Existence of positive solutions for third-order boundary value problems with integral boundary conditions on time scales
© Karaca and Tokmak; licensee Springer. 2013
Received: 16 August 2013
Accepted: 2 October 2013
Published: 8 November 2013
In this paper, four functionals fixed point theorem is used to verify the existence of at least one positive solution for third-order boundary value problems with integral boundary conditions for an increasing homeomorphism and homomorphism on time scales. We also provide an example to demonstrate our results.
The theory of time scales was introduced by Hilger  in his PhD thesis in 1988. Theoretically, this new theory has not only unified continuous and discrete equations, but has also exhibited much more complicated dynamics on time scales. Moreover, the study of dynamic equations on time scales has led to several important applications, for example, insect population models, biology, neural networks, heat transfer, and epidemic models, see [2–12].
Recently, scientists have noticed that the boundary conditions in many areas of applied mathematics and physics come down to integral boundary conditions. For instance, the models on chemical engineering, heat conduction, thermo-elasticity, plasma physics, and underground water flow can be reduced to the nonlocal problems with integral boundary conditions. For more information about this subject, we refer the readers to the excellent survey by Corduneanu , and Agarwal and O’Regan . In addition, such kind of boundary value problem in a Banach space has been studied by some researchers, we refer the readers to [15–18] and the references therein. However, to the best of our knowledge, little work has been done on the existence of positive solutions for third-order boundary value problem with integral boundary conditions on time scales. This paper attempts to fill this gap in literature.
The main tool is Guo-Krasnoselskii fixed point theorem.
By using Krasnoselskii’s fixed point theorem, he obtained the existence criteria of at least one positive solution.
By using Legget-Williams fixed point theorem, they obtained the existence criteria of at least three positive solutions.
They investigated the existence, nonexistence, and multiplicity of positive solutions for a class of nonlinear boundary value problems of third-order differential equations with integral boundary conditions in ordered Banach spaces by means of fixed-point principle in cone and the fixed-point index theory for strict set contraction operator.
The arguments were based upon the fixed-point principle in cone for strict set contraction operators.
where is p-Laplacian operator, i.e., , , , . By using fixed point theorems in cones, they obtained the existence of multiple positive solutions for singular nonlinear boundary value problem.
If , then for all ;
ϕ is a continuous bijection, and its inverse mapping is also continuous;
for all .
Throughout this paper, we assume that the following conditions hold:
(C1) with ,
(C3) q, and .
By using the four functionals fixed point theorem , we get the existence of at least one positive solution for BVP (1.1). In fact, our result is also new when (the differential case) and (the discrete case). Therefore, the result can be considered as a contribution to this field.
This paper is organized as follows. In Section 2, we provide some definitions and preliminary lemmas, which are the key tools for our main result. We give and prove our main result in Section 3. Finally, in Section 4, we give an example to demonstrate our result.
In this section, to state the main results of this paper, we need the following lemmas.
Lemma 2.1 Let (C1)-(C3) hold. Assume that
then u is a solution of the boundary value problem (1.1).
which implies that and satisfy (2.8) and (2.9), respectively. □
Lemma 2.2 Let (C1)-(C3) hold. Assume that
(C5) , , .
Proof It is an immediate subsequence of the facts that on and , . □
Lemma 2.3 Let (C1)-(C3) and (C5) hold. Assume that
Then the solution of problem (1.1) satisfies for .
According to Lemma 2.2, we have that . So, . However, this contradicts to condition (C6). Consequently, for . □
The proof is finalized. □
where G, and are defined as in (2.7), (2.8) and (2.9), respectively.
Lemma 2.5 Let (C1)-(C6) hold. Then is completely continuous.
Proof By Arzela-Ascoli theorem, we can easily prove that operator T is completely continuous. □
3 Main results
Lemma 3.1 
for all with and ;
for all with ;
for all with and ;
for all with .
Then T has a fixed point u in .
Let , and be defined by (3.1).
Theorem 3.1 Assume that (C1)-(C6) hold. If there exist constants r, j, l, R with , , and suppose that f satisfies the following conditions:
(C7) for ;
(C8) for .
Proof The boundary value problem (1.1) has a solution if and only if u solves the operator equation . Thus, we set out to verify that the operator T satisfies four functionals fixed point theorem, which will prove the existence of a fixed point of T.
which means that is a bounded set. According to Lemma 2.5, it is clear that is completely continuous.
So, , which means that (i) in Lemma 3.1 is satisfied.
So, . Hence, (ii) in Lemma 3.1 is fulfilled.
Thus, (iii) and (v) in Lemma 3.1 hold. We finally prove that (iv) in Lemma 3.1 holds.
The proof is completed. □
4 An example
The authors would like to thank the referees for their valuable suggestions and comments.
- Hilger, S: Ein masskettenkalkül mit anwendug auf zentrumsmanningfaltigkeiten. PhD thesis, Universitat Würzburg (1988)MATHGoogle Scholar
- Agarwal RP, Bohner M: Basic calculus on time scales and some of its applications. Results Math. 1999, 35: 3–22. 10.1007/BF03322019MathSciNetView ArticleMATHGoogle Scholar
- Agarwal RP, O’Regan D: Nonlinear boundary value problems on time scales. Nonlinear Anal. 2001, 44: 527–535. 10.1016/S0362-546X(99)00290-4MathSciNetView ArticleMATHGoogle Scholar
- Anderson DR, Karaca IY: Higher-order three-point boundary value problem on time scales. Comput. Math. Appl. 2008, 56: 2429–2443. 10.1016/j.camwa.2008.05.018MathSciNetView ArticleMATHGoogle Scholar
- Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston; 2001.View ArticleMATHGoogle Scholar
- Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston; 2002.MATHGoogle Scholar
- Han W, Kang S: Multiple positive solutions of nonlinear third-order BVP for a class of p -Laplacian dynamic equations on time scales. Math. Comput. Model. 2009, 49: 527–535. 10.1016/j.mcm.2008.08.002MathSciNetView ArticleMATHGoogle Scholar
- Lakshmikantham V, Sivasundaram S, Kaymakcalan B: Dynamic Systems on Measure Chains. Kluwer Academic, Dordrecht; 1996.View ArticleMATHGoogle Scholar
- Liu J, Xu F: Triple positive solutions for third-order m -point boundary value problems on time scales. Discrete Dyn. Nat. Soc. 2009., 2009: Article ID 123283Google Scholar
- Sun TT, Wang LL, Fan YH: Existence of positive solutions to a nonlocal boundary value problem with p -Laplacian on time scales. Adv. Differ. Equ. 2010., 2010: Article ID 809497Google Scholar
- Tokmak F, Karaca IY: Existence of positive solutions for third-order m -point boundary value problems with sign-changing nonlinearity on time scales. Abstr. Appl. Anal. 2012., 2012: Article ID 5067163Google Scholar
- Tokmak F, Karaca IY: Existence of symmetric positive solutions for a multipoint boundary value problem with sign-changing nonlinearity on time scales. Bound. Value Probl. 2013., 52: Article ID 12Google Scholar
- Corduneanu C: Integral Equations and Applications. Cambridge University Press, Cambridge; 1991.View ArticleMATHGoogle Scholar
- Agarwal RP, O’Regan D: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht; 2001.View ArticleMATHGoogle Scholar
- Boucherif A: Second-order boundary value problems with integral boundary conditions. Nonlinear Anal. 2009, 70: 364–371. 10.1016/j.na.2007.12.007MathSciNetView ArticleMATHGoogle Scholar
- Fu D, Ding W: Existence of positive solutions of third-order boundary value problems with integral boundary conditions in Banach spaces. Adv. Differ. Equ. 2013., 2013: Article ID 65Google Scholar
- Li Y, Zhang T: Multiple positive solutions for second-order p -Laplacian dynamic equations with integral boundary conditions. Bound. Value Probl. 2011., 2011: Article ID 867615Google Scholar
- Zhao J, Wang P, Ge W: Existence and nonexistence of positive solutions for a class of third order BVP with integral boundary conditions in Banach spaces. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 402–413. 10.1016/j.cnsns.2009.10.011MathSciNetView ArticleMATHGoogle Scholar
- Ma R: Multiple positive solutions for nonlinear m -point boundary value problems. Appl. Math. Comput. 2004, 148: 249–262. 10.1016/S0096-3003(02)00843-3MathSciNetView ArticleMATHGoogle Scholar
- Avery R, Henderson J, O’Regan D: Four functionals fixed point theorem. Math. Comput. Model. 2008, 48: 1081–1089. 10.1016/j.mcm.2007.12.013MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.