤𝐶₁𝜆∫^𝜎(1)_𝜌(0)𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)(𝑓(𝑡,[𝑥(𝑡)−𝑣(𝑡)]^∗)+𝑔(𝑡))∇𝑠.

On the other hand, we have𝑇𝑥(𝑡)=𝜆^𝜎(1)_𝜌(0)𝐺(𝑡,𝑠)𝑝(𝑠)𝑞(𝑠)(𝑓(𝑡,[𝑥(𝑡)−𝑣(𝑡)]^∗)+𝑔(𝑡))∇𝑠≥𝜆^𝜎(1)_𝜌(0)𝐶₂𝜙₁(𝑡)𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)(𝑓(𝑡,[𝑥(𝑡)−𝑣(𝑡)]^∗≥𝐶)+𝑔(𝑡))∇𝑠₂𝐶₁𝜙₁(𝑡)𝜆^𝜎(1)_𝜌(0)𝐶₁𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)(𝑓(𝑡,[𝑥(𝑡)−𝑣(𝑡)]^∗≥𝐶)+𝑔(𝑡))∇𝑠₂𝐶₁𝜙₁(𝑡)‖𝑇𝑥‖. (3.10) Thus, 𝑇(𝑃)⊂𝑃. In addition, standard arguments show that 𝑇(𝑃)⊂𝑃 and 𝑇 is a compact, and completely continuous.

Theorem 3.1. Suppose that (H1)-(H2) hold. Then there exists a constant 𝜆>0 such that, for any 0<𝜆≤𝜆, boundary value problem (1.1) has at least one positive solution.

Proof. Fix 𝛿∈(0,1). From (H2), let 0<𝜀<1 be such that 𝑓(𝑡,𝑧)≥𝛿𝑓(𝑡,0),for𝜌(0)≤𝑡≤𝜎(1),0≤𝑧≤𝜀. (3.11) Suppose that 𝜀0<𝜆<2𝑐𝑓(𝜀)∶=𝜆 (3.12) where 𝑓(𝜀)=max_{𝜌(0)≤𝑡≤𝜎(1),0≤𝑧≤𝜀}{𝑓(𝑡,𝑧)+𝑔(𝑡)} and 𝑐=𝐶₁∫^𝜎(1)_𝜌(0)𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)∇𝑠. Since lim_𝑧↓0𝑓(𝑧)𝑧=+∞,𝑓(𝜀)𝜀<1,2𝑐𝜆 (3.13) there exists a 𝑅₀∈(0,𝜀) such that 𝑓(𝑅₀)𝑅₀=1.2𝑐𝜆 (3.14)
Let 𝑥∈𝑃 and 𝜈∈(0,1) be such that 𝑥=𝜈𝑇(𝑥), we claim that ‖𝑥‖≠𝑅₀. In fact ‖𝑇𝑥(𝑡)‖≤𝜈𝜆^𝜎(1)_𝜌(0)𝐶₁𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)[𝑓(𝑠,[𝑥(𝑠)−𝑣(𝑠)]^∗)+𝑔(𝑠)]∇𝑠≤𝜆^𝜎(1)_𝜌(0)𝐶₁𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)[𝑓(𝑠,[𝑥(𝑠)−𝑣(𝑠)]^∗)+𝑔(𝑠)]∇𝑠≤𝜆^𝜎(1)_𝜌(0)𝐶₁𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)max_{0≤𝑠≤1;0≤𝑧≤𝑅0}[𝑓(𝑠,𝑧)+𝑔(𝑠)]∇𝑠≤𝜆^𝜎(1)_𝜌(0)𝐶₁𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)𝑓(𝑅₀)∇𝑠≤𝜆𝑐𝑓(𝑅₀), (3.15) that is, 𝑓(𝑅₀)𝑅₀≥1>1𝑐𝜆=2𝑐𝜆𝑓(𝑅₀)𝑅₀ (3.16) which implies that ‖𝑥‖≠𝑅₀. Let 𝑈={𝑥∈𝑃∶‖𝑥‖<𝑅₀}. By nonlinear alternative of Leray-Schauder type theorem, 𝑇 has a fixed point 𝑥∈𝑈. Moreover, combing (3.8), (3.28), and 𝑅₀<𝜀, we obtain that 𝑥(𝑡)=𝜆^𝜎(1)_𝜌(0)𝐺(𝑡,𝑠)𝑝(𝑠)𝑞(𝑠)[𝑓(𝑠,[𝑥(𝑠)−𝑣(𝑠)]^∗)+𝑔(𝑠)]∇𝑠≥𝜆^𝜎(1)_𝜌(0)𝐺(𝑡,𝑠)𝑝(𝑠)𝑞(𝑠)[𝛿𝑓(𝑠,0)+𝑔(𝑠)]∇𝑠≥𝜆[𝛿^𝜎(1)_𝜌(0)𝐺(𝑡,𝑠)𝑝(𝑠)𝑞(𝑠)𝑓(𝑠,0)∇𝑠+^𝜎(1)_𝜌(0)𝐺(𝑡,𝑠)𝑝(𝑠)𝑞(𝑠)𝑔(𝑠)∇𝑠]>𝜆^𝜎(1)_𝜌(0)𝐺(𝑡,𝑠)𝑝(𝑠)𝑞(𝑠)𝑔(𝑠)∇𝑠=𝑣(𝑡)for𝑡∈(𝜌(0),𝜎(1)). (3.17) Let 𝑢(𝑡)=𝑥(𝑡)−𝑣(𝑡)>0. Then (1.1) has a positive solution 𝑢 and ‖𝑢‖≤‖𝑥‖≤𝑅₀<1. This completes the proof of Theorem 3.1.

Theorem 3.2. Suppose that (H∗1) and (H3)-(H4) hold. Then there exists a constant 𝜆^∗>0 such that, for any 0<𝜆≤𝜆^∗, boundary value problem (1.1) has at least one positive solution.

Proof. Let Ω₁={𝑥∈𝑃∶‖𝑥‖<𝑅₁}, where 𝑅₁=max{1,𝑟} and 𝑟=(𝐶₁𝐶₃/𝐶₂)∫^𝜎(1)_𝜌(0)𝑞(𝑠)𝑔(𝑠)∇𝑠. Choose 𝜆^∗=min{1,𝑅₁[𝐶₁^𝜎(1)_𝜌(0)𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)[max_{0≤𝑧≤𝑅1}𝑓(𝑠,𝑧)+𝑔(𝑠)]∇𝑠]⁻¹}. (3.18) Then for any 𝑥∈𝑃∩𝜕Ω₁, then ‖𝑥‖=𝑅₁ and 𝑥(𝑠)−𝑣(𝑠)≤𝑥(𝑠)≤‖𝑥‖, we have ‖𝑇𝑥(𝑡)‖≤𝜆^𝜎(1)_𝜌(0)𝐶₁𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)[𝑓(𝑠,[𝑥(𝑠)−𝑣(𝑠)]^∗)+𝑔(𝑠)]∇𝑠≤𝜆^𝜎(1)_𝜌(0)𝐶₁𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)[𝑓(𝑠,[𝑥(𝑠)−𝑣(𝑠)]^∗)+𝑔(𝑠)]∇𝑠≤𝜆𝐶₁^𝜎(1)_𝜌(0)𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)[max_{0≤𝑧≤𝑅1}𝑓(𝑠,𝑧)+𝑔(𝑠)]∇𝑠≤𝑅₁=‖𝑥‖. (3.19) This implies that ‖𝑇𝑥‖≤‖𝑥‖,𝑥∈𝑃∩𝜕Ω₁. (3.20)
On the other hand, choose a constant 𝑁>0 such that 𝜆𝐶²₂𝑁2𝐶₁^𝜃2_𝜃1𝐻(𝑠,𝑠)𝜙₁(𝑠)𝑝(𝑠)𝑞(𝑠)∇𝑠min_{𝜃1≤𝑡≤𝜃2}𝜙₁(𝑡)≥1. (3.21) By assumption (H3), for any 𝑡∈[𝜃₁,𝜃₂], there exists a constant 𝐵>0 such that 𝑓(𝑡,𝑧)𝑧>𝑁,namely,𝑓(𝑡,𝑧)>𝑁𝑧,for𝑧>𝐵. (3.22) Choose 𝑅₂=max{𝑅₁+1,2𝜆𝑟,2𝐶₁(𝐵+1)/𝐶₂min_{𝜃1≤𝑡≤𝜃2}𝜙₁(𝑡)}, and let Ω₂={𝑥∈𝑃∶‖𝑥‖<𝑅₂}, then for any 𝑥∈𝑃∩𝜕Ω₂, we have 𝑥(𝑡)−𝑣(𝑡)=𝑥(𝑡)−𝜆^𝜎(1)_𝜌(0)𝐺(𝑡,𝑠)𝑝(𝑠)𝑞(𝑠)𝑔(𝑠)∇𝑠≥𝑥(𝑡)−𝜆𝐶₃𝜙₁(𝑡)^𝜎(1)_𝜌(0)𝐶𝑝(𝑠)𝑞(𝑠)𝑔(𝑠)∇𝑠≥𝑥(𝑡)−¹𝑥(𝑡)𝐶₂‖𝑥‖𝜆𝐶₃^𝜎(1)_𝜌(0)𝑝(𝑠)𝑞(𝑠)𝑔(𝑠)∇𝑠≥𝑥(𝑡)−𝑥(𝑡)𝑅₂𝜆𝑟≥(1−𝜆𝑟𝑅₂≥1)𝑥(𝑡)2𝑥(𝑡)≥0,𝑡∈[0,1]. (3.23) Then, min_{𝜃1≤𝑡≤𝜃2}{𝑥(𝑡)−𝑣(𝑡)}≥min_{𝜃1≤𝑡≤𝜃2}{12𝑥(𝑡)}≥min_{𝜃1≤𝑡≤𝜃2}{𝐶₂2𝐶₁𝜙₁=𝑅(𝑡)‖𝑥‖}₂𝐶₂min_{𝜃1≤𝑡≤𝜃2}𝜙₁(𝑡)2𝐶₁≥𝐵+1>𝐵. (3.24) Now, ‖𝑇𝑥(𝑡)‖≥max_0≤𝑡≤1𝜆^𝜎(1)_𝜌(0)𝐶₂𝜙₁(𝑡)𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)[𝑓(𝑠,[𝑥(𝑠)−𝑣(𝑠)]^∗)+𝑔(𝑠)]∇𝑠≥max_0≤𝑡≤1𝜆𝐶₂𝜙₁(𝑡)^𝜎(1)_𝜌(0)𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)𝑓(𝑠,[𝑥(𝑠)−𝑣(𝑠)]^∗)∇𝑠≥𝜆𝐶₂min_{𝜃1≤𝑡≤𝜃2}𝜙₁(𝑡)^𝜃2_𝜃1𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)𝑓(𝑠,𝑥(𝑠)−𝑣(𝑠))∇𝑠≥𝜆𝐶₂min_{𝜃1≤𝑡≤𝜃2}𝜙₁(𝑡)^𝜃2_𝜃1𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)𝑁(𝑥(𝑠)−𝑣(𝑠))∇𝑠≥𝜆𝐶₂min_{𝜃1≤𝑡≤𝜃2}𝜙₁(𝑡)^𝜃2_𝜃1𝑁𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)2𝑥(𝑠)∇𝑠≥𝜆𝐶²₂𝑁2𝐶₁min_{𝜃1≤𝑡≤𝜃2}𝜙₁(𝑡)^𝜃2_𝜃1𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)𝜙₁(𝑠)‖𝑥‖∇𝑠≥𝜆𝐶²₂𝑁2𝐶₁min_{𝜃1≤𝑡≤𝜃2}𝜙₁(𝑡)^𝜃2_𝜃1𝐻(𝑠,𝑠)𝜙₁(𝑠)𝑝(𝑠)𝑞(𝑠)∇𝑠‖𝑥‖≥‖𝑥‖.⇒‖𝑇𝑥‖≥‖𝑥‖,𝑥∈𝑃∩𝜕Ω₂. (3.25)(14)(15)(16)(17)(18)(19)(20)(21)
Condition (2.1) of Krasnosel'skii’s fixed-point theorem is satisfied. So 𝑇 has a fixed point 𝑥 with 𝑅₁≤‖𝑥‖<𝑅₂ such that 𝑥^Δ∇(𝑡)+𝑎(𝑡)𝑥^Δ(𝑡)+𝑏(𝑡)𝑥(𝑡)=−𝜆𝑞(𝑠)(𝑓(𝑠,[𝑥(𝑠)−𝑣(𝑠)]^∗)+𝑔(𝑠)),0<𝑡<1,𝑥(𝜌(0))=0,𝑥(𝜎(1))=^𝑚−2_𝑖=1𝛼_𝑖𝑥(𝜂_𝑖). (3.26) Since 𝑟<‖𝑥‖, 𝐶𝑥(𝑡)−𝑣(𝑡)≥²𝐶₁𝜙₁(𝑡)‖𝑥‖−𝜆^𝜎(1)_𝜌(0)≥𝐶𝐺(𝑡,𝑠)𝑝(𝑠)𝑞(𝑠)𝑔(𝑠)∇𝑠₂𝐶₁𝜙₁(𝑡)‖𝑥‖−𝜙₁(𝑡)𝜆𝐶₃^𝜎(1)_𝜌(0)≥𝐶𝑝(𝑠)𝑞(𝑠)𝑔(𝑠)∇𝑠₂𝐶₁𝜙₁𝐶(𝑡)‖𝑥‖−²𝐶₁𝜙₁≥𝐶(𝑡)𝜆𝑟₂𝐶₁𝜙₁𝐶(𝑡)𝑟−²𝐶₁𝜙₁𝐶(𝑡)𝜆𝑟≥(1−𝜆)²𝐶₁𝑟𝜙₁(𝑡)>0. (3.27) Let 𝑢(𝑡)=𝑥(𝑡)−𝑣(𝑡), then 𝑢(𝑡) is a positive solution of boundary value problem (1.1). This completes the proof of Theorem 3.2.

Since condition (H1) implies conditions (H∗1) and (H4), and from proof of Theorems 3.1 and 3.2, we immediately have the following theorem.

Theorem 3.3. Suppose that (H1)–(H3) hold. Then boundary value problem (1.1) has at least two positive solutions for 𝜆>0 sufficiently small.

Proof. On the hand, fix 𝛿∈(0,1). From (H2), let 0<𝜀<min{1,𝑟/2} be such that 𝑓(𝑡,𝑧)≥𝛿𝑓(𝑡,0),for𝜌(0)≤𝑡≤𝜎(1),0≤𝑧≤𝜀. (3.28) Choose 𝜀2𝑐𝑓(𝜀)∶=𝜆, (3.29) where 𝑓(𝜀)=max_{𝜌(0)≤𝑡≤𝜎(1),0≤𝑧≤𝜀}{𝑓(𝑡,𝑧)+𝑔(𝑡)}, 𝑐=𝐶₁∫^𝜎(1)_𝜌(0)𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)∇𝑠, and 𝑟=(𝐶₁𝐶₃/𝐶₂)∫^𝜎(1)_𝜌(0)𝑞(𝑠)𝑔(𝑠)∇𝑠.
On the other hand, set Ω₁={𝑥∈𝑃∶‖𝑥‖<𝑅₁}, where 𝑅₁=1+𝑟. Choose 𝜆^∗=min{1,𝑅₁[𝐶₁^𝜎(1)_𝜌(0)𝐻(𝑠,𝑠)𝑝(𝑠)𝑞(𝑠)[max_{0≤𝑧≤𝑅1}𝑓(𝑠,𝑧)+𝑔(𝑠)]∇𝑠]⁻¹}. (3.30)
So, let 0<𝜆<min{𝜆,𝜆^∗}. (3.31)
From 0<𝜆<min{𝜆,𝜆^∗}, we have 0<𝜆<𝜆, from proof of Theorem 3.1, we know that (1.1) has a positive solution 𝑢₁ and ‖𝑢₁‖≤‖𝑢₁‖≤𝑅₀<𝑟/2. Further, also from 0<𝜆<min{𝜆,𝜆^∗}, we have 0<𝜆<𝜆^∗, from proof of Theorem 3.2, we know that (1.1) has a positive solution 𝑢₂ and ‖𝑢₂‖≥𝑅₁/2>𝑟/2. Then (1.1) has at least two positive solutions 𝑢₁ and 𝑢₂. This completes the proof of Theorem 3.3.

4. Examples

To illustrate the usefulness of the results, we give some examples.

Example 4.1. Consider the boundary value problem 𝑢^Δ∇(𝑡)+𝑎(𝑡)𝑢^Δ(𝑡)+𝑏(𝑡)𝑢(𝑡)+𝜆(𝑢^𝑎1(𝑡)+(𝑡−𝑡²)^1/2cos(2𝜋𝑢(𝑡)))=0,𝜆>0,𝑢(𝜌(0))=0,𝑢(𝜎(1))=^𝑚−2_𝑖=1𝛼_𝑖𝑢(𝜂_𝑖), (4.1) where 𝑎>1. Then, if 𝜆>0 is sufficiently small, (4.1) has a positive solution 𝑢 with 𝑢(𝑡)>0 for 𝑡∈(0,1).
To see this, we will apply Theorem 3.2 with 𝑞(𝑡)=1,𝑓(𝑡,𝑢)=𝑢^𝑎1(𝑡)+(𝑡−𝑡²)^1/21cos(2𝜋𝑢(𝑡)),𝑔(𝑡)=(𝑡−𝑡²)^1/2. (4.2) Clearly 1𝑓(𝑡,0)=(𝑡−𝑡²)^1/2>0,𝑓(𝑡,𝑢)+𝑔(𝑡)≥𝑢^𝑎(𝑡)>0,lim_𝑢↑+∞𝑓(𝑡,𝑢)𝑢=+∞, (4.3) for 𝑡∈(𝜌(0),𝜎(1)),𝑢>0. Namely, (H∗1) and (H2)–(H4) hold. From ∫^𝜎(1)_𝜌(0)(1/((𝜎(1)−𝜌(0))𝑠−(𝑠−𝜌(0))²)^1/2)∇𝑠=𝜋, set 𝑅₁=𝐶₁𝜋 and 𝑚=max_{𝜌(0)≤𝑡≤𝜎(1)}{𝑝(𝑡)}+1, we have ^𝜎(1)_𝜌(0)𝐶₁𝐻(𝑠,𝑠)𝑝(𝑠)[max_{0≤𝑧≤𝑅1}≤𝑓(𝑠,𝑧)+𝑔(𝑠)]∇𝑠_{𝜎(1)𝜌(0)}𝑚𝐶₁‖𝜙₁‖‖𝜙₂‖𝜙^Δ₁(𝜌(0))[(𝐶₁𝜋)^𝑎+1((𝜎(1)−𝜌(0))𝑠−(𝑠−𝜌(0))²)^1/2≤]∇𝑠𝑚𝐶₁‖𝜙₁‖‖𝜙₂‖𝜙^Δ₁(𝜌(0))((𝐶₁𝜋)^𝑎+𝜋) (4.4) and 𝜆^∗=min{1,𝜙^Δ₁(𝜌(0))/𝑚‖𝜙₁‖‖𝜙₂‖((𝐶^𝑎₁𝜋^𝑎−1+1)}. Now, if 𝜆<𝜆^∗, Theorem 3.2 guarantees that (4.1) has a positive solution 𝑢 with ‖𝑢‖≥2.

Example 4.2. Consider the boundary value problem 𝑢^Δ∇(𝑡)+𝑎(𝑡)𝑢^Δ(𝑡)+𝑏(𝑡)𝑢(𝑡)+𝜆(𝑢²9(𝑡)−2𝑢(𝑡)+2)=0,0<𝑡<1,𝜆>0,𝑢(𝜌(0))=0,𝑢(𝜎(1))=^𝑚−2_𝑖=1𝛼_𝑖𝑢(𝜂_𝑖). (4.5) Then, if 𝜆>0 is sufficiently small, (4.5) has two solutions 𝑢_𝑖 with 𝑢_𝑖(𝑡)>0 for 𝑡∈(0,1),𝑖=1,2.
To see this, we will apply Theorem 3.3 with 𝑓(𝑡,𝑢)=𝑢²9(𝑡)−2𝑢(𝑡)+2,𝑔(𝑡)=4. (4.6) Clearly 𝑞(𝑡)=0,𝑓(𝑡,0)=2>0,𝑓(𝑡,𝑢)+𝑔(𝑡)≥1516>0,lim_𝑢↑+∞𝑓(𝑡,𝑢)𝑢=+∞. (4.7) Namely, (H1)–(H3) hold. Let 𝛿=1/4, 𝜀=1/4 and 𝑐=∫^𝜎(1)_𝜌(0)𝐶₁𝐻(𝑠,𝑠)𝑝(𝑠)∇𝑠, then we may have 1𝜆=8𝑐(max_{0≤𝑧≤𝜀}=1𝑓(𝑡,𝑧)+4).48𝑐 (4.8) Now, if 𝜆<𝜆, Theorem 3.2 guarantees that (4.5) has a positive solution 𝑢₁ with ‖𝑢₁‖≤1/4.
Next, let 𝑅₁=𝑝, where 𝑝=4𝑐𝐶₃/𝐶₂+1 is a constant, then we have ^𝜎(1)_𝜌(0)𝐶₁𝐻(𝑠,𝑠)𝑝(𝑠)[max_{0≤𝑧≤𝑅1}𝑓(𝑠,𝑧)+𝑔(𝑠)]∇𝑠=^𝜎(1)_𝜌(0)𝐶₁9𝐻(𝑠,𝑠)𝑝(𝑠)[2+4]∇𝑠=17𝑐2 (4.9) and 𝜆^∗=min{1,2𝑝/17𝑐}. Now, if 0<𝜆<𝜆^∗, Theorem 3.1 guarantees that (4.5) has a positive solution 𝑢₂ with ‖𝑢₂‖≥𝑝.
So, since all the conditions of Theorem 3.3 are satisfied, if 𝜆<min{𝜆,𝜆^∗}, Theorem 3.3 guarantees that (4.5) has two solutions 𝑢_𝑖 with 𝑢_𝑖(𝑡)>0  (𝑖=1,2).

Example 4.3. Consider the boundary value problem 𝑢^Δ∇(𝑡)+𝑎(𝑡)𝑢^Δ(𝑡)+𝑏(𝑡)𝑢(𝑡)+𝜆(𝑢^𝑎(𝑡)+cos(2𝜋𝑢(𝑡)))=0,0<𝑡<1,𝜆>0,𝑢(𝜌(0))=0,𝑢(𝜎(1))=^𝑚−2_𝑖=1𝛼_𝑖𝑢(𝜂_𝑖), (4.10) where 𝑎>1. Then, if 𝜆>0 is sufficiently small, (4.10) has two solutions 𝑢_𝑖 with 𝑢_𝑖(𝑡)>0 for 𝑡∈(0,1),𝑖=1,2.
To see this we will apply Theorem 3.3 with 𝑓(𝑡,𝑢)=𝑢^𝑎(𝑡)+cos(2𝜋𝑢(𝑡)),𝑔(𝑡)=2. (4.11) Clearly 𝑓(𝑡,0)=1>0,𝑓(𝑡,𝑢)+𝑔(𝑡)≥𝑢^𝑎(𝑡)+1>0,lim_𝑢↑+∞𝑓(𝑡,𝑢)𝑢=+∞,for𝑡∈(𝜌(0),𝜎(1)). (4.12) Namely, (H1)–(H3) hold. Let 𝛿=1/2, 𝜀=1/8 and 𝑐=∫^𝜎(1)_𝜌(0)𝐶₁𝐻(𝑠,𝑠)𝑝(𝑠)∇𝑠, then we may have 𝜀2𝑐(max_{0≤𝑥≤𝜀}≥1𝑓(𝑡,𝑥)+2)16𝑐((1/8)^𝑎+3)∶=𝜆. (4.13) Now, if 0<𝜆<𝜆 then 0<𝜆<𝜀/2𝑐(max_{0≤𝑥≤𝜀}𝑓(𝑡,𝑥)+2), Theorem 3.2 guarantees that (4.10) has a positive solution 𝑢₁ with ‖𝑢₁‖≤1/8.
Next, let 𝑅₁=𝑝, where 𝑝=4𝑐𝐶₃/𝐶₂+1 is a constant, then we have ^𝜎(1)_𝜌(0)𝐶₁𝐻(𝑠,𝑠)𝑝(𝑠)[max_{0≤𝑧≤𝑅1}𝑓(𝑠,𝑧)+𝑔(𝑠)]∇𝑠≤^𝜎(1)_𝜌(0)𝐶₁𝐻(𝑠,𝑠)𝑝(𝑠)[𝑝^𝑎+2]∇𝑠=(𝑝^𝑎+2)𝑐 (4.14) and 𝜆^∗=min{1,𝑝/(𝑝^𝑎+2)𝑐}. Now, if 0<𝜆<𝜆^∗, then 0<𝜆<𝑝(∫^𝜎(1)_𝜌(0)𝐶₁𝐻(𝑠,𝑠)𝑝(𝑠)[max_{0≤𝑧≤𝑅1}𝑓(𝑠,𝑧)+𝑔(𝑠)]∇𝑠)⁻¹, Theorem 3.1 guarantees that (4.10) has a positive solution 𝑢₂ with ‖𝑢₂‖≥1.
So, if 𝜆<min{𝜆,𝜆^∗}, Theorem 3.3 guarantees that (4.10) has two solutions 𝑢_𝑖 with 𝑢_𝑖(𝑡)>0  (𝑖=1,2).

Example 4.4. Let 𝕋={0,1/4,2/4,3/4,1,5/4,…}. We consider the following four point boundary value problem: 𝑢^Δ∇(𝑡)+125𝑢^Δ(𝑡)−165𝑢(𝑡)+𝜆(𝑢^𝑎5(𝑡)+cos(2𝜋𝑢(𝑡)))=0,𝜆>0,𝑢(0)=0,𝑢(41)=21𝑢(41)+41𝑢(2), (4.15) where 𝑎(𝑡)=12/5, 𝑏(𝑡)=−16/5, and 𝑞(𝑡)=1. Then, if 𝜆>0 is sufficiently small, (4.15) has two solutions 𝑢_𝑖 with 𝑢_𝑖>0 (𝑖=1,2).
Let 𝜙₁ and 𝜙₂ be the solutions of the following linear boundary value problems, respectively, 𝑢^Δ∇(𝑡)+125𝑢^Δ(𝑡)−1655𝑢(𝑡)=0,𝑢(0)=0,𝑢(4𝑢)=1,_Δ∇(𝑡)+125𝑢^Δ(𝑡)−1655𝑢(𝑡)=0,𝑢(0)=1,𝑢(4)=0. (4.16) It is evident (form the Corollaries 4.24 and 4.25 and Theorem 4.28 of [17]) that 𝜙₁(𝑡)=(5/4)^4𝑡−(1/2)^4𝑡(5/4)⁵−(1/2)⁵,𝜙₂(𝑡)=(5/4)⁵(1/2)^4𝑡−(1/2)⁵(5/4)^4𝑡(5/4)⁵−(1/2)⁵. (4.17) Also 𝜙₁ satisfies (C3). Green's function is 𝐻(𝑡,𝑠)=1024(592794)^4𝑠1−(2)^4𝑠(54)⁵(12)^4𝑡1−(2)⁵(54)^4𝑡(5,𝑠≤𝑡,4)^4𝑡1−(2)^4𝑡(54)⁵(12)^4𝑠1−(2)⁵(54)^4𝑠,𝑡≤𝑠, (4.18) and 𝑝(𝑡)=(2/5)^4𝑡−1 follows from 𝑒_𝛼(𝑡,𝑡₀)=(1+𝛼ℎ)^{(𝑡−𝑡0)/ℎ} on 𝕋=ℎℕ.
To see this, we will apply Theorem 3.3 with 𝑓(𝑡,𝑢)=𝑢^𝑎(𝑡)+cos(2𝜋𝑢(𝑡)),𝑔(𝑡)=2. (4.19) Clearly 𝑓(𝑡,0)=1>0,𝑓(𝑡,𝑢)+𝑔(𝑡)≥𝑢^𝑎(𝑡)+1>0,lim_𝑢↑+∞𝑓(𝑡,𝑢)𝑢=+∞,for𝑡∈(𝜌(0),𝜎(1)). (4.20) Namely, (H1)–(H3) hold. Let 𝛿=1/2, 𝜀=1/8 and 𝑐=∫^𝜎(1)_𝜌(0)𝐶₁𝐻(𝑠,𝑠)𝑝(𝑠)∇𝑠, then we may have 𝜀2𝑐(max_{0≤𝑥≤𝜀}≥1𝑓(𝑡,𝑥)+2)16𝑐((1/8)^𝑎+3)∶=𝜆. (4.21) Now, if 0<𝜆<𝜆 then 0<𝜆<𝜀/2𝑐(max_{0≤𝑥≤𝜀}𝑓(𝑡,𝑥)+2), Theorem 3.2 guarantees that (4.15) has a positive solution 𝑢₁ with ‖𝑢₁‖≤1/8.
Next, let 𝑅₁=4𝑐𝐶₃/𝐶₂+1 is a constant, then we have ^𝜎(1)_𝜌(0)[max_{0≤𝑧≤𝑅1}𝑓(𝑠,𝑧)+𝑔(𝑠)]∇𝑠≤^𝜎(1)_𝜌(0)𝐶₁𝐻(𝑠,𝑠)𝑝(𝑠)[𝑅^𝑎₁+2]∇𝑠=(𝑝^𝑎+2)𝑐 (4.22) and 𝜆^∗=min{1,𝑅₁/(𝑅^𝑎₁+2)𝑐}. Now, if 0<𝜆<𝜆^∗, then 0<𝜆<𝑅₁(∫^𝜎(1)_𝜌(0)𝐶₁𝐻(𝑠,𝑠)[max_{0≤𝑧≤𝑅1}𝑓(𝑠,𝑧)+𝑔(𝑠)]∇𝑠)⁻¹, Theorem 3.1 guarantees that (4.15) has a positive solution 𝑢₂ with ‖𝑢₂‖≥1.
So, if 𝜆<min{𝜆,𝜆^∗}, Theorem 3.3 guarantees that (4.15) has two solutions 𝑢_𝑖 with 𝑢_𝑖(𝑡)>0  (𝑖=1,2).

Acknowledgments

The work was supported by the Scientific Research Fund of Heilongjiang Provincial Education Department (no. 11544032) and the National Natural Science Foundation of China (no. 10871071).