Abstract
In this paper, we determine the eigenvalue intervals of the parameters for which there exist positive solutions of the iterative systems of mpoint boundary value problems on time scales. The method involves an application of GuoKrasnosel’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; mpoint; positive solution1 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 [27].
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 GuoKrasnosel’skii fixed point theorem. To mention a few papers along these lines, see [812]. 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 GuoKrasnosel’skii fixed point theorem for which there exist positive solutions of the mpoint 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 GuoKrasnosel’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 mpoint boundary conditions,
They used the GuoKrasnosel’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 mpoint boundary value problems on time scales,
satisfying the mpoint boundary conditions,
Throughout this paper we assume that following conditions hold:
(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 GuoKrasnosel’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:
Denote by θ and φ, the solutions of the corresponding homogeneous equation
under the initial conditions
Using the initial conditions (2.3), we can deduce from equation (2.2) for θ and φ the following equations:
Set
and
Lemma 2.1Let (C1) hold. Assume that
Ifis a solution of the equation
where
and
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
Since
we have
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
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
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
Proof We have from (2.8) that
which implies
for all . Applying (2.8), we have for ,
where
So, the proof is completed. □
We note that an ntuple 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 GuoKrasnosel’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
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
Notice from (C1)(C6) and Lemma 2.2 that, for , on . Also, we have from (2.8), that
so that
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 ArzelaAscoli 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
there exists anntuplesatisfying (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 ,
then
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
then
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
there exists anntuplesatisfying (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
Continuing with this bootstrapping argument, we get
Then
then
Since each is assumed to be a positive real number, it follows that , , is unbounded at ∞.
Then, for each , is a nondecreasing realvalued 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
then
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 ntuple 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.,
satisfying the following boundary conditions:
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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