# Dissipative sets and nonlinear perturbated equations in Banach spaces

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## Abstract

Dissipative sets and nonlinear perturbated equations in Banach spaces ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze VIORELBARBU Dissipative sets and nonlinear perturbated equations in Banach spaces Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e série, tome 26, no 2 (1972), p. 365-390. <http://www.numdam.org/item?id=ASNSP_1972_3_26_2_365_0> © Scuola Normale Superiore, Pisa, 1972, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ DISSIPATIVE SETS AND NONLINEAR PERTURBATED EQUATIONS IN BANACH SPACES by VIOREL BARBU ABSTRACT - Some existence results for abstract functional equations in Banach spaces are proved. Introduction. Let .~ be a real Banach space ~~ its dual space, (u, v) the paring between v in .~~ and r in X. The duality mapping of X in the subset F of X X .X~ defined by where )] ]) denotes the norm in X (respectively X*). Let A be a subset of X x ~X. We define and . where oc is real. If .~ is a subset of X then A subset ~1 of X x X is called dissipative if for every Ixi, yi] E A, i = 1, 2 there exists f E F (x~ - x2) such that or equivalently (see T. Kato [10], Lemma 1.1), Pervenuto alla Redazione il 31 Marzo 1971. 366 for 0 and If A is dissipative one can define for À &#x3E; 0 a single valued operator Ai = I-1 ((1- lA)-I - 1) with D (A,) == R (1 2013~A). We notice some pro- perties of Ai which will be used frequently in this paper (for the proof see T. Kato [11]). LEMMA 0

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