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Topological degree theories for densely defined mappings involving operators of type \((S_+)\). (English) Zbl 0959.47037

The authors define a topological degree for operators \(A: D(A)(\subset X)\to X^*\) satisfying the condition \((S_+)_{0,L}\). Here \(X\) is a real separable reflexive Banach space with dual spaces \(X^*\) and there is a subspace \(L\subset D(A)\) of \(X\) such that \(\overline L= X\). It is also shown that a topological degree can be defined for operators of the type \(M+A\) where \(M+A: D(M+ A)(\subset X)\to X^*\), \(L\subset D(M+ A)\) and \(\overline L= X\). Here \(X\) is not necessarily separable and \(M\) satisfies a variant of the maximal monotonicity condition. They apply the results obtained to nonlinear Dirichlet elliptic problems of the type \[ \sum^n_{i=1} {\partial\over\partial x_i} \Biggl[\rho^2(u){\partial u\over\partial x_i}+ a_i\Biggl(x, u, {\partial u\over\partial x}\Biggr)\Biggr]= \sum^n_{i=1} {\partial\over\partial x_i} f_i(x) \] as well as to Cauchy-Dirichlet parabolic problems of the type \[ {\partial u\over\partial t}- \sum^n_{i=1} {\partial\over\partial x_i} a_i\Biggl(x,t,u,{\partial u\over\partial x}\Biggr)+ \rho(x, t,u)= \sum^n_{i=1} {\partial\over\partial x_i} f_i(x,t). \]

MSC:

47H11 Degree theory for nonlinear operators
47H05 Monotone operators and generalizations
35J65 Nonlinear boundary value problems for linear elliptic equations
47F05 General theory of partial differential operators
47N20 Applications of operator theory to differential and integral equations
58C30 Fixed-point theorems on manifolds
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