Analytic Solutions of an Iterative Functional Differential Equation near Resonance

Abstract

We investigate the existence of analytic solutions of a class of second-order differential equations involving iterates of the unknown function 𝑥^′′(𝑧)+𝑐𝑥^′(𝑧)=𝑥(𝑎𝑧+𝑏𝑥(𝑧)) in the complex field ℂ. By reducing the equation with the Schröder transformation to the another functional differential equation without iteration of the unknown function 𝜆²𝑔^′′(𝜆𝑧)𝑔^′(𝑧)−𝜆𝑔^′(𝜆𝑧)𝑔^′′(𝑧) + 𝑐(𝑔^′(𝑧))²(𝜆𝑔^′(𝜆𝑧)−𝑎𝑔^′(𝑧)) = (𝑔^′(𝑧))³(𝑔(𝜆²𝑧)−𝑎𝑔(𝜆𝑧)), we get its local invertible analytic solutions.

1. Introduction

Functional differential equations with state dependent delay have attracted the attentions of many authors in the last years because of their extensive applications (e.g., [1–4] ). However, there are only a few papers dealing with functional differential equation with state derivative dependent delay. In [5] Eder studied the functional differential equation 𝑥^′(𝑡)=𝑥(𝑥(𝑡)). V. R. Petahov [6] proved the existence of solutions of equation 𝑥^′′(𝑡)=𝑎𝑥(𝑥(𝑡)). (1.1) In [7–9], the authors studied the existence of analytic solutions of the following second-order iterative functional differential equations: 𝑥^′′(𝑧)=𝑥(𝑎𝑧+𝑏𝑥^′𝑥(𝑧)),_′′(𝑥^[𝑟](𝑧))=𝑐₀𝑧+𝑐₁𝑥(𝑧)+⋯+𝑐_𝑚𝑥^[𝑚]𝑥(𝑧),_′′(𝑧)=𝑥(𝑎𝑧+𝑏𝑥(𝑧)), (1.2) respectively. Since such equations are quite different from the usual differential equations, the standard existence and uniqueness theorems cannot be applied directly. It is therefore of interest to find some or all of their solutions. In this paper, we will discuss the existence of analytic solutions for another second-order functional differential equation with a state derivative dependent delay: 𝑥^′′(𝑧)+𝑐𝑥^′(𝑧)=𝑥(𝑎𝑧+𝑏𝑥(𝑧)), (1.3) where 𝑎,𝑏, and 𝑐 are complex numbers. When 𝑐=0, (1.3) change into functional differential equation 𝑥^′′(𝑧)=𝑥(𝑎𝑧+𝑏𝑥(𝑧)), it has been studied in [9]. For the general equation (1.3) the same idea, however, cannot be applied. Therefore, in Section 3 we first reduce (1.3) to an iterative functional differential equation. Then, as in [9], the author reduces again this iterative functional differential equation with the Shchröder transformation, that is, 𝑥(𝑧)=(1/𝑏)(𝑔(𝜆𝑔⁻¹(𝑧))−𝑎𝑧) to a functional differential equation with proportional delay (which is called the auxiliary equation). Lastly, according to the position of an indeterminate constant 𝜆 in complex plane, we discuss the existence of analytic solutions.

In the next section, we will seek explicit analytic solutions of (1.3) in the form of power functions, in the case 𝑏=0.

2. Explicit Analytic Solutions

In case 𝑏=0, (1.3) changes into functional differential equation 𝑥^′′(𝑧)+𝑐𝑥^′(𝑧)=𝑥(𝑎𝑧). (2.1) For the above equation we have the following proposition.

Proposition 2.1. Suppose 0<|𝑎|≤1. Then (2.1) has an analytic solution 𝑥(𝑧) in a neighborhood of the origin, satisfying 𝑥(0)=𝜇 and 𝑥^′(0)=𝜂∈ℂ.

Proof. Let 𝑥(𝑧)=^∞_𝑛=0𝑎_𝑛𝑧^𝑛, (2.2) be the expansion of formal solution 𝑥(𝑧) of (2.1). Substituting (2.2) into (2.1), we get ^∞_𝑛=0(𝑛+2)(𝑛+1)𝑎_𝑛+2𝑧^𝑛=^∞_𝑛=0(𝑎^𝑛𝑎_𝑛−𝑐(𝑛+1)𝑎_𝑛+1)𝑧^𝑛. (2.3) By comparing the coefficients, we have 2𝑎₂=𝑎_𝑜−𝑐𝑎₁𝑎=𝜇−𝑐𝜂,_𝑛+2=𝑎^𝑛𝑎(𝑛+2)(𝑛+1)^𝑛−𝑐𝑎(𝑛+2)^𝑛+1. (2.4) In view of |𝑎|≤1 and 𝑐 is complex number, thus there exists a positive number 𝑚, so that |𝑎^𝑛𝑐(𝑛+2)(𝑛+1)|≤𝑚,|(𝑛+2)|≤𝑚. (2.5) Thus if we define recursively a sequence {𝐵_𝑛}^∞_𝑛=0 by 𝐵₀=|𝜇|,𝐵₁=|𝜂|,𝐵_𝑛+2=𝑚(𝐵_𝑛+𝐵_𝑛+1),𝑛=0,1,2,…, (2.6) then one can show that by induction |𝑎_𝑛|≤𝐵_𝑛,𝑛=0,1,2,…. (2.7) Now if we define 𝑀(𝑧)=^∞_𝑛=0𝐵_𝑛𝑧^𝑛, (2.8) then 𝑀(𝑧)=𝐵₀+𝐵₁𝑧+𝑧^2∞_𝑛=1𝐵_𝑛+2𝑧^𝑛=𝐵₀+𝐵₁𝑧+𝑚^∞_𝑛=1(𝐵_𝑛+𝐵_𝑛+1)𝑧^𝑛+2=(𝑚𝑧²+𝑚𝑧)𝑀(𝑧)+(|𝜂|−𝑚|𝜇|)𝑧+|𝜇|, (2.9) that is, (𝑚𝑧²+𝑚𝑧−1)𝑀(𝑧)+(|𝜂|−𝑚|𝜇|)𝑧+|𝜇|=0. (2.10) Let 𝑄(𝑧,𝑤)=(𝑚𝑧²+𝑚𝑧−1)𝑤+(|𝜂|−𝑚|𝜇|)𝑧+|𝜇| (2.11) for (𝑧,𝑤) from a neighborhood of (0,|𝜇|). Since 𝑄(0,|𝜇|)=0,𝑄^′_𝑤(0,|𝜇|)=−1≠0, there exists a unique function 𝑤(𝑧), analytic on a neighborhood of zero, such that 𝑤(0)=|𝜇|,𝑤^′(0)=|𝜂| and satisfying the equality 𝑄(𝑧,𝑤(𝑧))=0. According to (2.8) and (2.10), we have 𝑀(𝑧)=𝑤(𝑧). It follows that the power series (2.8) converges on a neighborhood of the origin, which implies that the power series (2.2) also converges in a neighborhood of the origin. The proof is complete.
Thus, the desired solution of (2.1) is 𝑥(𝑧)=𝜇+𝜂𝑧+𝜇−𝑐𝜂𝑧2!²+(𝑎+𝑐²)𝜂−𝑐𝜇𝑧3!³+⋯. (2.12)

3. Analytic Solutions of the Auxiliary Equation

A distinctive feature of the (1.3) when 𝑏≠0 is that the argument of the unknown function is dependent on the state 𝑥(𝑧), and this is the case we will emphasize in this paper. We now discuss the existence of analytic solution of (1.3) by locally reducing the equation to another functional differential equation with proportional delays. Let 𝑦(𝑧)=𝑎𝑧+𝑏𝑥(𝑧). (3.1) Then for any number 𝑧, we have 1𝑥(𝑧)=𝑏(𝑦(𝑧)−𝑎𝑧), (3.2) and so 𝑥(𝑦(𝑧))=(1/𝑏)(𝑦(𝑦(𝑧))−𝑎𝑦(𝑧)). Therefore, in view of (1.3) and 𝑥^′′(𝑧)=(1/𝑏)𝑦^′′(𝑧), we have 𝑦^′′(𝑧)+𝑐𝑦^′(𝑧)−𝑎=𝑦(𝑦(𝑧))−𝑎𝑦(𝑧). (3.3)

To find analytic solution of (3.3), we first seek an analytic solution 𝑔(𝑧) of the auxiliary equation 𝜆²𝑔^′′(𝜆𝑧)𝑔^′(𝑧)−𝜆𝑔^′(𝜆𝑧)𝑔^′′(𝑧)+𝑐(𝑔^′(𝑧))²(𝜆𝑔^′(𝜆𝑧)−𝑎𝑔^′(𝑧))=(𝑔^′(𝑧))³(𝑔(𝜆²𝑧)−𝑎𝑔(𝜆𝑧)) (3.4) satisfying the initial value conditions 𝑔(0)=𝜇,𝑔^′(0)=𝜂≠0, (3.5) where 𝜇 and 𝜂 are complex numbers, and 𝜆 satisfies one of the following conditions:

(H1)0<|𝜆|<1;(H2)𝜆=𝑒^{2𝜋𝑖𝜃}, where 𝜃∈ℝ⧵ℚ is a Brjuno number, that is, 𝐵(𝜃)=∑^∞_𝑘=0log𝑞_𝑘+1/𝑞_𝑘<∞, where {𝑝_𝑘/𝑞_𝑘} denotes the sequence of partial fraction of the continued fraction expansion of 𝜃;(H3)𝜆=𝑒^{2𝜋𝑖𝑞/𝑝} for some integers 𝑝∈ℕ with 𝑝≥2 and 𝑞∈ℤ⧵{0}, and 𝛼≠𝑒^{2𝜋𝑖𝑙/𝑘} for all 1≤𝑘≤𝑝−1 and 𝑙∈ℤ⧵{0}.

We observe that 𝜆 is inside the unit circle 𝑆¹ in case (H1) but on 𝑆¹ in the rest cases. More difficulties are encountered for 𝜆 on 𝑆¹ since the small divisor 𝜆^𝑛+2−𝜆 is involved in the latter (3.10). Under Diophantine condition: “𝜆=𝑒^{2𝜋𝑖𝜃}, where 𝜃∈ℝ⧵ℚ and there exist constants 𝜁>0 and 𝛿>0 such that |𝜆^𝑛−1|≥𝜁⁻¹𝑛^−𝛿 for all 𝑛≥1,” the number 𝜆∈𝑆¹ is “far” from all roots of the unity and was considered in different settings [7–12]. Since then, we have been striving to give a result of analytic solutions for those 𝜆 “near” a root of the unity, that is, neither being roots of the unity nor satisfying the Diophantine condition. The Brjuno condition in (H2) provides such a chance for us. Moreover, we also discuss the so-called the resonance case, that is, the case of (H3).

Theorem 3.1. If (H1) holds. Then, for the initial value conditions (3.5), (3.4) has an analytic solution of the form 𝑔(𝑧)=𝜇+𝜂𝑧+^∞_𝑛=2𝑏_𝑛𝑧^𝑛 (3.6) in a neighborhood of the origin.

Proof. Rewrite (3.4) in the form 𝜆²𝑔^′′(𝜆𝑧)𝑔^′(𝑧)−𝜆𝑔^′(𝜆𝑧)𝑔^′′(𝑧)(𝑔^′(𝑧))²+𝑐(𝜆𝑔^′(𝜆𝑧)−𝑎𝑔^′(𝑧))=𝑔^′(𝑧)(𝑔(𝜆²𝑧)−𝑎𝑔(𝜆𝑧)) (3.7) or 𝑔𝜆(^′(𝜆𝑧)𝑔^′)(𝑧)^′=𝑔^′(𝑧)(𝑔(𝜆²𝑧)−𝑎𝑔(𝜆𝑧))−𝑐(𝜆𝑔^′(𝜆𝑧)−𝑎𝑔^′(𝑧)). (3.8) Therefore, in view of 𝑔^′(0)=𝜂≠0, we obtain 𝑔^′(𝜆𝑧)=𝑔^′1(𝑧)(1+𝜆^𝑧₀𝑔^′(𝑠)(𝑔(𝜆²𝑠)−𝑎𝑔(𝜆𝑠))−𝑐𝜆𝑔^′(𝜆𝑠)+𝑎𝑐𝑔^′(𝑠)𝑑𝑠). (3.9) We now seek a solution of (3.4) in the form of a power series (3.6). By defining 𝑏₀=𝜇 and 𝑏₁=𝜂 and then substituting (3.6) into (3.9), we see that the sequence {𝑏_𝑛}^∞_𝑛=2 is successively determined by the condition (𝜆^𝑛+2−𝜆)(𝑛+2)𝑏_𝑛+2=^𝑛_𝑘=0(^𝑛−𝑘_𝑗=0(𝑗+1)(𝑘+1)(𝜆^{2(𝑛−𝑗−𝑘)}−𝑎𝜆^{𝑛−𝑗−𝑘})𝑏𝑛−𝑘+1^𝑗+1𝑏_𝑘+1𝑏_{𝑛−𝑗−𝑘}+𝑐(𝑘+1)(𝑎−𝜆^{𝑛+1−𝑘})𝑏_𝑘+1𝑏_{𝑛+1−𝑘}),𝑛=0,1,2,… (3.10) in a unique manner. Now we show that the resulting power series (3.6) converges in a neighborhood of the origin. First of all, we have lim_𝑛→∞|1/(𝜆^𝑛+2−𝜆)|=1/|𝜆|, thus there exists a positive number 𝑀, such that |(𝑗+1)(𝑘+1)(𝜆^{2(𝑛−𝑗−𝑘)}−𝑎𝜆^{𝑛−𝑗−𝑘})(𝑛−𝑘+1)(𝑛+2)(𝜆^𝑛+2|−𝜆)|≤𝑀,𝑐(𝑘+1)(𝑎−𝜆^{𝑛+1−𝑘})(𝑛+2)(𝜆^𝑛+2−𝜆)|≤𝑀. (3.11) If we define a sequence {𝐷_𝑛}^∞_𝑛=0 by 𝐷₀=|𝜇|,𝐷₁=|𝜂| and 𝐷_𝑛+2=𝑀^𝑛_𝑘=0(^𝑛−𝑘_𝑗=0𝐷_𝑗+1𝐷_𝑘+1𝐷_{𝑛−𝑗−𝑘}+𝐷_𝑘+1𝐷_{𝑛+1−𝑘}),𝑛=0,1,2,…, (3.12) then in view of (3.10), we can show by induction that |𝑏_𝑛|≤𝐷_𝑛,𝑛=0,1,2,…. (3.13) Now if we define 𝐺(𝑧)=^∞_𝑛=0𝐷_𝑛𝑧^𝑛, (3.14) then 𝐺²(𝑧)=(|𝜇|+^∞_𝑛=0𝐷_𝑛+1𝑧^𝑛+1)(^∞_𝑛=0𝐷_𝑛𝑧^𝑛)=|𝜇|^∞_𝑛=0𝐷_𝑛𝑧^𝑛+^∞𝑛_𝑛=0𝑘=0𝐷_𝑘+1𝐷_𝑛−𝑘𝑧^𝑛+1=|𝜇|²+2|𝜇|^∞_𝑛=0𝐷_𝑛+1𝑧^𝑛+1+^∞𝑛_𝑛=0𝑘=0𝐷_𝑘+1𝐷_{𝑛+1−𝑘}𝑧^𝑛+2,𝐺³(𝑧)=(|𝜇|+^∞_𝑛=0𝐷_𝑛+1𝑧^𝑛+1)(|𝜇|^∞_𝑛=0𝐷_𝑛𝑧^𝑛+^∞𝑛_𝑛=0𝑘=0𝐷_𝑘+1𝐷_𝑛−𝑘𝑧^𝑛+1)=|𝜇|^2∞_𝑛=0𝐷_𝑛𝑧^𝑛+2|𝜇|^∞𝑛_𝑛=0𝑘=0𝐷_𝑘+1𝐷_𝑛−𝑘𝑧^𝑛+1+^∞𝑛_𝑛=0𝑘=0^𝑛−𝑘_𝑗=0𝐷_𝑘+1𝐷_𝑗+1𝐷_{𝑛−𝑗−𝑘}𝑧^𝑛+2. (3.15) We get immediately 𝐺³(𝑧)+𝐺²(𝑧)=2|𝜇|𝐺²(𝑧)+(2|𝜇|−|𝜇|²)𝐺(𝑧)−|𝜇|²+1𝑀^∞_𝑛=0𝐷_𝑛+2𝑧^𝑛+2=2|𝜇|𝐺²(𝑧)+(2|𝜇|−|𝜇|²)𝐺(𝑧)−|𝜇|²+1𝑀(𝐺(𝑧)−|𝜇|−|𝜂|𝑧). (3.16) That is, 𝐺³(𝑧)−(2|𝜇|−1)𝐺²1(𝑧)−(𝑀+2|𝜇|−|𝜇|²1)𝐺(𝑧)+𝑀(|𝜂|𝑧+|𝜇|)+|𝜇|²=0. (3.17) Let 𝑅(𝑧,𝑤)=𝑤³−(2|𝜇|−1)𝑤²1−(𝑀+2|𝜇|−|𝜇|²1)𝑤+𝑀(|𝜂|𝑧+|𝜇|)+|𝜇|² (3.18) for (𝑧,𝑤) from a neighborhood of (0,|𝜇|). Since 𝑅(0,|𝜇|)=0 and 𝑅^′_𝑤(0,|𝜇|)=−1/𝑀≠0, and the implicit function theorem, there exists a unique function 𝑤(𝑧), analytic in a neighborhood of zero, such that 𝑤(0)=|𝜇|,𝑤^′(0)=|𝜂| and 𝑅(𝑧,𝑤)=0. According to (3.14) and (3.17), we have 𝐺(𝑧)=𝑤(𝑧). It follows that the power series (3.14) converges in a neighborhood of the origin. So does (3.6). The proof is complete.

Now, we discuss local invertible analytic solutions of auxiliary equation (3.4) in cases (H2). In order to study the existence of analytic solutions of (3.6) under the Brjuno condition, we first recall briefly the definition of Brjuno numbers and some basic facts. As stated in [13], for a real number 𝜃, we let [𝜃] denote its integer part and {𝜃}=𝜃−[𝜃] its fractional part. Then every irrational number 𝜃 has a unique expression of the Gauss’ continued fraction 𝜃=𝑑₀+𝜃₀=𝑑₀+1𝑑₁+𝜃₁=⋯, (3.19) denoted simply by 𝜃=[𝑑₀,𝑑₁,…,𝑑_𝑛,…], where 𝑑_𝑗’s and 𝜃_𝑗’s are calculated by the algorithm: (a) 𝑑₀=[𝜃],𝜃₀={𝜃} and (b) 𝑑_𝑛=[1/𝜃_𝑛−1],𝜃_𝑛={1/𝜃_𝑛−1} for all 𝑛≥1. Define the sequences {𝑝_𝑛}_𝑛∈ℕ and {𝑞_𝑛}_𝑛∈ℕ as follows: 𝑞₋₂=1,𝑞₋₁=0,𝑞_𝑛=𝑑_𝑛𝑞_𝑛−1+𝑞_𝑛−2,𝑝₋₂=0,𝑝₋₁=1,𝑝_𝑛=𝑑_𝑛𝑝_𝑛−1+𝑝_𝑛−2. (3.20) It is easy to show that 𝑝_𝑛/𝑞_𝑛=[𝑑₀,𝑑₁,…,𝑑_𝑛]. Thus, for every 𝜃∈ℝ⧵ℚ we associate, using its convergence, an arithmetical function 𝐵(𝜃)=∑_𝑛≥0log𝑞_𝑛+1/𝑞_𝑛. We say that 𝜃 is a Brjuno number or that it satisfies Brjuno condition if 𝐵(𝜃)<+∞. The Brjuno condition is weaker than the Diophantine condition. For example, if 𝑑_𝑛+1≤𝑐𝑒^𝑑𝑛 for all 𝑛≥0, where 𝑐>0 is a constant, then 𝜃=[𝑑₀,𝑑₁,…,𝑑_𝑛,…] is a Brjuno number but is not a Diophantine number. So, the case (H2) contains both Diophantine condition and a part of 𝛼 “near” resonance. Let 𝜃∈ℝ⧵ℚ and {𝑞_𝑛}_𝑛∈ℕ be the sequence of partial denominators of the Gauss’s continued fraction for 𝜃. As in [13], let 𝐴_𝑘1={𝑛≥0∣‖𝑛𝜃‖≤8𝑞_𝑘},𝐸_𝑘=max{𝑞_𝑘,𝑞_𝑘+14},𝜂_𝑘=𝑞_𝑘𝐸_𝑘. (3.21) Let 𝐴^∗_𝑘 be the set of integers 𝑗≥0 such that either 𝑗∈𝐴_𝑘 or for some 𝑗₁ and 𝑗₂ in 𝐴_𝑘, with 𝑗₂−𝑗₁<𝐸_𝑘, one has 𝑗₁<𝑗<𝑗₂ and 𝑞_𝑘 divides 𝑗−𝑗₁. For any integer 𝑛≥0, define 𝑙_𝑘(𝑛)=max{(1+𝜂_𝑘)𝑛𝑞_𝑘−2,(𝑚_𝑛𝜂_𝑘1+𝑛)𝑞_𝑘−1}, (3.22) where 𝑚_𝑛=max{𝑗∣0≤𝑗≤𝑛,𝑗∈𝐴^∗_𝑘}. We then define function ℎ_𝑘∶ℕ→ℝ₊ as follows: ℎ_𝑘𝑚(𝑛)=^𝑛+𝜂_𝑘𝑛𝑞_𝑘−1,if𝑚_𝑛+𝑞_𝑘∈𝐴^∗_𝑘,𝑙_𝑘(𝑛),if𝑚_𝑛+𝑞_𝑘∉𝐴^∗_𝑘. (3.23) Let 𝑔_𝑘(𝑛)∶=max{ℎ_𝑘(𝑛),[𝑛/𝑞_𝑘]}, and define 𝑘(𝑛) by the condition 𝑞_𝑘(𝑛)≤𝑛≤𝑞_𝑘(𝑛)+1. Clearly, 𝑘(𝑛) is nondecreasing. Now we are able to state the following result.

Lemma 3.2 (Davie’s lemma [14]). Let 𝐾(𝑛)=𝑛log2+∑^𝑘(𝑛)_𝑘=0𝑔_𝑘(𝑛)log(2𝑞_𝑘+1). Then(a)there is a universal constant 𝛾>0 (independent of 𝑛 and 𝜃) such that 𝐾(𝑛)≤𝑛(𝐵(𝜃)+𝛾); (3.24)(b)𝐾(𝑛₁)+𝐾(𝑛₂)≤𝐾(𝑛₁+𝑛₂) for all 𝑛₁ and 𝑛₂;(c)−log|𝛼^𝑛−1|≤𝐾(𝑛)−𝐾(𝑛−1).

Theorem 3.3. Suppose (H2) holds. Then (3.4) has an analytic solution of the form (3.6) in a neighborhood of the origin such that 𝑔(0)=𝜇,𝑔^′(0)=𝜂.

Proof. As in the proof of Theorem 3.1, we seek a power series solution of the form (3.6). Set 𝑏₀=𝜇 and 𝑏₁=𝜂, (3.10) holds again. From (3.10) we get |𝑏_𝑛+2𝑀|≤|𝜆^𝑛+1−1|^𝑛_𝑘=0(^𝑛−𝑘_𝑗=0|𝑏_𝑗+1||𝑏_𝑘+1||𝑏_{𝑛−𝑗−𝑘}|+|𝑏_𝑘+1||𝑏_{𝑛+1−𝑘}|),𝑛=0,1,2,…. (3.25)
To construct a governing series of (3.6), we consider the implicit functional equation 𝑅(𝑧,𝜑)=∶𝑅(𝑧,𝜑,𝜇,𝜂,𝑀)=0, (3.26) where 𝑅 is defined in (3.18). Similarly to the proof of Theorem 3.1, using the implicit function theorem we can prove that (3.26) has a unique analytic solution 𝜑(𝑧,𝜇,𝜂,𝑀) in a neighborhood of the origin such that 𝜑(0,𝜇,𝜂,𝑀)=|𝜇| and 𝜑^′_𝑧(0,𝜇,𝜂,𝑀)=|𝜂|. Thus 𝜑(𝑧,𝜇,𝜂,𝑀) in (3.26) can be expanded into a convergent series 𝜑(𝑧,𝜇,𝜂,𝑀)=^∞_𝑛=0𝐶_𝑛𝑧^𝑛 (3.27) in a neighborhood of the origin. Replacing (3.27) into (3.26) and comparing coefficients, we obtain that 𝐶₀=|𝜇|,𝐶₁=|𝜂| and 𝐶_𝑛+2=𝑀^𝑛_𝑘=0(^𝑛−𝑘_𝑗=0𝐶_𝑗+1𝐶_𝑘+1𝐶_{𝑛−𝑗−𝑘}+𝐶_𝑘+1𝐶_{𝑛+1−𝑘}),𝑛=0,1,2,…. (3.28)
Note the power series (3.27) converges in a neighborhood of zero. Hence there is a positive constant 𝑇 such that 𝐶_𝑛<𝑇^𝑛,𝑛=1,2,…. (3.29) Now by induction, we prove that |𝑏_𝑛|≤𝐶_𝑛𝑒^{𝐾(𝑛−1)},𝑛=1,2,…, (3.30) where 𝐾∶ℕ→ℝ is defined in Lemma 3.2. In fact, |𝑏₁|=𝐶₁. For inductive proof, we assume that |𝑏_𝑚|≤𝐶_𝑚𝑒^{𝐾(𝑚−1)}. From (3.10) and Lemma 3.2 ,we know |𝑏_𝑚+1𝑀|≤|𝜆^𝑚−1|^𝑚−1_𝑘=0(^{𝑚−1−𝑘}_𝑗=0|𝑏_𝑗+1||𝑏_𝑘+1||𝑏_{𝑚−1−𝑗−𝑘}|+|𝑏_𝑘+1||𝑏_𝑚−𝑘≤𝑀|)|𝜆^𝑚−1|^𝑚−1_𝑘=0(^{𝑚−1−𝑘}_𝑗=0𝐶_𝑗+1𝐶_𝑘+1𝐶_{𝑚−1−𝑗−𝑘}𝑒^{𝐾(𝑗)+𝐾(𝑘)+𝐾(𝑚−2−𝑗−𝑘)}+𝐶_𝑘+1𝐶_𝑚−𝑘𝑒^{𝐾(𝑘)+𝐾(𝑚−1−𝑘)}). (3.31) Note that 𝐾(𝑗)+𝐾(𝑘)+𝐾(𝑚−2−𝑗−𝑘)≤𝐾(𝑚−2)≤𝐾(𝑚−1)≤log|𝜆^𝑚−1|+𝐾(𝑚),𝐾(𝑘)+𝐾(𝑚−1−𝑘)≤𝐾(𝑚−1)≤log|𝜆^𝑚−1|+𝐾(𝑚). (3.32) Hence |𝑏_𝑚+1|≤𝑒^𝐾(𝑚)⋅𝑀^𝑚−1_𝑘=0(^{𝑚−1−𝑘}_𝑗=0𝐶_𝑗+1𝐶_𝑘+1𝐶_{𝑚−1−𝑗−𝑘}+𝐶_𝑘+1𝐶_𝑚−𝑘)=𝐶_𝑚+1𝑒^𝐾(𝑚) (3.33) as desired. In view of (3.29) and Lemma 3.2, we know that 𝐾(𝑛)≤𝑛(𝐵(𝜃)+𝛾) for some universal constant 𝛾>0. Then |𝑏_𝑛|≤𝑇^𝑛𝑒^{𝑛(𝐵(𝜃)+𝛾)}. (3.34) That is, lim_𝑛→∞sup(|𝑏_𝑛|)^1/𝑛≤lim_𝑛→∞sup(𝑇^𝑛𝑒^{𝑛(𝐵(𝜃)+𝛾)})^1/𝑛≤𝑇𝑒^{𝐵(𝜃)+𝛾}. (3.35) This implies that the convergence radius of (3.6) is at least (𝑇𝑒^{𝐵(𝜃)+𝛾})⁻¹. This concludes the proof.

In case (H3), the constant 𝜆 is not only the unit circle in ℂ, but also a root of unity. In such a case, the resonant case, both the Diophantine and the Brjuno conditions are not satisfied. Let {𝐷_𝑛}^∞_𝑛=0 be a sequence define by 𝐷₀=|𝜇|,𝐷₁=|𝜂| and 𝐷_𝑛+2=Γ𝑀^𝑛_𝑘=0(^𝑛−𝑘_𝑗=0𝐷_𝑗+1𝐷_𝑘+1𝐷_{𝑛−𝑗−𝑘}+𝐷_𝑘+1𝐷_{𝑛+1−𝑘}),𝑛=0,1,2,…, (3.36) where Γ=max{1,|𝜆^𝑖−1|⁻¹∶𝑖=1,2,…,𝑝−1} and 𝑀 is defined in Theorem 3.1.

Theorem 3.4. Suppose that (H3) holds, let {𝑏_𝑛}^∞_𝑛=0 be determined by 𝑏₀=𝜇,𝑏₁=𝜂 and (𝜆^𝑛+2−𝜆)(𝑛+2)𝑏_𝑛+2=Θ(𝑛,𝜆),𝑛=1,2,…, (3.37) where Θ(𝑛,𝜆)=^𝑛_𝑘=0(^𝑛−𝑘_𝑗=0(𝑗+1)(𝑘+1)(𝜆^{2(𝑛−𝑗−𝑘)}−𝑎𝜆^{𝑛−𝑗−𝑘})𝑏𝑛−𝑘+1^𝑗+1𝑏_𝑘+1𝑏_{𝑛−𝑗−𝑘}+𝑐(𝑘+1)(𝑎−𝜆^{𝑛+1−𝑘})𝑏_𝑘+1𝑏_{𝑛+1−𝑗}). (3.38) If Θ(𝑣𝑝−1,𝜆)=0,𝑣=1,2,…, then (3.4) has an analytic solution 𝑔(𝑧) in a neighborhood of the origin such that 𝑔(0)=𝜇,𝑔^′(0)=𝜂, and 𝑔^{(𝑣𝑝−1)}(0)=(𝑣𝑝−1)!𝜂_𝑣𝑝−1, where all 𝜂_𝑣𝑝−1’s are arbitrary constants satisfying the inequality |𝜂_𝑣𝑝−1|≤𝐷_𝑣𝑝−1, and sequence {𝐷_𝑛}^∞_𝑛=0 is defined in (3.36). Otherwise, if Θ(𝑣𝑝−1,𝜆)≠0,𝑣=1,2,…, then (3.4) has no any analytic solution 𝑔(𝑧) in a neighborhood of the origin.

Proof. We seek a power series solution of (3.4) of the form (3.6), as in the proof of Theorem 3.1, where the equality in (3.10) or (3.29) is indispensable. If Θ(𝑣𝑝−1,𝜆)≠0,𝑣=1,2,…, then the equality in (3.37) does not hold for 𝑛=𝑣𝑝−1, since𝜆^𝑣𝑝−1=0. In such a circumstance (3.4) has no formal solutions.
If Θ(𝑣𝑝−1,𝜆)=0 for all natural numbers 𝑣, the corresponding 𝑏_𝑣𝑝−1 in (3.29) has infinitely many choices in ℂ; this is, the formal series solution (3.6) defines a family of solutions with infinitely many parameters. Choose 𝑏_𝑣𝑝−1=𝜂_𝑣𝑝−1 arbitrarily such that |𝜂_𝑣𝑝−1|≤𝐷_𝑣𝑝−1,𝑣=1,2,…, (3.39) where 𝐷_𝑣𝑝−1 is defined by (3.36). In what follows, we prove that the formal series solution (3.6) converges in a neighborhood of the origin. Observe that |𝜆^𝑛+1−1|⁻¹≤Γ for 𝑛≠𝑣𝑝−1. It follows from (3.29) that |𝑏_𝑛+2|≤Γ𝑀^𝑛_𝑘=0(^𝑛−𝑘_𝑗=0|𝑏_𝑗+1||𝑏_𝑘+1||𝑏_{𝑛−𝑗−𝑘}|+|𝑏_𝑘+1||𝑏_{𝑛+1−𝑘}|),𝑛=0,1,2,⋯ (3.40) for 𝑛≠𝑣𝑝−1,𝑣=1,2,…. Further, we can prove that |𝑏_𝑛|≤𝐷_𝑛,𝑛=1,2,…. (3.41) Set 𝜓(𝑧,𝜇,𝜂,Γ𝑀)=^∞_𝑛=0𝐷_𝑛𝑍^𝑛. (3.42) It is easy to check that (3.42) satisfies the implicit functional equation 𝑅(𝑧,𝜓,𝜇,𝜂,Γ𝑀)=0, (3.43) where 𝑅 is defined in (3.18). Moreover, similarly to the proof of Theorem 3.1, we can prove that (3.43) has a unique analytic solution 𝜓(𝑧,𝜇,𝜂,Γ𝑀) in a neighborhood of the origin such that 𝜓(0,𝜇,𝜂,Γ𝑀)=|𝜇|, and 𝜓^′_𝑧(0,𝜇,𝜂,Γ𝑀)=|𝜂|. Thus (3.42) converges in a neighborhood of the origin. Therefore, the series (3.6) converges in a neighborhood of the origin. The proof is complete.

Theorem 3.5. Suppose the conditions of Theorem 3.1 or Theorem 3.3 or Theorem 3.4 are satisfied. Then (1.3) has an analytic solution 𝑥(𝑧)=(1/𝑏)(𝑔(𝜆𝑔⁻¹(𝑧))−𝑎𝑧) in a neighborhood of the number 𝜇, where 𝑔(𝑧) is an analytic solution of (3.4).

Proof. In view of Theorem 3.1 or Theorem 3.3 or Theorem 3.4, we may find a sequence {𝑏_𝑛}^∞_𝑛=0 such that the function 𝑔(𝑧) of the form (3.6) is an analytic solution of (3.4) in a neighborhood of the origin. Since 𝑔^′(0)=𝜂≠0, the function 𝑔⁻¹(𝑧) is analytic in a neighborhood of 𝑔(0)=𝜇. If we now define 𝑥(𝑧)=(1/𝑏)(𝑔(𝜆𝑔⁻¹(𝑧))−𝑎𝑧) then 𝑥^′1(𝑧)=𝑏(𝜆𝑔^′(𝜆𝑔⁻¹(𝑧))𝑔^′(𝑔⁻¹𝑥(𝑧))−𝑎)_′′𝜆(𝑧)=²𝑔^′′(𝜆𝑔⁻¹(𝑧))𝑔^′(𝑔⁻¹(𝑧))−𝜆𝑔^′(𝜆𝑔⁻¹(𝑧))𝑔^′′(𝑔⁻¹(𝑧))𝑏[𝑔^′(𝑔⁻¹(𝑧))]³, (3.44) then 𝑥^′′(𝑧)+𝑐𝑥^′𝜆(𝑧)=²𝑔^′′(𝜆𝑔⁻¹(𝑧))𝑔^′(𝑔⁻¹(𝑧))−𝜆𝑔^′(𝜆𝑔⁻¹(𝑧))𝑔^′′(𝑔⁻¹(𝑧))𝑏[𝑔^′(𝑔⁻¹(𝑧))]³−𝑐𝑔^′(𝑔⁻¹(𝑧))²[𝜆𝑔^′(𝜆𝑔⁻¹(𝑧))−𝑎𝑔^′(𝑔⁻¹(𝑧))]𝑏[𝑔^′(𝑔⁻¹(𝑧))]³=[𝑔^′(𝑔⁻¹(𝑧))]³[𝑔(𝜆²𝑔⁻¹(𝑧))−𝑎𝑔(𝜆𝑔⁻¹(𝑧))]𝑏[𝑔^′(𝑔⁻¹(𝑧))]³=𝑥(𝑎𝑧+𝑏𝑥(𝑧)) (3.45) as required.

4. Analytic Solution of (1.3)

In the previous section, we have shown that under the conditions of Theorem 3.1 or Theorem 3.3 or Theorem 3.4, (3.3) has an analytic solution 𝑦(𝑧)=𝑔(𝜆𝑔⁻¹(𝑧)) in a neighborhood of the number 𝜇, where 𝑔(𝑧) is an solution of (3.4). Since the function 𝑦(𝑧)=𝑔(𝜆𝑔⁻¹(𝑧)) can be determined by (3.10), it is possible to calculate, at last in theory, the explicit form of 𝑦(𝑧), an analytic solution of (1.3), in a neighborhood of the fixed point 𝜇 of 𝑦(𝑧) by means of (3.2). However, knowing that an analytic solution of (1.3) exists, we can take an alternative route as follows. Assume that 𝑥(𝑧) is of the form 𝑥(𝑧)=𝑥(𝜇)+𝑥^′𝑥(𝜇)(𝑧−𝜇)+^′′(𝜇)2!(𝑧−𝜇)²+⋯, (4.1) we need to determine the derivatives 𝑥^(𝑛)(𝜇), 𝑛=0,1,2,…. First of all, in view of (1.3) and (3.2), we have 𝑥(𝜇)=𝑦(𝜇)−𝑎𝜇𝑏=𝑔(𝜆𝑔⁻¹(𝜇))−𝑎𝜇𝑏=(1−𝑎)𝜇𝑏,𝑥^′𝜆(𝑧)=𝑏𝑔^′(𝜆𝑔⁻¹(𝑧))𝑔^′(𝑔⁻¹−𝑎(𝑧))𝑏,𝑥^′(𝜇)=𝜆−𝑎𝑏, (4.2) respectively. Furthermore, 𝑥^′′(𝑧)=𝑥(𝑎𝑧+𝑏𝑥(𝑧))−𝑐𝑥^′𝑥(𝑧),_′′(𝜇)=(1−𝑎)𝜇𝑏−(𝜆−𝑎)𝑐𝑏=(1−𝑎)𝜇−(𝜆−𝑎)𝑐𝑏. (4.3) Next by calculating the derivatives of both sides of (1.3), we obtain successively 𝑥^′′′(𝑧)=𝑥^′(𝑎𝑧+𝑏𝑥(𝑧))(𝑎+𝑏𝑥^′(𝑧))−𝑐𝑥^′′(𝑧), (4.4) so that 𝑥^′′′(𝜇)=𝜆(𝜆−𝑎)−𝑐(1−𝑎)𝜇+(𝜆−𝑎)𝑐²𝑏. (4.5)

Recall the formula for the higher derivatives of composition. Namely, for 𝑛≥1,𝑥^(𝑘+2)(𝑧)=𝑥^(𝑘)(ℎ(𝑧))−𝑐𝑥^(𝑘+1)=(𝑧)_𝑘𝑖=1𝑃_𝑖,𝑘(ℎ^′(𝑧),ℎ^′′(𝑧),…,ℎ^(𝑖)(𝑧))𝑥^(𝑖)(ℎ(𝑧))−𝑐𝑥^(𝑘+1)(𝑧), (4.6) where ℎ(𝑧)=𝑎𝑧+𝑏𝑥(𝑧), and 𝑃_𝑖,𝑘 is a polynomial with nonnegative coefficients.

We have 𝑥^(𝑘+2)(𝜇)=^𝑘_𝑖=1𝑃_𝑖,𝑘(ℎ^′(𝜇),ℎ^′′(𝜇),…,ℎ^(𝑖)(𝜇))𝑥^(𝑖)(ℎ(𝜇))−𝑐𝑥^(𝑘+1)(𝜇)=∶Δ_𝑘+2. (4.7) Thus the desired solutions is 𝑥(𝑧)=(1−𝑎)𝜇𝑏+𝜆−𝑎𝑏(𝑧−𝜇)+(1−𝑎)𝜇−(𝜆−𝑎)𝑐2!𝑏(𝑧−𝜇)²+^∞_𝑛=3Δ_𝑛𝑛!(𝑧−𝜇)^𝑛. (4.8)