- Open Access
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.
- fractional differential equations
- nonlocal boundary conditions
- positive solutions
- fixed point theorem
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. □
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 ;
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.
- Miller KS, Ross B: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York; 1993.MATHGoogle Scholar
- Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon; 1993.MATHGoogle Scholar
- Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.MATHGoogle Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
- Lakshmikantham V, Leela S, Vasundhara Devi J: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge; 2009.MATHGoogle Scholar
- Baleanu D, Diethelm K, Scalas E, Trujillo JJ Series on Complexity, Nonlinearity and Chaos. In Fractional Calculus Models and Numerical Methods. World Scientific, Boston; 2012.Google Scholar
- Diethelm K Lecture Notes in Mathematics 2004. In The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin; 2010.Google Scholar
- Guezane-Lakoud A, Khaldi R: Solvability of a three-point fractional nonlinear boundary value problem. Differ. Equ. Dyn. Syst. 2012, 20: 395-403. 10.1007/s12591-012-0125-7MATHMathSciNetView ArticleGoogle Scholar
- Guezane-Lakoud A, Khaldi R: Positive solution to a higher order fractional boundary value problem with fractional integral condition. Rom. J. Math. Comput. Sci. 2012, 2: 41-54.MATHMathSciNetGoogle Scholar
- Kaufmann E: Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem. Discrete Contin. Dyn. Syst. 2009, 2009: 416-423. suppl.MATHMathSciNetGoogle Scholar
- Wang J, Xiang H, Liu Z: Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations. Int. J. Differ. Equ. 2010., 2010: Article ID 186928Google Scholar
- Bai Z, Lü H: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 2005, 311: 495-505. 10.1016/j.jmaa.2005.02.052MATHMathSciNetView ArticleGoogle Scholar
- Bai Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 2010, 72: 916-924. 10.1016/j.na.2009.07.033MATHMathSciNetView ArticleGoogle Scholar
- Sudsutad W, Tariboon J: Boundary value problems for fractional differential equations with three-point fractional integral boundary conditions. Adv. Differ. Equ. 2012., 2012: Article ID 93Google Scholar
- Ntouyas SK: Existence results for nonlocal boundary value problems for fractional differential equations and inclusions with fractional integral boundary conditions. Discuss. Math., Differ. Incl. Control Optim. 2013, 33: 17-39. 10.7151/dmdico.1146MATHMathSciNetView ArticleGoogle Scholar
- Ntouyas SK: Boundary value problems for nonlinear fractional differential equations and inclusions with nonlocal and fractional integral boundary conditions. Opusc. Math. 2013, 33: 117-138. 10.7494/OpMath.2013.33.1.117MATHMathSciNetView ArticleGoogle Scholar
- Guezane-Lakoud A, Khaldi R: Solvability of a fractional boundary value problem with fractional integral condition. Nonlinear Anal. 2012, 75: 2692-2700. 10.1016/j.na.2011.11.014MATHMathSciNetView ArticleGoogle Scholar
- Yang W: Positive solutions for nonlinear Caputo fractional differential equations with integral boundary conditions. J. Appl. Math. Comput. 2014. 10.1007/s12190-013-0679-8Google Scholar
- Ahmad B, Ntouyas SK, Assolani A: Caputo type fractional differential equations with nonlocal Riemann-Liouville integral boundary conditions. J. Appl. Math. Comput. 2013, 41: 339-350. 10.1007/s12190-012-0610-8MATHMathSciNetView ArticleGoogle Scholar
- Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, New York; 1988.MATHGoogle Scholar
- Leggett RW, Williams LR: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J. 1979, 28: 673-688. 10.1512/iumj.1979.28.28046MATHMathSciNetView ArticleGoogle Scholar
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