Multiplicity Result of Positive Solutions for Nonlinear Differential Equation of Fractional Order

Abstract

We investigate the existence of multiple positive solutions for a class of boundary value problems of nonlinear differential equation with Caputo’s fractional order derivative. The existence results are obtained by means of the Avery-Peterson fixed point theorem. It should be point out that this is the first time that this fixed point theorem is used to deal with the boundary value problem of differential equations with fractional order derivative.

1. Introduction

In this paper, we consider the existence and multiple existence of positive solutions for following boundary value problem of differential equation involving the Caputo's fractional order derivative 𝐷^𝛼₀₊𝑢(𝑡)=𝑓𝑡,𝑢(𝑡),𝐷^𝛽₀₊𝑢(𝑡)(),𝑢(0),𝑡∈0,1+𝑢^′(0)=0,𝑢(1)+𝑢^′(1)=0, (1.1) where 1<𝛼<2, 0<𝛽<𝛼−1 and 𝑓∶𝐶([0,1]×𝑅⁺×𝑅,𝑅⁺). Here 𝐷^𝛼₀₊ is the Caputo's derivative of fractional order.

Due to the development of the theory of fractional calculus and its applications, such as in the fields of control theory, blood flow phenomena, Bode's analysis of feedback amplifiers, aerodynamics, and polymer rheology and many work on fractional calculus, fractional order differential equations has appeared [1–7]. Recently, there have been many results concerning the solutions or positive solutions of boundary value problems for nonlinear fractional differential equations, see [8–28] and references along this line.

For example, Bai and Lü [12] considered the following Dirichlet boundary value problem of fractional differential equation: 𝐷^𝛼₀₊𝑢(𝑡)+𝑓(𝑡,𝑢(𝑡))=0,𝑡∈(0,1),𝑢(0)=0=𝑢(1),1<𝛼≤2. (1.2) By means of different fixed-point theorems on cone, some existence and multiplicity results of positive solutions were obtained. Jiang and Yuan [20] improved the results in [12] by discussing some new positive properties of the Green function for problem (1.2). By using the fixed point theorem on a cone due to Krasnoselskii, the authors established the existence results of positive solution for problem (1.2). Recently, Caballero et al. [21] obtained the existence and uniqueness of positive solution for singular boundary value problem (1.2). The existence results were established in the case that the nonlinear term 𝑓 may be singular at 𝑡=0. As to positive solutions of problem (1.1), under the case that the nonlinear term was not involved with the derivative of the function 𝑢(𝑡), Zhang [13] obtained the existence and multiplicity results of positive solutions by means of a fixed-point theorem on cones.

There are also some results concerning multipoint boundary value problems for differential equations of fractional order. Bai [23] investigated the existence and uniqueness of positive solution for three-point boundary value problem 𝐷^𝛼₀₊𝑢(𝑡)+𝑓(𝑡,𝑢(𝑡))=0,𝑡∈(0,1),𝑢(0)=0,𝑢(1)=𝛽𝑢𝜂, (1.3) where 1<𝛼≤2, 𝜂∈(0,1), 0<𝛽𝜂^𝛼−1<1. In [23], the uniqueness of positive solution was obtained by the use of contraction map principle and some existence results of positive solutions were established by means of the fixed point index theory. Very recently, Wang et al. [26] considered the boundary value problem of fractional differential equation with integral condition 𝐷^𝛼₀₊𝑢(𝑡)(𝑡)𝑓((𝑡()𝑢(0)+𝑞𝑡,𝑢))=0,𝑡∈0,1,𝑛−1<𝛼<𝑛,=𝑢^′(0)=⋯=𝑢^(𝑛−2)(0)=0,𝑢(1)=¹₀𝑢(𝑠)𝑑𝐴(𝑠), (1.4) where 𝛼>2, ∫¹₀𝑢(𝑠)𝑑𝐴(𝑠) was given by Riemann-Stieltjes integral with a signed measure. By using the fixed point theorem, the existence of positive solution for this problem was established.

However, in this work, the derivative of the unknown function 𝑢(𝑡) was not involved in the nonlinear term explicitly. To our best knowledge, there are few papers considering the positive solution of boundary value problem of nonlinear fractional differential equations which the derivative of the unknown function 𝑢(𝑡) is involved in the nonlinear term. In [29], Guo and Ge proved a new fixed point theorem, which can be regarded as an extension of Krasnoselskii's fixed point theorem in a cone. By applying this new theorem, Guo and Ge obtained the existence of positive solutions for second-order three-point boundary value problem 𝑢^′′+𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡)[],𝑢(0)(1)𝜂(),=0,𝑡∈0,1=0,𝑢=𝛼𝑢,𝜂∈0,1 (1.5) where 𝑓 depended on the first order derivative of 𝑢. Very recently, Yang et al. [30] considered following boundary value problem 𝐷^𝛼₀₊𝑢(𝑡)=𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡)(),𝑢(0),𝑡∈0,1+𝑢^′(0)=0,𝑢(1)+𝑢^′(1)=0. (1.6) By means of Schauder's fixed point theorem and the fixed point theorem duo to Guo and Ge, some results on the existence of positive solutions were obtained.

In [31], Avery and Peterson gave an new triple fixed point theorem, which can be regarded as an extension of Leggett-Williams fixed point theorem. By using this method, many results concerning the existence of at least three positive solutions of boundary value problems of differential equation with integer order were established, see [32–37]. For example, by using the Avery-Peterson fixed point theorem, Yang et al. [32] established the existence of at least three positive solutions of second-order multipoint boundary value problem 𝑢^′′+𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡)[],𝑢=0,𝑡∈0,1_′(0)=^𝑚−2_𝑖=1𝛽_𝑖𝑢^′𝜉_𝑖,𝑢(1)=^𝑚−2_𝑖=1𝛼_𝑖𝑢𝜉_𝑖. (1.7)

But by using the Avery-Peterson fixed point theorem, the nonlinear terms are often assumed to be nonnegative to ensure the concavity or convexity of the unknown function. When the differential equations of fractional order are considered, we cannot derive the concavity or convexity of function 𝑢(𝑡) by the sign of its fractional order derivative. Thus the Avery-Peterson fixed point theorem cannot directly be used to consider the boundary value problem of nonlinear differential equation with fractional order where the derivative of the unknown function 𝑢(𝑡) is involved in the nonlinear term explicitly.

In this paper, by obtaining some new inequalities of the unknown function and defining a special cone, we overcome the difficulties brought by the lack of the concavity or convexity of unknown function 𝑢(𝑡). By an application of Avery-Peterson fixed point theorem, the existence of at least three positive solutions of problem (1.1) is established. It should be pointed out that it is the first time that the Avery-Peterson fixed point theorem is used to deal with the positive solutions of boundary value problem of differential equations with fractional order derivative.

2. Preliminary Results

Definition 2.1. The Riemann-Liouville fractional integral of order 𝛼>0 of a function 𝑢(𝑡)∶(0,∞)→𝑅 is given by 𝐼^𝛼₀₊𝑢(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑢(𝑠)𝑑𝑠 (2.1) provided the right side is point-wise defined on (0,∞).

Definition 2.2. The Caputo's fractional derivative of order 𝛼>0 of a continuous function 𝑢(𝑡)∶(0,∞)→𝑅 is given by 𝐷^𝛼₀₊𝑢(𝑡)=1Γ(𝑛−𝛼)^𝑡₀𝑢^(𝑛)(𝑠)(𝑡−𝑠)^{𝛼−𝑛+1}𝑑𝑠, (2.2) where 𝑛−1<𝛼≤𝑛, provided that the right side is point-wise defined on (0,∞).

Lemma 2.3. Let 𝛼>0. Then the fractional differential equation 𝐷^𝛼₀₊𝑢(𝑡)=0 has solutions 𝑢(𝑡)=𝑐₀+𝑐₁𝑡+𝑐₂𝑡²+⋯+𝑐_𝑛−1𝑡^𝑛−1,𝑐_𝑖∈𝑅,𝑖=0,1,…,𝑛−1. (2.3)

Lemma 2.4. Let 𝛼>0. Then 𝐼^𝛼₀₊𝐷^𝛼₀₊𝑢(𝑡)=𝑢(𝑡)𝑐₀+𝑐₁𝑡+𝑐₂𝑡²+⋯+𝑐_𝑛−1𝑡^𝑛−1,𝑐_𝑖∈𝑅,𝑖=1,2,…,𝑛−1. (2.4)

Definition 2.5. Let 𝐸 be a real Banach space over 𝑅. A nonempty convex closed set 𝑃⊂𝐸 is said to be a cone provided that(1)𝑎𝑢∈𝑃, for all 𝑢∈𝑃, 𝑎≥0,(2)𝑢,−𝑢∈𝑃 implies 𝑢=0.

Definition 2.6. An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.

Definition 2.7. The map 𝛼 is said to be a continuous nonnegative convex functional on cone 𝑃 of a real Banach space 𝐸 provided that 𝛼∶𝑃→[0,+∞) is continuous and 𝛼𝑡𝑥+(1−𝑡)𝑦≤𝑡𝛼(𝑥)+(1−𝑡)𝛼𝑦,𝑥,𝑦∈𝑃,𝑡∈[0,1]. (2.5)

Definition 2.8. The map 𝛽 is said to be a continuous nonnegative concave functional on cone 𝑃 of a real Banach space 𝐸 provided that 𝛽∶𝑃→[0,+∞) is continuous and 𝛽𝑡𝑥+(1−𝑡)𝑦≥𝑡𝛽(𝑥)+(1−𝑡)𝛽𝑦,𝑥,𝑦∈𝑃,𝑡∈[0,1]. (2.6)
Let 𝛾 and  𝜃 be nonnegative continuous convex functionals on 𝑃, 𝛼 a nonnegative continuous concave functional on 𝑃, and 𝜓 a nonnegative continuous functional on P. Then for positive numbers 𝑎, 𝑏, 𝑐, and 𝑑, we define the following convex sets: 𝑃=(𝑥),𝑃=(𝑥)(𝑥),𝑃=(𝑥)(𝑥)(𝑥),𝛾,𝑑𝑥∈𝑃∣𝛾<𝑑𝛾,𝛼,𝑏,𝑑𝑥∈𝑃∣𝑏≤𝛼,𝛾≤𝑑𝛾,𝜃,𝛼,𝑏,𝑐,𝑑𝑥∈𝑃∣𝑏≤𝛼,𝜃≤𝑐,𝛾≤𝑑 (2.7) and a closed set 𝑅𝛾,𝜓,𝑎,𝑑=𝑥∈𝑃∣𝑎≤𝜓(𝑥),𝛾(𝑥)≤𝑑. (2.8)

Lemma 2.9 (see [31]). Let 𝑃 be a cone in Banach space E. Let 𝛾, 𝜃 be nonnegative continuous convex functionals on 𝑃, 𝛼 a nonnegative continuous concave functional on 𝑃, and 𝜓 a nonnegative continuous functional on 𝑃 satisfying 𝜓(𝜆𝑥)≤𝜆𝜓(𝑥),for0≤𝜆≤1, (2.9) such that for some positive numbers 𝑙 and d, 𝛼(𝑥)≤𝜓(𝑥),‖𝑥‖≤𝑙𝛾(𝑥), (2.10) for all 𝑥∈𝑃(𝛾,𝑑). Suppose 𝑇∶𝑃(𝛾,𝑑)→𝑃(𝛾,𝑑) is completely continuous and there exist positive numbers 𝑎, 𝑏, 𝑐 with 𝑎<𝑏 such that(𝑆₁){𝑥∈𝑃(𝛾,𝜃,𝛼,𝑏,𝑐,𝑑)|𝛼(𝑥)>𝑏}≠∅ and 𝛼(𝑇𝑥)>𝑏 for 𝑥∈𝑃(𝛾,𝜃,𝛼,𝑏,𝑐,𝑑);(𝑆₂)𝛼(𝑇𝑥)>𝑏 for 𝑥∈𝑃(𝛾,𝛼,𝑏,𝑑) with 𝜃(𝑇𝑥)>𝑐;(𝑆₃)0∉𝑅(𝛾,𝜓,𝑎,𝑑) and 𝜓(𝑇𝑥)<𝑎 for 𝑥∈𝑅(𝛾,𝜓,𝑎,𝑑) with 𝜓(𝑥)=𝑎.Then 𝑇 has at least three fixed points 𝑥₁,𝑥₂,𝑥₃∈𝑃(𝛾,𝑑) such that 𝛾𝑥_𝑖≤𝑑,𝑖=1,2,3;𝑏<𝛼𝑥₁;𝑎<𝜓𝑥₂,𝛼𝑥₂<𝑏;𝜓𝑥₃<𝑎. (2.11)

3. Main Results

Lemma 3.1 (see [30]). Given 𝑦(𝑡)∈𝐶[0,1], then boundary value problem 𝐷^𝛼₀₊𝑢(𝑡)=𝑦(𝑡),𝑢(0)+𝑢^′(0)=0,𝑢(1)+𝑢′(1)=0 (3.1) is equivalent to 𝑢(𝑡)=^𝑡₀(1−𝑠)^𝛼−1(1−𝑡)+(1−𝑠)^𝛼−1Γ(𝛼)+(1−𝑠)^𝛼−2(1−𝑡)Γ()𝑦(𝑠)+𝛼−1𝑑𝑠_1𝑡(1−𝑠)^𝛼−1(1−𝑡)Γ(𝛼)+(1−𝑠)^𝛼−2(1−𝑡)Γ()𝑦(𝑠)𝛼−1𝑑𝑠=¹₀𝐺(𝑡,𝑠)𝑦(𝑠)𝑑𝑠, (3.2) where 𝐺()=()𝑡,𝑠1−𝑠^𝛼−1(1−𝑡)+(1−𝑠)^𝛼−1Γ(𝛼)+(1−𝑠)^𝛼−2(1−𝑡)Γ()()𝛼−1,0≤𝑠≤𝑡≤1,1−𝑠^𝛼−1(1−𝑡)Γ(𝛼)+(1−𝑠)^𝛼−2(1−𝑡)Γ(𝛼−1),0≤𝑡≤𝑠≤1. (3.3)

Lemma 3.2. Given 𝑦(𝑡)∈𝐶[0,1], assume that 𝑢(𝑡) is a solution of boundary value problem 𝐷^𝛼₀₊𝑢(𝑡)=𝑦(𝑡),𝑢(0)+𝑢^′(0)=0,𝑢(1)+𝑢^′(1)=0. (3.4) Then 𝐷^𝛽𝑢(𝑡)=^𝑡₀(𝑡−𝑠)^{𝛼−𝛽−1}Γ−𝑡𝛼−𝛽^1−𝛽(1−𝑠)^𝛼−2(𝛼−𝑠)Γ(𝛼)Γ𝑦(𝑠)+2−𝛽𝑑𝑠_1𝑡−𝑡^1−𝛽(1−𝑠)^𝛼−2(𝛼−𝑠)Γ(𝛼)Γ𝑦(𝑠)2−𝛽𝑑𝑠. (3.5)

Proof. From Lemmas 2.3 and 2.4, we get that 𝑢(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠−𝐶₁−𝐶₂𝑡,𝑢^′(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−2Γ()𝑦(𝑠)𝛼−1𝑑𝑠−𝐶₂. (3.6) The boundary condition 𝑢(0)+𝑢′(0)=0 implies that 𝐶₁+𝐶₂=0. Considering the boundary condition 𝑢(1)+𝑢′(1)=0, we have 𝐶₂=−𝐶₁=1Γ(𝛼)¹₀(1−𝑠)^𝛼−1𝑦(𝑠)1𝑑𝑠+Γ(𝛼−1)¹₀(1−𝑠)^𝛼−2𝑦(𝑠)𝑑𝑠. (3.7) From the definition of the Caputo derivative of fractional order, we see 𝐷^𝛽𝑢(𝑡)=𝐷^𝛽𝑡₀(1−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠−𝐶₁−𝐶₂𝑡=^𝑡₀(𝑡−𝑠)^{𝛼−𝛽−1}Γ𝑦(𝑠)𝐶𝛼−𝛽𝑑𝑠−²Γ𝑡2−𝛽^1−𝛽=^𝑡₀(𝑡−𝑠)^{𝛼−𝛽−1}Γ−𝑡𝛼−𝛽^1−𝛽(1−𝑠)^𝛼−2(𝛼−𝑠)Γ(𝛼)Γ𝑦(𝑠)+2−𝛽𝑑𝑠_1𝑡−𝑡^1−𝛽(1−𝑠)^𝛼−2(𝛼−𝑠)Γ(𝛼)Γ𝑦(𝑠)2−𝛽𝑑𝑠. (3.8)

Lemma 3.3 (see [30]). The function 𝐺(𝑡,𝑠) satisfies the following conditions:(1)𝐺(𝑡,𝑠)∈𝐶([0,1]×[0,1)), 𝐺(𝑡,𝑠)>0, for 𝑡,𝑠∈(0,1);(2)there exist a positive function 𝛾(𝑠),𝐾(𝑠)∈𝐶(0,1) such that max_0≤𝑡≤1≤𝐾(𝑠),𝑠∈(0,1),min_{[]𝑠∈1/3,2/3}𝐺(𝑡,𝑠)≥𝛾(𝑠)𝐾(𝑠),0<𝑠<1, (3.9) where 𝐾(𝑠)=2(1−𝑠)^𝛼−1Γ(𝛼)+(1−𝑠)^𝛼−2Γ(),𝛾(𝑠)=1𝛼−13(1−𝑠)^𝛼−1+(𝛼−1)(1−𝑠)^𝛼−22(1−𝑠)^𝛼−1+(𝛼−1)(1−𝑠)^𝛼−2>16,𝑠∈[0,1). (3.10)(3.11)

Lemma 3.4. Assume that 𝑦(𝑡)>0 and 𝑢(𝑡) is a solution of boundary value problem (3.1). There exists a positive constant 𝛾₀ such that max_0≤𝑡≤1|𝑢(𝑡)|≤𝛾₀max_0≤𝑡≤1𝐷^𝛽|𝑢(𝑡)|. (3.12)

Proof. From Lemma 3.2,max_0≤𝑡≤1𝐷^𝛽𝑢(𝑡)=max_0≤𝑡≤1^𝑡₀(𝑡−𝑠)^{𝛼−𝛽−1}Γ−𝑡𝛼−𝛽^1−𝛽(1−𝑠)^𝛼−2(𝛼−𝑠)Γ(𝛼)Γ𝑦(𝑠)−2−𝛽𝑑𝑠_1𝑡𝑡^1−𝛽(1−𝑠)^𝛼−2(𝛼−𝑠)Γ(𝛼)Γ𝑦(𝑠)2−𝛽𝑑𝑠≥max_0≤𝑡≤1𝐷^𝛽𝑢(1)=¹₀(1−𝑠)^𝛼−2(𝛼−𝑠)Γ(𝛼)Γ−()2−𝛽1−𝑠^{𝛼−𝛽−1}Γ𝑦(𝑠)𝛼−𝛽𝑑𝑠=¹₀𝑘(𝑠)𝑦(𝑠)𝑑𝑠. (3.13) Denote ℎ(𝑠)=𝐾(𝑠)𝑘(𝑠)=2(1−𝑠)^𝛼−1/Γ(𝛼)+(1−𝑠)^𝛼−2/Γ(𝛼−1)(1−𝑠)^𝛼−2(𝛼−𝑠)/Γ(𝛼)Γ2−𝛽−(1−𝑠)^{𝛼−𝛽−1}=2()(𝛼)()/Γ𝛼−𝛽1−𝑠/Γ+1/Γ𝛼−1(𝛼−𝑠)/Γ(𝛼)Γ2−𝛽−(1−𝑠)^1−𝛽./Γ𝛼−𝛽 (3.14) By a simple computation, we have ℎ(𝑠)≤2/Γ(𝛼)+1/Γ(𝛼−1)𝛼−𝑠₀/Γ(𝛼)Γ2−𝛽−1−𝑠₀^1−𝛽=(𝛼)()/Γ𝛼−𝛽2/Γ+1/Γ𝛼−1(𝛼−1)1−𝛽−𝛽1−𝛽Γ(𝛼)Γ2−𝛽/Γ𝛼−𝛽^1/𝛽/1−𝛽Γ(𝛼)Γ𝛼−𝛽=𝛾₀>0. (3.15) Thus, max_0≤𝑡≤1|𝑢(𝑡)|=max_0≤𝑡≤1¹₀𝐺(𝑡,𝑠)𝑦(𝑠)𝑑𝑠≤¹₀𝐾(𝑠)𝑦(𝑠)𝑑𝑠≤¹₀𝛾₀𝑘(𝑠)𝑦(𝑠)𝑑𝑠≤𝛾₀max_0≤𝑡≤1𝐷^𝛽𝑢(𝑡). (3.16)
Let the Banach space 𝐸={𝑢(𝑡)∈𝐶[0,1], 𝐷^𝛽𝑢(𝑡)∈𝐶[0,1]} be endowed with the norm ‖𝑢‖=maxmax_0≤𝑡≤1|𝑢(𝑡)|,max_0≤𝑡≤1𝐷^𝛽𝑢(𝑡),𝑢∈𝐸. (3.17) We define the cone 𝑃⊂𝐸 by 𝑃=𝑢∈𝐸∣𝑢(𝑡)≥0,min_{1/3≤𝑡≤2/3}𝑢(𝑡)≥16max_0≤𝑡≤1𝑢(𝑡),max_0≤𝑡≤1𝑢(𝑡)≤𝛾₀max_0≤𝑡≤1𝐷^𝛽𝑢(𝑡). (3.18) Denote the positive constants 𝑀=¹₀𝐾(𝑠)2𝑑𝑠=Γ()+1𝛼+1Γ(𝛼),1𝑁=Γ+1𝛼−𝛽+1Γ12−𝛽Γ(𝛼)+1Γ().𝛼+1 (3.19)

Lemma 3.5. Let 𝑇∶𝑃→𝐸 be the operator defined by 𝑇𝑢(𝑡)∶=¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢(𝑠),𝐷^𝛽₀₊𝑢(𝑠)𝑑𝑠. (3.20) Then 𝑇∶𝑃→𝑃 is completely continuous.

Proof. The operator 𝑇 is nonnegative and continuous in view of the nonnegativeness and continuity of functions 𝐺(𝑡,𝑠) and 𝑓(𝑡,𝑢(𝑡),𝐷^𝛽₀₊𝑢(𝑡)). Let Ω⊂𝐾 be bounded. Then there exists a positive constant 𝑅₁>0 such that ‖𝑢‖≤𝑅₁, 𝑢∈Ω. Denote 𝑅=max_{0≤𝑡≤1,𝑢∈Ω}𝑓𝑡,𝑢(𝑡),𝐷^𝛽₀₊𝑢(𝑡)+1. (3.21) Then for 𝑢∈Ω, we have |(𝑇𝑢)(𝑡)|≤¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢,𝐷^𝛽₀₊𝑢𝑑𝑠≤¹₀𝐾(𝑠)𝑓𝑠,𝑢,𝐷^𝛽₀₊𝑢𝑑𝑠≤𝑅¹₀𝐾(𝑠)𝐷𝑑𝑠=𝑀𝑅,_𝛽0+(𝑡=1𝑇𝑢)()Γ𝛼−𝛽^𝑡₀(𝑡−𝑠)^{𝛼−𝛽−1}𝑓𝑠,𝑢(𝑠),𝐷^𝛽₀₊𝑢(s−𝑡)𝑑𝑠_1−𝛽Γ12−𝛽Γ(𝛼−1)¹₀(1−𝑠)^𝛼−2𝑓𝑠,𝑢,𝐷^𝛽₀₊𝑢+1𝑑𝑠Γ(𝛼)¹₀(1−𝑠)^𝛼−1𝑓𝑠,𝑢,𝐷^𝛽₀₊𝑢≤1𝑑𝑠Γ+1𝛼−𝛽𝛼−𝛽Γ12−𝛽Γ(𝛼−1)¹₀(1−𝑠)^𝛼−21𝑑𝑠+Γ(𝛼)¹₀(1−𝑠)^𝛼−1𝑑𝑠×𝑅=𝑁𝑅. (3.22) Hence 𝑇(Ω) is bounded. On the other hand, for 𝑢∈Ω, 𝑡₁,𝑡₂∈[0,1], one has 𝑇𝑢𝑡₂−𝑇𝑢𝑡₁=¹₀𝐺𝑡₁,𝑠𝑓𝑠,𝑢(𝑠),𝐷^𝛽₀₊𝑢(𝑠)−𝑑𝑠₁₀𝐺𝑡₂,𝑠𝑓𝑠,𝑢(𝑠),𝐷^𝛽₀₊𝑢(𝑠)≤𝑑𝑠_𝑡10𝐺𝑡₂,𝑠−𝐺𝑡₁,𝑠𝑓𝑠,𝑢(𝑠),𝐷^𝛽₀₊𝑢(𝑠)+𝑑𝑠_𝑡2𝑡1𝐺𝑡₂,𝑠−𝐺𝑡₁,𝑠𝑓𝑠,𝑢(𝑠),𝐷^𝛽₀₊𝑢(𝑠)𝑑𝑠+¹_𝑡2𝐺𝑡₂,𝑠−𝐺𝑡₁,𝑠𝑓𝑠,𝑢(𝑠),𝐷^𝛽₀₊𝑢(𝑠)≤2𝑡𝑑𝑠₂−𝑡₁Γ(𝛼)+𝑡^𝛼₂−𝑡^𝛼₁Γ()𝐷𝛼+1×𝑅,_𝛽0+𝑇𝑢𝑡₂−𝐷^𝛽₀₊𝑇𝑢𝑡₁=1Γ𝛼−𝛽^𝑡2₀𝑡₂−𝑠^{𝛼−𝛽−1}𝑓𝑠,𝑢,𝐷^𝛽₀₊𝑢−𝑑𝑠_𝑡10𝑡₁−𝑠^{𝛼−𝛽−1}𝑓𝑠,𝑢,𝐷^𝛽₀₊𝑢+𝑡𝑑𝑠_11−𝛽−𝑡₂^1−𝛽Γ12−𝛽Γ(𝛼−1)¹₀(1−𝑠)^𝛼−2𝑓𝑠,𝑢,𝐷^𝛽₀₊𝑢+1𝑑𝑠Γ(𝛼)¹₀(1−𝑠)^𝛼−1𝑓𝑠,𝑢,𝐷^𝛽₀₊𝑢≤𝑡𝑑𝑠𝑅×₂^𝛼−𝛽−𝑡₁^𝛼−𝛽Γ+𝑅𝛼−𝛽𝛼−𝛽Γ12−𝛽Γ(𝛼−1)¹₀(1−𝑠)^𝛼−21𝑑𝑠+Γ(𝛼)¹₀(1−𝑠)^𝛼−1×𝑡₂^1−𝛽−𝑡₁^1−𝛽≤𝑅Γ𝑡𝛼−𝛽𝛼−𝛽^2𝛼−𝛽−𝑡₁^𝛼−𝛽+𝑅Γ12−𝛽Γ()+1𝛼+1Γ(𝛼)𝑡₂^1−𝛽−𝑡₁^1−𝛽. (3.23) Thus, 𝑇𝑢𝑡₂−𝑇𝑢𝑡₁⟶0,for𝑡₁⟶𝑡₂. (3.24) By means of the Arzela-Ascoli theorem, 𝑇 is completely continuous. Furthermore, for 𝑢∈𝑃, we have min_{1/3≤𝑡≤2/3}𝑇𝑢(𝑡)=min_{1/3≤𝑡≤2/3}¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢(𝑠),𝐷^𝛽₀₊𝑢(𝑠)≥1𝑑𝑠6¹₀𝐾(𝑠)𝑓𝑠,𝑢(𝑠),𝐷^𝛽₀₊𝑢(𝑠=1)𝑑𝑠6max_0≤𝑡≤1𝑇𝑢(𝑡),max_0≤𝑡≤1|𝑢(𝑡)|=¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢(𝑠),𝐷^𝛽₀₊𝑢(𝑠)≤𝑑𝑠₁₀𝐾(𝑠)𝑓𝑠,𝑢(𝑠),𝐷^𝛽₀₊𝑢(𝑠)≤𝑑𝑠₁₀𝛾₀𝑘(𝑠)𝑓𝑠,𝑢(𝑠),𝐷^𝛽₀₊𝑢(𝑠)𝑑𝑠=𝛾₀max_0≤𝑡≤1𝐷^𝛽𝑢(𝑡). (3.25) Thus, 𝑇∶𝑃→𝑃 is completely continuous.
Let the continuous nonnegative concave functional 𝛼, the the continuous nonnegative convex functionals 𝛾, 𝜃, and the continuous nonnegative functional 𝜓 be defined on the cone by 𝛾(𝑢)=max_0≤𝑡≤1𝐷^𝛽𝑢(𝑡),𝜃(𝑢)=𝜓(𝑢)=max_0≤𝑡≤1|𝑢(𝑡)|,𝛼(𝑢)=min_{1/3≤𝑡≤2/3}|𝑢(𝑡)|. (3.26) By Lemmas 3.3 and 3.4, the functionals defined above satisfy that 16𝜃(𝑢)≤𝛼(𝑢)≤𝜃(𝑢)=𝜓(𝑢),‖𝑢‖≤𝛾₁𝛾(𝑢),𝑢∈𝑃, (3.27) where 𝛾₁=max{𝛾₀,1}. Therefore condition (2.10) of Lemma 2.9 is satisfied.
Assume that there exist constants 0<𝑎,𝑏,𝑑 with 𝑎<𝑏<(𝑀/6𝑁)𝑑, 𝑐=6𝑏 such that(𝐴₁)𝑓(𝑡,𝑢,𝑣)≤𝑑/𝑁, (𝑡,𝑢,𝑣)∈[0,1]×[0,𝛾₁𝑑]×[−𝑑,𝑑],(𝐴₂)𝑓(𝑡,𝑢,𝑣)>6𝑏/𝑀, (𝑡,𝑢,𝑣)∈[0,1]×[𝑏,6𝑏]×[−𝑑,𝑑],(𝐴₃)𝑓(𝑡,𝑢,𝑣)<𝑎/𝑀, (𝑡,𝑢,𝑣)∈[0,1]×[0,𝑎]×[−𝑑,𝑑].

Theorem 3.6. Under assumptions (𝐴₁)–(𝐴₃), problem (1.1) has three positive solutions 𝑢₁, 𝑢₂, 𝑢₃ satisfying max_0≤𝑡≤1𝐷^𝛽𝑢(𝑡)≤𝑑,𝑖=1,2,3;𝑏<min_{1/3≤𝑡≤2/3}𝑢₁(𝑡);𝑎<max_0≤𝑡≤1𝑢₂(𝑡),min_{1/3≤𝑡≤2/3}𝑢₂(𝑡)<𝑏;max_0≤𝑡≤1𝑢₃(𝑡)≤𝑎. (3.28)

Proof. Problem (1.1) has a solution 𝑢=𝑢(𝑡) if and only if 𝑢 solves the operator equation 𝑢(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢(𝑠),𝐷^𝛽𝑢(𝑠)𝑑𝑠=(𝑇𝑢)(𝑡). (3.29) For 𝑢∈𝑃(𝛾,𝑑), we have 𝛾(𝑢)=max_0≤𝑡≤1|𝐷^𝛽𝑢(𝑡)|<𝑑. From assumption (𝐴₁), we obtain 𝑓𝑡,𝑢(𝑡),𝐷^𝛽𝑢(𝑡)≤𝑑𝑁. (3.30) Thus 𝛾(𝑇𝑢)=max_0≤𝑡≤1𝐷^𝛽(=1T𝑢)Γ𝛼−𝛽^𝑡₀(𝑡−𝑠)^{𝛼−𝛽−1}𝑓𝑠,𝑢,𝐷^𝛽₀₊𝑢−𝑡𝑑𝑠_1−𝛽Γ12−𝛽Γ(𝛼−1)¹₀(1−𝑠)^𝛼−2𝑓𝑠,𝑢,𝐷^𝛽₀₊𝑢+1𝑑𝑠Γ(𝛼)¹₀(1−𝑠)^𝛼−1𝑓𝑠,𝑢,𝐷^𝛽₀₊𝑢≤1𝑑𝑠Γ+1𝛼−𝛽𝛼−𝛽Γ12−𝛽Γ(𝛼−1)¹₀(1−𝑠)^𝛼−21𝑑𝑠+Γ(𝛼)¹₀(1−𝑠)^𝛼−1×𝑑𝑑𝑠𝑁=𝑑. (3.31) Hence, 𝑇∶𝑃(𝛾,𝑑)→𝑃(𝛾,𝑑).
The fact that the constant function 𝑢(𝑡)=6𝑏∈𝑃(𝛾,𝜃,𝛼,𝑏,𝑐,𝑑) and 𝛼(6𝑏)>𝑏 implies that {𝑢∈𝑃(𝛾,𝜃,𝛼,𝑏,𝑐,𝑑|𝛼(𝑢)>𝑏)}≠∅.
For 𝑢∈𝑃(𝛾,𝜃,𝛼,𝑏,𝑐,𝑑), we have 𝑏≤𝑢(𝑡)≤6𝑏 and |𝐷^𝛽𝑢(𝑡)|<𝑑 for 0≤𝑡≤1. From assumption (𝐴₂), 𝑓𝑡,𝑢(𝑡),𝐷^𝛽𝑢(𝑡)>6𝑏𝑀. (3.32) Thus 𝛼(𝑇𝑢)=min_{1/3≤𝑡≤2/3}¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢(𝑠),𝐷^𝛽₀₊𝑢(𝑠)≥1𝑑𝑠6¹₀𝐾(𝑠)𝑓𝑠,𝑢(𝑠),𝐷^𝛽𝑢(𝑠≥1)𝑑𝑠6¹₀𝐾(𝑠)𝑑𝑠×6𝑏𝑀=𝑏, (3.33) which means 𝛼(𝑇𝑢)>𝑏, for  all 𝑢∈𝑃(𝛾,𝜃,𝛼,𝑏,6𝑏,𝑑). These ensure that condition (𝑆₁) of Lemma 2.9 is satisfied. Secondly,  for all 𝑢∈𝑃(𝛾,𝛼,𝑏,𝑑) with 𝜃(𝑇𝑢)>6𝑏, 𝛼()≥1𝑇𝑢6𝜃()>1𝑇𝑢61×𝑐=6×6𝑏=𝑏. (3.34) Thus, condition (𝑆₂) of Lemma 2.3 holds. Finally we show that (𝑆₃) also holds. We see that 𝜓(0)=0<𝑎 and 0∉𝑅(𝛾,𝜓,𝑎,𝑑). Suppose that 𝑢∈𝑅(𝛾,𝜓,𝑎,𝑑) with 𝜓(𝑥)=𝑎. Then by assumption (𝐴₃), 𝜓(𝑇𝑢)=max_0≤𝑡≤1|(𝑇𝑢)(𝑡)|=¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢(𝑠),𝐷^𝛽𝑢(𝑠)≤𝑑𝑠₁₀𝐾(𝑠)𝑎𝑑𝑠×𝑀=𝑎. (3.35) Thus, all conditions of Lemma 2.9 are satisfied. Hence problem (1.1) has at least three positive concave solutions 𝑢₁, 𝑢₂, 𝑢₃ satisfying (3.28).

4. Example

Consider the nonlinear FBVPs 𝐷^𝛼₀₊𝑢(𝑡)=𝑓𝑡,𝑢(𝑡),𝐷^𝛽₀₊𝑢(𝑡)(),𝑢(0),𝑡∈0,1+𝑢^′(0)=0,𝑢(1)+𝑢^′(1)=0, (4.1) where 𝛼=1.6, 𝛽=0.5, 𝑛=2 and 𝑓()=1𝑡,𝑢,𝑣𝜋⁴𝑒^𝑡+1𝑢10⁴+1𝑣100sin110000,0≤𝑢≤10,𝜋⁴𝑒^𝑡1+1000+𝑣100sin10000,𝑢>10. (4.2) Choose 𝑎=1, 𝑏=3, 𝑑=10000. By a simple computation, we obtain that 𝛾₀≈8.7149,𝛾₁≈8.7149,𝑀≈2.5181,𝑁≈2.8672. (4.3) We can check that the nonlinear term 𝑓(𝑡,𝑢,𝑣) satisfies (1)𝑓(𝑡,𝑢,𝑣)<𝑑/𝑁, (𝑡,𝑢,𝑣)∈[0,1]×[0,87149]×[−10000,10000],(2)𝑓(𝑡,𝑢,𝑣)>6𝑏/𝑀, (𝑡,𝑢,𝑣)∈[0,1]×[3,18]×[−10000,10000],(3)𝑓(𝑡,𝑢,𝑣)<𝑎/𝑀, (𝑡,𝑢,𝑣)∈[0,1]×[0,1]×[−10000,10000]. Then all assumptions of Theorem 3.6 are satisfied. Thus problem (4.1) has at least three positive solutions 𝑢₁(𝑡), 𝑢₂(𝑡), 𝑢₃(𝑡) satisfying max_0≤𝑡≤1𝐷^𝛽₀₊𝑢(𝑡)≤10000,𝑖=1,2,3;3<min_{1/2≤𝑡≤1}𝑢₁(𝑡),1<max_0≤𝑡≤1𝑢₂(𝑡),min_{1/2≤𝑡≤1}𝑢₂(𝑡)<3,max_0≤𝑡≤1𝑢₃(𝑡)≤1. (4.4)

Acknowledgments

This work was supported by the Anhui Provincial Natural Science Foundation (10040606Q50) and Innovation Program of Shanghai Municipal Education Commission (13zz118).