In this talk, we present results on the existence and localization of solutions to nonlocal differential problems in abstract spaces. In particular, we will discuss an approach based on fixed point theorems associated with the so-called transversality conditions. Special attention will be devoted to techniques that allow one to weaken the classical compactness assumptions that frequently appear in the literature on differential equations in abstract spaces studied by topological methods.

This approach provides a unified framework for the analysis of models describing diffusion processes in a variety of settings. It allows the treatment of periodic conditions as well as more general nonlocal initial conditions, such as multipoint and integral conditions, and it can accommodate nonlinearities with superlinear growth, including cubic polynomials and nonlinear terms depending on the integral of the solution. In this way, models exhibiting genuinely nonlocal diffusion effects can also be covered.