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Regular version of the site

Article

Fixed points of modular contractive maps

Доклады Академии наук. 2012. Vol. 86. No. 1. P. 515-518.

In the framework of modular metric spaces, introduced by the author in 2006, we define a new notion of modular convergence, which is more weak than the metric convergence, and establish the necessary and sufficient condition on the modular under consideration, under which the modular convergence is equivalent to the metric one. We introduce the notion of modular contractive maps, study their relationship with Lipschitz continuous maps with respect to the corresponding metrics and present the main result of the paper concerning the existence of fixed points of modular contractive maps. As an application of the fixed point theorem, we prove the existence of solutions to a Caratheodory-type differential equation with the right hand side from the Orlicz space.