Abstract
Let X be an infinite‐dimensional real reflexive Banach space with dual space X∗ and G ⊂ X open and bounded. Assume that X and X∗ are locally uniformly convex. Let be maximal monotone and C : X⊃D(C) → X∗ quasibounded and of type . Assume that L ⊂ D(C), where L is a dense subspace of X, and 0 ∈ T(0). A new topological degree theory is introduced for the sum T + C. Browder′s degree theory has thus been extended to densely defined perturbations of maximal monotone operators while results of Browder and Hess have been extended to various classes of single‐valued densely defined generalized pseudomonotone perturbations C. Although the main results are of theoretical nature, possible applications of the new degree theory are given for several other theoretical problems in nonlinear functional analysis.
Highlights
Introduction and preliminariesIn what follows, the symbol X stands for an infinite-dimensional real reflexive Banach space which has been renormed so that it and its dual X∗ are locally uniformly convex
We denote by G(T) the graph of T, that is, G(T) = {(x, y) : x ∈ D(T), y ∈ Tx}
Using the fact that the operator C satisfies condition (S+), we conclude that xn → x0, x0 ∈ D(C) ∩ ∂G, and Cx0 = h∗
Summary
The symbol X stands for an infinite-dimensional real reflexive Banach space which has been renormed so that it and its dual X∗ are locally uniformly convex. We. note that the operator T is single-valued and the operator C satisfies two basic conditions (quasiboundedness and generalized (S+)) involving the dense subspace L ⊂ D(T) ∩ D(C) of the space X as well as the operator T itself. Note that the operator T is single-valued and the operator C satisfies two basic conditions (quasiboundedness and generalized (S+)) involving the dense subspace L ⊂ D(T) ∩ D(C) of the space X as well as the operator T itself This introduction is instructive in view of the degree theory that we are going to develop later in this paper. It can be seen that the present situation is sufficient for the development of our degree after a careful study of the construction in [15]
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