Positive solutions for fractional differential equations with three-point multi-term fractional integral boundary conditions
© Tariboon et al.; licensee Springer. 2014
Received: 19 August 2013
Accepted: 6 January 2014
Published: 22 January 2014
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.
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.
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 . 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.
provided the right-hand side is point-wise defined on , where Γ is the gamma function.
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 )
where , , and .
Lemma 2.2 (see )
where , , and .
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 ;
(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 . Therefore, , , which implies that , , for all .
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 .
for . This completes the proof. □
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 .
Therefore, , and so is uniformly bounded.
whenever , we have the following two cases.
Case 1. .
Case 2. , .
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. □
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 ()
, , and , ; or
, , 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 .
Proof We shall show that the first part of Theorem 3.1 is satisfied. By Lemma 2.5, the operator is completely continuous.
The proof is complete. □
Theorem 3.3 Let all the assumptions of Theorem 3.2 hold. In addition, assume that
It follows from (3.2), (3.3) and the second part of Theorem 3.1 that A has a fixed point in .
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 ;
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.
for all and .
Let be constants. We define , and .
Theorem 4.1 ()
and for ;
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 .
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).
which implies . Hence, .
Thus . Therefore, condition (ii) of Theorem 4.1 holds.
Thus for all . This shows that condition (i) of Theorem 4.1 is also satisfied.
The proof is complete. □
In this section, we present some examples to illustrate our results.
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 .
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 .
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 .
Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) - Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
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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