Monotone operators. A survey directed to applications to differential equations. (English) Zbl 0724.47025

Franců, Jan

Apl. Mat. 35, No. 4, 257-301 (1990).

This is a self-contained and well-written survey on the theory and some applications of monotone operators between a Banach space V and its dual \(V'\). A central point is of course G. J. Minty’s celebrated theorem on the surjectivity of a monotone hemicontinuous coercive operator [Duke Math. J. 29, 341-346 (1962; Zbl 0111.312)] which was respeated later by F. E. Browder [Duke Math. J. 30, 557-566 (1963; Zbl 0119.325)]. Moreover, the author gives a nice scheme on the “hierarchy” between various monotonicity and continuity concepts arising in the theory of monotone operators. The last section is concerned with several applications to both ordinary and partial differential equations, mainly in the spirit of S. Fučik’s and A. Kufner’s book on nonlinear differential equations [Amsterdam (1980; Zbl 0474.35001)]. This paper will be very useful to anyone who wants to get a first orientation on the notions, results, methods, and applications of monotone operators.

Reviewer: J.Appell (Würzburg)

MSC:

47H05

Monotone operators and generalizations

Keywords:

monotone operators; surjectivity of a monotone hemicontinuous coercive operator

Citations:

Zbl 0111.312; Zbl 0119.325; Zbl 0474.35001

References:

[1]	K. Deimling: Nonlinear functional analysis. Springer 1985. · Zbl 0559.47040
[2]	P. Doktor: Modern methods of solving partial differential equations. (Czech), Lecture Notes, SPN, Prague, 1976. · Zbl 0342.41025
[3]	S. Fučík: Solvability of nonlinear equations and boundary value problems. D. Reidel Publ. Comp., Dordrecht; JČSMF, Prague, 1980.
[4]	S. Fučík A. Kufner: Nonlinear differential equations. Czech edition - SNTL, Prague 1978;
[5]	S. Fučík J. Milota: Mathematical analysis II. (Czech), Lecture Notes, SPN, Prague 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. · Zbl 0268.47056
[7]	R. I. Kačurovskij: Nonlinear monotone operators in Banach spaces. (Russian), Uspechi Mat. Nauk 23 (1968), 2, 121-168.
[8]	D. Kinderlehrer G. Stampacchia: An introduction to variational inequalities and their applications. Academic Press, New York 1980; Russian translation - Mir, Moscow 1983. · Zbl 0457.35001
[9]	A. N. Kolmogorov S. V. Fomin: Introductory real analysis. (Russian), Moscow 1954, · Zbl 0213.07305
[10]	A. Kufner O. John S. Fučík: Function spaces. Academia, Prague 1977.
[11]	J. Nečas: Introduction to the theory of nonlinear elliptic equations. Teubner-Texte zur Math. 52, Leipzig, 1983. · Zbl 0526.35003
[12]	D. Pascali S. Sburlan: Nonlinear mappings of monotone type. Editura Academiei, Bucuresti 1978. · Zbl 0423.47021
[13]	A. Pultr: Subspaces of Euclidean spaces. (Czech), Matematický seminář - 22, SNTL, Prague 1987.
[14]	E. Zeidler: Lectures on nonlinear functional analysis II - Monotone operators. (German), Teubner-Texte zur Math. 9, Leipzig 1977; Revised extended
[15]	J. Nečas: Nonlinear elliptic equations. (French), Czech. Math. J. 19 (1969), 252-274. · Zbl 0193.39202
[16]	M. Feistauer A. Ženíšek: Compactness method in the finite element theory of nonlinear elliptic problems. Numer. Math. 52 (1988), 147-163. · Zbl 0642.65075