Asymptotic Behavior of Resolvents of a Convergent Sequence of Convex Functions on Complete Geodesic Spaces

Yasunori Kimura,Keisuke Shindo

doi:10.3390/axioms11010021

Abstract

The asymptotic behavior of resolvents of a proper convex lower semicontinuous function is studied in the various settings of spaces. In this paper, we consider the asymptotic behavior of the resolvents of a sequence of functions defined in a complete geodesic space. To obtain the result, we assume the Mosco convergence of the sets of minimizers of these functions.

Highlights

IntroductionIn a Hilbert space H, for a proper lower semicontinuous convex function f : H → ]−∞, ∞], a resolvent J f : X → X is defined by the following:
This paper considers the asymptotic behavior of the resolvents of a given convergent sequence of convex functions on a complete CAT(0) space and a complete admissible
We consider the asymptotic behavior of a resolvent on CAT(0) space

Summary

Introduction

In a Hilbert space H, for a proper lower semicontinuous convex function f : H → ]−∞, ∞], a resolvent J f : X → X is defined by the following:. A complete CAT(0) space, which is an example of a geodesic space, is a generalization of Hilbert spaces In this space, the following resolvent is proposed (see [2]). We can consider asymptotic behavior at infinity and have results similar to Theorem 1 (see [3]) In these cases, a convex function f is fixed. This paper considers the asymptotic behavior of the resolvents of a given convergent sequence of convex functions on a complete CAT(0) space and a complete admissible.

Preliminaries

Main Results

Applications to Hilbert Spaces