Kok-Keong Tan

Fixed point iteration processes for asymptotically nonexpansive mappings

Let X be a uniformly convex Banach space which satisfies Opials condition or has a Frechet differentiable norm, C a bounded closed convex subset of X, and T: C -C an asymptotically nonexpansive mapping. It is then shown that the modified Mann and Ishikawa iteration processes defined by xn+, = tnTnxn+(I-tn)xn and xn+ i = tnTn(snTnxn+(I-Sn)xn) +(-tn)xn 7 respectively, converge weakly to a fixed point of T.

Generalized quasi-variational inequalities in locally convex topological vector spaces

Abstract Let E be a Hausdorff topological vector space and X ⊂ E an arbitrary nonempty set. Denote by E′ the dual space of E and the pairing between E′ and E by 〈w, x〉 for w ϵ E′ and x ϵ E. Given a point-to-set map S: X → 2X and a point-to-set map T: X → 2E′, the generalized quasi-variational inequality problem (GQVI) is to find a point y ϵ S( y ) and a point u ϵ T( y ) such that Re 〈 u , y − x〉 ⩽ 0 for all x ϵ S( y ) . By using the Ky Fan minimax principle or its generalized version as a tool, some general theorems on solutions of the GQVI in locally convex Hausdorff topological vector spaces are obtained which include a fixed point theorem due to Ky Fan and I. L. Glicksberg, and two different multivalued versions of the Hartman-Stampacchia variational inequality.

Equilibria of non-compact generalized games with l∗-majorized preference correspondences

An existence theorem of maximal elements in a non-compact set for L∗-majorized correspondences is obtained. Next an existence theorem of an equilibrium in a qualitative game is proved and is then applied to achieve an existence theorem of equilibrium in a non-compact abstract economy with L∗-majorized preference correspondences. These results are either very closely related to or generalizations of those recent results of Borglin-Keiding, Yannelis-Prabhakar, Toussaint, and Tulcea.

The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces

Let X be a uniformly convex Banach space with a Frechet differentiable norm, C a bounded closed convex subset of X, and T: C -+ C an asymptotically nonexpansive mapping. It is shown that for each x in C, the sequence {Tnx} is weakly almost-convergent to a fixed point y of T, i.e., (1/n)%n-1 Tk+ix -*. y weakly as n tends to infinity uniformly in k=O, 1,2,....

A selection theorem and its applications

In this paper, we first prove an improved version of the selection theorem of Yannelis-Prabhakar and next prove a fixed point theorem in a non-compact product space. As applications, an intersection theorem and two equilibrium existence theorems for a non-compact abstract economy are given.

Browder-Hartman-Stampacchia variational inequalities for multi-valued monotone operators☆

Abstract An existence theorem of Browder-Hartman-Stampacchia variational inequalities is extended to multi-valued monotone operators. Surjectivity for multi-valued monotone operators and properties of solution sets are discussed.

A minimax inequality with applications to existence of equilibrium points

A new minimax inequality is first proved. As a consequence, five equivalent fixed point theorems are formulated. Next a theorem concerning the existence of maximal elements for an L c -majorised correspondence is obtained. By the maximal element theorem, existence theorems of equilibrium points for a non-compact one-person game and for a non-compact qualitative game with L c -majorised correspondences are given. Using the latter result and employing an “approximation” technique used by Tulcea, we deduce equilibrium existence theorems for a non-compact generalised game with L C correspondences in topological vector spaces and in locally convex topological vector spaces. Our results generalise the corresponding results due to Border, Borglin-Keiding, Chang, Ding-Kim-Tan, Ding-Tan, Shafer-Sonnenschein, Shih-Tan, Toussaint, Tulcea and Yannelis-Prabhakar.

Generalized variational inequalities and generalized quasi-variational inequalities☆

Abstract A very general minimax inequality is first established. Three generalized variational inequalities are then derived, which improve those obtained by Tan and Browder. By applying a fixed point theorem of Himmelberg, two generalized quasi-variational inequalities are also proved, one of which generalizes those of Shih-Tan to the non-compact case with much weaker hypotheses and in a more general setting.

Linear preservers on matrices

Abstract Let U denote either the vector space of n×n matrices or the vector space of n×n symmetric matrices over an infinite field F. In this paper we characterize linear mappings L on U that satisfy one of the following properties: (i) L(adjA)=adjL(A) for all A in U; (ii) L preserves idempotent matrices, and L(In)=In, where F is the real field R or the complex field C ; (iii) L(eA)=eL(A) for all A in U, where F= R or C .

Generalized Bi-quasi-variational inequalities

Abstract Let E, F be Hausdorff topological vector spaces over the field Φ (which is either the real field or the complex field), let 〈 , 〉: F × E → Φ be a bilinear functional, and let X be a non-empty subset of E. Given a multi-valued map S: X → 2x and two multi-valued maps M, T: X → 2F, the generalized bi-quasi-variational inequality (GBQVI) problem is to find a point y ϵ X such that y ϵ S( y ) and inf w ϵ T( y ) Re 〈ƒ − w, y − x〉 ⩽ 0 for all x ϵ S( y ) and for all ƒ ϵ M( y ) . In this paper two general existence theorems on solutions of GBQVIs are obtained which simultaneously unify, sharpen, and extend existence theorems for multi-valued versions of Hartman-Stampacchia variational inequalities proved by Browder and by Shih and Tan, variational inequalities due to Browder, existence theorems for generalized quasi-variational inequalities achieved by Shih and Tan, theorems for monotone operators obtained by Debrunner and Flor, Fan, and Browder, and the Fan-Glicksberg fixed-point theorem.