Monotone Iterative Technique for First-Order Nonlinear Periodic Boundary Value Problems on Time Scales
© Ya-Hong Zhao and Jian-Ping Sun. 2010
Received: 12 February 2010
Accepted: 26 June 2010
Published: 13 July 2010
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.
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  and the references therein.
2. Some Results on Time Scales
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,
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 ).
where is the graininess function.
Lemma 2.5 (see ).
If then is increasing.
and for let
and for let
where is the principal logarithm function. For we define for all
The set of all regressive and rd-continuous functions will be denoted by
where the cylinder transformation is defined as in Definition 2.8.
Lemma 2.12 (see ).
Lemma 2.13 (see ).
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.
is completely continuous.
This shows that . Furthermore, with similar arguments as in , we can prove that is completely continuous by Arzela-Ascoli theorem.
where and for .
where and for .
where and for .
This work was supported by the National Natural Science Foundation of China (10801068).
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