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Determining equations and the relatedness principle. (English. Russian original) Zbl 0542.47054

Sib. Math. J. 24, 65-72 (1983); translation from Sib. Mat. Zh. 24, No. 1(137), 79-88 (1983).
This article is devoted to a very common method in nonlinear analysis of studying the operator equation \((1)\quad Fx=0\) with a nonlinear operator F mapping one Banach space X into another Y. If the operator U maps X into a finite dimensional subspace \(L\subseteq Y\) and the operator \(F+U\) has an inverse then the equation (1) is equivalent to the equation \((2)\quad z=U(F+U)^{-1}z=0\) with unknown \(z\in L\); the equation (2) is called determination equation for (1). The Lyapunov and Schmidt bifurcation equations, the determination equations by Cesari and the Poincaré and A. M. Samoilenko equations in the oscillation theory are particular cases of determination equations for suitable U. In the article the general relatedness principle binding the rotations of \(F:X\to Y\) and \(I-U(F+U)^{-1}:L\to L\) on the boundaries of corresponding domains in X and L are given; this principle generalized known relatedness principles stated by M. A. Krasnoselskij, V. V. Strygin, E. A. Lifsic and some new analoguous results.

MSC:

47J25 Iterative procedures involving nonlinear operators
47J05 Equations involving nonlinear operators (general)
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References:

[1] M. A. Krasnosel’skii and P. P. Zabreiko, Geometric Methods Methods of Nonlinear Analysis [in Russian], Nauka, Moscow (1975).
[2] M. A. Krasnosel’skii, G. M. Vainikko, P. P. Zabreiko, and Ya. B. Rutitskii, Approximate Methods of Solving Operator Equations [in Russian], Nauka, Moscow (1969).
[3] L. Cesari, ?Functional analysis and periodic solutions of nonlinear differential equations,? in: Contributions to Differential Equations, New York (1963). · Zbl 0132.07101
[4] P. P. Zabreiko and S. O. Strugina, ?On periodic solutions to evolution equations,? Mat. Zametki,9, No. 6, 651-662 (1971).
[5] M. A. Krasnosel’skii, Translation alogn Trajectories of Differential Equations, Translation of Mat. Monographs, Vol. 19, Amer. Math. Soc. Providence, Rhode Island (1968).
[6] A. M. Samoilenko and N. I. Ronto, Numerical-Analytic Methods of Investigation of Periodic Solutions [in Russian], Vishcha Shkola, Kiev (1976).
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