Portugaliae Mathematica   EMIS ELibM Electronic Journals PORTUGALIAE
Vol. 61, No. 4, pp. 439-459 (2004)

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Semilinear problems with a non-symmetric linear part having an infinite dimensional kernel

J. Berkovits and C. Fabry

Department of Mathematical Sciences, University of Oulu,
P.O.Box 3000, FIN-90014 Oulu -- FINLAND
E-mail: jberkovi@sun3.oulu.fi
Institut de Mathématique Pures et Appliquée, Université Catholique de Louvain,
Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve -- BELGIUM
E-mail: fabry@math.ucl.ac.be

Abstract: We consider semilinear equations, where the linear part $L$ is non-symmetric and has a possibly infinite dimensional kernel. We shall show that, under certain monotonicity conditions for the nonlinearity, a generalized Leray--Schauder degree can be defined for these problems. In order to build the degree theory, we introduce, for the nonlinearity $N$, monotonicity properties with respect to a linear map $T$, e.g. $T$-pseudomonotonicity or maps of class $(S_+)_{T}$. As applications, we obtain new existence results for semilinear equations, in particular in resonance situations. In this latter case, we modify the standard inequalities of Landesman--Lazer type by replacing the identity map $I$ by a linear homeomorphism ${\mathcal J}$, which will then appear in the monotonicity conditions.

Keywords: non-symmetric\,linear\,operators; topological\,degree; resonance; Landesman--Lazer conditions.

Classification (MSC2000): 47J25, 47H11.

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Electronic version published on: 7 Mar 2008.

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