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On the Existence of Common Fixed Points for Semigroups of Nonlinear Mappings in Modular Function Spaces

100%

Kozlowski W. M.

Commentationes Mathematicae

2011

tom 51

nr 1

Let \(C\) be a \(\rho\)-bounded, \(\rho\)-closed, convex subset of a modular function space \(L_\rho\). We investigate the existence of common fixed points for semigroups of nonlinear mappings \(T_t\colon C\to C\), i.e. a family such that \(T_0(x) = x\), \(T_{s+t} = T_s (T_t (x))\), where each \(T_t\) is either \(\rho\)-contraction or \(\rho\)-nonexpansive. We also briefly discuss existence of such semigroups and touch upon applications to differential equations.

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

80%

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.

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2 Commentationes Mathematicae

2 Kozlowski W. M.

1 2012

1 2011

Wyniki wyszukiwania

On the Existence of Common Fixed Points for Semigroups of Nonlinear Mappings in Modular Function Spaces

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