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The solutions of the Feigenbaum-like functional equation
Advances in Difference Equations volume 2013, Article number: 231 (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.
Specifically, in 1985, to give a feasible method, the second kind of the Feigenbaum functional equation,
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].
In this paper, by using Schauder’s fixed point theorem, and constructing the special functional 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,
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 , , .
Definition 2.1 is a quasi-convex (or quasi-concave) function  if for and , we have
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).
Lemma 2.3 If , then
Proof Note that
Let . Then , and
Thus, (2.1) holds. □
Lemma 2.4 Suppose that . If the positive constants m, M and λ satisfy
then Lφ, defined by
is an orientation-preserving homeomorphism from I onto itself, and
Proof Because , by the paper , . Thus, for any , by (2.3) and (2.5)
On the other hand,
Therefore, . This implies that Lφ is strictly increasing and invertible on I, and . □
Lemma 2.5 Suppose that and . If
are well defined, and , .
From Lemma 2.4, is well defined and is an orientation-preserving homeomorphism from I onto itself, and . Then is well defined and by (2.6) and Lemma 2.4. If
is well defined and is an orientation-preserving homeomorphism from I onto itself, then . We similarly see that
is 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)
Given a positive constant and . If there exist constants M and λ such that
then equation (1.1) has a solution f in .
Proof Define by
Because f, and g are continuous for all , we obtain that Tf is continuous for all , and . By (3.1), for any x, y in I,
Thus, . Now we prove the continuity of T under the norm . For arbitrary ,
where Lemma 2.3 is used and
Thus, T is continuous under the norm . Summarizing all the above, we see that T is a continuous mapping from the compact convex subset of the Banach space into itself. By Schauder’s fixed point theorem, we assert that there is a mapping such that
This completes the proof. □
Theorem 3.2 (Uniqueness)
Suppose that (3.1) is satisfied and
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 .
Proof Define a mapping as in (3.2). Note that each is an increasing function. In fact, if in I, there exists such that , and by the quasi-convexity, we get
Thus, for , , and , we get
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 .
Proof By Lemma 3.1 and (3.2), we have
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 .
Proof For , Theorem 3.1 implies that there are functions such that
Thus, by Lemma 2.3
Inequality (3.5) yields that the solution of (1.1) in is continuously dependent on the given function g in . □
Definition 3.2 The functional equation
Theorem 3.6 (Hyers-Ulam stability)
Suppose that , and satisfy
where is a positive constant. If (2.6), (3.1) and (3.5) are satisfied, there exists a unique continuous solution of (1.1) such that
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).
Now we claim that
for all and .
The case is trivial. Assume that (3.13) and (3.14) hold for k. Since
where Lemma 2.3 is applied. From the hypothesis of induction, it follows that
Thus, (3.13) and (3.14) hold.
On the other hand, for any positive integers k and l with , by (3.14)
Note that , so from (3.15), it follows that
This implies that is a Cauchy sequence. Hence, uniformly converges in the Banach space . Let . Clearly, . From (3.13),
i.e., φ is a solution of (1.1). Furthermore, from (3.14)
Thus, . Then (3.12) holds.
Concerning the uniqueness, we assume that there is another continuous solution (), such that
where only depends on δ. Then
The assumption of Theorem 3.6 yields , i.e., , which contradicts with the assumption. The proof is completed. □
Example 1 Consider the equation
where and . Let
Then is quasi-convex and nonconvex (see Figure 1). Note that, for and
Similarly, we can show that for any , . Thus, .
For and , we have
If , then ; if , then
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.
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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.
The authors declare that they have no competing interests.
All the authors have contributed in all the parts, and they have read and approved the final manuscript.
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