Eigenvalues for iterative systems of nonlinear m-point boundary value problems on time scales
© Karaca and Tokmak; licensee Springer. 2014
Received: 3 October 2013
Accepted: 3 March 2014
Published: 21 March 2014
In this paper, we determine the eigenvalue intervals of the parameters for which there exist positive solutions of the iterative systems of m-point boundary value problems on time scales. The method involves an application of Guo-Krasnosel’skii fixed point theorem. We give an example to demonstrate our main results.
KeywordsGreen’s function iterative system eigenvalue interval time scales boundary value problem fixed point theorem m-point positive solution
The study of dynamic equations on time scales goes back to Stefan Hilger . Theoretically, this new theory has not only unify continuous and discrete equations, but it 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 [2–7].
There has been much interest shown in obtaining optimal eigenvalue intervals for the existence of positive solutions of the boundary value problems on time scales, often using Guo-Krasnosel’skii fixed point theorem. To mention a few papers along these lines, see [8–12]. On the other hand, there is not much work concerning the eigenvalues for iterative system of nonlinear boundary value problems on time scales; see [13, 14].
The method involves application of Guo-Krasnosel’skii fixed point theorem for operators on a cone in a Banach space.
They used the Guo-Krasnosel’skii fixed point theorem.
where is a time scale, , .
Throughout this paper we assume that following conditions hold:
(C1) with ; , for ,
(C2) is continuous, for ,
(C3) and does not vanish identically on any closed subinterval of , for ,
(C4) each of and , , exists as positive real number.
In fact, our results are also new when (the differential case) and (the discrete case). Therefore, the results can be considered as a contribution to this field.
This paper is organized as follows. In Section 2, we construct the Green’s function for the homogeneous problem corresponding to (1.1)-(1.2) and estimate bounds for the Green’s function. In Section 3, we determine the eigenvalue intervals for which there exist positive solutions of the boundary value problem (1.1)-(1.2) by using the Guo-Krasnosel’skii fixed point theorem for operators on a cone in a Banach space. Finally, in Section 4, we give an example to demonstrate our main results.
We need the auxiliary lemmas that will be used to prove our main results.
Lemma 2.1 Let (C1) hold. Assume that
then is a solution of the boundary value problem (2.1).
which implies that and satisfy (2.9) and (2.10), respectively. □
Lemma 2.2 Let (C1) hold. Assume
(C6) , , .
Proof It is an immediate subsequence of the facts that on and , . □
Lemma 2.3 Let (C1) and (C6) hold. Assume
Then the solution of the problem (2.1) satisfies for .
According to Lemma 2.2, we have . So, . However, this contradicts to condition (C7). Consequently, for . □
So, the proof is completed. □
To determine the eigenvalue intervals of the boundary value problem (1.1)-(1.2), we will use the following Guo-Krasnosel’skii fixed point theorem .
Theorem 2.1 
, , and , , or
, , and , .
Then T has a fixed point in .
3 Positive solutions in a cone
where γ is given in (2.13).
Hence, and . In addition, the operator T is completely continuous by an application of the Arzela-Ascoli theorem.
Now, we investigate suitable fixed points of T belonging to the cone . For convenience we introduce the following notations.
there exists an n-tuple satisfying (1.1)-(1.2) such that , , on .
The proof is completed. □
there exists an n-tuple satisfying (1.1)-(1.2) such that , , on .
Since each is assumed to be a positive real number, it follows that , , is unbounded at ∞.
Applying Theorem 2.1 to (3.7) and (3.8), we see that T has a fixed point , which in turn with , we obtain an n-tuple satisfying (1.1)-(1.2) for the chosen values of , . The proof is completed. □
4 An example
Applying Theorem 3.1, we get the optimal eigenvalue interval , , for which the boundary value problem (4.1)-(4.2) has a positive solution.
The authors would like to thank the referees for their valuable suggestions and comments.
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