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Weak compactness in the space of operator valued measures $M_ba(Σ,𝓛(X,Y))$ and its applications

100%

Ahmed N.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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2011

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tom 31

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nr 2

231-247

EN

In this note we present necessary and sufficient conditions characterizing conditionally weakly compact sets in the space of (bounded linear) operator valued measures $M_{ba}(Σ,𝓛(X,Y))$. This generalizes a recent result of the author characterizing conditionally weakly compact subsets of the space of nuclear operator valued measures $M_{ba}(Σ,𝓛₁(X,Y))$. This result has interesting applications in optimization and control theory as illustrated by several examples.

2

Set-valued stochastic integrals and stochastic inclusions in a plane

100%

Sosulski W.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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2001

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tom 21

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nr 2

249-259

EN

We present the concepts of set-valued stochastic integrals in a plane and prove the existence of a solution to stochastic integral inclusions of the form $z_{s,t} ∈ φ_{s,t} + ∫_{0}^{s} ∫_{0}^{t} F_{u,v}(z_{u,v})dudv + ∫_{0}^{s} ∫_{0}^{t}G_{u,v}(z_{u,v})dw_{u,v}$

3

On the construction of common fixed points for semigroups of nonlinear mappings in uniformly convex and uniformly smooth Banach spaces

51%

Kozlowski W. M.

Commentationes Mathematicae

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2012

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tom 52

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nr 2

EN

Let $C$ be a bounded, closed, convex subset of a uniformly convex and uniformly smooth Banach space $X$. We investigate the weak convergence of the generalized Krasnosel'skii-Mann and Ishikawa iteration processes to common fixed points of semigroups of nonlinear mappings $T_t\colon C \to C$. Each of $T_t$ is assumed to be pointwise Lipschitzian, that is, there exists a family of functions $\alpha_t\colon C \to [0, \infty)$ such that $\|T_t(x) - T_t (y)\| \leq\alpha_t (x)\|x -y\|$ for $x, y \in C$. The paper demonstrates how the weak compactness of $C$ plays an essential role in proving the weak convergence of these processes to common fixed points.

Ograniczanie wyników

2 Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1 Commentationes Mathematicae

1 Ahmed N.

1 Kozlowski W. M.

1 Sosulski W.

1 2012

1 2011

1 2001

Weak compactness in the space of operator valued measures $M_ba(Σ,𝓛(X,Y))$ and its applications

Set-valued stochastic integrals and stochastic inclusions in a plane

On the construction of common fixed points for semigroups of nonlinear mappings in uniformly convex and uniformly smooth Banach spaces