Locally Lipschitz Composition Operators in Space of the Functions of Bounded<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">Φ</mml:mi></mml:math>-Variation
Locally Lipschitz Composition Operators in Space of the Functions of Bounded<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">Φ</mml:mi></mml:math>-Variation
We give a necessary and sufficient condition on a function<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>h</mml:mi><mml:mo>:</mml:mo><mml:mi mathvariant="double-struck">R</mml:mi><mml:mo>→</mml:mo><mml:mi mathvariant="double-struck">R</mml:mi></mml:math>under which the nonlinear composition operator<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi>H</mml:mi></mml:mrow></mml:math>, associated with the function<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:math>,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mi>H</mml:mi><mml:mi>u</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math>, acts in the space<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mi>κ</mml:mi><mml:mi mathvariant="normal">Φ</mml:mi><mml:mi>B</mml:mi><mml:mi>V</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:math>and satisfies a local Lipschitz condition.