Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2009, Article ID 173028, 7 pages
doi:10.1155/2009/173028
Research Article
The C1 Solutions of the Series-Like Iterative
Equation with Variable Coefficients
Yuzhen Mi, Xiaopei Li, and Ling Ma
Mathematics and Computational School, Zhanjiang Normal University, Zhanjiang,
Guangdong 524048, China
Correspondence should be addressed to Xiaopei Li, lixp27333@sina.com
Received 23 March 2009; Revised 11 June 2009; Accepted 6 July 2009
Recommended by Tomas Domı́nguez Benavides
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 C1 solutions of the series-like
iterative equations with variable coefficients are discussed.
Copyright q 2009 Yuzhen Mi et al. 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.
1. Introduction
An important form of iterative equations is the polynomial-like iterative equation
λ1 fx λ2 f 2 x · · · λn f n x Fx,
x ∈ I : a, b,
1.1
where F is a given function, f is an unknown function, λi ∈ R1 i 1, 2, . . . , n, and f k k 1, 2, . . . , n is the kth iterate of f, that is, f 0 x x, f k x f ◦ f k−1 x. The case of all
constant λi s 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 λi λi x
which are all continuous in interval a, b. 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
∞
i1
λi xf i x Fx,
x ∈ I : a, b,
1.2
2
Fixed Point Theory and Applications
where λi x : I → 0, 1 are given continuous functions and ∞
i1 λi x 1, λ1 x ≥ c >
0 ∀x ∈ I, maxx∈I λi x ci . 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 C0 I, R {f : I → R, f is continuous}, clearly C0 I, R, · c0 is a Banach space, where
fc0 maxx∈I |fx|, for f in C0 I, R.
Let C1 I, R {f : I → R, f is continuous and continuously differentiable}, then
1
C I, R is a Banach space with the norm · c1 , wherefc1 fc0 f c0 , for f in C1 I, R.
Being a closed subset, C1 I, I defined by
C1 I, I f ∈ C1 I, R, fI ⊆ I, ∀x ∈ I
2.1
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 ϕ ∈ C1 I, I and
ϕ x ≤ M,
∀x ∈ I,
ϕ x1 − ϕ x2 ≤ M |x1 − x2 |, ∀x1 , x2 ∈ I,
2.2
2.3
where M and M are positive constants.Then
n
ϕ x1 − ϕn x2 ≤ M
2n−2
M
i
|x1 − x2 |,
2.4
in−1
for any x1 , x2 in I, where ϕn denotes dϕn /dx.
Lemma 2.2. Suppose that ϕ1 , ϕ2 ∈ C1 I, I satisfy 2.2.Then
ϕn1
−
ϕn2 c0
n
i−1
ϕ1 − ϕ2 c0 .
≤
M
2.5
i1
Lemma 2.3. Suppose that ϕ1 , ϕ2 ∈ C1 I, I satisfy 2.2 and 2.3.Then
k1 k1 ϕ
≤ k 1Mk ϕ − ϕ 0
−
ϕ
2 c
1
2
1
0
c
k
ki−1
ϕ1 − ϕ2 c0 ,
Qk 1M
k − i 1M
i1
for k 0, 1, 2, . . . , where Qs 0 as s 1 and Qs 1 as s 2, 3, . . ..
2.6
Fixed Point Theory and Applications
3
3. Main Results
For given constants M1 > 0 and M2 > 0, let
AM1 , M2 ϕ ∈ C1 I, I : ϕ x ≤ M1 , ∀x ∈ I ,
ϕ x1 − ϕ x2 ≤ M2 |x1 − x2 |, ∀x1 , x2 ∈ I .
3.1
Theorem 3.1 existence. Given positive constants M1 , M2 and F ∈ AM1 , M2 , if there exists
constants N1 ≥ 1 and N2 > 0, such that
P1 c −
P2 c −
∞
i−1
i2 ci N1
∞
i2 ci ≥ M1 /N1 ,
2i−2
j
ji−1 N1 ≥ M2 /N2 ,
then 1.2 has a solution f in AN1 , N2 .
Proof. For convenience, let d max{|a|, |b|}.
Define K : AN1 , N2 → C1 I, I such that K : f → Kf , where
Kf t ∞
λi xf i t,
∀x, t ∈ I.
3.2
i1
Since f ∈ AN1 , N2 , it is easy to see that |f i t| ≤ d for all t ∈ I, and |λi xf i t| ≤ d|λi x| for
∞
i
all x, t ∈ I. It follows from ∞
i1 λi x 1 that
i1 λi xf t is uniformly convergent. Then
Kf t is continuous for t ∈ I. Also we have
a
∞
λi xa ≤
i1
∞
∞
λi xf i t ≤
λi xb b,
i1
3.3
i1
thus Kf ∈ C0 I, I.
For any f ∈ AN1 , N2 , we have
d
i−1 i−1 i
≤ ci N i .
λ
f
f
f
λ
f
x
t
x
t
t
i
1
dt i
3.4
∞
i
i
By condition P1 , we see that ∞
i1 ci N1 is convergent, therefore
i1 ci f t is uniformly
convergent for t ∈ I, this implies that Kf t is continuously differentiable for t ∈ I. Moreover
∞
d
∞
i Kf t ≤
f t ≤
λ
ci N1i : μ1 .
x
i
dt
i1
i1
3.5
4
Fixed Point Theory and Applications
By Lemma 2.1,
∞
d
i
d
i
λi x f t1 − f t2 dt Kf t1 − dt Kf t2 ≤
i1
⎛
⎞
∞
2i−2
j
≤
ci ⎝N2
N1 ⎠|t1 − t2 | : μ2 |t1 − t2 |.
i1
3.6
ji−1
Thus Kf ∈ Aμ1 , μ2 .
Define T : AN1 , N2 → C1 I, I as follows:
T ft 1
1
Ft −
Kf t ft,
λ1 x
λ1 x
∀t, x ∈ I,
3.7
where f ∈ AN1 , N2 . Because Kf , F, and f are continuously differentiable for all t ∈ I, T f
is continuously differentiable for all t ∈ I. By conditions P1 and P2 , for any t1 , t2 in I, we
have
∞
∞
i 1
d
f t ≤ M1 1 ci N i
≤ 1 F t 1
λ
T
ft
x
i
1
c
dt
λ x
λ1 x i2
c i2
1
3.8
1
1
≤ M1 cN1 − M1 N1 .
c
c
We furthermore have
∞ i
d
d
1 1 i
dt T ft1 − dt T ft2 ≤ λ x F t1 − F t2 λ x ci f t1 − f t2 1
1
i2
⎛
⎞
∞
2i−2
j
1
1
≤ M2 |t1 − t2 | c i N2 ⎝
N1 ⎠|t1 − t2 |
c
c i2
ji−1
≤ N2 |x1 − x2 |.
3.9
Thus T : AN1 , N2 → AN1 , N2 is a self-diffeomorphism.
Now we prove the continuity of T under the norm · c1 . For arbitrary f1 , f2 ∈
AN1 , N2 ,
Fixed Point Theory and Applications
5
1
1
T f1 − T f2 c0 max−
Kf1 t f1 t Kf2 t − f2 t
t∈I
λ1 x
λ1 x
∞
∞
1
i
i
≤ max λi xf1 t − λi xf2 t
c t∈I i2
i2
∞ 1
ci f1i − f2i 0
c
c i2
∞
i
1
k−1 f1 − f2 c0 ,
≤
ci
N1
c i2
k1
1 1 max−
Kf1 t f1 t Kf2 t − f2 t t∈I
λ1 x
λ1 x
∞
∞
1
i
i
≤ max λi x f1 t − λi x f2 t c t∈I i2
i2
∞ i i 1
≤
f
ci −
f
2 c i2 1
c0
∞
i−1
1
i−1 ik−2 f1 − f2 c0 .
≤
ci iN1 f1 − f2 c0 QiN2
i − kN1
c i2
k1
3.10
≤
d
T f1 − d T f2 0
dt
dt
c
Let
∞
i
i−1
1
k−1
ik−2
E1 ,
ci
N1 QiN2 i − kN1
c i2
k1
k1
∞
1
E2 ci iN1i−1 ,
c i2
3.11
E max{E1 , E2 }.
Then we have
T f1 − T f2 1 T f1 − T f2 0 T f1 − T f2 0 ≤ E1 f1 − f2 c0 E2 f1 − f2 c0
c
c
c
≤ Ef1 − f2 c0 Ef1 − f2 c0 Ef1 − f2 c1 ,
3.12
which gives continuity of T .
It is easy to show that AN1 , N2 is a compact convex subset of C1 I, I. By Schauder’s
fixed point theorem, we assert that there is a mapping f ∈ AN1 , N2 such that
ft T ft 1
1
Ft −
Kf t ft,
λ1 x
λ1 x
∀t ∈ I.
Let t x, we have fx as a solution of 1.2 in AN1 , N2 . This completes the proof.
3.13
6
Fixed Point Theory and Applications
Theorem 3.2 Uniqueness. Suppose that (P1 ) and (P2 ) are satisfied, also one supposes that
P3 E < 1,
then for arbitrary function F in AM1 , M2 , 1.2 has a unique solution f ∈ AN1 , N2 .
Proof. The existence of 1.2 in AN1 , N2 is given by Theorem 3.1, from the proof of
Theorem 3.1, we see that AN1 , N2 is a closed subset of C1 I, I, by 3.12 and P3 , we see
that T : AN1 , N2 → AN1 , N2 is a contraction. Therefore T has a unique fixed point fx
in AN1 , N2 , that is, 1.2 has a unique solution in AN1 , N2 , this proves the theorem.
4. Example
Consider the equation
∞
1
λi xf i x x2 ,
4
i1
x ∈ I : −1, 1,
4.1
where λ1 x 33/36 1/36 cos2 πx/2, λ2 x 1/36 1/36 sin2 πx/2, λ3 x 1/36,
λ4 x λ5 x · · · 0. It is easy to see that 0 ≤ λi x ≤ 1, ∞
i1 λi x 1, c 33/36, c2 2/36, c3 1/36, c4 c5 · · · 0.
For any x, y in −1, 1,
F x |0.5x| ≤ 0.5,
F x − F y ≤ |0.5x| 0.5y ≤ 1,
4.2
thus F ∈ A 0.5, 1. By condition P1 , we can choose N1 1.1, and by condition P1 , we can
choose N2 1.5. Then by Theorem 3.1, there is a continuously differentiable solution of 4.1
in A1.1, 1.5.
Remark 4.1. Here Fx is not monotone for x ∈ −1, 1, hence it cannot be concluded by 11,
12.
Acknowledgments
This work was supported by Guangdong Provincial Natural Science Foundation 07301595
and Zhan-jiang Normal University Science Research Project L0804.
References
1 J. Z. Zhang and L. Yang, “Disscussion on iterative roots of continuous and piecewise monotone
functions,” Acta Mathematica Sinica, vol. 26, no. 4, pp.
398–412, 1983 Chinese.
2 W. N. Zhang, “Discussion on the iterated equation ni1 λi f i x Fx,” Chinese Science Bulletin, vol.
32, pp. 1441–1451, 1987 Chinese.
3 W. N. Zhang, “Stability of the solution of the iterated equation ni1 λi f i x Fx,” Acta Mathematica
Scientia, vol. 8, no. 4, pp. 421–424, 1988.
4 W. N. Zhang, “Discussion on the differentiable solutions of the iterated equation ni1 λi f i x Fx,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 15, no. 4, pp. 387–398, 1990.
Fixed Point Theory and Applications
7
5 W. Zhang, “Solutions of equivariance for a polynomial-like iterative equation,” Proceedings of the Royal
Society of Edinburgh. Section A, vol. 130, no. 5, pp. 1153–1163, 2000.
6 M. Kulczycki and J. Tabor, “Iterative functional equations in the class of Lipschitz functions,”
Aequationes Mathematicae, vol. 64, no. 1-2, pp. 24–33, 2002.
7 X. P. Li, “A class of iterative equation on a Banach space,” Journal of Sichuan University. Natural Science
Edition, vol. 41, no. 3, pp. 505–510, 2004 Chinese.
8 W. Zhang, K. Nikodem, and B. Xu, “Convex solutions of polynomial-like iterative equations,” Journal
of Mathematical Analysis and Applications, vol. 315, no. 1, pp. 29–40, 2006.
9 B. Xu and W. Zhang, “Decreasing solutions and convex solutions of the polynomial-like iterative
equation,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 483–497, 2007.
10 X. P. Li and S. F. Deng, “Differentiability for the high dimensional polynomial-like iterative equation,”
Acta Mathematica Scientia B, vol. 25, no. 1, pp. 130–136, 2005.
11 W. Zhang and J. A. Baker, “Continuous solutions of a polynomial-like iterative equation with variable
coefficients,” Annales Polonici Mathematici, vol. 73, no. 1, pp. 29–36, 2000.
12 J.-G. Si and X.-P. Wang, “Differentiable solutions of a polynomial-like iterative equation with variable
coefficients,” Publicationes Mathematicae Debrecen, vol. 58, no. 1-2, pp. 57–72, 2001.