Open Access

Monotone Iterative Technique for First-Order Nonlinear Periodic Boundary Value Problems on Time Scales

Advances in Difference Equations20102010:620459

https://doi.org/10.1155/2010/620459

Received: 12 February 2010

Accepted: 26 June 2010

Published: 13 July 2010

Abstract

We investigate the following nonlinear first-order periodic boundary value problem on time scales: , , . Some new existence criteria of positive solutions are established by using the monotone iterative technique.

1. Introduction

Recently, periodic boundary value problems (PBVPs for short) for dynamic equations on time scales have been studied by several authors by using the method of lower and upper solutions, fixed point theorems, and the theory of fixed point index. We refer the reader to [110] for some recent results.

In this paper we are interested in the existence of positive solutions for the following first-order PBVP on time scales:
(1.1)

where will be defined in Section 2, is a time scale, is fixed and . For each interval of we denote by By applying the monotone iterative technique, we obtain not only the existence of positive solution for the PBVP (1.1), but also give an iterative scheme, which approximates the solution. It is worth mentioning that the initial term of our iterative scheme is a constant function, which implies that the iterative scheme is significant and feasible. For abstract monotone iterative technique, see [11] and the references therein.

2. Some Results on Time Scales

Let us recall some basic definitions and relevant results of calculus on time scales [1215].

Definition 2.1.

For we define the forward jump operator by
(2.1)
while the backward jump operator is defined by
(2.2)

In this definition we put and where denotes the empty set If we say that is right scattered, while if we say that is left scattered. Also, if and then is called right dense, and if and then is called left dense. We also need below the set which is derived from the time scale as follows. If has a left-scattered maximum then Otherwise,

Definition 2.2.

Assume that is a function and let Then is called differentiable at if there exists a such that, for any given there is an open neighborhood of such that
(2.3)
In this case, is called the delta derivative of at and we denote it by If then we define the integral by
(2.4)

Definition 2.3.

A function is called rd-continuous provided that it is continuous at right-dense points in and its left-sided limits exist at left-dense points in The set of rd-continuous functions will be denoted by

Lemma 2.4 (see [13]).

If and then
(2.5)

where is the graininess function.

Lemma 2.5 (see [13]).

If then is increasing.

Definition 2.6.

For we define the Hilger complex numbers as
(2.6)

and for let

Definition 2.7.

For let be the strip
(2.7)

and for let

Definition 2.8.

For we define the cylinder transformation by
(2.8)

where is the principal logarithm function. For we define for all

Definition 2.9.

A function is regressive provided that
(2.9)

The set of all regressive and rd-continuous functions will be denoted by

Definition 2.10.

We define the set of all positively regressive elements of by
(2.10)

Definition 2.11.

If , then the generalized exponential function is given by
(2.11)

where the cylinder transformation is defined as in Definition 2.8.

Lemma 2.12 (see [13]).

If , then
  1. (i)

    ,

     
  2. (ii)

    ,

     
  3. (iii)

    ,

     
  4. (iv)

    for and

     

Lemma 2.13 (see [13]).

If and , then
(2.12)

3. Main Results

For the forthcoming analysis, we assume that the following two conditions are satisfied.

(H1) is rd-continuous, which implies that .

(H2) is continuous and is nondecreasing on .

If we denote that
(3.1)

then we may claim that , which implies that

In fact, in view of (H1) and Lemmas 2.12 and 2.13, we have
(3.2)
which together with Lemma 2.5 shows that is increasing on And so,
(3.3)

This indicates that .

Let
(3.4)

be equipped with the norm Then is a Banach space.

First, we define two cones and in as follows:
(3.5)
and then we define an operator
(3.6)

It is obvious that fixed points of are solutions of the PBVP (1.1).

Since is nondecreasing on , we have the following lemma.

Lemma 3.1.

is nondecreasing.

Lemma 3.2.

is completely continuous.

Proof.

Suppose that Then
(3.7)
so,
(3.8)
Therefore,
(3.9)

This shows that . Furthermore, with similar arguments as in [7], we can prove that is completely continuous by Arzela-Ascoli theorem.

Theorem 3.3.

Assume that there exist two positive numbers such that
(3.10)
Then the PBVP (1.1) has positive solutions and , which may coincide with
(3.11)

where and for .

Proof.

First, we define
(3.12)
Then we may assert that
(3.13)
In fact, if , then
(3.14)
which together with (H2) and (3.10) implies that
(3.15)
which shows that
(3.16)
Now, if we denote that for , then . Let
(3.17)
In view of , we have Since the set is bounded and the operator is compact, we know that the set is relatively compact, which implies that there exists a subsequence such that
(3.18)
Moreover, since
(3.19)
it follows from Lemma 3.1 that ; that is, . By induction, it is easy to know that
(3.20)
which together with (3.18) implies that
(3.21)
Since is continuous, it follows from (3.17) and (3.21) that
(3.22)
which shows that is a solution of the PBVP (1.1). Furthermore, we get from that
(3.23)
On the other hand, if we denote that for and that then we can obtain similarly that and there exists a subsequence such that
(3.24)
Moreover, since
(3.25)
it is also easy to know that
(3.26)
With the similar arguments as above, we can prove that is a solution of the PBVP (1.1) and satisfies
(3.27)

Corollary 3.4.

If the following conditions are fulfilled:
(3.28)
then there exist two positive numbers such that (3.10) is satisfied, which implies that the PBVP (1.1) has positive solutions and , which may coincide with
(3.29)

where and for .

Example 3.5.

Let We consider the following PBVP on :
(3.30)
Since and we can obtain that
(3.31)
Thus, if we choose and , then all the conditions of Theorem 3.3 are fulfilled. So, the PBVP (3.30) has positive solutions and , which may coincide with
(3.32)

where and for .

Declarations

Acknowledgment

This work was supported by the National Natural Science Foundation of China (10801068).

Authors’ Affiliations

(1)
Department of Applied Mathematics, Lanzhou University of Technology

References

  1. Cabada A: Extremal solutions and Green's functions of higher order periodic boundary value problems in time scales. Journal of Mathematical Analysis and Applications 2004,290(1):35-54. 10.1016/j.jmaa.2003.08.018MathSciNetView ArticleMATHGoogle Scholar
  2. Dai Q, Tisdell CC: Existence of solutions to first-order dynamic boundary value problems. International Journal of Difference Equations 2006,1(1):1-17.MathSciNetMATHGoogle Scholar
  3. Eloe PW: The method of quasilinearization and dynamic equations on compact measure chains. Journal of Computational and Applied Mathematics 2002,141(1-2):159-167. 10.1016/S0377-0427(01)00443-5MathSciNetView ArticleMATHGoogle Scholar
  4. Topal SG: Second-order periodic boundary value problems on time scales. Computers & Mathematics with Applications 2004,48(3-4):637-648. 10.1016/j.camwa.2002.04.005MathSciNetView ArticleMATHGoogle Scholar
  5. Stehlík P: Periodic boundary value problems on time scales. Advances in Difference Equations 2005, 1: 81-92. 10.1155/ADE.2005.81MathSciNetMATHGoogle Scholar
  6. Sun J-P, Li W-T: Positive solution for system of nonlinear first-order PBVPs on time scales. Nonlinear Analysis: Theory, Methods & Applications 2005,62(1):131-139. 10.1016/j.na.2005.03.016MathSciNetView ArticleMATHGoogle Scholar
  7. Sun J-P, Li W-T: Existence of solutions to nonlinear first-order PBVPs on time scales. Nonlinear Analysis: Theory, Methods & Applications 2007,67(3):883-888. 10.1016/j.na.2006.06.046MathSciNetView ArticleMATHGoogle Scholar
  8. Sun J-P, Li W-T: Existence and multiplicity of positive solutions to nonlinear first-order PBVPs on time scales. Computers & Mathematics with Applications 2007,54(6):861-871. 10.1016/j.camwa.2007.03.009MathSciNetView ArticleMATHGoogle Scholar
  9. Sun J-P, Li W-T: Positive solutions to nonlinear first-order PBVPs with parameter on time scales. Nonlinear Analysis: Theory, Methods & Applications 2009,70(3):1133-1145. 10.1016/j.na.2008.02.007MathSciNetView ArticleMATHGoogle Scholar
  10. Wu S-T, Tsai L-Y: Periodic solutions for dynamic equations on time scales. Tamkang Journal of Mathematics 2009,40(2):173-191.MathSciNetMATHGoogle Scholar
  11. Ahmad B, Nieto JJ:The monotone iterative technique for three-point second-order integrodifferential boundary value problems with -Laplacian. Boundary Value Problems 2007, 2007:-9.Google Scholar
  12. Agarwal RP, Bohner M: Basic calculus on time scales and some of its applications. Results in Mathematics 1999,35(1-2):3-22.MathSciNetView ArticleMATHGoogle Scholar
  13. Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleMATHGoogle Scholar
  14. Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics 1990,18(1-2):18-56.MathSciNetView ArticleMATHGoogle Scholar
  15. Kaymakcalan B, Lakshmikantham V, Sivasundaram S: Dynamic Systems on Measure Chains, Mathematics and Its Applications. Volume 370. Kluwer Academic Publishers, Boston, Mass, USA; 1996:x+285.MATHGoogle Scholar

Copyright

© Ya-Hong Zhao and Jian-Ping Sun. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.