Eigenvalues and ranges for perturbations of nonlinear accretive and monotone operators in Banach spaces
The paper concerns perturbation results for \(m\)-accretive and maximal monotone operators. First, \(X\) is an infinite dimensional Banach space, \(T:D(T)\subseteq X\to X\) is an \(m\)-accretive multifunction, and \(C:D(C)\subseteq X\to X\) is a compact function. Suppose \(0\in T(0)\) and \(D\subseteq X\) is an open, bounded neighborhood of \(0\). Under suitable hypotheses, \(0\not\in C(\partial D)\) implies that, for every \(c>0\), there exists \(\lambda>0\) such that \(0\in(T+cI-\lambda C)(\partial D)\). In addition, \(0\not\in T(\partial D)\) implies that there exists \(\lambda>0\) such that \(0\in(T-\lambda C)(\partial D)\) [cf. \textit{Z. Guan} and \textit{A. G. Kartsatos}, Nonlinear Anal., Theory Methods Appl. 27, No. 2, 125-141 (1996; Zbl 0864.47028)]. Second, \(X\) is a reflexive Banach space, \(T:D(T)\subseteq X\to X^*\) ia a maximal monotone multifunction, and \(C:D(C)\subseteq X\to X^*\) is a function. Suppose \(0\in T(0)\) and \(G\subseteq X\) is an open, bounded neighborhood of \(0\). Under suitable hypotheses, \(0\in(T+C)(\overline{G})\) without the assumption that \(0\not\in T(\partial G)\) [cf. \textit{A. G. Kartsatos}, Proc. Am. Math. Soc. 124, No. 6, 1811-1820 (1996; Zbl 0857.47032)].
scientific article; zbMATH DE number 1370650