An Alternative Way of Utilizing Fixed Point Theory

The author proposes an alternative way of using fixed point theory to get the existence for semilinear equations. As an example, a nonlocal ordinary differential equation is considered. The idea is to solve homogeneous equations in the linearization. One feature of this method is that it does not need the equation to have special structures, for instance, variational structures, maximum principle, etc. Roughly speaking, the existence comes from good properties of the suitably linearized equation. The idea may have wider application.