On semi-continuous mappings, equations and inclusions in a Banach space

Abstract

We investigate the description of the image of a continuous mappings acting in a Banach space, and the solvability of equations and inclu- sions. The results obtained can be applied to the Cauchy problem for a nonlinear differential equation (or inclusion). In particular, a solvabil- ity theorem of the mixed problem for a nonlinear hyperbolic equation is proved, and one Nirenberg problem is studied.