Metric regularity, fixed points and some associated problems of variational analysis

Abstract

The paper is basically concerned with interrelations between metric regularity and metric critical point theories. At the early stage of developments of regularity theory there was a feeling that every regularity criterion can be obtained with the help of a suitable (set-valued) contraction mapping principle. On the other hand, the assumptions of the well-known metric fixed point theorems (e.g., of Nadler (1969) or of Dontchev and Hager (1994)) imply metric regularity of the inverse mapping. The message of the paper is that, nonetheless, the theories are independent although the fundamental principles on which they are based have much in common. The first main result of the paper guarantees the existence of a fixed point if the contraction property holds only along the orbits of the mapping (rather than for arbitrary pairs of points in the feasible area) and actually only on some bounded pieces of the orbits determined by a certain “regularity horizon” condition. The bulk of the paper is devoted to the study of “two maps paradigm” when we have two set-valued mappings acting in opposite directions and we are interested in the existence of the so-called double fixed point. We consider two types of iteration procedures to get a fixed point, one based on simple Picard’s iterations and the other on the Lyusternik–Graves version of Newton’s method, that in principle may lead to different fixed points, although the impression is that estimates provided by the first procedure may be better. The paper is concluded by a discussion of the regularity problem for compositions of set-valued mappings. There are also quite a few examples in the paper.