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

- Ya-Hong Zhao
^{1}and - Jian-Ping Sun
^{1}Email author

**2010**:620459

**DOI: **10.1155/2010/620459

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

**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 [1–10] for some recent results.

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 [12–15].

Definition 2.1.

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.

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]).

where is the graininess function.

Lemma 2.5 (see [13]).

If then is increasing.

Definition 2.6.

and for let

Definition 2.7.

and for let

Definition 2.8.

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

Definition 2.9.

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

Definition 2.10.

Definition 2.11.

where the cylinder transformation is defined as in Definition 2.8.

Lemma 2.12 (see [13]).

- (i)
,

- (ii)
,

- (iii)
,

- (iv)
for and

Lemma 2.13 (see [13]).

## 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 .

then we may claim that , which implies that

This indicates that .

be equipped with the norm Then is a Banach space.

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.

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

Theorem 3.3.

where and for .

Proof.

Corollary 3.4.

where and for .

Example 3.5.

where and for .

## Declarations

### Acknowledgment

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

## Authors’ Affiliations

## References

- 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 - 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 - 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 - 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 - Stehlík P:
**Periodic boundary value problems on time scales.***Advances in Difference Equations*2005,**1:**81-92. 10.1155/ADE.2005.81MathSciNetMATHGoogle Scholar - 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 - 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 - 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 - 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 - 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 - 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 - 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 - Bohner M, Peterson A:
*Dynamic Equations on Time Scales: An Introduction with Applications*. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleMATHGoogle Scholar - 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 - 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

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