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1
Content available Ergodic properties of random infinite products of nonexpansive mappings
Reich S., Zaslavski A. J.
Journal of Mathematics and Applications
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2017
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Vol. 40
149--159
EN
In this paper we are concerned with the asymptotic behavior of random (unrestricted) infinite products of nonexpansive selfmappings of closed and convex subsets of a complete hyperbolic space. In contrast with our previous work in this direction, we no longer assume that these subsets are bounded. We first establish two theorems regarding the stability of the random weak ergodic property and then prove a related generic result. These results also extend our recent investigations regarding nonrandom infinite products.
2
Content available Structure of solutions of nonautonomous optimal control problems in metric spaces
Zaslavski A. J.
Journal of Mathematics and Applications
|
2015
|
Vol. 38
151--169
EN
We establish turnpike results for a nonautonomous discrete-time optimal control system describing a model of economic dynamics.
3
Porosity and the bounded linear regularity property
Reich S., Zaslavski A. J.
Journal of Applied Analysis
|
2014
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Vol. 20, nr 1
1--6
EN
H. H. Bauschke and J. M. Borwein showed that in the space of all tuples of bounded, closed, and convex subsets of a Hilbert space with a nonempty intersection, a typical tuple has the bounded linear regularity property. This property is important because it leads to the convergence of infinite products of the corresponding nearest point projections to a point in the intersection. In the present paper we show that the subset of all tuples possessing the bounded linear regularity property has a porous complement. Moreover, our result is established in all normed spaces and for tuples of closed and convex sets, which are not necessarily bounded.
4
Content available remote Structure of approximate solutions for a class of optimal control systems
Zaslavski A. J.
Journal of Mathematics and Applications
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2011
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Vol. 34
135--149
EN
We study a turnpike property of approximate solutions of a discrete-time control system with a compact metric space of states which arises in economic dynamics. To have this property means that the approximate solutions of the optimal control problems are determined mainly by an objective function, and are essentially independent of the length of the interval, for all sufficiently large intervals. We show that the turnpike property is stable under perturbations of an objective function.
5
Content available remote Convergence of approximate solutions of variational problems
Zaslavski A. J.
Control and Cybernetics
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2009
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Vol. 38, no 4B
1607-1629
EN
We study the structure of approximate solutions of autonomous variational problems on large finite intervals. In our previous research, which was summarized in Zaslavski (2006b), we showed that approximate solutions are determined mainly by the integrand, and are essentially independent of the choice of time interval and data, except in regions close to the endpoints of the time interval. In the present paper we establish convergence of approximate solutions in regions close to the endpoints of the time intervals.
6
Content available remote Stability of solutions for a class of convex minimization problems on reflective Banach spaces
Zaslavski A. J.
Journal of Mathematics and Applications
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2008
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Vol. 30
151-160
EN
In this paper we study stability of solutions of minimization problems �(x) → min, x ∈ C, where � is a convex lower semicontinuous function and a set C is the countable intersection of a decreasing sequence of closed sets Ci in a reflexive Banach space X.
7
Content available remote Turnpike properties of approximate solutions of autonomous variational problems
Zaslavski A. J.
Control and Cybernetics
|
2008
|
Vol. 37, no 2
491-512
EN
In this work we study the structure of approximate solutions of autonomous variational problems with vector-valued functions. We are interested in turnpike properties of these solutions, which are independent of the length of the interval, for all sufficiently large intervals. We show that the turnpike properties are stable under small perturbations of integrands.
8
Content available remote Two results on Jachymski-Schörder-Stein contractions
Reich S., Zaslavski A. J.
Bulletin of the Polish Academy of Sciences. Mathematics
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2008
|
Vol. 56, no 1
53-58
EN
We establish two fixed point theorems for certain mappings of contractive type.
9
Content available remote On extremals of a variational problem arising in crystallography
Zaslavski A. J.
Journal of Mathematics and Applications
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2007
|
Vol. 29
157-166
EN
In this paper we study the structure of minirnizers of variational problems which were introduced by Hannon, Marcus and Mizel (ESAIM Control Optim. Calc. Var., 2003) to describe step-terraces on surfaces of so-called "unorthodox" crystals. These variational problems are associated with two positive parameters. We will show that if one of these parameters is not smali and the second parameter is large, then the rainimizer is a constant function.
10
Asymptotic behavior of inexact orbits for a class of operators in complete metric spaces
Butnariu D., Reich S., Zaslavski A. J.
Journal of Applied Analysis
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2007
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Vol. 13, nr 1
1-11
EN
We exhibit a class of nonlinear operators with the property that their iterates converge to their unique fixed points even when com- putational errors are present. We also showthat most (in the sense of the Baire category) elements in an appropriate complete metric space of operators do, in fact, possess this property.
11
Content available remote Existence of solutions in parametric optimization and porosity
Zaslavski A. J.
Journal of Mathematics and Applications
|
2006
|
Vol. 28
161-184
EN
In this work we study the parametric family of the minimization problems f (b, x) → min, x ∈ X on a complete metric space X with a parameter b which belongs to a Hausdorff compact space Β. Here f(•,•) belongs to a space of functions on Β x X, say Μ, endowed with an appropriate metric. We study the set of all functions f(•,•) ∈ Μ for which the corresponding parametric family of the minimization problems has solutions for all parameters b ∈ Β. We show that the complement of this set is not only of the first category but also a σ-porous set. This result and its extensions are obtained as realizations of a variational principle.
12
Asymptotic behavior of relatively nonexpansive operators in Banach spaces
Butnariu D., Reich S., Zaslavski A. J.
Journal of Applied Analysis
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2001
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Vol. 7, nr 2
151-174
EN
Let K be a closed convex subset of a Banach space X and let F be a nonempty closed convex subset of K. We consider complete metric spaces of self-mappings of K which fix all the points of F and are relatively nonexpansive with respect to a given convex function f on X. We prove (under certain assumptions on f) that the iterates of a generic mapping in these spaces converge strongly to a retraction onto F.
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