Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2008, Article ID 636843, 13 pages
doi:10.1155/2008/636843
Research Article
Differentiable Solutions of Equations Involving
Iterated Functional Series
Wei Song
Deparment of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Correspondence should be addressed to Wei Song, dawenhxi@126.com
Received 12 October 2008; Accepted 28 December 2008
Recommended by John Rassias
The nonmonotonic differentiable solutions of equations involving iterated functional series are
investigated. Conditions for the existence, uniqueness, and stability of such solutions are given.
These extend earlier results due to Murugan and Subrahmanyam.
Copyright q 2008 Wei Song. 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
Let f be a self-mapping on a topological space X and f m denote the mth iterate of f, that is,
f m f ◦ f m−1 , f 0 id, m 1, 2, . . . . Let CX, X be the set of all continuous self-mappings
on X. Equations having iteration as their main operation, that is, including iterates of the
unknown mapping, are called iterative equations. It is one of the most interesting classes
of functional equations 1–4, because it concludes the problem of iterative roots 1, 5, 6,
that is, finding f ∈ CX, X such that f n is identical to a given F ∈ CX, X. As a natural
generalization of the problem of iterative roots, a class of iterative equations named as
polynomial-like iterative equation
λ1 fx λ2 f 2 x · · · λn f n x Fx,
x ∈ I a, b,
1.1
had fascinated many scholars, such as Dhombres 7, Zhao 8, Mukherjea and Ratti
9. Despite their nice constructive proofs, the classical methods prevented them from
obtaining more fruitful results. In 1986, Zhang 10 constructed an interesting operator called
“structural operator” for 1.1 and used the fixed point theory in Banach space to get the
solutions of 1.1. Hence he overcame the difficulties encountered by the formers. By means of
this method, Zhang and Si made a series of work concerning these qualitative problems, such
2
Abstract and Applied Analysis
as 11–15. In 2002, Kulczycki and Tabor 16 improved Zhang’s method and investigated the
existence of Lipschitzian solutions of the iterative functional equation
∞
λn f n x Fx,
x ∈ B,
1.2
n1
where B is a compact convex subset of Rn and F : B → B are a given Lipschitz function. It is
easy to see that 1.1 is the special case of 1.2 with λi 0, i n 1, . . . and B a, b.
Recently Zhang et al. 17 and Xu et al. 18 developed this method and they have got
the nonmonotonic, convex, and decreasing continuous solutions of 1.1. In fact they have
answered the open problem 2 which was proposed by J. Zhang et al. 19.
The problem of differentiable solutions of iterative equation had also fascinated many
scholars’ attentions. In Zhang 12 and Si 15, the C1 and C2 solutions of 1.1 are considered.
In Wang and Si 20 the differentiable solutions of the below equation
H x, φn1 x, . . . , φni x Fx,
x ∈ I a, b
1.3
are considered. Murugan and Subrahmanyam 21, 22 discussed the existence and
uniqueness of C1 solutions of the more general equations
∞
λi Hi f i x Fx,
x ∈ I a, b,
1.4
i1
∞
λi Hi x, φai1 x, . . . , φaini x Fx,
x ∈ I a, b,
1.5
i1
which involve iterated functional series. All the above references only got the increasing
differentiable solutions for the above equations because they only considered the case that
F is increasing. Li and Deng 23 considered the C1 solutions of the 1.2. In 24 C1 solutions
of the equation
∞
λn xf n x Fx,
x ∈ B,
1.6
n1
where B is a compact convex subset of Rn and λn x : B → R are discussed. Li and Deng 23
and Li 24 work in higher dimensional case, they do not require monotonicity. It should be
pointed out that Mai and Liu 25 made an important contribution to Cm solutions of iterative
equations. Mai and Liu proved the existence, uniqueness of Cm solutions of a relatively
general kind of iterative equations
G x, fx, . . . , f n x 0,
x ∈ J,
1.7
where J is a connected closed subset of R and G ∈ Cm J n1 , R, n ≥ 2. Here Cm J n1 , R
denotes the set of all Cm mappings from J n1 to R.
Wei Song
3
Inspired by the above work, we will investigate 1.4 and extend earlier results due
to Murugan and Subrahmanyam in two directions. In 21 the authors only get increasing
solutions of 1.4, so the present paper will investigate the nonmonotonic differentiable
solutions of 1.4 and give conditions for the existence, uniqueness, and stability of such
solutions. In 21 the authors require not only all coefficients are nonnegative but also
Hi , i 2, 3, . . . are all increasing, but we will find that those conditions are not necessary.
2. Preliminaries
Let I a, b and J c, d be two compact intervals. Let C1 I, J be the set of all
continuously differentiable functions from I to J. Then C1 I, J is a closed subset of the
Banach space C1 I, R consisting of all continuously differentiable functions from I to R with
norm ·c1 . Here the norm ·c1 is defined by ϕc1 ϕc0 ϕ c0 , ϕ ∈ C1 I, R, where
ϕc0 maxx∈I |ϕx| and ϕ is the derivative of ϕ. Following Zhang 12, we define the
families of functions
AI, J, m, M, N ϕ ∈ C1 I, J : ϕa c, ϕb d, m ≤ ϕ x ≤ M,
ϕ x1 − ϕ x2 ≤ N x1 − x2 , ∀x, x1 , x2 ∈ I ,
A I, J, m, M, N ϕ ∈ C1 I, J : ϕa d, ϕb c, m ≤ ϕ x ≤ M,
ϕ x1 − ϕ x2 ≤ N x1 − x2 , ∀x, x1 , x2 ∈ I ,
2.1
where 0 ≤ m < M, N > 0 are all constants.
Lemma 2.1. Both AI, J, m, M, N and A I, J, m, M, N are compact convex subsets of C1 I, J.
The Lemma above can be proved by a method which is contained in the proof of
Theorem 3.1 in 12.
Lemma 2.2 see 12. Suppose that ϕ, φ ∈ AI, J, m, M, N (or ϕ, φ ∈ A I, J, m, M, N). Then
for n 1, 2, . . . ,
n ϕ x ≤ Mn ,
n ϕ x1 − ϕn x2 ≤ N
∀x ∈ I,
2n−2
Mi x1 − x2 ,
∀x1 , x2 ∈ I,
in−1
n
ϕ − φn 0 ≤
c
n
i−1 ϕ − φ
c0 ,
M
2.2
i1
n−1
n n
n−1
ni−2
ϕ − φ 0 ≤ nM ϕ − φ 0 QnN
ϕ − φ
0 ,
n − iM
c
c
c
i1
where Q1 0, Qm 1 as m 2, 3, . . . and ϕn denotes dϕn /dx.
We can get the following Lemma from 12. In 12 the author proved that the lemma
is valid for C1 I, I, but we find it is also valid for C1 I, J.
4
Abstract and Applied Analysis
Lemma 2.3 see 12. Suppose that f ∈ C1 I, J satisfies that
0 < δ ≤ f x, ∀x ∈ I,
f x1 − f x2 ≤ M∗ x1 − x2 , ∀x1 , x2 ∈ I,
2.3
where δ, M∗ are positive constants. Then
∗
−1 f y1 − f −1 y2 ≤ M y1 − y2 ,
3
δ
∀y1 , y2 ∈ J.
2.4
Lemma 2.4 see 18. If both fi : I → J, i 1, 2 are homeomorphisms from I to J such that
fi x1 − fi x2 ≤ K x1 − x2 ,
∀x1 , x2 ∈ I,
2.5
where K is a positive constant. Then
f1 − f2 0 ≤ K f −1 − f −1 0 .
2
1
c
c
2.6
3. Differentiable solutions of 1.4
3.1. Existence of solutions
of 1.4 and I a, b. For any f ∈ AI, I, 0, M, N or A I, I, 0,
Let {λi }∞
i1 be coefficients
∞
i−1
b. It is easy to see that both the
M, N define A i1 λi Hi f i−1 a and B ∞
i1 λi Hi f
convergence and the value of A, B have nothing to do with the choice of f.
Theorem 3.1. Suppose M > 1, L are positive constants and H1 ∈ AI, I, m1 , M1 , N1 , Hi ∈ AI, I,
0, Mi , Ni or Hi ∈ A I, I, 0, Mi , Ni for i 2, 3, . . . , where m1 > 0 and Mi ≥ 1, Ni are positive
constants for i 1, 2, . . . . Assume further the following conditions:
K0 λ1 m1 −
∞ λi Mi Mi−1 > 0,
i2
K1 ∞ 1
λi1 Mi1 Mi−1 Mi − 1 < ∞,
M − 1 i1
K0 − K1 M2 > 0,
K2 3.1
∞ λi Ni Mi−1 Mi − 1 < ∞,
i1
−∞ < A < B < ∞,
hold. Then for any given F ∈ AI, J, 0, K0 M, L or A I, J, 0, K0 M, L, 1.4 has a solution
f ∈ AI, I, 0, M, M∗ or A I, I, 0, M, M∗ , where M∗ ≥ L K4 M2 /K0 − K1 M2 , K4 ∞
2i−1
and I a, b, J A, B.
i1 |λi |Ni M
Wei Song
5
As in 22, firstly we give the following three Lemmas which lead directly to the proof
of Theorem 3.1. In the sequel we denote
L K4 M2
N
K0 − K1 M2
3.2
.
Lemma 3.2. Under the assumptions of Theorem 3.1 the following series:
K3 λ1 M1 ∞ λi Mi Mi−1 ,
i2
K4 ∞ λi Ni M2i−1 ,
i1
∞ k
i−1 λk1 Mk1
,
K5 M
i1
k1
K6 3.3
∞ λk1 Mk1 kMk−1 ,
k1
∞ k
k
i−1
λk1 Nk1 M
,
K7 M
i1
k1
K8 ∞ k−1
λk1 Mk1 k − jMkj−2 ,
j1
K2
are all convergent.
Proof. The convergence of K3 , K4 , K5 , K6 , and K7 is easy to be verified. As mentioned in 21,
the equality
n−1
n − ixni−2 xn−1
i1
xn − 1
n
−
2
x
−1
x − 1
,
x
/1
3.4
holds. We get that
K8 ∞
|λk1 |Mk1 M
K2
k−1
Mk − 1
k
−
2
M−1
M − 1
.
3.5
By the convergence of K1 and K6 , K8 is also convergent.
Lemma 3.3. Under the assumptions of Theorem 3.1, for each f ∈ AI, I, 0, M, N or A I, I, 0,
M, N the mapping Lf : I → R defined by
Lf x ∞
i1
λi Hi f i−1 x ,
x∈I
3.6
6
Abstract and Applied Analysis
has the following properties:
i Lf ∈ AI, J, K0 , K3 , K1 N K4 ;
ii L−1
∈ AJ, I, 1/K3 , 1/K0 , K1 N K4 /K0 3 ,
f
where I a, b and J A, B.
Proof. For any f ∈ AI, I, 0, M, N or A I, I, 0, M, N, we have
Lf a ∞
∞
λi Hi f i−1 a < Lf b λi Hi f i−1 b .
i1
3.7
i1
It is easy to see that for any x ∈ I
0 < λ1 m1 −
∞ ∞
∞ λi Mi Mi−1 ≤
λi Hi f i−1 x f i−1 x ≤ λ1 M1 λi Mi Mi−1 .
i2
3.8
i2
i1
We have for any x ∈ I
0 < K0 ≤ L f x ≤ K3 ,
3.9
and for any y ∈ J
0<
1
1
≤ L−1
y ≤
.
f
K3
K0
3.10
Thus Lf : I → J is an orientation-preserving diffeomorphism.
By Lemma 2.2 we can see that for any x1 , x2 ∈ I,
∞
∞
L f x1 − L f x2 λi H f i−1 x1 f i−1 x1 − λi H f i−1 x2 f i−1 x2 i
i
i1
i1
≤
∞ λi H f i−1 x1 · f i−1 x1 − f i−1 x2 i
i1
Hi f i−1 x1 − Hi f i−1 x2 · f i−1 x2 2i−2 ∞ ∞ j
2i−1
λi Mi N
x1 − x2 ≤
M λi Ni M
i2
ji−2
K1 N K4 x1 − x2 .
3.11
i1
By 3.9, 3.11, and Lemma 2.3 we get for any y1 , y2 ∈ J:
−1 K1 N K4 L
y1 − y2 .
y1 − L−1
y2 ≤
f
f
3
K0 3.12
Wei Song
7
Lemma 3.4. Under the assumptions of Theorem 3.1, for each f1 , f2 ∈ AI, I, 0, M, N or A I, I, 0,
M, N the mappings Lf1 , Lf2 : I → J satisfy the following inequalities:
≤ K5 /K0 f1 − f2 0 ;
− L−1
i L−1
c
f1
f2 c0
−1 −1 3 ii Lf1 − Lf2 c0 ≤
K1 N K4 K5 K7 NK8 K0 / K0 f1 − f2 c0 2 K6 / K0 f − f 0 .
2 c
1
Proof. Firstly we have
∞ Lf − Lf 0 ≤
λk1 ·
Hk1 ◦ f k − Hk1 ◦ f k 0
1
2 c
2 c
1
k1
k
i−1
λk1 Mk1
·
f1 − f2 c0 ,
M
∞ 2.2 ≤
k1
3.13
i1
and thus
−1
L − L−1 0
f1
f2 c
Lemma 2.4
≤
1 Lf − Lf 0 ≤ K5 f1 − f2 0 .
1
2 c
c
K0
K0
3.14
Secondly we get
−1 −1 L
− L
f1
f2
c0
1
1
−
max L f1 L−1 y L f2 L−1 y y∈J
f1
f2
−1 L f2 Lf2 y − L f1 L−1
y 3.9
f1
≤ max
y∈J
K0 2
≤
1
K0 3.11
≤
2
−1 · max L f2 L−1
f2 y − L f2 Lf1 y
y∈J
1
K0 2
−1 · max L f2 L−1
f1 y − L f1 Lf1 y
y∈J
K1 N K4
K0 2
−1
· max L−1
f2 y−Lf1 y y∈J
1
K0 2
· max L f2 x−L f1 x
x∈I
K1 N K4 L−1 − L−1 0 1 · L f − L f 0 .
·
2
1
f
f
c
c
2
1
K0 2
K0 2
3.15
Notice that
∞ λk1 · Hk1 ◦ f k − Hk1 ◦ f k 0
L f − L f 0 ≤
2
1 c
2
1
c
k1
3.16
8
Abstract and Applied Analysis
and for k 1, 2, . . .,
Hk1 ◦ f k − Hk1 ◦ f k 0 H ◦ f k · f k − H ◦ f k · f k 0
2
2
2
1
1
1
k1
k1
c
c
≤ Hk1 ◦ f2k − Hk1
◦ f1k c0 · f2k c0
Hk1
◦ f1k c0 f2k − f1k c0
≤ Mk Nk1
k
Mi−1 f2 − f1 c0 Mk1 kMk−1 f2 − f1 c0
i1
Mk1 QkN
k−1
k − iMki−2 f2 − f1 c0 ,
i1
3.17
then
L f − L f 0 ≤ K7 NK8 f2 − f1 0 K6 f − f 0 .
2
1 c
2
1 c
c
3.18
Finally we get
−1 −1 L
− Lf2 c0 ≤
f1
K1 N K4 K5 K7 NK8 K0 f1 − f2 0 K6 f − f 0 .
3
2 c
1
c
2
K0
K0
3.19
Proof of Theorem 3.1. For any f ∈ AI, I, 0, M, N or A I, I, 0, M, N we define Θf as
follows:
Θf L−1
f ◦ F,
3.20
and denote Θf g for convenience. Clearly g ∈ C1 I, I, ga a, gb b, or ga b, gb a, and 3.10 yields that for any x ∈ I,
K0 M
−1 g x L
Fx · F x ≤
M.
f
K0
3.21
Furthermore by 3.12 we get that for any x1 , x2 ∈ I,
g x1 − g x2 ≤ L−1 F x1 · F x1 − F x2 f
L−1 F x1 − L−1 F x2 · F x2 f
f
K1 N K4
1 L x1 − x2 K0 M · 3 · K0 Mx1 − x2 K0
K0
N x1 − x2 .
≤
3.22
Wei Song
9
So g ∈ AI, I, 0, M, N or A I, I, 0, M, N, which means that ΘAI, I, 0, M, N ⊂ AI, I, 0,
M, N or ΘA I, I, 0, M, N ⊂ A I, I, 0, M, N.
Secondly we prove that
Θ : AI, I, 0, M, N −→ AI, I, 0, M, N
or Θ : A I, I, 0, M, N −→ A I, I, 0, M, N
3.23
is continuous. For any f1 , f2 ∈ AI, I, 0, M, N or A I, I, 0, M, N we denote gi Θfi , i 1, 2. It is easy to see that
g1 − g2 0 L−1 ◦ F − L−1 ◦ F 0 ≤ L−1 − L−1 0 ≤ K5 f1 − f2 0 .
c
c
c
f1
f2
f1
f2 c
K0
3.24
By Lemma 3.4 we get
g − g 0 L−1 ◦ F ·F − L−1 ◦ F · F 0
1
2 c
f1
f2
c
−1 −1 ≤ K0 M · Lf1 − Lf2 c0
K1 N K4 K5 M K7 NK8 K0 M f1 − f2 0 MK6 f − f 0 .
≤
2
1
2 c
c
K
0
K0
3.25
By the discussion above we get
g1 − g2 c1 g1 − g2 c0 g 1 − g 2 c0
K1 N K4 K5 M K7 NK8 K0 M K5
f2 − f1 0 MK6 f − f 0
≤
2
1 c
c
2
K0
K0
K0
≤ E
f1 − f2 c1 ,
3.26
where
E max
K1 N K4 K5 M K7 NK8 K0 M MK6
K5
.
,
2
K0
K0
K0
3.27
Hence Θ : AI, I, 0, M, N → AI, I, 0, M, N or Θ : A I, I, 0, M, N → A I, I, 0,
M, N is continuous. By Schauder fixed point theorem, there exists a function f ∈ AI, I, 0,
M, N or A I, I, 0, M, N such that
f Θf L−1
f ◦ F.
That means f is a solution of 1.4 in AI, I, 0, M, N or A I, I, 0, M, N.
3.28
10
Abstract and Applied Analysis
4. Uniqueness and stability of solutions
Theorem 4.1. Let {λi }∞
i1 be coefficient of 1.4 and M > 1, N > 0 positive constants. Suppose that
the conditions in 3.1 are valid. Further one assumes that
E max
K1 N K4 K5 M K7 NK8 K0 M MK6
K5
< 1.
,
2
K0
K0
K0
4.1
Then for any F ∈ AI, J, 0, K0 M, L or A I, J, 0, K0 M, L, there exists a unique function f ∈
AI, I, 0, M, M∗ or A I, I, 0, M, M∗ satisfying 1.4, where M∗ ≥ L K4 M2 /K0 − K1 M2 .
Furthermore the solution f depends continuously on the given function F.
Proof. If E < 1, then by 3.26 the map Θ defined in Theorem 3.1 becomes a strict contraction.
The fix point of Θ, which is a solution of 1.4, is unique by Banach’s contraction principle.
Let f1 , f2 be the solutions of 1.4 for the corresponding functions F1 , F2 . First, since
fi L−1
fi ◦ Fi ,
i 1, 2,
4.2
we get
f1 − f2 0 L−1 ◦ F1 − L−1 ◦ F2 0
f1
f2
c
c
−1
−1
−1
≤ Lf1 ◦ F1 − Lf1 ◦ F2 c0 L−1
f1 ◦ F2 − Lf2 ◦ F2 c0
Lemma 3.4
≤
4.3
1 F1 − F2 0 K5 f1 − f2 0 .
c
c
K0
K0
Second, we have
f − f 0 L−1 ◦ F1 · F 1 − L−1 ◦ F2 · F 2 0
1
2 c
f1
f2
c
−1 −1 ≤ Lf1 ◦ F1 · F 1 − Lf1 ◦ F1 · F 2 c0
L−1
◦ F1 · F 2 − L−1
◦ F2 · F 2 c0
f1
f1
L−1
◦ F2 · F 2 − L−1
◦ F2 · F 2 c0
f1
f2
3.12 1 M K1 N K4 F1 − F2 0
F 1 − F 2 c0 ≤
c
2
K0
K0
−1 K0 M
L−1
− Lf2 c0
f1
Lemma 3.4 1 M K1 N K4 F1 − F2 0
F 1 − F 2 c0 ≤
2
c
K0
K0
K1 N K4 K5 M K7 NK8 K0 M f1 − f2 0
2
c
K0
MK6 f − f 0 .
2
1 c
K0
4.4
Wei Song
11
By the above discussion we get
f1 − f2 1 f1 − f2 0 f − f 0
1
2 c
c
c
M K1 N K4
1 F1 − F2 1 ,
≤ E f1 − f2 c1 2
c
K0
K0
4.5
which means that
f1 − f2 1 ≤
c
1
1−E
M K1 N K4
1 F1 − F2 1 .
2
c
K
0
K0
4.6
So the solution f depends continuously on the given function F.
4.1. Examples
Example 4.2. Let M 10, L 20 and I 0, 1. The equation
2 2
1001 efx − 1
f x
1
−
x sin2πx,
1000e − 1
1000
2
4.7
where x ∈ 0, 1, has a unique solution f ∈ A0, 1, 0, 1, 0, 10, 900.
Proof. It is easy to see that H1 x ex − 1/e − 1 ∈ AI, I, 1/2, 2, 2, H2 x x2 ∈
AI, I, 0, 2, 2 and Fx x 1/2 sin2πx ∈ AI, I, 0, 4.5, 20. By simple calculation we get
that
K0 961
,
2000
K1 K5 K6 1
,
500
1
,
500
K2 K7 K0 − K1 M2 N
9999
,
500
1
,
50
561
,
2000
K3 K8 0,
2022
,
1000
K0 M K4 961
,
200
1101
,
500
4.8
A 0 < B 1,
L K4 M2 480400
< 900,
561
K0 − K1 M2 by Theorem 3.1 the equation has a solution f ∈ A0, 1, 0, 1, 0, 10, 900. Further we get that
K1 N K4 K5 M K7 NK8 K0 M 804
K5
,
<
2
K0
961
K0
MK6
40
,
K0
961
this means E < 1. By Theorem 4.1 the solution is unique.
4.9
12
Abstract and Applied Analysis
By similar discussion we have the following example.
Example 4.3. Let M 10, L 20, and I 0, 1. The equation
2
1 − f 2 x
1001 efx − 1
1
−
1 − x − sin2πx,
1000e − 1
1000
2
4.10
where x ∈ 0, 1 has a unique solution f ∈ A 0, 1, 0, 1, 0, 10, 900.
Acknowledgements
The author would like to thank the referees for their detailed comments and helpful
suggestions. He is supported by Project HITC200706 supported by Science Research
Foundation in Harbin Institute of Technology.
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