The author presents an exhausting description of Lipschitz continuous and some other classes of nonlinear superposition operators acting in modular metric spaces of functions of a real variable of finite generalized variation in the sense of M. Schramm
[Trans. Am. Math. Soc. 287, 49–63 (1985; Zbl 0567.26009
)] with values in metric semigroups and cones. A complete description of generators for Lipschitz continuous, bounded and some other classes of superposition Nemytskii operators mapping in these semigroups and cones is given, which extends recent results by J. Matkowski
and J. Miś
[Math. Nachr. 117, 155–159 (1984; Zbl 0566.47033
)] and some other authors.