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The
Solutions of the Series-Like Iterative Equation with Variable Coefficients
Fixed Point Theory and Applications volume 2009, Article number: 173028 (2009)
Abstract
By constructing a structure operator quite different from that ofZhang and Baker (2000) and using the Schauder fixed point theory, the existence and uniqueness of the solutions of the series-like iterative equations with variable coefficients are discussed.
1. Introduction
An important form of iterative equations is the polynomial-like iterative equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ1_HTML.gif)
where is a given function,
is an unknown function,
and
is the
th iterate of
that is,
The case of all constant
was considered in [1–10]. In 2000, W. N. Zhang and J. A. Baker first discussed the continuous solutions of such an iterative equation with variable coefficients
which are all continuous in interval
In 2001, J. G. Si and X. P. Wang furthermore gave the continuously differentiable solution of such equation in the same conditions as in [11]. In this paper, we continue the works of [11, 12], and consider the series-like iterative equation with variable coefficients
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ2_HTML.gif)
where are given continuous functions and
We improve the methods given by the authors in [11, 12], and the conditions of [11, 12] are weakened by constructing a new structure operator.
2. Preliminaries
Let , clearly
is a Banach space, where
, for
in
.
Let , then
is a Banach space with the norm
, where
, for
in
.
Being a closed subset, defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ3_HTML.gif)
is a complete space.
The following lemmas are useful, and the methods of proof are similar to those of paper [4], but the conditions are weaker than those of [4].
Lemma 2.1.
Suppose that and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ4_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ5_HTML.gif)
where and
are positive constants.Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ6_HTML.gif)
for any in
, where
denotes
.
Lemma 2.2.
Suppose that satisfy (2.2).Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ7_HTML.gif)
Lemma 2.3.
Suppose that satisfy (2.2) and (2.3).Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ8_HTML.gif)
for where
as
and
as
.
3. Main Results
For given constants and
, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ9_HTML.gif)
Theorem 3.1 (existence).
Given positive constants and
if there exists constants
and
, such that
,
,
then (1.2) has a solution in
.
Proof.
For convenience, let
Define such that
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ10_HTML.gif)
Since , it is easy to see that
for all
, and
for all
It follows from
that
is uniformly convergent. Then
is continuous for
. Also we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ11_HTML.gif)
thus .
For any , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ12_HTML.gif)
By condition , we see that
is convergent, therefore
is uniformly convergent for
, this implies that
is continuously differentiable for
. Moreover
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ13_HTML.gif)
By Lemma 2.1,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ14_HTML.gif)
Thus .
Define as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ15_HTML.gif)
where . Because
,
and
are continuously differentiable for all
,
is continuously differentiable for all
. By conditions
and
, for any
in
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ16_HTML.gif)
We furthermore have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ17_HTML.gif)
Thus is a self-diffeomorphism.
Now we prove the continuity of under the norm
. For arbitrary
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ18_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ19_HTML.gif)
Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ20_HTML.gif)
which gives continuity of .
It is easy to show that is a compact convex subset of
. By Schauder's fixed point theorem, we assert that there is a mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ21_HTML.gif)
Let we have
as a solution of (1.2) in
. This completes the proof.
Theorem 3.2 (Uniqueness).
Suppose that (P1) and (P2) are satisfied, also one supposes that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_IEq98_HTML.gif)
then for arbitrary function in
, (1.2) has a unique solution
.
Proof.
The existence of (1.2) in is given by Theorem 3.1, from the proof of Theorem 3.1, we see that
is a closed subset of
, by (3.12) and
, we see that
is a contraction. Therefore
has a unique fixed point
in
, that is, (1.2) has a unique solution in
, this proves the theorem.
4. Example
Consider the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ22_HTML.gif)
where It is easy to see that
For any in
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F173028/MediaObjects/13663_2009_Article_1119_Equ23_HTML.gif)
thus By condition
, we can choose
and by condition
, we can choose
. Then by Theorem 3.1, there is a continuously differentiable solution of (4.1) in
.
Remark 4.1.
Here is not monotone for
, hence it cannot be concluded by [11, 12].
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Acknowledgments
This work was supported by Guangdong Provincial Natural Science Foundation (07301595) and Zhan-jiang Normal University Science Research Project (L0804).
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Mi, Y., Li, X. & Ma, L. The Solutions of the Series-Like Iterative Equation with Variable Coefficients.
Fixed Point Theory Appl 2009, 173028 (2009). https://doi.org/10.1155/2009/173028
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DOI: https://doi.org/10.1155/2009/173028