21,854 research outputs found
New kinds of generalized variational-like inequality problems in topological vector spaces
AbstractIn this work, we consider a generalized nonlinear variational-like inequality problem, in topological vector spaces, and, by using the KKM technique, we prove an existence theorem. Our result extends a theorem of Ahmad and Irfan [R. Ahmad, S.S. Irfan, On the generalized nonlinear variational-like inequality problems, Appl. Math. Lett. 19 (2006) 294–297]
Variational Inclusions with General Over-relaxed Proximal Point and Variational-like Inequalities with Densely Pseudomonotonicity
This dissertation focuses on the existence and uniqueness of the solutions of variational inclusion and variational inequality problems and then attempts to develop efficient algorithms to estimate numerical solutions for the problems. The dissertation consists a total of five chapters. Chapter 1 is an introduction to variational inequality problems, variational inclusion problems, monotone operators, and some basic definitions and preliminaries from convex analysis. Chapter 2 is a study of a general class of nonlinear implicit inclusion problems. The objective of this study is to explore how to omit the Lipschitz continuity condition by using an alternating approach to the proximal point algorithm to estimate the numerical solution of the implicit inclusion problems. In chapter 3 we introduce generalized densely relaxed ƞ - α pseudomonotone operators and generalized relaxed ƞ - α proper quasimonotone operators as well as relaxed ƞ - α quasimonotone operators. Using these generalized monotonicity notions, we establish the existence results for the generalized variational-like inequality in the general setting of Banach spaces. In chapter 4, we use the auxiliary principle technique to introduce a general algorithm for solutions of the densely relaxed pseudomonotone variational-like inequalities. Chapter 5 is the chapter concluding remarks and scope for future work
Local strong maximal monotonicity and full stability for parametric variational systems
The paper introduces and characterizes new notions of Lipschitzian and
H\"olderian full stability of solutions to general parametric variational
systems described via partial subdifferential and normal cone mappings acting
in Hilbert spaces. These notions, postulated certain quantitative properties of
single-valued localizations of solution maps, are closely related to local
strong maximal monotonicity of associated set-valued mappings. Based on
advanced tools of variational analysis and generalized differentiation, we
derive verifiable characterizations of the local strong maximal monotonicity
and full stability notions under consideration via some positive-definiteness
conditions involving second-order constructions of variational analysis. The
general results obtained are specified for important classes of variational
inequalities and variational conditions in both finite and infinite dimensions
An induction theorem and nonlinear regularity models
A general nonlinear regularity model for a set-valued mapping , where and are metric spaces, is considered
using special iteration procedures, going back to Banach, Schauder, Lusternik
and Graves. Namely, we revise the induction theorem from Khanh, J. Math. Anal.
Appl., 118 (1986) and employ it to obtain basic estimates for studying
regularity/openness properties. We also show that it can serve as a
substitution of the Ekeland variational principle when establishing other
regularity criteria. Then, we apply the induction theorem and the mentioned
estimates to establish criteria for both global and local versions of
regularity/openness properties for our model and demonstrate how the
definitions and criteria translate into the conventional setting of a
set-valued mapping .Comment: 28 page
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