Abstract
A family of maximally monotone operators on a separable Hilbert space is considered. The domains of these operators depend on time ranging over an interval of the real line. The space of square-integrable functions on this interval taking values in the same Hilbert space is also considered. On the space of square-integrable functions a superposition (Nemytskii) operator is constructed based on a family of maximally monotone operators. Under fairly general assumptions, the maximal monotonicity of the Nemytskii operator is proved. This result is applied to the family of maximally monotone operators endowed with a pseudodistance in the sense of A. A. Vladimirov, to the family of subdifferential operators generated by a proper convex lower semicontinuous function depending on time, and to the family of normal cones of a moving closed convex set.
Similar content being viewed by others
References
V. Niemytzki, “Théorèmes d’existence et d’unicité des solutions de quelques équations intégrales non-linéaires”, Mat. Sb., 41:3 (1934), 421–452.
C. J. Himmelberg, “Measurable relations”, Fund. Math., 87 (1975), 53–72.
V. Barbu, Nonlinear differential equations of monotone types in Banach spaces, Springer-Verlag, New York–Dordrecht–Heidelberg–London, 2010.
J.-P. Aubin and A. Cellina, Differential inclusions. Set-valued maps and viability theory, Springer-Verlag, Berlin, 1984.
R. T. Rockafellar, “Convex integral functions and duality”, Contributions to Nonlinear Functional Analysis, Academic Press, New York–London, 1971, 215–236.
H. Attouch, Variational convergence for functions and operators, Pitman (Advanced Publishing Program), Boston–London–Melbourne, 1984.
A. A. Vladimirov, “Nonstationary dissipative evolution equations in a Hilbert space”, Nonlinear Anal., Theory, Meth., Appl., 17:6 (1991), 499–518.
M. Kunze, M. D. P. Monteiro Marques, “BV solutions to evolution problems with time dependent domains”, Set-valued Anal., 5:1 (1997), 57–72.
H. Attouch, “Familles d’operateurs maximaux monotones et mesurabilité”, Ann. Math. Pura Appl., 120:1 (1979), 35–111.
A. A. Tolstonogov, “BV continuous solutions of an evolution inclusion with maximal monotone operator and nonconvex-valued perturbation. Existence theorem”, Set-valued Var. Anal., 29:1 (2021), 29–60.
A. A. Tolstonogov, “Compactness of BV solutions of a convex sweeping process of measurable differential inclusion”, J. Convex Anal., 27:2 (2020), 673–695.
G. Minty, “Monotone (nonlinear) operators in Hilbert spaces”, Duke Math. J., 29 (1962), 341–346.
Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976.
D. Azzam-Laouir, Ch. Castaing, and M. D. P. Monteiro Marques, “Perturbed evolution problem with continuous bounded variation in time and applications”, Set-valued Var. Anal., 26:3 (2018), 693–728.
D. Azzam-Laouir and I. Boutana-Harid, “Mixed semicontinuous perturbation to an evolution problem with time-dependent maximal monotone operator”, J. Nonlinear and Convex Anal., 20:1 (2018), 39–52.
C. Castaing, C. Godet-Thobie, and Truong Xuan Le, “Fractional order of evolution inclusion coupled with a time and state dependent maximal monotone operator”, Mathematics, 8 (2020).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2021, Vol. 55, pp. 51–61 https://doi.org/10.4213/faa3892.
Translated by O. V. Sipacheva
Rights and permissions
About this article
Cite this article
Tolstonogov, A.A. Maximal Monotonicity of a Nemytskii Operator. Funct Anal Its Appl 55, 217–225 (2021). https://doi.org/10.1134/S0016266321030047
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0016266321030047