A range and existence theorem for pseudomonotone perturbations of maximal monotone operators

Le, Vy

doi:10.1090/S0002-9939-2010-10594-4

A range and existence theorem for pseudomonotone perturbations of maximal monotone operators
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by Vy Khoi Le PDF

Proc. Amer. Math. Soc. 139 (2011), 1645-1658 Request permission

Abstract:

In this paper, we prove a range and existence theorem for multivalued pseudomonotone perturbations of maximal monotone operators. We assume a general coercivity condition on the sum of a maximal monotone and a pseudomonotone operator instead of a condition on the pseudomonotone operator only. An illustrative example of a variational inequality in a Sobolev space with variable exponent is given.

References

Haïm Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble) 18 (1968), no. fasc. 1, 115–175 (French). MR 270222
H. Brézis and L. Nirenberg, Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 2, 225–326. MR 513090
Felix E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R.I., 1976, pp. 1–308. MR 0405188
Felix E. Browder and Peter Hess, Nonlinear mappings of monotone type in Banach spaces, J. Functional Analysis 11 (1972), 251–294. MR 0365242, DOI 10.1016/0022-1236(72)90070-5
Xianling Fan, Boundary trace embedding theorems for variable exponent Sobolev spaces, J. Math. Anal. Appl. 339 (2008), no. 2, 1395–1412. MR 2377096, DOI 10.1016/j.jmaa.2007.08.003
Xianling Fan and Dun Zhao, On the spaces $L^{p(x)}(\Omega )$ and $W^{m,p(x)}(\Omega )$, J. Math. Anal. Appl. 263 (2001), no. 2, 424–446. MR 1866056, DOI 10.1006/jmaa.2000.7617
Z. Guan, A. G. Kartsatos, and I. V. Skrypnik, Ranges of densely defined generalized pseudomonotone perturbations of maximal monotone operators, J. Differential Equations 188 (2003), no. 1, 332–351. MR 1954518, DOI 10.1016/S0022-0396(02)00066-9
Athanassios G. Kartsatos and Igor V. Skrypnik, A new topological degree theory for densely defined quasibounded $(\~S_+)$-perturbations of multivalued maximal monotone operators in reflexive Banach spaces, Abstr. Appl. Anal. 2 (2005), 121–158. MR 2179439, DOI 10.1155/AAA.2005.121
Nobuyuki Kenmochi, Nonlinear operators of monotone type in reflexive Banach spaces and nonlinear perturbations, Hiroshima Math. J. 4 (1974), 229–263. MR 383162
N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities, Handbook of differential equations: stationary partial differential equations. Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007, pp. 203–298. MR 2569333, DOI 10.1016/S1874-5733(07)80007-6
David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 567696
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris; Gauthier-Villars, Paris, 1969 (French). MR 0259693
Z. Naniewicz and P. D. Panagiotopoulos, Mathematical theory of hemivariational inequalities and applications, Monographs and Textbooks in Pure and Applied Mathematics, vol. 188, Marcel Dekker, Inc., New York, 1995. MR 1304257
Eberhard Zeidler, Nonlinear functional analysis and its applications. II/B, Springer-Verlag, New York, 1990. Nonlinear monotone operators; Translated from the German by the author and Leo F. Boron. MR 1033498, DOI 10.1007/978-1-4612-0985-0