| Abstract: |
| The talk deals with multivalued differential equations in abstract spaces. Nonlocal initial conditions are assumed. The model includes an m-dissipative multioperator which generates a nonlinear semigroup that is equicontinuous, but not necessarily compact. The existence of integral solutions is discussed, with a topological index argument and it is based on the regularity of the nonlinear term with respect to the Hausdorff measure of noncompactness and on a transversality condition. The motivation for these studies is that nonlocal Cauchy problems may have better effects to describe real life phenomena than the classical initial value problem. For example, it is used to represent mathematical models for evolution of various phenomena, such as nonlocal neural networks, nonlocal pharmacokinetics, nonlocal pollution and nonlocal combustion. Moreover, the presence of a multivalued nonlinearity allows to consider optimal control problems as applications. The discussion is completed by an example of how to use these results in the study of the existence of solutions for partial differential inclusions of parabolic type in a bounded domain in $\mathbb{R}^n$, with a nonlinear term described by a subdifferential of a suitable map and with nonlocal integral initial conditions. |
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