The concept of mapped coercivity for nonlinear operators in Banach spaces

We provide a concise proof of existence for nonlinear operator equations in separable Banach spaces. Notably, the operator is not assumed to be monotone. Instead, our main hypotheses consist of a continuity assumption and a generalized coercivity property. Mapped coercivity is a generalization of the usual coercivity property for nonlinear operators. In the case of linear operators, we recover linear coercivity and the traditional inf-sup condition. To illustrate the applicability of this general concept, we apply it to semi-linear elliptic problems and the Navier-Stokes equations.