Homotopy invariance of parameter-dependent domains and perturbation theory for maximal monotone and \(m\)-accretive operators in Banach spaces.
The authors study perturbations of maximal monotone and \(m\)-accretive ope\-rators on parameter-dependent domains. For a multivalued operator \(T\), one denotes by \(D(T)\) the set of all \(x\) such that \(Tx\not=\emptyset\). Let then \(X\) and \(Y\) be real Banach spaces, \(G\subset X\) open and \(A:[0,1]\times X\to 2^Y\) such that each \(D(A(t,\cdot)\cap G)\not=\emptyset\), \(A_t:=A(t,\cdot)\) is invertible with a single-valued inverse and such that \((t,u)\mapsto A_t^{-1}u\) is a continuous mapping \([0,1]\times Y\to X\). Let \(H:[0,1]\times Y\to Y\) be a continuous mapping such that \(H([0,1]\times\overline{B}(0;m))\) is relatively compact for each \(m>0\). Moreover, one assumes the following growth condition: There are \(b>0\), \(0<c<1\) such that \(\| H(t,y)\| \leq b+c\| y\| \) whenever \(t\in[0,1]\) and \(y\in A_t(\overline{G})\) and \(u-H(t,u)\not=0\) whenever \(t\in[0,1]\) and \(u\in A_t(\partial G)\). The conclusion is that \(\deg(I-H(t,\cdot), A_t(G)\cap B(0;m),0)\) is well defined and constant for \(t\in[0,1]\) provided \(m>b/(1-c)\). The authors use this result to obtain a localized version of Hirano's surjectivity theorem [\textit{N. Hirano}, ``Theory and Applications of nonlinear operators of accretive and monotone type'' (New York: Marcel Dekker), 131--140 (1996; Zbl 1050.47504)] and they define a degree for demicontinuous accretive operators.
scientific article; zbMATH DE number 2002526