Application of Integrals of Scalar Functions with Respect to a Vector Measure to Some Problems of Functional Analysis and the Theory of Linear Integral Equations

We prove some statements on the decomposition of indefinite integrals of scalar functions with respect to a vector measure. We also consider continuous linear operators acting from the fundamental Banach space to a Hilbert space H. This gives a representation theorem for continuous linear operators from X to H. These results are applied to most general linear integral equations of the form . Such equations are equivalent to certain infinite systems of scalar integral equations and to infinite systems of linear algebraic equations. Bibliography: 11 titles.     

Nonconvex second-order differential inclusions with memory

We prove several existence theorems for the second-order differential inclusion of the form in the case whenF or bothG andF are maps with nonconvex values in an Euclidean or Hilbert space andF(t, T(t)x) is a memory term ([T(t)x]()=x(t+)).     

一类算子值解析函数族的极值点

设 H 是一个Hilbert空间. B(H) 表示所有H 到 H 的有界线性算子构成的Banach空间. 设 T= {f(z): f(z)=zI-∑^∞_n=2 zⁿA_n 在单位圆盘|z|<1上解析, 其中系数A_n是 H 到 H 的紧正Hermitian算子, I 表示 H 上的恒等算子, ∑^∞_n=2 n(A_n x, x) ≤1 对所有x ∈H, ∣|x∣∣=1 成立. 该文研究了函数族 T 的极值点.     

The uniqueness of the adjoint operation I

Let H be a complex, infinite-dimensional Hilbert space. Let B(H) denote the set of bounded linear operators on H. This paper contains a nonlinear characterization of the adjoint operation on B(H). The statement of this result is:THEOREM:Let h: B(H) B(H)be a function such that h(I)0.Then h(ST)=h(T)h(S)and h(S)S0for all elements Sand Tof B(H)if and only if h(S)=S^* for all S B(H).     

Boundary values of generalized solutions of a homogeneous sturm — Liouville equation in a space of vector functions

We consider a differential equation of the form ?y” + A²y=0, where A is a self-adjoint operator in a Hilbert space H. We show that each generalized solution of this equation inw _?m (0, b) (0 < b < ∞, m ≥ 0) has boundary values in the spaceH _?m?1/2, where H_J (?∞ ?m(0, b) is the space of continuous linear functionals on ?W_m(0, b), the completion of the space of infinitely differentiable vector functions with compact support with respect to the norm \(\left\| u \right\|_{W_m (0, b) = (\left\| u \right\|_{L_2 (H_{m,} (0, b))} + \left\| u \right\|_{L_2 (H, (0, b))}^{(m)} )} \) . It follows that each function u(t, x) which is harmonic in the strip G = [0, b] x (?∞, ∞) and which is in the space that is dual to ? ₂ ^m (G) has limiting values as t→0 and t→b in the space \(W_2^{ - m - {\raise0.7ex\hbox{\(1\)} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0em}\!\lower0.7ex\hbox{\(2\)}}} ( - \infty ,\infty )\) .     

On the boundedness of solutions of nonlinear differential equations in Hilbert spaces

LetH be a complex Hilbert space and letB be the space of all bounded linear operators fromH intoH with the strong operator topology. We will give a boundedness result for the solutions of the differential equationx′=A(t)x+f(t,x) whereA: I=[t ₀, ∞)→B is continuous,f: I×H→H is also continuous and for every bounded setS⊂I×H there exists a constantM(S)>0 such that |f(t,x)−f(t,y)|≤M(S)|x−y|,(t,x), (t,y)∈S.

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Sunto SiaH uno spazio di Hilbert complesso e siaB lo spazio degli operatori lineari limitati daH inH, con la topologia forte. In questo lavoro si prova un risultato di limitatezza per le soluzioni dell'equazione differenzialex′=A(t)x+f(t,x), doveA: I=[t ₀, ∞)→B è continua,f: I×H→H è continua e per ogni insieme limitatoS⊂I×H esiste una costanteM(S)>0 tale che |f(t,x)−f(t,y)|≤M(S)|x−y| per ogni(t,x), (t,y)∈S.

     

Self-adjoint abstract differential operators

Let H be an abstract separable Hilbert space. We will consider the Hilbert space H₁ whose elements are functionsf(x) with domain H and we will also consider the set of self-adjoint operators Q(x) in H of the form Q(x)=A+B(x). In this formula AE, B(x)0, and the operator B(x) is bounded for all x. An operator L₀ is defined on the set of finite, infinitely differentiable (in the strong sense) functions y(x) H₁ according to the formula: L₀y=–y + Q(x)y (–0 is a self-adjoint operator in H₁ under the given assumptions.Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 65–72, July, 1969.     

关于Moore-Penrose逆T+连续性的一个注

设X为拓扑空间.H1和H2为Hilbert空间,T(·)为X到B(H1,H2)的连续映射.本文主要利用Tikhonov正则化算子给出了Moore-Penrose逆T x连续的充分必要条件.这个结果在计算数学中是很重要的.     

An existence result for variational problems and applications to some nonlinear hyperbolic and elliptic equations involving critical Sobolev exponents

Summary We prove the existence of nontrivial solutions for nonlinear equations of the type Lu=g(x, u) + ¦u¦^¯p–2u, ¯ p > 2, where L is a continuous self- adjoint linear operator in a Hilbert space H and ug(x, u) is a lower order perturbation of ¦u¦^¯p–1. We assume that ¯p is the critical exponent in the sense that the embedding H L^p, If is compact for 1p<¯p and is continuous (not necessarily compact) for p=¯p. From this result we deduce, for example, that u_tt -u- u=¦u¦^2/Nu, u L²(S^N×S¹) has at least one pair (–u, u) of solutions nonconstant with respect to t, provided that is sufficiently close to some eigenvalue of _tt–.Work supported by M.P.I. Italy (fondi 40%, 60%) and by G.N.A.F.A. of C.N.R.     

Conditions for the negative spectrum of the Schrodinger equation operator to be discrete and finite

Let H be a separable Hilbert space and let H₁ be a Hilbert space whose elements are vector functions f(x) (0x<) with domain H. Define the scalar product in H₁ to be (f(x), g(x))₁= . This paper concerns the study of the negative spectrum of the operatorl(y) =–y +Q(x)y, y'(0)–hy(0)=0, y(x) H₁, where Q(x) is a strongly measurable operator function in H which is completely continuous for almost all x, and where h is a fixed completely continuous operator in H. There is 1 reference.Translated from Matematicheskie Zametki, Vol. 2, No. 5, pp. 531–538, November, 1967.     

Evolution inclusions governed by subdifferentials in reflexive Banach spaces

The existence, uniqueness and regularity of strong solutions for Cauchy problem and periodic problem are studied for the evolution equation: where is the so-called subdifferential operator from a real Banach space V into its dual V*. The study in the Hilbert space setting (V = V* = H: Hilbert space) is already developed in detail so far. However, the study here is done in the V–V* setting which is not yet fully pursued. Our method of proof relies on approximation arguments in a Hilbert space H. To assure this procedure, it is assumed that the embeddings are both dense and continuous.     

Homomorphisms of weighted algebras of continuous functions

Summary A characterization is given of all continuous algebra homomorphisms H from the weighted space of continuous functions CV(X, E) into ℂ, where X is a completely regular Hausdorff space, E a locally convex algebra and V a family of weight functions that vanish at infinity. These homomorphisms are represented in the form H(f) = h(f(x)) with x in X and h a continuous algebra homomorphism on E. We then consider the space of homomorphisms of CV(X, E) and give some counterexamples to possible further generalizations. Entrata in Redazione il 18 febbraio 1977. ? Aspirant ? of the Belgian ? Nationaal Fonds voor Wetenschappelijk Onderzoek ?.     

On a continuous analog of the gradient method

In a real Hilbert space H we consider the nonlinear operator equation P(x)=0 and the continuous gradient methodx (t)= –P (x)^* P (x), x (0) = x₀. Two theorems on the convergence of the process (*) to the solution of the equation P(x)=0 are proved.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 421–426, April, 1968.     

Canonical State/Signal Shift Realizations of Passive Continuous Time Behaviors

This work is devoted to the construction of canonical passive and conservative state/signal shift realizations of arbitrary passive continuous time behaviors. By definition, a passive future continuous time behavior is a maximal nonnegative right-shift invariant subspace of the Kreĭn space L²([0,￥);W){L^2([0,\infty);\mathcal W)}, where W{\mathcal W} is a Kreĭn space, and the inner product in L²([0,￥);W){L^2([0,\infty);\mathcal W)} is the one inherited from W{\mathcal W}. A state/signal system S = (V;X,W){\Sigma=(V;\mathcal X,\mathcal W)}, with a Hilbert state space X{\mathcal X} and a Kreĭn signal space W{\mathcal W}, is a dynamical system whose classical trajectories (x, w) on [0, ∞) satisfy x ? C¹([0,￥);X){x\in C^1([0,\infty);\mathcal X)}, w ? C([0,￥);W){w \in C([0,\infty);\mathcal W)}, and

Function Spaces Associated with Schrödinger Operators: The Pöschl-Teller Potential

We address the function space theory associated with the Schrödinger operator H = ?d²/dx² + V. The discussion is featured with potential V (x) = ?n(n + 1) sech²x, which is called in quantum physics Pöschl-Teller potential. Using a dyadic system, we introduce Triebel-Lizorkin spaces and Besov spaces associated with H. We then use interpolation method to identify these spaces with the classical ones for a certain range of p, q > 1. A physical implication is that the corresponding wave function ψ(t, x) = e^?itHf(x) admits appropriate time decay in the Besov space scale.     

Cauchy problem for an essentially infinite-dimensional parabolic equation with variable coefficients

The Cauchy problem for the equation with positive essentially infinite-dimensional functionalsj(x) is studied in a properly chosen Banach space of functions on an infinite-dimensional separable real Hilbert space. Kiev Polytechnic Institute, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 663–670, June, 1994.     

Generalized kinetic equations

We study the abstract differential equation on a Hilbert space H, which represents a variety of different kinetic equations. T is assumed bounded and self-adjoint on H, and A (unbounded) positive self-adjoint and Fredholm. For partial range boundary conditions and 0x<, we prove existence and (non-) uniqueness theorems and give representations of the solution. Various examples from neutron transport, radiative transfer of polarized and unpolarized light, and electron transport are given.This paper is dedicated to K.M. Case on the occasion of his sixtieth birthday     

Existence Results on the Second-Order Nonconvex Sweeping Processes with Perturbations

We prove several existence theorems for second-order differential inclusions of the form , when K and F are two convex or nonconvex set-valued mappings taking their values in a Hilbert space.     

Homogenization of Hamilton‐Jacobi‐Bellman equations with respect to time‐space shifts in a stationary ergodic medium

We consider a family {u_? (t, x, ω)}, ? < 0, of solutions to the equation ?u_?/?t + ?Δu_?/2 + H (t/?, x/?, ?u_?, ω) = 0 with the terminal data u_?(T, x, ω) = U(x). Assuming that the dependence of the Hamiltonian H(t, x, p, ω) on time and space is realized through shifts in a stationary ergodic random medium, and that H is convex in p and satisfies certain growth and regularity conditions, we show the almost sure locally uniform convergence, in time and space, of u_?(t, x, ω) as ? → 0 to the solution u(t, x) of a deterministic averaged equation ?u/?t + H?(?u) = 0, u(T, x) = U(x). The “effective” Hamiltonian H? is given by a variational formula. © 2007 Wiley Periodicals, Inc.     

BV Solutions to Evolution Problems with Time-Dependent Domains

In a Hilbert space H we consider evolution problems -du(t) A(t)u(t) on some interval [0, T], where every A(t): D(A(t)) 2 H is a maximal monotone operator, and the correspondence t A(t) is – in a suitable sense – of bounded variation or absolutely continuous.