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1.
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   

2.
It is shown that every almost linear Pexider mappings f, g, h from a unital C*-algebra into a unital C*-algebra ℬ are homomorphisms when f(2 n uy) = f(2 n u)f(y), g(2 n uy) = g(2 n u)g(y) and h(2 n uy) = h(2 n u)h(y) hold for all unitaries u ∈ , all y ∈ , and all n ∈ ℤ, and that every almost linear continuous Pexider mappings f, g, h from a unital C*-algebra of real rank zero into a unital C*-algebra ℬ are homomorphisms when f(2 n uy) = f(2 n u)f(y), g(2 n uy) = g(2 n u)g(y) and h(2 n uy) = h(2 n u)h(y) hold for all u ∈ {v ∈ : v = v* and v is invertible}, all y ∈ and all n ∈ ℤ. Furthermore, we prove the Cauchy-Rassias stability of *-homomorphisms between unital C*-algebras, and ℂ-linear *-derivations on unital C*-algebras. This work was supported by Korea Research Foundation Grant KRF-2003-042-C00008. The second author was supported by the Brain Korea 21 Project in 2005.  相似文献   

3.
Let A , B be two unital C*-algebras. By using fixed pint methods, we prove that every almost unital almost linear mapping h : A → B which satisfies h(2 n uy) = h(2 n u)h(y) for all u ∈ U(A), all y ∈ A, and all n = 0, 1, 2, … , is a homomorphism. Also, we establish the generalized Hyers-Ulam-Rassias stability of *-homomorphisms on unital C*-algebras.  相似文献   

4.
Let X and Y be vector spaces. It is shown that a mapping f : XY satisfies the functional equation
(‡)
if and only if the mapping f : XY is additive, and prove the Cauchy–Rassias stability of the functional equation (‡) in Banach modules over a unital C*-algebra. Let and be unital C*-algebras, Poisson C*-algebras, Poisson JC*-algebras or Lie JC*-algebras. As an application, we show that every almost homomorphism h : → of into is a homomorphism when h((d + 2)nuy) = h((d + 2)nu)h(y) or h((d + 2)nuy) = h((d + 2)nu) ∘ h(y) for all unitaries u ∈ , all y ∈ , and n = 0, 1, 2, • • • . Moreover, we prove the Cauchy–Rassias stability of homomorphisms in C*-algebras, Poisson C*-algebras, Poisson JC*-algebras or Lie JC*-algebras. Supported by Korea Research Foundation Grant KRF-2004-041-C00023.  相似文献   

5.
It is shown that every almost linear bijection of a unital C-algebra A onto a unital C-algebra B is a C-algebra isomorphism when h(n2uy)=h(n2u)h(y) for all unitaries uA, all yA, and n=0,1,2,…, and that almost linear continuous bijection of a unital C-algebra A of real rank zero onto a unital C-algebra B is a C-algebra isomorphism when h(n2uy)=h(n2u)h(y) for all , all yA, and n=0,1,2,…. Assume that X and Y are left normed modules over a unital C-algebra A. It is shown that every surjective isometry , satisfying T(0)=0 and T(ux)=uT(x) for all xX and all unitaries uA, is an A-linear isomorphism. This is applied to investigate C-algebra isomorphisms between unital C-algebras.  相似文献   

6.
Let A and B be strongly separating linear subspaces of C0(X) and C0(Y), respectively, and assume that ?A ≠ ?? (?A stands for the set of generalized peak points for A) and ?B ≠ ??. Let T: A × BC0(Z) be a bilinear isometry. Then there exist a nonempty subset Z0 of Z, a surjective continuous mapping h: Z0 → ?A × ?B and a norm‐one continuous function a: Z0K such that T (f, g)(z) = a (z)f (πx (h (z))g (πy (h (z)) for all zZ0 and every pair (f, g) ∈ A × B. These results can be applied, for example, to non‐unital function algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

7.
We consider the non‐local singular boundary value problem (1) where qC0([0,1]) and f, hC0((0,∞)), limf(x)=?∞, limh(x)=∞. We present conditions guaranteeing the existence of a solution xC1([0,1]) ∩ C2((0,1]) which is positive on (0,1]. The proof of the existence result is based on regularization and sequential techniques and on a non‐linear alternative of Leray–Schauder type. Copyright © 2005 John Wiley & Sons, Ltd.  相似文献   

8.
Let (X, d X ) and (Y,d Y ) be pointed compact metric spaces with distinguished base points e X and e Y . The Banach algebra of all $\mathbb{K}$ -valued Lipschitz functions on X — where $\mathbb{K}$ is either?or ? — that map the base point e X to 0 is denoted by Lip0(X). The peripheral range of a function f ∈ Lip0(X) is the set Ranµ(f) = {f(x): |f(x)| = ‖f} of range values of maximum modulus. We prove that if T 1, T 2: Lip0(X) → Lip0(Y) and S 1, S 2: Lip0(X) → Lip0(X) are surjective mappings such that $Ran_\pi (T_1 (f)T_2 (g)) \cap Ran_\pi (S_1 (f)S_2 (g)) \ne \emptyset $ for all f, g ∈ Lip0(X), then there are mappings φ1φ2: Y $\mathbb{K}$ with φ1(y2(y) = 1 for all y ∈ Y and a base point-preserving Lipschitz homeomorphism ψ: YX such that T j (f)(y) = φ j (y)S j (f)(ψ(y)) for all f ∈ Lip0(X), yY, and j = 1, 2. In particular, if S 1 and S 2 are identity functions, then T 1 and T 2 are weighted composition operators.  相似文献   

9.
Assume that X is a left Banach module over a unital C*-algebra A. It is shown that almost every n-sesquilinear-quadratic mapping h:X×X×XnA is an n-sesquilinear-quadratic mapping when holds for all x,y,z1,…,znX.Moreover, we prove the generalized Hyers–Ulam–Rassias stability of an n-sesquilinear-quadratic mapping on a left Banach module over a unital C*-algebra.  相似文献   

10.
Let A be a complex commutative Banach algebra and let MA be the maximal ideal space of A. We say that A has the bounded separating property if there exists a constant C > 0 such that for every two distinct points ?1, ?2MA, there is an element aA for which , and ‖a‖ ? C, where is the Gelfand transform of aA. We show that if A is a strongly regular Banach algebra with the bounded separating property, then every compact homomorphism from A into another Banach algebra is of finite dimensional range. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim  相似文献   

11.
LetA denote a unital Banach algebra, and letB denote aC *-algebra which is contained as a unital subalgebra inA. We prove thatB is inverse closed inA if the norms ofA andB coincide. This generalizes well known result about inverse closedness ofC *-subalgebras inC *-algebras.  相似文献   

12.
In this paper, we consider a family of finite difference operators {Ah }h >0 on discrete L q ‐spaces L q (?N h ). We show that the solution u h to uh (t) – A h u h(t) = f h (t), t > 0, u h (0) = 0 satisfies the estimate ‖A h u h ‖equation/tex2gif-inf-15.gif ≤ Cf h ‖equation/tex2gif-inf-21.gif, where C is independent of h and f h . In this case, the family {A h }h >0 is said to have discrete maximal L p regularity on the discrete L q ‐space. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

13.
Let X and Y be completely regular spaces and E and F be Hausdorff topological vector spaces. We call a linear map T from a subspace of C(X, E) into C(Y, F) a Banach–Stone map if it has the form T f (y) =  S y (f (h(y))) for a family of linear operators S y : EF, \({y \in Y}\) , and a function h: YX. In this paper, we consider maps having the property:
$\bigcap^{k}_{i=1}Z(f_{i}) \neq\emptyset \iff \bigcap^{k}_{i=1}Z(Tf_{i})\neq\emptyset , \quad({\rm Z}) $
where Z(f) =  {f =  0}. We characterize linear bijections with property (Z) between spaces of continuous functions, respectively, spaces of differentiable functions (including C ), as Banach–Stone maps. In particular, we confirm a conjecture of Ercan and Önal: Suppose that X and Y are realcompact spaces and E and F are Hausdorff topological vector lattices (respectively, C *-algebras). Let T: C(X, E) → C(Y, F) be a vector lattice isomorphism (respectively, *-algebra isomorphism) such that
$ Z(f) \neq\emptyset\iff Z(Tf) \neq\emptyset. $
Then X is homeomorphic to Y and E is lattice isomorphic (respectively, C *-isomorphic) to F. Some results concerning the continuity of T are also obtained.
  相似文献   

14.
We consider the evolution of microstructure under the dynamics of the generalized Benjamin–Bona–Mahony equation (1) with u: ?2 → ?. If we model the initial microstructure by a sequence of spatially faster and faster oscillating classical initial data vn, we obtain a sequence of spatially highly oscillatory classical solutions un. By considering the Young measures (YMs) ν and µ generated by the sequences vn and un, respectively, as n → ∞, we derive a macroscopic evolution equation for the YM solution µ, and show exemplarily how such a measure‐valued equation can be exploited in order to obtain classical evolution equations for effective (macroscopic) quantities of the microstructure for suitable initial data vn and non‐linearities f. Copyright © 2005 John Wiley & Sons, Ltd.  相似文献   

15.
We study different notions of subsolutions for an abstract evolution equation du/dt+Auf where A is an m-accretive nonlinear operation in an ordered Banach space X with order-preserving resolvents. A first notion is related to the operator d/dt+A in the ordered Banach space L 1(0, T; X); a second one uses the evolution equation du/dt+A uf where A :x{y;zy for some zAx}; other notions are also considered.  相似文献   

16.
It is shown that every almost *-homomorphism h : A→B of a unital JC*-algebra A to a unital JC*-algebra B is a *-homomorphism when h(rx) = rh(x) (r 〉 1) for all x∈A, and that every almost linear mapping h : A→B is a *-homomorphism when h(2^nu o y) - h(2^nu) o h(y), h(3^nu o y) - h(3^nu) o h(y) or h(q^nu o y) = h(q^nu) o h(y) for all unitaries u ∈A, all y ∈A, and n = 0, 1,.... Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings. We prove that every almost *-homomorphism h : A→B of a unital Lie C*-algebra A to a unital Lie C*-algebra B is a *-homomorphism when h(rx) = rh(x) (r 〉 1) for all x ∈A.  相似文献   

17.
In this paper, we study the existence of infinitely many solutions to p‐Kirchhoff‐type equation (0.1) where f(x,u) = λh1(x)|u|m ? 2u + h2(x)|u|q ? 2u,a≥0,μ > 0,τ > 0,λ≥0 and . The potential function verifies , and h1(x),h2(x) satisfy suitable conditions. Using variational methods and some special techniques, we prove that there exists λ0>0 such that problem 0.1 admits infinitely many nonnegative high‐energy solutions provided that λ∈[0,λ0) and . Also, we prove that problem 0.1 has at least a nontrivial solution under the assumption f(x,u) = h2|u|q ? 2u,p < q< min{p*,p(τ + 1)} and has infinitely many nonnegative solutions for f(x,u) = h1|u|m ? 2u,1 < m < p. Copyright © 2015 John Wiley & Sons, Ltd.  相似文献   

18.
Let fL2, ? µ(?3), where where x = (x1, x2, x3) is the Cartesian system in ?3, x′ = (x1, x2), , µ∈?+\?. We prove the decomposition f = ? ?u + g, with g divergence free and u is a solution to the problem in ?3 Given fL2, ? µ(?3) we show the existence of uH(?3) such that where Since f, u, g are defined in ?3 we need a sufficiently fast decay of these functions as |x|→∞. Copyright © 2010 John Wiley & Sons, Ltd.  相似文献   

19.
We prove the following statement, which is a quantitative form of the Luzin theorem on C-property: Let (X, d, μ) be a bounded metric space with metric d and regular Borel measure μ that are related to one another by the doubling condition. Then, for any function f measurable on X, there exist a positive increasing function η ∈ Ω (η(+0) = 0 and η(t)t a decreases for a certain a > 0), a nonnegative function g measurable on X, and a set EX, μE = 0 , for which
| f(x) - f(y) | \leqslant [ g(x) + g(y) ]h( d( x,y ) ), x,y ? X / E \left| {f(x) - f(y)} \right| \leqslant \left[ {g(x) + g(y)} \right]\eta \left( {d\left( {x,y} \right)} \right),\,x,y \in {{X} \left/ {E} \right.}  相似文献   

20.
Mahdi Boukrouche  Ionel Ciuperca 《PAMM》2007,7(1):4080023-4080024
Let (m, n) ∈ ℕ2, Ω an open bounded domain in ℝm , Y = [0, 1]m ; uε in (L2(Ω))n which is two-scale converges to some u in (L2(Ω × Y))n . Let φ: Ω × ℝm × ℝn → ℝ such that: φ(x, ·, ·) is continuous a.e. x ∈ Ω φ(·, y, z) is measurable for all (y, z) in ℝm × ℝn , φ(x, ·, z) is 1-periodic in y, φ(x, y, ·) is convex in z. Assume that there exist a constant C1 > 0 and a function C2L2(Ω) such that

  相似文献   


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