R E S E A R C H
Open Access
The
C
solutions of the Feigenbaum-like
functional equation
Ying Liang, Xiaopei Li
*and Yuzhen Mi
*Correspondence:
[email protected] Mathematics and Computational School, Zhanjiang Normal University, Zhanjiang, Guangdong 524048, China
Abstract
By using the Schauder’s fixed point theorem, and constructing the special functional space and the construction operator, the existence, uniqueness, quasi-convexity (or quasi-concavity), symmetry and stability of theC0solutions of the Feigenbaum-like
functional equations are discussed.
MSC: 39B22; 39B12; 39A10
Keywords: Feigenbaum-like functional equations; quasi-convex (or quasi-concave); symmetry; stability
1 Introduction
As early as , Feigenbaum found the period-doubling bifurcation phenomenon by re-searching the iteration of a single-peak function class []. To reveal the mechanism of the Feigenbaum phenomenon, many years ago, the Feigenbaum functional equation had been researched extensively. McCarthy [] obtained the general continuous exact bijec-tive solutions. Epstein [] gave a new proof of the existence of analytic, unimodal solutions by taking advantage of the normality properties of Herglotz functions and the Schauder-Tikhonov theorem. Eokmann and Wittwer [] studied the Feigenbaum fixed point by us-ing the computer. Thompson [] investigated an essentially sus-ingular solution by expressus-ing Feigenbaum’s equation as a singular Schroder functional equation whose solution was ob-tained using a scaling ansatz, and so on. Thus, some solutions in specific cases were found. Specifically, in , to give a feasible method, the second kind of the Feigenbaum func-tional equation,
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
f(x) =λf(f(λx)), <λ< ,
f() = ,
≤f(x)≤, x∈[, ]
a kind of the equivalent equation, was given by Yang and Zhang []. The continuous valley-unimodal solutions were shown by using the constructive method. Recently, there have been a lots of results about the polynomial-like iterative equation. In , by using Schauder’s fixed point theorem to an operator defined by a linear combination of iter-ates of the unknown mappingf, a result on the existence of continuous solutions of the polynomial-like iterative equation was given in []. Furthermore, the results were given
for its differentiable solutions []. Then the convex solutions and concave solutions [, ], the analytic solutions [–], the symmetric solutions [], the higher-dimensional solutions [], and the results on the unit circle [] were obtained. In order to understand the dynamics of a second order delay differential equation with a piecewise constant ar-gument, the derived planar mappings and their invariant curves were studied []. Based on the iterative root theory for monotone functions, an algorithm for computing polygo-nal iterative roots of increasing polygopolygo-nal functions was given []. Else, a problem about the Hyers-Ulam stability was raised first by Ulam in and solved for Cauchy equa-tion by Hyers []. Later, many papers on the Hyers-Ulam stability have been published, especially, for the polynomial-like iterative equation [–].
In this paper, by using Schauder’s fixed point theorem, and constructing the special func-tional space and the construction operator, we consider the properties of the solutions of the Feigenbaum-like functional equation, which is a non-extended iterative equation,
⎧ ⎨ ⎩
f(x) = λ+ f(λx) +λλ+g(x), <λ< ,
a≤f(x)≤b, x∈I, (.)
whereg(x) is a given disturbance function,f(x) is an unknown function, andf(x) =f(f(x)),
I= [a,b]. It is clear thata≤≤b, sinceλx∈Ifor allx∈I. We give not only the existence of continuous solutions of (.) but also their uniqueness, stability (the continuous depen-dence and the Hyers-Ulam stability), quasi-convexity (or quasi-concavity), symmetry by applying fixed point theorems. Finally, we give an example to verify those conditions given in theorems.
2 Preliminary
In this section, we give several important definitions, lemmas and notions.
LetC(I,R) ={f :I→R,f is continuous}. Obviously,C(I,R) is a Banach space with the
norm · c, where the normfc=maxx∈I|f(x)|forf ∈C(I,R).
LetC(I) ={f ∈C(I,R) :a≤f(x)≤b,a≤f(λx)≤b,f is continuous}. ThenC(I) is a
complete metric space.
LetX(I;M) ={f ∈C(I) :|f(x) –f(y)| ≤M|y–x|,∀x,y∈I}, whereMis a positive
con-stant.
LetX(I;m,M) ={f ∈X(I;M) :|f(x) –f(y)| ≥m|x–x|,∀x∈I, <m≤M}, wheremis a
positive constant.
Letf(λx) :=f(λ)(x),f(λx) :=f(f(λx)) :=f(λ)(x),fk(λx) :=fk(λ)(x).
Definition . f:I→Ris a quasi-convex (or quasi-concave) function [] if for∀x,y∈I
andλ∈[, ], we have
fλx+ ( –λ)y≤maxf(x),f(y) or fλx+ ( –λ)y≥minf(x),f(y) .
Let Xσ(I;M) denote the families consisting of all convex functions or
quasi-concave ones inX(I;M), whereσ=qcvorσ=qcc.
Lemma . X(I;M),Xqcv(I;M),and Xqcc(I;M)are compact convex subsets of C(I,R).
Lemma . The composition f ◦g is quasi-convex (or quasi-concave)if f is increasing and g is quasi-convex(or quasi-concave).In particular,for an increasing quasi-convex(or quasi-concave)function f,fkis also quasi-convex(or quasi-concave).
Lemma . If f,h∈X(I;M),then
f(λ)–h(λ)c≤(M+ )f –hc. (.)
Proof Note that
f(λ)–h(λ)c=max
x∈I f
(λx) –h(λx)
≤max
x∈I
ff(λx)–fh(λx)+max
x∈I
fh(λx)–hh(λx)
≤Mf(λ)–h(λ)c+f–hc.
Lety=λx. Theny∈λI, and
f(λ)–h(λ)c=max
y∈λIf(y) –h(y) ≤max
y∈I f(y) –h(y) ≤ f–hc.
Then
f(λ)–h(λ)
c≤Mf –hc+f–hc
= (M+ )f–hc.
Thus, (.) holds.
Lemma . Suppose thatϕ∈X(I;m,M).If the positive constants m,M andλsatisfy
λ+ > M
m , (.)
then Lϕ,defined by
Lϕ(x) =
+
λ
x–
λϕ
λϕ–(x), (.)
is an orientation-preserving homeomorphism from I onto itself,and
(Lϕ)–∈X
I;
ξ
,
ξ
From Lemma .,Lϕis well defined and is an orientation-preserving homeomorphism
is well defined and is an orientation-preserving homeomorphism fromIonto itself, then (Lϕk)–∈X(I;ξ,
This implies that the results in Lemma . are also true fork+ , which completes the
proof.
3 Main results
In this section, we give several important theorems on the existence, uniqueness, quasi-convex (quasi-concave), symmetry and stability of equation (.).
Theorem .(Existence) Given a positive constant M and g(x)∈X(I;M).If there exist constants M andλsuch that
Thus,Tf∈X(I,M). Now we prove the continuity ofTunder the norm · c. For arbitrary f,f∈X(I,M),
Tf–Tfc=max
x∈I
Tf(x) –Tf(x)
=max
x∈I
λ+ f(λx) –
λ+ f
(λx)
=
λ+ maxx∈I
f(λx) –f(λx)
=
λ+ f
(λ) –f
(λ)
≤Ef–fC,
where Lemma . is used and
E=M+
λ+ . (.)
Thus,T is continuous under the norm · c. Summarizing all the above, we see thatT
is a continuous mapping from the compact convex subsetX(I,M) of the Banach space
C(I,R) into itself. By Schauder’s fixed point theorem, we assert that there is a mapping
f ∈X(I,M) such that
f(x) =Tf(x) =
λ+ f
(λx) + λ
λ+ g(x), ∀x∈I. (.) This completes the proof.
Theorem .(Uniqueness) Suppose that(.)is satisfied and
M<λ. (.)
For any function g∈X(I,M),equation(.)has a unique solution f ∈X(I,M).
Proof The existence of equation (.) inX(I;M) is given by Theorem .. Note thatX(I;M) is a closed subset ofC(I). By (.) and (.), we see thatT :X(I;M)→X(I;M) is a
con-traction mapping. ThereforeThas a unique fixed pointf(x) inX(I;M), that is, equation (.) has a unique solutionf(x) inX(I;M).
Below, we discuss the quasi-convex (or quasi-concave) solutions of equation (.).
Definition . Suppose thatis a Lie group of all linear transformations onR. A mapping
f :R→R, is said to be-equivariant [] iff(γx) =γf(x),∀γ ∈,∀x∈R. This implies that fi is the-equivariant. LetG
(I) ={g∈C(I)|g(γx) =γg(x),∀γ ∈
,∀x∈I}andG(I;M) =G(I)∩X(I;M).
Lemma . G(I)is a closed convex subset of C(I),and G(I;M)is a compact convex
subset of C(I).
Theorem .(Quasi-convexity (Quasi-concavity)) If g∈Xσ(I;M), (.)and(.)are sat-isfied,then(.)has a solution f ∈Xσ(I;M).
Proof Define a mappingT:Xqcv(I;M)→C(I) as in (.). Note that eachf∈Xqcv(I;M) is
an increasing function. In fact, ifx<yinI, there existst∈(, ) such thatx=ta+ ( –t)y,
and by the quasi-convexity, we get
f(x)≤maxf(a),f(y)=f(y). (.)
Thus, forf ∈Xqcv(I,M),x,y∈I, andt∈[, ], we get
Tftx+ ( –t)y=
λ+ f
λtx+ ( –t)y+ λ λ+ g
tx+ ( –t)y
≤
λ+ f
maxf(λx),f(λy) + λ
λ+ max
g(x),g(y)
≤
λ+ max
f(λx),f(λy) + λ
λ+ max
g(x),g(y)
≤maxTf(x),Tf(y) .
Thus, T maps Xqcv(I,M) into itself. Similarly, we can prove that T is continuous. By
Lemma ., Xqcv(I,M) is a compact convex subset of the Banach space C(I). Then
Schauder’s fixed point theorem guarantees the existence of a fixed point f of T in
Xqcv(I;M). In the same way, the proof of the quasi-concave solution of equation (.) is
similarly obtained.
Now, we study the symmetric solutions of (.).
Theorem .(Symmetry) If(.)and(.)are satisfied,and g∈G(I;M),then equation
(.)has a unique-equivariant solution f∈G(I;M). Proof By Lemma . and (.), we have
Tf(γx) =
λ+ f
(γ λx) + λ λ+ g(γx) =γ
λ+ f
(λx) +γ λ λ+ g(x) =γTf(x).
From Theorem ., we can find that T is a contraction mapping in G(I;M). Since G(I;M) is a compact convex subset ofC(I), by Banach’s fixed point theorem, we
as-sert that there is a unique fixed pointf ∈G(I;M).
In the following, we give the conditions to guarantee two kinds of stability: the contin-uous dependence and the Hyers-Ulam stability [].
Theorem .(Continuous dependence) If(.)and(.)are satisfied,the solutions of(.)
Proof Forg,g∈X(I;M), Theorem . implies that there are functionsf,f∈X(I;M)
Inequality (.) yields that the solution of (.) inX(I;M) is continuously dependent on the given functionginX(I;M).
Definition . The functional equation
Proof Construct a sequence{ϕk} of functions as follows. Takeϕ=ϕs, and then define
ϕk as in (.) andLϕk as in (.). By Lemma ., bothϕk andLϕk are well defined for
any integerk≥. Lemma . and Lemma . also imply thatϕkorLϕkis an
orientation-preserving homeomorphism fromIinto itself with (Lϕk)–∈X(I;ξ,
ξ), whereξandξ
are given in (.). Now we claim that
g–Lϕk–◦ϕk– ≤ηk–δ, (.)
ϕk–ϕk– ≤ηk– δ ξ
(.)
for allx∈Iandk= , , . . . .
The casek= is trivial. Assume that (.) and (.) hold fork. Since
g(x) –Lϕk◦ϕk≤ Lϕk–◦ϕk–Lϕk◦ϕk
≤
λ–ϕ
k–
λϕk––(ϕk)
+ϕkλϕ–k (ϕk) ≤
λϕ
k
λϕk––ϕk–λϕ–k–
≤
λϕ
k
λϕk––ϕkλϕ–k–+ϕkλϕk–––ϕk–λϕk––
≤
λ
λMϕk––ϕk––+ (M+ )ϕk–ϕk–
,
where Lemma . is applied. From the hypothesis of induction, it follows that
g(x) –Lϕk◦ϕk≤Mϕ–k –ϕk––+
(M+ )
λ ϕk–ϕk–
≤
M m
+M+
λξ
k
δ=ηkδ.
Moreover,
ϕk+–ϕk=(Lϕk)–◦g– (Lϕk)–◦Lϕk◦ϕk ≤
ξ
g–Lϕk◦ϕk
≤ηkδ ξ
.
Thus, (.) and (.) hold.
On the other hand, for any positive integerskandlwithk>l, by (.)
ϕk–ϕl ≤ ϕk–ϕk–+ϕk––ϕk–+· · ·+ϕl+–ϕl
≤ηk–δ ξ
+ηk–δ ξ
+· · ·+ηlδ ξ
≤ηk–ηl
–η δ ξ
Note thatη< , so from (.), it follows that
ϕk–ϕl → ask,l→+∞.
This implies that{ϕk(x)}is a Cauchy sequence. Hence,{ϕk(x)}uniformly converges in the
Banach spaceC(I). Letϕ=lim
k→+∞ϕk(x). Clearly,ϕ∈X(I;m,M). From (.), g–Lϕ◦ϕ= lim
k→+∞g–Lϕk◦ϕk ≤k→lim+∞η
kδ= , (.)
i.e.,ϕis a solution of (.). Furthermore, from (.)
ϕ–ϕs= lim
k→+∞ϕk–ϕ
≤ lim
k→+∞
ϕk–ϕk–+ϕk––ϕk–+· · ·+ϕ–ϕ
≤ lim
k→+∞
ηk–δ ξ
+ηk–δ ξ
+· · ·+ δ
ξ
≤ δ ξ
–η=
ξ( –η)δ. Thus,ϕ–ϕs<ζ δ. Then (.) holds.
Concerning the uniqueness, we assume that there is another continuous solutionφ∈ X(I;m,M) (φ=ϕ), such that
φ(x) –ϕs(x)≤ε,
whereε> only depends onδ. Then
ϕ–φ=(Lϕ)–◦g– (Lφ)–◦g
≤(Lϕ)–– (Lφ)–
≤(Lϕ)–– (Lϕ)–◦(Lϕ)◦(Lφ)–
≤
ξ
(Lφ)◦(Lφ)–– (Lϕ)◦(Lφ)–
≤
ξ
Lϕ–Lφ
≤
λξ
ϕλϕ––φλϕ–+φλϕ––φλφ–
≤M+
λξ ϕ–φ+ M
ξ ϕ
––φ–
≤M+
λξ
ϕ–φ+M
ξ
ϕ––ϕ–◦ϕ◦φ–
≤M+
λξ
ϕ–φ+ M
mξ
φ◦φ––ϕ◦φ–
≤M+
λξ
ϕ–φ+ M
mξ
that is, assumption. The proof is completed.
4 Example
Example Consider the equation
If ≥y≥, then|f(x) –f(y)| ≥|y–x| ≥|y–x|; if≥y≥, then
f(x) –f(y)≥
|y–x|– ≥
|y–x| ≥ |y–x|, since
≤ |y–x| ≤
. Similarly, we can show that for any x,y∈[, ], |f(x) –f(y)| ≥
|y–x|. Thus,m=
. Therefore, we can get a quasi-convex solutionf(x) of equation (.)
by Theorem ., which is continuously dependent on the given functiong(x)∈X(I;M)
withM=–√
by Theorem .. Moreover, equation (.) satisfies the Hyers-Ulam stability
inX(I;–√ –√,
–√
) by Theorem ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors have contributed in all the parts, and they have read and approved the final manuscript.
Acknowledgements
This work was supported by the PhD Start-up Fund of the Natural Science Foundation of Guangdong Province, China (S2011040000464), the Project of Department of Education of Guangdong Province, China (2012KJCX0074), the Natural Science Funds of Zhanjiang Normal University (QL1002, LZL1101), and the Doctoral Project of Zhanjiang Normal University (ZL1109). The authors would like to thank Dr. Shengfu Deng for his very helpful comments and suggestions.
Received: 21 March 2013 Accepted: 23 July 2013 Published: 6 August 2013 References
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