Abstract
In the paper we prove a variant of the well known Filippov–Pliss lemma for evolution inclusions given by multivalued perturbations of m-dissipative differential equations in Banach spaces with uniformly convex dual. The perturbations are assumed to be almost upper hemicontinuous with convex weakly compact values and to satisfy one-sided Peron condition. The result is then applied to prove the connectedness of the solution set of evolution inclusions without compactness and afterward the existence of attractor of autonomous evolution inclusion when the perturbations are one-sided Lipschitz with negative constant.
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References
Barbu V.: Semigroups and Differential Equations in Banach Spaces. Nordhoff, Leyden (1976)
Barbu V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer, New York (2010)
Benilan P.: Solutions integrales d’equations d’evolution dans un espace de Banach. C.R. Acad. Sci. Paris 274, 47–50 (1972)
Anguiano M., Caraballo T., Real J.: Pullback attractors for reaction–diffusion equations in some unbounded domains with an H −1-valued non-autonomous forcing term and without uniqueness of solutions. Discret. Contin. Dyn. Syst. B 14, 307–326 (2010)
Bothe D.: Multivalued Perturbations of m-Accretive Differential Inclusions. Israel J. Math. 108, 109–138 (1998)
Bothe D.: Nonlinear Evolutions in Banach Spaces. Habilitationsschaft, Paderborn (1999)
Cârjă O., Necula M., Vrabie I.: Viability, Invariance and Applications, North-Holland Mathematics Studies, vol. 207. Elsevier Science B.V., Amsterdam (2007)
Deimling K.: Multivalued Differential Equations. DeGruyter, Berlin (1992)
Donchev T.: Multi-valued perturbations of m-dissipative differential inclusions in uniformly convex spaces. N. Z. J. Math. 31, 19–32 (2002)
Donchev T., Farkhi E.: On the theorem of Filippov–Plis and some applications. Control Cybern 38, 1–21 (2009)
Donchev T., Farkhi E., Reich S.: Fixed set iterations for relaxed Lipschitz multi-maps. Nonlinear Anal. 53, 997–1015 (2003)
Donchev T., Farkhi E., Reich S.: Discrete approximations and fixed set iterations in Banach spaces. SIAM J. Optim. 18, 895–906 (2007)
Donchev T., Rios V., Wolenski P.: Strong invariance and one-sided Lipschitz multifunctions. Nonlinear Anal. 60, 849–862 (2005)
Hu S., Papageorgiou N.: Handbook of Multivalued Analysis, vol. II Applications. Kluwer, Dodrecht (2000)
Kapustyan O., Valero J.: Attractors of multivalued semiflows generated by differential inclusions and their approximations. Abstr. Appl. Anal. 5(1), 33–46 (2000)
Kloeden P., Valero J.: Attractors of set valued partial differential equations under discretization. IMA J. Numer. Anal. 29, 690–711 (2009)
Lakshmikantham V., Leela S.: Nonlinear Differential Equations in Abstract Spaces. Pergamon, Oxford (1981)
Morillas F., Valero J.: Assymptotic compactness and attrectors for phase–field equations in \({\mathbb{R}^3}\). Set Valued Anal. 16, 861–897 (2008)
Roubiček T.: Nonlinear Partial Differential Equations with Applications. Birkhäuser, Basel (2005)
Showalter, R.: Monotone operators in Banach space and nonlinear partial differential equations. Math. Surv. Monogr. 49 AMS (1997)
Tolstonogov A.: Differential Inclusions in a Banach Space. Kluwer, Dordrecht (2000)
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Din, Q., Donchev, T. & Kolev, D. Filippov–Pliss lemma and m-dissipative differential inclusions. J Glob Optim 56, 1707–1717 (2013). https://doi.org/10.1007/s10898-012-9963-7
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DOI: https://doi.org/10.1007/s10898-012-9963-7