Analytic Solutions for a Functional Differential Equation Related to a Traffic Flow Model

Abstract

We study the existence of analytic solutions of a functional differential equation (𝑧(𝑠)+𝛼)²𝑧^′(𝑠)=𝛽(𝑧(𝑠+𝑧(𝑠))−𝑧(𝑠)) which comes from traffic flow model. By reducing the equation with the Schröder transformation to an auxiliary equation, the author discusses not only that the constant 𝜆 at resonance, that is, at a root of the unity, but also those 𝜆 near resonance under the Brjuno condition.

1. Introduction

Traffic flow models have found much attention [1–13] in the last few years. They mostly fall into two types: one is “macroscopic” which was introduced by Aw and Rascle [14], and Zhang [13], we also refer the reader to [2–5, 9, 12], and the other is called “microscopic” which has been discussed in [8, 10, 11, 15]. In particular, Illner et al. [4, 7] investigated kinetic models which can be seen as a bridge between macroscopic and microscopic models.

Recently, Illner and McGregor [6] studied (𝑧(𝑠)+𝛼)²𝑧^′(𝑠)=𝛽(𝑧(𝑠+𝑧(𝑠))−𝑧(𝑠)), (1.1) where 𝛼, 𝛽 are positive parameters arising from a nonlocal traffic flow model in a travelling wave approximation. Analytical and numerical studies of (1.1) exist, in particular on the existence and properties of nonconstant travelling wave solutions.

In this paper, we prove the existence of analytic solutions for (1.1) by locally reducing the equation to another functional differential equation, which we called auxiliary equation. In fact, if we let 𝑔(𝑠)=𝑠+𝑧(𝑠), then 𝑧(𝑠)=𝑔(𝑠)−𝑠, and (1.1) can be written in 𝑔(𝑠)−𝑠+𝛼²𝑔^′(𝑠)−1=𝛽𝑔𝑔(𝑠)−2𝑔(𝑠)+𝑠. (1.2) As in [16, 17], we reduce (1.2) with 𝑔(𝑠)=ℎ(𝜆ℎ⁻¹(𝑠)) to the auxiliary equation (ℎ(𝜆𝑠)−ℎ(𝑠)+𝛼)²ℎ^′(𝜆𝑠)−ℎ^′(𝑠)=𝛽ℎ^′(𝑠)ℎ𝜆²𝑠−2ℎ(𝜆𝑠)+ℎ(𝑠),𝑠∈ℂ. (1.3) If we can prove the existence of analytic solutions for (1.3), then the analytic solutions of (1.1) can be obtained.

Throughout this paper, we will assume that 𝛼,𝛽>0, and 𝜆 in (1.3) satisfies one of the following conditions:(C1)0<|𝜆|<1; (C2)𝜆=𝑒^{2𝜋𝑖𝜃}, 𝜃∈ℝ⧵ℚ, and 𝜃 is a Brjuno number [18, 19]: 𝐵(𝜃)=∑^∞_𝑛=0(log𝑞_𝑛+1/𝑞_𝑛)<∞, where {𝑝_𝑛/𝑞_𝑛} denotes the sequence of partial fraction of the continued fraction expansion of 𝜃;(C3)𝜆=𝑒^{2𝜋𝑖𝑞/𝑝} for some integer 𝑝∈ℕ with 𝑝≥2 and 𝑞∈ℤ⧵{0}, and 𝜆≠𝑒^{2𝜋𝑖𝜉/𝑣} for all 1≤𝑣≤𝑝−1 and 𝜉∈ℤ⧵{0}.

We observe that 𝜆 is inside the unit circle 𝑆¹ in case (C1) but on 𝑆¹ in the rest of cases. More difficulties are encountered for 𝜆 on 𝑆¹ since the small divisor 𝜆^𝑛−1 is involved in the latter (2.24). 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. 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 (C2) provides such a chance for us. Moreover, we also discuss the so-called resonance case, that is, the case of (C3).

2. Analytic Solutions of the Auxiliary Equation

In this section, we discuss local invertible analytic solutions of (1.3) with the initial condition ℎ(𝑠)=^∞_𝑛=1𝑎_𝑛𝑠^𝑛,ℎ(0)=0,ℎ^′(0)=𝜂≠0,𝜂∈ℂ. (2.1)

Lemma 2.1. Equation (1.3) has a formal solution of the form ℎ(𝑠)=𝜂𝑠+^∞_𝑛=2𝑎_𝑛𝑠^𝑛, (2.2) where 𝜂 is as in (2.1).

Proof. If we let ℎ(𝑠)=∑^∞_𝑛=1𝑎_𝑛𝑠^𝑛 and substituting into (1.3), we have 𝛼^2∞_𝑛=0(𝑛+1)𝜆^𝑛−1𝑎_𝑛+1𝑠^𝑛+2𝛼^∞_𝑛=1^𝑛−1_𝑖=0(𝑖+1)𝜆^𝑛−𝑖−1𝜆^𝑖−1𝑎_𝑛−𝑖𝑎_𝑖+1𝑠^𝑛+^∞_𝑛=2^𝑛−2_𝑖=0^{𝑛−𝑖−1}_𝑘=1(𝑖+1)𝜆^{𝑛−𝑘−𝑖}−1𝜆^𝑘−1𝜆^𝑖−1𝑎_{𝑛−𝑘−𝑖}𝑎_𝑘𝑎_𝑖+1𝑠^𝑛=𝛽^∞_𝑛=1^𝑛−1_𝑖=0(𝑖+1)𝜆^𝑛−𝑖−1²𝑎_𝑛−𝑖𝑎_𝑖+1𝑠^𝑛. (2.3) Comparing coefficients we obtain 𝛼²𝜆⁰−1𝑎₁𝛼=0,₂(𝑛+1)𝜆^𝑛−1𝑎_𝑛+1=𝛽^𝑛−1_𝑖=0(𝑖+1)𝜆^𝑛−𝑖−1²𝑎_𝑛−𝑖𝑎_𝑖+1−2𝛼^𝑛−1_𝑖=0(𝑖+1)𝜆^𝑛−𝑖−1𝜆^𝑖−1𝑎_𝑛−𝑖𝑎_𝑖+1−^𝑛−2_𝑖=0^{𝑛−𝑖−1}_𝑘=1(𝑖+1)𝜆^{𝑛−𝑘−𝑖}−1𝜆^𝑘−1𝜆^𝑖−1𝑎_{𝑛−𝑘−𝑖}𝑎_𝑘𝑎_𝑖+1,𝑛≥1. (2.4)(2.5) Then for arbitrarily chosen 𝑎₁=𝜂≠0, the sequence {𝑎_𝑛}^∞_𝑛=2 is successively determined by (2.5) in a unique manner.

This shows that (1.3) has a formal power series solution of the form (2.2).

Theorem 2.2. Suppose that (C1) holds, then (1.3) in a neighborhood of the origin has an analytic solution of the form (2.2).

Proof. From (C1), we have lim_𝑛→∞1𝜆^𝑛−1=1. (2.6) There exists 𝐿>0 such that 1/|𝜆^𝑛−1|≤𝐿, for all 𝑛≥1. It follows from (2.5) that 𝑎_𝑛+1≤8𝐿𝛼²𝛼+𝛽^𝑛−1_𝑖=0𝑎_𝑛−𝑖𝑎_𝑖+1+^𝑛−2_𝑖=0^{𝑛−𝑖−1}_𝑘=1𝑎_{𝑛−𝑘−𝑖}𝑎_𝑘𝑎_𝑖+1 (2.7) for 𝑛≥1.
We consider the implicit function equation: 𝐵(𝑠)=𝜂𝑠+8𝐿𝛼²𝛼+𝛽𝐵²(𝑠)+𝐵³(𝑠). (2.8) Define the function Θ𝑠,𝜔;𝐿,𝜂,𝛼,𝛽=𝜂𝑠−𝜔+8𝐿𝛼²𝛼+𝛽𝜔²+𝜔³ (2.9) for (𝑠,𝜔) from a neighborhood of (0,0), then the function 𝐵(𝑠) satisfies Θ𝑠,𝐵(𝑠);𝐿,𝜂,𝛼,𝛽=0. (2.10) In view of Θ(0,0;𝐿,𝜂,𝛼,𝛽)=0, Θ^′_𝜔0,0;𝐿,𝜂,𝛼,𝛽=−1≠0, (2.11) and the implicit function theorem, there exists a unique function Φ(𝑠), analytic in a neighborhood of zero, such that Φ(0)=0,Φ^′(0)Θ=−^′𝑠0,0;𝐿,𝜂,𝛼,𝛽Θ^′_𝜔=𝜂,0,0;𝐿,𝜂,𝛼,𝛽 (2.12) and Θ(𝑠,Φ(𝑠);𝐿,𝜂,𝛼,𝛽)=0. According to (2.10), we have 𝐵(𝑠)=Φ(𝑠).
If we assume that the power series expansion of 𝐵(𝑠) is as follows: 𝐵(𝑠)=^∞_𝑛=1𝐵_𝑛𝑠^𝑛,𝐵₁=𝜂, (2.13) substituting the series in (2.10) and comparing coefficients, we obtain 𝐵₁=|𝜂| and 𝐵_𝑛+1=8𝐿𝛼²𝛼+𝛽^𝑛−1_𝑖=0𝐵_𝑛−𝑖𝐵_𝑖+1+^𝑛−2_𝑖=0^{𝑛−𝑖−1}_𝑘=1𝐵_{𝑛−𝑘−𝑖}𝐵_𝑘𝐵_𝑖+1,𝑛≥2. (2.14) From (2.7) we obtain immediately that |𝑎_𝑛|≤𝐵_𝑛 for all 𝑛 by induction. This implies that (2.2) converges in a neighborhood of the origin. This completes the proof.

Next we devote to the existence of analytic solutions of (1.3) under the Brjuno condition. First, we recall briefly the definition of Brjuno numbers and some basic facts. As stated in [20], 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=⋯, (2.15) 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. (2.16) 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≥0(log𝑞_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 (C2) contains both Diophantine condition and a part of 𝜆 “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 [20], let 𝐴_𝑘=‖‖≤1𝑛≥0∣𝑛𝜃8𝑞_𝑘,𝐸_𝑘=max𝑞_𝑘,𝑞_𝑘+14,𝜂_𝑘=𝑞_𝑘𝐸_𝑘. (2.17) 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)=max1+𝜂_𝑘𝑛𝑞_𝑘−2,𝑚_𝑛𝜂_𝑘1+𝑛𝑞_𝑘−1, (2.18) where 𝑚_𝑛=max{𝑗∣0≤𝑗≤𝑛,𝑗∈𝐴^∗_𝑘}. We then define function ℎ_𝑘∶ℕ→ℝ₊ as follows: ℎ_𝑘(𝑛)=𝑚_𝑛+𝜂_𝑘𝑛𝑞_𝑘−1,if𝑚_𝑛+𝑞_𝑘∈𝐴^∗_𝑘,𝑙_𝑘(𝑛),if𝑚_𝑛+𝑞_𝑘∉𝐴^∗_𝑘. (2.19) 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. Then we are able to state the following result.

Lemma 2.3 (Davie's lemma [21]). Let 𝐾(𝑛)=𝑛log2+∑^𝑘(𝑛)_𝑘=0𝑔_𝑘(𝑛)log(2𝑞_𝑘+1). Then(a)there is a universal constant 𝜚>0 (independent of 𝑛 and 𝜃) such that 𝐾(𝑛)≤𝑛^𝑘(𝑛)_𝑘=0log𝑞_𝑘+1𝑞_𝑘+𝜚, (2.20)(b)𝐾(𝑛₁)+𝐾(𝑛₂)≤𝐾(𝑛₁+𝑛₂) for all 𝑛₁ and 𝑛₂, and(c)−log|𝜆^𝑛−1|≤𝐾(𝑛)−𝐾(𝑛−1).

Theorem 2.4. Suppose that (C2) holds, then (1.3) has an analytic solution of the form (2.2) in a neighborhood of the origin.

Proof. As in the Theorem 2.2, we seek a power series solution of the form (2.2). First, we have 𝑎_𝑛+1≤1𝜆^𝑛8−1𝛼²𝛼+𝛽^𝑛−1_𝑖=0𝑎_𝑛−𝑖𝑎_𝑖+1+^𝑛−2_𝑖=0^{𝑛−𝑖−1}_𝑘=1𝑎_{𝑛−𝑘−𝑖}𝑎_𝑘𝑎_𝑖+1 (2.21) for 𝑛≥2.
To construct a majorant series, we consider the implicit functional equation: Θ𝑠,𝜓;𝐿,𝜂,𝛼,𝛽=0, (2.22) where Θ is defined in (2.9) and 𝐿=1. Similarly to the proof of Theorem 2.2, using the implicit function theorem we can prove that (2.22) has a unique analytic solution 𝜓(𝑠) in a neighborhood of the origin such that 𝜓(0)=0,  𝜓^′(0)=|𝜂| and Θ(𝑠,𝜓(𝑠);𝐿,𝜂,𝛼,𝛽)=0. Thus 𝜓(𝑠) in (2.22) can be expanded into a convergent series: 𝜓(𝑠)=^∞_𝑛=1𝐵_𝑛𝑠^𝑛, (2.23) in a neighborhood of the origin. Replacing (2.23) into (2.22) and comparing coefficients, we obtain that 𝐵₁=|𝜂| and 𝐵_𝑛=8𝛼²𝛼+𝛽^𝑛−1_𝑖=0𝐵_𝑛−𝑖𝐵_𝑖+1+^𝑛−2_𝑖=0^{𝑛−𝑖−1}_𝑘=1𝐵_{𝑛−𝑘−𝑖}𝐵_𝑘𝐵_𝑖+1,𝑛≥2. (2.24) Note that the series (2.23) converges in a neighborhood of the origin. Now, we can deduce, by induction, that |𝑎_𝑛|≤𝐵_𝑛𝑒^{𝐾(𝑛−1)} for 𝑛≥1, where 𝐾∶ℕ→ℝ is defined in Lemma 2.3.
In fact, |𝑎₁|=|𝜂|=𝐵₁. For inductive proof we assume that |𝑎_𝑗|≤𝐵_𝑗𝑒^{𝐾(𝑗−1)}, for 𝑗=1,2,…,𝑛. From (2.21) we know 𝑎_𝑛+1≤1𝜆^𝑛8−1𝛼²𝛼+𝛽^𝑛−1_𝑖=0𝑎_𝑛−𝑖𝑎_𝑖+1+^𝑛−2_𝑖=0^{𝑛−𝑖−1}_𝑘=1𝑎_{𝑛−𝑘−𝑖}𝑎_𝑘𝑎_𝑖+1≤1𝜆^𝑛8−1𝛼²𝛼+𝛽^𝑛−1_𝑖=0𝐵_𝑛−𝑖𝐵_𝑖+1𝑒^{𝐾(𝑛−𝑖−1)+𝐾(𝑖)}+^𝑛−2_𝑖=0^{𝑛−𝑖−1}_𝑘=1𝐵_{𝑛−𝑘−𝑖}𝐵_𝑘𝐵_𝑖+1𝑒^{𝐾(𝑛−𝑘−𝑖−1)+𝐾(𝑘−1)+𝐾(𝑖)}. (2.25)
Note that 𝐾()(𝑖)(),𝐾()()(𝑖)().𝑛−𝑖−1+𝐾≤𝐾𝑛−1𝑛−𝑘−𝑖−1+𝐾𝑘−1+𝐾≤𝐾𝑛−1 (2.26) Then from Lemma 2.3, we have 𝑎_𝑛+1≤𝑒^{𝐾(𝑛−1)}𝜆^𝑛8−1𝛼²𝛼+𝛽^𝑛−1_𝑖=0𝐵_𝑛−𝑖𝐵_𝑖+1+^𝑛−2_𝑖=0^{𝑛−𝑖−1}_𝑘=1𝐵_{𝑛−𝑘−𝑖}𝐵_𝑘𝐵_𝑖+1,𝑛≥1. (2.27) Since ∑^∞_𝑛=1𝐵_𝑛𝑠^𝑛 is convergent in a neighborhood of the origin, there exists a constant Λ>0 such that 𝐵_𝑛<Λ^𝑛,𝑛≥1. (2.28) Moreover, from Lemma 2.3, we know that 𝐾(𝑛)≤𝑛(𝐵(𝜃)+𝜚) for some universal constant 𝜚>0. Then 𝑎_𝑛≤𝐵_𝑛𝑒^{𝐾(𝑛−1)}≤Λ^𝑛𝑒^{(𝑛−1)(𝐵(𝜃)+𝜚)}, (2.29) that is, lim_𝑛→∞sup𝑎_𝑛^1/𝑛≤lim_𝑛→∞supΛ^𝑛𝑒^{(𝑛−1)(𝐵(𝜃)+𝜚)1/𝑛}=Λ𝑒^{𝐵(𝜃)+𝜚}. (2.30) This implies that the convergence radius of (2.2) is at least (Λ𝑒^{𝐵(𝜃)+𝜚})⁻¹. This completes the proof.

In the case (C3) both the Diophantine condition and Brjuno condition are not satisfied. We need to define a sequence {𝐶_𝑛}^∞_𝑛=1 by 𝐶₁=|𝜂| and 𝐶_𝑛+1=8Γ𝛼²𝛼+𝛽^𝑛−1_𝑖=0𝐶_𝑛−𝑖𝐶_𝑖+1+^𝑛−2_𝑖=0^{𝑛−𝑖−1}_𝑘=1𝐶_{𝑛−𝑘−𝑖}𝐶_𝑘𝐶_𝑖+1,𝑛≥1, (2.31) where Γ∶=max{1,1/|1−𝜆|,1/|1−𝜆²|,…,1/|1−𝜆^(𝑝−1)|}, and 𝑝 is defined in (C3).

Theorem 2.5. Assume that (C3) holds. Let {𝑎_𝑛}^∞_𝑛=0 be determined by 𝑎₁=𝜂 and 𝛼²(𝑛+1)𝜆^𝑛−1𝑎_𝑛+1=Ξ(𝑛,𝜆),𝑛≥1, (2.32) where Ξ(𝑛,𝜆)=𝛽^𝑛−1_𝑖=0(𝑖+1)𝜆^𝑛−𝑖−1²𝑎_𝑛−𝑖𝑎_𝑖+1−2𝛼^𝑛−1_𝑖=0(𝑖+1)𝜆^𝑛−𝑖−1𝜆^𝑖−1𝑎_𝑛−𝑖𝑎_𝑖+1−^𝑛−2_𝑖=0^{𝑛−𝑖−1}_𝑘=1(𝑖+1)𝜆^{𝑛−𝑘−𝑖}−1𝜆^𝑘−1𝜆^𝑖−1𝑎_{𝑛−𝑘−𝑖}𝑎_𝑘𝑎_𝑖+1. (2.33) If Ξ(𝑙𝑝,𝜆)=0 for all 𝑙=1,2,…, then (1.3) has an analytic solution of the form ℎ(𝑠)=𝜂𝑠+_{𝑛=𝑙𝑝,𝑙∈ℕ}𝜇_𝑙𝑝+1𝑠^𝑙𝑝+1+_{𝑛≠𝑙𝑝,𝑙∈ℕ}𝑎_𝑛+1𝑠^𝑛+1,ℕ={1,2,3,…} (2.34) in a neighborhood of the origin, where all 𝜇^,_𝑙𝑝+1𝑠 are arbitrary constants satisfying the inequality |𝜇_𝑙𝑝+1|≤𝐶_𝑙𝑝+1 and the sequence {𝐶_𝑛}^∞_𝑛=1 is defined in (2.31). Otherwise, if Ξ(𝑙𝑝,𝛼)≠0 for some 𝑙=1,2,…, then (1.3) has no analytic solutions in any neighborhood of the origin.

Proof. Analogously to the proof of Lemma 2.1, let (2.2) be the expansion of a formal solution ℎ(𝑠) of (1.3); we also have (2.5) or (2.32). If Ξ(𝑙𝑝,𝜆)≠0 for some natural number 𝑙, then the equality in (2.32) does not hold for 𝑛=𝑙𝑝 since 𝜆^𝑙𝑝−1=0. In such a circumstance (1.3) has no formal solutions.
If Ξ(𝑙𝑝,𝜆)=0 for all natural numbers 𝑙, then there are infinitely many choices of corresponding 𝑎_𝑙𝑝+1 in (2.32) and the formal solutions (2.2) form a family of functions of infinitely many parameters. We can arbitrarily choose 𝑎_𝑙𝑝+1=𝜇_𝑙𝑝+1 such that |𝜇_𝑙𝑝+1|≤𝐶_𝑙𝑝+1,  𝑙=1,2,…. In what follows we prove that the formal solution (2.2) converges in a neighborhood of the origin. First of all, note that |𝜆^𝑛−1|⁻¹≤Γ, for 𝑛≠𝑙𝑝. It follows from (2.32) that 𝑎_𝑛+1≤8Γ𝛼²𝛼+𝛽^𝑛−1_𝑖=0𝑎_𝑛−𝑖𝑎_𝑖+1+^𝑛−2_𝑖=0^{𝑛−𝑖−1}_𝑘=1𝑎_{𝑛−𝑘−𝑖}𝑎_𝑘𝑎_𝑖+1, (2.35) for all 𝑛≠𝑙𝑝,  𝑙=1,2,…. Further, we can prove that 𝑎_𝑛≤𝐶_𝑛,𝑛=1,2,…. (2.36) In fact, for inductive proof we assume that |𝑎_𝑟|≤𝐶_𝑟 for all 1≤𝑟≤𝑛. When 𝑛=𝑙𝑝, we have |𝑎_𝑛+1|=|𝜇_𝑛+1|≤𝐶_𝑛+1. On the other hand, when 𝑛≠𝑙𝑝, from (2.36) we get 𝑎_𝑛+1≤8Γ𝛼²𝛼+𝛽^𝑛−1_𝑖=0𝐶_𝑛−𝑖𝐶_𝑖+1+^𝑛−2_𝑖=0^{𝑛−𝑖−1}_𝑘=1𝐶_{𝑛−𝑘−𝑖}𝐶_𝑘𝐶_𝑖+1=𝐶_𝑛+1 (2.37) as desired. Set 𝐹(𝑠)=^∞_𝑛=1𝐶_𝑛𝑠^𝑛,𝐶₁=𝜂. (2.38) It is easy to check that (2.38) satisfies Θ𝑥,𝐹;Γ,𝜂,𝛼,𝛽=0, (2.39) where the function Θ is defined in (2.9). Moreover, similarly to the proof of Theorem 2.2, we can prove that (2.39) has a unique analytic solution 𝐹(𝑠) in a neighborhood of the origin such that 𝐹(0)=0 and 𝐹^′(0)=|𝜂|≠0. Thus, (2.38) converges in a neighborhood of the origin. By the convergence of (2.38) and inequality (2.36), the series (2.2) converges in a neighborhood of the origin. This completes the proof.

3. Analytic Solutions of (1.2)

Theorem 3.1. Suppose that conditions of Theorems 2.2, 2.4, or 2.5 are fulfilled. Then (1.2) has an invertible analytic solution of the form 𝑔(𝑠)=ℎ𝜆ℎ⁻¹(𝑠) (3.1) in a neighborhood of the origin, where ℎ(𝑠) is an analytic solutions of (1.3) satisfying the initial conditions (2.1).

Proof. In a view of Theorems 2.2–2.5, we may find an analytic solution ℎ(𝑠) of the auxiliary equation (1.3) in the form of (2.2) such that ℎ(0)=0 and ℎ^′(0)=𝜂≠0. Clearly the inverse ℎ⁻¹(𝑠) exists and is analytic in a neighborhood of the ℎ(0)=0. Define 𝑔(𝑠)∶=ℎ𝜆ℎ⁻¹(𝑠). (3.2) Then 𝑔(𝑠) is invertible analytic in a neighborhood of 𝑠=0. From (3.2) it is easy to see 𝑔(0)=ℎ𝜆ℎ⁻¹(0(0)𝑔)=ℎ=0,_′(0)=𝜆ℎ^′𝜆ℎ⁻¹(0)ℎ^−1′(0)=𝜆ℎ^′𝜆ℎ⁻¹(0)ℎ^′ℎ⁻¹(0)=𝜆ℎ^′(0)ℎ^′(0)=𝜆≠0. (3.3)
From (1.3), we have 𝑔(𝑠)−𝑠+𝛼²𝑔^′(𝑠)=ℎ−1𝜆ℎ⁻¹(𝑠)−ℎℎ⁻¹(𝑠)+𝛼²ℎ^′𝜆ℎ⁻¹(𝑠)ℎ^′ℎ⁻¹(𝑠)ℎ𝜆−1=𝛽²ℎ⁻¹(𝑠)−2ℎ𝜆ℎ⁻¹(𝑠)+ℎℎ⁻¹(𝑠𝑔𝑔(𝑠)(𝑠))=𝛽−2𝑔+𝑠 (3.4) as required. This completes the proof.

By Theorem 3.1, we have shown that under the conditions of Theorems 2.2, 2.4, or 2.5, (1.2) has an analytic solution 𝑔(𝑠)=ℎ(𝜆ℎ⁻¹(𝑠)) in a neighborhood of the number 0, where ℎ(𝑠) is an analytic solution of (1.3). Since the function ℎ(𝑠) in (2.2) can be determined by (2.5), it is possible to calculate, at least in theory, the explicit form of ℎ(𝑠), an analytic solution of (1.3), in a neighborhood of the fixed point 0 of ℎ(𝑠). However, knowing that an analytic solution of (1.3) exists, we can take an alternative route as follows.

Example 3.2. Consider (𝑧(𝑠)+1)²𝑧^′(𝑠)=𝑧(𝑠+𝑧(𝑠))−𝑧(𝑠), (3.5) where 𝛼=𝛽=1. (3.6) If taking 𝜆=1/2, then by (1.2) and (1.3), we have 𝑔(𝑠)−𝑠+1²𝑔^′(𝑠)−1=𝑔𝑔(𝑠)−2𝑔(𝑠)+𝑠, (3.7) and the auxiliary equation is ℎ12𝑠−ℎ(𝑠)+1²ℎ^′12𝑠−ℎ^′(𝑠)=ℎ^′(𝑠)ℎ14𝑠1−2ℎ2𝑠+ℎ(𝑠),𝑠∈ℂ. (3.8)

From Lemma 2.1, 𝑎₁=𝜂≠0 is given arbitrarily, and 𝑎_𝑛,  𝑛≥2 can be determined by (1𝑛+1)2^𝑛−1𝑎_𝑛+1=^𝑛−1_𝑖=0()1𝑖+12^𝑛−𝑖−1²𝑎_𝑛−𝑖𝑎_𝑖+1−2^𝑛−1_𝑖=0()1𝑖+12^𝑛−𝑖1−12^𝑖−1𝑎_𝑛−𝑖𝑎_𝑖+1−^𝑛−2_𝑖=0^{𝑛−𝑖−1}_𝑘=1()1𝑖+12^{𝑛−𝑘−𝑖}1−12^𝑘1−12^𝑖−1𝑎_{𝑛−𝑘−𝑖}𝑎_𝑘𝑎_𝑖+1,𝑛≥1. (3.9)

Now, by (3.9), 𝑎₂=ℎ^′′(0)12!=−4𝜂²,𝑎₃=ℎ^′′′(0)13!=−𝑎36¹𝑎₂=1𝜂144³,⋯. (3.10) Because ℎ(0)=0, ℎ^′(0)=𝜂≠0, and the inverse ℎ⁻¹(𝑠) is analytic near the origin, we can calculate that ℎ^−1′(0)=1ℎ^′ℎ⁻¹(0)=1𝜂,ℎ^−1′′(0)ℎ=−^′′ℎ⁻¹(0)ℎ^−1′(0)ℎ^′ℎ⁻¹(0)²=1,ℎ2𝜂_{−1′′′}(0)ℎ=−^′′′ℎ⁻¹(0)ℎ^−1′(0)²+ℎ^′′ℎ⁻¹(0)ℎ^−1′′(0)ℎ^′ℎ⁻¹(0)²ℎ^′ℎ⁻¹(0)⁴+ℎ^′′ℎ⁻¹(0)ℎ^−1′(0)⋅2ℎ^′ℎ⁻¹(0)ℎ^′′ℎ⁻¹(0)ℎ^−1′(0)ℎ^′ℎ⁻¹(0)⁴=(5/24)𝜂³+(1/2)𝜂³𝜂⁴=17⋅124𝜂⋯. (3.11) Now, we have 𝑔(0)1=ℎ2ℎ⁻¹(0)=ℎ(0)=0,𝑔^′(0)=ℎ^′12ℎ⁻¹(0)12ℎ^−1′(0)=12ℎ^′(0)ℎ^−1′(0)=12,𝑔^′′(0)=14ℎ^′′12ℎ⁻¹(0)ℎ^−1′(0)²+12ℎ^′12ℎ⁻¹(0)ℎ^−1′′(0)=18,𝑔^′′′(0)=18ℎ^′′′12ℎ⁻¹(0)ℎ^−1′(0)³+12ℎ^′′12ℎ⁻¹(0)ℎ^−1′(0)ℎ^−1′′(0)+14ℎ^′′12ℎ⁻¹(0)ℎ^−1′(0)ℎ^−1′′(0)+12ℎ^′12ℎ⁻¹(0)ℎ^{−1′′′}(0)=18ℎ^′′′(0)ℎ^−1′(0)³+34ℎ^′′(0)ℎ^−1′(0)ℎ^−1′′(0)+12ℎ^′(0)ℎ^{−1′′′}(0)=11,64⋯. (3.12)

Thus, near the origin, (3.7) has an analytic solution: 𝑔(𝑠)=121𝑠−𝑠16²+11𝑠384³𝑧(𝑠)(𝑠)1+⋯,=𝑔−𝑠=−21𝑠−𝑠16²+11𝑠384³+⋯ (3.13)is the analytic solution of (3.5).

Remark 3.3. If we restrict our arguments to the real number field, then by Theorem 3.1, (1.1) has an invertible analytic real solution. We can define a real sequence {𝑏_𝑚}^∞_𝑚=0 and obtain a solution ℎ(𝑠) of the form of (2.2) with real coefficients. Restricted on ℝ both the function ℎ(𝑠) and its inverse are valued in ℝ. Hence, the function 𝑔(𝑠)=ℎ(𝜆ℎ⁻¹(𝑠)) is also a real function and Theorem 3.1 implies its invertible analyticity.

Acknowledgment

This work was partially supported by the Natural Science Foundation of Chongqing Normal University (Grant no. 12XLB003).