Evolution inclusions with state-dependent maximal monotone operators

A. A. Tolstonogov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 3, pp. 241-254

Résumé

The existence of an absolutely continuous solution of a differential inclusion whose right-hand side contains a time- and state-dependent maximal monotone operator and a nonconvex perturbation is proved in a Hilbert space. The proofs are based on our comparison theorems for inclusions with maximal monotone operators and a fixed point theorem for multivalued mappings. This approach allows us to extend the class of inclusions with maximal monotone operators for which existence theorems are valid and, as a result, to obtain significant results of this kind.

Détail

Keywords: maximal monotone operator, comparison theorem.
Mots-clés : $G$-convergence

Bibliographie

[1] Brezis H., Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Elsevier, North–Holland, Amsterdam–London, 1973, 183 pp. | MR | Zbl

[2] Vladimirov A.A., “Nonstationary dissipative evolution equations in a Hilbert space”, Nonlinear Anal.: Theory, Method Appl, 17:6 (1991), 499–518 | DOI | MR | Zbl

[3] Selamnia F., Azzam-Laouir D., Monteiro Marques M.D.P., “Evolution problems involving state-dependent maximal monotone operators”, Appl. Anal., 101:1 (2022), 297–313 | DOI | MR | Zbl

[4] Amiour F., Sene M., Haddad T., “Existence Results for State-Dependent Maximal Monotone Differential Inclusions: Fixed Point Approach”, Numer. Funct. Analysis Optimiz., 43:7 (2022), 1–22 | DOI | MR

[5] Tolstonogov A.A., “Teoremy sravneniya dlya evolyutsionnykh vklyuchenii s maksimalno monotonnymi operatorami. $L^2$-teoriya”, Mat. sb., 214:6 (2023), 110–135 | DOI | MR

[6] Fan K., “Fixed point and minimax theorems in locally convex topological linear spaces”, Proc. Nat. Acad Sci. USA, 38 (1952), 121–126 | DOI | MR | Zbl

[7] Le B.K., “Well-posedness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions”, Optimization, 69:6 (2020), 1187–1217 | DOI | MR | Zbl

[8] Le B.K., “On a class of Lur'e dynamical systems with state-dependent set-valued feedback”, Set-Valued and Variational Analysis, 28:1 (2020), 537–557 | DOI | MR | Zbl

[9] Attouch H., “Families d'operateurs maximax monotone et measurabilite”, Ann. Math. Pura Appl., 120:4 (1979), 35–111 | DOI | MR | Zbl

[10] Attouch H., Variational convergence for functions and operators, Pitman Advanced Publishing program, Boston–London–Melbourne, 1984, 423 pp. | MR | Zbl

[11] Shui-Hung Hou, “On property $(Q)$ and other semicontinuity properties of multifunctions”, Pacific J. Math., 103:2 (1982), 39–56 | MR | Zbl

[12] Denkowski Z., Migorski S., Papageorgiou N.S., An introduction to nonlinear analysis: theory, Kluwer Acad., Plenum Publ., NY, 2003 | DOI | MR | Zbl

[13] Kunze M., Monteiro Marques M.D.P., “BV solutions to evolution problems with time dependent domains”, Set-valued Anal., 5 (1994), 57–72 | DOI | MR

[14] Tolstonogov A.A., “Upper semicontinuous convex-valued selectors of a Nemytskii operator with nonconvex values and evolution inclusions with maximal monotone operators”, J. Math. Anal. Appl., 526:1 (2023), 127197 | DOI | MR

[15] Moreau J.J., “Evolution problem associated with a moving convex set in a Hilbert space”, J. Diff. Equat., 26:3 (1997), 347–374 | DOI | MR