 +7 (3412) 91 60 92 imi@udsu.ru

## Archive of Issues

Russia Nizhni Novgorod
Year
2012
Issue
2
Pages
84-99
 Section Mathematics Title On Volterra type generalization of monotonization method for nonlinear functional operator equations Author(-s) Chernov A.V.a Affiliations Nizhni Novgorod State Universitya Abstract Let $n,m,\ell,s\in\mathbb{N}$ be given numbers, $\Pi\subset\mathbb{R}^n$ be a set measurable by Lebesgue and $\mathcal{X}$, $\mathcal{Z}$ be some Banach ideal spaces of functions measurable on $\Pi$. We consider a nonlinear operator equation of the form as follows: $$x=\theta+AF[x], \quad x\in\mathcal{X}^\ell, \qquad \quad(1)$$ where $A:\mathcal{Z}^m\to\mathcal{X}^\ell$ is bounded linear operator, $F:\mathcal{X}^\ell\to\mathcal{Z}^m$ is some operator. Equation $(1)$ is natural form of lumped and distributed parameter systems from a wide enough class. Formerly, by V.P. Polityukov it was suggested monotonization method for justification of solvability of equation $(1)$ and obtaining pointwise estimations for solutions. The matter of this method consisted in that solvability of equation $(1)$ was proved (besides other conditions) under following: I) operator $F$ allows some correction of the form $G=\lambda I$ to monotone operator $\mathcal{F}[x]=F[\theta+x]+G[x]$ such that II) $(I+A G)^{-1}A\geqslant0$ ($\lambda>0$, $I$ is identity operator). As our examples show, conditions I) and II) may be contradictory to each other, that narrows a sphere of application of the method. The main result of the paper is that for the case of operator $A$, possessing the Volterra property, which is natural for evolutionary equations, the requirement I) of ability to be monotonized can be replaced by the requirement of some upper and lower estimates for operator $F$ on some cone segment through linear operator $G$ and additional fixed element. We prove that for global solvability of a boundary value problem associated with a semilinear evolutionary equation it is sufficient that analogous boundary value problem associated with linear equation, derived from the original equation by estimating of a right-hand side on some cone segment, have a positive solution. The application of results obtained is illustrated by Goursat-Darboux system, Cauchy problem associated with wave equation and first boundary value problem associated with diffusion equation. Keywords nonlinear operator equation, solvability, monotonization method, Volterra property UDC 517.988.63 MSC 47J05, 47J35 DOI 10.20537/vm120208 Received 15 February 2012 Language Russian Citation Chernov A.V. On Volterra type generalization of monotonization method for nonlinear functional operator equations, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 2, pp. 84-99. References Polityukov V.P. Solution of some nonlinear operator equations and their application to integrodifferential equations, Sov. Math. Dokl., 1980, vol. 21, pp. 229-233. Original Russian text published in Dokl. Akad. Nauk SSSR, 1980, vol. 250, no. 4, pp. 818-822. Polityukov V.P. Method of monotonization of nonlinear equations in Banach spaces, Mathematical Notes, 1988, vol. 44, no. 6, pp. 938--944. Original Russian text published in Mat. Zametki, 1988, vol. 44, no. 6, pp. 814-822. DOI: 10.1007/BF01158033. Sumin V.I., Chernov A.V. On sufficient conditions of existence stability of global solutions of Volterra operator equations, Vestn. Nizhegorod. Gos. Univ. Ser. Mat. Model. Optim. Upr., 2003, no. 1 (26), pp.39-49. Chernov A.V. Volterra functional operator games on a given set, Mat. teoriya igr i ee prilozheniya, 2011, vol. 3, no. 1, pp. 91-117. Chernov A.V. A majorant criterion for the total preservation of global solvability of controlled functional operator equation, Russian Mathematics (Iz. VUZ), 2011, vol. 55, no. 3, pp. 85-95. Original Russian text published in Izv. Vyssh. Uchebn. Zaved. Mat., 2011, no. 3, pp. 95-107. DOI: 10.3103/S1066369X11030108. Chernov A.V. On convergence of fixed point iteration method for solving nonlinear functional operator equations, Vestn. Nizhegorod. Gos. Univ., 2011, no. 4 (1), pp. 149-155. Birkhoff G. Lattice theory, Providence, Rhode Island: American Mathematical Society Colloquium Publications, 1967, vi+418 p. Translated under the title Teoriya reshetok, Moscow: Nauka, 1984, 568 p. Kantorovich L.V., Akilov G.P. Funktsional'nyi analiz (Functional Analysis), Moscow: Nauka, 1984, 752 p. Sumin V.I., Chernov A.V. Volterra operator equations in Banach spaces: existence stability of global solutions, NNSU, Nizhni Novgorod, 2000, 75 p. Deposited in VINITI 25.04.2000, no. 1198-V00. Sumin V.I. The features of gradient methods for distributed optimal control problems, USSR Comput. Math. Math. Phys., 1990, vol. 30, no. 1, pp. 1-15. Original Russian text published in Zh. Vychisl. Mat. Mat. Fiz., 1990, vol. 30, no. 1, pp. 3-21. Sumin V.I. Controlled functional Volterra equations in Lebesgue spaces, Vestn. Nizhegorod. Gos. Univ. Ser. Mat. Model. Optim. Upr., 1998, no. 2 (19), pp. 138-151. Mordukhovich B.Sh. Metody approksimatsii v zadachakh optimizatsii i upravleniya (Approximation methods in optimization and control problems), Moscow: Nauka, 1988, 360 p. Sumin V.I., Chernov A.V. Operators in spaces of measurable functions: the Volterra property and quasinilpotency, Differential Equations, 1998, vol. 34, no. 10, pp. 1403-1411. Original Russian text published in Differ. Uravn., 1998, vol. 34, no. 10, pp. 1402-1411. Pugachev V.S. Lektsii po funktsional'nomu analizu (Lecture notes on Functional Analysis), Moscow: Moscow Aviation Institute, 1996, 744 p. Lisachenko I.V., Sumin V.I. The maximum principle for terminal optimization problem connected with Goursat-Darboux system in the class of functions having summable mixed derivatives, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 2, pp. 52-67. Sabitov K.B. Uravneniya matematicheskoi fiziki (The Equations of Mathematical Physics), Moscow: Vysshaya Shkola, 2003, 255 p. Korn G., Korn T. Mathematical handbook for scientists and engineers. Definitions, theorems and formulas for reference and review, New York-San Francisco-Toronto-London-Sydney: McGraw-Hill, 1968, 572p. Translated under the title Spravochnik po matematike dlya nauchnykh rabotnikov I inzhenerov. Opredeleniya, teoremy, formuly, Moscow: Nauka, 1970, 720 p. Sumin V.I. Funktsional'nye vol'terrovy uravneniya v teorii optimal'nogo upravleniya raspredelennymi sistemami. Chast' I. Vol'terrovy uravneniya i upravlyaemye nachal'no-kraevye zadachi (Functional Volterra equations in the theory of optimal control of distributed systems. Part I. Volterra equations and controlled initial boundary value problems), Nizhni Novgorod: Nizhni Novgorod State University, 1992, 110 p. Sobolev S.L. Nekotorye primeneniya funktsional'nogo analiza v matematicheskoi fizike (Some applications of Functional Analysis in Mathematical Physics), Moscow: Nauka, 1988, 334 p. Mikhailov V.P. Differentsial'nye uravneniya v chastnykh proizvodnykh (Partial differential equations), Moscow: Nauka, 1976, 392 p. Budak B.M., Tikhonov A.N., Samarskii A.A. Sbornik zadach po matematicheskoi fizike (A collection of problems in mathematical physics), Moscow: Nauka, 1972, 686 p. Polyanin A.D. Spravochnik po lineinym uravneniyam matematicheskoi fiziki (Handbook on linear equations of mathematical physics), Moscow: Fizmatlit, 2001, 576 p. Ladyzhenskaya O.A., Solonnikov V.A., Ural'tseva N.N. Lineinye i kvazilineinye uravneniya parabolicheskogo tipa (Linear and quasilinear equations of the parabolic type), Moscow: Nauka, 1967, 736 p. Fedorov V.M. Kurs funktsional'nogo analiza (A course of functional analysis), St.-Petersburg: Lan', 2005, 352 p. Vladimirov V.S., Zharinov V.V. Uravneniya matematicheskoi fiziki (The equations of mathematical physics), Moscow: Fizmatlit, 2000, 400 p. Full text