| Abstract: |
| In the talk we investigate the problem of the existence and properties of solution sets of two types of differential inclusions with Aumann and Young set-valued integrals, presented in their integral form
$$x(t)-x(s)\in \int_{s}^{t}F(\tau ,x(\tau ))d\tau +\int_{s}^{t}cl_{\mathcal C}({\mathcal H}\circ x)(\tau)dg(\tau ), \;\;x( 0) =x_0$$
and
$$x(t)-x(s)\in \int_{s}^{t}F(\tau ,x(\tau ))d\tau+ \int_{s}^{t}G(\tau ,x(\tau ))dg(\tau ) ,\;\; x(0)=x_0.$$
The existence of solutions, boundedness, compactness of the set of solutions and continuous dependence type results are considered. The obtained results are applied to some optimality problems driven by Young differential inclusions. The properties of optimal solutions and their reachable sets are discussed.
These inclusions contain as a particular case set-valued stochastic inclusions with respect to fractional Brownian motion (fBm), and therefore, their properties are crucial for investigation the properties of solutions of (fBm) stochastic differential inclusions. The controlled cancer tumor growth model under the influence of drugs will be presented also.
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References
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[1] M. Michta, J. Motyl, Selection Properties and Set-Valued Young Integrals of Set-Valued Functions, Results Math. {\bf 75}, 164 (2020).
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[2] M. Michta, J. Motyl, Set-Valued Functions of Bounded Generalized Variation and Set-Valued Young Integrals, J. Theor. Probab. {\bf 35}, 528--549 (2022).
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[3] M. Michta, J. Motyl, Solution Sets for Young Differential Inclusions, Qual. Theory Dyn. Syst. {\bf 22}, 132 (2023).
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[4] M. Michta, J. Motyl, Properties of set-valued Young integrals and Young differential inclusions generated by sets of H\older functions, Nonlinear Differ. Equ. Appl., Vol. 31, art. no. 70, 1--33, published online June 4, 2024
https://doi.org/10.1007/s00030-024-00963-2 |
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