Modular metric spaces, II: Application to superposition operators

Abstract

Applying the theory of modular metric spaces developed in the first part of this paper [V.V. Chistyakov (2009) [1]] we define a metric semigroup and an abstract convex cone of functions of finite generalized variation in the approach of Schramm [M. Schramm, Functions of Φ -bounded variation and Riemann–Stieltjes integration, Trans. Amer. Math. Soc. 287 (1) (1985) 49–63], which are significantly larger as compared to the spaces of bounded variation in the sense of Jordan, Wiener–Young and Waterman. We present a complete description of generators of Lipschitz continuous, bounded and some other classes of superposition Nemytskii operators mapping in these semigroups and cones, which extends recent results by Matkowski and Miś [J. Matkowski, J. Miś, On a characterization of Lipschitzian operators of substitution in the space BV 〈 a , b 〉 , Math. Nachr. 117 (1984) 155–159], Maligranda and Orlicz [L. Maligranda, W. Orlicz, On some properties of functions of generalized variation, Monatsh. Math 104 (1987) 53–65], Zawadzka [G. Zawadzka, On Lipschitzian operators of substitution in the space of set-valued functions of bounded variation, Rad. Mat. 6 (1990) 279–293] and Chistyakov [V.V. Chistyakov, Mappings of generalized variation and composition operators, J. Math. Sci. (New York) 110 (2) (2002) 2455–2466, V.V. Chistyakov, Lipschitzian Nemytskii operators in the cones of mappings of bounded Wiener φ -variation, Folia Math. 11 (1) (2004) 15–39].