- Open Access
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.
- Green’s function
- iterative system
- eigenvalue interval
- time scales
- boundary value problem
- fixed point theorem
- 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 .
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. □
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.
- Hilger, S: Ein maßkettenkalkül mit anwendug auf zentrumsmanningfaltigkeite. PhD thesis, Universität Würzburg (1988)Google Scholar
- Agarwal RP, Bohner M: Basic calculus on time scales and some of its applications. Results Math. 1999, 35: 3-22. 10.1007/BF03322019MATHMathSciNetView ArticleGoogle Scholar
- 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.2008.05.018MATHMathSciNetView ArticleGoogle Scholar
- Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston; 2001.View ArticleGoogle Scholar
- Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston; 2003.MATHView ArticleGoogle Scholar
- Bohner M, Luo H: Singular second-order multipoint dynamic boundary value problems with mixed derivatives. Adv. Differ. Equ. 2006., 2006: Article ID 54989Google Scholar
- 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., 2013: Article ID 52Google Scholar
- Agarwal RP, Bohner M, Wong P: Strum-Liouville eigenvalue problems on time scale. Appl. Math. Comput. 1999, 99: 153-166. 10.1016/S0096-3003(98)00004-6MATHMathSciNetView ArticleGoogle Scholar
- Anderson DR: Eigenvalue intervals for even order Strum-Liouville dynamic equations. Commun. Appl. Nonlinear Anal. 2005, 12: 1-13.MATHGoogle Scholar
- Benchohra M, Henderson J, Ntouyas SK: Eigenvalue problems for systems of nonlinear boundary value problems on time scales. Adv. Differ. Equ. 2007., 2007: Article ID 31640Google Scholar
- Karaca IY: Multiple positive solutions for dynamic m -point boundary value problems. Dyn. Syst. Appl. 2008, 17: 25-42.MATHMathSciNetGoogle Scholar
- Karaca IY: Existence and nonexistence of positive solutions to a right-focal boundary value problem on time scales. Adv. Differ. Equ. 2006., 2006: Article ID 43039Google Scholar
- Benchohra M, Berhoun F, Hamani S, Henderson J, Ntouyas SK, Ouahab A, Purnaras IK: Eigenvalues for iterative systems of nonlinear boundary value problems on time scales. Nonlinear Dyn. Syst. Theory 2009, 9: 11-22.MATHMathSciNetGoogle Scholar
- Prasad KR, Sreedhar N, Narasimhulu Y: Eigenvalue intervals for iterative systems of nonlinear m -point boundary value problems on time scales. Differ. Equ. Dyn. Syst. 2013. 10.1007/s12591-013-0183-5Google Scholar
- Ma R, Thompson B: Positive solutions for nonlinear m -point eigenvalue problems. J. Math. Anal. Appl. 2004, 297: 24-37. 10.1016/j.jmaa.2003.12.046MATHMathSciNetView ArticleGoogle Scholar
- Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, Orlando; 1988.MATHGoogle Scholar
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