Topological degree theories for densely defined mappings involving Operators of Type $(\mathrm{S}_+)^*$.

Abstract

Let $X$ be a real separable reflexive Banach space with dual space $X^*.$ Assume that the operator $A:X\supset\mathcal{D}(A)\to X^*$ is such that $L\subset \mathcal{D}(A)$ and $L=X,$ where $L$ is a subspace of $X.$ It is shown that it is possible to define a topological degree for such operators $A$ that satisfy, mainly, Condition $(S_+)_{0,L}.$ It is also shown that a topological degree can be defined for operators of the type $M+A,$ where $M+A: X\supset\mathcal{D}(M+A)\to X^*, L\subset\mathcal{D}(M+A),$ and $ L=X.$ Here, $X$ is not necessarily separable and $M$ satisfies a variant of the maximal monotonicity condition (with respect to the space $L$) as well as an approximation condition. The operator $A$ satisfies, mainly, analogues of the quasi-boundedness condition and the $(S_+)$ condition (with respect to the operator $M).$ Properties of these degrees are studied and applications are given for nonlinear Dirichlet elliptic problems of the type $$ \sum_{i=1}^n\frac{\partial}{\partial x_i}\left\{\rho^2(u)\frac{\partial u}{\partial x_i}+ a_i\left(x,u,\frac{\partial u}{\partial x}\right)\right\}=\sum_{i=1}^n\frac{\partial}{\partial x_i}f_i(x), $$ as well as Cauchy-Dirichlet parabolic problems of the type $$ \frac{\partial u}{\partial t}-\sum_{i=1}^n\frac{\partial}{\partial x_i}a_i\left(x,t,u,\frac{\partial u}{\partial x}\right) +\rho(x,t,u)=\sum_{i=1}^n\frac{\partial}{\partial x_i}f_i(x,t). $$