非齐次线性差分方程的Cm解[英文]

设c0,c1,…,cn均为实的常数，F（x)是个从R到R的C^m映射。本文讨论了非齐次线性差分方程∑i=1ncif(x i)=F(x)的C^m(m≥0）的存在性和唯一性。     

Analytic solutions of an iterative functional differential equation

This paper is concerned with a functional differential equation x^′(z)=1/x(az+bx(z)), where a, b are two complex numbers. By constructing a convergent power series solution y(z) of a auxiliary equation of the form b²y^′(z)=(y(μ²z)−ay(μz))(μy^′(μz)−ay^′(z)), analytic solutions of the form for the original differential equation are obtained.     

Existence of analytic solutions of an iterative functional equation

This paper is concern analytic solutions of an iterative functional equation of the form

f(p(z)+q(f(z)))=h(f(z)),z∈C.     

Existence, uniqueness and stability of Cm solutions of iterative functional equations

In this paper we discuss a relatively general kind of iterative functional equation G(x,f(x),..., fn(x))=0 (for all x∈J), where J is a connected closed subset of the real number axis R, G∈Cm(Jn+1, R), and n≥2. Using the method of approximating fixed points by small shift of maps, choosing suitable metrics on functional spaces and finding a relation between uniqueness and stability of fixed points of maps of general spaces, we prove the existence, uniqueness and stability of Cm solutions of the above equation for any integer m≥0 under relatively weak conditions, and generalize related results in reference in different aspects.     

Analytic solutions of a general nonlinear functional equations near resonance

Existence of analytic solutions of a general class of nonlinear functional equations is discussed. This general class includes some specific functional equations studied recently. Moreover, we can generalize this problem to finding analytic solutions of a general class of iterative equations.     

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

In this paper a second-order nonautonomous iterative functional differential equation is considered. By reducing the equation with the Schröder transformation to another functional differential equation without iteration of the unknown function, we give existence of its local analytic solutions. We first discuss the case that the constant α given in the Schröder transformation does not lie on the unit circle in C and the case that the constant lies on the circle but fulfills the Diophantine condition. Then we further study the case that the constant is a unit root in C but the Diophantine condition is offended. Finally, we investigate analytic solutions of the form of power functions.     

Convex solutions of polynomial-like iterative equations

In this paper convex solutions and concave solutions of polynomial-like iterative equations are investigated. A result for non-monotonic solutions is given first and applied then to prove the existence of convex continuous solutions and concave ones. Furthermore, another condition for convex solutions, which is weaker in some aspects, is also given. The uniqueness and stability of those solutions are also discussed.     

Analytic solutions of a nonlinear iterative equation near neutral fixed points and poles

In this paper existence of analytic solutions of a nonlinear iterative equations is studied when given functions are all analytic and when given functions have poles. As well as in many previous works, we reduce this problem to finding analytic solutions of a functional equation without iteration of the unknown function f. For technical reasons, in previous works an indeterminate constant related to the eigenvalue of the linearized f at its fixed point O is required to fulfill the Diophantine condition that O is an irrationally neutral fixed point of f. In this paper the case of rationally neutral fixed points is also discussed, where the Diophantine condition is not required.     

Existence and Uniqueness of Analytic Solutions of Systems of Iterative Functional Equations

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　Ａｌｏｎｇｗｉｔｈｔｈｅｄｅｖｅｌｏｐｍｅｎｔｏｆｎｏｎｌｉｎｅａｒｄｙｎａｍｉｃａｌｓｙｓｔｅｍｓ,ｏｎｅｈａｓｐａｉｄａｔｔｅｎｔｉｏｎｔｏｎｏｔｏｎｌｙｔｈｅｌｏｎｇ ｔｉｍｅｂｅｈａｖｉｏｕｒｏｆａｍｏｖｅｍｅｎｔｂｕｔａｌｓｏｉｔｓｐｒｏｃｅｓｓ,ｗｈｉｃｈｔｏｕｃｈｅｓｕｐｏｎｔｈｅｐｒｏｂｌｅｍｏｆｔｈｅｉｎｖｅｒｓｅｏｐｅｒａｔｉｏｎｏｆａｎｉｔｅｒａｔｉｖｅｏｐｅｒａｔｉｏｎ .Ｂｙｉｔｅｒａｔｉｖｅｆｕｎｃｔｉｏｎａｌｅｑｕａｔｉｏｎｓｗｅｍｅａｎｔ…     

Positive periodic solutions of higher-dimensional nonlinear functional difference equations

By using a well-known fixed point index theorem, we study the existence, multiplicity and nonexistence of positive T-periodic solution(s) to the higher-dimensional nonlinear functional difference equations of the form

     

Existence of positive solutions for higher-order functional differential equations

Under suitable conditions on f(t,y(t+θ)), the boundary value problem of higher-order functional differential equation (FDE) of the form

     

Iterative equations in Banach spaces

Let X be a Banach space, let

     

On solvability of functional equations and system of functional equations arising in dynamic programming

The purpose of this paper is to study solvability of two classes of functional equations and a class of system of functional equations arising in dynamic programming of multistage decision processes. By using fixed point theorems, a few existence and uniqueness theorems of solutions and iterative approximation for solving these classes of functional equations are established. Under certain conditions, some existence theorems of coincidence solutions for the class of system of functional equations are shown. Some examples are given to demonstrate the advantage of our results than existing ones in the literature.     

Positive periodic solutions of higher-dimensional functional difference equations with a parameter

By using Krasnoselskii's fixed point theorem and upper and lower solutions method, we find some sets of positive values λ determining that there exist positive T-periodic solutions to the higher-dimensional functional difference equations of the form where A(n)=diag[a₁(n),a₂(n),…,a_m(n)], h(n)=diag[h₁(n),h₂(n),…,h_m(n)], a_j,h_j :Z→R⁺, τ :Z→Z are T -periodic, j=1,2,…,m, T1, λ>0, x :Z→R^m, f :R₊^m→R₊^m, where R₊^m={(x₁,…,x_m)^TR^m, x_j0, j=1,2,…,m}, R⁺={xR, x>0}.     

Analytic solutions to integral equations in the complex domain

Schauder's fixed point theorem and the Leray–Schauder alternative are used to establish the existence of analytic solutions to integral equations in the complex domain.     

Two periodic solutions of second-order neutral functional differential equations

In this paper, we consider a type of second-order neutral functional differential equations. We obtain some existence results of multiplicity and nonexistence of positive periodic solutions. Our approach is based on a fixed point theorem in cones.     

泛函方程组的解析解

§ 1　IntroductionThe Feigenbaum functional equation plays an importantrole in the theory concerninguniversal properties of one-parameter families of maps of the interval that has the formf2 (λx) +λf(x) =0 ,0 <λ=-f(1 ) <1 ,f(0 ) =1 ,(1 .1 )where f is a map ofthe interval[-1 ,1 ] into itself.Lanford[1 ] exhibited a computer-assist-ed proof for the existence of an even analytic solution to Eq.(1 .1 ) .It was shown in[2 ]that Eq.(1 .1 ) does not have an entire solution.Si[3] discussed the it…     

Analytic solutions of a second order iterative functional differential equation

This paper is concerned with an iterative functional differential equation

c₁x(z)+c₂x^′(z)+c₃x^″(z)=x(az+bx^′(z))