The Bolzano-Poincar\'e-Miranda theorem in infinite dimensional Banach spaces

TL;DR

This paper extends the Bolzano-Poincaré-Miranda theorem to infinite dimensional Banach spaces and explores the existence of periodic solutions to differential equations in such spaces.

Contribution

It provides a novel extension of a classical fixed point theorem to infinite-dimensional Banach spaces and applies it to differential equations.

Findings

Extended Bolzano-Poincaré-Miranda theorem to Banach spaces

Proved existence of periodic solutions in Banach spaces

Established new fixed point results in infinite dimensions

Abstract

We study the existence of zeroes of mappings defined in Banach spaces. We obtain, in particular, an extension of the well-known Bolzano-Poincar\'e-Miranda theorem to infinite dimensional Banach spaces. We also establish a result regarding the existence of periodic solutions to differential equations posed in an arbitrary Banach space.