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# Eigenvalues for iterative systems of nonlinear m-point boundary value problems on time scales

Ilkay Y Karaca1* and Fatma Tokmak12

Author Affiliations

1 Department of Mathematics, Ege University, Bornova, Izmir, 35100, Turkey

2 Department of Mathematics, Gazi University, Teknikokullar, Ankara, 06500, Turkey

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Boundary Value Problems 2014, 2014:63  doi:10.1186/1687-2770-2014-63

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/63

 Received: 3 October 2013 Accepted: 3 March 2014 Published: 21 March 2014

© 2014 Karaca and Tokmak; licensee Springer.

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 credited.

### Abstract

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.

MSC: 34B18, 34N05.

##### Keywords:
Green’s function; iterative system; eigenvalue interval; time scales; boundary value problem; fixed point theorem; m-point; positive solution

### 1 Introduction

The study of dynamic equations on time scales goes back to Stefan Hilger [1]. 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].

In [15], Ma and Thompson are concerned with determining values λ, by using the Guo-Krasnosel’skii fixed point theorem for which there exist positive solutions of the m-point boundary value problem

In [13], Benchohra et al. studied the eigenvalues for iterative system of nonlinear boundary value problems on time scales,

satisfying the boundary conditions,

The method involves application of Guo-Krasnosel’skii fixed point theorem for operators on a cone in a Banach space.

In [14], Prasad et al. studied the eigenvalues for iterative system of nonlinear boundary value problems on time scales,

satisfying the m-point boundary conditions,

They used the Guo-Krasnosel’skii fixed point theorem.

Motivated by the above results, in this study, we are concerned with determining the eigenvalue intervals of , , for which there exist positive solutions for the iterative system of nonlinear m-point boundary value problems on time scales,

(1.1)

satisfying the m-point boundary conditions,

(1.2)

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.

### 2 Preliminaries

We need the auxiliary lemmas that will be used to prove our main results.

We define , which is a Banach space with the norm

Let , then we consider the following boundary value problem:

(2.1)

Denote by θ and φ, the solutions of the corresponding homogeneous equation

(2.2)

under the initial conditions

(2.3)

Using the initial conditions (2.3), we can deduce from equation (2.2) for θ and φ the following equations:

(2.4)

Set

(2.5)

and

(2.6)

Lemma 2.1Let (C1) hold. Assume that

(C5) .

Ifis a solution of the equation

(2.7)

where

(2.8)

(2.9)

and

(2.10)

thenis a solution of the boundary value problem (2.1).

Proof Let satisfy the integral equation (2.7), then we have

i.e.,

Hence

Since

we have

(2.11)

Since

we have

(2.12)

From (2.11) and (2.12), we get

which implies that and satisfy (2.9) and (2.10), respectively. □

Lemma 2.2Let (C1) hold. Assume

(C6) , , .

Then forwith, the solutionof the problem (2.1) satisfies

Proof It is an immediate subsequence of the facts that on and , . □

Lemma 2.3Let (C1) and (C6) hold. Assume

(C7) .

Then the solutionof the problem (2.1) satisfiesfor.

Proof Assume that the inequality holds. Since is nonincreasing on , one can verify that

From the boundary conditions of the problem (2.1), we have

The last inequality yields

Therefore, we obtain

i.e.,

According to Lemma 2.2, we have . So, . However, this contradicts to condition (C7). Consequently, for . □

Lemma 2.4Let (C1) andhold. Letbe a constant. Then the unique solutionof the problem (2.1) satisfies

whereand

(2.13)

Proof We have from (2.8) that

(2.14)

which implies

(2.15)

for all . Applying (2.8), we have for ,

(2.16)

where

Thus for ,

So, the proof is completed. □

We note that an n-tuple is a solution of the boundary value problem (1.1)-(1.2) if and only if

and

where

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 [16].

Theorem 2.1[16]

Letbe a Banach space, and letbe a cone in. Assumeandare open subsets ofwithand, and let

be a completely continuous operator such that either

(i) , , and, , or

(ii) , , and, .

ThenThas a fixed point in.

### 3 Positive solutions in a cone

In this section, we establish criteria to determine the eigenvalue intervals for which the boundary value problem (1.1)-(1.2) has at least one positive solution in a cone. We construct a cone by

where γ is given in (2.13).

Now, we define an integral operator , for , by

(3.1)

Notice from (C1)-(C6) and Lemma 2.2 that, for , on . Also, we have from (2.8), that

so that

(3.2)

Next, if , we have from Lemma 2.4 and (3.2) that

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.

Let

and

Theorem 3.1Suppose conditions (C1)-(C7) are satisfied. Then, for eachsatisfying

(3.3)

there exists ann-tuplesatisfying (1.1)-(1.2) such that, , on.

Proof Let , , be as in (3.3). Now, let be chosen such that

and

We investigate fixed points of the completely continuous operator defined by (3.1). Now, from the definitions of , , there exists an such that, for each ,

Let with . We have from (2.14) and the choice of ϵ, for ,

It follows in a similar manner from (2.14), for , that

Continuing with this bootstrapping argument, we have, for ,

so that, for ,

Hence, . If we set

then

(3.4)

Next, from the definitions of , , there exists such that, for each ,

Let

Let and . Then, we have from Lemma 2.4

Consequently, from Lemma 2.4 and the choice of ϵ, for , we have

It follows in a similar manner from Lemma 2.4 and the choice of ϵ, for ,

Again, using a bootstrapping argument, we have

so that

Hence, . So if we set

then

(3.5)

Applying Theorem 2.1 to (3.4) and (3.5), we see that T has a fixed point . Therefore, setting , we obtain a positive solution of (1.1)-(1.2) given iteratively by

The proof is completed. □

For our next result, we define the positive numbers and by

and

Theorem 3.2Suppose conditions (C1)-(C7) are satisfied. Then, for eachsatisfying

(3.6)

there exists ann-tuplesatisfying (1.1)-(1.2) such that, , on.

Proof Let , , be as in (3.6). Now, let be chosen such that

and

Let T be the cone preserving, completely continuous operator that was defined by (3.1). From the definition of , , there exists such that, for each ,

Also, from the definition of , it follows that , , and so there exist such that

and

Choose with . Then we have

Continuing with this bootstrapping argument, we get

Then

So, . If we put

then

(3.7)

Since each is assumed to be a positive real number, it follows that , , is unbounded at ∞.

For each , set

Then, for each , is a nondecreasing real-valued function, , and

Next, by definition of , , there exists such that, for each ,

It follows that there exists such that, for each ,

Choose with . Then, using the bootstrapping argument, we have

So we have

Hence, . So, if we set

then

(3.8)

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

Example 4.1 In BVP (1.1)-(1.2), suppose that , , , , , , , and i.e.,

(4.1)

satisfying the following boundary conditions:

(4.2)

where

It is easy to see that (C1)-(C7) are satisfied. By simple calculation, we get , , , , , , and

We obtain

and

Applying Theorem 3.1, we get the optimal eigenvalue interval , , for which the boundary value problem (4.1)-(4.2) has a positive solution.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally to the manuscript and typed, read and approved the final manuscript.

### Acknowledgements

The authors would like to thank the referees for their valuable suggestions and comments.

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