Exact and explicit solutions for a nonlinear extended iterative differential equation

Abstract

This paper is concerned with a nonlinear iterative functional differential equation x′(z) = 1/x(p(z) + bx′(z)). By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. We discuss not only in the general case, but also in critical cases, especially for α given in Schröder transformation is a root of the unity. And in case (H4), we dealt with the equation under the Brjuno condition, which is weaker than the Diophantine condition. Moreover, the exact and explicit solution of the original equation has been investigated for the first time. Such equations are important in both applications and the theory of iterations.

Introduction

The famous functional differential equation of the form

$$x^{\prime }(z)=f[z\text{,}x(z-\tau (z))]$$

is 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 form

$$x^{\prime }(z)=\frac{1}{x(p(z)+{\mathit{bx}}^{\prime }(z))}\text{,}z\in D\text{,}$$

where 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 = e^2πiθ, where θ ∈ R⧹Q is a Brjuno number [6], [7], [8], i.e., $\text{B}(\theta )={\sum }_{k=0}^{\infty }\frac{{\mathit{logq}}_{k+1}}{q_k}<\infty$, where {p_k/q_k} denotes the sequence of partial fraction of the continued fraction expansion of θ, said to satisfy the Brjuno condition;

• (H5) α = e^2πiq/p for some integer p ∈ N with p ⩾ 2 and q ∈ Z⧹{0}, and α ≠ e^2πil/k for all 1 ⩽ k ⩽ p − 1 and l ∈ Z⧹{0}.

Writing that

$$y(z)=p(z)+{\mathit{bx}}^{\prime }(z)\text{,}$$

so we have

$$x^{\prime }(z)=\frac{1}{b}[y(z)-p(z)]\text{,}$$

that is

$$x(z)=x(z_0)+\frac{1}{b}{\int }_{z_0}^z[y(s)-p(s)]\mathit{ds}\text{,}\forall z_0\in D\text{.}$$

From (5), we obtain

$$x(y(z))=x(z_0)+\frac{1}{b}{\int }_{z_0}^{y(z)}[y(s)-p(s)]\mathit{ds}\text{.}$$

By substituting (4), (6) into (2), we get

$$x(z_0)+\frac{1}{b}{\int }_{z_0}^{y(z)}[y(s)-p(s)]\mathit{ds}=\frac{b}{y(z)-p(z)}$$

and from (7), we can get again

$$\left[y(y(z))-p(y(z))\right][y(z)-p(z){]}^2{y}^{\prime }(z)+b^2[y^{\prime }(z)-p^{\prime }(z)]=0\text{.}$$

Let

$$y(z)=g(\alpha g^{-1}(z))\text{,}$$

then Eq. (8) may be reduced to its auxiliary equation

$$\alpha \left[g({\alpha }^{2}z)-p(g(\alpha z))\right]{\left[g(\alpha z)-p(g(z))\right]}^2{g}^{\prime }(\alpha z)+b^2\left[\alpha g^{\prime }(\alpha z)-p^{\prime }(g(z))g^{\prime }(z)\right]=0\text{,}$$

with 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.