Exact and explicit solutions for a nonlinear extended iterative differential equation☆
Introduction
The famous functional differential equation of the formis very important in both the theory of iterations and dynamical systems. Such an equation has been studied in paper [1], [2]. In [3], [4], [5], [6], [7], [8], [15], analytic solutions of the state dependent functional differential equations are found. In [9], [10], [11], [12], [13], Liu et al. studied the existence of analytic solutions of the equations x′(z) = 1/x(p(z) + bx(z)), x″(z) = ax(p(z) + bx(z)) and c0x″(z) + c1x′(z) + c2x(z) = x(az + bx(z)) + h(z), etc. Taking f(z, x) = 1/x and τ(z) = z − p(z) − bx′(z) in (1), we deduce the equation with a state derivative dependent delay of the formwhere x(z) denotes the unknown complex function, the prime is the derivative with respect to z, p(z) is a given complex function, b is a constant complex number, D = {z∣x(z) ≠ 0, z ∈ C} is the domain of definition of x(z). The purpose of this paper is to investigate the existence of analytic solutions of Eq. (2) and give its exact and explicit solution in the complex field.
Actually, Eq. (2) is very complicated for two reasons. Firstly, the right hand is an iteration of the unknown function x(z) and its derivative x′(z) rather than x(z) itself only, which is called the extended iteration sometimes, and p is any analytic function. Secondly, the domain of definition of the unknown function is not C. In other words, there are singular points for Eq. (2) in complex field C.
Clearly, Eq. (2) includes the equation x′(z) = 1/(x(az + bx′(z))) (see [6]) in case p(z) = az. In [6], Si et al. had studied the equation thoroughly and got some detailed results. But we will see that Eq. (2) differs greatly from the equation above and far more complicated than it.
We note that such iterated equations are exhibited largely in applications. For example, in many physical systems, economical and financial problems and mathematical biology modelling, etc., which exhibit a large number of iterated phenomena.
A form of functional differential equation (2) is quite different from ordinary iterative differential equation (1), and the iteration of the unknown function affects properties of solutions very much. So the known theorems of existence and uniqueness for ordinary iterative differential equations (such as Eq. (1)) cannot be applied directly. Therefore, it seems inevitable to find some or all of their solutions under appropriate conditions.
In view of Eq. (2), it is expected that additional conditions are needed to guarantee the existence of nontrivial analytic solutions. Now, we list the basic conditions as follows:
(H1) p(z) is analytic in a neighborhood of the origin, and p(0) = p0 = r ≠ 0, p′(0) = p1 = s;
(H2.1) The complex number α is not roots of the equation zn − s = 0, for all n ∈ N = {1, 2, …}, α ≠ 0 and α ≠ 1;
(H2.2) The complex number α is a root of the equation z − s = 0, but it is not roots of the equation zn − s = 0, for all n = 2, 3, …, α ≠ 0 and α ≠ 1;
(H3) α is not on the unit circle in C, more precisely, we suppose that ∣α∣ < 1;
(H4) α = e2πiθ, where θ ∈ R⧹Q is a Brjuno number [6], [7], [8], i.e., , where {pk/qk} denotes the sequence of partial fraction of the continued fraction expansion of θ, said to satisfy the Brjuno condition;
(H5) α = e2πiq/p for some integer p ∈ N with p ⩾ 2 and q ∈ Z⧹{0}, and α ≠ e2πil/k for all 1 ⩽ k ⩽ p − 1 and l ∈ Z⧹{0}.
Writing thatso we havethat isFrom (5), we obtainBy substituting (4), (6) into (2), we getand from (7), we can get againLetthen Eq. (8) may be reduced to its auxiliary equationwith the initial condition g(0) = 0, where g is the unknown complex function.
A rough description of this paper is as follows. Firstly, we will construct an analytic solution g(z) of Eq. (10). Then, based on the auxiliary equation (10), we will prove that Eq. (2) has a nontrivial analytic solution in a neighborhood of the origin. Finally, the exact and explicit solution of Eq. (2) based on the procedure of proof of Theorem 1A is investigated.
Section snippets
Analytic solution of the auxiliary equation in general case
In this section, we discuss Eq. (10) for the case where ∣α∣ < 1. Theorem 1A Assume that (H1), (H2.1) and (H3) hold. Then for any η ∈ C, the auxiliary equation (10) has an analytic solution g(z) in a neighborhood of the origin such that . Proof Clearly, in view of above assumptions, Eq. (10) has a trivial solution g(z) ≡ 0, if η = 0. So we assume that η ≠ 0. In view of (H1), we supposeSo (see [4], [11]), without loss of generality,
Analytic solutions of the auxiliary equation in critical cases
In this section, we suppose that (H2.2) holds and r = 0 in (11a). So we haveIn this case, we can getwhere .
Generally, for n = 1, 2, …, from (14) we have
The main results and remarks
Summarizing the theorems obtained in our previous discussions, we get the main result as following: Theorem 4 Suppose one of the conditions of Theorem 1A, Theorem 1B, Theorem 2, Theorem 3 is fulfilled, then Eq. (2) has an analytic solution of the formin a neighborhood of the origin, where y(z) = g(αg−1(z)) satisfies Eq. (8) and g(z) is an analytic solution of the auxiliary equation (10). Proof By Theorem 1A, Theorem 1B, Theorem 2, Theorem 3, we can find an analytic solution g(z) of
Acknowledgements
The authors are grateful to the referees and editors for their precious comments and suggestions, which helped to improve the paper greatly. We also express our sincere thanks to Professor W. Li and Professor J. Si for their enthusiastic guidance.
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This work is supported by the Natural Science Foundation of Shandong Province (No. ZR2010AM029) and the Doctoral Foundation of Binzhou University (No. 2009Y01).