The Solutions of the Series-Like Iterative Equation with Variable Coefficients

Volume 2009, Article ID 173028,7pages doi:10.1155/2009/173028

Research Article

The

C

1

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,[email protected]

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 Baker2000and using the Schauder fixed point theory, the existence and uniqueness of theC1_{solutions of the series-like}

iterative equations with variable coefficients are discussed.

Copyrightq2009 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

λ1fx λ2f2x · · ·λnfnx Fx, x∈I: a, b , 1.1

whereFis a given function,fis an unknown function,λi ∈ R1i1,2, . . . , n,andfk k 1,2, . . . , n is the kth iterate of f, that is, f0x x, fkx f ◦ fk−1x. The case of all constantλ_iswas considered in1–10 . In 2000, W. N. Zhang and J. A. Baker first discussed the continuous solutions of such an iterative equation with variable coefficients λi λix 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 in11 . In this paper, we continue the works of11,12 , and consider the series-like iterative equation with variable coefficients

∞

i1

whereλix : I → 0,1 are given continuous functions and∞_i1λix 1, λ1x ≥ c >

0∀x∈I, maxx∈Iλix ci.We improve the methods given by the authors in11,12, and the conditions of11,12 are weakened by constructing a new structure operator.

2. Preliminaries

LetC0_{I,R {f:I → R, f is continuous}, clearlyC}0_{I,R, ·}

c0is a Banach space, where

f _c0maxx∈I|fx|, forfinC0I,R.

Let C1I,R {f : I → R, f is continuous and continuously differentiable}, then

C1_{I,Ris a Banach space with the norm ·}

c1, where f _c1 f _c0 f _c0, forfinC1I,R. Being a closed subset,C1_{I, Idefined 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 of4 .

Lemma 2.1. Suppose thatϕ∈C1_{I, Iand}

_{ϕx≤M, ∀x∈I, 2.2}

_{ϕx1−ϕx2≤M|x}

1−x2|, ∀x1, x2∈I, 2.3

whereMandMare positive constants.Then

ϕn_x1−ϕn_x2≤M 2n−2 in−1

Mi

|x1−x2|, 2.4

for anyx1, x2inI, whereϕndenotesdϕn/dx.

Lemma 2.2. Suppose thatϕ1, ϕ2∈C1I, Isatisfy2.2.Then

ϕn₁−ϕn_{2 c}0≤

n

i1

Mi−1

ϕ1−ϕ2 c0. 2.5

Lemma 2.3. Suppose thatϕ1, ϕ2∈C1I, Isatisfy2.2and2.3.Then

ϕk₁1−ϕk₂1

c0≤k

1Mkϕ₁−ϕ_2c0

Qk1M

k

i1

k−i1Mki−1

ϕ1−ϕ2 c0,

2.6

3. Main Results

For given constantsM1>0 andM2>0, let

AM1, M2

ϕ∈C1I, I:ϕx≤M1, ∀x∈I ,

_{ϕx1−ϕx2≤M}

2|x1−x2|, ∀x1, x2∈I

.

3.1

Theorem 3.1existence. Given positive constantsM1, M2 andF ∈ AM1, M2,if there exists constantsN1≥1andN2>0, such that

P1c−∞i2ciN₁i−1≥M1/N1,

P2c−∞_i2ci2i_j−_i−2₁N₁j≥M2/N2,

then1.2has a solutionfinAN1, N2.

Proof. For convenience, letdmax{|a|,|b|}.

DefineK:AN1, N2 → C1I, Isuch thatK:f → Kf, where

Kft ∞

i1

λixfit, ∀x, t∈I. 3.2

Sincef ∈ AN1, N2, it is easy to see that|fit| ≤dfor allt∈I, and|λixfit| ≤d|λix|for allx, t ∈ I.It follows from∞_i1λix 1 that∞_i1λixfi_{tis uniformly convergent. Then}

Kftis continuous fort∈I. Also we have

a

∞

i1

λixa≤

∞

i1

λixfit≤

∞

i1

λixbb, 3.3

thusKf ∈C0I, I.

For anyf ∈ AN1, N2, we have

_dtdλixfitλixffi−1tfi−1t≤ciN₁i. 3.4

By condition P1, we see that ∞_i1ciN₁i is convergent, therefore∞i1cifit

is uniformly convergent fort∈I, this implies thatKftis continuously differentiable fort∈I. Moreover

_dtdKft

≤∞

i1

λixfit≤

∞

i1

ByLemma 2.1,

_dtdKft1

− d

dt

Kft2≤ ∞

i1

λixfit1−fit2

≤∞

i1

ci

⎛ ⎝_N22i−2

ji−1

N₁j ⎞

⎠_{|t1−t2|:μ2|t1−t2|.}

3.6

ThusKf ∈ Aμ1, μ2.

DefineT :AN1, N2 → C1I, Ias follows:

Tft 1

λ1xFt−

1

λ1xKft ft, ∀t, x∈I, 3.7

wheref ∈ AN1, N2. BecauseKf,F,andf are continuously differentiable for allt∈ I,Tf is continuously differentiable for allt∈I. By conditionsP1andP2, for anyt1, t2inI,we have

_dtdTft≤ 1 λ1xF

_t 1

λ1x

∞

i2

λixfit≤ 1

cM1

1

c

∞

i2

ciN₁i

≤ 1

cM1

1

ccN1−M1 N1.

3.8

We furthermore have

_dtdTft1− d dt

Tft2≤ 1

λ1xF

_t1−Ft2 1

λ1x

∞

i2

ci

fit1−fit2

≤ 1

cM2|t1−t2|

1

c

∞

i2

ciN2

⎛ ⎝2i−2

ji−1

N₁j ⎞ ⎠_|t1−t2|

≤N2|x1−x2|.

3.9

ThusT :AN1, N2 → AN1, N2is a self-diffeomorphism.

Now we prove the continuity of T under the norm · _c1. For arbitrary f1, f2 ∈

Tf1−Tf2 _c0 max

t∈I

− 1

λ1xKf1t f1t 1

λ1xKf2t−f2t

≤ 1

cmaxt∈I

∞

i2

λixf₁it−

∞

i2

λixf2it

≤ 1 c ∞

i2

cif1i−f2i_c0

≤ 1

c

∞

i2

ci i

k1

N₁k−1

_f1−f2

c0,

_dtdTf1− d dtTf2

c0

max t∈I

− 1

λ1x

Kf1t

f1t 1

λ1x

Kf2t

₋

f2t

≤ 1

cmaxt∈I

∞

i2

λixf₁it−

∞

i2

λixf₂it

≤ 1

c

∞

i2

ci

fi

1

−fi 2 c0 ≤ 1 c ∞

i2

ci

iN₁i−1f₁−f_2c0QiN2 i−1

k1

i−kN₁ik−2

_f1−f2

c0

.

3.10

Let

E1 1

c

∞

i2

ci i

k1

N₁k−1QiN2 i−1

k1

i−kNi₁k−2 , E2 1 c ∞

i2

ciiN₁i−1, Emax{E1, E2}.

3.11

Then we have

_Tf1−Tf2

c1Tf1−Tf2_c0

Tf1

₋

Tf2

c0≤E1f1−f2c0E2f 1−f2c0

≤Ef1−f2_c0Ef₁−f_2c0 Ef1−f2_c1,

3.12

which gives continuity ofT.

It is easy to show thatAN1, N2is a compact convex subset ofC1I, I. By Schauder’s fixed point theorem, we assert that there is a mappingf ∈ AN1, N2such that

ft Tft 1

λ1xFt−

1

λ1xKft ft, ∀t∈I. 3.13

Theorem 3.2Uniqueness. Suppose that (P1) and (P2) are satisfied, also one supposes that

P3E <1,

then for arbitrary functionFinAM1, M2,1.2has a unique solutionf∈ 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 thatAN1, N2is a closed subset ofC1I, I, by3.12andP3, we see

thatT :AN1, N2 → AN1, N2is a contraction. ThereforeT has a unique fixed pointfx inAN1, N2, that is,1.2has a unique solution inAN1, N2, this proves the theorem.

4. Example

Consider the equation

∞

i1

λixfi_x 1 4x

2_{, x∈I: −1,1 , 4.1}

whereλ1x 33/36 1/36cos2_{πx/2, λ2x 1/36 1/36sin}2_{πx/2, λ3x 1/36,}

λ4x λ5x · · · 0.It is easy to see that 0 ≤λix≤ 1, ∞i1λix 1, c 33/36, c2 2/36, c31/36, c4c5· · ·0.

For anyx, yin−1,1 ,

_{Fx|0.5x| ≤0.5, Fx−Fy≤ |0.5x|0.5y≤1, 4.2}

thusF∈ A0.5,1.By conditionP1, we can chooseN11.1,and by conditionP1, we can chooseN21.5. Then byTheorem 3.1, there is a continuously differentiable solution of4.1 inA1.1,1.5.

Remark 4.1. HereFxis not monotone forx∈−1,1 , hence it cannot be concluded by11,

12 .

Acknowledgments

This work was supported by Guangdong Provincial Natural Science Foundation07301595 and Zhan-jiang Normal University Science Research ProjectL0804.

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