(CK)映射及其不动点

定义1 距离空间(E,ρ)到自身的映射 T 称为(CK)映射,若0≤α,β<1,对任意x,y∈E,ρ(Tx,Ty)≤α·ρ(x,y).+β·[ρ(x,Tx)+ρ(y,Ty)].显然,这类映射,当β=0时,为 Banach 压缩映射;当α=0,β=1/2时,为 Kannan映射([1],[2]),当α=0时,为 Kannan —型映射([3]).距离空间(E,ρ)到自身的映射     

关于压缩映象的一个未解决的问题及一个新的不动点定理

§1.引言 设(X,d)是一完备度量空间,T是映X到X的映象.按照Rhoades的说法,T称为第(25)类的压缩映象,如果T满足下面的条件:对一切的x,y∈X,x≠y, d(Tx,Ty)相似文献   

一个新的压缩型映象的不动点定理

设X是完备的度量空间,T:X→X是连续映象,若存在X×X上对称实函数φ(x,y).使T满足φ(Tx,Ty)≤aφ(x,y), x,y∈X及d(Tx,Ty)≤cd(x,y)+φ(x,y), x,y∈X,其中d是X上的度量,两个常数a,c满足0≤a<1,0≤c<1.本文证明了映象T有唯一的不动点x_*,且 x∈X,有T_x~n→x_*(n→∞).在φ(x,x)=0,φ是一元连续的条件下,证明了在X上一定存在一个拓扑等价的度量d~*使T关于d~*是X上的Banach压缩映象.     

凸度量空间中非扩张映象的不动点定理

设X是一凸度量空间,并且它的每一直径趋于零的非空闭子集的逆减序列具有非空交的性质.本文证明了,如果X的非空闭子集K的自映象T满足不等式:d(Tx,Ty)≤ad(x,y) b{d(x,Tx) d(y,Ty)} c{d(x,Ty) d(y,Tx)},(?)x,y∈K其中0≤a<1,b≥0,c≥0,使得a c≠0且a 2b 3c≤1.则T在K中存在唯一不动点.     

SOME FIXED POINT THEOREMS OF CONTRACTIVE TYPE MAPFINGS

1. Let X be the conjugate of a separable Banach space satifying the *-Opial condition, i. e., if \[\{ {x_n}\} \subset x,{x_n}\mathop \to \limits^{{w^*}} {x_\infty },{x_\infty } \ne y\], then\[\mathop {\overline {\lim } }\limits_{n \to \infty } ||{x_n} - {x_\infty }|| < \mathop {\overline {\lim } }\limits_{n \to \infty } ||{x_n} - y||\] for rxample \[X = {l_1}\] Let K be a nonempty weak* closed convex subset of X. The main results are: Theorem 1. Suppose T is a ooniinuons mappings of K into itself such that for every \[x,y \in K\]，\[||Tx - Ty|| \le a||x - y|| + b\{ ||x - Tx|| + ||y - Ty||\} + c\{ ||x - Ty|| + ||y - Tx||\} \] where real numbers \[a,b,c \ge 0\] and \[a + 2b + 2c = 1\]. Suppose also K is bounded.Then T has at least one fixed point in K. Theorem 2. Let T be a mapping of K into itself, and \[a(x,y),b(x,y),c(x,y)\]be real functions such that for all\[x,y \in K\] \[||Tx - Ty|| \le a(x,y)||x - y|| + b(x,y)\{ ||x - Tx|| + ||y - Ty||\} + c(x,y)\{ ||x - Ty|| + ||y - Tx||\} \] and \[a(x{\rm{y}},y){\rm{ + }}2b(x,y){\rm{ + }}2c(x,y) \le 1\] Suppose there exists \[x \in K\] such that \[O(x) = \{ {T^n}x\} _{n = 1}^\infty \] is bounded and \[\mathop {\inf }\limits_{y,z \in o(x)} c(y,z) > 0\] Then T has at least one fixed point z in K and \[{T^n}x\mathop \to \limits^{{w^*}} z\]. 2. We denote \[CL(x) = \{ A;nonempty{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} closed{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} subset{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} of{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} X\} \] \[K(x) = A;nonempty{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} closed{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} subset{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} of{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x\} \] here X is a complete metric space with metric d. On \[CL(x)\] and \[K(x)\] we introduce the generalized Hausdorff distance \[H(,)\], The main results are: Theorem 3. Suppose \[\{ T,S\} \] is a pair of set-valued mappings of X into \[CL(x)\],which satisfies the following condition: \[H(Tx,Sy) \le hMax\{ d(x,y),D(x,Tx),D(y,Sy),\frac{1}{2}[D(x,Sy) + D(y,Tx)]\} \] for each \[x,y \in K\], where 0相似文献   

一类可加和二次混合泛函方程的稳定性

本文研究了一类新的源自可加映射和二次映射的混合泛函方程f(x+2y)+f(x-2y)=f(x+y)+f(x-y)+3f(2y)-6f(y),及其在拟β赋范线性空间中的Hyers-Ulam稳定性.     

On an Open Question and a Fixed Point Theorem for a Class of Contractive Mappings

Let X be a complete metric space with, distance fcmction d, and T a mapping from X into X. T is said to be a contractive mapping (25), if it satisfies tbe following condition d(Tx, Ty) 相似文献   

QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN BANACH SPACES

In this paper,we solve the quadratic p-functional inequalities‖f(x+y)+f(x-y)-2f(x)-2f(y)‖≤‖ρ(2f((x+y)/2)+2f((x-y)/2)-f(x)-f(y))‖,(0.1)where ρ is a fixed complex number with |ρ| 1,and‖2f((x+y)/2)+2f((x-y)/2)-f(x)-f(y)‖≤‖ρ(f(x+y)+f(x-y)-2f(x)-2f(y))‖,(0.2)where ρ is a fixed complex number with |ρ| 1/2.Using the direct method,we prove the Hyers-Ulam stability of the quadratic ρ-functional inequalities(0.1) and(0.2) in complex Banach spaces and prove the Hyers-Ulam stability of quadratic ρ-functional equations associated with the quadratic ρ-functional inequalities(0.1)and(0.2) in complex Banach spaces.     

伸张函数增长阶的估计

本文建立了Beurling-Ahlfors扩张的伸张函数的一个新的估计式,并对当ρ-函数在递减函数控制下伸张函数的增长阶进行了估计,改进了已有的结果,得到估计式: 当ρ(y/2)≥2时,D(x,y)≤2ρ(y/2); 当1≤ρ(y/2)<2时,D(x,y)2≤ρ(y/2)+1/2.     

一类源自可加、二次、三次和四次映射的泛函方程的模糊稳定性

在模糊Banach空间中研究了混合泛函方程f(x+ky)+f(x-ky)=k~2f(x+y)+k~2f(x-y)+2(1-k~2)f(x)+(k~2(k~2-1))/12(f(2y)-4f(y))的Hyers-Ulam稳定性,这里k>1是固定的一个整数,f(y)=f(y)+f(-y).     

A New Generalization of a Theorem of Jung for the Orthogonality Equation

The main result of this paper states as follows: Assume that for a closed ball D @ 0 and with center at the origin, a mapping T : D M D satisfies $$ T(0) = 0\ \hbox{and}\ \vert \langle Tx, Ty \rangle - \langle x, y \rangle \vert \leq \varepsilon \eqno (1) $$ for some 0 h l < min { 1/4, d 2 /17} and for all x , y ] D . Then, there exists an isometry I : D M D with $$ \vert Tx - Ix \vert \le \left\{ {\matrix{ {13\sqrt \varepsilon } \hfill &{{\rm for}\; d     

On Semitransitive Collections of Operators

A collection F of operators on a vector space V is said to be semitransitive if for every pair of nonzero vectors x and y in V there exists a member T of F such that either Tx = y or Ty = x (or both). We study semitransitive algebras and semigroups of operators. One of the main results is that if the underlying field is algebraically closed, then every semitransitive algebra of operators on a space of dimension n contains a nilpotent element of index n. Among other results on semitransitive semigroups, we show that if the rank of nonzero members of such a semigroup acting on an n-dimensional space is a constant k, then k divides n.     

有关Euler函数(n)的方程的正整数解

设(n)是Euler函数.主要研究了方程(xy)=3((x)+(y))的可解性问题,利用初等的方法给出了这一方程的所有的35组正整数解.对于任意素数k>3,(x,y)=(3k,4k),(4k,3k)是方程(xy)=k((x)+(y))的2个正整数解.证明了更为一般的结论:对于任意奇数k>3,当gcd(k,3)=1时,(x,y)=(3k,4k),(4k,3k)是方程(xy)=k((x)+(y))的2个正整数解.     

The weakest contractive conditions for Edelstein’s mappings to have a fixed point in complete metric spaces

In this paper, we give the answer to the following problem: Let (X, d) be a complete metric space and let T be a mapping on X satisfying \(d(Tx, Ty) < d(x, y)\) for any \(x, y \in X\) with \(x \ne y\). Then what are the weakest additional assumptions to imply the same conclusion as in the Banach contraction principle?     

具正規核的積分方程

<正> §1.引言 凡合條件即是說凡合條件kk[x,y]=kk[x,y](1.1)的核k(x,y)叫做正規核(normal kernel).這種核顯然包括實對稱核、實畸對稱核、艾氏核及畸艾氏核等為特例。在本文中,我們將討論具此種核之積分方程之性質及解法尤其是關於此種核之特值及奇值(即希米特(E.Schmidt)的特值)之性質     

Banach空间中的平均非扩张映象:不动点的存在定理

<正> 设X是Banach空间,E是X中的集合,T是映集合E到自身的映象.若T满足条件(称为平均非扩张条件)其中x,y∈E,a,b,c≥0且a+2b+2c≤1,则称T是平均非扩张映象. 文[1]概括了近年来研究关于平均非扩张映象不动点的一些主要结果.本文进     

一类非线性积分方程非负解的存在性

本文利用凝聚映象的不动点定理得到了如下形式的积分方程：ψ０（ｘ）＝ψ（ｘ）＋ｆ（ψ（ｘ））∫１０Ｒ（ｘ,ｙ）ｘ２－ｙ２ｇ（ψ（ｙ））ｄｙ非负连续解的存在性．     

具非线性边界条件的Volterra型时滞微分方程边值问题奇摄动

该文研究一类时滞微分方程边值问题〖JB({〗εx″(t)=f(t,x(t),x(t-τ(t)),\[Tx\](t),x′(t),ε),t∈(0,1),\=x(t)=φ(t,ε),t∈\[-τ,0\],h(x(1),x′(1),ε)=A(ε),[JB)]其中ε>0为小参数，τ(t)≥τ\-0>0,τ=\%\{max\}\%[DD(X]t∈\[0,1\][DD)]τ(t)<1,\[Tx\](t)=ψ(t)+∫\+t\-0k(t,x)x(s）ds为Volterra型算子。利用微分不等式理论证明了边值问题解的存在性，并给出了解的一 致有效渐近展开式。     

BOUNDARY REGULARITY FOR WEAK HEAT FLOWS

61. IntroductionLet (M, g) be a compact smooth foemannian manifOld of dimension n with C2 boundary0M, and (N, h) be a smooth compact Riemannian manifolds of dimension k. Assume that(N, h) without boundary is isometrically embedded into the Euclidean space (Rm, (., .)).We assume that Sobolev spaceHl (M, N) = {u E Hl (M; R',.)lu(x) E N for a.e.x E M}and for every u E H1 (M; N), define the energy of u,E(u) = / lVuI'dv, (1.1)j. lVuI'dv, (1.1)where in local coordinate 1VuI' = g"pff 3, …     

基于样条的高阶矩阵微分方程近似解

矩阵微分方程经常出现在许多物理模型和工程技术模型中.利用矩阵样条构造形如{y(p)(x)=Ap-1(x)y(p-1)(x)+Ap-2(x)y(p-2)(x)+…+A1(x)y(1)(x)+A0(x)y(x)+B0(x),y(a)=ya,…,y(p-1)(a)=y(p-1)a,x∈[a,b];Ai(x),B0(x)∈C4[a,b],0≤i≤p-烅烄烆1的高阶矩阵线性微分方程初值问题的数值解.给出实现算法和数值解的近似误差估计以及数值实例.先将高阶矩阵微分方程转化为一阶矩阵微分方程,然后利用三次矩阵样条求出一阶矩阵线性微分方程的数值解,从而解决高阶微分方程问题.