Orientation

Many of the results that hold for linear maps can be generalized to nonlinear maps, although there are important differences and added difficulties. While boundedness and continuity are equivalent for linear operators, this is no longer the case for nonlinear maps. Another issue is that a nonlinear map is typically only defined on a subset of a vector space. Likewise, the image of the nonlinear map may only be a subset of the range space. For these reasons it is usually necessary to analyze applied nonlinear problems one by one. Nevertheless, there are general principles also in nonlinear analysis, where fixed point theorems, coercivity and monotonicity are of fundamental importance.