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We show that the recent result of Bełdziński et al. on continuous dependence of fixed points on parameters, which extends a classical parametrized version of the Banach fixed point theorem, may be treated as a particular case of the contraction principle for a Nemytskiĭ operator on a space of continuous functions with values in a metric space. Next, we establish a generalization of the result of Bełdziński et al. involving an equicontinuous family of mappings having contractive p$$ p $$th iterates for some p∈ℕ$$ p\in \mathbb{N} $$. As an application, we obtain a theorem on continuous dependence of solutions of Cauchy's problem on both the initial values and the parameters. Finally, we establish two remetrization theorems and we discuss a possibility of deriving our main result from the Banach contraction principle by using a remetrization of the space. [ABSTRACT FROM AUTHOR]

We revisit Perov's fixed point theorem for selfmaps of a set endowed with a vector metric taking values in the Euclidean space Rm. In particular, we show that this result is subsumed by the classical Banach contraction principle. We also obtain a generalization of Perov's theorem by con- sidering mappings on K-metric spaces satisfying a nonlinear Lipschitz condition. Two applications are presented and some characterizations of convergence in K-metric spaces are given. [ABSTRACT FROM AUTHOR]