We give an example of an unbounded, convex, and closed set <mml:math alttext="$C$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> in the Hilbert space <mml:math alttext="$l^2$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>l</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math> with the following two properties: (i) <mml:math alttext="$C$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> has the approximate fixed-point property for nonexpansive
mappings, (ii) <mml:math alttext="$C$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> is not contained in a block for every orthogonal basis in <mml:math alttext="$l^2$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>l</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math>.

Abstract and Applied Analysis

A remark on the approximate fixed-point property

Abstract

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