- Open Access
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.
- Green’s function
- time scales
- fixed point theorem
- increasing homeomorphism and positive homomorphism
- integral boundary conditions
- positive solution
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. □
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. □
The authors would like to thank the referees for their valuable suggestions and comments.
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