Opuscula Math. 32, no. 2 (2012), 239-247
http://dx.doi.org/10.7494/OpMath.2012.32.2.239

Opuscula Mathematica

# Uniformly continuous composition operators in the space of bounded Φ-variation functions in the Schramm sense

Tomás Ereú
Nelson Merentes
José L. Sánchez
Małgorzata Wróbel

Abstract. We prove that any uniformly continuous Nemytskii composition operator in the space of functions of bounded generalized $$\Phi$$-variation in the Schramm sense is affine. A composition operator is locally defined. We show that every locally defined operator mapping the space of continuous functions of bounded (in the sense of Jordan) variation into the space of continous monotonic functions is constant.

Keywords: $$\Phi$$-variation in the sense of Schramm, uniformly continuous operator, regularization, Jensen equation, locally defined operators.

Mathematics Subject Classification: 47H30.

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• Tomás Ereú
• Universidad Nacional Abierta, Centro Local Lara (Barquisimeto)-Venezuela
• Nelson Merentes
• Universidad Central de Venezuela, Escuela de Matemáticas, Caracas-Venezuela
• José L. Sánchez
• Universidad Central de Venezuela, Escuela de Matemáticas, Caracas-Venezuela
• Małgorzata Wróbel
• Jan Długosz University, Institute of Mathematics and Computer Science, 42-200 Częstochowa, Poland
• Revised: 2011-04-12.
• Accepted: 2011-04-14.