Abstract
We study a certain second-order evolution inclusion defined by a family of linear closed operators which is the generator for an evolution system of operators and by a set-valued map with nonconvex values in a separable Banach space. We provide results concerning the differentiability of mild solutions with respect to the initial conditions of the problem considered. Certain variational inclusions are obtained in terms of set-valued derivatives defined by the contingent cone, the quasitangent cone and Clarke’s tangent cone. Our results may be interpreted as extensions to a special class of second-order differential inclusions of the classical Bendixson–Picard–Lindelőf theorem concerning the differentiability of the maximal flow of a differential equation.
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References
Aubin, J.P., Frankowska, H.: Set-valued Analysis. Birkhauser, Basel (1990)
Baliki, A., Benchohra, M., Graef, J.R.: Global existence and stability of second order functional evolution equations with infinite delay. Electron. J. Qual. Theory Differ. Equ. 2016(23), 1–10 (2016)
Baliki, A., Benchohra, M., Nieto, J.J.: Qualitative analysis of second-order functional evolution equations. Dyn. Syst. Appl. 24, 559–572 (2015)
Benchohra, M., Medjadj, I.: Global existence results for second order neutral functional differential equations with state-dependent delay. Comment. Math. Univ. Carolin. 57, 169–183 (2016)
Benchohra, M., Rezzoug, N.: Measure of noncompactness and second-order evolution equations. Gulf J. Math. 4, 71–79 (2016)
Cernea, A.: Variational inclusions for a nonconvex second-order differential inclusion. Mathematica (Cluj) 50(73), 169–176 (2008)
Cernea, A.: Variational inclusions for a Sturm-Liouville type differential inclusion. Math. Bohemica 135, 171–178 (2010)
Cernea, A.: A note on the solutions of a second-order evolution inclusion in non separable Banach spaces. Comment. Math. Univ. Carolin. 58, 307–314 (2017)
Cernea, A.: Some remarks on the solutions of a second-order evolution inclusion. Dyn. Syst. Appl. 27, 319–330 (2018)
Cernea, A.: Continuous selections of solution sets of a second-order integro-differential inclusion. In: Pinelas, S., Caraballo, T., Kloeden, P., Graef, J. (eds.) Differential and Difference Equations with Application. Springer Proceedings in Mathematics and Statistics, vol. 230, pp. 53–65. Springer, Cham (2018)
Dunford, N.S., Schwartz, J.T.: Linear Operator Part I. General Theory. Wiley Interscience, New York (1958)
Filippov, A.F.: Classical solutions of differential equations with multivalued right hand side. SIAM J. Control 5, 609–621 (1967)
Henriquez, H.R.: Existence of solutions of nonautonomous second order functional differential equations with infinite delay. Nonlinear Anal. 74, 3333–3352 (2011)
Henriquez, H.R., Poblete, V., Pozo, J.C.: Mild solutions of non-autonomous second order problems with nonlocal initial conditions. J. Math. Anal. Appl. 412, 1064–1083 (2014)
Kozak, M.: A fundamental solution of a second-order differential equation in a Banach space. Univ. Iagel. Acta. Math. 32, 275–289 (1995)
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Cernea, A. (2019). Differentiability Properties of Solutions of a Second-Order Evolution Inclusion. In: Area, I., et al. Nonlinear Analysis and Boundary Value Problems . NABVP 2018. Springer Proceedings in Mathematics & Statistics, vol 292. Springer, Cham. https://doi.org/10.1007/978-3-030-26987-6_2
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DOI: https://doi.org/10.1007/978-3-030-26987-6_2
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