The solutions of the Feigenbaum-like functional equation
© Liang et al.; licensee Springer 2013
Received: 21 March 2013
Accepted: 23 July 2013
Published: 6 August 2013
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 the solutions of the Feigenbaum-like functional equations are discussed.
MSC:39B22, 39B12, 39A10.
As early as 1978, Feigenbaum found the period-doubling bifurcation phenomenon by researching 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 bijective 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 using the computer. Thompson  investigated an essentially singular solution by expressing Feigenbaum’s equation as a singular Schroder functional equation whose solution was obtained using a scaling ansatz, and so on. Thus, some solutions in specific cases were found.
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 1987, by using Schauder’s fixed point theorem to an operator defined by a linear combination of iterates of the unknown mapping f, 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 [9, 10], the analytic solutions [11–13], 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 argument, the derived planar mappings and their invariant curves were studied . Based on the iterative root theory for monotone functions, an algorithm for computing polygonal iterative roots of increasing polygonal functions was given . Else, a problem about the Hyers-Ulam stability was raised first by Ulam in 1940 and solved for Cauchy equation by Hyers . Later, many papers on the Hyers-Ulam stability have been published, especially, for the polynomial-like iterative equation [20–22].
where is a given disturbance function, is an unknown function, and , . It is clear that , since for all . We give not only the existence of continuous solutions of (1.1) but also their uniqueness, stability (the continuous dependence 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.
In this section, we give several important definitions, lemmas and notions.
Let . Obviously, is a Banach space with the norm , where the norm for .
Let . Then is a complete metric space.
Let , where M is a positive constant.
Let , where m is a positive constant.
Let , , .
Let denote the families consisting of all quasi-convex functions or quasi-concave ones in , where or .
The following Lemma 2.1 and Lemma 2.2 are useful, and the methods of their proofs are similar to ones in the paper .
Lemma 2.1 , , and are compact convex subsets of .
Lemma 2.2 The composition 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, is also quasi-convex (or quasi-concave).
Thus, (2.1) holds. □
Therefore, . This implies that Lφ is strictly increasing and invertible on I, and . □
are well defined, and , .
This implies that the results in Lemma 2.5 are also true for , 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 (1.1).
Theorem 3.1 (Existence)
then equation (1.1) has a solution f in .
This completes the proof. □
Theorem 3.2 (Uniqueness)
For any function , equation (1.1) has a unique solution .
Proof The existence of equation (1.1) in is given by Theorem 3.1. Note that is a closed subset of . By (3.3) and (3.5), we see that is a contraction mapping. Therefore T has a unique fixed point in , that is, equation (1.1) has a unique solution in . □
Below, we discuss the quasi-convex (or quasi-concave) solutions of equation (1.1).
Definition 3.1 Suppose that Γ is a Lie group of all linear transformations on R. A mapping , is said to be Γ-equivariant  if , , .
This implies that is the Γ-equivariant. Let and .
Lemma 3.1 is a closed convex subset of , and is a compact convex subset of .
The methods of the proofs are similar to the paper .
Theorem 3.3 (Quasi-convexity (Quasi-concavity))
If , (3.1) and (3.5) are satisfied, then (1.1) has a solution .
Thus, T maps into itself. Similarly, we can prove that T is continuous. By Lemma 2.1, is a compact convex subset of the Banach space . Then Schauder’s fixed point theorem guarantees the existence of a fixed point f of T in . In the same way, the proof of the quasi-concave solution of equation (1.1) is similarly obtained. □
Now, we study the symmetric solutions of (1.1).
Theorem 3.4 (Symmetry)
If (3.1) and (3.5) are satisfied, and , then equation (1.1) has a unique Γ-equivariant solution .
From Theorem 3.1, we can find that T is a contraction mapping in . Since is a compact convex subset of , by Banach’s fixed point theorem, we assert that there is a unique fixed point . □
In the following, we give the conditions to guarantee two kinds of stability: the continuous dependence and the Hyers-Ulam stability .
Theorem 3.5 (Continuous dependence)
If (3.1) and (3.5) are satisfied, the solutions of (1.1) in is continuously dependent on the given function in .
Inequality (3.5) yields that the solution of (1.1) in is continuously dependent on the given function g in . □
Theorem 3.6 (Hyers-Ulam stability)
where , .
Proof Construct a sequence of functions as follows. Take , and then define as in (2.7) and as in (2.8). By Lemma 2.4, both and are well defined for any integer . Lemma 2.4 and Lemma 2.5 also imply that or is an orientation-preserving homeomorphism from I into itself with , where and are given in (2.5).
for all and .
Thus, (3.13) and (3.14) hold.
Thus, . Then (3.12) holds.
The assumption of Theorem 3.6 yields , i.e., , which contradicts with the assumption. The proof is completed. □
since . Similarly, we can show that for any , . Thus, . Therefore, we can get a quasi-convex solution of equation (4.1) by Theorem 3.3, which is continuously dependent on the given function with by Theorem 3.5. Moreover, equation (4.1) satisfies the Hyers-Ulam stability in by Theorem 3.6.
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.
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