Strongly quasibounded maximal monotone perturbations for the Berkovits–Mustonen topological degree theory

Abstract

Keywords:Berkovits-Mustonen degree theory Browder degree theory Maximal monotone operator Bounded demicontinuous operator of type (S + ) Let X be a real reflexive Banach space with dual X * . Let L : X ⊃ D(L) → X * be densely defined, linear and maximal monotone. Let T : X ⊃ D(T ) → 2 X * , with 0 ∈ D(T ) and 0 ∈ T (0), be strongly quasibounded and maximal monotone, and C : X ⊃ D(C ) → X * bounded, demicontinuous and of type (S + ) w.r.t. D(L). A new topological degree theory has been developed for the sum L + T + C . This degree theory is an extension of the Berkovits-Mustonen theory (for T = 0) and an improvement of the work of Addou and Mermri (for T : X → 2 X * bounded). Unbounded maximal monotone operators with 0 ∈D(T ) are strongly quasibounded and may be used with the new degree theory.

Topological degree theories for continuous perturbations of resolvent compact maximal monotone operators, existence theorems and applications

“…But this is impossible, i.e., the claim holds. Next we show that 1] is an admissible Nagumo homotopy, i.e., we obtain that…”

“…The proof requires the regularity property of a topological space. 1], G and sufficiently small ε > 0.…”

“…A number of generalizations and/or improvements of Browder's theory have been developed over the past 45 years. The main contributions are due to Hu and Papageorgiou [18], Berkovits and Mustonnen [10], Kobayashi and Otani [22], Kartsatos and Skrypnik [19], Adhikari and Kartsatos [1], Asfaw and Kartsatos [7] and, Asfaw [3][4][5]. It is worth mentioning that Browder's theory and those in [3,5,7,10,14,18,19,22] are for (S + ) (or variant of (S + )) perturbations of maximal monotone operators.…”

Topological Degrees on Unbounded Domains

“…The theory of topological degrees has progressed significantly in recent years because of its applicability to the analysis of ordinary and partial differential equations and continuation methods in nonlinear analysis in general (e.g. see [1,2,3,4,5,6,7,8,9]). The classical topological degree theory developed by Brouwer [10] in 1912 for continuous functions on finite-dimensional spaces and the Leray-Schauder degree [11] in 1934 for compact displacements of the identity in Banach spaces both assume the boundedness of the domains over which the degrees are defined.…”

“…In this paper, we construct topological degrees on unbounded domains both in finite-dimensional and infinite dimensional spaces and discuss their properties. For the development of degree theories for operators of monotone type that involve (S + )-operators and their generalized and/or multivalued versions, the reader is referred to Kartsatos and Skrypnik [1,2], Berkovits [13], Berkovits and Mustonen [14], Kartsatos and the first author [5,7], Kartsatos and Kerr [8], Hu and Papageorgiou [15], Kittilä [16] and the references therein. For the coincidence degree developed by Mawhin for nonlinear perturbations of certain Fredholm operators in normed spaces, the reader is referred to [17].…”