Abstract
We are concerned with the existence of at least one, two or three positive solutions for the boundary value problem with three-point multi-term fractional integral boundary conditions:
where is the standard Riemann-Liouville fractional derivative. Our analysis relies on the Krasnoselskii fixed point theorem and the Leggett-Williams fixed point theorem. Some examples are also given to illustrate the main results.
MSC:26A33, 34A08, 34B18.
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1 Introduction
In recent years, the interest in the study of fractional differential equations has been growing rapidly. Fractional differential equations have arisen in mathematical models of systems and processes in various fields such as aerodynamics, acoustics, mechanics, electromagnetism, signal processing, control theory, robotics, population dynamics, finance, etc.
We refer a reader interested in the systematic development of the topic to the books [1–7]. A variety of results on initial and boundary value problems of fractional differential equations and inclusions can easily be found in the literature on the topic. For some recent results, we can refer to [8–19] and references cited therein.
In this paper, we concentrate on the study of positive solutions to the boundary value problems of fractional differential equations. More precisely, we consider the nonlinear fractional differential equation
subject to three-point multi-term fractional integral boundary conditions
where is the standard Riemann-Liouville fractional derivative of order q, is the Riemann-Liouville fractional integral of order , , and , , are real constants such that .
We mention that integral boundary conditions are encountered in various applications such as population dynamics, blood flow models, chemical engineering, cellular systems, heat transmission, plasma physics, thermoelasticity, etc. Nonlocal conditions come up when values of the function on the boundary is connected to values inside the domain.
One of the most frequently used tools for proving the existence of positive solutions to the integral equations and boundary value problems is the Krasnoselskii theorem on cone expansion and compression and its norm-type version due to Guo and Lakshmikantham [20]. The main idea is to construct a cone in a Banach space and a completely continuous operator defined on this cone based on the corresponding Green’s function and then find fixed points of the operator. See [9, 18] and references therein for recent development.
The rest of this paper is organized as follows. In Section 2 we present some necessary basic knowledge and definitions for fractional calculus theory and give the corresponding Green’s function of boundary value problem (1.1)-(1.2). Moreover, some properties of the Green’s function are also proved. In Section 3 we use the properties of the corresponding Green’s function and the Guo-Krasnoselskii fixed point theorem to show the existence of at least one or two positive solutions of (1.1)-(1.2) under the condition that the nonlinear f is either sublinear or superlinear. In Section 4 we prove the existence of at least three positive solutions via the Leggett-Williams fixed point theorem. Finally, illustrative examples are presented in Section 5.
2 Preliminaries
In this section, we introduce some notations and definitions of fractional calculus [3, 4] and present preliminary results needed in our proofs later.
Definition 2.1 The Riemann-Liouville fractional integral of order of a function is defined by
provided the right-hand side is point-wise defined on , where Γ is the gamma function.
Definition 2.2 The Riemann-Liouville fractional derivative of order of a continuous function is defined by
where , denotes the integer part of a real number α, provided the right-hand side is point-wise defined on .
From the definition of the Riemann-Liouville fractional derivative, we can obtain the following lemmas.
Lemma 2.1 (see [4])
Let and . Then the fractional differential equation has a unique solution
where , , and .
Lemma 2.2 (see [4])
Let . Then for it holds
where , , and .
The following property (Dirichlet’s formula) of the fractional calculus is well known [1]:
which has the form
For convenience we put
Lemma 2.3 Let , , , , and . The unique solution of the boundary value problem
is the integral equation
where is the Green’s function given by
where
and
Proof Using Lemmas 2.1-2.2, problem (2.2)-(2.3) can be expressed as an equivalent integral equation
for . The first condition of (2.3) implies that . Taking the Riemann-Liouville fractional integral of order for (2.8) and using Dirichlet’s formula, we get that
The second condition of (2.3) yields
Then we have that
Therefore, the unique solution of boundary value problem (2.2)-(2.3) is written as
Hence, by taking into account (2.1), we have
The proof is completed. □
Lemma 2.4 The Green’s function in (2.5) satisfies the following conditions:
(P1) is continuous on ;
(P2) for all ;
(P3) for all ;
(P4) ;
(P5) for .
Proof It is easy to check that (P1) holds. To prove (P2), we will show that and , , for all .
Let for , then we have
Let for , then we get . Therefore, for all . Now, let for , then we get
Let for , then we have . Therefore, , , which implies that , , for all .
To prove (P3), we will show that for . For , by the definition of , we have
Hence, is decreasing with respect to t. Then we have for . For , by the definition of , we have that is increasing with respect to t. Thus for . Therefore, for .
From the above analysis, we have for that
To prove (P4), by direct integration, we have
To prove (P5), from and , , for all , we have
for . This completes the proof. □
Let be the Banach space endowed with the supremum norm . Define the cone by
and the operator by
In view of Lemma 2.3, the positive solutions of problem (1.1)-(1.2) are given by the operator equation .
Lemma 2.5 Suppose that is continuous. The operator is completely continuous.
Proof Since for , we have for all . Hence, .
For a constant , we define .
Let
Then, for , from Lemma 2.4, one has
Therefore, , and so is uniformly bounded.
Now, we shall show that is equicontinuous. For , , , we have
where L is defined by (2.10). Since is continuous on , therefore is uniformly continuous on . Hence, for any , there exists a positive constant
whenever , we have the following two cases.
Case 1. .
Therefore,
Case 2. , .
Therefore,
Thus, is equicontinuous. In view of the Arzelá-Ascoli theorem, we have that is compact, i.e., is a completely continuous operator. This completes the proof. □
For convenience, we set
3 Existence of at least one or two positive solutions
For the main results of this section, we use the well-known Guo-Krasnoselskii fixed point theorem.
Theorem 3.1 ([20])
Let E be a Banach space, and let be a cone. Assume that , are open subsets of E with , , and let be a completely continuous operator such that:
-
(i)
, , and , ; or
-
(ii)
, , and , .
Then T has a fixed point in .
Theorem 3.2 Let be a continuous function. Assume that there exist constants , and such that:
(H1) , for ;
(H2) , for .
Then boundary value problem (1.1)-(1.2) has at least one positive solution u such that
Proof We shall show that the first part of Theorem 3.1 is satisfied. By Lemma 2.5, the operator is completely continuous.
Let , then for any , we have for all . From (H1), it follows for that
which yields
Let , then for any , we have for all . For , assumption (H2) yields
one has
Therefore, from (3.1), (3.2) and the first part of Theorem 3.1, it follows that A has a fixed point in which is a positive solution of boundary value problem (1.1)-(1.2). Hence, problem (1.1)-(1.2) has at least one positive solution u such that
The proof is complete. □
Theorem 3.3 Let all the assumptions of Theorem 3.2 hold. In addition, assume that
(H3) .
Then boundary value problem (1.1)-(1.2) has at least two positive solutions and such that
Proof It follows from Theorem 3.2 that there exists a positive solution such that . From (H3), there exists such that for any and for any ,
Let . Then, for any and for , we have
This implies that
It follows from (3.2), (3.3) and the second part of Theorem 3.1 that A has a fixed point in .
Therefore, we conclude that boundary value problem (1.1)-(1.2) has at least two positive solutions such that
□
Similarly to the previous theorems, we can prove the following.
Theorem 3.4 Let be a continuous function. Assume that there exist constants and , such that:
(H4) for ;
(H5) for ;
(H6) .
Then boundary value problem (1.1)-(1.2) has at least two positive solutions and such that
Corollary 3.1 Assume that conditions (H4)-(H5) are satisfied. Then boundary value problem (1.1)-(1.2) has at least one positive solution u such that
4 Existence of at least three positive solutions
In this section we use the Leggett-Williams fixed point theorem to prove the existence of at least three positive solutions.
Definition 4.1 A continuous mapping is said to be a nonnegative continuous concave functional on the cone of a real Banach space E provided that
for all and .
Let be constants. We define , and .
Theorem 4.1 ([21])
Let be a cone in the real Banach space E and be a constant. Assume that there exists a concave nonnegative continuous functional θ on with for all . Let be a completely continuous operator. Suppose that there exist constants such that the following conditions hold:
-
(i)
and for ;
-
(ii)
for ;
-
(iii)
for with .
Then A has at least three fixed points , and in .
Furthermore, , , with .
We now prove the following result.
Theorem 4.2 Let be a continuous function. Suppose that there exist constants such that the following assumptions hold:
(H7) for ;
(H8) for ;
(H9) for .
Then boundary value problem (1.1)-(1.2) has at least three positive solutions , and with
and
Proof We will show that all the conditions of the Leggett-Williams fixed point theorem are satisfied for the operator A defined by (2.9).
For , we have . From condition (H9), we have for . Therefore,
which implies . Hence, .
If , then condition (H7) yields
Thus . Therefore, condition (ii) of Theorem 4.1 holds.
Define a concave nonnegative continuous functional θ on by . To check condition (i) of Theorem 4.1, we choose for . It is easy to see that and ; consequently, the set . Hence, if , then for . From condition (H8), we have
Thus for all . This shows that condition (i) of Theorem 4.1 is also satisfied.
We finally show that condition (iii) of Theorem 4.1 also holds. Assume that with , then we have for all . From (H8) and Lemma 2.4, one has
So, condition (iii) of Theorem 4.1 is satisfied. Therefore, an application of Theorem 4.1 implies that boundary value problem (1.1)-(1.2) has at least three positive solutions , and such that
The proof is complete. □
5 Examples
In this section, we present some examples to illustrate our results.
Example 5.1 Consider the following three-point three-term fractional integral boundary value problem:
where
Set , , , , , , , , , and we can show that
Through a simple calculation we can get
Choose , , and , then satisfies
and
Thus, (H1) and (H2) hold. By Theorem 3.2, we have that boundary value problem (5.1)-(5.2) has at least one positive solution u such that .
Example 5.2 Consider the following three-point four-term fractional integral boundary value problem:
where
Here , , , , , , , , , , , and we can show that
Through a simple calculation we can get
Choose , , and , then satisfies
and
and
Thus, (H1), (H2) and (H3) hold. By Theorem 3.3, we have that boundary value problem (5.3)-(5.4) has at least two positive solutions and such that .
Example 5.3 Consider the following three-point five-term fractional integral boundary value problem:
where
Set , , , , , , , , , , , , , and we can show that
Through a simple calculation we can get
Choose , and , then satisfies
and
and
Thus, (H7), (H8) and (H9) hold. By Theorem 4.2, we have that boundary value problem (5.5)-(5.6) has at least three positive solutions , and such that , and with .
Authors’ information
Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) - Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
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Acknowledgements
We would like to thank the reviewers for their valuable comments and suggestions on the manuscript. This research of J Tariboon and W Sudsutad is supported by King Mongkut’s University of Technology North Bangkok, Thailand.
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Tariboon, J., Ntouyas, S.K. & Sudsutad, W. Positive solutions for fractional differential equations with three-point multi-term fractional integral boundary conditions. Adv Differ Equ 2014, 28 (2014). https://doi.org/10.1186/1687-1847-2014-28
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DOI: https://doi.org/10.1186/1687-1847-2014-28