scholarly journals Eigenvalue for Densely Defined Perturbations of Multivalued Maximal Monotone Operators in Reflexive Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Boubakari Ibrahimou
Keyword(s):  
Banach Space ◽  
Reflexive Banach Space ◽  
Degree Theory ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Infinite Length ◽  
Topological Degree Theory ◽  
Reflexive Banach Spaces ◽  
Densely Defined

Let be a real reflexive Banach space and let be its dual. Let be open and bounded such that . Let be maximal monotone with and . Using the topological degree theory developed by Kartsatos and Quarcoo we study the eigenvalue problem where the operator is a single-valued of class . The existence of continuous branches of eigenvectors of infinite length then could be easily extended to the case where the operator is multivalued and is investigated.

Eigenvalue results for pseudomonotone perturbations of maximal monotone operators

2013 ◽  
Vol 11 (5) ◽  
Author(s):  
In-Sook Kim ◽  
Jung-Hyun Bae
Keyword(s):  
Reflexive Banach Space ◽  
Degree Theory ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Dense Subspace ◽  
Uniformly Convex ◽  
Infinite Dimensional ◽  
Surjectivity Result ◽  
Densely Defined

AbstractLet X be an infinite-dimensional real reflexive Banach space such that X and its dual X* are locally uniformly convex. Suppose that T: X⊃D(T) → 2X* is a maximal monotone multi-valued operator and C: X⊃D(C) → X* is a generalized pseudomonotone quasibounded operator with L ⊂ D(C), where L is a dense subspace of X. Applying a recent degree theory of Kartsatos and Skrypnik, we establish the existence of an eigensolution to the nonlinear inclusion 0 ∈ T x + λ C x, with a regularization method by means of the duality operator. Moreover, possible branches of eigensolutions to the above inclusion are discussed. Furthermore, we give a surjectivity result about the operator λT + C when λ is not an eigenvalue for the pair (T, C), T being single-valued and densely defined.

scholarly journals A new topological degree theory for densely defined quasibounded(S˜+)-perturbations of multivalued maximal monotone operators in reflexive Banach spaces

2005 ◽  
Vol 2005 (2) ◽  
pp. 121-158 ◽  
Author(s):  
Athanassios G. Kartsatos ◽  
Igor V. Skrypnik
Keyword(s):  
Topological Degree ◽  
Degree Theory ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Dense Subspace ◽  
Topological Degree Theory ◽  
Nonlinear Functional ◽  
Infinite Dimensional ◽  
Densely Defined

LetXbe an infinite-dimensional real reflexive Banach space with dual spaceX∗andG⊂Xopen and bounded. Assume thatXandX∗are locally uniformly convex. LetT:X⊃D(T)→2X∗be maximal monotone andC:X⊃D(C)→X∗quasibounded and of type(S˜+). Assume thatL⊂D(C), whereLis a dense subspace ofX, and0∈T(0). A new topological degree theory is introduced for the sumT+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 perturbationsC. 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.

scholarly journals The sum theorem for maximal monotone operators in reflexive Banach spaces revisited

2021 ◽  
Vol 66 (1) ◽  
pp. 123-126
Author(s):  
Mircea D. Voisei
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Reflexive Banach Space ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Minkowski Sum ◽  
Reflexive Banach Spaces ◽  
Maximal Monotonicity ◽  
Sum Theorem

The goal of this note is to present a new shorter proof for the maximal monotonicity of the Minkowski sum of two maximal monotone multi-valued operators defined in a reflexive Banach space under the classical interiority condition involving their domains.

scholarly journals Convergence theorems for generalized projections and maximal monotone operators in Banach spaces

2003 ◽  
Vol 2003 (10) ◽  
pp. 621-629 ◽  
Author(s):  
Takanori Ibaraki ◽  
Yasunori Kimura ◽  
Wataru Takahashi
Keyword(s):  
Banach Space ◽  
Pointwise Convergence ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Convex Banach Space ◽  
Limit Set ◽  
Strong And Weak Convergence ◽  
Strictly Convex Banach Space ◽  
Generalized Projections

We study a sequence of generalized projections in a reflexive, smooth, and strictly convex Banach space. Our result shows that Mosco convergence of their ranges implies their pointwise convergence to the generalized projection onto the limit set. Moreover, using this result, we obtain strong and weak convergence of resolvents for a sequence of maximal monotone operators.

scholarly journals Strong and Weak Convergence of Modified Mann Iteration for New Resolvents of Maximal Monotone Operators in Banach Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-20 ◽  
Author(s):  
Somyot Plubtieng ◽  
Wanna Sriprad
Keyword(s):  
Banach Space ◽  
Weak Convergence ◽  
Convergence Rate ◽  
Minimization Problem ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Convex Minimization Problem ◽  
Mann Iteration ◽  
Strong And Weak Convergence

We prove strong and weak convergence theorems for a new resolvent of maximal monotone operators in a Banach space and give an estimate of the convergence rate of the algorithm. Finally, we apply our convergence theorem to the convex minimization problem. The result present in this paper extend and improve the corresponding result of Ibaraki and Takahashi (2007), and Kim and Xu (2005).

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