Monotone and Concave Positive Solutions to a Boundary Value Problem for Higher-Order Fractional Differential Equation

Abstract

We consider boundary value problem for nonlinear fractional differential equation 𝐷^𝛼0+𝑢(𝑡)+𝑓(𝑡,𝑢(𝑡))=0,0<𝑡<1,𝑛−1<𝛼≤𝑛,𝑛>3,𝑢(0)=𝑢′(1)=𝑢^(0)=⋯=𝑢^(𝑛−1)(0)=0, where 𝐷^𝛼0+ denotes the Caputo fractional derivative. By using fixed point theorem, we obtain some new results for the existence and multiplicity of solutions to a higher-order fractional boundary value problem. The interesting point lies in the fact that the solutions here are positive, monotone, and concave.

1. Introduction

In this paper, we deal with the following boundary value problem for higher-order fractional differential equation: 𝐷^𝛼0+𝑢(𝑡)𝑢+(𝑓0)(=𝑡,𝑢𝑢(_𝑡))=0,0<𝑡<1,𝑛−1<𝛼⩽𝑛,𝑛>3,(1)=𝑢^(0)=⋯=𝑢^(𝑛−1)(0)=0,(1.1) where 𝐷^𝛼0+ denotes the Caputo fractional derivative and 𝑓∶[0,1]×[0,+∞)→[0,+∞) is a real function. By using fixed point theorem, some sufficient conditions for existence and multiplicity of solutions to the above boundary value problem are obtained. Moreover, we will show that the solutions obtained here are positive, monotone, and concave.

Fractional differential equations are valuable tools in the modelling of many phenomena in various fields of science and engineering [1–5]. Due to their applications, fractional differential equations have gained considerable attentions and there has been a significant development in the study of existence of solutions, and positive solutions to boundary value problems for fractional differential equations (e.g., [6–9] and references therein).

Some papers are devoted to study the existence of solutions for higher-order fractional boundary value problem. Salem [10] investigated the existence of pseudosolutions for the nonlinear 𝑚-point boundary value problem of fractional type 𝑥𝐷(0^𝛼)𝑥=(𝑡)𝑥+_𝑞(𝑡)𝑓(𝑡,𝑥(𝑡))=0,0<𝑡<1,𝛼∈(𝑛−1,𝑛],𝑛⩾2,(0)=𝑥^(0)=⋯=𝑥^(𝑛−2)(0)=0,𝑥(1)=^𝑚−2_𝑖=1𝜉_𝑖𝑥𝜂_𝑖.(1.2) Zhang [11] considered the existence of positive solutions to the singular boundary value problem for fractional differential equations 𝐷^𝛼0+𝑢(𝑡)+𝑞(𝑡𝑢)𝑓(0)𝑢=,𝑢𝑢^,_…,𝑢^(𝑛−2)=0,0<𝑡<1,𝛼∈(𝑛−1,𝑛],𝑛⩾2,(0)=⋯=𝑢^(𝑛−2)(0)=𝑢^(𝑛−2)(1)=0,(1.3) where 𝐷^𝛼0+ is the Riemann-Liouville fractional derivative of order 𝛼. In another paper, Zhang [9] studied the existence, multiplicity, and nonexistence of positive solutions for the following higher-order fractional boundary value problem: 𝐷^𝛼𝑢+𝑢𝜆(ℎ1)(𝑡=)𝑓𝑢_(𝑢)=0,0<𝑡<1,𝛼∈(𝑛−1,𝑛],𝑛⩾2,(0)=⋯=𝑢^(𝑛−2)(0)=𝑢^(𝑛−1)(0)=0,(1.4) where 𝐷^𝛼 is the Caputo fractional derivative of order 𝛼.

It seems that the authors of the papers only studied the existence of the solutions or positive solutions. No one consider the qualities of the solutions for boundary value problems of fractional differential equation. Motivated by all the above works, the aim of this paper is to study the monotone, concave, and positive solutions of a fractional differential equation.

The rest of the paper is organized as follows. In Section 2, we will introduce some lemmas and definitions which will be used later. In Section 3, the existence and multiplicity of positive solutions for the boundary value problem (1.1) will be discussed. In Section 4, examples are given to check our results.

2. Basic Definitions and Preliminaries

In this section, we introduce some necessary definitions and lemmas, which will be used in the proofs of our main results.

Definition 2.1 (see [12]). The integral 𝐼^𝛼0+𝑦(𝑡)=1Γ(𝛼)^𝑡0(𝑡−𝑠)^𝛼−1𝑦(𝑠)𝑑𝑠,(2.1) where 𝛼>0 is called the Riemann-Liouville fractional integral of order 𝛼.

Definition 2.2 (see [12]). The Caputo fractional derivative for a function 𝑦∶(0,∞)→𝑅 can be written as 𝐷^𝛼0+𝑦(𝑡)=1Γ(𝑛−𝛼)^𝑡0(𝑡−𝑠)^{𝑛−𝛼−1}𝑦^(𝑛)(𝑠)𝑑𝑠,(2.2) where 𝑛=[𝛼]+1, [𝛼] denotes the integer part of real number 𝛼.

According to the definitions of fractional calculus, we can obtain that the fractional integral and the Caputo fractional derivative satisfy the following Lemma.

Lemma 2.3 (see [13]). Assume that 𝑢∈𝐶^𝑚[0,1] and 𝜌∈(𝑚−1,𝑚),𝑚∈𝑁 and 𝑣∈𝐶¹[0,1]. Then, for 𝑡∈[0,1], (a)𝐷^𝜌0+𝐼^𝜌0+𝑣(𝑡)=𝑣(𝑡), (b)𝐼^𝜌0+𝐷^𝜌0+𝑢(𝑡)=𝑢(𝑡)−∑^{𝑚−1𝑘=0}((𝑢^(𝑘)(0))/𝑘!)𝑡^𝑘, (c)lim_𝑡→0⁺𝐷^𝜌0+𝑢(𝑡)=lim_𝑡→0⁺𝐼^𝜌0+𝑢(𝑡)=0.

Definition 2.4. Let 𝐸 be a real Banach space over 𝑅. A nonempty convex closed set 𝑝⊂𝐸 is said to be a cone, provided that(a)𝑎𝑢∈𝑃,forall𝑢∈𝑃,𝑎⩾0, (b)𝑢,−𝑢∈𝑃,implies𝑢=0.

Definition 2.5. Let 𝐸 be a real Banach space and 𝑃⊂𝐸 a cone. A function 𝜑∶𝑃→[0,∞) is called a nonnegative continuous concave functional if 𝜑 is continuous and 𝜑(𝜆𝑥+(1−𝜆)𝑦)⩾𝜆𝜑(𝑥)+(1−𝜆)𝜑(𝑦),(2.3) for all 𝑥,𝑦∈𝑃 and 0⩽𝜆⩽1.

Lemma 2.6 (see [14]). Let 𝐸 be a Banach space, 𝐾⊆𝐸 a cone in 𝐸, and Ω₁, Ω₂ two bounded open subsets of 𝐸 with 0∈Ω₁ and Ω₁⊂Ω₂. Suppose that 𝑇∶𝐾∩(Ω₂⧵Ω₁)→𝐾 is continuous and completely continuous such that either (i)‖𝑇𝑢‖⩽‖𝑢‖for𝑢∈𝐾∩𝜕Ω₁,‖𝑇𝑢‖⩾‖𝑢‖for𝑢∈𝐾∩𝜕Ω₂,or (ii)‖𝑇𝑢‖⩾‖𝑢‖for𝑢∈𝐾∩𝜕Ω₁,‖𝑇𝑢‖⩽‖𝑢‖for𝑢∈𝐾∩𝜕Ω₂holds. Then, 𝑇 has a fixed point in 𝐾∩(Ω₂⧵Ω₁).
Let 𝑏,𝑑,𝑟>0 be constants, 𝑃_𝑟={𝑢∈𝑃∶‖𝑢‖<𝑟}, 𝑃(𝜑,𝑏,𝑑)={𝑢∈𝑃∶𝑏⩽𝜑(𝑢),‖𝑢‖⩽𝑑}.

Lemma 2.7 (see [15]). Let 𝑃 be a cone in real Banach space 𝐸. Let 𝑇∶𝑃_𝑐→𝑃_𝑐 be a completely continuous map and 𝜑 a nonnegative continuous concave functional on 𝑃 such that 𝜑(𝑢)⩽‖𝑢‖, for all 𝑢∈𝑃_𝑐. Suppose that there exist constants 𝑎,𝑏,𝑑 with 0<𝑎<𝑏<𝑑⩽𝑐 such that ((ii)i){‖𝑢𝑇∈𝑢‖𝑃<(𝜑𝑎,𝑏,∀𝑑𝑢)∈∶𝜑(𝑢)>𝑏}≠∅,𝜑(𝑇𝑢)>𝑏∀𝑢∈𝑃(𝜑,𝑏,𝑑),(iii)𝜑(𝑇𝑢)>𝑏,∀𝑢∈𝑃_𝑎𝑃,(𝜑,𝑏,𝑐)with‖𝑇𝑢‖>𝑑.(2.4) Then, 𝑇 has at least three fixed points 𝑢₁, 𝑢₂, and 𝑢₃ satisfying ‖‖𝑢₁‖‖<𝑎,𝑏<𝜑𝑢₂,‖‖𝑢₃‖‖>𝑎,𝜑𝑢₃<𝑏.(2.5)

Lemma 2.8. Assume that 𝑓(𝑡,𝑢)∈𝐶([0,1]×[0,+∞),[0,+∞)), then 𝑢∈𝐶[0,1] be a solution of fractional boundary value problem (1.1) if and only if 𝑢∈𝐶[0,1] is a solution of integral equation 𝑢(𝑡)=¹⁰𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠,(2.6) where 𝐺(𝑡,𝑠)=⎧⎪⎨⎪⎩(𝛼−1)𝑡(1−𝑠)^𝛼−2−(𝑡−𝑠)^𝛼−1(𝛼−1)𝑡(1−Γ𝑠)(𝛼)_𝛼−2,0⩽𝑠⩽𝑡⩽1,Γ(𝛼),0⩽𝑡⩽𝑠⩽1.(2.7)

Proof. Firstly, we prove the necessity. Let 𝑢∈𝐶[0,1] is a solution of fractional boundary value problem (1.1). By Lemma 2.3, we have 𝑢(𝑡)=−𝐼^𝛼0+𝑓(𝑡,𝑢(𝑡))+𝑢(0)+𝑢^(0)𝑡+⋯+𝑢^(𝑛−1)(0)(𝑛−1)!𝑡^𝑛−1=−1Γ(𝛼)^𝑡0(𝑡−𝑠)^𝛼−1𝑓(𝑠,𝑢(𝑠))𝑑𝑠+𝑢^(0)𝑡.(2.8) Therefore, 𝑢^(𝑡)=−1Γ(𝛼)^𝑡0(𝛼−1)(𝑡−𝑠)^𝛼−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠+𝑢^(0).(2.9) By the boundary value condition 𝑢^(1)=0, we have 𝑢^(0)=1Γ(𝛼)¹⁰(𝛼−1)(1−𝑠)^𝛼−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠.(2.10) Hence, we obtain 𝑢(𝑡)=−1Γ(𝛼)^𝑡0(𝑡−𝑠)^𝛼−1𝑓(𝑠,𝑢(𝑠))𝑑𝑠+1Γ(𝛼)¹⁰=(𝛼−1)𝑡(1−𝑠)^𝛼−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠₁₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠.(2.11) The necessity is proved.
Now, we prove the sufficiency. Let 𝑢∈𝐶[0,1] be a solution of integral equation (2.6). Then, we have 𝐷^𝛼0+𝑢(𝑡)=−𝐷^𝛼0+^𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+¹⁰(𝛼−1)(1−𝑠)^𝛼−2Γ(𝛼)𝑓(𝑠,𝑢(𝑠))𝑑𝑠𝐷^𝛼0+=−𝐷_𝛼0+𝑡𝐼^𝛼0+𝑓(𝑡,𝑢(𝑡))=−𝑓(𝑡,𝑢(𝑡)).(2.12) By direct computation, we obtain that 𝑢(0)=𝑢^(1)=𝑢^(0)=⋯=𝑢^(𝑛−1)(0)=0.(2.13) That is to say, 𝑢 is a solution of fractional boundary value problem (1.1). Thus, the sufficiency is proved.

Lemma 2.9. Let 𝑓(𝑡,𝑢(𝑡))∈𝐶([0,1]×[0,∞),[0,∞)). Then, the solution 𝑢(𝑡) of fractional boundary value problem (1.1) satisfies (1)𝑢(𝑡)isconcaveon(0,1), (2)𝑢(𝑡)⩾0isincreasingfor𝑡∈[0,1].

Proof. Suppose that 𝑢(𝑡) is a solution of fractional boundary value problem (1.1). By (2.11), we know that 𝑢(𝑡)=−1Γ(𝛼)^𝑡0(𝑡−𝑠)^𝛼−1𝑓(𝑠,𝑢(𝑠))𝑑𝑠+1Γ(𝛼)¹⁰(𝛼−1)𝑡(1−𝑠)^𝛼−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠.(2.14) Therefore, 𝑢^(𝑡)=−1Γ(𝛼)^𝑡0(𝛼−1)(𝑡−𝑠)^𝛼−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠+1Γ(𝛼)¹⁰𝑢(𝛼−1)(1−𝑠)^𝛼−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠,(2.15)_(𝑡)=−1Γ(𝛼)^𝑡0(𝛼−1)(𝛼−2)(𝑡−𝑠)^𝛼−3𝑓(𝑠,𝑢(𝑠))𝑑𝑠⩽0,for𝑡∈[0,1],𝑛−1<𝛼⩽𝑛,𝑛>3,(2.16) which implies that 𝑢(𝑡) is concave on (0,1). The statement (1) is proved.
Since 𝑢^(𝑡)⩽0, we know that 𝑢′(𝑡) is nonincreasing. By 𝑢^(1)=0, we have 𝑢^(𝑡)⩾0, 𝑡∈[0,1]. Thus, 𝑢(𝑡) is increasing. Noting 𝑢(0)=0, we obtain that 𝑢(𝑡)⩾0 for 𝑡∈[0,1]. The statement (2) is proved.

Lemma 2.10. The Green's function 𝐺(𝑡,𝑠), defined by (2.7), satisfies (1)max_0⩽𝑡⩽1𝐺(𝑡,𝑠)=𝐺(1,𝑠),𝑠∈[0,1], (2)𝐺(𝑡,𝑠)⩾0,𝑡,𝑠∈[0,1], (3)min_{𝜉⩽𝑡⩽𝜂}𝐺(𝑡,𝑠)⩾𝜉^𝛼−1𝐺(1,𝑠),𝑠∈[0,1],forall𝜉,𝜂∈(0,1),𝜉<𝜂.

Proof. By (2.7), we have 𝐺^𝑡(𝑡,𝑠)=⎧⎪⎨⎪⎩(𝛼−1)(1−𝑠)^𝛼−2−(𝛼−1)(𝑡−𝑠)^𝛼−2(𝛼−1)(1−𝑠)Γ_𝛼−2(𝛼),0⩽𝑠⩽𝑡⩽1,Γ(𝛼),0⩽𝑡⩽𝑠⩽1.(2.17) It is clear that 𝐺^𝑡(𝑡,𝑠)⩾0,𝑡,𝑠∈[0,1]. Therefore, 𝐺(𝑡,𝑠) is increasing respect to 𝑡 for 𝑠∈[0,1]. Thus, max_0⩽𝑡⩽1𝐺(𝑡,𝑠)=𝐺(1,𝑠). The statement (1) holds.
If 0⩽𝑡⩽𝑠⩽1, then, 𝐺(𝑡,𝑠)=(𝛼−1)𝑡(1−𝑠)^𝛼−2Γ(𝛼)⩾0.(2.18) Since 𝐺(𝑡,𝑠) is increasing respect to 𝑡 for 𝑠∈[0,1], it is easy to see that 𝐺(𝑡,𝑠)⩾0 for 0⩽𝑠⩽𝑡⩽1. We get the statement (2).
On the other hand, we have min_{𝜉⩽𝑡⩽𝜂}𝐺(𝑡,𝑠)=⎧⎪⎪⎪⎨⎪⎪⎪⎩(𝛼−1)𝜉(1−𝑠)^𝛼−2−(𝜉−𝑠)^𝛼−1(𝛼−1)𝜉(1−Γ𝑠)(𝛼)_𝛼−2,𝑠∈[0,𝜉],(𝛼−1)Γ𝜉((1𝛼)−𝑠)_𝛼−2,𝑠∈[𝜉,𝜂],=⎧⎪⎨⎪⎩(𝛼−1)Γ𝜉((1𝛼)−𝑠)_𝛼−2,𝑠∈[𝜂,1]−(𝜉−𝑠)^𝛼−1(𝛼−1)𝜉(1−Γ𝑠)(𝛼)_𝛼−2,𝑠∈[0,𝜉],Γ(𝛼),𝑠∈[𝜉,1].(2.19) If 𝑠∈[0,𝜉], then (𝛼−1)𝜉(1−𝑠)^𝛼−2−(𝜉−𝑠)^𝛼−1=𝜉(𝛼−1)(1−𝑠)^𝛼−2−𝜉^𝛼−1(1−(𝑠/𝜉))^𝛼−1⩾𝜉^𝛼−1(𝛼−1)(1−𝑠)^𝛼−2−𝜉^𝛼−1(1−𝑠)^𝛼−1=𝜉^𝛼−1(𝛼−1)(1−𝑠)^𝛼−2−(1−𝑠)^𝛼−1.(2.20) If 𝑠∈[𝜉,1], then (𝛼−1)𝜉(1−𝑠)^𝛼−2⩾(𝛼−1)𝜉^𝛼−1(1−𝑠)^𝛼−2⩾𝜉^𝛼−1(𝛼−1)(1−𝑠)^𝛼−2−(1−𝑠)^𝛼−1.(2.21) Thus, min_{𝜉⩽𝑡⩽𝜂}𝐺(𝑡,𝑠)⩾𝜉^𝛼−1(𝛼−1)(1−𝑠)^𝛼−2−(1−𝑠)^𝛼−1=𝜉_𝛼−1Γ(𝛼)𝐺(1,𝑠),𝑠∈[0,1].(2.22) This yields the statement (3). The proof is finished.

3. Main Results

In this section, we establish the results for the existence and multiplicity of monotone and concave positive solutions for fractional boundary value problem (1.1).

Let 𝐸=𝐶[0,1] with ‖𝑢‖=max_0⩽𝑡⩽1|𝑢(𝑡)|. We define the cone 𝑃⊂𝐸 by 𝑃=𝑢∈𝐸∶𝑢(𝑡)⩾0isconcaveon[0,1],min_{𝜉⩽𝑡⩽𝜂}𝑢(𝑡)⩾𝜉^𝛼−1‖𝑢‖.(3.1)

And denote the operator 𝑇 by 𝑇𝑢(𝑡)=¹⁰𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠.(3.2)

Lemma 3.1. Assume that 𝑓(𝑡,𝑢)∈𝐶([0,1]×[0,+∞),[0,+∞)), then 𝑇∶𝑃→𝑃 is completely continuous.

Proof. In view of non-negativeness and continuity of 𝐺(𝑡,𝑠) and 𝑓(𝑡,𝑢(𝑡)), we know that the operator 𝑇 is continuous and 𝑇𝑢(𝑡)⩾0, for 𝑢∈𝑃.
By (2.16), we have (𝑇𝑢)^(𝑡)=−1Γ(𝛼)^𝑡0⩽0,for𝑡∈(𝛼[−0,11)](,𝛼𝑛−−21)(<𝑡−𝛼𝑠⩽)𝑛^𝛼−3,𝑓𝑛(𝑠>,𝑢3.(𝑠))𝑑𝑠(3.3)
Moreover, it follows from Lemma 2.10 that for 𝑢∈𝑃, min_{𝜉⩽𝑡⩽𝜂}𝑇𝑢(𝑡)=min_{𝜉⩽𝑡⩽𝜂}¹⁰⩾𝜉_𝛼−1𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠¹⁰=𝜉_𝛼−1𝐺(1,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠‖𝑇𝑢‖.(3.4) Therefore, the operator 𝑇∶𝑃→𝑃 is well defined.
Assume that Ω∈𝑃 is bounded; that is, there exists a positive constant 𝑀>0 such that ‖𝑢‖⩽𝑀 for all 𝑢∈Ω. Let 𝑁=max_{0⩽𝑡⩽1,‖𝑢‖⩽𝑀}|𝑓(𝑡,𝑢(𝑡))|+1. For all 𝑢∈Ω, we have ||𝑇𝑢(𝑡)||=||||¹⁰𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠||||⩽𝑁¹⁰𝐺(1,𝑠)𝑑𝑠,(3.5) which shows that 𝑇(Ω) is uniformly bounded.
In addition, for each 𝑢∈Ω, 𝑡₁,𝑡₂∈[0,1] such that 𝑡₁<𝑡₂, we have ||𝑇𝑢𝑡₂−𝑇𝑢𝑡₁||⩽=|||||||||¹⁰𝐺𝑡₂,𝑠𝑓(𝑠,𝑢(𝑠))𝑑𝑠−¹⁰𝐺𝑡₁,𝑠𝑓(𝑠,𝑢(𝑠))𝑑𝑠_𝑡10||||𝑡₁−𝑠^𝛼−1−𝑡₂−𝑠^𝛼−1Γ(𝛼)𝑓(𝑠,𝑢(𝑠))𝑑𝑠−^𝑡2_𝑡1𝑡₂−𝑠^𝛼−1+||||Γ(𝛼)𝑓(𝑠,𝑢(𝑠))𝑑𝑠₁₀|||||(𝛼−1)𝑡₂−𝑡₁(1−𝑠)^𝛼−2⩽𝑁Γ(𝛼)𝑓(𝑠,𝑢(𝑠))𝑑𝑠||||Γ(𝛼+1)𝑡^𝛼2−𝑡^𝛼1+𝑁Γ(𝛼)𝑡²−𝑡₁.(3.6) Thus, by the standard arguments, we obtain that 𝑇(Ω) is equicontinuous. The Arzela-Ascoli theorem implies that 𝑇∶𝑃→𝑃 is completely continuous. The proof is completed.

Let 𝑀₁=¹⁰𝐺(1,𝑠)𝑑𝑠,𝑁₁=𝜉^𝛼−1^𝜂𝜉𝐺(1,𝑠)𝑑𝑠.(3.7)

Theorem 3.2. Let 𝑓(𝑡,𝑢)∈𝐶([0,1]×[0,∞),[0,∞)). Assume that there exist two positive constants 𝑟₂>𝑟₁>0 such that (𝐻₁)𝑓(𝑡,𝑢)≤𝑟₂/𝑀₁for(𝑡,𝑢)∈[0,1]×[0,𝑟₂], (𝐻₂)𝑓(𝑡,𝑢)≥𝑟₁/𝑁₁for(𝑡,𝑢)∈[0,1]×[0,𝑟₁]. Then, fractional boundary value problem (1.1) has at least one positive, increasing, and concave solution 𝑢 such that 𝑟₁≤‖𝑢‖≤𝑟₂.

Proof. Lemmas 2.8 and 3.1 imply that 𝑇∶𝑃→𝑃 is completely continuous and fractional boundary problem (1.1) has a solution 𝑢=𝑢(𝑡) if and only if 𝑢 satisfies the operator equation 𝑢=𝑇𝑢.
Let Ω₁∶={𝑢∈𝑃∶‖𝑢‖<𝑟₁}. By (𝐻₂), for 𝑢∈𝜕Ω₁ and 𝑡∈[0,1], we have 𝑇𝑢(𝑡)=⩾𝑟¹⁰𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠₁𝑁₁^𝜂𝜉⩾𝑟min_{𝜉≤𝑡≤𝜂}𝐺(𝑡,𝑠)𝑑𝑠₁𝑁₁𝜉^𝛼−1^𝜂𝜉𝐺(1,𝑠)𝑑𝑠=𝑟₁=‖𝑢‖.(3.8) So, ‖𝑇𝑢‖⩾‖𝑢‖,for𝑢∈𝜕Ω₁.(3.9)
Let Ω₂∶={𝑢∈𝑃∶‖𝑢‖<𝑟₂}. For 𝑢∈𝜕Ω₂, and 𝑡∈[0,1], it follows from (𝐻₁) that ‖𝑇𝑢(𝑡)‖=max_0≤𝑡≤1||||¹⁰⩽𝑟𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠₂||||𝑀₁¹⁰=𝑟𝐺(1,𝑠)𝑑𝑠₂𝑀₁¹⁰𝐺(1,𝑠)𝑑𝑠=𝑟₂=‖𝑢‖.(3.10) Lemma 2.6 implies that the fractional boundary value problem (1.1) has at least one positive solution 𝑢 such that 𝑟₁≤‖𝑢‖≤𝑟₂. By Lemma 2.9, the solution is also increasing and concave.

In order to use Lemma 2.7, we define the nonnegative continuous concave functional 𝜑 by 𝜑(𝑢)=min_{𝜉⩽𝑡⩽𝜂}𝑢(𝑡),forall𝑢∈𝑃.

Theorem 3.3. Suppose that 𝑓(𝑡,𝑢)∈𝐶([0,1]×[0,∞),[0,∞)) and there exist constants 0<𝑎<𝑏<𝑐 such that the following conditions hold: (𝐻₃)𝑓(𝑡,𝑢)<𝑎𝑀₁for(𝑡,𝑢)∈[0,1]×[0,𝑎],(𝐻₄)𝑓(𝑡,𝑢)>𝑏𝑁₁for(𝑡,𝑢)∈[𝜉,𝜂]×[𝑏,𝑐],(𝐻₅)𝑓(𝑡,𝑢)⩽𝑐𝑀₁for(𝑡,𝑢)∈[0,1]×[0,𝑐].Then, the fractional boundary problem (1.1) has at least three positive, increasing, and concave solutions 𝑢₁, 𝑢₂, and 𝑢₃ such that max_0⩽𝑡⩽1||𝑢₁(𝑡)||<𝑎,𝑏<min_{𝜉⩽𝑡⩽𝜂}||𝑢₂m(𝑡i)n||_{𝜉⩽𝑡⩽𝜂}<max_0⩽𝑡⩽1||𝑢₂(𝑡)||⩽𝑐,𝑎<max_0⩽𝑡⩽1||𝑢₃(𝑡)||⩽𝑐,||𝑢₃(𝑡)||<𝑏.(3.11)

Proof. By Lemmas 2.8 and 3.1, 𝑇∶𝑃→𝑃 is completely continuous, and fractional boundary value problem (1.1) has a solution 𝑢=𝑢(𝑡) if and only if 𝑢 satisfies the operator equation 𝑢=𝑇𝑢.
First of all, we will prove the following assertions.
Assertion 3.4 (𝑇(𝑃_𝑐)⊆𝑃_𝑐 and 𝑇(𝑃_𝑎)⊆𝑃_𝑎). Firstly, Lemma 3.1 guarantees 𝑇(𝑃_𝑐)⊆𝑃. Secondly, for all 𝑢∈𝑃_𝑐, we have ‖𝑢‖⩽𝑐. By (𝐻₅), ‖𝑇𝑢(𝑡)‖=max_0≤𝑡≤1||||¹⁰⩽𝑐𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠||||𝑀₁¹⁰=𝑐,𝐺(1,𝑠)𝑑𝑠(3.12) which implies that 𝑇(𝑃_𝑐)⊆𝑃_𝑐. In the same way, 𝑇(𝑃_𝑎)⊆𝑃_𝑎.Assertion 3.5 ({𝑢∈𝑃(𝜑,𝑏,𝑑)∣𝜑(𝑢)>𝑏}≠∅, and 𝜑(𝑇𝑢)>𝑏, for all 𝑢∈𝑃(𝜑,𝑏,𝑑)). Let 𝑑=𝑐, 𝑢=(𝑏+𝑐)/2. Then, ‖𝑢‖<𝑑 and 𝜑(𝑢)=𝜑((𝑏+𝑐)/2)>𝑏. Consequently, {𝑢∈𝑃(𝜑,𝑏,𝑑)∣𝜑(𝑢)>𝑏}≠∅.
If 𝑢∈𝑃(𝜑,𝑏,𝑑), then from (𝐻₄) and Lemma 2.10, we obtain that 𝜑(𝑇𝑢)=⩾min_{𝜉⩽𝑡⩽𝜂}𝑇𝑢(𝑡)_𝜂𝜉>𝑏min_{𝜉⩽𝑡⩽𝜂}𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠𝑁₁𝜉^𝛼−1^𝜂𝜉=𝑏.𝐺(1,𝑠)𝑑𝑠(3.13) That is, 𝜑(𝑇𝑢)>𝑏, for all 𝑢∈𝑃(𝜑,𝑏,𝑑).Assertion 3.6 (𝜑(𝑢)>𝑏, for all 𝑢∈𝑃(𝜑,𝑏,𝑐) with ‖𝑢‖>𝑑). If 𝑢∈𝑃(𝜑,𝑏,𝑐) and ‖𝑇𝑢‖>𝑑=𝑐, similar to the above, we also have 𝜑(𝑇𝑢)>𝑏.
Assertions 1∼3 imply that all conditions of Lemma 2.7 hold. Therefore, the fractional boundary value problem (1.1) has at least three positive solutions 𝑢₁,𝑢₂, and 𝑢₃ satisfying max_0⩽𝑡⩽1||𝑢₁(𝑡)||<𝑎,𝑏<min_{𝜉⩽𝑡⩽𝜂}||𝑢₂m(𝑡i)n||_{𝜉⩽𝑡⩽𝜂}<max_0⩽𝑡⩽1||𝑢₂(𝑡)||⩽𝑐,𝑎<max_0⩽𝑡⩽1||𝑢₃(𝑡)||⩽𝑐,||𝑢₃(𝑡)||<𝑏.(3.14) By Lemma 2.9, the positive solutions are also increasing and concave. The proof is completed.

4. Example

In this section, we will present some examples to show the effectiveness of our work.

Example 4.1. Consider the fractional boundary value problem 𝐷𝑢^7/20+(𝑢0()𝑡=)+𝑢𝑡²_+√𝑢+4=0,0<𝑡<1,(0)=𝑢^(0)=𝑢^(1)=0.(4.1) Setting 𝜉=1/2,𝜂=3/4, we obtain 𝑀₁=¹⁰𝐺(1,𝑠)𝑑𝑠=(7/2)−1Γ(7/2)¹⁰(1−𝑠)^(7/2)−2𝑑𝑠−1Γ(7/2)¹⁰=1(1−𝑠)^(7/2)−1𝑑𝑠𝑁(5/2)Γ(5/2)−(7/2)Γ1(7/2)≈0.2149,₁=12^(7/2)−1^3/41/2=(1/2)_5/2𝐺(1,𝑠)𝑑𝑠Γ(7/2)^3/41/252(1−𝑠)^(7/2)−2𝑑𝑠−^3/41/2(1−𝑠)^(7/2)−1𝑑𝑠≈0.0065.(4.2) Let 𝑟₁=1/40,𝑟₂=4. We have 𝑓(𝑡,𝑢)=𝑡²+√𝑢+4⩽𝑟²𝑀₁𝑓(𝑡,𝑢)=𝑡₂≈18.613,for(𝑡,𝑢)∈[0,1]×[0,4],+√𝑢+4⩾𝑟¹𝑁₁≈3.846,for(𝑡,𝑢)∈[0,1]×0,140.(4.3) Theorem 3.2 implies that fractional boundary value problem (4.1) has at least one positive, increasing, and concave solution. The approximate solution is obtained by the Adams-type predictor-corrector method [16], which is displayed in Figure 1 for the step size ℎ=0.01.

Figure 1 

Transient response of state variable 𝑢(𝑡).

Example 4.2. Consider the fractional boundary value problem 𝐷𝑢(0^7/20+)𝑢=(𝑡)𝑢+_𝑓(𝑡,𝑢)=0,0<𝑡<1,(0)=𝑢^(0)=𝑢^(1)=0,(4.4) where 𝑓(𝑡,𝑢)=⎧⎪⎨⎪⎩1𝑡0+155𝑢³𝑡,0⩽𝑢⩽1,𝑡∈[0,1],10+𝑢+154,𝑢>1,𝑡∈[0,1].(4.5)
Setting 𝜉=1/2,𝜂=3/4, we know that 𝑀₁≈0.2149,𝑁₁≈0.0065. Choosing 𝑎=1/10,𝑏=1,𝑐=45, we obtain that 𝑓(𝑡,𝑢)=𝑡10+155𝑢³<𝑎𝑀₁≈0.466for(𝑡,𝑢)∈[0,1]×0,1𝑓(𝑡,𝑢)=𝑡10,10+𝑢+154>𝑁𝑏₁≈153.846for(𝑡,𝑢)∈1𝑓(𝑡,𝑢)=𝑡2,34×[1,45],10+𝑢+154<𝑀𝑐₁≈209.4for(𝑡,𝑢)∈[0,1]×[0,45].(4.6) Theorem 3.3 implies that fractional boundary value problem (4.4) has three positive, increasing, and concave solutions such that max_0⩽𝑡⩽1||𝑢₁(𝑡)||<110,1<min^{1/2⩽𝑡⩽3/4}||𝑢₂(𝑡)||<max_0⩽𝑡⩽1||𝑢₂(𝑡)||⩽45,110<max^0⩽𝑡⩽1min_{1/2⩽𝑡⩽3/4}||𝑢₃(𝑡)||⩽45,||𝑢₃(𝑡)||<1.(4.7) For numerical simulation case 1, Figure 2 depicts the phase responses state variables of 𝑢(𝑡) with the step size ℎ=0.01.

Figure 2 

Transient response of state variable 𝑢(𝑡).

Acknowledgments

This work was jointly supported by Natural Science Foundation of China (no. 10871214), Natural Science Foundation of Hunan Provincial under Grants nos. 11JJ3005, 10JJ6007, and 2010GK3008.