A new topological degree theory for densely defined \((S_+)_L\)-perturbations of multivalued maximal monotone operators in reflexive separable Banach spaces
The authors introduce a topological degree theory for densely defined perturbations of multivalued maximal monotone operators in reflexive separable Banach spaces, which is based on the degree theory of \textit{A.\,G.\thinspace Kartsatos} and \textit{I.\,V.\thinspace Skrypnik} [Adv.\ Differ.\ Equ.\ 4, No.\,3, 413--456 (1999; Zbl 0959.47037)]. More precisely, let \(X\) be a real reflexive separable Banach space with its dual \(X^*\). Suppose that \(T: D(T)\subset X\to 2^{X^*}\) is maximal monotone such that \(0\) is an interior point of \(D(T)\) and \(0\in T(0)\), and that \(C: D(C)\subset X\to X^*\) is of class \((S_{+})_L\) such that \(\langle Cx,x\rangle \geq - \psi(| \!| x | \!| )\) for \(x\in D(C)\), where \(L\subset D(C)\) is a dense linear subspace of \(X\) and \(\psi: [0,\infty) \to [0,\infty)\) is nondecreasing. The authors define a topological degree for the sum \(T+C\) as the limit of the Kartsatos--Skrypnik degree of single-valued mappings \(T_t+C\), where each \(T_t=(T^{-1}+tJ^{-1})^{-1}: X\to X^*\) is maximal monotone, \(J\) being the normalized duality mapping. This degree theory extends the theory by the first author and \textit{I.\,V.\thinspace Skrypnik} [Abstr.\ Appl.\ Anal.\ 2005, No.\,2, 121--158 (2005; Zbl 1110.47049)] in some sense. Moreover, this degree is applied to establish an abstract result on the existence of zeros for inclusions.
densely defined \((S_{+})_L\)-perturbations
scientific article; zbMATH DE number 5354984
scientific article