Monotone operators. A survey directed to applications to differential equations

Jan Franců

Aplikace matematiky (1990)

  • Volume: 35, Issue: 4, page 257-301
  • ISSN: 0862-7940

Abstract

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The paper deals with the existence of solutions of the form A u = b with operators monotone in a broader sense, including pseudomonotone operators and operators satisfying conditions S and M . The first part of the paper which has a methodical character is concluded by the proof of an existence theorem for the equation on a reflexive separable Banach space with a bounded demicontinuous coercive operator satisfying condition ( M ) 0 . The second part which has a character of a survey compares various types of continuity and monotony and introduces further results. Application of this theory to proofs of existence theorems for boundary value problems for ordinary and partial differential equations is illustrated by examples.

How to cite

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Franců, Jan. "Monotone operators. A survey directed to applications to differential equations." Aplikace matematiky 35.4 (1990): 257-301. <http://eudml.org/doc/15631>.

@article{Franců1990,
abstract = {The paper deals with the existence of solutions of the form $Au=b$ with operators monotone in a broader sense, including pseudomonotone operators and operators satisfying conditions $S$ and $M$. The first part of the paper which has a methodical character is concluded by the proof of an existence theorem for the equation on a reflexive separable Banach space with a bounded demicontinuous coercive operator satisfying condition $(M)_0$. The second part which has a character of a survey compares various types of continuity and monotony and introduces further results. Application of this theory to proofs of existence theorems for boundary value problems for ordinary and partial differential equations is illustrated by examples.},
author = {Franců, Jan},
journal = {Aplikace matematiky},
keywords = {monotone; pseudomonotone operators; operators satisfying $S$; $M$ conditions; existence theorems for boundary value problems for differential equations; monotone operators; surjectivity of a monotone hemicontinuous coercive operator},
language = {eng},
number = {4},
pages = {257-301},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Monotone operators. A survey directed to applications to differential equations},
url = {http://eudml.org/doc/15631},
volume = {35},
year = {1990},
}

TY - JOUR
AU - Franců, Jan
TI - Monotone operators. A survey directed to applications to differential equations
JO - Aplikace matematiky
PY - 1990
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 35
IS - 4
SP - 257
EP - 301
AB - The paper deals with the existence of solutions of the form $Au=b$ with operators monotone in a broader sense, including pseudomonotone operators and operators satisfying conditions $S$ and $M$. The first part of the paper which has a methodical character is concluded by the proof of an existence theorem for the equation on a reflexive separable Banach space with a bounded demicontinuous coercive operator satisfying condition $(M)_0$. The second part which has a character of a survey compares various types of continuity and monotony and introduces further results. Application of this theory to proofs of existence theorems for boundary value problems for ordinary and partial differential equations is illustrated by examples.
LA - eng
KW - monotone; pseudomonotone operators; operators satisfying $S$; $M$ conditions; existence theorems for boundary value problems for differential equations; monotone operators; surjectivity of a monotone hemicontinuous coercive operator
UR - http://eudml.org/doc/15631
ER -

References

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  2. P. Doktor, Modern methods of solving partial differential equations, (Czech), Lecture Notes, SPN, Prague, 1976. (1976) 
  3. S. Fučík, Solvability of nonlinear equations and boundary value problems, D. Reidel Publ. Соmр., Dordrecht; JČSMF, Prague, 1980. (1980) MR0620638
  4. S. Fučík A. Kufner, Nonlinear differential equations;, Czech edition - SNTL, Prague 1978; English translation - Elsevier, Amsterdam 1980. (1978) MR0558764
  5. S. Fučík J. Milota, Mathematical analysis II, (Czech), Lecture Notes, SPN, Prague 1980. (1980) 
  6. S. Fučík J. Nečas J. Souček V. Souček, Spectral analysis of nonlinear operators, Lecture Notes in Math. 346, Springer, Berlin 1973; JCSMF, Prague 1973. (1973) MR0467421
  7. R. I. Kačurovskij, Nonlinear monotone operators in Banach spaces, (Russian), Uspechi Mat. Nauk 23 (1968), 2, 121-168. (1968) MR0226455
  8. D. Kinderlehrer G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, New York 1980; Russian translation - Mir, Moscow 1983. (1980) MR0738631
  9. A. N. Kolmogorov S. V. Fomin, Introductory real analysis, (Russian), Moscow 1954, English translation - Prentice Hall, New York 1970, Czech translation - SNTL, Prague 1975. (1954) MR0267052
  10. A. Kufner O. John S. Fučík, Function spaces, Academia, Prague 1977. (1977) MR0482102
  11. J. Nečas, Introduction to the theory of nonlinear elliptic equations, Teubner-Texte zur Math. 52, Leipzig, 1983. (1983) MR0731261
  12. D. Pascali S. Sburlan, Nonlinear mappings of monotone type, Editura Academiei, Bucuresti 1978. (1978) MR0531036
  13. A. Pultr, Subspaces of Euclidean spaces, (Czech), Matematický seminář - 22, SNTL, Prague 1987. (1987) 
  14. E. Zeidler, Lectures on nonlinear functional analysis II - Monotone operators, (German), Teubner-Texte zur Math. 9, Leipzig 1977; Revised extended English translation: Nonlinear functional analysis and its application II, Springer, New York (to appear). (1977) MR0628004
  15. J. Nečas, Nonlinear elliptic equations, (French), Czech. Math. J. 19 (1969), 252-274. (1969) 
  16. M. Feistauer A. Ženíšek, 10.1007/BF01398687, Numer. Math. 52 (1988), 147-163. (1988) MR0923708DOI10.1007/BF01398687

Citations in EuDML Documents

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  1. Petr Harasim, On the worst scenario method: a modified convergence theorem and its application to an uncertain differential equation
  2. Chang-Ho Song, Yong-Gon Ri, Cholmin Sin, Properties of a quasi-uniformly monotone operator and its application to the electromagnetic $p$-$\text {curl}$ systems
  3. Miroslav Pospíšek, Nonlinear boundary value problems with application to semiconductor device equations
  4. Ivan Hlaváček, Finite element analysis of a static contact problem with Coulomb friction
  5. Jan Franců, Weak convergence in infinite dimensional spaces
  6. Michal Křížek, Liping Liu, On a comparison principle for a quasilinear elliptic boundary value problem of a nonmonotone type
  7. Petr Harasim, On the worst scenario method: Application to a quasilinear elliptic 2D-problem with uncertain coefficients
  8. Ondřej Bartoš, Miloslav Feistauer, Filip Roskovec, On the effect of numerical integration in the finite element solution of an elliptic problem with a nonlinear Newton boundary condition
  9. Jiří Vala, The method of Rothe and two-scale convergence in nonlinear problems
  10. Miloslav Feistauer, Karel Najzar, Veronika Sobotíková, On the finite element analysis of problems with nonlinear Newton boundary conditions in nonpolygonal domains

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