Volume 2009, Article ID 173028, 7 pagesdoi:10.1155/2009/173028 Research Article Equation with Variable Coefficients Yuzhen Mi, Xiaopei Li, and Ling Ma Mathematics and Computational Schoo
Trang 1Volume 2009, Article ID 173028, 7 pages
doi:10.1155/2009/173028
Research Article
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 Dom´ınguez 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 C1solutions of the series-like iterative equations with variable coefficients are discussed
Copyrightq 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
λ1f x λ2f2x · · · λ n f n x Fx, x ∈ I : a, b , 1.1
where F is a given function, f is an unknown function, λ i ∈ R1i 1, 2, , n, and f k k
1, 2, , n is the kth iterate of f, that is, f0x 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 λ i x
which are all continuous in intervala, 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 of11,12 , and consider the series-like iterative equation with variable coefficients
∞
i1
λ i xf i x Fx, x ∈ I : a, b , 1.2
Trang 2where λ i x : I → 0, 1 are given continuous functions and∞i1 λ i x 1, λ1x ≥ c >
0∀x ∈ I, max x∈I λ i x c i We improve the methods given by the authors in 11,12 , and the conditions of11,12 are weakened by constructing a new structure operator
2 Preliminaries
Let C0I, R {f : I → R, f is continuous}, clearly C0I, R, · c0 is a Banach space, where
f c0 maxx∈I |fx|, for f in C0I, R.
Let C1I, R {f : I → R, f is continuous and continuously differentiable}, then
C1I, R is a Banach space with the norm · c1, where f c1 f c0 f c0, for f in C1I, R Being a closed subset, C1I, I defined by
C1I, I f ∈ C1I, 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 paper4 , but the conditions are weaker than those of 4
Lemma 2.1 Suppose that ϕ ∈ C1I, I and
ϕx ≤ M, ∀x ∈ I, 2.2
ϕx1 − ϕx2 ≤ M|x1− x2|, ∀x1, x2∈ I, 2.3
ϕ n x1−ϕ n x2 ≤ M
2n−2
in−1
M i |x1− x2|, 2.4
for any x1, x2in I, where ϕ ndenotes dϕ n /dx.
Lemma 2.2 Suppose that ϕ1, ϕ2∈ C1I, I satisfy 2.2.Then
ϕ n
1− ϕ n
2 c0≤
n
i1
M i−1 ϕ1− ϕ2 c0. 2.5
Lemma 2.3 Suppose that ϕ1, ϕ2∈ C1I, I satisfy 2.2 and 2.3.Then
ϕ k11 − ϕ k12
c0≤ k 1M k ϕ
1− ϕ
2
c0
Qk 1M
k
i1
k − i 1M ki−1 ϕ1− ϕ2 c0,
2.6
for k 0, 1, 2, , where Qs 0 as s 1 and Qs 1 as s 2, 3,
Trang 33 Main Results
For given constants M1> 0 and M2> 0, let
AM1, M2 ϕ ∈ C1I, I :ϕx ≤ M1, ∀x ∈ I ,
ϕx1 − ϕx2 ≤ 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 −∞
i2 c i N i−1
1 ≥ M1/N1,
P2 c −∞
i2 c i2i−2
ji−1 N1j ≥ M2/N2, then1.2 has a solution f in AN1, N2.
Proof For convenience, let d max{|a|, |b|}.
Define K : AN1, N2 → C1I, I such that K : f → K f, where
K f t ∞
i1
λ i xf i t, ∀x, t ∈ I. 3.2
Since f ∈ AN1, N2, it is easy to see that |f i t| ≤ d for all t ∈ I, and |λ i xf i t| ≤ d|λ i x| for all x, t ∈ I It follows from∞
i1 λ i x 1 that∞
i1 λ i xf i t is uniformly convergent Then
K f t is continuous for t ∈ I Also we have
a
∞
i1
λ i xa ≤∞
i1
λ i xf i t ≤∞
i1
λ i xb b, 3.3
thus K f ∈ C0I, I.
For any f ∈ AN1, N2, we have
dt d λ i x f i t λ i x
f
f i−1 t f i−1 t
≤ c i N1i 3.4
By condition P1, we see that∞
i1 c i N1i is convergent, therefore∞
i1 c i f i tis uniformly
convergent for t ∈ I, this implies that K f t is continuously differentiable for t ∈ I Moreover
dt d K f t
≤∞
i1
λ i x
f i t
≤∞
i1
c i N1i : μ1. 3.5
Trang 4ByLemma 2.1,
dt dK f t1− d
dt
K f t2 ≤∞
i1
λ i x
f i t1 − f i t2
≤∞
i1
c i
⎛
⎝N2
2i−2
ji−1
N1j
⎞
⎠|t1− t2| : μ2|t1− t2|.
3.6
Thus K f ∈ Aμ1, μ2
Define T : AN1, N2 → C1I, I as follows:
Tf t 1
λ1x F t −
1
λ1x K f t ft, ∀t, x ∈ I, 3.7
where f ∈ AN1, N2 Because K f , F, and f are continuously differentiable for all t ∈ I, Tf
is continuously differentiable for all t ∈ I By conditions P1 and P2, for any t1, t2in I, we
have
dt dTf t ≤ 1
λ1xFt 1
λ1x
∞
i2
λ i x
f i t
≤ 1c M11
c
∞
i2
c i N1i
≤ 1
c cN1− M1 N1.
3.8
We furthermore have
dt dTf t1− d
dt
Tf t2 ≤ 1
λ1xFt1 − Ft2 1
λ1x
∞
i2
c i
f i t1 − f i t2
≤ 1
c M2|t1− t2| 1
c
∞
i2
c i N2
⎛
⎝2i−2
ji−1
N1j
⎞
⎠|t1− t2|
≤ 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,
Trang 5Tf1− Tf2 c0 max
t∈I
−λ11x K f1t f1t 1
λ1x K f2t − f2t
≤ 1
cmaxt∈I
∞
i2
λ i xf i
1t −∞
i2
λ i xf i
2t
≤ 1
c
∞
i2
c i f i
1− f i
2
c0
≤ 1
c
∞
i2
c i
i
k1
N1k−1 f1− f2
c0,
dt d Tf1 − d
dt Tf2
c0
max
t∈I
−λ11xK f1tf1t 1
λ1x
K f2t−f2t
≤ 1
cmaxt∈I
∞
i2
λ i x f1i t −∞
i2
λ i x f2i t
≤ 1
c
∞
i2
c i
f i
1
− f i
2
c0
≤ 1
c
∞
i2
c i
iN1i−1 f
1− f
2
c0 QiN2
i−1
k1
i − kN ik−2
1 f1− f2
c0
.
3.10 Let
E1 1
c
∞
i2
c i
i
k1
N1k−1 QiN2
i−1
k1
i − kN ik−2
E2 1
c
∞
i2
c i iN1i−1 , E max {E1, E2}.
3.11
Then we have
Tf1− Tf2
c1 Tf1− Tf2
c0
Tf1
−Tf2
c0≤ E1 f1− f2
c0 E2 f
1− f
2
c0
≤ E f1− f2
c0 E f
1− f
2
c0 E f1− f2
c1,
3.12
which gives continuity of T.
It is easy to show thatAN1, N2 is a compact convex subset of C1I, I By Schauder’s fixed point theorem, we assert that there is a mapping f ∈ AN1, N2 such that
f t Tft 1
λ1x F t −
1
λ1x K f t ft, ∀t ∈ I. 3.13
Let t x, we have fx as a solution of 1.2 in AN1, N2 This completes the proof
Trang 6Theorem 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.
Theorem 3.1, we see thatAN1, N2 is a closed subset of C1I, 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
inAN1, N2, that is, 1.2 has a unique solution in AN1, N2, this proves the theorem
4 Example
Consider the equation
∞
i1
λ i xf i x 1
where λ1x 33/36 1/36 cos2πx/2, λ2x 1/36 1/36 sin2πx/2, λ3x 1/36,
λ4x λ5x · · · 0 It is easy to see that 0 ≤ λ i x ≤ 1, ∞i1 λ i x 1, c 33/36, c2
2/36, c3 1/36, c4 c5 · · · 0.
For any x, y in −1, 1 ,
Fx |0.5x| ≤ 0.5, Fx − 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 byTheorem 3.1, there is a continuously differentiable solution of 4.1
inA1.1, 1.5.
12
Acknowledgments
This work was supported by Guangdong Provincial Natural Science Foundation07301595 and Zhan-jiang Normal University Science Research ProjectL0804
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