Nonlocal Cauchy problems close to an asymptotically stable equilibrium point

In this work we investigate the existence of solutions for semilinear Cauchy problems with nonlocal initial conditions in the neighborhood of an asymptotically stable equilibrium point of the evolution equation. Using Granas' continuation principle for contractive maps and the qualitative theory of differential equations in Banach spaces, under mild assumptions, we prove the existence of a unique solution. We also show that the main abstract result can be applied to nonlocal initial boundary value problems for reaction–diffusion equations with non-convex nonlinearities.