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Working paper

A fixed point theorem for contractions in modular metric spaces

arxiv.org. math. Cornell University, 2011. No. 1112.5561v1.
The notion of a (metric) modular on an arbitrary set and the corresponding modular space, more general than a metric space, were introduced and studied recently by the author [V.V. Chistyakov, Metric modulars and their application, Dokl. Math. 73 (1) (2006) 32–35, and Modular metric spaces, I: Basic concepts, Nonlinear Anal. 72 (1) (2010) 1–14]. In this paper we establish a fixed point theorem for contractive maps in modular spaces. It is related to contracting rather “generalized average velocities” than metric distances, and the successive approximations of fixed points converge to the fixed points in a weaker sense as compared to the metric convergence.