Measures of noncompactness in locally convex spaces and fixed point theory for the sum of two operators on unbounded convex sets

Józef Banaś; Afif Ben Amar

Commentationes Mathematicae Universitatis Carolinae (2013)

  • Volume: 54, Issue: 1, page 21-40
  • ISSN: 0010-2628

Abstract

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In this paper we prove a collection of new fixed point theorems for operators of the form T + S on an unbounded closed convex subset of a Hausdorff topological vector space ( E , Γ ) . We also introduce the concept of demi- τ -compact operator and τ -semi-closed operator at the origin. Moreover, a series of new fixed point theorems of Krasnosel’skii type is proved for the sum T + S of two operators, where T is τ -sequentially continuous and τ -compact while S is τ -sequentially continuous (and Φ τ -condensing, Φ τ -nonexpansive or nonlinear contraction or nonexpansive). The main condition in our results is formulated in terms of axiomatic τ -measures of noncompactness. Apart from that we show the applicability of some our results to the theory of integral equations in the Lebesgue space.

How to cite

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Banaś, Józef, and Ben Amar, Afif. "Measures of noncompactness in locally convex spaces and fixed point theory for the sum of two operators on unbounded convex sets." Commentationes Mathematicae Universitatis Carolinae 54.1 (2013): 21-40. <http://eudml.org/doc/252455>.

@article{Banaś2013,
abstract = {In this paper we prove a collection of new fixed point theorems for operators of the form $T+S$ on an unbounded closed convex subset of a Hausdorff topological vector space $(E,\Gamma )$. We also introduce the concept of demi-$\tau $-compact operator and $\tau $-semi-closed operator at the origin. Moreover, a series of new fixed point theorems of Krasnosel’skii type is proved for the sum $T+S$ of two operators, where $T$ is $\tau $-sequentially continuous and $\tau $-compact while $S$ is $\tau $-sequentially continuous (and $\Phi _\{\tau \}$-condensing, $\Phi _\{\tau \}$-nonexpansive or nonlinear contraction or nonexpansive). The main condition in our results is formulated in terms of axiomatic $\tau $-measures of noncompactness. Apart from that we show the applicability of some our results to the theory of integral equations in the Lebesgue space.},
author = {Banaś, Józef, Ben Amar, Afif},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$\tau $-measure of noncompactness; $\tau $-sequential continuity; $\Phi _\{\tau \}$-condensing operator; $\Phi _\{\tau \}$-nonexpansive operator; nonlinear contraction; fixed point theorem; demi-$\tau $-compactness; operator $\tau $-semi-closed at origin; Lebesgue space; integral equation; -measure of noncompactness; -sequential continuity; -condensing operator; -nonexpansive operator; demi--compactness; -semi-closed operator},
language = {eng},
number = {1},
pages = {21-40},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Measures of noncompactness in locally convex spaces and fixed point theory for the sum of two operators on unbounded convex sets},
url = {http://eudml.org/doc/252455},
volume = {54},
year = {2013},
}

TY - JOUR
AU - Banaś, Józef
AU - Ben Amar, Afif
TI - Measures of noncompactness in locally convex spaces and fixed point theory for the sum of two operators on unbounded convex sets
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2013
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 54
IS - 1
SP - 21
EP - 40
AB - In this paper we prove a collection of new fixed point theorems for operators of the form $T+S$ on an unbounded closed convex subset of a Hausdorff topological vector space $(E,\Gamma )$. We also introduce the concept of demi-$\tau $-compact operator and $\tau $-semi-closed operator at the origin. Moreover, a series of new fixed point theorems of Krasnosel’skii type is proved for the sum $T+S$ of two operators, where $T$ is $\tau $-sequentially continuous and $\tau $-compact while $S$ is $\tau $-sequentially continuous (and $\Phi _{\tau }$-condensing, $\Phi _{\tau }$-nonexpansive or nonlinear contraction or nonexpansive). The main condition in our results is formulated in terms of axiomatic $\tau $-measures of noncompactness. Apart from that we show the applicability of some our results to the theory of integral equations in the Lebesgue space.
LA - eng
KW - $\tau $-measure of noncompactness; $\tau $-sequential continuity; $\Phi _{\tau }$-condensing operator; $\Phi _{\tau }$-nonexpansive operator; nonlinear contraction; fixed point theorem; demi-$\tau $-compactness; operator $\tau $-semi-closed at origin; Lebesgue space; integral equation; -measure of noncompactness; -sequential continuity; -condensing operator; -nonexpansive operator; demi--compactness; -semi-closed operator
UR - http://eudml.org/doc/252455
ER -

References

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