Characterization of latticial cones in Hilbert spaces by isotonicity and generalized infimum

A mapping is called isotone if it is monotone increasing with respect to the order defined by a pointed closed convex cone. Finding the pointed closed convex generating cones for which the projection mapping onto the cone is isotone is a difficult problem which was analyzed in [1, 2, 3, 4, 5]. Such cones are called isotone projection cones. In particular it was shown that any isotone projection cone is latticial [2]. This problem is extended by replacing the projection mapping with a continuous isotone retraction onto the cone. By introducing the notion of sharp mappings, it is shown that a pointed closed convex generating cone is latticial if and only if there is a continuous isotone retraction onto the cone whose complement is sharp. This result is used for characterizing a subdual latticial cone by the isotonicity of a generalization of the positive part mapping x ↦ x ⁺. This generalization is achieved by generalizing the infimum for subdual cones. The theoretical results of this paper exhibit fundamental properties of the lattice structure of the space which were not analysed before.     

Generalized isotone projection cones

This article extends the notion of isotone projection cones to generalized isotone projection cones by replacing the usual metric projection with a generalized one. It is shown that all such cones are simplicial.     

For cones in Hilbert spaces, regularity is equivalent to self-duality

The theorem stating that a cone in a Hilbert space is regular if and only if it is self-dual is proved and applied to obtain new proofs of earlier results.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 616–621, October, 1998.     

Finding solutions of implicit complementarity problems by isotonicity of the metric projection

Isac and Németh [G. Isac and A. B. Németh, Projection method, isotone projection cones and the complementarity problem, J. Math. Anal. App., 153, 258-275(1990)] proved that solving a coincidence point equation (fixed point problem) in turn solves the corresponding implicit complementarity problem (nonlinear complementarity problem) and they exploited the isotonicity of the metric projection onto isotone projection cones to solve implicit complementarity problems (nonlinear complementarity problems) defined by these cones. In this paper, the notion of *-isotone projection cones is employed and an iterative algorithm is presented in connection with an implicit complementarity problem on *-isotone projection cones. It is proved that if the sequence generated through the defined algorithm is convergent, then its limit is a solution of the coincidence point equation and thus solves the implicit complementarity problem. Sufficient conditions are given for this sequence to be convergent for implicit complementarity problems defined by *-isotone projection cones. The question of finding nonzero solutions of these problems is also studied.     

CHARACTERIZATION OF BEST APPROXIMATIONS IN METRIC LINEAR SPACES

Let (X,d) be a real metric linear space, with translation-invariant metric d and G a linear subspace of X. In this paper we use functionals in the Lipschitz dual of X to characterize those elements of G which are best approximations to elements of X. We also give simultaneous characterization of elements of best approximation and also consider elements of ε-approximation.     

Single projection method for pseudo-monotone variational inequality in Hilbert spaces

ABSTRACT

In this paper, a projection-type approximation method is introduced for solving a variational inequality problem. The proposed method involves only one projection per iteration and the underline operator is pseudo-monotone and L-Lipschitz-continuous. The strong convergence result of the iterative sequence generated by the proposed method is established, under mild conditions, in real Hilbert spaces. Sound computational experiments comparing our newly proposed method with the existing state of the art on multiple realistic test problems are given.     

Isotone retraction cones in Hilbert spaces

A mapping is called isotone if it is monotone increasing with respect to the order induced by a pointed closed convex cone. Finding the pointed closed convex generating cones for which the projection mapping onto the cone is isotone is a difficult problem which was analyzed in Isac and Németh (1986, 1990, 1992) [1], [2], [3], [4] and [5]. Such cones are called isotone projection cones. In particular it was shown that any isotone projection cone is latticial (Isac (1990) [2]). This problem is extended by replacing the projection mapping with continuous retractions onto the cone. By introducing the notion of sharp mappings, it is shown that a pointed closed convex generating cone is latticial if and only if there is a continuous retraction onto the cone whose complement is sharp. Several particular cases are considered and examples are given.     

Solving nonlinear complementarity problems by isotonicity of the metric projection

The main motivation for introducing the notion of isotone projection cones was to solve nonlinear complementarity problems. The notion of *-isotone projection cones is introduced by this paper in a similar fashion. Iterative methods for finding solutions of complementarity problems on *-isotone projection cones are presented. The problem of finding nonzero solutions of these problems is also considered.     

锥度量空间中c-距离下的不动点定理

在锥度量空间中,用压缩性函数代替具体实数,获得了c-距离下的映射的新的不动点定理.所得结果在条件上不要求映射的非减性,且第一个定理去掉了锥的正规性,第二个定理去掉了映射的连续性,改进了原有的许多重要结论,并给出了相应的例子.     

Simultaneous metric projection onto closed sets

We examine simultaneous metric projection by closed sets in a class of ordered normed spaces. First, we study simultaneous metric projection onto downward and upward sets and separation properties of these sets. The results obtained are used for examination of simultaneous metric projection by arbitrary closed sets, and we examine the minimization of the distance from a bounded set to an arbitrary closed set in a class of ordered normed spaces.     

Regular exceptional family of elements with respect to isotone projection cones in Hilbert spaces and complementarity problems

Conditions for the non-existence of a regular exceptional family of elements with respect to an isotone projection cone in a Hilbert space will be presented. The obtained results will be used for generating existence theorems for a complementarity problem with respect to an isotone projection cone in a Hilbert space.     

Conditions of regularity in cone metric spaces

In this paper we prove some fixed points results on cone metric spaces for maps satisfying general contractive type conditions. Among other things, we extend some results of Nguyen [11] from metric spaces to cone metric spaces. The example is included.     

Fixed point theorems in fuzzy metric spaces

In this paper, we state and prove some common fixed point theorems in fuzzy metric spaces. These theorems generalize and improve known results (see [1]).     

Isotone cones in Banach spaces and applications to best approximations of operators without continuity conditions

In this paper, we introduce the concept of isotone cones in Banach spaces. Then, we apply the order monotonic property of the metric projection operator to prove the existence of best approximations for some operators without continuity conditions in partially ordered Banach spaces.     

Fixed point of contractive mappings in generalized metric spaces

We prove a fixed point theorem for contractive mappings of Boyd and Wong type in generalized metric spaces, a concept recently introduced in [BRANCIARI, A.: A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. Debrecen 57 (2000), 31–37].      

锥度量空间中扩张映射的一个新的不动点定理

为了研究完备的锥度量空间中扩张型映象不动点的存在性和唯一性问题,对满足不同条件的扩张型映象,采用不同的迭代方法,得到了锥度量空间中扩张映射的一个新的不动点定理.这些结果是度量空间中某些经典结果在锥度量空间的进一步推广和发展.     

Continuous homogeneous selections of set-valued metric generalized inverses of linear operators in Banach spaces

In this paper,continuous homogeneous selections for the set-valued metric generalized inverses T of linear operators T in Banach spaces are investigated by means of the methods of geometry of Banach spaces.Necessary and sufficient conditions for bounded linear operators T to have continuous homogeneous selections for the set-valued metric generalized inverses T are given.The results are an answer to the problem posed by Nashed and Votruba.     

Common fixed points for four maps in cone metric spaces

The existence of coincidence points and common fixed points for four mappings satisfying generalized contractive conditions without exploiting the notion of continuity of any map involved therein, in a cone metric space is proved. These results extend, unify and generalize several well known comparable results in the existing literature.