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On Nonlinear Differential Equations in Generalized Musielak-Orlicz Spaces

100%

Kozlowski W. M.

Commentationes Mathematicae

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2013

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tom 53

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nr 2

EN

We consider ordinary differential equations $u'(t)+(I-T)u(t)=0$, where an unknown function takes its values in a given modular function space being a generalization of Musielak-Orlicz spaces, and $T$ is nonlinear mapping which is nonexpansive in the modular sense. We demonstrate that under certain natural assumptions the Cauchy problem related to this equation can be solved. We also show a process for the construction of such a solution. This result is then linked to the recent results of the fixed point theory in modular function spaces.

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Strong convergence of implicit iteration processes for nonexpansive semigroups in Banach spaces

100%

Kozlowski W. M.

Commentationes Mathematicae

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2014

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tom 54

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nr 2

EN

Let $C$ be a convex compact subset of a uniformly convex Banach space. Let $\{T_t\}_{t \geq0}$ be a strongly-continuous nonexpansive semigroup on $C$. Consider the iterative process defined by the sequence of equations $$x_{k+1} =c_k T_{t_{k+1}}(x_{k+1})+(1-c_k)x_k.$$ We prove that, under certain conditions on $\{c_k\}$ and $\{t_k\}$, the sequence $\{x_k\}_{n=1}^\infty$ converges strongly to a common fixed point of the semigroup $\{T_t\}_{t \geq0}$. There are known results on convergence of such iterative processes for nonexpansive semigroups in Hilbert spaces and Banach spaces with the Opial property, and also weak convergence results in Banach spaces that are simultaneously uniformly convex and uniformly smooth. In this paper, we do not assume the Opial property or uniform smoothness of the norm.

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

100%

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

3 Commentationes Mathematicae

3 Kozlowski W. M.

1 2014

1 2013

1 2012

On Nonlinear Differential Equations in Generalized Musielak-Orlicz Spaces

Strong convergence of implicit iteration processes for nonexpansive semigroups in Banach spaces

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