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Volume 2009, Article ID 892691, 14 pagesdoi:10.1155/2009/892691 Research Article On Series-Like Iterative Equation with a General Boundary Restriction Wei Song,1 Guo-qiu Yang,1 and Feng-

Volume 2009, Article ID 892691, 14 pages

doi:10.1155/2009/892691

Research Article

On Series-Like Iterative Equation with

a General Boundary Restriction

Wei Song,1 Guo-qiu Yang,1 and Feng-chun Lei2

1 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

2 Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China

Correspondence should be addressed to Wei Song,dawenhxi@126.com

Received 26 August 2008; Revised 19 November 2008; Accepted 4 February 2009

Recommended by Tomas Dominguez Benavides

By means of Schauder fixed point theorem and Banach contraction principle, we investigate the existence and uniqueness of Lipschitz solutions of the equationPf ◦ f  F Moreover, we get that the solution f depends continuously on F As a corollary, we investigate the existence and

uniqueness of Lipschitz solutions of the series-like iterative equation∞

n1a n f n x  Fx, x ∈B

with a general boundary restriction, where F : B → A is a given Lipschitz function, and B, A are

compact convex subsets ofRNwith nonempty interior

Copyrightq 2009 Wei Song 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

denotes the composition of mappings Let CX, X be the set of all continuous self-mappings

on X An equation with iteration as its main operation is simply called an iterative equation.

the problem of iterative roots2,5,6, that is, finding f ∈ CX, X such that f nis identical

bifurcations11,12, is also an iterative equation

As a natural generalization of the problem of iterative roots, a class of iterative equations which is called polynomial-like iterative equation:

always fascinates many scholars’ attentions3,13 It is more difficult than the analogous

differentiation is a linear operator but iteration is not

f → Lf for 1.1 and used the fixed point theory in Banach spaces to get the solutions of 1.1

By means of this method, Zhang and Si made a series of works concerning these qualitative

solutions of1.1, and the convexity of solutions is also considered

class of Lipschitz functions In 2004, Tabor and ˙Zołdak 23 studied the iterative equations in Banach spaces In the above references, the authors first gave theorems for the existence of solutions of

solutions of the iterative functional equation:

∞



n1

Lipschitz function In23, the existence of solutions of



i

i

considered the case F|∂B id|∂B.

It is easy to see that1.1 is the special case of 1.3 with ai  0, i  n  1, n  2, ,

andB  a, b Since the left-hand side of 1.3 is a functional series, in this paper we call it series-like iterative equation In14–20, the authors considered the solutions of 1.1 with

F : I → I, Fa  a, Fb  b In 22, the authors considered the solutions of 1.3 with

∞



n1

a n f n x  Fx, x ∈ B,

1.5

whereB ⊂ RNis a convex compact set with nonempty interior In21, the authors considered

the more general case is

∞



n1

a n f n x  Fx, x ∈ B,

1.6

series-like iterative equation with a general boundary restriction It is easy to see that14–22 all considered one special case of1.6

The problem of differentiable solutions of iterative equations has also fascinated many scholars’ attentions In Zhang16 and Si 19, the C1and C2solutions of1.1 are considered

H

uniqueness of differentiable solutions to the iterative equations involving iterated functional series:

∞



i1

λ i H if i x  Fx, x ∈ I  a, b,

∞



i1

λ i H i

x, φ a i1 x, , φ a ini x Fx, x ∈ I  a, b.

1.8

But the references above only considered the case that Fa  a, Fb  b.

The problem of differentiable solutions of higher dimensional iterative equations is

of solutions of1.3 In 29, C1solutions of

∞



n1

continuous The boundary restrictions are not considered in the two references above because

they only consider the case that FB ⊆ B.

small shift of maps, choosing suitable metrics, and finding a relation between uniqueness

and stability of fixed points of maps of general spaces, Mai and Liu proved the existence,

G

investigate the existence and uniqueness of Lipschitz solution of this equation

and uniqueness of solution of

theorem

2 Preliminary

x∈B fx , for f ∈ CB, R N

LipB, A, m, M : {f : B −→ A | f is continuous, fB  A, ∀x, y ∈ B,

Let g : ∂B → ∂A be a continuous surjective map, and let CB, A, m, M, g denote the

Lemma 2.2 Let m ∈ 0, 1, M ∈ 1, ∞, and f ∈ LipB, A, m, M be arbitrary, then f−1 ∈ LipA, B, 1/M, 1/m

Lemma 2.3 For every m > 0, the mapping



A, B, 0, 1 m



2.3

is well defined and Lipschitz with constant 1/m.

Lemma 2.4 For 1 ≤ K, M < ∞ and F ∈ LipB, A, 0, K, the mapping

is Lipschitz with constant 1.

Proposition 1 in23

Lemma 2.5 If H, G are homeomorphisms from B to A with Lipschitz constant L, then H − G B≤

L H−1− G−1 A.

Lemma 2.6 If f, g ∈ LipB, B, m, M, then f k − g k B≤k−1

j0M j f − g B.

Lemma 2.7 For any M ∈ 1, ∞ and g : ∂B → ∂B, which is a surjective map, CB, B, 0, M, g is

a compact subset of C B, R N .

{fn k}∞

k1of{fn}∞

Noticing that

y − fx  fn

x n

− fx ≤ fn

x n

we can get fx  y Then, CB, A, 0, M, g is compact.

Lemma 2.8 If f : D N → D N is continuous and f S N−1 ⊂ S N−1 Let f0 denote f|SN−1 If

degf0 / 0, then f is surjective, where degf0 denotes the degree of f0.

Proof Suppose that f is not surjective Let x0∈ D N \fD N  If x0∈ S N−1, then f0is homotopic

to a constant and degf0  0, a contradiction So x0/ ∈ S N−1, then there exists a retraction

f0 r ◦ f| S N−1 This means thatf0∗N−1is trivial, then degf0  0 So f is surjective.

Lemma 2.9 Let M ∈ 1, ∞ and CB, B, 0, M, g be defined as above, where the surjective map g :

∂ B → ∂B is the restriction of the elements of CB, B, 0, M, g If degg / 0, then CB, B, 0, M, g

is a convex subset of C B, R N .

Proof For ∀t ∈ 0, 1 and ∀f, h ∈ CB, B, 0, M, g, tf  1 − th is continuous and

It is easy to see that

C B, B, 0, M, g is convex.

3 Main Result

Theorem 3.1 Give M, K ∈ 1, ∞ and A, B which are compact convex subsets of R N with

maps and deg g / 0 If there exist a decreasing function α : 1, ∞ → 0, 1 and a continuous map

P defined on CB, B, 0, M, g such that

Pf ∈ CB, A, αM, ∞, T, ∀f ∈ CB, B, 0, M, g

Then, for any F ∈ CB, A, 0, K, T ◦ g, there exists a f ∈ CB, B, 0, M, g such that

Furthermore, if P is Lipschitz with a Lipschitz constant d which satisfies d/αM < 1, then f is unique, and f depends continuously on F.





,





−→ f ◦ F ∈ Lip



are both well defined and continuous

From the above discussions, for∀f ∈ CB, B, 0, M, g, we can get that

Pf ∈ CB, A, αM, ∞, T,



α M , T−1



,



α M , g



⊂ CB, B, 0, M, g.

3.4

continuous

f in C B, B, 0, M, g Then,

which implies

This means that f satisfies the assertion of the theorem.

For f1, f2∈ CB, B, 0, M, g, byLemma 2.5,

S F◦ L ◦ P

f1

− SF◦ L ◦ Pf2

B

Pf1

−1

◦ F − Pf2−1

◦ F

B

≤Pf1

−1

− Pf2−1

A

1



− Pf2

B

≤



d

f

1− f2B.

3.7

contraction principle, f is unique.

Suppose F1, F2∈ CB, A, 0, K, T ◦ g and f1, f2∈ CB, B, 0, M, g such that Pf1 ◦ f1 

F1andPf2 ◦ f2 F2, then,

f1− f2

BS F

1◦ L ◦ Pf1

− SF2◦ L ◦ Pf2

B

Pf1

−1

◦ F1− Pf2−1

◦ F2

B

≤Pf1

−1

◦ F1− Pf2−1

◦ F1

BPf2

−1

◦ F1− Pf2−1

◦ F2

B

≤



d

f

1− f2B 1

F

1− F2B.

3.8

This means that

f1− f2B≤ 1



−1

then f depends continuously on F.

Theorem 3.2 Let the sequence {a k}∞

k0 ⊂ R satisfy that∞

k0a k is absolutely convergent, then for all M ∈ 1, ∞ the mapping

k0

a k f k ∈ CB, R N

3.10

is well defined and continuous.

Proof Since∞

k0a kis absolutely convergent and{f k}∞k0is a uniformly bounded sequence of

and continuous

ByLemma 2.6and the absolute convergence of∞

k0a k, the continuity of the mapping

P can be easily got

4 Iterative Equation in R

antipodal maps on ∂I Let g1, g2: ∂I → ∂J satisfy g1a  c, g1b  d and g2a  d, g2b 

c Obviously, g1 g1◦ h1and g2 g1◦ h2 Obviously, deg h1  1 and degh2  −1

Theorem 4.1 Suppose that the sequence {a k}∞k1 ⊂ R satisfy a1 > 0 and ∞

k1a k is absolutely convergent and M, K ≥ 1, if

1≥ a1−∞

n2

a n M n−1≥ K

∞



i1

a i h i k−1a <∞

i1

Then, for any F ∈ CI, J, 0, K, gk, 1.6 has a solution in CI, I, 0, K, hk, where I  a, b and

J ∞i1a i h i−1

k a,∞i1a i h i−1

k b, k  1, 2 Moreover, if

k2 a k k−2

j0M j

f is unique and depends continuously on F.

Proof For t ∈ 1, ∞ define αt by αt  min{maxa1−∞i2|ai|t i−1, 0 , 1} Since 0 ≤ αt ≤ 1,

i1

i1a i h i−1

Pfb ∞i1a i h i k−1b For x, y ∈ I with y > x, one can check that

The above discussions imply that

Pf ∈ CI, J, α M, ∞, g1



x ∈I

∞



i1

a i f i−1x −∞

i1

a i g i−1x

≤ sup

x ∈I

∞



i1

a i · f i−1x − g i−1x

i2

a i · f i−1− g i−1

I

i2

a i ·i−2

j0

M j f − g I

4.7

ByTheorem 3.1, the assertion is true

Example 4.2.

54

55f x ∞

i2

−1i−2

54i−1 f

i x  x2, x ∈ I  0, 1,

F1x  x2: I −→ I, F1

∂I  g1.

4.8

Obviously, F1x  x2∈ CI, I, 0, 2, g1 Let M  4 since

α M  a1−∞

n2

a n M n−1 54

55−∞

i2

4i−1

54i−1  248

275 >

2

4,

k2 a k k−2

j0M j

k2

1/54 k−1 k−2

j04j

k2



2k−1· 4k−2

/54 k−1

275

23× 248.

4.9

4, h1 For

F2x 

⎧

⎪

⎪

⎪

⎪

⎪

⎪



0,1

4



,

1

 1

4,

3 4



,

4, 1



,

4.10

it is easy to see that F2∈ CI, I, 0, 2, g1 Then,

54

55f x ∞

i2

−1i−2

54i−1 f

i x  F2x, x ∈ I  0, 1,

F2: I −→ I, F2

∂I  g1

4.11

has an unique increasing solution in CI, I, 0, 4, h1

x2in0, 1 The equation

210

211f x ∞

i2

−1i−2

210i−1f

i x  F3x, x ∈ I  0, 1,

F3: I −→ I, F3

∂I  g1

4.12

Example 4.3 For convenience, we only consider {ak}∞k1with a 2i  0, i  1, 2, Obviously, for I  0, 1, F1x  1 − x2∈ CI, I, 0, 2, g2 ByTheorem 4.1,

254

255f x ∞

i2

−1i−2

256i−1f 2i−1 x  F1x, x ∈ 0, 1,

F1: I −→ I, F1

∂I  g2

4.13

has an unique strictly decreasing solution in CI, I, 0, 4, h2 For

F2x 

⎧

⎪

⎪

⎪

⎪

⎪

⎪



0,1

4



,

1

4,

3 4



,

 3

4, 1



,

4.14

it is easy to see that F2∈ CI, I, 0, 2, g2 Then,

254

255f x ∞

i2

−1i−2

256i−1f 2i−1 x  F2x, x ∈ 0, 1,

F2: I −→ I, F2

∂I  g2

4.15

has an unique decreasing solution in CI, I, 0, 4, h2

equation

1000

1001f x ∞

i2

−1i−2

1000i−1f

2i−1 x  F3x, x ∈ 0, 1,

F3: I −→ I, F3

∂I  g2

4.16

5 Iterative Equation in RN N ≥ 2

ball ofRN

degξ  ±1

Theorem 5.1 Let {a i}∞i1⊂ 0, 1 with∞

i1a i  1, a1> 0 and there exist two constants M, K≥ 1

with

a1−∞

i2

a i M 2i−2≥ K

Then, for any F ∈ CD N , D N , 0, K, ξ ,

∞



i1

a i f 2i−1 x  Fx, x ∈ D N ,

F : D N −→ D N , F

S N−1  ξ

5.2

has a solution f ∈ CD N , D N , 0, M, ξ  Moreover, if ∞k2a k2k−3

j0 M j /αM < 1, then f is unique and depends continuously on F.

Proof For t ∈ 1, ∞, define αt  max{a0−∞

i1a i t 2i−2 , 0 }, then 0 ≤ αt ≤ a0 ≤ 1 Define

P : CD N , D N , 0, M, ξ  → CD N ,RN by

i1

Then, we get





∞



i1

a i f 2i−2 x −∞

i1

a i f 2i−2 y





≥ a1 x − y −∞

i2

a if 2i−2 x − f 2i−2 y

≥



a1−∞

i2

a i M 2i−2



x − y

 αM x − y , Pfx − Pfy ≤



a1∞

i2

a i M 2i−2



x − y

5.4

∞



i1

SincePf| S N−1 ∞i1a i ξ 2i−2|SN−1  id|SN−1 and degid|SN−1  1, then PfD N   D N

i1a i ξ 2i−1  ξ For f, g ∈

C D N , D N , 0, M, ξ, byLemma 2.6, we get that

k2

a k

2k−3

j0

ByTheorem 3.1, the assertion is true

Example 5.2 For F x  −x1, x2  distx, S11, 0  1 − x1−x2

1 x2

2, −x2, where x 

x1, x2 ∈ D2, and distx, S1 denotes the distance of the point x from S1 Obviously, F|S1  r1,

D2,

Fx − Fy ≤ 2 x − y , Fx ≤ x  distx, S1

hold By the above discussion, we get that F ∈ CD2, D2, 0, 2, r1 Then, byTheorem 5.1,

254

255f x ∞

i2

1

256i−1f

2i−1 x  Fx, x ∈ D2,

F : D2−→ D2, F|S1  r1

5.8

has a unique solution in CD2, D2, 0, 4, r1

Acknowledgments

The authors would like to thank the referees for their valuable comments and suggestions that led to truly significant improvement of the manuscript Project HITC200706 is supported

by Science Research Foundation in Harbin Institute of Technology

References

1 M Kuczma, Functional Equations in a Single Variable, vol 46 of Monografie Matematyczne, PWN-Polish

Scientific, Warsaw, Poland, 1968

2 M Kuczma, B Choczewski, and R Ger, Iterative Functional Equations, vol 32 of Encyclopedia of

Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1990.

3 K Baron and W Jarczyk, “Recent results on functional equations in a single variable, perspectives

and open problems,” Aequationes Mathematicae, vol 61, no 1-2, pp 1–48, 2001.

4 S Nabeya, “On the functional equation fp  qx  rfx  a  bx  cfx,” Aequationes Mathematicae,

vol 11, no 2-3, pp 199–211, 1974

5 N Abel, Oeuvres Compl`etes, Vol II, Grundahl & Son, Christiania, France, 1881.

6 J M Dubbey, The Mathematical Work of Charles Babbage, Cambridge University Press, Cambridge, UK,

1978

7 C T Ng and W Zhang, “Invariant curves for planar mappings,” Journal of Difference Equations and

Applications, vol 3, no 2, pp 147–168, 1997.

8 E Fontich, “Transversal homoclinic points of a class of conservative diffeomorphisms,” Journal of

Differential Equations, vol 87, no 1, pp 1–27, 1990.

9 D Bessis, S Marmi, and G Turchetti, “On the singularities of divergent majorant series arising from

normal form theory,” Rendiconti di Matematica e delle sue Applicazioni, vol 9, no 4, pp 645–659, 1989.

10 S J Greenfield and R D Nussbaum, “Dynamics of a quadratic map in two complex variables,”

Journal of Differential Equations, vol 169, no 1, pp 57–141, 2001.

11 P Collet, J P Eckmann, and O E Lanford, “Universal properties of maps on an interval,”

Communications in Mathematical Physics, vol 91, no 3, pp 211–254, 1986.

12 P J McCarthy, “The general exact bijective continuous solution of Feigenbaum’s functional equation,”

Communications in Mathematical Physics, vol 91, no 3, pp 431–443, 1983.

13 J Zhang, L Yang, and W Zhang, “Some advances on functional equations,” Advances in Mathematics,

vol 24, no 5, pp 385–405, 1995

14 W Zhang, “Discussion on the iterated equationn

i1λ i f i x  Fx,” Chinese Science Bulletin, vol 32,

no 21, pp 1444–1451, 1987

15 W Zhang, “Stability of the solution of the iterated equationn

i1λ i f i x  Fx,” Acta Mathematica

Scientia, vol 8, no 4, pp 421–424, 1988.

16 W Zhang, “Discussion on the differentiable solutions of the iterated equationn

i1λ i f i x  Fx,”

Nonlinear Analysis: Theory, Methods & Applications, vol 15, no 4, pp 387–398, 1990.

17 W Zhang, “An application of Hardy-B¨oedewadt’s theorem to iterated functional equations,” Acta

Mathematica Scientia, vol 15, no 3, pp 356–360, 1995.

18 J G Si, “The existence and uniqueness of solutions to a class of iterative systems,” Pure and Applied

Mathematics, vol 6, no 2, pp 38–42, 1990Chinese

19 J G Si, “The C2-solutions to the iterated equationn

i1λ i f i x  Fx,” Acta Mathematica Sinica, vol.

36, no 3, pp 348–357, 1993Chinese

20 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.

21 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.

22 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.

23 J Tabor and M ˙Zołdak, “Iterative equations in Banach spaces,” Journal of Mathematical Analysis and

Applications, vol 299, no 2, pp 651–662, 2004.

24 X Wang and J G Si, “Differentiable solutions of an iterative functional equation,” Aequationes

Mathematicae, vol 61, no 1-2, pp 79–96, 2001.

25 V Murugan and P V Subrahmanyam, “Existence of solutions for equations involving iterated

functional series,” Fixed Point Theory and Applications, vol 2005, no 2, pp 219–232, 2005.

26 V Murugan and P V Subrahmanyam, “Special solutions of a general iterative functional equation,”

Aequationes Mathematicae, vol 72, no 3, pp 269–287, 2006.

27 V Murugan and P V Subrahmanyam, “Erratum to: “Special solutions of a general iterative functional equation”Aequationes Mathematicae, vol 72, pp 269—287, 2006,” Aequationes Mathematicae, vol 76,

no 3, pp 317–320, 2008

28 X Li and S Deng, “Differentiability for the high dimensional polynomial-like iterative equation,”

Acta Mathematica Scientia, vol 25, no 1, pp 130–136, 2005.

29 X Li, “The C1solution of the high dimensional iterative equation with variable coefficients,” College

Mathematics, vol 22, no 3, pp 67–71, 2006.

30 J Mai and X Liu, “Existence, uniqueness and stability of C m solutions of iterative functional

equations,” Science in China Series A, vol 43, no 9, pp 897–913, 2000.