Topological Degree and Variational Inequality Theories for Pseudomonotone Perturbations of Maximal Monotone Operators (original) (raw)
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Journal of Mathematical Analysis and Applications, 2008
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.
A new topological degree theory for perturbations of the sum of two maximal monotone operators
Nonlinear Analysis: Theory, Methods & Applications, 2011
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 T : X ⊃ D(T) → 2 X * be maximal monotone and strongly quasibounded, S : X ⊃ D(S) → X * maximal monotone, and C : X ⊃ D(C) → X * strongly quasibounded w.r.t. S and such that it satisfies a generalized (S +)-condition w.r.t. S. Assume that D(S) = L ⊂ D(T) ∩ D(C), where L is a dense subspace of X , and 0 ∈ T (0), S(0) = 0. A new topological degree theory is introduced for the sum T + S + C , with degree mapping d(T + S + C , G, 0). The reason for this development is the creation of a useful tool for the study of a class of time-dependent problems involving three operators. This degree theory is based on a degree theory that was recently developed by Kartsatos and Skrypnik just for the single-valued sum S + C , as above.
Journal of Nonlinear Sciences and Applications, 2020
Let X be a real locally uniformly convex reflexive Banach space. Let T : X ⊇ D(T) → 2 X * and A : X ⊇ D(A) → 2 X * be maximal monotone operators such that T is of compact resolvents and A is strongly quasibounded, and C : X ⊇ D(C) → X * be a bounded and continuous operator with D(A) ⊆ D(C) or D(C) = U. The set U is a nonempty and open (possibly unbounded) subset of X. New degree mappings are constructed for operators of the type T + A + C. The operator C is neither pseudomonotone type nor defined everywhere. The theory for the case D(C) = U presents a new degree mapping for possibly unbounded U and both of these theories are new even when A is identically zero. New existence theorems are derived. The existence theorems are applied to prove the existence of a solution for a nonlinear variational inequality problem.
Journal of function spaces, 2016
Let be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space *. Let : ⊇ () → 2 * be maximal monotone of type Γ (i.e., there exist ≥ 0 and a nondecreasing function : [0, ∞) → [0, ∞) with (0) = 0 such that ⟨V * , − ⟩ ≥ − ‖ ‖ − (‖ ‖) for all ∈ (), V * ∈ , and ∈), : ⊃ () → * be linear, surjective, and closed such that −1 : * → is compact, and : → * be a bounded demicontinuous operator. A new degree theory is developed for operators of the type + +. The surjectivity of can be omitted provided that () is closed, is densely defined and self-adjoint, and = , a real Hilbert space. The theory improves the degree theory of Berkovits and Mustonen for + , where is bounded demicontinuous pseudomonotone. New existence theorems are provided. In the case when is monotone, a maximality result is included for and +. The theory is applied to prove existence of weak solutions in = 2 (0, ; 1 0 (Ω)) of the nonlinear equation given by / −∑ =1 ((/) (
On the translations of quasimonotone maps and monotonicity
Journal of Inequalities and Applications, 2012
We show that given a convex subset K of a topological vector space X and a multivalued map T : K ⇒ X * , if there exists a nonempty subset S of X * with the surjective property on K and T + w is quasimonotone for each w ∈ S, then T is monotone. Our result is a new version of the result obtained by N. Hadjisavvas (Appl.
On Quasimonotone Variational Inequalities
Journal of Optimization Theory and Applications, 2000
The purpose of this paper is to prove the existence of solutions of the Stampacchia variational inequality for a quasimonotone multivalued operator without any assumption on the existence of inner points. Moreover, the operator is not supposed to be bounded valued. The result strengthens a variety of other results in the literature.
Journal of Convex Analysis, 1994
The generalized topological degree theory is based on the Brouwer and Leray-Schauder degrees. It can be defined for general classes of mappings. The purpose of this article is twofold. One goal is to define the topological degree for maximal monotone operators. Particular attention is paid to the continuation methods for this kind of operators and real functions of convex type. This allows us to extend some recent results (see [5], [6]) by withdrawing the compactness assumptions.