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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, Issue 2, Pages 84–99 (Mi vuu324)

MATHEMATICS

On Volterra type generalization of monotonization method for nonlinear functional operator equations

A. V. Chernov

Department of Mathematical Physics, Nizhni Novgorod State University, Nizhni Novgorod, Russia

Abstract: Let $n,m,\ell,s\in\mathbb N$ be given numbers, $\Pi\subset\mathbb R^n$ be a set measurable by Lebesgue and $\mathcal{X,Z}$ be some Banach ideal spaces of functions measurable on $\Pi$. We consider a nonlinear operator equation of the form as follows:
$$x=\theta+AF[x],\quad x\in\mathcal X^\ell, \tag{1}$$
where $A\colon\mathcal Z^m\to\mathcal X^\ell$ is bounded linear operator, $F\colon\mathcal X^\ell\to\mathcal Z^m$ is some operator. Equation (1) is a 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) $(I+A G)^{-1}A\geq0$ ($\lambda>0$, $I$ 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.

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UDC: 517.988.63
MSC: 47J05, 47J35

Citation: A. V. Chernov, “On Volterra type generalization of monotonization method for nonlinear functional operator equations”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, no. 2, 84–99

Citation in format AMSBIB
\Bibitem{Che12} \by A.~V.~Chernov \paper On Volterra type generalization of monotonization method for nonlinear functional operator equations \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2012 \issue 2 \pages 84--99 \mathnet{http://mi.mathnet.ru/vuu324} \elib{https://elibrary.ru/item.asp?id=17790056} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. V. Chernov, “Ob $\varepsilon$-ravnovesii v beskoalitsionnykh funktsionalno-operatornykh igrakh so mnogimi uchastnikami”, Tr. IMM UrO RAN, 19, no. 1, 2013, 316–328
2. A. V. Chernov, “Uniformly continuous dependence of a solution to a controlled functional operator equation on a shift of control”, Russian Math. (Iz. VUZ), 57:5 (2013), 29–41
3. A. V. Chernov, “Ob upravlyaemosti nelineinykh raspredelennykh sistem na mnozhestve konechnomernykh approksimatsii upravleniya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2013, no. 1, 83–98
4. A. V. Chernov, “On the structure of a solution set of controlled initial-boundary value problems”, Russian Math. (Iz. VUZ), 60:2 (2016), 62–71
5. A. V. Chernov, “On a majorant-minorant criterion for the total preservation of global solvability of distributed controlled systems”, Differ. Equ., 52:1 (2016), 111–121
6. A. V. Chernov, “On total preservation of solvability for a controlled Hammerstein type equation with non-isotone and non-majorized operator”, Russian Math. (Iz. VUZ), 61:6 (2017), 72–81
7. A. V. Chernov, “Differentiation of the functional in a parametric optimization problem for a coefficient of a semilinear elliptic equation”, Differ. Equ., 53:4 (2017), 551–562
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