Open mapping theorem for nonlinear operators

Abstract

Many authors gave definitions of a new kind of derivative and integral, moving the roles of subtraction and addition to division and multiplication, and thus established a new calculus, called multiplicative calculus. In this paper, our aim is to bring up this calculus to the attention of researchers and demonstrate its usefulness. We first discussed multiplicative normed spaces by giving some topological properties of the relevant multiplicative normed spaces. Also, we give a generalization of the open mapping theorem and the uniform boundedness principle for nonlinear operators with a number of important examples.