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International Journal of Differential Equations
Volume 2009 (2009), Article ID 145213, 14 pages
Analytic Solutions of an Iterative Functional Differential Equation near Resonance
Department of Mathematics, Shandong University of Technology, Zibo, Shandong 255049, China
Received 6 January 2009; Accepted 14 April 2009
Academic Editor: Peiguang Wang
Copyright © 2009 Tongbo Liu and Hong Li. 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.
We investigate the existence of analytic solutions of a class of second-order differential equations involving iterates of the unknown function in the complex field . By reducing the equation with the Schr####^~^~^~^~^~^####xF6;der transformation to the another functional differential equation without iteration of the unknown function + = , we get its local invertible analytic solutions.
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####^~^~^~^~^~^####x2013;4] ). However, there are only a few papers dealing with functional differential equation with state derivative dependent delay. In  Eder studied the functional differential equation . V. R. Petahov  proved the existence of solutions of equation In [7####^~^~^~^~^~^####x2013;9], the authors studied the existence of analytic solutions of the following second-order iterative functional differential equations: 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: where and are complex numbers. When , (1.3) change into functional differential equation it has been studied in . 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 , the author reduces again this iterative functional differential equation with the Shchr####^~^~^~^~^~^####xf6;der transformation, that is, to a functional differential equation with proportional delay (which is called the auxiliary equation). Lastly, according to the position of an indeterminate constant 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 .
2. Explicit Analytic Solutions
In case (1.3) changes into functional differential equation For the above equation we have the following proposition.
Proposition 2.1. Suppose . Then (2.1) has an analytic solution in a neighborhood of the origin, satisfying and
be the expansion of formal solution of (2.1). Substituting (2.2) into (2.1), we get
By comparing the coefficients, we have
In view of and is complex number, thus there exists a positive number , so that
Thus if we define recursively a sequence by
then one can show that by induction
Now if we define
for from a neighborhood of . Since , there exists a unique function , analytic on a neighborhood of zero, such that and satisfying the equality . According to (2.8) and (2.10), we have . 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
3. Analytic Solutions of the Auxiliary Equation
A distinctive feature of the (1.3) when is that the argument of the unknown function is dependent on the state , 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 Then for any number , we have and so Therefore, in view of (1.3) and , we have
To find analytic solution of (3.3), we first seek an analytic solution of the auxiliary equation satisfying the initial value conditions where and are complex numbers, and satisfies one of the following conditions:(H1)(H2), where is a Brjuno number, that is, where denotes the sequence of partial fraction of the continued fraction expansion of (H3) for some integers with and and for all and
We observe that is inside the unit circle in case (H1) but on in the rest cases. More difficulties are encountered for on since the small divisor is involved in the latter (3.10). Under Diophantine condition: ####^~^~^~^~^~^####x201c; where and there exist constants and such that for all ####^~^~^~^~^~^####x201d; the number is ####^~^~^~^~^~^####x201c;far####^~^~^~^~^~^####x201d; from all roots of the unity and was considered in different settings [7####^~^~^~^~^~^####x2013;12]. Since then, we have been striving to give a result of analytic solutions for those ####^~^~^~^~^~^####x201c;near####^~^~^~^~^~^####x201d; 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).
Proof. Rewrite (3.4) in the form or Therefore, in view of , we obtain We now seek a solution of (3.4) in the form of a power series (3.6). By defining and and then substituting (3.6) into (3.9), we see that the sequence is successively determined by the condition 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 , thus there exists a positive number , such that If we define a sequence by and then in view of (3.10), we can show by induction that Now if we define then We get immediately That is, Let for from a neighborhood of . Since and , and the implicit function theorem, there exists a unique function , analytic in a neighborhood of zero, such that and . According to (3.14) and (3.17), we have . 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 , 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####^~^~^~^~^~^####x2019; continued fraction denoted simply by , where ####^~^~^~^~^~^####x2019;s and ####^~^~^~^~^~^####x2019;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 (H2) contains both Diophantine condition and a part of ####^~^~^~^~^~^####x201c;near####^~^~^~^~^~^####x201d; resonance. Let and be the sequence of partial denominators of the Gauss####^~^~^~^~^~^####x2019;s continued fraction for As in , 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. Now we are able to state the following result.
Lemma 3.2 (Davie####^~^~^~^~^~^####x2019;s lemma ). Let Then(a)there is a universal constant (independent of and ) such that (b) for all and ;(c)
Proof. As in the proof of Theorem 3.1, we seek a power series solution of the form (3.6). Set and , (3.10) holds again. From (3.10) we get
To construct a governing series of (3.6), we consider the implicit functional equation where 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 in a neighborhood of the origin such that and . Thus in (3.26) can be expanded into a convergent series in a neighborhood of the origin. Replacing (3.27) into (3.26) and comparing coefficients, we obtain that and
Note the power series (3.27) converges in a neighborhood of zero. Hence there is a positive constant such that Now by induction, we prove that where is defined in Lemma 3.2. In fact, For inductive proof, we assume that . From (3.10) and Lemma 3.2 ,we know Note that Hence as desired. In view of (3.29) and Lemma 3.2, we know that for some universal constant . Then That is, This implies that the convergence radius of (3.6) is at least . This concludes the proof.
In case (H3), the constant is not only the unit circle in , but also a root of unity. In such a case, the resonant case, both the Diophantine and the Brjuno conditions are not satisfied. Let be a sequence define by and where and is defined in Theorem 3.1.
Theorem 3.4. Suppose that (H3) holds, let be determined by and where If then (3.4) has an analytic solution in a neighborhood of the origin such that and where all ####^~^~^~^~^~^####x2019;s are arbitrary constants satisfying the inequality and sequence is defined in (3.36). Otherwise, if then (3.4) has no any analytic solution 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 then the equality in (3.37) does not hold for , since. In such a circumstance (3.4) has no formal solutions.
If for all natural numbers , the corresponding in (3.29) has infinitely many choices in ; this is, the formal series solution (3.6) defines a family of solutions with infinitely many parameters. Choose arbitrarily such that where 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 for . It follows from (3.29) that for Further, we can prove that Set It is easy to check that (3.42) satisfies the implicit functional equation where 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 in a neighborhood of the origin such that , and . 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 in a neighborhood of the number , where 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 such that the function of the form (3.6) is an analytic solution of (3.4) in a neighborhood of the origin. Since , the function is analytic in a neighborhood of . If we now define then then 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 in a neighborhood of the number , where is an solution of (3.4). Since the function can be determined by (3.10), it is possible to calculate, at last in theory, the explicit form of , an analytic solution of (1.3), in a neighborhood of the fixed point of 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 is of the form we need to determine the derivatives , First of all, in view of (1.3) and (3.2), we have respectively. Furthermore, Next by calculating the derivatives of both sides of (1.3), we obtain successively so that
Recall the formula for the higher derivatives of composition. Namely, for where and is a polynomial with nonnegative coefficients.
We have Thus the desired solutions is
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