A duality theorem in Hilbert space

The distance from a point to a linear variety defined by a finite intersection of hyperplanes is equal to the maximum of the distances from the point to all the hyperplanes which contain the variety.     

Vitali covering theorem in Hilbert space

It is shown that the statement of the Vitali Covering Theorem does not hold for a certain class of measures in a Hilbert space. This class contains all infinite-dimensional Gaussian measures.

     

A viability theorem for set-valued states in a Hilbert space

Applications in robust control problems and shape evolution motivate the mathematical interest in control problems whose states are compact (possibly non-convex) sets rather than vectors. This leads to evolutions in a basic set which can be supplied with a metric (like the well-established Pompeiu–Hausdorff distance), but it does not have an obvious linear structure. This article extends differential inclusions with state constraints to compact-valued states in a separable Hilbert space H. The focus is on sufficient conditions such that a given constraint set (of compact subsets) is viable a.k.a. weakly invariant. Our main result extends the tangential criterion in the well-known viability theorem (usually for differential inclusions in a vector space) to the metric space of non-empty compact subsets of H.     

Regularized continuous analog of the newton method for monotone equations in a Hilbert space

In a Hilbert space we construct a regularized continuous analog of the Newton method for nonlinear equation with a Fréchet differentiable and monotone operator. We obtain sufficient conditions of its strong convergence to the normal solution of the given equation under approximate assignment of the operator and the right-hand of the equation.     

A representation theorem for Schauder bases in Hilbert space

A sequence of vectors in a separable Hilbert space is said to be a Schauder basis for if every element has a unique norm-convergent expansion

If, in addition, there exist positive constants and such that

then we call a Riesz basis. In the first half of this paper, we show that every Schauder basis for can be obtained from an orthonormal basis by means of a (possibly unbounded) one-to-one positive self adjoint operator. In the second half, we use this result to extend and clarify a remarkable theorem due to Duffin and Eachus characterizing the class of Riesz bases in Hilbert space.

     

Strong convergence theorem for nonexpansive semigroups in Hilbert space

The purpose of this paper is to prove the strong convergence of a method combining the descent method and the hybrid method in mathematical programming for finding a point in the common fixed point set of a semigroup of nonexpansive mappings in Hilbert space.     

Fubini theorem for anticipating stochastic integrals in Hilbert space

Let (X,l,) be a measure space, letW be a cylindrical Hilbert-Wiener process, and let be an anticipating integrable process-valued function onX. We prove, under natural assumptions on, that there exists a measurable version Y_x,x X, of the anticipating integral of(x) such that the integral _x Y_x(dx) is a version of the anticipating integral of _X (x)(dx). We apply this anticipating Fubini theorem to study solutions of a class of stochastic evolution equations in Hilbert space.     

An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space

J. B. Baillon [C. R. Acad. Sci. Paris Ser. A.280 (1975), 1511–1514] proved an ergodic theorem for a single nonexpansive mapping in a Hilbert space, which is a nonlinear version of von Neumann's mean ergodic theorem. In this paper, we study the ergodic behavior of a semigroup of nonexpansive mappings. We try to find a sequence of means on the semigroup, generalizing the Cesàro means on $\text{N}$, such that the corresponding sequence of nonexpansive mappings converges to a projection onto the set of common fixed-points. Our method of proof is an appropriate modification of A. Pazy's proof [Israel J. Math.26 (1977), 197–204] of Baillon's theorem.     

Sequential stable Kuhn-Tucker theorem in nonlinear programming

A general parametric nonlinear mathematical programming problem with an operator equality constraint and a finite number of functional inequality constraints is considered in a Hilbert space. Elements of a minimizing sequence for this problem are formally constructed from elements of minimizing sequences for its augmented Lagrangian with values of dual variables chosen by applying the Tikhonov stabilization method in the course of solving the corresponding modified dual problem. A sequential Kuhn-Tucker theorem in nondifferential form is proved in terms of minimizing sequences and augmented Lagrangians. The theorem is stable with respect to errors in the initial data and provides a necessary and sufficient condition on the elements of a minimizing sequence. It is shown that the structure of the augmented Lagrangian is a direct consequence of the generalized differentiability properties of the value function in the problem. The proof is based on a “nonlinear” version of the dual regularization method, which is substantiated in this paper. An example is given illustrating that the formal construction of a minimizing sequence is unstable without regularizing the solution of the modified dual problem.     

Peano's theorem in an infinite-dimensional Hilbert space is false even in a weakened formulation

We formulate a continuous function FR×HH, where H is a separable Hilbert space such that the Cauchy problem. x(t)=F(t, x(t)), x(t₀)=x₀ has no solution in any neighborhood of the point t₀, no matter what t₀ R and x₀ H are considered.Translated from Matematicheskie Zametki, Vol. 15, No. 3, pp. 467–477, March, 1974.In conclusion, the author thanks O. G. Smolyanov and V. I. Averbukh for their constant interest and for a number of useful remarks.     

A short proof of the levy continuity theorem in Hilbert space

A short proof of the Levy continuity theorem in Hilbert space. In the theory of the normal distribution on a real Hilbert spaceH, certain functionsφ have been shown by L. Gross to give rise to random variablesφ∼ in a natural way; in particular, this is the case for functions which are “uniformly τ-continuous near zero”. Among such functions are the characteristic functionsφ of probability distributionsm onH, given byφ(y)=∫e ^i(y,x)dm(x). The following analogue of the Levy continuity theorem has been proved by Gross: Letφ _j be the characteristic function of the probability measurem _j onH, Then necessary and sufficient that ∫f dm _j → ∫f dm for some probability measurem and all bounded continuousf, is that there exists a functionφ, uniformly τ-continuous near zero, withφ _j∼ →φ∼ in probability.φ turns out, of course, to be the characteristic function ofm. In the present paper we give a short proof of this theorem. Research supported by National Science Foundation Grant GP-3977.     

Convergence theorem for I-asymptotically quasi-nonexpansive mapping in Hilbert space

Let H be a Hilbert space with inner product (⋅,⋅) and ‖⋅‖ norm, and let K be weakly compact a subset of H. Let be nonlinear mapping and be a nonlinear bounded mapping. In this paper, we define the I-asymptotically quasi-nonexpansive mapping in Hilbert space. If T is an I-asymptotically quasi-nonexpansive mapping, then we prove that , for u∈K as n→∞, is weakly almost convergent to its asymptotic center.