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

and Applied Analysis 3 Ge obtained the existence of positive solutions for second-order three-point boundary value problem u′′ f ( t, u t , u′ t ) 0, t ∈ 0, 1 , u 0 0, u 1 αu ( η ) , 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 0 u t f ( t, u t , u′ t ) , t ∈ 0, 1 , u 0 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′′ f ( t, u t , u′ t ) 0, t ∈ 0, 1 , u′ 0 m−2 ∑ i 1 βiu ′ ξi , u 1 m−2 ∑ i 1 αiu ξi . 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. 4 Abstract and Applied Analysis 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 0 u t 1 Γ α ∫ t 0 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 0 u t 1 Γ n − α ∫ t 0 u 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 0 u t 0 has solutions u t c0 c1t c2t · · · cn−1tn−1, ci ∈ R, i 0, 1, . . . , n − 1. 2.3 Lemma 2.4. Let α > 0. Then I 0 D α 0 u t u t c0 c1t c2t 2 · · · cn−1tn−1, ci ∈ 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 α ( tx 1 − t y) ≤ tα x 1 − t α(y), 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 β ( tx 1 − t y) ≥ tβ x 1 − t β(y), x, y ∈ P, t ∈ 0, 1 . 2.6 Abstract and Applied Analysis 5 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:and Applied Analysis 5 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 ( γ, d ) { x ∈ P | γ x < d}, P ( γ, α, b, d ) { x ∈ P | b ≤ α x , γ x ≤ d}, P ( γ, θ, α, b, c, d ) { x ∈ P | b ≤ α x , θ x ≤ c, γ x ≤ d}, 2.7


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 where 1 < α < 2, 0 < β < α−1 and f : C 0, 1 ×R ×R, R . Here D α 0 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 2 Abstract and Applied Analysis 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: 0 u t f t, u t 0, t ∈ 0, 1 , u 0 0 u 1 , 1 < α ≤ 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 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 where α > 2, 1 0 u 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

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.

Preliminary Results
Definition 2.1. The Riemann-Liouville fractional integral of order α > 0 of a function u t : 0, ∞ → R is given by 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 where n − 1 < α ≤ n, provided that the right side is point-wise defined on 0, ∞ . Then 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 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 Abstract and Applied Analysis 5 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: and a closed set Then T has at least three fixed points x 1 , x 2 , x 3 ∈ P γ, d such that Abstract and Applied Analysis is equivalent to

3.3
Lemma 3.2. Given y t ∈ C 0, 1 , assume that u t is a solution of boundary value problem

3.5
Proof. From Lemmas 2.3 and 2.4, we get that The boundary condition u 0 u 0 0 implies that C 1 C 2 0. Considering the boundary condition u 1 u 1 0, we have From the definition of the Caputo derivative of fractional order, we see 3.8

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 γ 0 such that Abstract and Applied Analysis k s y s ds.

3.13
Denote h s K s k s 3.14 By a simple computation, we have

3.16
Let the Banach space E {u t ∈ C 0, 1 , D β u t ∈ C 0, 1 } be endowed with the norm We define the cone P ⊂ E by

3.18
Abstract and Applied Analysis 9 Denote the positive constants

3.19
Lemma 3.5. Let T : P → E be the operator defined by

Abstract and Applied Analysis
Hence T Ω is bounded. On the other hand, for u ∈ Ω, t 1 , t 2 ∈ 0, 1 , one has

3.24
Abstract and Applied Analysis

11
By means of the Arzela-Ascoli theorem, T is completely continuous. Furthermore, for u ∈ P , we have

3.26
By Lemmas 3.3 and 3.4, the functionals defined above satisfy that where γ 1 max{γ 0 , 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

3.35
Thus, all conditions of Lemma 2.9 are satisfied. Hence problem 1.1 has at least three positive concave solutions u 1 , u 2 , u 3 satisfying 3.28 .

Example
Consider the nonlinear FBVPs