The Epi-Continuation Method for Minimization Problems. Relation with the Degree Theory of F. Browder for Maximal Monotone Operators

1989 ◽  
pp. 29-58 ◽  
Author(s):  
Hedy Attouch ◽  
Hassan Riahi
Keyword(s):  
Continuation Method ◽  
Degree Theory ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Minimization Problems

scholarly journals Iterative Methods for Equilibrium Problems and Monotone Inclusion Problems in Hilbert Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Huan-chun Wu ◽  
Cao-zong Cheng
Keyword(s):  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Inclusion Problems ◽  
Minimization Problems ◽  
Convex Minimization Problems ◽  
New Iterative Method

We introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of zeros of the sum of maximal monotone operators, and we obtain strong convergence theorems in Hilbert spaces. We also apply our results to the variational inequality and convex minimization problems. Our results extend and improve the recent result of Takahashi et al. (2012).

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 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.

scholarly journals Topological degree and application to a parabolic variational inequality problem

2001 ◽  
Vol 25 (4) ◽  
pp. 273-287 ◽  
Author(s):  
A. Addou ◽  
B. Mermri
Keyword(s):  
Variational Inequality ◽  
Topological Degree ◽  
Variational Inequality Problem ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Parabolic Variational Inequality ◽  
Inequality Problem ◽  
Monotone Map ◽  
New Formulation

We are interested in constructing a topological degree for operators of the formF=L+A+S, whereLis a linear densely defined maximal monotone map,Ais a bounded maximal monotone operators, andSis a bounded demicontinuous map of class(S+)with respect to the domain ofL. By means of this topological degree we prove an existence result that will be applied to give a new formulation of a parabolic variational inequality problem.

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