`Abstract and Applied AnalysisVolume 2012 (2012), Article ID 180595, 13 pageshttp://dx.doi.org/10.1155/2012/180595`
Research Article

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

School of Mathematics, Chongqing Normal University, Chongqing 401331, China

Received 19 July 2012; Accepted 23 October 2012

Copyright © 2012 Houyu Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the existence of analytic solutions of a functional differential equation 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 at resonance, that is, at a root of the unity, but also those near resonance under the Brjuno condition.

1. Introduction

Traffic flow models have found much attention [113] 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 [25, 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 where , 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 , then , and (1.1) can be written in As in [16, 17], we reduce (1.2) with to the auxiliary equation 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 , and in (1.3) satisfies one of the following conditions:(C1); (C2), , and is a Brjuno number [18, 19]: , where denotes the sequence of partial fraction of the continued fraction expansion of ;(C3) for some integer with and , and for all and .

We observe that is inside the unit circle in case (C1) but on in the rest of cases. More difficulties are encountered for on since the small divisor is involved in the latter (2.24). Under Diophantine condition, “, where and there exist constants and such that for all ,” the number is “far” from all roots of the unity. Since then, we have been striving to give a result of analytic solutions for those “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

Lemma 2.1. Equation (1.3) has a formal solution of the form where is as in (2.1).

Proof. If we let and substituting into (1.3), we have Comparing coefficients we obtain Then for arbitrarily chosen , the sequence 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 There exists such that , for all . It follows from (2.5) that for .
We consider the implicit function equation: Define the function for from a neighborhood of , then the function satisfies In view of , and the implicit function theorem, there exists a unique function , analytic in a neighborhood of zero, such that and . According to (2.10), we have .
If we assume that the power series expansion of is as follows: substituting the series in (2.10) and comparing coefficients, we obtain and From (2.7) we obtain immediately that for all 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 , we let denote its integer part and its fractional part. Then every irrational number has a unique expression of the Gauss' continued fraction: denoted simply by , where 's and 's are calculated by the algorithm: (a) , , and (b) ,   for all . Define the sequences and as follows: It is easy to show that . Thus, for every we associate, using its convergence, an arithmetical function . We say that is a Brjuno number or that it satisfies Brjuno condition if . The Brjuno condition is weaker than the Diophantine condition. For example, if for all , where is a constant, then is a Brjuno number but is not a Diophantine number. So the case (C2) contains both Diophantine condition and a part of “near” resonance. Let and be the sequence of partial denominators of the Gauss's continued fraction for . As in [20], let Let be the set of integers such that either or for some and in , with , one has and divides . For any integer , define where . We then define function as follows: Let , and define by the condition . Clearly, is nondecreasing. Then we are able to state the following result.

Lemma 2.3 (Davie's lemma [21]). Let . Then(a)there is a universal constant (independent of and ) such that (b) for all and , and(c).

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 for .
To construct a majorant series, we consider the implicit functional equation: where is defined in (2.9) and . Similarly to the proof of Theorem 2.2, using the implicit function theorem we can prove that (2.22) has a unique analytic solution in a neighborhood of the origin such that ,   and . Thus in (2.22) can be expanded into a convergent series: in a neighborhood of the origin. Replacing (2.23) into (2.22) and comparing coefficients, we obtain that and Note that the series (2.23) converges in a neighborhood of the origin. Now, we can deduce, by induction, that for , where is defined in Lemma 2.3.
In fact, . For inductive proof we assume that , for . From (2.21) we know
Note that Then from Lemma 2.3, we have Since is convergent in a neighborhood of the origin, there exists a constant such that Moreover, from Lemma 2.3, we know that for some universal constant . Then that is, This implies that the convergence radius of (2.2) is at least . This completes the proof.

In the case (C3) both the Diophantine condition and Brjuno condition are not satisfied. We need to define a sequence by and where , and is defined in (C3).

Theorem 2.5. Assume that (C3) holds. Let be determined by and where If for all , then (1.3) has an analytic solution of the form in a neighborhood of the origin, where all are arbitrary constants satisfying the inequality and the sequence is defined in (2.31). Otherwise, if for some , 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 of (1.3); we also have (2.5) or (2.32). If for some natural number , then the equality in (2.32) does not hold for since . In such a circumstance (1.3) has no formal solutions.
If for all natural numbers , then there are infinitely many choices of corresponding in (2.32) and the formal solutions (2.2) form a family of functions of infinitely many parameters. We can arbitrarily choose such that ,  . In what follows we prove that the formal solution (2.2) converges in a neighborhood of the origin. First of all, note that , for . It follows from (2.32) that for all ,  . Further, we can prove that In fact, for inductive proof we assume that for all . When , we have . On the other hand, when , from (2.36) we get as desired. Set It is easy to check that (2.38) satisfies where the function 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 in a neighborhood of the origin such that and . 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 in a neighborhood of the origin, where is an analytic solutions of (1.3) satisfying the initial conditions (2.1).

Proof. In a view of Theorems 2.22.5, we may find an analytic solution of the auxiliary equation (1.3) in the form of (2.2) such that and . Clearly the inverse exists and is analytic in a neighborhood of the . Define Then is invertible analytic in a neighborhood of . From (3.2) it is easy to see
From (1.3), we have 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 in a neighborhood of the number , where is an analytic solution of (1.3). Since the function in (2.2) can be determined by (2.5), it is possible to calculate, at least in theory, the explicit form of , an analytic solution of (1.3), in a neighborhood of the fixed point of . However, knowing that an analytic solution of (1.3) exists, we can take an alternative route as follows.

Example 3.2. Consider where If taking , then by (1.2) and (1.3), we have and the auxiliary equation is

From Lemma 2.1, is given arbitrarily, and ,   can be determined by

Now, by (3.9), Because , , and the inverse is analytic near the origin, we can calculate that Now, we have

Thus, near the origin, (3.7) has an analytic solution: 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 and obtain a solution of the form of (2.2) with real coefficients. Restricted on both the function and its inverse are valued in . Hence, the function 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).

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