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1.
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   

2.
We present a nine-point fourth-order finite difference method for the nonlinear second-order elliptic differential equation Auxx + Buyy = f(x, y, u, ux, uy) on a rectangular region R subject to Dirichlet boundary conditions u(x, y) = g(x, y) on ?R. We establish, under appropriate conditions O(h4)-convergence of the finite difference scheme. Numerical examples are given to illustrate the method and its fourth-order convergence.  相似文献   

3.
It is shown that every almost linear mapping of a unital Poisson JC*-algebra to a unital Poisson JC*-algebra is a Poisson JC*-algebra homomorphism when h(2 n uy) = h(2 n u) h(y), h(3 n u y) = h(3 n u) h(y) or h(q n u y) = h(q n u) h(y) for all , all unitary elements and n = 0, 1, 2, · · · , and that every almost linear almost multiplicative mapping is a Poisson JC*-algebra homomorphism when h(2x) = 2h(x), h(3x) = 3h(x) or h(qx) = qh(x) for all . Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings or in the almost linear almost multiplicative mappings.Moreover, we prove the Cauchy–Rassias stability of Poisson JC*-algebra homomorphisms in Poisson JC*-algebras.*This work was supported by grant No. R05-2003-000-10006-0 from the Basic Research Program of the Korea Science & Engineering Foundation.  相似文献   

4.
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   

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7.
The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form
f( t,uy,wy + uuz ) = f( x,y,z )u2 u+ g( t,x,u,u,w )uz + h( t,x,u,u,w )y + 2uwzf\left( {t,\upsilon y,wy + u\upsilon z} \right) = f\left( {x,y,z} \right)u^2 \upsilon + g\left( {t,x,u,\upsilon ,w} \right)\upsilon z + h\left( {t,x,u,\upsilon ,w} \right)y + 2uwz  相似文献   

8.
It is established that the linear problemu u a 2 u xx =g(x,t),u(0,t) =u(x, t + T) =u(x,t) is always solvable in the function spaceA = {g:g(x,t) =g(x,t+T) =g( –x,t) = –g(–x,t)} provided thataTq = (2p – 1) and (2p – 1,q) = 1, wherepandq are integer numbers. To prove this statement, an exact solution is constructed in the form of an integral operator, which is used to prove the existence of a solution of a periodic boundary-value problem for a nonlinear second-order wave equation. The results obtained can be used when studying the solutions to nonlinear boundary-value problems by asymptotic methods.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1115–1121, August, 1993.  相似文献   

9.
Preference optimality is an optimality concept in multicriteria problems, that is, in problems where several criteria are to beoptimized simultaneously. Formally, one wishes to optimizeN criteriag i(·) or, equivalently, a criterion vectorg(·) N , subject to either functional constraints in programming or to side conditions which are differential equations in optimal control. Subject to these constraints, one obtains forg(·) a set of attainable values in N . This set is preordered by the introduction of a complete preordering ; a controlu*(·) or a decisionx*, then, is preference-optimal if it results ing(u*(·))g(u(·)) for all admissible controlsu(·) or ifg(x*)g(x) for all feasible decisionsx. The present paper concerns sufficient conditions for preference-optimal control and for preference-optimal decisions.  相似文献   

10.
New solutions of the wave equation with three space variables of the form u = g(x,y,z,t)f(), where the functions g and = (x,y,z,t) are some specified functions and f is an arbitrary function of one variable, are presented. Bibliography: 4 titles.  相似文献   

11.
Suppose that X is a topological space with preorder , and that –g, f are bounded upper semicontinuous functions on X such that g(x) f(y) whenever x y. We consider the question whether there exists a bounded increasing continuous function h on X such that g h f, and obtain an existence theorem that gives necessary and sufficient conditions. This result leads to an extension theorem giving conditions that allow a bounded increasing continuous function defined on an open subset of X to be extended to a function of the same type on X. The application of these results to extremally disconnected locally compact spaces is studied.Received: 26 May 2004  相似文献   

12.
A spectral collocation method with collocation at the Legendre Gauss points is discussed for solving the Helmholtz equation –u+(x,y)u=f(x,y) on a rectangle with the solution u subject to inhomogeneous Robin boundary conditions. The convergence analysis of the method is given in the case of u satisfying Dirichlet boundary conditions. A matrix decomposition algorithm is developed for the solution of the collocation problem in the case the coefficient (x,y) is a constant. This algorithm is then used in conjunction with the preconditioned conjugate gradient method for the solution of the spectral collocation problem with the variable coefficient (x,y).  相似文献   

13.
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   

14.
We consider the problem of finding, from the final data u(x,y,T)=g(x,y), the initial data u(x,y,0) of the temperature function u(x,y,t),(x,y)I=(0,π)×(0,π),t[0,T] satisfying the following system
The problem is severely ill-posed. In this paper a simple and convenient new regularization method for solving this problem is considered. Meanwhile, some quite sharp error estimates between the approximate solution and exact solution are provided. A numerical example also shows that the method works effectively.  相似文献   

15.
The linear equation Δ2u = 1 for the infinitesimal buckling under uniform unit load of a thin elastic plate over ?2 has the particularly interesting nonlinear generalization Δg2u = 1, where Δg = e?2u Δ is the Laplace‐Beltrami operator for the metric g = e2ug0, with g0 the standard Euclidean metric on ?2. This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order Δu(x)+Kg(x) exp(2u(x)) = 0 and Δ Kg(x) + exp(2u(x)) = 0, with x ∈ ?2, describing a conformally flat surface with a Gauss curvature function Kg that is generated self‐consistently through the metric's conformal factor. We study this conformal plate buckling equation under the hypotheses of finite integral curvature ∫ Kg exp(2u)dx = κ, finite area ∫ exp(2u)dx = α, and the mild compactness condition K+L1(B1(y)), uniformly w.r.t. y ∈ ?2. We show that asymptotically for |x|→∞ all solutions behave like u(x) = ?(κ/2π)ln |x| + C + o(1) and K(x) = ?(α/2π) ln|x| + C + o(1), with κ ∈ (2π, 4π) and . We also show that for each κ ∈ (2π, 4π) there exists a K* and a radially symmetric solution pair u, K, satisfying K(u) = κ and maxK = K*, which is unique modulo translation of the origin, and scaling of x coupled with a translation of u. © 2001 John Wiley & Sons, Inc.  相似文献   

16.
For a selected family of Lagrange-type control problems involving a nonnegative integral costJ T (y,u) over the interval [0,T], 0<T<, with system conditions consisting of differential inequalities and/or equalities, the following material is treated: (i) a resumé of relevant necessary conditions and sufficient conditions for a pair (y T ,u T ) to minimizeJ T (y,u); (ii) conditions sufficient for the convergence asT of minimizing pairs (y T ,u T ) over [0,T] to a limit pair (y ,u ) over the infinite-time interval [0, ); (iii) conditions sufficient for (y ,u ) to minimize the costJ (y,u) over [0, ); and (iv) conditions sufficient for the optimal cost per unit timeJ T (y T ,u T )/T to have a limit asT.  相似文献   

17.
We consider the nonlinear Sturm-Liouville problem –u = f(u) + h in (0, 1), u(0) = u(1) = 0, where h L2(0,1) and f is a positive convex nonlinearity with superlinear growth at infinity. Our main result establishes that the above boundary value problem admits a finite number of solutions but it cannot have infinitely many solutions.Received: 8 July 2004  相似文献   

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19.
The nonlinear hyperbolic equation ∂2u(x, y)/∂xy + g(x, y)f(u(x, y)) = 0 with u(x, 0) = φ(x) and u(0, y) = Ψ(y), considered by [1.], 31–45) under appropriate smoothness conditions, is solvable by the author's decomposition method (“Stochastic Systems,” Academic Press, 1983 and “Nonlinear Stochastic Operator Equations,” Academic Press, 1986).  相似文献   

20.
The functional equationf(x,y)+g(x)h(y)F(u/1?x,ν/1?y)=f(u,ν)+g(u)h(ν)F(x/1?u,y/1?ν) ... (1) forx, y, u, ν ∈ [0, 1) andx+u,y+ν ∈ [0,1) whereg andh satisfy the functional equationφ (x+y?xy)=φ(x)φ(y)... (2) has been solved for some non-constant solution of (2) in [0, 1] withφ (0)=1,φ(1)=0 and the solution is used in characterising some measures of information.  相似文献   

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