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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.

tom 31

nr 2

231-247

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.

Set-valued stochastic integrals and stochastic inclusions in a plane

100%

Sosulski W.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2001

tom 21

nr 2

249-259

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 $z_{s,t} ∈ φ_{s,t} + ∫_{0}^{s} ∫_{0}^{t} F_{u,v}(z_{u,v})dudv + ∫_{0}^{s} ∫_{0}^{t}G_{u,v}(z_{u,v})dw_{u,v}$

On weak compactness and lower closure results for Pettis integrable (multi) functions

75%

Balder E. J. , Sambucini A. R.

tom Vol. 52, nr 1

53-61

In [4, 5, 7] an abstract, versatile approach was given to sequential weak compactness and lower closure results for scalarly integrable functions and multifunctions. Its main tool is an abstract version of the Komlos theorem, which applies to scalarly integrable functions. Here it is shown that this same approach also applies to Pettis integrable multifunctions, because the abstract Komlos theorem can easily be extended so as to apply to generalized Pettis integrable functions. Some results in the literature are thus unified.

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

2012

tom 52

nr 2

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.

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

2012

tom Vol. 52, [Z] 2

113--136

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 Tt: C → C. Each of Tt: is assumed to be pointwise Lipschitzian, that is, there exists a family of functions αt: C → [0, ∞) such that ||Tt(x) — Tt(y)\\ ≤ αt:(x) || - y|| for x,y € 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.