Existence of Positive Solutions for m-Point Boundary Value Problem for Nonlinear Fractional Differential Equation

Abstract

We investigate an m-point boundary value problem for nonlinear fractional differential equations. The associated Green function for the boundary value problem is given at first, and some useful properties of the Green function are obtained. By using the fixed point theorems of cone expansion and compression of norm type and Leggett-Williams fixed point theorem, the existence of multiple positive solutions is obtained.

1. Introduction

In recent years, the existence of positive solutions multipoint boundary value problems of fractional order differential equations has been studied by many authors using various methods (see [1–7]).

The study of multipoint boundary value problems for linear second-order ordinary differential equations was initiated by II'in and Moiseev [8, 9].

Since then, nonlinear multipoint boundary value problems have been studied by several authors (see [10–14]). Recently, in [15], the authors have studied the existence of at least one positive solution for the following 𝑛th-order three-point boundary value problem:𝑢(𝑎)=𝛼𝑢(𝜂),𝑢𝑢^(𝑛)_(𝑡)+ℎ(𝑡)𝑓(𝑡,𝑢(𝑡))=0,𝑡∈[𝑎,𝑏],(𝑎)=𝑢^(𝑎)=⋯=𝑢^(𝑛−2)(𝑎)=0,𝑢(𝑏)=𝛽𝑢(𝜂),(1.1) where 𝑎<𝜂<𝑏,0≤𝛼<1,𝑓∈𝐶([𝑎,𝑏]×[0,∞),[0,∞)) and ℎ∈𝐶([𝑎,𝑏]×[0,∞)) may be singular at 𝑡=𝑎 and 𝑡=𝑏.

Goodrich [16] considered the BVP for thehigher-dimensional fractional differential equation as follows:−𝐷^𝜈0₊𝑦(𝑡)=𝑦𝑓(𝑡,𝑦(𝑡)),0<𝑡<1,𝑛−1<𝜈≤𝑛,_(𝑖)𝐷(0)=0,0≤𝑖≤𝑛−2,_𝛼0+𝑦(𝑡)_𝑡=1=0,1≤𝛼≤𝑛−2,(1.2) and a Harnack-like inequality associated with the Green function related to the above problem is obtained improving the results in [17].

Motivated by the aforementioned results and techniques in coping with those boundary value problems of fractional differential equations, we then turn to investigate the existence and multiplicity of positive solutions for the following BVP:𝑢𝐷^𝐶𝛼𝑎₊𝑢(𝑡)+𝑓(𝑡,𝑢(𝑡))=0,𝑎≤𝑡≤𝑏,𝑛−1≤𝛼<𝑛,𝑛>2,(1.3)_(𝑎)=^𝑚−2_𝑖=1𝛽_𝑖𝑢^𝜂_𝑖,𝑢^(𝑎)=𝑢^(𝑎)=⋯=𝑢^(𝑛−1)(𝑎)=0,𝑢(𝑏)=^𝑚−2_𝑖=1𝛾_𝑖𝑢𝜂_𝑖,(1.4) where 𝑎<𝜂₁<𝜂₂<⋯<𝜂_𝑚−2<𝑏,∑^{𝑚−2𝑖=1}𝛽_𝑖<1,∑^{𝑚−2𝑖=1}𝛾_𝑖<1 and ^𝐶𝐷^𝛼𝑎₊ are the Caputo fractional derivative.

In this paper, we study the existence of at least one positive solution, existence of two positive solutions associated with the BVP (1.3)-(1.4) by applying the fixed point theorems of cone expansion and compression of norm type, and the existence of at least three positive solutions for BVP (1.3)-(1.4) by using Leggett-Williams fixed point theorem.

The rest of the paper is organized as follows. In Section 2, we introduce some basic definitions and preliminaries used later. In Section 3, the existence of multipoint boundary value problem (1.3)-(1.4) will be discussed.

2. Preliminaries

In this section, we introduce definitions and preliminary facts which are used throughout this paper.

Definition 2.1 (see [18]). For a function 𝑦∶(𝑎,∞)→𝑅, the Caputo derivative of fractional order 𝛼>0 is defined as ^𝐶𝐷^𝛼𝑎₊𝑦(𝑡)=1Γ(𝑛−𝛼)^𝑡𝑎(𝑡−𝑠)^{𝑛−𝛼−1}𝑦^(𝑛)(𝑠)𝑑𝑠,𝑛−1<𝛼≤𝑛.(2.1)

Definition 2.2 (see [18]). The standard Riemann-Liouville fractional derivative of order 𝛼>0 of a continuous function 𝑦∶(𝑎,∞)→𝑅 is given by 𝐷^𝛼𝑎₊𝑦(𝑡)=1Γ(𝑛−𝛼)𝑑𝑑𝑡^𝑛^𝑡𝑎(𝑡−𝑠)^{𝑛−𝛼−1}𝑦(𝑠)𝑑𝑠,(2.2) where 𝑛=[𝛼]+1, provided that the integral on the right-hand side converges.

Definition 2.3 (see [18]). The Riemann-Liouville fractional integral of order 𝛼>0 of a function 𝑦∶(𝑎,∞)→𝑅 is given by 𝐼^𝛼𝑎₊𝑦(𝑡)=1Γ(𝛼)^𝑡𝑎(𝑡−𝑠)^𝛼−1𝑦(𝑠)𝑑𝑠(2.3) provided that the integral on the right-hand side converges.

Definition 2.4 (see [19]). Let 𝐸 be a real Banach space. A nonempty closed convex set 𝐾⊂𝐸 is called cone of 𝐸 if it satisfies the following conditions:(1)𝑥∈𝐾,𝜎≥0implies𝜎𝑥∈𝐾; (2)𝑥∈𝐾,−𝑥∈𝐾implies𝑥=0.

Definition 2.5. An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.

Theorem 2.6 (see [20]). Let 𝐸 be a Banach space and 𝐾⊂𝐸 is a cone in 𝐸. Assume that Ω₁ and Ω₂ are open subsets of 𝐸 with 0∈Ω₁ and Ω₁⊂Ω₂. Let 𝑇∶𝐾∩(Ω₂⧵Ω₁)→𝐾 be completely continuous operator. In addition, suppose either (i)‖𝑇𝑢‖≤‖𝑢‖,forall𝑢∈𝐾∩𝜕Ω₁,and‖𝑇𝑢‖≥‖𝑢‖,forall𝑢∈𝐾∩𝜕Ω₂or(ii)‖𝑇𝑢‖≤‖𝑢‖,forall𝑢∈𝐾∩𝜕Ω₂,and‖𝑇𝑢‖≥‖𝑢‖,forall𝑢∈𝐾∩𝜕Ω₁holds. Then, 𝑇 has a fixed point in 𝐾∩(Ω₂⧵Ω₁).

Lemma 2.7. For 𝛼>0, the general solution of the fractional differential equation ^𝐶𝐷^𝛼𝑎₊𝑢(𝑡)=0 is given by 𝑢(𝑡)=𝑐₀+𝑐₁(𝑡−𝑎)+𝑐₂(𝑡−𝑎)²+⋯+𝑐_𝑛−1(𝑡−𝑎)^𝑛−1,(2.4) where 𝑐_𝑖∈𝑅,𝑖=0,1,2,…,𝑛−1.

Remark 2.8 (see [18]). In view of Lemma 2.7, it follows that 𝐼^𝛼𝑎₊^𝐶𝐷^𝛼𝑎₊𝑢(𝑡)=𝑢(𝑡)+𝑐₀+𝑐₁(𝑡−𝑎)+𝑐₂(𝑡−𝑎)²+⋯+𝑐_𝑛−1(𝑡−𝑎)^𝑛−1,(2.5) for some 𝑐_𝑖∈𝑅,𝑖=0,1,2,…,𝑛−1.

Definition 2.9. The map 𝜃 is said to be a nonnegative continuous concave functional on a cone 𝑃 of a real Banach space 𝐸 provided that 𝜃∶𝑃→[0,∞) is continuous and 𝜃(𝜆𝑥+(1−𝜆)𝑦)≥𝜆𝜃(𝑥)+(1−𝜆)𝜃(𝑦),(2.6) for all 𝑥,𝑦∈𝑃,0≤𝜆≤1.

Lemma 2.10 (see [21]). Let 𝑃 be a cone in a real Banach space 𝐸,𝑃_𝑐={𝑥∈𝑃∶‖𝑥‖<𝑐},𝜃 is a nonnegative continuous concave functional on 𝑃 such that 𝜃(𝑥)≤‖𝑥‖, for all 𝑥∈𝑃_𝑐, and 𝑃(𝜃,𝑏,𝑑)={𝑥∈𝑃∶𝑏≤𝜃(𝑥),𝑥≤𝑑}. Suppose that 𝑇∶𝑃_𝑐→𝑃_𝑐 is completely continuous and there exist positive constants 0<𝑎<𝑏<𝑑≤𝑐 such that (𝐶1){𝑥∈𝑃(𝜃,𝑏,𝑑)∶𝜃(𝑥)>𝑏}≠𝜙and𝜃(𝑥)>𝑏for𝑥∈𝑃(𝜃,𝑏,𝑑), (𝐶2)‖𝑇𝑥‖<𝑎for𝑥∈𝑃_𝑎, (𝐶3)𝜃(𝑇𝑥)>𝑏for𝑥∈𝑃(𝜃,𝑏,𝑑)with‖𝑇𝑥‖>𝑑, then 𝑇 has at least three fixed points 𝑥₁,𝑥₂, and 𝑥₃ with ‖‖𝑥₁‖‖<𝑎,𝑏<𝜃𝑥₂,𝑎<‖‖𝑥₃‖‖with𝜃𝑥₃<𝑏.(2.7)

Lemma 2.11. For a given 𝑦(𝑡)∈𝐶[𝑎,𝑏] and 𝑛−1≤𝛼<𝑛, the unique solution of the boundary value problem 𝑢𝐷^𝐶𝛼𝑎₊𝑢(𝑡)+𝑦(𝑡)=0,𝑎≤𝑡≤𝑏,𝑛−1≤𝛼<𝑛,𝑛>2,𝑛∈𝑁,(2.8)_(𝑎)=^𝑚−2_𝑖=1𝛽_𝑖𝑢^𝜂_𝑖,𝑢^(𝑎)=𝑢^(𝑎)=⋯=𝑢^(𝑛−1)(𝑎)=0,𝑢(𝑏)=^𝑚−2_𝑖=1𝛾_𝑖𝑢𝜂_𝑖,(2.9) is given by 𝑢(𝑡)=^𝑏𝑎𝐺(𝑡,𝑠)𝑦(𝑠)𝑑𝑠+^𝑏𝑎𝐻𝑡,𝑠;𝜂₁,…,𝜂_𝑚−2𝑦(𝑠)𝑑𝑠,(2.10) where 𝐺(𝑡,𝑠)=1Γ(𝛼)⎧⎪⎨⎪⎩(𝑏−𝑠)^𝛼−1(𝑏−𝑠)_𝛼−1−(𝑡−𝑠)^𝛼−1,𝑎≤𝑠≤𝑡≤𝑏,𝐻𝑡,𝑠;𝜂₁,𝑎≤𝑡≤𝑠≤𝑏,,…,𝜂_𝑚−2=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩∑^{𝑚−2𝑖=1}𝛾_𝑖(𝑏−𝑠)^𝛼−1−𝜂_𝑖−𝑠^𝛼−1+𝜇(𝑡)∑_{𝑚−2𝑖=1}𝛿₂Γ(𝛼)𝛽_𝑖𝜂_𝑖−𝑠^𝛼−2𝛿₁𝛿₂Γ(𝛼−1),𝑎≤𝑠≤𝜂^𝑖∑,𝑖=1,2,…,𝑚−2,_{𝑚−2𝑖=1}𝛾_𝑖(𝑏−𝑠)^𝛼−1𝛿₂Γ(𝛼),𝜂^𝑖𝛿≤𝑠≤𝑏,𝑖=1,2,…,𝑚−2,₁=1−^𝑚−2_𝑖=1𝛽_𝑖,𝛿₂=1−^𝑚−2_𝑖=1𝛾_𝑖,𝜇(𝑡)=𝑏−^𝑚−2_𝑖=1𝛾_𝑖𝜂_𝑖−𝛿²𝑡.(2.11)

Proof. Using Remark 2.8, for arbitrary constants 𝑐_𝑖∈𝑅,𝑖=0,1,2,…,𝑛−1, we have 𝑢(𝑡)=−1Γ(𝛼)^𝑡𝑎(𝑡−𝑠)^𝛼−1𝑦(𝑠)𝑑𝑠+𝑐₀+𝑐₁(𝑡−𝑎)+𝑐₂(𝑡−𝑎)²+⋯+𝑐_𝑛−1(𝑡−𝑎)^𝑛−1=−𝐼^𝛼𝑎₊𝑦(𝑡)+𝑐₀+𝑐₁(𝑡−𝑎)+𝑐₂(𝑡−𝑎)²+⋯+𝑐_𝑛−1(𝑡−𝑎)^𝑛−1.(2.12) In view of the relations ^𝐶𝐷^𝛼𝑎₊𝐼^𝛼𝑎₊𝑢(𝑡)=𝑢(𝑡) and 𝐼^𝛼𝑎₊𝐼^𝛽𝑎₊𝑢(𝑡)=𝐼^{𝑎𝛼+𝛽}₊𝑢(𝑡) for 𝛼,𝛽>0, we obtain 𝑢𝑢^_(𝑡)=−𝐼^𝑎𝛼−1₊𝑦(𝑡)+𝑐₁+2𝑐₂(𝑡−𝑎)+⋯+(𝑛−1)𝑐_𝑛−1(𝑡−𝑎)^𝑛−2,𝑢_(𝑛−1)(𝑡)=⋮−𝐼^𝑎𝛼−2₊𝑦(𝑡)+2𝑐₂+⋯+(𝑛−1)(𝑛−2)𝑐_𝑛−1(𝑡−𝑎)^𝑛−3,(𝑡)=−𝐼^{𝑎𝛼−𝑛+1}₊𝑦(𝑡)+(𝑛−1)!𝑐_𝑛−1.(2.13) Applying the boundary conditions (2.9), we find that 1−𝑐_2𝑚−2=𝑐₃=⋯=𝑐_𝑛−1=0,_𝑖=1𝛽_𝑖𝑐¹=−^𝑚−2_𝑖=1𝛽_𝑖𝐼^𝑎𝛼−1₊𝑦𝜂_𝑖,(2.14) then 𝑐₁=−∑^{𝑚−2𝑖=1}𝛽_𝑖𝐼^𝑎𝛼−1₊𝑦(𝜂_𝑖)/𝛿₁, and 𝑐₀=𝐼^𝛼𝑎+𝑦(𝑏)1−∑^{𝑚−2𝑖=1}𝛾_𝑖−∑^{𝑚−2𝑖=1}𝛾_𝑖𝐼^𝛼𝑎₊𝑦𝜂_𝑖1−∑^{𝑚−2𝑖=1}𝛾_𝑖+(𝑏−𝑎)−∑^{𝑚−2𝑖=1}𝛾_𝑖𝜂_𝑖−𝑎∑^{𝑚−2𝑖=1}𝛽_𝑖𝐼^𝑎𝛼−1₊𝑦𝜂_𝑖=𝐼_𝛼𝑎+1−∑^{𝑚−2𝑖=1}𝛽_𝑖1−∑^{𝑚−2𝑖=1}𝛾_𝑖𝑦(𝑏)𝛿₂−∑^{𝑚−2𝑖=1}𝛾_𝑖𝐼^𝛼𝑎₊𝑦𝜂_𝑖𝛿₂+(𝑏−𝑎)−∑^{𝑚−2𝑖=1}𝛾_𝑖𝜂_𝑖−𝑎∑^{𝑚−2𝑖=1}𝛽_𝑖𝐼^𝑎𝛼−1₊𝑦𝜂_𝑖𝛿₁𝛿₂.(2.15) Substituting the values of the constants 𝑐_𝑖,𝑖=0,1,2,…,𝑛−1, in (2.12), we obtain 𝑢(𝑡)=−𝐼^𝛼𝑎₊𝑦(𝑡)+𝐼^𝛼𝑎+𝑦(𝑏)𝛿₂−∑^{𝑚−2𝑖=1}𝛾_𝑖𝐼^𝛼𝑎₊𝑦𝜂_𝑖𝛿₂+(𝑏−𝑎)−∑^{𝑚−2𝑖=1}𝛾_𝑖𝜂_𝑖−𝑎∑^{𝑚−2𝑖=1}𝛽_𝑖𝐼^𝑎𝛼−1₊𝑦𝜂_𝑖𝛿₁𝛿₂−∑^{𝑚−2𝑖=1}𝛽_𝑖𝐼^𝑎𝛼−1₊𝑦𝜂_𝑖𝛿₁=−𝐼_𝛼𝑎+(𝑡−𝑎)𝑦(𝑡)+𝐼^𝛼𝑎+𝑦(𝑏)𝛿₂−∑^{𝑚−2𝑖=1}𝛾_𝑖𝐼^𝛼𝑎₊𝑦𝜂_𝑖𝛿₂+𝜇(𝑡)𝛿₁𝛿_{2𝑚−2𝑖=1}=−𝐼_𝛼𝑎+𝛽_𝑖𝐼^𝑎𝛼−1₊𝑦𝜂_𝑖𝑦(𝑡)+𝐼^𝛼𝑎₊𝑦(𝑏)+1−𝛿²𝛿₂𝐼^𝛼𝑎₊𝑦(𝑏)−∑^{𝑚−2𝑖=1}𝛾_𝑖𝐼^𝛼𝑎₊𝑦𝜂_𝑖𝛿₂+𝜇(𝑡)𝛿₁𝛿_{2𝑚−2𝑖=1}=−_𝑡𝑎𝛽_𝑖𝐼^𝑎𝛼−1₊𝑦𝜂_𝑖(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+^𝑏𝑎(𝑏−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+∑^{𝑚−2𝑖=1}𝛾_𝑖𝛿₂^𝑏𝑎(𝑏−𝑠)^𝛼−1−∑Γ(𝛼)𝑦(𝑠)𝑑𝑠_{𝑚−2𝑖=1}𝛾_𝑖𝛿₂^𝜂𝑖_𝑎𝜂_𝑖−𝑠^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝛿𝜇₁(𝛿𝑡)_{2𝑚−2𝑖=1}𝛽_𝑖^𝜂𝑖_𝑎𝜂_𝑖−𝑠^𝛼−2=−_𝑡𝑎Γ(𝛼−1)𝑦(𝑠)𝑑𝑠(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+^𝑏𝑎(𝑏−𝑠)^𝛼−1+∑Γ(𝛼)𝑦(𝑠)𝑑𝑠_{𝑚−2𝑖=1}𝛾_𝑖𝛿₂^𝜂𝑖_𝑎(𝑏−𝑠)^𝛼−1−𝜂_𝑖−𝑠^𝛼−1+∑Γ(𝛼)𝑦(𝑠)𝑑𝑠_{𝑚−2𝑖=1}𝛾_𝑖𝛿₂^𝑏𝜂_𝑖(𝑏−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝛿𝜇₁(𝛿𝑡)_{2𝑚−2𝑖=1}𝛽_𝑖^𝜂𝑖_𝑎𝜂_𝑖−𝑠^𝛼−2Γ(𝛼−1)𝑦(𝑠)𝑑𝑠.(2.16)

Lemma 2.12. 𝜇(𝑡)=(𝑏−∑^{𝑚−2𝑖=1}𝛾_𝑖𝜂_𝑖)−𝛿₂𝑡≥0, for 𝑡,𝜂_𝑖∈[𝑎,𝑏],𝑖=1,2,…,𝑚−2.

Proof. Wehave𝜇=𝑏−^𝑚−2_𝑖=1𝛾_𝑖𝜂_𝑖−𝛿²≥𝑏−_𝑚−2𝑡_𝑖=1𝛾_𝑖𝜂_𝑖−1−^𝑚−2_𝑖=1𝛾_𝑖≥_𝑚−2𝑏_𝑖=1𝛾_𝑖𝑏−𝜂_𝑖>0.(2.17)

Lemma 2.13. The functions 𝐺(𝑡,𝑠),𝐻(𝑡,𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2) defined by (2.11) satisfy (i)𝐺(𝑡,𝑠)≥0,𝐻(𝑡,𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2)≥0,forall𝑡,𝑠∈[𝑎,𝑏], (ii)min_{𝜏1≤𝑡≤𝜏2}𝐺(𝑡,𝑠)≥𝜏₀max_{𝑎≤𝑡≤𝑏}𝐺(𝑡,𝑠)=𝜏₀𝐺(𝑠,𝑠), for all 𝑡,𝑠∈(𝑎,𝑏),𝑎<𝜏₁<𝜏₂<𝑏,𝜏₀=min_{𝜏1≤𝑡≤𝜏2}𝜑(𝑡)=(𝑏−𝜏₂)/(𝑏−𝑎),(iii)𝑁₂𝑞(𝑠)≤𝐻(𝑡,𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2)≤𝑁₁𝑞(𝑠), where𝑞(𝑠)=(𝑏−𝑠)^𝛼−2𝛿₁𝛿₂Γ(𝛼),𝑁¹=(𝛼−1)𝛿₁(𝑏−𝑎)^𝑚−2_𝑖=1𝛾_𝑖+𝑏^𝑚−2_𝑖=1𝛽_𝑖𝑁₂,=min𝛿¹∑^{𝑚−2𝑖=1}𝛾_𝑖(𝛼−1),𝛿^1𝑚−2_𝑖=1𝛾_𝑖𝑏−𝜂_𝑖,(2.18)(iv)min_{𝜏1≤𝑡≤𝜏2}𝐻(𝑡,𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2)≥𝜏^∗max_{𝑎≤𝑡≤𝑏}𝐻(𝑡,𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2),𝑠∈(𝑎,𝑏), where 𝜏^∗=𝑏−∑^{𝑚−2𝑖=1}𝛾_𝑖𝜂_𝑖−𝛿²𝜏₂𝑏−∑^{𝑚−2𝑖=1}𝛾_𝑖𝜂_𝑖−𝛿²𝑎<1,𝑎<𝜏¹<𝜏₂<𝑏.(2.19)

Proof. It is clear that (i) holds. So, we prove that (ii) is true.
(ii) For 𝛼>1, in view of the expression for 𝐺(𝑡,𝑠), it follows that 𝐺(𝑡,𝑠)≤𝐺(𝑠,𝑠) for all 𝑠,𝑡∈[𝑎,𝑏], where 𝐺(𝑠,𝑠)=(𝑏−𝑠)^𝛼−1/Γ(𝛼).
If 𝑎≤𝑠≤𝑡≤𝑏, we have 𝐺(𝑡,𝑠)𝐺(𝑠,𝑠)=(𝑏−𝑠)^𝛼−1−(𝑡−𝑠)^𝛼−1(𝑏−𝑠)^𝛼−1=(𝑏−𝑠)^𝛼−2(𝑏−𝑠)−(𝑡−𝑠)^𝛼−2(𝑡−𝑠)(𝑏−𝑠)^𝛼−1≥(𝑏−𝑠)^𝛼−2[(𝑏−𝑠)−(𝑡−𝑠)](𝑏−𝑠)^𝛼−1=(𝑏−𝑡)(𝑏−𝑎)∶=𝜑(𝑡).(2.20) If 𝑎≤𝑡≤𝑠≤𝑏, then we have 𝐺(𝑡,𝑠)𝐺(𝑠,𝑠)=1≥((𝑏𝑏−−𝑎𝑡))∶=𝜑(𝑡).(2.21) Thus max_{𝑎≤𝑡≤𝑏}𝐺(𝑡,𝑠)=𝐺(𝑠,𝑠),𝜑(𝑡)𝐺(𝑠,𝑠)≤𝐺(𝑡,𝑠)≤𝐺(𝑠,𝑠),∀𝑡,𝑠∈(𝑎,𝑏).(2.22) Therefore, min_{𝜏1≤𝑡≤𝜏2}𝐺(𝑡,𝑠)≥𝜏₀max_{𝑎≤𝑡≤𝑏}𝐺(𝑡,𝑠)=𝜏₀𝐺(𝑠,𝑠),∀𝑡,𝑠∈(𝑎,𝑏),𝑎<𝜏₁<𝜏₂<𝑏.(2.23)(iii) If 𝑎≤𝑠≤𝜂_𝑖,𝑖=1,2,…,𝑚−2, then 𝐻𝑡,𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2=∑^{𝑚−2𝑖=1}𝛾_𝑖(𝑏−𝑠)^𝛼−1−𝜂_𝑖−𝑠^𝛼−1𝛿₂Γ(𝛼)+𝜇(𝑡)∑^{𝑚−2𝑖=1}𝛽_𝑖𝜂_𝑖−𝑠^𝛼−2≤∑𝛿₁𝛿₂Γ(𝛼−1)_{𝑚−2𝑖=1}𝛾_𝑖(𝑏−𝑠)^𝛼−1𝛿₂Γ(𝛼)+(𝛼−1)𝑏∑^{𝑚−2𝑖=1}𝛽_𝑖𝜂_𝑖−𝑠^𝛼−2=1𝛿₁𝛿₂Γ(𝛼)𝛿₁𝛿₂Γ(𝛼)𝛿^1𝑚−2_𝑖=1𝛾_𝑖(𝑏−𝑠)^𝛼−1+(𝛼−1)𝑏^𝑚−2_𝑖=1𝛽_𝑖𝜂_𝑖−𝑠^𝛼−2≤(𝛼−1)(𝑏−𝑠)_𝛼−2𝛿₁𝛿₂Γ(𝛼)𝛿^1𝑚−2_𝑖=1𝛾_𝑖(𝑏−𝑠)+𝑏^𝑚−2_𝑖=1𝛽_𝑖≤(𝛼−1)(𝑏−𝑠)_𝛼−2𝛿₁𝛿₂Γ(𝛼)𝛿¹(𝑏−𝑎)^𝑚−2_𝑖=1𝛾_𝑖+𝑏^𝑚−2_𝑖=1𝛽_𝑖∶=𝑁¹𝐻𝑡,𝑠;𝜂₁𝑞(𝑠),,𝜂₂,…,𝜂_𝑚−2=∑^{𝑚−2𝑖=1}𝛾_𝑖(𝑏−𝑠)^𝛼−1−𝜂_𝑖−𝑠^𝛼−1𝛿₂Γ(𝛼)+𝜇(𝑡)∑^{𝑚−2𝑖=1}𝛽_𝑖𝜂_𝑖−𝑠^𝛼−2≥∑𝛿₁𝛿₂Γ(𝛼−1)_{𝑚−2𝑖=1}𝛾_𝑖(𝑏−𝑠)^𝛼−1−𝜂_𝑖−𝑠^𝛼−1=∑𝛿₂Γ(𝛼)_{𝑚−2𝑖=1}𝛾_𝑖(𝑏−𝑠)^𝛼−2(𝑏−𝑠)−𝜂_𝑖−𝑠^𝛼−2𝜂_𝑖−𝑠=(𝑏−𝑠)_𝛼−2𝛿₂Γ(𝛼)𝛿₂Γ(𝛼)^𝑚−2_𝑖=1𝛾_𝑖𝑏−𝜂_𝑖≥𝑁₂𝑞(𝑠).(2.24) If 𝜂_𝑖≤𝑠≤𝑏,𝑖=1,2,…,𝑚−2, then we have 𝐻𝑡,𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2=∑^{𝑚−2𝑖=1}𝛾_𝑖(𝑏−𝑠)^𝛼−1=𝛿_1𝑚−2𝛿₂Γ(𝛼)_𝑖=1𝛾_𝑖(𝑏−𝑠)(𝑏−𝑠)^𝛼−2≤𝛿_1𝑚−2𝛿₁𝛿₂Γ(𝛼)_𝑖=1<𝑁₁𝛾_𝑖(𝑏−𝑎)𝑞(𝑠)𝐻𝑡,𝑠;𝜂₁𝑞(𝑠),,𝜂₂,...,𝜂_𝑚−2=∑^{𝑚−2𝑖=1}𝛾_𝑖(𝛼−1)(𝑏−𝑠)^𝛼−1≥∑(𝛼−1)𝛿₂Γ(𝛼)_{𝑚−2𝑖=1}𝛾_𝑖(𝑏−𝑠)^𝛼−2≥𝛿(𝛼−1)𝛿₂Γ(𝛼)₁∑^{𝑚−2𝑖=1}𝛾_𝑖(𝛼−1)𝑞(𝑠)≥𝑁²𝑞(𝑠).(2.25)
(iv) Since 𝜕𝐻(𝑡,𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2)/𝜕𝑡=−(∑^{𝑚−2𝑖=1}𝛽_𝑖(𝜂_𝑖−𝑠)^𝛼−2/𝛿₁Γ(𝛼−1))≤0, then 𝐻(𝑡,𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2) is nonincreasing in 𝑡, so max_{𝑎≤𝑡≤𝑏}𝐻𝑡,𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2==𝐻∑𝑎_{𝑚−2𝑖=1},𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2𝛾_𝑖(𝑏−𝑠)^𝛼−1−𝜂_𝑖−𝑠^𝛼−1+𝑏−∑_{𝑚−2𝑖=1}𝛿₂Γ(𝛼)𝛾_𝑖𝜂_𝑖−𝛿²𝑎∑^{𝑚−2𝑖=1}𝛽_𝑖𝜂_𝑖−𝑠^𝛼−2min_{𝜏1≤𝑡≤𝜏2}𝛿₁𝛿₂Γ(𝛼−1),𝐻𝑡,𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2=∑^{𝑚−2𝑖=1}𝛾_𝑖(𝑏−𝑠)^𝛼−1−𝜂_𝑖−𝑠^𝛼−1+𝑏−∑_{𝑚−2𝑖=1}𝛿₂Γ(𝛼)𝛾_𝑖𝜂_𝑖−𝛿²𝜏₂∑^{𝑚−2𝑖=1}𝛽_𝑖𝜂_𝑖−𝑠^𝛼−2=∑𝛿₁𝛿₂Γ(𝛼−1)_{𝑚−2𝑖=1}𝛾_𝑖(𝑏−𝑠)^𝛼−1−𝜂_𝑖−𝑠^𝛼−1+𝜏_∗𝛿₂Γ(𝛼)𝑏−∑^{𝑚−2𝑖=1}𝛾_𝑖𝜂_𝑖−𝛿²𝑎∑^{𝑚−2𝑖=1}𝛽_𝑖𝜂_𝑖−𝑠^𝛼−2>𝜏_∗𝛿₁𝛿₂Γ(𝛼−1)max_{𝑎≤𝑡≤𝑏}𝐻𝑡;𝜂₁,𝜂₂,…,𝜂_𝑚−2,𝑠.(2.26)

3. Main Results

Let us denote by 𝐸=𝐶[𝑎,𝑏] the Banach space of all continuous real functions on [𝑎,𝑏] endowed with the norm ‖𝑢‖=max_{𝑎≤𝑡≤𝑏}|𝑢(𝑡)| and 𝑃 the cone 𝑃=𝑢∈𝐸∶𝑢≥0,min_{𝜏1≤𝑡≤𝜏2}𝑢(𝑡)≥𝜏‖𝑢‖,𝑡∈[𝑎,𝑏],(3.1) where 𝜏=min{𝜏₀,𝜏^∗}, since 𝜏₀, 𝜏^∗ are constants do not depend on 𝑡.

Let the nonnegative continuous concave functional 𝜃 on the cone 𝑃 be defined by 𝜃(𝑢)=min_{𝜏1≤𝑡≤𝜏2}𝑢(𝑡).

Set 𝑇∶𝑃→𝐸 by𝑇𝑢(𝑡)=^𝑏𝑎𝐺(𝑡,𝑠)+𝐻𝑡,𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠,𝑎≤𝑡≤𝑏,(3.2) where 𝐺(𝑡,𝑠),𝐻(𝑡,𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2) are defined as in Lemma 2.11.

From (3.2) and Lemma 2.13, we havemin_{𝜏1≤𝑡≤𝜏2}(𝑇𝑢(𝑡))≥^𝑏𝑎𝜏⁰≥𝜏‖𝑇𝑢‖𝐺.(𝑠,𝑠)+𝜏^∗max_{𝑎≤𝑡≤𝑏}𝐻𝑡,𝑠;𝜂₁,…,𝜂_𝑚−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠(3.3) Hence, we have 𝑇(𝑃)⊂𝑃.

By standard argument, one can prove that 𝑇∶𝑃→𝑃 is a completely continuous operator.

The Existence of One Positive Solution
We introduce the following definitions: 𝑓𝑓(𝑢)∶=sup_{[]𝑡∈𝑎,𝑏}𝑓(𝑡,𝑢),𝑓(𝑢)∶=inf_{[]𝑡∈𝑎,𝑏}𝑓(𝑡,𝑢),₀=limsup_𝑢→0⁺𝑓(𝑢)𝑢,𝑓⁰=liminf_𝑢→0⁺𝑓(𝑢)𝑓_∞𝑢,=limsup_𝑢→∞𝑓(𝑢)𝑢,𝑓^∞=liminf_𝑢→∞𝑓(𝑢)𝑀=_𝑏𝑎𝑢,𝐺(𝑠,𝑠)+𝑁₁𝑞(𝑠)𝑑𝑠⁻¹,𝑁=^𝜏2_𝜏1𝜏𝐺(𝑠,𝑠)+𝑁₂𝑞(𝑠)𝑑𝑠⁻¹.(3.4)
Theorem 3.1. Let 𝑓(𝑡,𝑢) be continuous on [𝑎,𝑏]×[0,∞)→[0,∞). If there exist two positive constants 𝑟₂>𝑟₁>0 such that (𝐻₁)𝑓(𝑡,𝑢)≤𝑀𝑟₂,for(𝑡,𝑢)∈[𝑎,𝑏]×[0,𝑟₂], (𝐻₂)𝑓(𝑡,𝑢)≥𝑁𝑟₁,for(𝑡,𝑢)∈[𝑎,𝑏]×[0,𝑟₁], then the BVP (1.3)-(1.4) has at least a positive solution.Proof. We know that the operator 𝑇∶𝑃→𝑃 defined by (3.2) is completely continuous.
(a) Let Ω₂={𝑢∈𝐸∶‖𝑢‖<𝑟₂}. For any 𝑢∈𝑃∩𝜕Ω₂, we have ‖𝑢‖=𝑟₂ which implies that 0≤𝑢(𝑡)≤𝑟₂ for every 𝑡∈[𝑎,𝑏]: 𝑇𝑢(𝑡)=≤𝑀^𝑏𝑎𝑟₂𝐺(𝑡,𝑠)+𝐻𝑡,𝑠;𝜂₁,…,𝜂_𝑚−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠^𝑏𝑎𝐺(𝑠,𝑠)+𝑁₁≤𝑟₂𝑞(𝑠)𝑑𝑠=‖𝑢‖,(3.5) which implies that ‖𝑇𝑢‖≤‖𝑢‖,𝑢∈𝑃∩𝜕Ω₂.(3.6)
(b) Let Ω₁={𝑢∈𝐸∶‖𝑢‖<𝑟₁}. For any 𝑡∈[𝜏₁,𝜏₂],𝑢∈𝑃∩𝜕Ω₁. We have 𝑇𝑢(𝑡)=≥^𝑏𝑎𝐺(𝑡,𝑠)+𝐻𝑡,𝑠;𝜂₁,…,𝜂_𝑚−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠_𝑏𝑎≥𝑁𝑟₁𝜑(𝑡)𝐺(𝑠,𝑠)+𝑁₂𝑞(𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠^𝜏2_𝜏1=𝑟₁𝜏𝐺(𝑠,𝑠)+𝑁₂𝑞(𝑠)𝑑𝑠=‖𝑢‖,(3.7) which implies that ‖𝑇𝑢‖≥‖𝑢‖,𝑢∈𝑃∩𝜕Ω₁.(3.8) In view of Theorem 2.6, 𝑇 has a fixed point 𝑢₀∈𝑃∩(Ω₂⧵Ω₁) which is a solution of the BVP (1.3)-(1.4).

The Existence of Two Positive Solutions
Theorem 3.2. Assume that all assumptions of Theorem 3.1, hold. Moreover, one assumes that 𝑓(𝑡,𝑢) also satisfies (𝐻₃)𝑓_∞=∞. Then, the BVP (1.3)-(1.4) has at least two positive solutions.Proof. At first, it follows from condition (𝐻₁) that ‖𝑇𝑢‖≤‖𝑢‖,𝑢∈𝑃∩𝜕Ω₂.(3.9) Further, it follows from condition (𝐻₂) that ‖𝑇𝑢‖≥‖𝑢‖,𝑢∈𝑃∩𝜕Ω₁.(3.10) Finally, since 𝑓_∞=∞, there exists 𝜓>(𝜏²∫^𝜏2_𝜏1[𝐺(𝑠,𝑠)+𝑁₂𝑞(𝑠)]𝑑𝑠)⁻¹ and 𝑟₃>𝑟₂ such that 𝑓(𝑡,𝑢)≥𝜓𝑢(𝑡),𝑡∈[𝑎,𝑏],𝑢≥𝑟₃.(3.11) Let 𝑟^∗=max{2𝑟₂,𝜏⁻¹𝑟₃} and set Ω₃={𝑢∈𝐸∶‖𝑢‖<𝑟^∗}, then 𝑢∈𝑃∩𝜕Ω₃impliesmin_{𝜏1≤𝑡≤𝜏2}𝑢(𝑡)≥𝜏‖𝑢‖≥𝜏𝑟^∗≥𝑟₃, 𝑇𝑢(𝑡)=≥^𝑏𝑎𝐺(𝑡,𝑠)+𝐻𝑡,𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠_𝑏𝑎≥𝜓𝜑_𝑏𝑎(𝑡)𝐺(𝑠,𝑠)+𝑁₂𝑞(𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠≥𝑟_∗𝜑(𝑡)𝐺(𝑠,𝑠)+𝑁₂𝑞(𝑠)𝑢(𝑠)𝑑𝑠𝜓^𝜏2_𝜏1𝜏²𝐺(𝑠,𝑠)+𝑁₂𝑞(𝑠)𝑑𝑠>𝑟^∗=‖𝑢‖.(3.12) Therefore, we have ‖𝑇𝑢‖≥‖𝑢‖,𝑢∈𝑃∩𝜕Ω₃.(3.13) Thus, from (3.6), (3.8), (3.13), and Theorem 2.6, 𝑇 has a fixed point 𝑢₁, in 𝑃∩(Ω₂⧵Ω₁) and a fixed point 𝑢₂, in 𝑃∩(Ω₃⧵Ω₂). Both are positive solutions of BVP (1.3)-(1.4) and satisfy 0<‖‖𝑢₁‖‖<𝑟₂<‖‖𝑢₂‖‖.(3.14)Theorem 3.3. Assume that 𝑓(𝑡,𝑢) be continuous on [𝑎,𝑏]×[0,∞)→[0,∞). If the following assumptions hold: (𝐻₁)𝑓₀>𝜓, (𝐻₂)𝑓_∞>𝜓, (𝐻₃) there exists a constant 𝜌>0 such that 𝑓(𝑡,𝑢)≤𝜌𝑀,(𝑡,𝑢)∈[𝑎,𝑏]×[0,𝜌],then the BVP (1.3)-(1.4) has at least two positive solutions 𝑢₁ and 𝑢₂ such that 0<‖‖𝑢₁‖‖<𝜌<‖‖𝑢₂‖‖.(3.15)Proof. At first, it follows from condition (𝐻₁) that we may choose 𝜌₁∈(0,𝜌) such that 𝑓(𝑡,𝑢)>𝜓𝑢,0<𝑢≤𝜌₁,(3.16) where 𝜓 is defined as in Theorem 3.2. Set Ω₁={𝑢∈𝐸∶‖𝑢‖<𝜌₁}, and 𝑢∈𝑃∩𝜕Ω₁; from (3.2) and Lemma 2.13, for 𝑎≤𝑡≤𝑏, we have 𝑇𝑢(𝑡)=≥^𝑏𝑎𝐺(𝑡,𝑠)+𝐻𝑡,𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠_𝑏𝑎≥𝜓𝜑_𝑏𝑎(𝑡)𝐺(𝑠,𝑠)+𝑁₂𝑞(𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠≥𝜌₁𝜑(𝑡)𝐺(𝑠,𝑠)+𝑁₂𝑞(𝑠)𝑢(𝑠)𝑑𝑠>𝜌₁𝜓^𝜏2_𝜏1𝜏²𝐺(𝑠,𝑠)+𝑁₂𝑞(𝑠)𝑑𝑠=‖𝑢‖.(3.17) Therefore, we have ‖𝑇𝑢‖≥‖𝑢‖,𝑢∈𝑃∩𝜕Ω₁.(3.18) Further, it follows from condition (𝐻₂) that there exists 𝜌₂>𝜌 such that 𝑓(𝑡,𝑢)>𝜓𝑢(𝑡),𝑢≥𝜌₂.(3.19) Let 𝜌^∗=max{2𝜌,𝜏⁻¹𝜌₂}, set Ω₂={𝑢∈𝐸∶‖𝑢‖<𝜌^∗}, then 𝑢∈𝑃∩𝜕Ω₂impliesmin_{𝜏1≤𝑡≤𝜏2}𝑢(𝑡)≥𝜏‖𝑢‖≥𝜏𝜌^∗≥𝜌₂, 𝑇𝑢(𝑡)=≥^𝑏𝑎𝐺(𝑡,𝑠)+𝐻𝑡,𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠_𝑏𝑎≥𝜓𝜑_𝑏𝑎(𝑡)𝐺(𝑠,𝑠)+𝑁₂𝑞(𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠≥𝜌_∗𝜑(𝑡)𝐺(𝑠,𝑠)+𝑁₂𝑞(𝑠)𝑢(𝑠)𝑑𝑠𝜓^𝜏2_𝜏1𝜏²𝐺(𝑠,𝑠)+𝑁₂𝑞(𝑠)𝑑𝑠>𝜌^∗=‖𝑢‖.(3.20) Therefore, we have ‖𝑇𝑢‖≥‖𝑢‖,𝑢∈𝑃∩𝜕Ω₂.(3.21) Finally, let Ω₃={𝑢∈𝐸∶‖𝑢‖<𝜌} and 𝑢∈𝑃∩𝜕Ω₃. By condition (𝐻₃), we have 𝑇𝑢(𝑡)≤≤𝑀^𝑏𝑎𝜌𝐺_𝑏𝑎(𝑠,𝑠)+𝑁₁𝑞(𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠=𝜌=‖𝑢‖𝐺,(𝑠,𝑠)+𝑁₁𝑞(𝑠)𝑑𝑠(3.22) which implies ‖𝑇𝑢‖≤‖𝑢‖,𝑢∈𝑃∩𝜕Ω₃.(3.23) Thus, from (3.18), (3.21), (3.23), and Theorem 2.6, 𝑇 has a fixed point 𝑢₁ in 𝑃∩(Ω₃⧵Ω₁) and a fixed point 𝑢₂, in 𝑃∩(Ω₂⧵Ω₃). Both are positive solutions of BVP (1.3)-(1.4) and satisfy 0<‖‖𝑢₁‖‖<𝜌<‖‖𝑢₂‖‖.(3.24)Theorem 3.4. Assume that 𝑓(𝑡,𝑢) be continuous on [𝑎,𝑏]×[0,∞)→[0,∞). If the following assumptions hold: (𝐻^1)𝑓₀=∞, (𝐻^2)𝑓_∞=∞, (𝐻^3)there exists a constant 𝜌^>0 such that 𝑓(𝑡,𝑢)≤𝜌^𝑀,(𝑡,𝑢)∈[𝑎,𝑏]×[0,𝜌^],then the BVP (1.3)-(1.4) has at least two positive solutions 𝑢₁ and 𝑢₂ such that 0<‖‖𝑢₁‖‖<𝜌^<‖‖𝑢₂‖‖.(3.25) The proof of Theorem 3.4 is very similar to that of Theorem 3.3 and therefore is omitted.Theorem 3.5. Assume that 𝑓(𝑡,𝑢) be continuous on [𝑎,𝑏]×[0,∞)→[0,∞). If the following assumptions hold: (𝐻₁)𝑓⁰<𝑀, (𝐻₂)𝑓^∞<𝑀, (𝐻₃)there exists a constant 𝑙>0 such that 𝑓(𝑡,𝑢)≥𝑁𝑙,(𝑡,𝑢)∈[𝑎,𝑏]×[𝜏𝑙,𝑙],then the BVP (1.3)-(1.4) has at least two positive solutions 𝑢₁ and 𝑢₂ such that 0<‖‖𝑢₁‖‖<𝑙<‖‖𝑢₂‖‖.(3.26)Proof. It follows from condition (𝐻₁) that we may choose 𝜌₃∈(0,𝑙) such that 𝑓(𝑡,𝑢)<𝑀𝑢,0<𝑢≤𝜌₃.(3.27) Set Ω₄={𝑢∈𝐸∶‖𝑢‖<𝜌₃}, and 𝑢∈𝑃∩𝜕Ω₄; from (3.2) and Lemma 2.13, for 𝑎≤𝑡≤𝑏, we have 𝑇𝑢(𝑡)=<𝑀^𝑏𝑎𝐺_𝑏𝑎(𝑡,𝑠)+𝐻𝑡,𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠𝐺(𝑠,𝑠)+𝑁₁𝑞(𝑠)𝑑𝑠⋅‖𝑢‖=𝑀⋅𝑀⁻¹‖𝑢‖=‖𝑢‖.(3.28) Therefore, we have ‖𝑇𝑢‖≤‖𝑢‖,𝑢∈𝑃∩𝜕Ω₄.(3.29) It follows from condition (𝐻₂) that there exists 𝜌₄>𝑙 such that 𝑓(𝑡,𝑢)<𝑀𝑢,𝑢≥𝜌₄,(3.30) and we consider two cases.Case 1. Suppose that 𝑓 is unbounded, there exists 𝑙^∗>𝜌₄ such that 𝑓(𝑡,𝑢)≤𝑓(𝑡,𝑙^∗) for 0<𝑢≤𝑙^∗.Then, for 𝑢∈𝑃 and ‖𝑢‖=𝑙^∗, we have 𝑇𝑢(𝑡)=≤^𝑏𝑎𝐺(𝑡,𝑠)+𝐻𝑡,𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠_𝑏𝑎<𝑀𝑙_∗𝐺(𝑠,𝑠)+𝑁₁𝑞(𝑠)𝑓𝑠,𝑙^∗𝑑𝑠^𝑏𝑎𝐺(𝑠,𝑠)+𝑁₁𝑞(𝑠)𝑑𝑠=𝑙^∗=‖𝑢‖.(3.31)Case 2. If 𝑓 is bounded, that is, 𝑓(𝑡,𝑢)≤𝑘 for all 𝑢∈[0,∞), taking 𝑙^∗≥max{2𝑙,𝑘𝑀⁻¹}, for 𝑢∈𝑃 and ‖𝑢‖=𝑙^∗, then we have 𝑇𝑢(𝑡)=≤𝑘^𝑏𝑎_𝑏𝑎𝐺(𝑡,𝑠)+𝐻𝑡,𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠𝐺(𝑠,𝑠)+𝑁₁𝑞(𝑠)𝑑𝑠=𝑘𝑀⁻¹≤𝑙^∗=‖𝑢‖.(3.32) Hence, in either case, we always may set Ω₅={𝑢∈𝐸∶‖𝑢‖<𝑙^∗} such that ‖𝑇𝑢‖≤‖𝑢‖,𝑢∈𝑃∩𝜕Ω₅.(3.33) Finally, set Ω₆={𝑢∈𝐸∶‖𝑢‖<𝑙}, then 𝑢∈𝑃∩𝜕Ω₆ and min_{𝜏1≤𝑡≤𝜏2}𝑢(𝑡)≥𝜏‖𝑢‖=𝜏𝑙,(3.34) and by condition (𝐻₃) and (3.2), we have 𝑇𝑢(𝑡)=≥𝑁^𝑏𝑎𝑙𝐺_𝑏𝑎(𝑡,𝑠)+𝐻𝑡,𝑠;𝜂₁,𝜂₂,…,𝜂_𝑚−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠≥𝑁𝑙_𝜏2𝜏1𝜑(𝑡)𝐺(𝑠,𝑠)+𝑁₂𝑞(𝑠)𝑑𝑠𝜏𝐺(𝑠,𝑠)+𝑁₂𝑞(𝑠)𝑑𝑠=𝑙=‖𝑢‖.(3.35) Hence, we have ‖𝑇𝑢‖≥‖𝑢‖,𝑢∈𝑃∩𝜕Ω₆.(3.36) Thus, from (3.29), (3.33), (3.36) and Theorem 2.6, 𝑇 has a fixed point 𝑢₁ in 𝑃∩(Ω₆⧵Ω₄) and a fixed point 𝑢₂ in 𝑃∩(Ω₅⧵Ω₆).Both are positive solutions of BVP (1.3)-(1.4) and satisfy 0<‖‖𝑢₁‖‖<𝑙<‖‖𝑢₂‖‖.(3.37)Theorem 3.6. Assume that 𝑓(𝑡,𝑢) be continuous on [𝑎,𝑏]×[0,∞)→[0,∞). If the following assumptions hold:(𝐻^1)𝑓⁰=0, (𝐻^2)𝑓^∞=0,(𝐻^3)there exists a constant 𝜌^>0 such that 𝑓(𝑡,𝑢)≥𝑁𝜌^,(𝑡,𝑢)∈[𝑎,𝑏]×[𝜏𝜌^,𝜌^],then the BVP (1.3)-(1.4) has at least two positive solutions 𝑢₁ and 𝑢₂ such that 0<‖‖𝑢₁‖‖<𝜌^<‖‖𝑢₂‖‖.(3.38) The proof of Theorem 3.6 is very similar to that of Theorem 3.5 and therefore omitted.

The Existence of Three Positive Solutions
Theorem 3.7. Let 𝑓(𝑡,𝑢) be continuous on [𝑎,𝑏]×[0,∞)→[0,∞). If there exist constants 0<𝑎₁<𝑎₂≤𝑎₃ such that the following assumptions (i)𝑓(𝑡,𝑢)<𝑀𝑎₁,(𝑡,𝑢)∈[𝑎,𝑏]×[0,𝑎₁], (ii)𝑓(𝑡,𝑢)≤𝑀𝑎₃,(𝑡,𝑢)∈[𝑎,𝑏]×[0,𝑎₃], (iii)𝑓(𝑡,𝑢)≥𝑁𝑎₂,(𝑡,𝑢)∈[𝜏₁,𝜏₂]×[𝑎₂,𝑎₂/𝜏], hold, then BVP (1.3)-(1.4) has at least three positive solutions 𝑢₁,𝑢₂, and 𝑢₃ with ‖‖𝑢₁‖‖<𝑎₁𝑎₁,𝑎₂<𝜃𝑢₂<‖‖𝑢₂‖‖≤𝑎₃,<‖‖𝑢₃‖‖,𝜃𝑢₃<𝑎₂.(3.39)Proof. We will show that all conditions of Lemma 2.10, are satisfied.
First, if 𝑢∈𝑃_𝑎3, then ‖𝑢‖≤𝑎₃. So, 0≤𝑢(𝑡)≤𝑎₃,𝑡∈[𝑎,𝑏].
By condition (ii), we have 𝑇𝑢(𝑡)=≤^𝑏𝑎𝐺(𝑡,𝑠)+𝐻𝑡,𝑠;𝜂₁,…,𝜂_𝑚−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠_𝑏𝑎=𝑀𝑎₃𝐺(𝑠,𝑠)+𝑁₁𝑞(𝑠)𝑀𝑎₃𝑑𝑠^𝑏𝑎𝐺(𝑠,𝑠)+𝑁₁𝑞(𝑠)𝑑𝑠=𝑎₃,(3.40) which implies that ‖𝑇𝑢‖≤𝑎₃,𝑢∈𝑃_𝑎3. Hence 𝑇∶𝑃_𝑎3→𝑃_𝑎3.
Next, by using the analogous argument, it follows from condition (i) that if 𝑢∈𝑃_𝑎1, then ‖𝑇𝑢‖<𝑎₁.
Choose 𝑢(𝑡)=(𝑎₂+𝑎₂/𝜏)/2,𝑡∈[𝑎,𝑏], it is easy to see that 𝑢(𝑡)=(𝑎₂+𝑎₂/𝜏)/2∈𝑃(𝜃,𝑎₂,𝑎₃),𝜃(𝑢)=(𝑎₂+𝑎₂/𝜏)/2>𝑎₂.
Therefore, {𝑢∈𝑃(𝜃,𝑎₂,𝑎₂/𝜏)∣𝜃(𝑢)>𝑎₂}≠𝜙. On the other hand, if 𝑢∈𝑃(𝜃,𝑎₂,𝑎₂/𝜏), then 𝑎₂≤𝑢(𝑡)≤𝑎₂/𝜏,𝑡∈[𝜏₁,𝜏₂]. By condition (iii), we have 𝑓(𝑡,𝑢(𝑡))≥𝑁𝑎₂.
Hence, 𝜃(𝑇𝑢(𝑡))==mmiinn_{𝜏1≤𝑡≤𝜏2𝜏1≤𝑡≤𝜏2}𝑇𝑢(𝑡)^𝑏𝑎𝐺(𝑡,𝑠)+𝐻𝑡,𝑠;𝜂₁≥𝑁𝑎₂,…,𝜂_𝑚−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠^𝑏𝑎>𝑁𝑎₂𝜏𝐺(𝑠,𝑠)+𝑁₂𝑞(𝑠)𝑑𝑠^𝜏2_𝜏1𝜏𝐺(𝑠,𝑠)+𝑁₂𝑞(𝑠)𝑑𝑠=𝑎₂,(3.41) which implies that 𝜃(𝑇𝑢)>𝑎₂, for 𝑢∈𝑃(𝜃,𝑎₂,𝑎₂/𝜏).
Finally, if 𝑢∈𝑃(𝜃,𝑎₂,𝑎₃) and ‖𝑇𝑢‖>𝑎₂/𝜏, then 𝜃(𝑢)=min_{𝜏1≤𝑡≤𝜏2}𝑇𝑢(𝑡)≥𝜏‖𝑇𝑢‖>𝑎₂.(3.42) Thus, all the conditions of the Leggett-Williams fixed point theorem are satisfied by taking 𝑑=𝑎₂/𝜏. Hence, the BVPs have at least three solutions in 𝑃, that is, three positive solutions 𝑢_𝑖(𝑖=1,2,3) such that ‖‖𝑢₁‖‖<𝑎₁𝑎₁,𝑎₂<𝜃𝑢₂<‖‖𝑢₂‖‖≤𝑎₃,<‖‖𝑢₃‖‖,𝜃𝑢₃<𝑎₂.(3.43)Example 3.8. Consider the problem 𝐷^0(4.2)₊𝑢(𝑡)+13(1+𝑢𝑒^𝑢𝑢)=0,𝑡∈(0,1),_(0)=14𝑢^21,𝑢^(0)=𝑢^(0)=𝑢^(0)=0,𝑢(1)=34𝑢12,(3.44) where 𝛼=4.2, 𝑎=0, 𝑏=1, 𝛽=0.25, 𝛾=0.75, 𝜂=0.5, 𝜏₁=0.25, 𝜏₂=0.75,𝑁₁=2.6,𝑁₂=0.1758, 𝑁=168.9596, and 𝑀=1.6968,𝑓(𝑡,𝑢)=(1/3)(1+𝑢𝑒^𝑢).
Since 𝑓(𝑡,𝑢)=(1/3)(1+𝑢𝑒^𝑢) is a monotone increasing function on [0,∞), we take 𝑟₁=0.001,𝑟₂=0.8. We can get 𝑓𝑓(𝑡(,𝑡𝑢,𝑢))≤≥𝑓𝑓(0(0.8))==00.3.9323638><𝑁𝑀𝑟₁𝑟₂,.(3.45) So, conditions (𝐻₁) and (𝐻₂) hold. By Theorem 3.1, the BVP (3.44) has at least one positive solution.