Abstract
An evolution inclusion with the right-hand side containing the difference of subdifferentials of proper convex lower semicontinuous functions and a multivalued perturbation whose values are nonconvex closed sets is considered in a separable Hilbert space. In addition to the original inclusion, we consider an inclusion with convexified perturbation and a perturbation whose values are extremal points of the convexified perturbation that also belong to the values of the original perturbation. Questions of the existence of solutions under various perturbations are studied and relations between solutions are established. The primary focus is on the weakening of assumptions on the perturbation as compared to the known assumptions under which existence and relaxation theorems are valid. All our assumptions, in contrast to the known assumptions, concern the convexified rather than original perturbation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Y. Koi and J. Watanabe, “On nonlinear evolution equation with a difference term of subdifferentials,” Proc. Japan. Acad. 52 (8), 413–446 (1976).
M. Otani, “On existence of strong solutions for du(t)/dt+1(u(t))2(u(t))f(t),” J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 24 (3), 575–605 (1977).
G. Akagi and M. Otani, “Evolution inclusions governed by the difference of two subdifferentials in reflexive Banach spaces,” J. Differential Equations 209 (2), 392–415 (2005).
R. D. Bourgin, Geometric Aspects of Convex Sets with the Radon–Nikodym Property (Springer, Berlin, 1983), Ser. Lecture Notes in Mathematics, Vol. 993.
C. J. Himmelberg, “Measurable relations,” Fund. Math. 87, 53–72 (1975).
H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert (North- Holland, Amsterdam, 1973).
N. Kenmochi, “Solvability of nonlinear evolution equations with time-dependent constraints and applications,” Bull. Fac. Educ. Chiba Univ. 30, 1–87 (1981).
A. A. Tolstonogov, “Relaxation in non-convex optimal control problems described by first-order evolution equations,” Sb. Math. 190 (11), 1689–1714 (1999).
A. A. Tolstonogov, “Strongly exposed points of decomposable sets in spaces of Bochner integrable functions,” Math. Notes 71 (2), 267–275 (2002).
F. Hiai and H. Umegaki, “Integrals, conditional expectations, and martingales of multivalued functions,” J. Multivariate Anal. 7 (1), 149–182 (1977).
N. Dunford and J. Schwartz, Linear Operators: General Theory (Interscience, New York, 1958; Inostrannaya Lit., Moscow, 1962).
A. A. Tolstonogov and D. A. Tolstonogov, “Lp-continuous extreme selectors of multifunctions with decomposable values: Existence theorems,” Set-Valued Anal. 4 (2), 173–203 (1996).
A. A. Tolstonogov, “Scorza-Dragoni’s theorem for multi-valued mappings with variable domain of definition,” Math. Notes 48 (5), 1151–1158 (1990).
A. Fryszkowski, “Continuous selections for a class of nonconvex multivalued maps,” Studia Math. 76 (2), 163–174 (1983).
A. A. Tolstonogov and D. A. Tolstonogov, “Lp-continuous extreme selectors of multifunctions with decomposable values: relaxation theorems,” Set-Valued Anal. 4 (3), 237–269 (1996).
A. F. Filippov, “Classical solutions of differential equations with multivalued right-hand side,” Vestn. Mosk. Univ., Ser. Mat. Mekh. 3, 16–26 (1967).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A. A.Tolstonogov, 2015, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2015, Vol. 21, No. 2.
Rights and permissions
About this article
Cite this article
Tolstonogov, A.A. Solutions of evolution inclusions generated by a difference of subdifferentials. Proc. Steklov Inst. Math. 293 (Suppl 1), 199–215 (2016). https://doi.org/10.1134/S0081543816050187
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543816050187