- Open Access
Existence of positive solutions for third-order boundary value problems with integral boundary conditions on time scales
Journal of Inequalities and Applications volume 2013, Article number: 498 (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.
In , Ma considered the existence and multiplicity of positive solutions for the m-boundary value problems
The main tool is Guo-Krasnoselskii fixed point theorem.
In , Boucherif considered the second-order boundary value problem with integral boundary conditions
By using Krasnoselskii’s fixed point theorem, he obtained the existence criteria of at least one positive solution.
In , Li and Zhang were concerned with the second-order p-Laplacian dynamic equations with integral boundary conditions on time scales
By using Legget-Williams fixed point theorem, they obtained the existence criteria of at least three positive solutions.
In , Zhao et al. considered the third-order differential equation
subject to one of the following integral boundary conditions
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.
In , Fu and Ding considered the third-order boundary value problems with integral boundary conditions
The arguments were based upon the fixed-point principle in cone for strict set contraction operators.
In , Han and Kang were concerned with the existence of multiple positive solutions of the third-order p-Laplacian dynamic equation on time scales
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.
Motivated by the results above, in this study, we consider the following third-order boundary value problem (BVP) on time scales:
where is a time scale, , , is an increasing homeomorphism and positive homomorphism with . A projection is called an increasing homeomorphism and positive homomorphism if the following conditions are satisfied:
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.
We define , which is a Banach space with the norm
Define the cone by
Denote by θ and φ, the solutions of the corresponding homogeneous equation
under the initial conditions,
Using the initial conditions (2.2), we can deduce from equation (2.1) for θ and φ the following equations:
Lemma 2.1 Let (C1)-(C3) hold. Assume that
If is a solution of the equation
then u is a solution of the boundary value problem (1.1).
Proof Let u satisfy the integral equation (2.6), then u is a solution of problem (1.1). Then we have
So, we get
From (2.11) and (2.12), we get
which implies that and satisfy (2.8) and (2.9), respectively. □
Lemma 2.2 Let (C1)-(C3) hold. Assume that
(C5) , , .
Then for , the solution u of problem (1.1) satisfies
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 .
Proof Assume that the inequality holds. Since is nonincreasing on , one can verify that
From the boundary conditions of problem (1.1), we have
The last inequality yields
Therefore, we obtain that
According to Lemma 2.2, we have that . So, . However, this contradicts to condition (C6). Consequently, for . □
Lemma 2.4 If (C1)-(C6) hold, then for , where
Proof For , since is nonincreasing on , one arrives at
i.e., . Hence,
By , we get
The proof is finalized. □
From Lemma 2.4, we obtain
Now, define an operator by
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
Let α and Ψ be nonnegative continuous concave functionals on , and let β and Φ be nonnegative continuous convex functionals on , then for positive numbers r, j, l and R, we define the sets
Lemma 3.1 
If is a cone in a real Banach space , α and Ψ are nonnegative continuous concave functionals on , β and Φ are nonnegative continuous convex functionals on , and there exist positive numbers r, j, l and R such that
is a completely continuous operator, and is a bounded set. If
for all with and ;
for all with ;
for all with and ;
for all with .
Then T has a fixed point u in .
Suppose that with . For the convenience, we take the notations
and define the maps
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 .
Then BVP (1.1) has at least one positive solution such that
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.
We first show that is bounded, and is completely continuous. For all with Lemma 2.4, we have
which means that is a bounded set. According to Lemma 2.5, it is clear that is completely continuous.
Clearly, . By direct calculation,
So, , which means that (i) in Lemma 3.1 is satisfied.
For all with and , since is nonincreasing on , we have
So, . Hence, (ii) in Lemma 3.1 is fulfilled.
For all with ,
and for all with ,
Thus, (iii) and (v) in Lemma 3.1 hold. We finally prove that (iv) in Lemma 3.1 holds.
For all with and , we have
Thus, all conditions of Lemma 3.1 are satisfied. T has a fixed point u in . Therefore, BVP (1.1) has at least one positive solution such that
The proof is completed. □
4 An example
Example 4.1 In BVP (1.1), suppose that , , , , , and , i.e.,
By simple calculation, we get , , , , , , , and
Set , , then we get
Choose , , and , it is easy to check that , , and conditions (C1)-(C6) are satisfied. Now, we show that (C7) and (C8) are satisfied:
So, all conditions of Theorem 3.1 hold. Thus, by Theorem 3.1, BVP (4.1) has at least one positive solution u such that
Hilger, S: Ein masskettenkalkül mit anwendug auf zentrumsmanningfaltigkeiten. PhD thesis, Universitat Würzburg (1988)
Agarwal RP, Bohner M: Basic calculus on time scales and some of its applications. Results Math. 1999, 35: 3–22. 10.1007/BF03322019
Agarwal RP, O’Regan D: Nonlinear boundary value problems on time scales. Nonlinear Anal. 2001, 44: 527–535. 10.1016/S0362-546X(99)00290-4
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.2008.05.018
Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston; 2001.
Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston; 2002.
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.2008.08.002
Lakshmikantham V, Sivasundaram S, Kaymakcalan B: Dynamic Systems on Measure Chains. Kluwer Academic, Dordrecht; 1996.
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 123283
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 809497
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 5067163
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 12
Corduneanu C: Integral Equations and Applications. Cambridge University Press, Cambridge; 1991.
Agarwal RP, O’Regan D: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht; 2001.
Boucherif A: Second-order boundary value problems with integral boundary conditions. Nonlinear Anal. 2009, 70: 364–371. 10.1016/j.na.2007.12.007
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 65
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 867615
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.2009.10.011
Ma R: Multiple positive solutions for nonlinear m -point boundary value problems. Appl. Math. Comput. 2004, 148: 249–262. 10.1016/S0096-3003(02)00843-3
Avery R, Henderson J, O’Regan D: Four functionals fixed point theorem. Math. Comput. Model. 2008, 48: 1081–1089. 10.1016/j.mcm.2007.12.013
The authors would like to thank the referees for their valuable suggestions and comments.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and typed, read and approved the final manuscript.
About this article
Cite this article
Karaca, I.Y., Tokmak, F. Existence of positive solutions for third-order boundary value problems with integral boundary conditions on time scales. J Inequal Appl 2013, 498 (2013). https://doi.org/10.1186/1029-242X-2013-498
- Green’s function
- time scales
- fixed point theorem
- increasing homeomorphism and positive homomorphism
- integral boundary conditions
- positive solution