Abstract
For a family of operator equations of the first kind with nonlinear nonmonotone hemicontinuous operators in a reflexive Banach space, we prove a theorem on the solvability and a uniform estimate of the solution in the norm of the space. Our approach is related to the method of semimonotone operators but has some essential differences from the latter. In a specific example, we show that our theorem can be used to prove the total (with respect to the set of admissible controls) preservation of solvability for distributed control systems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Kachurovskii, R.I., Non-Linear Monotone Operators in Banach Spaces, Russ. Math. Surv., 1968, vol. 23, no. 2, pp. 117–165.
Vainberg, M.M., Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, New York; Toronto: John Wiley & Sons, 1973; Jerusalem, London: Israel Program for Scientific Translations, 1973.
Dubinskii, Yu.A., Nonlinear Elliptic and Parabolic Equations, Sovr. Probl. Mat., 1976, vol. 9, pp. 5–130.
Skrypnik, I.V., Metody issledovaniya nelineinykh ellipticheskikh granichnykh zadach (Methods for Studying Nonlinear Elliptic Boundary Value Problems), Moscow: Nauka, 1990.
Milojević, P.S., Solvability of Strongly Nonlinear Operator Equations and Applications, Differ. Equations, 1995, vol. 31, no. 3, pp. 466–479.
Karamardian, S. and Schaible, S., Seven Kinds of Monotone Maps, J. Optim. Theory Appl., 1990, vol. 66, pp. 37–46.
Brezis, H. and Browder, F., Partial Differential Equations in the 20th Century, Adv. Math., 1998, vol. 135, pp. 76–144.
Ryazantseva, I.P., Equations with Semimonotone Discontinuous Mappings, Math. Notes Acad. Sci. USSR, 1981, vol. 30,iss. 1, pp. 558–563.
Ryazantseva, I.P., On the Resolvability of Variational Inequalities with Unbounded Semimonotone Maps, Russ. Math., 1999, vol. 43, no. 7, pp. 46–50.
Pavlenko, V.N., Existence of Solutions of Nonlinear Equations with Discontinuous Semimonotonic Operators, Ukrain. Math. J., 1982, vol. 33, pp. 419–422.
Pavlenko, V.N., The Monotone Operators’ Method for Equations with Discontinuous Nonlinearities, Sov. Math., 1991, vol. 35, no. 6, pp. 38–44.
Chernov, A.V., On Total Preservation of the Global Solvability of Functional-Operator Equations, Vestn. Nizhegor. Univ., 2009, no. 3, pp. 130–137.
Chernov, A.V., A Majorant Criterion for the Total Preservation of Global Solvability of Controlled Functional Operator Equation, Russ. Math., 2011, vol. 55, no. 3, pp. 85–95.
Yosida, K., Functional Analysis, Berlin, 1965. Translated under the title Funktsional’nyi analiz, Moscow, 1967.
Pugachev, V.S., Lektsii po funktsional’nomu analizu (Lectures on Functional Analysis), Moscow, 1996.
Vulikh, B.Z., Kratkii kurs teorii funktsii veshchestvennoi peremennoi (Brief Course of Theory of Functions of Real Variable), Moscow, 1965.
Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites non linéaires, Paris: Dunod, 1969. Translated under the title Nekotorye metody resheniya nelineinykh kraevykh zadach, Moscow: Mir, 197
Author information
Authors and Affiliations
Additional information
Original Russian Text © A. V. Chernov, 2013, published in Differentsial’nye Uravneniya, 2013, Vol. 49, No. 4, pp. 535–544.
Rights and permissions
About this article
Cite this article
Chernov, A.V. On a generalization of the method of monotone operators. Diff Equat 49, 517–527 (2013). https://doi.org/10.1134/S0012266113040125
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0012266113040125