Quasimonotone Maps Research Articles

Strong convergent algorithm for finding minimum-norm solutions of quasimonotone variational inequalities with fixed point constraint and application

The class of quasimonotone mappings are known to be more general and applicable than the classes of pseudomonotone and monotone mappings. However, only very few results can be found in the literature on quasimonotone variational inequality problems and most of these results are on weak convergent algorithms. In this paper, we study the quasimonotone variational inequality problem (VIP) with constraint of fixed point problem (FPP) of quasi-pseudocontractive mappings. We introduce a new inertial Tseng’s extragradient method with self-adaptive step size for approximating the minimum-norm solutions of the aforementioned problem in the framework of Hilbert spaces. We prove that the sequence generated by the proposed method converges strongly to a common (minimum-norm) solution of the quasimonotone VIP and FPP of quasi-pseudocontractive mappings without the knowledge of the Lipschitz constant of the cost operator. We provide several numerical experiments for the proposed method in comparison with existing methods in the literature. Finally, we applied our result to image restoration problem. Our result improves, extends and generalizes several of the recently announced results in this direction.

Relatively Monotone and Relatively Generalized Monotone Maps in Nonsmooth Setting and Relative r-Monotonicity

The aim of this article is twofold. First, we consider relatively monotone, relatively pseudomonotone, and relatively quasimonotone maps in nonsmooth setting. For continuous maps, we describe -monotonicity when the relative parameters are different and prove that -quasimonotonicity implies -quasimonotonicity as well as -pseudomonotonicity implies -pseudomonotonicity. Then, under local Lipschitzianity assumption, we establish first-order criteria for -monotonicity, -pseudomonotonicity, and -quasimonotonicity. Second, we introduce and study the new notion of relatively r-monotone operators. We provide characterizations and properties. We give in particular criteria of first order for relative r-monotonicity, in smooth and nonsmooth locally Lipschitz settings, when the relative parameters are equal.

A topological duality for posets

In this paper, we present a topological duality for partially ordered sets. We use the duality to give a topological construction of the canonical extension of a poset, and we also topologically represent the quasi-monotone maps, that is, maps from a finite product of posets to a poset that are order-preserving or order-reversing in each coordinate.

Strong vector variational like inequality problems with properly quasimonotone bifunctions

In this paper, we introduce two types of proper quasimonotone maps over cones for a vector-valued bifunction and discuss their relations with generalized monotone maps, namely cone pseudomonotone and cone quasimonotone maps. Strong vector variational like inequality problems of the Stampacchia and the Minty type have been defined. These problems are a generalization of the classical Stampacchia and Minty problems and encompass many problems studied in the literature. A generalization of celebrated Minty lemma, relating the solutions of the two problems, has been proved. Existence results for strong Stampacchia and Minty type vector variational like inequality problems have been established using the notions of proper quasimonotone maps over cones. Gap functions have also been proposed for both problems.

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).

On the translations of quasimonotone maps and monotonicity

Abstract 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. Math. Lett. 19:913-915, 2006).

Differential-operator inclusions and multivariational inequalities with pseudomonotone mappings

Differential-operator inclusions and multivariational inequalities in Banach spaces with quasimonotone mappings are considered. The functional-topological properties of resolving operators are investigated. The Faedo---Galerkin method for noncoercive mappings and the finite-difference and Dubinsky methods for differential-operator inclusions in Banach spaces with pseudomonotone mappings are substantiated. Important a priori estimates are obtained.

Single-Directional Property of Multivalued Maps and Variational Systems

Dontchev and Hager [Math. Oper. Res., 19 (1994), pp. 753-768] have shown that a monotone set-valued map defined from a Banach space to its dual which satisfies the Aubin property around a point $(x,y)$ of its graph is actually single-valued in a neighborhood of $x$. We prove a result which is the counterpart of the above for quasi-monotone set-valued maps, based on the concept of single-directional property. As applications, we provide sufficient conditions for this single-valued property to hold for the solution map of general variational systems and quasi-variational inequalities. We also investigate the single-directionality property for the normal operator to the sublevel sets of a quasi-convex function.

On non-negative quasiconvex functions with quasimonotone gradients and prescribed zero sets

Let $M^{N\times n}$ be the space of real $N\times n$ matrices. We construct non-negative quasiconvex functions $F:M^{N\times n}\to R_+$ of quadratic growth whose zero sets are the graphs $\Gamma_f$ of certain Lipschitz mappings $f:K\subset E\to$ $E^$⊥, where $E\subset M^{N\times n}$ is a linear subspace without rank-one matrices, $K$ a compact subset of $E$ with $E^$⊥ its orthogonal complement. We show that the gradients $DF:M^{N\times n}\to M^{N\times n}$are strictly quasimonotone mappings and satisfy certain growth and coercivity conditions so that the variational integrals $u\to \int_{\Omega}F(Du(x))dx$ satisfythe Palais-Smale compactness condition in $W^{1,2}$.If $K$ is a smooth compact manifold of $E$ withoutboundary and the Lipschtiz mapping $f$ is of class $C^2$, then the closed $\epsilon$-neighbourhoods $(\Gamma_f)_\epsilon$ for small $\epsilon>0$ are quasiconvex sets.

Translations of quasimonotone maps and monotonicity

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 straight line S which is not perpendicular to K and such that T + w is quasimonotone for each w ∈ K , then T is monotone. No differentiability or even continuity assumption is imposed; thus we generalize some recent results in the literature.

Variational inequalities for generalized quasi-monotone maps

In this paper, we introduce a generalized quasi-monotone map and consider existence of solutions to generalized variational inequality problems for generalized quasi-monotone maps. Our result generalizes some theorems in [1].

Properties of a homotopy solution path for complementarity problems with quasi-monotone mappings

We study several properties of a homotopy solution path associated with continuous nonlinear quasi-monotone complementarity problems. We use Poincare-Bohn’s homotopy invariance theorem of degree to derive an alternative theorem. Based on this result, a sufficient condition is established to assure the existence and boundedness of this homotopy solution path. Our results provide a theoretical basis to develop a new computational method for quasi-monotone complementarity problems.

A first order characterization of generalized monotonicity

It is shown that pseudomonotone and quasimonotone maps can be characterized by a first order property provided they are regular. This result extends the well known characterization of nonvanishing generalized monotone maps to an essentially larger class. The paper supplements a recent contribution by Crouzeix and Ferland (1996) and solves a related open problem concerning homogeneous excess demand functions which occur in general equilibrium theory.

Quasi-P*-Maps, P(τ, α, β)-Maps, Exceptional Family of Elements, and Complementarity Problems

Quasi-P*-maps and P(τ, α, β)-maps defined in this paper are two large classes of nonlinear mappings which are broad enough to include P*-maps as special cases. It is of interest that the class of quasi-P*-maps also encompasses quasimonotone maps (in particular, pseudomonotone maps) as special cases. Under a strict feasibility condition, it is shown that the nonlinear complementarity problem has a solution if the function is a nonlinear quasi-P*-map or P(τ, α, β)-map. This result generalizes a classical Karamardian existence theorem and a recent result concerning quasimonotone maps established by Hadjisawas and Schaible, but restricted to complementarity problems. A new existence result under an exceptional regularity condition is also established. Our method is based on the concept of exceptional family of elements for a continuous function, which is a powerful tool for investigating the solvability of complementarity problems.

An alternative theorem for generalized variational inequalities and solvability of nonlinear quasi- [formula omitted]-complementarity problems

In this paper, an alternative theorem, and hence a sufficient solution condition, is established for generalized variational inequality problems. The concept of exceptional family for generalized variational inequality is introduced. This concept is general enough to include as special cases the notions of exceptional family of elements and the D-orientation sequence for continuous functions. Particularly, we apply the alternative theorem for investigating the solvability of the nonlinear complementarity problems with so-called quasi- P M ∗ -maps, which are broad enough to encompass the quasi-monotone maps and P ∗ -maps as the special cases. An existence theorem for this class of complementarity problems is established, which significantly generalizes several previous existence results in the literature.

On Quasimonotone Variational Inequalities

In this paper, we study variational inequalities with multivalued mappings. By employing Fan's lemma, we establish the existence result for the dual formulation of variational inequalities with semistrictly quasimonotone mappings. We also show that similar results for quasimonotone variational inequalities do not hold.

A remark on generalized variational inequalities in locally convex topological vector spaces

In this note, a general existence theorem of generalized variational inequalities for quasi-monotone set-valued mappings in locally convex topological vector spaces has been established. Our result includes corresponding results in recent literature as special cases.

Characterisations of quasiconvex functions

In this paper we introduce the concept of quasimonotone maps and prove that a lower semicontinuous function on an infinite dimensional space is quasiconvex if and only if its generalised subdifferential or its directional derivative is quasimonotone.

Fixed Points for Discontinuous Quasimonotone Maps in Sequence Spaces

In [2] Hu gives a fixed point theorem for discontinuous quasimonotone increasing maps in X = Rn . We will answer the question in [2] as to whether this result can be extended to X = /P, 1 O}, where I = {1, ..., n} or I = N, respectively. Then (X, K) is order complete, i.e., if a subset A of X has an upper bound, then A has a least upper bound, which we denote by sup A. For u,veX, u Mx} are fixed points of M. For a function f = (fi)iEI: X -* X we define for x E X, in analogy to the notation in [2], D?f1(x) = liminf I(fi (x + te') fi(x)), t-40? t D?fi(x) = limsup -(fi(x + te1) fi(x)), t-0? t where ei, i E I, are the elements of X with components ej = 1 for i = j, ej=0 for i j]. Theorem. Let u, v E X, u -oo forx E [u,,v], i E I; (3) ui fi(x + (vi xi)e) forx E [u, v], i E I. Then f has a greatest fixed point Y and a smallest fixed point x, and (4) x-= sup{x E [u, v]: x f(x)}. Received by the editors July 26, 1990 and, in revised form, November 27, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 47H10, 47H05. ( 1992 American Mathematical Society 0002-9939/92 $1.00 + $.25 per page

Fixed points for discontinuous quasi-monotone maps in sequence spaces

In [2] Hu gives a fixed point theorem for discontinuous quasimonotone increasing maps in X = R n X = {\mathbb {R}^n} . We will answer the question in [2] as to whether this result can be extended to X = l p , 1 ≤ p ≤ ∞ X = {l^p},1 \leq p \leq \infty .