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1.
In this paper we obtain a Douglas type factor decomposition theorem about certain important bounded module maps. Thus, we come to the discussion of the topological continuity of bounded generalized inverse module maps. Let X be a topological space, x →Tx : X→L(E) be a continuous map, and each R(Tx) be a closed submodule in E, for every fixed x C X. Then the map x→ Tx^+: X→L(E) is continuous if and only if ||Tx^+|| is locally bounded, where Tx^+ is the bounded generalized inverse module map of Tx. Furthermore, this is equivalent to the following statement: For each x0 in X, there exists a neighborhood ∪0 at x0 and a positive number λ such that (0, λ^2)lohtatn in ∩x∈∪0C/σ(Tx^+Tx), where a(T) denotes the spectrum of operator T.  相似文献   

2.
3.
Let X, Y be vector spaces. It is shown that if a mapping f : X → Y satisfies f((x+y)/2+z)+f((x-y)/2+z=f(x)+2f(z),(0.1) f((x+y)/2+z)-f((x-y)/2+z)f(y),(0.2) or 2f((x+y)/2+x)=f(x)+f(y)+2f(z)(0.3)for all x, y, z ∈ X, then the mapping f : X →Y is Cauchy additive. Furthermore, we prove the Cauchy-Rassias stability of the functional equations (0.1), (0.2) and (0.3) in Banach spaces. The results are applied to investigate isomorphisms between unital Banach algebras.  相似文献   

4.
This paper is a study of the oscillatory and asymptotic behaviour of solutions of the second order nonhomogeneous linear differential equation y"+P(x)y=f(x), and the associated homogeneous equation. Conditions are determined, under which the nonhomogeneous equation is oscillatory if and only if the homogeneous equation is oscillatory.  相似文献   

5.
Let X and Y be vector spaces. The authors show that a mapping f : X →Y satisfies the functional equation 2d f(∑^2d j=1(-1)^j+1xj/2d)=∑^2dj=1(-1)^j+1f(xj) with f(0) = 0 if and only if the mapping f : X→ Y is Cauchy additive, and prove the stability of the functional equation (≠) in Banach modules over a unital C^*-algebra, and in Poisson Banach modules over a unital Poisson C*-algebra. Let A and B be unital C^*-algebras, Poisson C^*-algebras or Poisson JC^*- algebras. As an application, the authors show that every almost homomorphism h : A →B of A into is a homomorphism when h((2d-1)^nuy) =- h((2d-1)^nu)h(y) or h((2d-1)^nuoy) = h((2d-1)^nu)oh(y) for all unitaries u ∈A, all y ∈ A, n = 0, 1, 2,.... Moreover, the authors prove the stability of homomorphisms in C^*-algebras, Poisson C^*-algebras or Poisson JC^*-algebras.  相似文献   

6.
In this paper, we determine the general solution of the functional equation f1 (2x + y) + f2(2x - y) = f3(x + y) + f4(x - y) + f5(x) without assuming any regularity condition on the unknown functions f1,f2,f3, f4, f5 : R→R. The general solution of this equation is obtained by finding the general solution of the functional equations f(2x + y) + f(2x - y) = g(x + y) + g(x - y) + h(x) and f(2x + y) - f(2x - y) = g(x + y) - g(x - y). The method used for solving these functional equations is elementary but exploits an important result due to Hosszfi. The solution of this functional equation can also be determined in certain type of groups using two important results due to Szekelyhidi.  相似文献   

7.
An equation involving a derivative is called a differential equation.Such as,dy/dx=2x,and the function y=f(x)satisfies this equation.When we know the additional condition that y=2 when x=-1,the function y=f(x)will be find exactly.The additional condition is called the initial condition.It is used to evaluate constant of integration.  相似文献   

8.
Let G = (V, E) be a connected graph. X belong to V(G) is a vertex set. X is a 3-restricted cut of G, if G- X is not connected and every component of G- X has at least three vertices. The 3-restricted connectivity κ3(G) (in short κ3) of G is the cardinality of a minimum 3-restricted cut of G. X is called κ3-cut, if |X| = κ3. A graph G is κ3-connected, if a 3-restricted cut exists. Let G be a graph girth g ≥ 4, κ3(G) is min{d(x) + d(y) + d(z) - 4 : xyz is a 2-path of G}. It will be shown that κ3(G) = ξ3(G) under the condition of girth.  相似文献   

9.
For a Banach Space X Garcia-Falset introduced the coefficient R(X) and showed that if R(X) 〈 2 then X has a fixed point. In this paper, we define a mean non-expansive mapping T on X in the sense that ||Tx - TY|| ≤ a||x - y|| + b||x - Ty|| for any x,y E X, where a,b ≥ 0, a + b ≤ 1. We show that if R(X) 〈 1/1+b then T has a fixed point in X.  相似文献   

10.
Let G =(V, E) be a connected simple graph. A labeling f : V → Z2 induces an edge labeling f* : E → Z2 defined by f*(xy) = f(x) +f(y) for each xy ∈ E. For i ∈ Z2, let vf(i) = |f^-1(i)| and ef(i) = |f*^-1(i)|. A labeling f is called friendly if |vf(1) - vf(0)| ≤ 1. For a friendly labeling f of a graph G, we define the friendly index of G under f by if(G) = e(1) - el(0). The set [if(G) | f is a friendly labeling of G} is called the full friendly index set of G, denoted by FFI(G). In this paper, we will determine the full friendly index set of every Cartesian product of two cycles.  相似文献   

11.
LetX be a real linear normed space, (G, +) be a topological group, andK be a discrete normal subgroup ofG. We prove that if a continuous at a point or measurable (in the sense specified later) functionf:XG fulfils the condition:f(x +y) -f(x) -f(y) ∈K whenever ‖x‖ = ‖y‖, then, under some additional assumptions onG,K, andX, there esists a continuous additive functionA :XG such thatf(x) -A(x) ∈K.  相似文献   

12.
Suppose that(T t )t>0 is aC 0 semi-group of contractions on a Banach spaceX, such that there exists a vectorxX, ‖x‖=1 verifyingJ −1(Jx)={x}, whereJ is the duality mapping fromX toP(X *). If |<T t x,f>|→1, whent→+∞ for somefX *, ‖f‖≤1 thenx is an eigenvector of the generatorA, associated with a purcly imaginary eigenvalue. Because of Lin's example [L], the hypothesis onxX is the best possible. If the hypothesisJ −1(Jx)={x} is not verified, we can prove that ifJx is a singleton and ifJ −1(Jx) is weakly compact, then if |<T t x, f>|→1, whent→+∞ for somefX *, ‖f‖≤1, there existsyJ −1(Jx) such thaty is an eigenvector of the generatorA, associated with a purely imaginary eigenvalue. We give also a counter-example in the case whereX is one of the spaces ℓ1 orL 1.  相似文献   

13.
It is well known that a function f of the real variable x is convex if and only if (x,y)→yf(y -1 x),y>0 is convex. This is used to derive a recursive proof of the convexity of the multiplicative potential function. In this paper, we obtain a conjugacy formula which gives rise, as a corollary, to a new rule for generating new convex functions from old ones. In particular, it allows to extend the aforementioned property to functions of the form (x,y)→g(y)f(g(y)-1 x) and provides a new tool for the study of the multiplicative potential and penalty functions. Received: June 3, 1999 / Accepted: September 29, 2000?Published online January 17, 2001  相似文献   

14.
We prove that there exists a Lipschitz function froml 1 into ℝ2 which is Gateaux-differentiable at every point and such that for everyx, y εl 1, the norm off′(x) −f′(y) is bigger than 1. On the other hand, for every Lipschitz and Gateaux-differentiable function from an arbitrary Banach spaceX into ℝ and for everyε > 0, there always exist two pointsx, y εX such that ‖f′(x) −f′(y)‖ is less thanε. We also construct, in every infinite dimensional separable Banach space, a real valued functionf onX, which is Gateaux-differentiable at every point, has bounded non-empty support, and with the properties thatf′ is norm to weak* continuous andf′(X) has an isolated pointa, and that necessarilya ε 0. This work has been initiated while the second-named author was visiting the University of Bordeaux. The second-named author is supported by grant AV 1019003, A1 019 205, GA CR 201 01 1198.  相似文献   

15.
We are going to discuss special cases of a conditional functional inequality
whereX is a real inner product space. In particular, we will give conditions which force the representationf(x)=c‖x2+a(x) for x ∈X, where c ∈ R anda:x→ℝ is an additive functional.  相似文献   

16.
Letf(t) = ∑a k e ikt be infinitely differentiable on R, |f(t)|<1. It is known that under these assumptions ‖n‖ converges to a finite limitl asn → ∞ (l 2 = sec(arga),a = (f′(0))2 -f″(0)). We obtain here more precise results: (i) an asymptotic series (in powers ofn -1/2) for the Fourier coefficientsa nk off n , which holds uniformly ink asn → ∞; (ii) an asymptotic series (this time only powers ofn -1 are present!) for ‖f n ‖; (iii) the fact that ifi j f (j)(0) is real forj = 1,2,..., 2h + 2 then ‖f n ‖ = l + o(n -h ),n → ∞. More generally, we obtain analogous finite asymptotic expansions whenf is assumed to be differentiable only finitely many times.  相似文献   

17.
LetT be a nonexpansive mapping on a normed linear spaceX. We show that there exists a linear functional.f, ‖f‖=1, such that, for allxX, limn→x f(T n x/n)=limn→xT n x/n ‖=α, where α≡inf y∈c Ty-y‖. This means, ifX is reflexive, that there is a faceF of the ball of radius α to whichT n x/n converges weakly for allx (infz∈f g(T n x/n-z)→0, for every linear functionalg); ifX is strictly conves as well as reflexive, the convergence is to a point; and ifX satisfies the stronger condition that its dual has Fréchet differentiable norm then the convergence is strong. Furthermore, we show that each of the foregoing conditions on X is satisfied if and only if the associated convergence property holds for all nonexpansiveT. Supported by National Science Foundation Grant MCS-79-066.  相似文献   

18.
We prove that if X is a separable Banach space, then a measurable multifunction Γ : [0, 1] → ck(X) is Henstock integrable if and only if Γ can be represented as Γ = G + f, where G : [0, 1] → ck(X) is McShane integrable and f is a Henstock integrable selection of Γ.  相似文献   

19.
Several results about convolution and about Fourier coefficients for X-valued functions defined on t he torus satisfying the condition sup ||y||=1∫-π^π|| B (f (e^iθ), y)||dθ/2π〈 ∞ for a bounded bilinear map B : X × Y → Z are presented and some applications are given.  相似文献   

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