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Uspekhi Mat. Nauk, 1968, Volume 23, Issue 2(140), Pages 121–168 (Mi umn5611)  

This article is cited in 20 scientific papers (total in 20 papers)

Non-linear monotone operators in Banach spaces

R. I. Kachurovskii


Abstract: The article is a survey of work on non-linear monotone operators on Banach spaces. Let $F(x)$ be an operator acting from a Banach space into its adjoint space. If on the whole space the scalar product inequality $(F(x) - F(y), x - y)\ge 0$ holds, then $F(x)$ is said to be a monotone operator. It turns out that monotonicity, in conjunction with some other conditions, makes it possible to obtain existence theorems for solutions of operator equations. The results obtained have applications to boundary-value problems of partial differential equations, to differential equations in Banach spaces, and to integral equations.
Here is a list of questions touched upon in the article. General properties of monotone operators. Existence theorems for solutions of equations with operators defined on the whole space or on an everywhere dense subset of the space. Fixed point principles. Approximate methods of solution of equations with monotone operators. Examples that illustrate the possibility of applying the methods of monotonicity to some problems of analysis. In conclusion, the article gives a bibliography of over one hundred papers.

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English version:
Russian Mathematical Surveys, 1968, 23:2, 121–168

Bibliographic databases:

UDC: 517.4
MSC: 47H07, 47J20, 35A05, 35K60, 47N20
Received: 25.01.1967

Citation: R. I. Kachurovskii, “Non-linear monotone operators in Banach spaces”, Uspekhi Mat. Nauk, 23:2(140) (1968), 121–168; Russian Math. Surveys, 23:2 (1968), 121–168

Citation in format AMSBIB
\Bibitem{Kac68}
\by R.~I.~Kachurovskii
\paper Non-linear monotone operators in Banach spaces
\jour Uspekhi Mat. Nauk
\yr 1968
\vol 23
\issue 2(140)
\pages 121--168
\mathnet{http://mi.mathnet.ru/umn5611}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=226455}
\zmath{https://zbmath.org/?q=an:0162.20102}
\transl
\jour Russian Math. Surveys
\yr 1968
\vol 23
\issue 2
\pages 121--168
\crossref{https://doi.org/10.1070/RM1968v023n02ABEH001239}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. P. P. Mosolov, “Variational methods in nonstationary problems (parabolic case)”, Math. USSR-Izv., 4:2 (1970), 431–463  mathnet  crossref  mathscinet  zmath
    2. J. Lions, “Partial differential inequalities”, Russian Math. Surveys, 27:2 (1972), 91–159  mathnet  crossref  mathscinet  zmath
    3. Yu. A. Dubinskii, “On some noncoercive nonlinear equations”, Math. USSR-Sb., 16:3 (1972), 323–332  mathnet  crossref  mathscinet  zmath
    4. Peter Hess, “On nonlinear mappings of monotone type homotopic to odd operators”, Journal of Functional Analysis, 11:2 (1972), 138  crossref
    5. V. V. Zhikov, “Monotonicity in the theory of almost periodic solutions of nonlinear operator equations”, Math. USSR-Sb., 19:2 (1973), 209–223  mathnet  crossref  mathscinet  zmath
    6. Yu. A. Dubinskii, “Sobolev spaces of infinite order and the behavior of solutions of some boundary value problems with unbounded increase of the order of the equation”, Math. USSR-Sb., 27:2 (1975), 143–162  mathnet  crossref  mathscinet  zmath
    7. V. V. Zhikov, B. M. Levitan, “Favard theory”, Russian Math. Surveys, 32:1 (1977), 129–180  mathnet  crossref  mathscinet  zmath
    8. P.S Milojević, W.V Petryshyn, “Continuation theorems and the approximation-solvability of equations involving multivalued A-proper mappings”, Journal of Mathematical Analysis and Applications, 60:3 (1977), 658  crossref
    9. A. A. Pankov, “Bounded and almost periodic solutions of evolutionary variational inequalities”, Math. USSR-Sb., 36:4 (1980), 519–533  mathnet  crossref  mathscinet  zmath  isi
    10. G. G. Kazaryan, G. A. Karapetyan, “On the convergence of Galerkin approximations to the solution of the Dirichlet problem for some general equations”, Math. USSR-Sb., 52:2 (1985), 285–299  mathnet  crossref  mathscinet  zmath
    11. M.A. Efendiev, W.L. Wendland, “Nonlinear Riemann–Hilbert problems for multiply connected domains”, Nonlinear Analysis: Theory, Methods & Applications, 27:1 (1996), 37  crossref
    12. Lan Shen, YingQian Wang, “Total colorings of planar graphs with maximum degree at least 8”, Sci China Ser A, 2009  crossref  mathscinet  isi
    13. Belavkin R.V., “Bounds of Optimal Learning”, Adprl: 2009 IEEE Symposium on Adaptive Dynamic Programming and Reinforcement Learning, IEEE, 2009, 199–204  isi
    14. Roman V. Belavkin, “On evolution of an information dynamic system and its generating operator”, Optim Lett, 2011  crossref
    15. Roman V. Belavkin, “Optimal measures and Markov transition kernels”, J Glob Optim, 2012  crossref
    16. S. N. Askhabov, “Approximate solution of nonlinear discrete equations of convolution type”, Journal of Mathematical Sciences, 201:5 (2014), 566–580  mathnet  crossref  mathscinet
    17. A. V. Chernov, “On a generalization of the method of monotone operators”, Diff Equat, 49:4 (2013), 517  crossref
    18. S. N. Askhabov, “Periodic solutions of convolution type equations with monotone nonlinearity”, Ufa Math. J., 8:1 (2016), 20–34  mathnet  crossref  isi  elib
    19. I. P. Ryazantseva, “Regularized continuous analog of the Newton method for monotone equations in the Hilbert space”, Russian Math. (Iz. VUZ), 60:11 (2016), 45–57  mathnet  crossref  isi
    20. S. N. Askhabov, “Nelineinye integralnye uravneniya s yadrami tipa potentsiala na otrezke”, Trudy Sedmoi Mezhdunarodnoi konferentsii po differentsialnym i funktsionalno-differentsialnym uravneniyam (Moskva, 22–29 avgusta, 2014). Chast 3, SMFN, 60, RUDN, M., 2016, 5–22  mathnet
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