Strictly Decreasing Solutions of a Class of Iterative Equations on the Unit Circle

Abstract

We discuss the strictly decreasing solutions of a class of iterative equations on the unit circle 𝕋¹. The conditions for the existence, uniqueness, and stability of such solutions are presented.

1. Introduction

Let 𝐶⁰(𝑋,𝑋) be the set of all continuous self-mappings on a topological space 𝑋. Let 𝑓∈𝐶⁰(𝑋,𝑋). For integer 𝑛≥0, define the 𝑛th iterate of 𝑓 by 𝑓^𝑛=𝑓∘𝑓^𝑛−1 and 𝑓⁰=id, where id denotes the identity mapping on 𝑋 and ∘ denotes the composition of mappings. An equation with iteration as its main operation is simply called an iterative equation. Founded on the problem of iterative roots, the problem of invariant curves and some problems originating from dynamical systems, and the iterative equationΦ(𝑓(𝑥),𝑓²(𝑥),…,𝑓^𝑛(𝑥))=𝐹(𝑥),𝑥∈𝑋, (1.1) where 𝐹 and Φ are given functions and 𝑓 is the unknown function, have been investigated extensively [1, 2].

Let 𝕋¹={𝑧∈ℂ∶𝑧=𝑒^{2𝜋𝑖𝑡},𝑡∈ℝ} denote the unit circle and 𝐶⁰(𝕋¹,𝕋¹) denote the set of all continuous maps from 𝕋¹ to itself. Let 𝟏 denote the point (1,0) in the complex plane ℂ so as to distinguish it from 1∈ℝ. The iteration on the unit circle 𝕋¹ is also important. Many scholars had got a lot of results on iterative roots and iteration groups, such as [3–7]. In 2007, Zdun and Zhang [8] discussed the solutions of (1.1) on 𝕋¹, that is, the equationΦ(𝑓(𝑧),𝑓²(𝑧),…,𝑓^𝑛(𝑧))=𝐹(𝑧),𝑧∈𝕋¹, (1.2) in the class of homeomorphisms 𝐻⁰_𝟏(𝕋¹,𝕋¹)={𝑓∈𝐶⁰(𝕋¹,𝕋¹)∶𝑓(𝕋¹)=𝕋¹homeomorphicallyand𝑓(𝟏)=𝟏}. (1.3) In [8], the authors lift 𝐹 from the circle 𝕋¹ to ℝ and Φ from the 𝑛-dimensional torus 𝕋^𝑛 to ℝ^𝑛. They apply ingenious techniques of restricting and extending to these lifts so that the problem can be reduced and discussed on the compact interval [0,1]. However, they only discussed the strictly increasing solutions of (1.2). In this paper, we will discuss the strictly decreasing solutions of a special form of (1.2), that is, the following equation:Φ(𝑓(𝑧),𝑓³(𝑧),…,𝑓^2𝑛−1(𝑧))=𝐹(𝑧),𝑧∈𝕋¹. (1.4)

2. Basic Definitions and Lemmas

As in [8], let ℎ∶𝑡∈ℝ↦𝑒^{2𝜋𝑖𝑡}∈𝕋¹. If 𝑣,𝑤,𝑧∈𝕋¹, there exist unique 𝑡₁,𝑡₂∈[0,1) such that 𝑤ℎ(𝑡₁)=𝑧 and 𝑤ℎ(𝑡₂)=𝑣. The cyclic order on 𝕋¹ is defined in [9], that is, 𝑣≺𝑤≺𝑧⇔0<𝑡₁<t₂,𝑣⪯𝑤⪯𝑧⇔𝑡₁≤𝑡₂or𝑡₂=0. (2.1)(2.2) Obviously, the relations 𝑣≺𝑤≺𝑧, 𝑤≺𝑧≺𝑣 and 𝑧≺𝑣≺𝑤 are equivalent.

Considering a nonempty subset 𝐴⊆𝕋¹, a map 𝑓∶𝐴→𝕋¹ is called to be increasing (strictly increasing) if 𝑓(𝑣)⪯𝑓(𝑤)⪯𝑓(𝑧) (𝑓(𝑣)≺𝑓(𝑤)≺𝑓(𝑧), resp.) for all 𝑣,𝑤,𝑧∈𝐴 with 𝑣≺𝑤≺𝑧. Similarly, a map 𝑓∶𝐴→𝕋¹ is called to be decreasing (strictly decreasing) if 𝑓(𝑧)⪯𝑓(𝑤)⪯𝑓(𝑣) (𝑓(𝑧)≺𝑓(𝑤)≺𝑓(𝑣), resp.) for all 𝑣,𝑤,𝑧∈𝐴 with 𝑣≺𝑤≺𝑧. A homeomorphism ℎ∶𝕋¹→𝕋¹ is said to be orientation preserving (orientation reversing) if it is strictly increasing (strictly decreasing, resp.).

If 𝑣,𝑧∈𝕋¹ with 𝑣≠𝑧, there exist 𝑡_𝑣,𝑡_𝑧∈ℝ with 𝑡_𝑣<𝑡_𝑧<𝑡_𝑣+1 and ℎ(t_𝑣)=𝑣,ℎ(𝑡_𝑧)=𝑧. Define the oriented arc (𝑣,𝑧)={𝑤=ℎ(𝑡)∶𝑡∈(𝑡_𝑣,𝑡_𝑧)}. (2.3) Obviously, the map 𝑓 is strictly increasing (strictly decreasing) if 𝑤∈(𝑣,𝑧) yields 𝑓(𝑤)∈(𝑓(𝑣),𝑓(𝑧)) (𝑓(𝑤)∈(𝑓(𝑧),𝑓(𝑣)), resp.).

2.1. The Lifts of Maps on 𝕋¹

Let 𝑓∈𝐶⁰(𝕋¹,𝕋¹). A continuous map 𝑓∶ℝ→ℝ is said to be a lift of 𝑓 if ℎ∘𝑓=𝑓∘ℎ. (2.4) The following properties are well known.

Lemma 2.1 (see [10]). (i) Each 𝑓∈𝐶⁰(𝕋¹,𝕋¹) has a lift 𝑓. (ii) There exists a constant 𝑘∈ℤ such that each lift 𝑓 of 𝑓 satisfies 𝑓(𝑡+1)−𝑓(𝑡)=𝑘,forall𝑡∈ℝ. The constant 𝑘 is said to be the degree of 𝑓 and denoted by deg(𝑓), that is, deg(𝑓)=𝑓(𝑡+1)−𝑓(𝑡),forall𝑡∈ℝ. (iii) For 𝑓,𝑔∈𝐶⁰(𝕋¹,𝕋¹) and 𝑖𝑑∶𝕋¹→𝕋¹, one has deg(𝑔∘𝑓)=deg(𝑔)⋅deg(𝑓) and deg(𝑖𝑑)=1. If 𝑓∈𝐶⁰(𝕋¹,𝕋¹) is a homeomorphism, then |deg(𝑓)|=1. (iv) If 𝑓 is a lift of 𝑓, then for each 𝑗∈ℤ, the map 𝑓+𝑗 is a lift of 𝑓 and every lift of 𝑓 can be expressed in this form, where ℤ denotes the set of integers.

Lemma 2.2 (see [10]). (i) If 𝑔∶[0,1]→ℝ is a continuous map such that 𝑔(1)−𝑔(0) is an integer 𝑘, then 𝑔 can be extended to a map 𝑔∶ℝ→ℝ with 𝑔(𝑡+1)−𝑔(𝑡)=𝑘,∀𝑡∈ℝ. (2.5)
(ii) Suppose 𝑓∶ℝ→ℝ is continuous, 𝑘 is an integer, and 𝑓(𝑡+1)−𝑓(𝑡)=𝑘,forall𝑡∈ℝ. Then there exists a unique 𝑓∈𝐶⁰(𝕋¹,𝕋¹) such that 𝑓 is a lift of 𝑓.

A map 𝑓∈𝐶⁰(𝕋¹,𝕋¹) is said to be Lipschitzian if there exists a lift 𝑓 of 𝑓 satisfying |𝑓(𝑡₁)−𝑓(𝑡₂)|≤𝑀|𝑡₁−𝑡₂|,∀𝑡₁,𝑡₂∈ℝ (2.6)

for a constant 𝑀>0. By (iv) of Lemma 2.1, the constant 𝑀 is independent of the choice of 𝑓, and 𝑀 is said to be a Lipschitzian constant of 𝑓.

In the sequel one will discuss the strictly increasing (strictly decreasing) maps 𝑓∈𝐶⁰(𝕋¹,𝕋¹) by means of their lifts.

Lemma 2.3. A lift 𝑓 of 𝑓∈𝐶⁰(𝕋¹,𝕋¹) is strictly monotonic (monotonic) if and only if there exists a 𝑡₀∈ℝ such that 𝑓|_{[𝑡0,𝑡0+1)} is strictly monotonic (monotonic).

The proof is trivial. So we omit it.

Lemma 2.4 (see [11]). A map 𝑓∈𝐶⁰(𝕋¹,𝕋¹) is strictly increasing (strictly decreasing) if and only if deg(𝑓)=1 (deg(𝑓)=−1) and its lift 𝑓 is strictly increasing (strictly decreasing) in ℝ.

Thus we get the following theorem.

Theorem 2.5. A map 𝑓∈𝐶⁰(𝕋¹,𝕋¹) is strictly increasing (strictly decreasing) if and only if it is an orientation-preserving (orientation-reversing) homeomorphism.

Proof. We only need to show that 𝑓 has an inverse mapping. Let 𝑓 be the lift of 𝑓. The map 𝑓⁻¹=ℎ∘𝑓⁻¹ is the inverse mapping, where 𝑓⁻¹ is the inverse function of 𝑓.

2.2. Induced Maps of Maps on 𝕋¹

Firstly, we define 𝐻⁰_𝟏(𝕋¹,𝑘)={𝑓∣𝑓∈𝐻⁰_𝟏(𝕋¹,𝕋¹),deg(𝑓)=𝑘}, (2.7) where 𝑘=1 or −1. Let ℎ_∗∶=ℎ|_(0,1); the map ℎ_∗ is a homeomorphism from (0,1) to 𝕋¹⧵{𝟏}. For 𝑓∈𝐻⁰_𝟏(𝕋¹,−1), define 𝑓_∗∶=ℎ_∗⁻¹∘𝑓∘ℎ_∗ which is a self-map on (0,1). We extend 𝑓_∗ to the compact interval 𝐼=[0,1]. Let𝐺(𝑡)=𝑓1,𝑡=0,_∗(𝑡),𝑡∈(0,1),0,𝑡=1. (2.8) For convenience, we call 𝐺 the induced map of 𝑓, which is a self-map on [0,1].

Lemma 2.6. The induced map 𝐺 of an orientation-reversing homeomorphism 𝑓∈𝐻⁰_𝟏(𝕋¹,−1) is continuous and strictly decreasing on 𝐼. It can be extended to a lift 𝑓 of 𝑓. This lift satisfies 𝑓(0)=1,𝑓(1)=0 and is unique on this property.

Proof. Obviously, 𝐺 is continuous on the interval (0,1). We only need to prove lim_𝑡→0⁺𝐺(𝑡)=1 and lim_𝑡→1⁻𝐺(𝑡)=0.
Concerning the continuity at 0, we note that lim_𝑡→0+𝐺(𝑡)=lim_𝑡→0+ℎ_∗⁻¹∘𝑓∘ℎ_∗(𝑡)=lim_𝑡→0+ℎ_∗⁻¹∘𝑓(𝑒^{2𝜋𝑖𝑡}). (2.9) By the continuity of 𝑓 at 𝟏, we have lim_𝑡→0⁺𝑓(𝑒^{2𝜋𝑖𝑡})=𝟏. Let 𝐹 be any lift of 𝑓. By ℎ∘𝐹=𝑓∘ℎ, we must have 𝐹(0)=𝑘₀∈ℤ, where ℤ denotes the set of integers. By (iv) of Lemma 2.1, let 𝐹 be a lift of 𝑓 with 𝐹(0)=1. Since 𝑓∘ℎ_∗(𝑡)=ℎ_∗∘𝐹(𝑡),𝑡∈(0,1), we have lim_𝑡→0+ℎ_∗⁻¹∘𝑓∘ℎ_∗(𝑡)=lim_𝑡→0+𝐹(𝑡)=1. (2.10) We can easily verify lim_𝑡→1⁻𝐺(𝑡)=0 in the same way. Note that 𝐺 is strictly decreasing on (0,1), so 𝐺 is strictly decreasing on [0,1].
Let 𝑡∈ℝ, and 𝑘 is the integer with 𝑡∈[𝑘,𝑘+1). Define 𝑓∶=𝐺(𝑡−𝑘)+𝑘. We can easily get that 𝑓∘ℎ=ℎ∘𝑓. Suppose that 𝑔 is another lift of 𝑓 which maps [0,1] into itself and 𝑔(0)=1,𝑔(1)=0. By (iv) of Lemma 2.1, we have 𝑓=𝑔+𝑗 for some integer 𝑗. Obviously, 𝑗=𝑓(0)−𝑔(0)=0. Then 𝑓=𝑔.

The following lemma is a converse of Lemma 2.6.

Lemma 2.7. Suppose that 𝐺∈𝐶⁰(𝐼,𝐼) is strictly decreasing and satisfies 𝐺(0)=1,𝐺(1)=0. Then the map ℎ𝑓∶=^∗∘𝐺∘ℎ_∗⁻¹(𝑧),𝑧∈𝕋¹⧵{𝟏},𝟏,𝑧=𝟏. (2.11) belongs to the class 𝐻⁰_𝟏(𝕋¹,−1). Moreover, 𝐺 can be extended to a lift of 𝑓.

Proof. Clearly, 𝑓(𝟏)=𝟏, and 𝑓|_{𝕋1⧵{𝟏}} is an orientation-reversing homeomorphism of 𝕋¹⧵{𝟏}. Then 𝑓 is orientation reversing on the whole 𝕋¹. We only need to show the continuity of 𝑓 at 𝟏. Let 𝑧→𝟏⁺ (𝑧→𝟏⁻) denote 𝑧 approaches to 𝟏 in clockwise (counter-clockwise) direction. Note that lim_𝑧→𝟏+𝑓(𝑧)=lim_𝑡→0+ℎ_∗∘𝐺∘ℎ_∗⁻¹(𝑒^{2𝜋𝑖𝑡})=lim_𝑡→0+ℎ_∗∘𝐺(𝑡)=𝟏. (2.12) By the same argument, we have lim_𝑧→𝟏⁻𝑓(𝑧)=𝟏.
Given 𝑡∈ℝ and integer 𝑘 with 𝑡∈[𝑘,𝑘+1), define 𝑓(𝑡)=𝐺(𝑡−𝑘)+𝑘. (2.13) One can easily show that 𝑓∘ℎ=ℎ∘𝑓.

2.3. Induced Maps of Maps on 𝕋^𝑛

For simplicity, let 𝕋^𝑛=𝕋¹×⋯×^{𝑛0𝑥0𝑓𝑒37}𝕋¹,𝟏^𝑛=(𝟏,…,^{𝑛0𝑥0𝑓𝑒37}𝟏). (2.14)

For 𝑓∈𝐻⁰_𝟏(𝕋¹,−1), define 𝐻_𝑓(𝑧)∶=(𝑓(𝑧),𝑓³(𝑧),…,𝑓^2𝑛−1(𝑧)). (2.15) Then (1.4) can be written as Φ∘𝐻_𝑓(𝑧)=𝐹(𝑧),𝑧∈𝕋¹. (2.16)

As in [8], we also have the following two remarks.

Remark 2.8. 𝐻_𝑓 maps 𝕋¹ into (𝕋¹⧵𝟏)^𝑛∪{𝟏^𝑛}.

Remark 2.9. Let Rang(Φ) denote the range of Φ. If 𝐹∈𝐻⁰_𝟏(𝕋¹,−1) and (1.4) has a solution 𝑓∈𝐻⁰_𝟏(𝕋¹,−1) on 𝐹, then RangΦ=𝕋¹ and Φ(𝟏^𝑛)=𝟏.

Let Dom(Φ) denote the domain of Φ. As in [8], Dom(Φ) is very important for (1.4). Now, we discuss the relationship of Dom(Φ) and (1.4). The first case is that the domain of Φ is 𝕋^𝑛.

2.3.1. The Case: Dom(Φ)=𝕋^𝑛

The following important and interesting lemma is proved in [8].

Lemma 2.10 (see [8]). If Φ∶𝕋^𝑛→𝕋¹ is continuous and Φ(𝟏^𝑛)=𝟏, then there exists a unique continuous function Φ∶ℝ^𝑛→ℝ such that Φ(ℎ(𝑡₁),…,ℎ(𝑡_𝑛))=𝑒^2𝜋𝑖Φ_{(𝑡1,…,𝑡𝑛)},Φ(0,…,0)=0. (2.17) Moreover, for each 𝑘∈{1,…,n}, there exists an 𝑚_𝑘∈ℤ such that Φ(𝑡₁,…,𝑡_𝑘+1,…,𝑡_𝑛)=Φ(𝑡₁,…,𝑡_𝑘,…,𝑡_𝑛)+𝑚_𝑘,∀𝑡₁,…,𝑡_𝑘∈ℝ. (2.18)

The map Φ is said to be the lift of Φ, and (𝑚₁,…,𝑚_𝑛) is defined to be the degree of Φ and is denoted by deg(Φ).

Lemma 2.11. Let 𝐹 be the lift of 𝐹 with 𝐹(1)=0, and let Φ be the lift of Φ with Φ(0,…,0)=0. Let 𝑓∈𝐻⁰_𝟏(𝕋¹,−1) be a solution of (1.4), and let 𝑓 be a lift of 𝑓 with 𝑓(1)=0. Then (1.4) is equivalent to Φ(𝑓(𝑡),𝑓³(𝑡),…,𝑓^2𝑛−1(𝑡))=𝐹(𝑡),𝑡∈𝐼. (2.19)

Proof. Note that 𝑓^𝑗(ℎ(𝑡))=ℎ(𝑓^𝑗(𝑡)) for all 𝑡∈ℝ. For 𝑧=𝑒^{2𝜋𝑖𝑡},𝑡∈𝐼 and (1.4) is equivalent to Φ(𝑒^2𝜋𝑖𝑓_(𝑡),𝑒^2𝜋𝑖𝑓³_(𝑡),…,𝑒^2𝜋𝑖𝑓^2𝑛−1_(𝑡))=𝑒^2𝜋𝑖𝐹_(𝑡). (2.20) By Lemma 2.10, 𝑒^2𝜋𝑖Φ₍𝑓_(𝑡),𝑓³_(𝑡),…,𝑓^2𝑛−1_(𝑡))=𝑒^2𝜋𝑖𝐹_(𝑡). This means that for each 𝑡∈𝐼, we have Φ(𝑓(𝑡),𝑓³(𝑡),…,𝑓^2𝑛−1(𝑡))=𝐹(𝑡)+𝑘(𝑡), (2.21) where 𝑘(𝑡)∈ℤ. Since 𝑓(1)=0,𝐹(1)=0, and Φ(0,…,0)=0, we get 𝑘(𝑡)=0. By the continuity of 𝑓,𝐹, and Φ, we get that 𝑘(𝑡)≡0.

Theorem 2.12. Suppose that Φ is continuous, Φ(𝟏^𝑛)=𝟏, 𝐹 is continuous, 𝐹(𝟏)=𝟏, and (1.4) has a solution 𝑓∈𝐻⁰_𝟏(𝕋¹,−1). Let deg(Φ)=(m₁,…,𝑚_𝑛). Then deg(𝐹)=−(𝑚₁+⋯+𝑚_𝑛).

Proof. Let 𝐹 be the lift of 𝐹 with 𝐹(1)=0, and let Φ be the lift of Φ with Φ(0,…,0)=0. Let 𝑓∈𝐻⁰_𝟏(𝕋¹,−1) be the solution of (1.4), and let 𝑓 be its lift with 𝑓(1)=0. Note that 𝑓(𝑡+1)=𝑓(𝑡)−1,𝑡∈ℝ. (2.22) Let 𝐻(𝑡)=(𝑓(𝑡),𝑓³(𝑡),…,𝑓^2𝑛−1(𝑡)). Since deg(Φ)=(𝑚₁,…,𝑚_𝑛), then by (2.18), we have Φ(−1,…,−1)=Φ(0,…,0)−𝑚₁−⋯−𝑚_𝑛. (2.23) Thus, we get that =𝐹(𝑡+1)=𝜙(𝐻(𝑡+1))=Φ(𝐻(𝑡)+(−1,…,−1))Φ(𝐻(𝑡))−𝑚₁−⋯−𝑚_𝑛=𝐹(𝑡)−𝑚₁−⋯−𝑚_𝑛. (2.24)

Definition 2.13. Suppose Φ∶𝕋^𝑛→𝕋¹ is continuous. Let 1≤𝑖≤𝑛, and let 𝑤₁,…,𝑤_𝑖−1,𝑤_𝑖+1,…,𝑤_𝑛∈ℂ be fixed.(1) One says Φ is increasing with respect to the 𝑖th variable if the map Φ(𝑤₁,…,𝑤_𝑖−1,𝑧_𝑖,𝑤_𝑖+1,…,𝑤_𝑛) (2.25) is increasing.(2) One says Φ is constant with respect to the 𝑖th variable if there is a 𝑧₀∈𝕋¹ such that the map Φ(𝑤₁,…,𝑤_𝑖−1,𝑧_𝑖,𝑤_𝑖+1,…,𝑤_𝑛)≡𝑧₀. (2.26)

Corollary 2.14. Let 𝐹∈𝐻⁰_𝟏(𝕋¹,−1), and Φ is continuous with Φ(𝟏^𝑛)=𝟏. If Φ is increasing with respect to each variable and Φ is nonconstant in at least two variables, then (1.4) has no solutions in 𝐻⁰_𝟏(𝕋¹,−1).

Proof. Let deg(Φ)=(𝑚₁,…,𝑚_𝑛). Hence, by Lemma 2.3, we have 𝑚₁≥0,…,𝑚_𝑛≥0 and 𝑚_𝑘=0 if and only if Φ is constant with respect to the 𝑘th variable. If (1.4) has a solution 𝑓 in 𝐻⁰_𝟏(𝕋¹,−1), then by Theorem 2.12, we get that deg(𝐹)=−(𝑚₁+⋯+𝑚_𝑛)≤−2. This a contradiction.

From the discussion above, we know that deg(𝐹) and deg(Φ) are interrelated. This is too severe for (1.4). So as in [8], by Remark 2.8, it is also natural to suppose that Dom(Φ)=(𝕋¹⧵𝟏)^𝑛∪{𝟏^𝑛}.

2.3.2. The Case Dom(Φ)=(𝕋¹⧵1)^𝑛∪{1^𝑛}

The following assumptions (H1) and (A) are cited from [8].(H1)Φ∶(𝕋¹⧵{𝟏})^𝑛∪{𝟏^𝑛}→𝕋¹ is continuous, Φ((𝕋¹⧵{𝟏})^𝑛)=𝕋¹⧵{𝟏}.(A) There exists a constant 𝛿>0 such that for 0<𝑡_𝑘<𝛿,𝑘=1,…,𝑛, we have Φ(ℎ_∗(𝑡₁),…,ℎ_∗(𝑡_𝑛))∈(𝟏,𝑖), and for 1−𝛿<𝑡_𝑘<1,𝑘=1,…,𝑛, we have Φ(ℎ_∗(𝑡₁),…,ℎ_∗(𝑡_𝑛))∈(−𝑖,𝟏).

Under assumptions (H1) and (A), as in [8], defineΨ(𝑡₁,…,𝑡_𝑛)∶=ℎ_∗⁻¹(Φ(ℎ_∗(𝑡₁),…,ℎ_∗(𝑡_𝑛))),𝑡_𝑗∈(0,1),𝑗=1,…,𝑛,Ψ(0,…,0)=0,Ψ(1,…,1)=1. (2.27) The function Ψ defined on (0,1)^𝑛∪{(0,…,0),(1,…,1)} is said to be the induced map of Φ. Let us note that lim_{𝑡𝑗→0,𝑗=1,…,𝑛}Ψ(𝑡₁,…,𝑡_𝑛)=0,lim_{𝑡𝑗→1,𝑗=1,…,𝑛}Ψ(𝑡₁,…,𝑡_𝑛)=1. (2.28)

Lemma 2.15 (see [8]). Under assumptions (H1) and (A), the induced map Ψ of Φ is continuous.

Lemma 2.16. Let 𝐹 be the lift of 𝐹 with 𝐹(1)=0, and let Ψ be the induced map of Φ. Let 𝑓∈𝐻⁰_𝟏(𝕋¹,−1) be a solution of (1.4), and let 𝑓 be a lift of 𝑓 with 𝑓(1)=0. Then (1.4) is equivalent to Ψ(𝑓(𝑡),𝑓³(𝑡),…,𝑓^2𝑛−1(𝑡))=𝐹(𝑡),𝑡∈𝐼. (2.29)

Lemma 2.16 can be proved by means of the method which is used to prove Lemma 2.11.

As in [8], we also need the following assumption.(H2) [8] There are nonnegative real constants 𝛼_𝑗,𝛽_𝑗,𝑗=1,…,𝑛 with 𝛽₁≥𝛼₁>0,𝛽_𝑗≥𝛼_𝑗≥0,𝑗=2,…,𝑛, such that ^𝑛_𝑗=1𝛼_𝑗(𝑡_𝑗−𝑠_𝑗)≤Ψ(𝑡₁,…,𝑡_𝑛)−Ψ(𝑠₁,…,𝑠_𝑛)≤^𝑛_𝑗=1𝛽_𝑗(𝑡_𝑗−𝑠_𝑗) (2.30) for all 𝑡_𝑗≥𝑠_𝑗 in (0,1),𝑗=1,…,𝑛.

Lemma 2.17 (see [8]). If Ψ is differentiable in (0,1)^𝑛 with respect to each variable and for every 𝑘 there exist 𝛼_𝑘,𝛽_𝑘 such that 𝛼₁>0,0≤𝛼_𝑘≤𝜕Ψ/𝜕𝑡_𝑘≤𝛽_𝑘, then Ψ satisfies (H2).

3. Existence of Strictly Decreasing Solutions

Let 𝐶⁰(𝐼) denote the set of all continuous maps on 𝐼=[0,1]. It is well known that C⁰(𝐼) equipped with the supremum norm ‖𝑔‖=sup{|𝑔(𝑡)|∶𝑡∈𝐼} is a Banach space. Let 𝐶⁰(𝐼,𝐼) denote the set of all continuous self-mappings on 𝐼; 𝐶⁰(𝐼,𝐼) is a compact closed subset of 𝐶⁰(𝐼).

The following lemma is useful in the proof of Theorem 3.2.

Lemma 3.1 (see [12]). Let 𝑖=1,2, and suppose that 𝑔_𝑖 is a self-homeomorphism of 𝐼 such that |𝑔_𝑖(𝑥)−𝑔_𝑖(𝑦)|≤𝑁|𝑥−𝑦| for all 𝑥,𝑦∈𝐼, where 𝑁>0 is a constant. Then(i)‖𝑔^𝑛₁−𝑔^𝑛₂‖≤(∑^𝑛−1_𝑖=0𝑁^𝑖)‖𝑔₁−𝑔₂‖ for all 𝑛∈ℤ;(ii)‖𝑔₁−𝑔₂‖≤𝑁‖𝑔₁⁻¹−𝑔₂⁻¹‖.

For 0≤𝑙≤𝐿, define 𝔉(𝐼,𝑙,𝐿)={𝑔∈𝐶⁰𝔉(𝐼,𝐼)∣𝑔(0)=0,𝑔(1)=1,𝑙(𝑡−𝑠)≤𝑔(𝑡)−𝑔(𝑠)≤𝐿(𝑡−𝑠),∀𝑠≤𝑡∈𝐼},_′(𝐼,𝑙,𝐿)={𝑔∈𝐶⁰(𝐼,𝐼)∣𝑔(0)=1,𝑔(1)=0,𝑙(𝑡−𝑠)≤𝑔(𝑠)−𝑔(𝑡)≤𝐿(𝑡−𝑠),∀𝑠≤𝑡∈𝐼}. (3.1) Both 𝔉(𝐼,𝑙,𝐿) and 𝔉^′(𝐼,𝑙,𝐿) are compact convex subsets of 𝐶⁰(𝐼) [1, 13].

Theorem 3.2. Suppose (H1), (H2) hold and 𝐹∈𝐻⁰_𝟏(𝕋¹,−1) with a Lipschitz constant 𝑀>𝛼₁. Equation (1.4) has a solution 𝑓∈𝐻⁰_𝟏(𝕋¹,−1) with a Lipschitz constant 𝑀/𝛼₁.

Proof. Let 𝐺 and Ψ be the induced maps of 𝐹 and Φ defined in Section 2. By Lemma 2.16 (1.4) is equivalent to the following equation: Ψ(𝑔(𝑡),𝑔³(𝑡),…,𝑔^2𝑛−1(𝑡))=𝐺(𝑡),𝑡∈𝐼. (3.2) By Lemma 2.3, the map 𝐺∈𝐶⁰(𝐼) is strictly decreasing 𝐺(0)=1,and𝐺(1)=0. The map 𝐺 can be extended to a lift of 𝐹, and |𝐺(𝑡₁)−𝐺(𝑡₂)|≤𝑀|𝑡₁−𝑡₂|,∀𝑡₁,𝑡₂∈𝐼. (3.3)
Let 𝑀₀=∑^𝑛_𝑗=1𝛽_𝑗(𝛼₁⁻¹𝑀)^2𝑗−2. Note that 𝔉^′(𝐼,0,𝛼₁⁻¹𝑀)≠𝜙, since 𝑀>𝛼₁. Define an operator 𝔏∶𝔉^′(𝐼,0,𝛼₁⁻¹𝑀)→𝐶⁰(𝐼) (3.4) by 𝑔↦𝔏_𝑔, where 𝔏_𝑔(𝑡)=Ψ(𝑡,𝑔²(𝑡),…,𝑔^2𝑛−2(𝑡)),𝑡∈𝐼. (3.5) By (H2), we have for any 𝑡≥𝑠∈𝐼,𝔏_𝑔(𝑡)−𝔏_𝑔(𝑠)=Ψ(𝑡,𝑔²(𝑡),…,𝑔^2𝑛−2(𝑡))−Ψ(𝑠,𝑔²(𝑠),…,𝑔^2𝑛−2(𝑠))≥𝛼₁(𝑡−𝑠)+^𝑛_𝑗=2𝛼_𝑗(𝑔^2𝑗−2(𝑡)−𝑔^2𝑗−2(𝑠))≥𝛼₁𝔏(𝑡−𝑠),_𝑔(𝑡)−𝔏_𝑔(𝑠)=Ψ(𝑡,𝑔²(𝑡),…,𝑔^2𝑛−2(𝑡))−Ψ(𝑠,𝑔²(𝑠),…,𝑔^2𝑛−2(𝑠))≤𝛽₁(𝑡−𝑠)+^𝑛_𝑗=2𝛽_𝑗(𝑔^2𝑗−2(𝑡)−𝑔^2𝑗−2(𝑠))≤𝑀₀(𝑡−𝑠). (3.6) Hence, 𝔏_𝑔 is increasing, 𝔏_𝑔(0)=0,𝔏_𝑔(1)=1, and 𝔏_𝑔∈𝔉(𝐼,𝛼₁,𝑀₀). Thus 𝔏_𝑔⁻¹∈𝔉(𝐼,𝑀₀⁻¹,𝛼₁⁻¹).
Define 𝔗∶𝔉^′(𝐼,0,𝛼₁⁻¹𝑀)→𝐶⁰(𝐼) by 𝔗(𝑔)=𝔏_𝑔⁻¹∘𝐺, (3.7) where 𝔏_𝑔 is defined in (3.5) and 𝐺 is the induced map of 𝐹. For convenience, denote 𝔗(𝑔)=𝔗_𝑔. Then, we have 𝔗_𝑔(0)=1,𝔗_𝑔(1)=0. Since 𝔏_𝑔⁻¹ is increasing, we have that for all 𝑠,𝑡∈𝐼 with 𝑡≥𝑠,0≤𝔗_𝑔(𝑠)−𝔗_𝑔(𝑡)=𝔏_𝑔⁻¹∘𝐺(𝑠)−𝔏_𝑔⁻¹∘𝐺(𝑡)≤𝛼₁⁻¹(𝐺(𝑠)−𝐺(𝑡))≤𝛼₁⁻¹𝑀(𝑡−𝑠). (3.8) Hence, 𝔗(𝑔)∈𝔉^′(𝐼,0,𝛼₁⁻¹𝑀).
Furthermore, for any 𝑔₁,𝑔₂∈𝔉^′(𝐼,0,𝛼₁⁻¹𝑀), we have ‖𝔗_𝑔1−𝔗_𝑔2‖=‖𝔏_𝑔⁻¹₁∘𝐺−𝔏_𝑔⁻¹₂∘𝐺‖=‖𝔏_𝑔⁻¹₁−𝔏_𝑔⁻¹₂‖≤𝛼₁⁻¹‖𝔏_𝑔1−𝔏_𝑔2‖,‖𝔏_𝑔1−𝔏_𝑔2‖≤sup_𝑡∈𝐼|Ψ(𝑡,𝑔²₁(𝑡),…,𝑔₁^2𝑛−2(𝑡))−Ψ(𝑡,𝑔²₂(𝑡),…,𝑔₂^2𝑛−2≤(𝑡))|_𝑛𝑗=2𝛽_𝑗‖𝑔₁^2𝑗−2−𝑔₂^2𝑗−2‖≤^𝑛_𝑗=2𝛽_𝑗^2𝑗−2_𝑖=1(𝛼₁⁻¹𝑀)^𝑖−1‖𝑔₁−𝑔₂‖, (3.9) where Lemma 3.1 and (H2) are applied. Finally we get that ‖𝔗_𝑔1−𝔗_𝑔2‖≤^𝑛_𝑗=2𝛽_𝑗^2𝑗−2_𝑖=1𝛼₁^−𝑖𝑀^𝑖−1‖𝑔₁−𝑔₂‖. (3.10) Hence 𝔗 maps 𝔉^′(𝐼,0,𝛼₁⁻¹𝑀) continuously into itself. By Schauder's fixed point theorem, 𝔗 has a fixed 𝑔∈𝔉^′(𝐼,0,𝛼₁⁻¹𝑀). By Lemma 2.2, the map 𝑔 can be extended to a map 𝑔 such that 𝑔(𝑡+1)−𝑔(𝑡)=−1,forall𝑡∈ℝ, and there exists a unique 𝑓∈𝐶⁰(𝕋¹,𝕋¹) such that 𝑔 is a lift of 𝑓. Note that for any 𝑡,𝑠∈ℝ with 𝑡<𝑠, there is an integer 𝑘 and a nonnegative integer 𝑚 such that 𝑡∈[𝑘,𝑘+1) and 𝑠∈[𝑘+𝑚,𝑘+𝑚+1). Thus |𝑔(𝑡)−𝑔(𝑠)|≤|𝑔(𝑡)−𝑔(𝑘+1)|+^𝑚_𝑗=1|≤𝑀𝑔(𝑘+𝑗)−𝑔(𝑘+𝑗+1)|+|𝑔(𝑠)−𝑔(𝑘+𝑚)|≤|𝑔(𝑡−𝑘)−𝑔(1)|+(𝑚−1)|𝑔(0)−𝑔(1)|+|𝑔(𝑠−𝑘−𝑚)−𝑔(0)|𝛼₁𝑀[1−(𝑡−𝑘)+𝑚−1+𝑠−𝑘−𝑚]=𝛼₁(𝑠−𝑡). (3.11) Obviously 𝑓∈𝐻⁰_𝟏(𝕋¹,−1) and 𝑓 is a solution of (1.4) with a Lipschitz constant 𝑀/𝛼₁.

4. Uniqueness and Stability

Let 𝐹₁,𝐹₂∈𝐶⁰(𝕋¹,𝕋¹) and 𝐹₁,𝐹₂ be their lifts, respectively. By (iv) of Lemma 2.1, we can define ‖𝐹₁−𝐹₂‖_ℝ∶=sup_𝑡∈ℝ|𝐹₁(𝑡₁)−𝐹₂(𝑡₂)|. (4.1)

Definition 4.1. For a given small constant 𝜖>0, one says that 𝐹₁ is 𝜖−𝐶⁰—close to 𝐹₂ if ‖𝐹₁−𝐹₂‖_ℝ<𝜖. (4.2)

Now as in [8], we give the definition that (1.4) is stable.

Definition 4.2. Let 𝐹₁,𝐹₂∈𝐶⁰(𝕋¹,𝕋¹) and 𝑓₁,𝑓₂ be corresponding solutions of (1.4) on 𝐹₁,𝐹₂. One says that (1.4) is stable if for all 𝜀>0 there is a 𝛿>0 such that if 𝐹₁ is 𝛿−𝐶⁰—close to 𝐹₂, then 𝑓₁ is 𝜀−𝐶⁰— close to 𝑓₂.

Theorem 4.3. Suppose that the conditions in Theorem 3.2 still hold, and 𝑟=^𝑛_𝑗=2𝛽_𝑗^2𝑗−2_𝑖=1𝛼₁^−𝑖𝑀^𝑖−1<1. (4.3) Then (1.4) has a unique solution 𝑓∈𝐻⁰_𝟏(𝕋¹,−1) with a Lipschitz constant 𝑀/𝛼₁. Moreover, (1.4) is stable.

Proof. By Banach contraction theorem and (3.10), the solution 𝑓 is unique.
Suppose that 𝐹₁,𝐹₂∈𝐻⁰_𝟏(𝕋¹,−1) both satisfy the conditions in Theorem 3.2 and 𝑓₁,𝑓₂ are the unique solutions of (1.4) corresponding to the given 𝐹_𝑗,𝑗=1,2, respectively. Assume that 𝐹_𝑖,𝑓_𝑖 be the lifts of 𝐹_𝑖,𝑓_𝑖,𝑖=1,2, respectively. Let 𝐺_𝑖,𝑔_𝑖 be the restrictions on [0,1] of 𝐹_𝑖,𝑓_𝑖,𝑖=1,2. Let Ψ be the induced map of Φ defined in (2.27). By the proof of Theorem 3.2, we have 𝑔_𝑖=𝔏_𝑔⁻¹_𝑖∘𝐺_𝑖,𝑖=1,2, (4.4) where 𝔏_𝑔𝑖 is defined as (3.5) and 𝔏_𝑔⁻¹_𝑖∈𝔉(𝐼,𝑀₀⁻¹,𝛼₁⁻¹),𝑖=1,2. By (3.10), we have ‖𝑔₁−𝑔₂‖=‖𝔏_𝑔⁻¹₁∘𝐺₁−𝔏_𝑔⁻¹₂∘𝐺₂‖≤‖𝔏_𝑔⁻¹₁∘𝐺₁−𝔏_𝑔⁻¹₁∘𝐺₂‖+‖𝔏_𝑔⁻¹₁∘𝐺₂−𝔏_𝑔⁻¹₂∘𝐺₂‖≤𝛼₁⁻¹‖𝐺₁−𝐺₂‖+𝑟‖𝑔₁−𝑔₂‖. (4.5) Thus ‖𝑔₁−𝑔₂‖≤[(1−𝑟)𝛼₁]⁻¹‖𝐺₁−𝐺₂‖. (4.6)
For any 𝑡∈ℝ, there is an integer 𝑘 with 𝑡∈[𝑘,𝑘+1) such that |𝑓₁(𝑡)−𝑓₂(𝑡)|=|𝑓₁(𝑡−𝑘)+𝑘−𝑓₂(𝑡−𝑘)−𝑘|=|𝑔₁(𝑡−𝑘)−𝑔₂(𝑡−𝑘)|. (4.7) This means that ‖𝑓₁−𝑓₂‖_ℝ=‖𝑔₁−𝑔₂‖. (4.8) By the same argument, we have ‖𝐹₁−𝐹₂‖_ℝ=‖𝐺₁−𝐺₂‖. (4.9) Thus ‖𝑓₁−𝑓₂‖_ℝ≤[(1−𝑟)𝛼₁]⁻¹‖𝐹₁−𝐹₂‖_ℝ. (4.10) By the discussion above, we know that (1.4) is stable.

5. Example

Example 5.1. Consider the equation (𝑓(𝑧))^359/360(𝑓³(𝑧))^{(1/720𝜋𝑖)ln(𝑓3(𝑧))}=exp(2𝜋𝑖(𝑒−𝑧^1/2𝜋𝑖)𝑒−1). (5.1) Obviously 𝐹(𝑧)=exp(2𝜋𝑖(𝑒−𝑧^1/2𝜋𝑖)𝑒−1),Φ(𝑧₁,𝑧₂)=𝑧₁^359/360𝑧^{(1/720𝜋𝑖)ln(𝑧2)}₂. (5.2) The map 𝐹 has a lift 𝐹 and 𝐹|_[0,1]=(𝑒−𝑒^𝑡)/(𝑒−1). Moreover, 𝐹 has a Lipschitz constant 2. The induced map of Φ is Ψ(𝑡₁,𝑡₂)=359𝑡360¹+1𝑡360²²,0<𝑡_𝑗<1,𝑗=1,2,Ψ(0,0)=0,Ψ(1,1)=1. (5.3) It is easy to verify that both (H1) and (H2) hold. The constants are 𝛼₁=𝛽₁=359/360,𝛼₂=0,𝛽₂=1/180. By Theorem 3.2, (5.1) has a solution 𝑓∈𝐻⁰_𝟏(𝕋¹,−1) with a Lipschitz constant 720/359.
Note that 𝛼₁⁻¹=359/360<1. We have 𝑟=𝛽₂²_𝑖=1𝛼₁^−𝑖𝑀^𝑖−1<1180²_𝑖=12^𝑖−1=1.60 (5.4) Thus by Theorem 4.3, the solution is unique and the equation is stable.

Acknowledgment

The authors would like to thank the referees for their valuable comments and suggestions which led to truly significant improvement of the paper. This work is part of Project 11001064 supported by National Natural Science Foundation of China, Project HITC200706 and HITC200701 supported by Science Research Foundation in Harbin Institute of Technology.