The Study of Minimax Inequalities, Abstract Economics and Applications to Variational Inequalities and Nash Equilibria

In this survey, a new minimax inequality and one equivalent geometricform are proved. Next, a theorem concerning the existence of maximalelements for an LC-majorized correspondence is obtained.By the maximal element theorem, existence theorems of equilibrium point fora noncompact one-person game and for a noncompact qualitative game withLC-majorized correspondences are given. Using the lastresult and employing 'approximation approach', we prove theexistence of equilibria for abstract economies in which the constraintcorrespondence is lower (upper) semicontinuous instead of having lower(upper) open sections or open graphs in infinite-dimensional topologicalspaces. Then, as the applications, the existence theorems of solutions forthe quasi-variational inequalities and generalized quasi-variationalinequalities for noncompact cases are also proven. Finally, with theapplications of quasi-variational inequalities, the existence theorems ofNash equilibrium of constrained games with noncompact are given. Our resultsinclude many results in the literature as special cases.     

Covering theorems of simplexes and systems of linear inequalities

A selection lemma for Ky Fan's coverings is first proved, and then applied together with Ky Fan's generalization of the classical Knaster-Kuratowski Mazurkiewicz theorem to obtain a new covering theorem of simplexes. As applications, results on the consistency for certain systems of linear inequalities are obtained.     

Generalized quasi-variational inequalities in locally convex topological vector spaces

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?E′ and x?E. Given a point-to-set map S: X → 2^X and a point-to-set map T: X → 2^E′, the generalized quasi-variational inequality problem (GQVI) is to find a point $\text{y}\text{?}\text{? S(}\text{y}\text{?}\text{)}$ and a point $\text{u}\text{?}\text{? T(}\text{y}\text{?}\text{)}$ such that $\text{Re}\text{〈}\text{u}\text{?}\text{,}\text{y}\text{?}\text{? x〉 ? 0}$ for all $\text{x ? S(}\text{y}\text{?}\text{)}$. 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.     

Photoluminescence emission behavior on the reduced band gap of Fe doping in CeO2-SiO2 nanocomposite and photophysical properties

Fe/CeO₂-SiO₂ nanocomposite was synthesized via hydrothermal method. Bond length of nanocomposite was determined through FTIR analysis, while Raman analysis showed lattice relaxation of CeO₂ phase in Fe/CeO₂-SiO₂. TEM, XRD and DLS-PSA revealed an increase in size of Fe/CeO₂-SiO₂ as compared to CeO₂-SiO₂ which was attributed to have more oxygen vacancies in CeO₂ after doping of iron. Lattice contraction was also observed in some phases of CeO₂ in Fe/CeO₂-SiO₂ nanocomposite as compared to CeO₂-SiO₂ nanocomposite. This contraction was used for determination of Fe content incorporated in CeO₂ [1 1 1] phase. The band gap values of Fe/CeO₂-SiO₂ nanocomposite were found reduced after doping of Fe by factors of 0.62 and 0.55 eV, respectively. Photoluminescence study of the materials was carried out to study the different type of transitions occurring after absorption of electromagnetic radiation. Photoluminescence intensity at 2.12 eV was found enhanced after doping of Fe due to increased oxygen vacancy. Photocatalytic activity of the nanocomposites was studied with the degradation of chlorpyrifos pesticide.     

Generalized variational-like inequalities for pseudo-monotone type III operators

Our aim in this paper is mainly to prove some existence results for solutions of generalized variational-like inequalities with (η,h)-pseudo-monotone type III operators defined on non-compact sets in topological vector spaces.     

Application of Upper Hemi-Continuous Operators on Generalized Bi-quasi-variational Inequalities in Locally Convex Topological Vector Spaces

Let and be Hausdorff topological vector spaces over the field , let be a bilinear functional, and let be a non-empty subset of . Given a set-valued map and two set-valued maps , the generalized bi-quasi-variational inequality (GBQVI) problem is to find a point and a point such that and for all and for all or to find a point a point and a point such that and for all . The generalized bi-quasi-variational inequality was introduced first by Shih and Tan [8] in 1989. In this paper we shall obtain some existence theorems of generalized bi-quasi-variational inequalities as application of upper hemi-continuous operators [4] in locally convex topological vector spaces on compact sets.     

Fixed points of semigroups of Lipschitzian mappings defined on nonconvex domains

Certain fixed point theorems are established for nonlinear semigroups of Lipschitzian mappings defined on nonconvex domains in Hilbert and Banach spaces. Some known results are thus generalized.