Existence Results for Multivalued Compact Perturbations of \({m}\)-Accretive Operators

• Published in: arXiv.org
• Main Authors: Adhikari, Dhruba R, Asfaw, Teffera M
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 12.03.2023
• Subjects: Banach spaces; Coercivity; Operators (mathematics); Perturbation

Let \(X\) be a real Banach space with its dual \(X^*\) and \(G\) be a nonempty, bounded and open subset of \(X\) with \(0\in G\). Let \(T: X\supset D(T)\to 2^{X}\) be an \(m\)-accretive operator with \(0\in D(T)\) and \(0\in T(0)\), and let \(C\) be a compact operator from \(X\) into \(X\) with \(D(T)\subset D(C)\). We prove that \(f\in \overline{R(T)}+\overline{R(C)}\) if \(C\) is multivalued and \(f\in \overline{R(T+C)}\) if \(C\) is single-valued, provided \(Tx+Cx+\varepsilon x\not\ni f\) for all \(x\in D(T)\cap \partial G\) and \(\varepsilon >0.\) The surjectivity of \(T+C\) is proved if \(T\) is expansive and \(T+C\) is weakly coercive. Analogous results are given if \(T\) has compact resolvents and \(C\) is continuous and bounded. Various results by Kartsatos, and Kartsatos and Liu are improved, and a result by Morales is generalized.