DEMONSTRATIO MATHEMATICA Vol. XLVII No 1 2014 Mireya Bracamonte, Jurancy Ereú, José Giménez and Nelson Merentes UNIFORMLY CONTINUOUS SUPERPOSITION OPERATORS IN THE SPACE OF FUNCTIONS OF BOUNDED n-DIMENSIONAL Φ-VARIATION Abstract. We prove that if a superposition operator maps a subset of the space of all metric-vector-space-valued-functions of bounded n-dimensional Φ-variation into another such space, and is uniformly continuous, then the generating function of the operator is an affine function in the functional variable. 1. Introduction Given two (non-empty) sets A and B, the notation B A will stand for the set of all functions from A to B. As usual, if M,N are linear spaces, the notation LpM,N q stands for the set of all linear maps from M to N. Let A, B and C be non-empty sets. If h : A ˆ C Ñ B is a given function, X Ă C A and Y Ă B A are linear spaces then the nonlinear superposition (Nemytskij) operator H : X Ñ Y , generated by the function h, is defined as pHf qptq :“ hpt,f ptqq, t P A. This operator plays a central role in various mathematical fields, e.g. in the theory of nonlinear integral equations, and has been studied thoroughly. Perhaps, the most important problem concerning the theory of the superposi- tion operator is to establish necessary and sufficient conditions guaranteeing that this operator maps a given function space into itself. These conditions are called acting conditions (e.g., (non-linear) boundedness, continuity, lo- cal or global Lipschitz conditions, etc.). On the other hand, superposition operators being the simplest operators between function spaces, another im- portant problem is to determine if a certain given operator, that acts between some given function spaces, can be redefined via the notion of superposition, 2010 Mathematics Subject Classification : 26B30, 26B40. Key words and phrases : superposition operator, bounded Φ-variation, metric semi- group. DOI: 10.2478/dema-2014-0005 c  Copyright by Faculty of Mathematics and Information Science, Warsaw University of Technology Unauthenticated Download Date | 11/10/19 12:38 AM