Quasimonotone Operator Research Articles

Relaxed η-α quasimonotone and application to the generalized variational-like inequality problem

In this paper, some new mappings called relaxed η-α quasimonotone and a relaxed η-α properly quasimonotone operator are first introduced. The relationships between them are obtained. After this, the variational-like inequality problem and the relaxed Minty variational-like inequality problem are discussed by use of the proposed generalized monotone operators. Furthermore, we give the gap function of the two variational-like inequalities and two kinds of optimization problems. Finally, we point out that the two optimization problems are equivalent under some conditions.

Single-Directional Properties of Quasi-Monotone Operators

A large supply of quasi-monotone multivalued mappings with values in a weak* fragmentable dual Banach space is shown to be generically single-directional. This is of some interest in analysis of quasi-convex functions. The paper extends/strengthens some results of a recent paper by Aussel and Eberhard (Math Program 30, 2012).

Relaxed Quasimonotone Operators and Relaxed Quasiconvex Functions

In this paper, we introduce the class of multivalued relaxed μ quasimonotone operators and establish the existence of solutions of variational inequalities for such operators. This result is compared with a recent result of Bai et al. on densely relaxed pseudomonotone operators. A similar comparison regarding an existence result of Luc on densely pseudomonotone operators is provided. Also, we introduce a broad class of functions, called relaxed quasiconvex functions, and show that they are characterized by the relaxed μ quasimonotonicity of their subdifferentials. The results strengthen a variety of other results in the literature.

A note on a paper by Su Ke and He Zhen

In this note we show that a multivalued bounded quasimonotone operator of type ( M ) between a reflexive Banach space and its dual is pseudomonotone. Hence we deduce that a result of Su Ke and He Zhen [Su Ke, He Zhen, Solutions of nonlinear evolution inclusions, J. Math. Anal. Appl. 313 (2006) 84–97] follows from a theorem of Bian and Webb [W. Bian, J.R.L. Webb, Solutions of nonlinear evolution inclusions, Nonlinear Anal. 37 (1999) 915–932].

On the generalized monotonicity of variational inequalities

In this paper, three new classes of generalized monotone operators are introduced: the relaxed μ pseudomonotone operator, relaxed μ quasimonotone operator and densely relaxed pseudomonotone operator. The relations between these three classes and the relations between them and some other well known classes of generalized monotone operators are discussed in detail. Various existence results for variational inequalities in normed spaces are derived. The results generalize some known results.

Existence conditions in general quasimonotone variational inequalities

In this paper we study a general variational inequality model with set-valued quasimonotone operators, a model which includes several variational inequalities and equilibrium problems. We establish unifying conditions for existence of solutions in a topological vector space setting. Applications to parametric equilibrium models and to a contact problem are given.

Non-compact generalized variational inequalities for quasi-monotone and hemi-continuous operators with applications

Some results are obtained for non-compact cases in topological vector spaces for the existence problem of solutions for some set-valued variational inequalities with quasi-monotone and lower hemi-continuous operators, and with quasi-semi-monotone and upper hemi-continuous operators. Some applications are given in non-reflexive Banach spaces for these existence problems of solutions and for perturbation problems for these set-valued variational inequalities with quasi-monotone and quasi-semi-monotone operators.

An Appropriate Subdifferential for Quasiconvex Functions

In this paper we introduce and study a subdifferential that is related to the quasi- convex functions, much as the Fenchel--Moreau subdifferential is related to the convex ones. It is defined for any lower semicontinuous function, through an appropriate combination of an abstract subdifferential and the normal cone to sublevel sets. We show that this quasiconvex subdifferential is always a cyclically quasimonotone operator that coincides with the Fenchel--Moreau subdifferential whenever the function is convex, and that under mild assumptions, the density of its domain in the domain of the function is equivalent to the quasiconvexity of the function. We also show that the quasiconvex subdifferential of a lower semicontinuous function contains the derivatives of its differentiable quasiaffine supports. As a consequence, it contains the subdifferential introduced by Martinez-Legaz and Sach in a recent paper [J. Convex Anal., 6 (1999), pp. 1--12]. Several other properties and calculus rules are also established.

On generalized cyclically monotone operators and proper quasimonotonictty

In this paper we consider an abstract subdifferential that fulfills a prioria weak type of a mean value property. We survey and extend some recent results connecting the gener-alized convexity of nonsmooth functions with the generalized cyclic monotonidty of their subdifferentials. It is shown that, for a large class of subdifferentials, a Isc function is quasiconvex if and only if its subdifferential is a cyclically quasimonotone operator. An analogous property holds for pseudoconvexity. It is also shown that the subdiffer-ential of a quasiconvex function is properly quasimonotone. This property is slightly stronger than quasimonotonicity, and is more useful in applications connected with variational inequalities

Random operator equations and inequalities in banach spaces

In this paper we discuss the existence of solutions of nonlinear random equations and inequalities. Two existence theorems on the solution of random equations with quasi–monotone operators and complex–monotone operators, and a general theorem on random inequalities are obtained.