We study two aspects of the stability problem for various types of branching fractions using the domains of the elements, the region of convergence, and limit-periodic fractions.Despite the fact that one of the most important properties of continued fractions and their multi-dimensional generalizations is the property of computational stability, only a comparatively small number of papers has been-devoted to this problem [1,3,[5][6][7].

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In this paper, we consider the initial-value problem for parabolic variational inequalities (subdifferential inclusions) with Volterra type operators. We prove the existence and the uniqueness of the solution. Furthermore, the estimates of the solution are obtained. The results are achieved using the Banach's fixed point theorem (the principle of compression mappings). The motivation for this work comes from the evolutionary variational inequalities arising in the study of frictionless contact problems for linear viscoelastic materials with long-term memory. Also, such kind of problems have their application in constructing different models of the injection molding processes.

By regarding a two-dimensional continued fraction as a function of its elements and applying recursion relations for its tails, we establish formulas for the first partial derivatives of the fraction, on the basis of which we construct linear approximations of limit-periodic two-dimensional continued fractions.

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