A note on iterative solutions of an iterative functional differential equation

Analytic Solutions of an Iterative Functional Differential Equation with Delays Depending On State

This This paper is concerned with an iterative functional differential equation with the form C z z x b az x z x     ,)) ((1) (.By constructing a convergent power series solution of an auxiliary equation C z z g z ag z g z ag z g z g z g a b        ), ()] () ()][ () ([)] () ([ 2 2 2      the analytic solutions for the original equation are obtained. We not only discuss the cons...

Local Analytic Solutions of an Iterative Functional Differential Equation Depend on the State Derivative

Analytic Solutions for Iterative Functional Differential Equations

Because of its technical difficulties the existence of analytic solutions to the iterative differential equation x′(z) = x(az + bx(z) + cx′(z)) is a source of open problems. In this article we obtain analytic solutions, using Schauder’s fixed point theorem. Also we present a unique solution which is a nonconstant polynomial in the complex field.

The Exist Local Analytic Solutions of an Iterative Functional Differential Equation

This paper is concerned with an iterative functional differential equation x′ x r z = c0z+ c1x z + c2x x z + · · · + cmx m z , where r and m are nonnegative integers, x 0 z = z x 1 z = x z x 3 z = x x x z , etc. are the iterates of the function x z , and ∑mj=0 cj = 0. By constructing a convergent power series solution y z of a companion equation of the form αy ′ αr+1z = y ′ αz ∑mj=0 cjy αz , an...

Smooth Solutions of a Class of Iterative Functional Differential Equations

and Applied Analysis 3 where i, j, and k are nonnegative integers. Let I be a closed interval in R. By induction, we may prove that x∗jk t Pjk ( x10 t , . . . , x1,j−1 t ; . . . ;xk0 t , . . . , xk,j−1 t ) , 1.11 βjk Pjk ⎛ ⎜⎝ j terms { }} { x′ ξ , . . . , x′ ξ ; . . . ; j terms { }} { x k ξ , . . . , x k ξ ⎞ ⎟⎠, 1.12 Hjk Pjk ⎛ ⎜⎝ j terms { }} { 1, . . . , 1; j terms { }} { M2, . . . ,M2; . . . ...