Isotone Cones in Banach Spaces and Applications to Best Approximations of Operators without Continuity Conditions

TL;DR

This paper introduces isotone cones in Banach spaces and uses their properties to establish the existence of best approximations for certain operators without requiring continuity, expanding approximation theory.

Contribution

It presents a novel concept of isotone cones and applies order monotonicity of metric projections to prove approximation existence without continuity assumptions.

Findings

Existence of best approximations proved without continuity conditions

Introduction of isotone cones in Banach spaces

Application of order monotonicity in approximation theory

Abstract

In this paper, we introduce the concept of isotone cones in Banach spaces. Then we apply the order monotonic property of the metric projection operator to prove the existence of best approximations for some operators without continuity conditions in partially ordered Banach spaces.