Xin-Ping Wang

Analytic Solutions of a Second-order Functional Differential Equation with a State Dependent Delay

This paper is concerned with a second-order functional differential equation of the form x″(z) = x(az + bx(z)). By constructing a convergent power series solution of an auxiliary equation of the form α2y″ (αz) y′ (z) = αy′(z)y″(z)+(y′(z))3[y(α2z)−ay(αz)], analytic solutions of the form (y(αyt - 1(z)) − az)/b for the original differential equation are obtained.

Analytic solutions of a second-order iterative functional differential equation☆

Abstract This paper is concerned with a second-order iterative functional differential equation x ″( z ) = ( x m ( z )) 2 . Its analytic solutions are discussed by locally reducing the equation to another functional differential equation with proportional delays μ 2 y ″(μ z ) y ′( z ) = μ y ′(μ z ) y ″( z ) + [ y ′( z )] 3 [ y (μ m z )] 2 and by constructing a convergent power series solutions for the latter equation.

Analytic invariant curves for a planar map

Abstract In this paper, we study the existence of analytic invariant curves for two-dimensional maps of the form F(x,y) = (x + y, y + G(x) + H(x + y)).

Analytic solutions of a polynomial-like iterative functional equation

In this paper existence of local analytic solutions of a polynomial-like iterative func-tional equation is studied.As well as in previous work,we reduce this problem with the Schrder transformation to finding analytic solutions of a functional equation without iteration of the un-known function f.For technical reasons,in previous work the constant α given in the Schrder transformation,i.e.,the eigenvalue of the linearized f at its fixed point O,is required to fulfill that α is off the unit circle S1 or lies on the circle with the Diophantine condition.In this paper,we obtain results of analytic solutions in the case of α at resonance,i.e.,at a root of the unity and the case of α near resonance under the Brjuno condition.

Analytic solutions of a second-order iterative functional differential equation

Abstract This paper is concerned with a second-order iterative functional differential equation x ″( x [ r ] ( z ))= c 0 z + c 1 x ( z )+⋯+ c m x [ m ] ( z ), where r and m are nonnegative integers, x [0] (z)=z, x [1] (z)=x(z), x [2] (z)=x(x(z)) , etc., are the iterates of the function x ( z ). By constructing a convergent power series solution y ( z ) of a companion equation of form α 2 y″(α r+1 z)y′(α r z)=αy′(α r+1 z)y″(α r z)+[y′(α r z)] 3 ∑ i=0 m c i y(α i z) , analytic solutions of the form y ( αy −1 ( z )) for the original differential equation are obtained.