| Abstract: |
| Let $X$ be a Banach space while $(Y,\leq )$ a Banach lattice. One of valuable problems in set valued analysis is concerned in the existence of regular selections of set-valued functions acting from $X$ into nonempty subsets of $Y$. Investigating nonlinear dynamical systems, continuous, Lipschitz, differentiable or bounded variation selections are considered most often.
In the talk we introduce the class of upper separated set-valued functions and investigate the problem of the existence of order-convex selections of $F$. First, we present necessary and sufficient conditions for the existence of such selections. Next we discuss the problem of the existence of the Carath\`eodory-order-convex type selections and apply obtained results to investigation of the existence and properties of solutions of set-valued deterministic and stochastic dynamical systems, like stability or lower-upper bounds. In second part of the talk we will discuss the applicability of obtained selection results to optimal control problems. Some examples will be presented also.
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References
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[1] M. Michta, J. Motyl, {\it Locally Lipschitz selections in Banach lattices}, Nonlinear Analysis {\bf 71} (2009), 2331-2342,
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[2] J. Motyl, {\it Carath\`eodory convex selections of set-valued functions in Banach lattices},
Topological Methods in Nonlinear Analysis {\bf 43} (2014), 1-10,
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[3] J. Motyl, {\it Stochastic retarded inclusion with Carath\`eodory-upper separated multifunctions}, Set-Valued and Variational Analysis {\bf 24} (2016), 191-205,
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[4] J. Motyl, {\it Carath\`eodory-convex selections of multifunctions and their applications}, Journal of Nonlinear and Convex Analysis {\bf 18} (2017), 535-551. |
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