Multiplicity Result of Positive Solutions for Nonlinear Differential Equation of Fractional Order
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.
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 , and . Here 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  considered the following Dirichlet boundary value problem of fractional differential equation: By means of different fixed-point theorems on cone, some existence and multiplicity results of positive solutions were obtained. Jiang and Yuan  improved the results in  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.  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 may be singular at . As to positive solutions of problem (1.1), under the case that the nonlinear term was not involved with the derivative of the function , Zhang  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  investigated the existence and uniqueness of positive solution for three-point boundary value problem where , , . In , 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.  considered the boundary value problem of fractional differential equation with integral condition where , 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 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 is involved in the nonlinear term. In , 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 where depended on the first order derivative of . Very recently, Yang et al.  considered following boundary value problem 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 , 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.  established the existence of at least three positive solutions of second-order multipoint boundary value problem
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 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 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 . 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 of a function is given by provided the right side is point-wise defined on .
Definition 2.2. The Caputo's fractional derivative of order of a continuous function is given by where , provided that the right side is point-wise defined on .
Lemma 2.3. Let . Then the fractional differential equation has solutions
Lemma 2.4. Let . Then
Definition 2.5. Let be a real Banach space over . A nonempty convex closed set is said to be a cone provided that(1), for all , ,(2) implies .
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 is said to be a continuous nonnegative convex functional on cone of a real Banach space provided that is continuous and
Definition 2.8. The map is said to be a continuous nonnegative concave functional on cone of a real Banach space provided that is continuous and
Let and be nonnegative continuous convex functionals on , a nonnegative continuous concave functional on , and a nonnegative continuous functional on P. Then for positive numbers , , , and , we define the following convex sets: and a closed set
Lemma 2.9 (see ). Let be a cone in Banach space E. Let , be nonnegative continuous convex functionals on , a nonnegative continuous concave functional on , and a nonnegative continuous functional on satisfying such that for some positive numbers and d, for all . Suppose is completely continuous and there exist positive numbers , , with such that and for ; for with ; and for with .Then has at least three fixed points such that
3. Main Results
Lemma 3.1 (see ). Given , then boundary value problem is equivalent to where
Lemma 3.2. Given , assume that is a solution of boundary value problem Then
Lemma 3.3 (see ). The function satisfies the following conditions:(1), , for ;(2)there exist a positive function such that where
Lemma 3.4. Assume that and is a solution of boundary value problem (3.1). There exists a positive constant such that
Proof. From Lemma 3.2,
By a simple computation, we have
Let the Banach space , be endowed with the norm We define the cone by Denote the positive constants
Lemma 3.5. Let be the operator defined by Then is completely continuous.
Proof. The operator is nonnegative and continuous in view of the nonnegativeness and continuity of functions and . Let be bounded. Then there exists a positive constant such that , . Denote
Then for , we have
Hence is bounded. On the other hand, for , , one has
By means of the Arzela-Ascoli theorem, is completely continuous. Furthermore, for , we have
Thus, is completely continuous.
Let the continuous nonnegative concave functional , the the continuous nonnegative convex functionals , , and the continuous nonnegative functional be defined on the cone by By Lemmas 3.3 and 3.4, the functionals defined above satisfy that where . Therefore condition (2.10) of Lemma 2.9 is satisfied.
Assume that there exist constants with , such that, ,, ,, .
Theorem 3.6. Under assumptions ()–(), problem (1.1) has three positive solutions , , satisfying
Proof. Problem (1.1) has a solution if and only if solves the operator equation
For , we have . From assumption , we obtain
The fact that the constant function and implies that .
For , we have and for . From assumption , Thus which means , for all . These ensure that condition of Lemma 2.9 is satisfied. Secondly, for all with , Thus, condition of Lemma 2.3 holds. Finally we show that also holds. We see that and . Suppose that with . Then by assumption , Thus, all conditions of Lemma 2.9 are satisfied. Hence problem (1.1) has at least three positive concave solutions , , satisfying (3.28).
Consider the nonlinear FBVPs where , , and Choose , , . By a simple computation, we obtain that We can check that the nonlinear term satisfies (1), ,(2), ,(3), . Then all assumptions of Theorem 3.6 are satisfied. Thus problem (4.1) has at least three positive solutions , , satisfying
This work was supported by the Anhui Provincial Natural Science Foundation (10040606Q50) and Innovation Program of Shanghai Municipal Education Commission (13zz118).
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.