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| The interest for multivalued equations in abstract spaces is motivated by the study of control problems. Different nonlocal boundary conditions, including periodic, anti-periodic, mean value and multi-point conditions, are needed for different applications. The existence of solutions for these problems is frequently studied with topological techniques based on fixed point theorems for a suitable solution operator. This requires strong compactness conditions, which are very hard to check in an infinite dimensional framework. To weaken these hypotheses, several approaches can be used. A first technique is based on the concept of measure of non-compactness combined with a topological degree theory. Alternatively, weak topologies can be exploited, joined with the classical Ky Fan Fixed Point Theorem. Finally, one can consider the multivalued problem in an Hilbert space compactly embedded into a Banach space, in connection with Hartman-type conditions. The purpose of this talk is to compare these different approaches, discussing their positive sides and drawbacks. |
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