Recession cones and asymptotically compact sets

The present note is concerned with the study of the relations between the notions of asymptotic cones introduced by Dedieu and that of recession cones introduced by Luc. Conditions under which these notions coincide are given, as well as the fact that the compactness condition used by Luc is related (more restrictively) to asymptotic compactness. As an application of these notions, a result on proper efficiency in the sense of Lampe, established by Luc in finite dimensions, is extended to the infinite-dimensional case.This work was partially done while the author was visiting the Department of Mathematics of the University of Pisa, Pisa, Italy.     

Existence of solutions for a vector variational inequality: An extension of the Hartmann-Stampacchia theorem

A vector variational inequality is studied. The paper deals with existence theorems for solutions under convexity assumptions and without convexity assumptions.This research was partially supported by the Italian Research Council (CNR), Group for Functional Analysis and Applications (GNAFA), and was carried out while the author was Visiting Professor at the Department of Mathematics, University of Pisa, September–November, 1989.     

A variational inequality with mixed boundary conditions

In this paper we study a variational inequality for a second order uniformly elliptic operator on a bounded domain, the solution of which is required to lie above a given obstacle and to assume assigned values on a part of the boundary of the domain. We are mainly concerned with the regularity of the solution in relation to the regularity of the data. During the preparation of the paper the authors were partially supported by the Italian Consiglio Nazionale delle Ricerche, the first as visiting professor at the Scuola Normale Superiore (Pisa) on deputation from the Tata Institute of Fundamental Research (Bombay) and the second through the Istituto per l’Elaborazione dell’Informazione (Pisa).     

On the connectedness of the solution set to nonlinear complementarity systems

This paper establishes sufficient conditions for the connectedness of nontrivial subsets of the solution set to nonlinear complementarity systems with twice continuously differentiable operators. In geometrical terms, the intersection of a Riemannian manifold, a pointed cone, and the level sets determined by nonlinear inequalities are studied.In the case of variational and quasivariational inequalities, sufficient conditions ensure the geodesic connectedness of a nontrivial solution set. The image representation of the results shows the close connections between these three topics.This paper was prepared mainly while the author was visiting the Department of Mathematics at the University of Pisa. This research was partially supported by the Hungarian National Research Foundation, Grant No. OTKA-2568.     

Existence of Solutions and Variational Principles for Generalized Vector Systems

By means of generalized KKM theory, we prove a result on the existence of solutions and we establish general variational principles, that is, vector optimization formulations of set-valued maps for vector generalized systems. A perturbation function is involved in general variational principles. We extend the theory of gap functions for vector variational inequalities to vector generalized systems and we prove that the solution sets of the related vector optimization problems of set-valued maps contain the solution sets of vector generalized systems. A further vector optimization problem is defined in such a way that its solution set coincides with the solution set of a weak vector generalized system. Research carried on within the agreement between National Sun Yat-Sen University of Kaohsiung, Taiwan and Pisa University, Pisa, Italy, 2007. L.C. Ceng research was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118). J.C. Yao research was partially supported by the National Science Center for Theoretical Sciences at Tainan.     

Hybrid Proximal-Point Methods for Common Solutions of Equilibrium Problems and Zeros of Maximal Monotone Operators

The purpose of this paper is to introduce and study two hybrid proximal-point algorithms for finding a common element of the set of solutions of an equilibrium problem and the set of solutions to the equation 0∈Tx for a maximal monotone operator T in a uniformly smooth and uniformly convex Banach space X. Strong and weak convergence results of these two hybrid proximal-point algorithms are established, respectively. The research of L.C. Ceng was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), Science and Technology Commission of Shanghai Municipality Grant (075105118), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Shanghai Leading Academic Discipline Project (S30405). The research of J.C. Yao was partially supported by Grant NSC 97-2115-M-110-001. Research was carried on within the agreement between National Sun Yat-Sen University of Kaohsiung, Taiwan and Pisa University, Pisa, Italy, 2008.     

Boundary value problems for some degenerate-elliptic operators

Summary Sono studiati alcuni problemi di valori al contorno per operatori ellittici che possono degenerare. During the preparation of this paper the first author, on deputation from the ? Tata Institute of Foundamental Research ? held a visiting assignement at the University of Pisa under the sponsorship of the ? Consiglio Nazionale delle Ricerche ?, and the second author was partially supported by the ? Air Force Office of Scientific Research ? trough grant ? AF EOAR 67–38 ?.     

Finite algorithm for generalized linear multiplicative programming

The nonconvex problem of minimizing the sum of a linear function and the product of two linear functions over a convex polyhedron is considered. A finite algorithm is proposed which either finds a global optimum or shows that the objective function is unbounded from below in the feasible region. This is done by means of a sequence of primal and/or dual simplex iterations.The first author gratefully acknowledges the research support received as Visiting Professor of the Dipartimento di Statistica e Matematica Applicata all' Economia, Universitá di Pisa, Pisa, Italy, Spring 1992.     

On the generation of certain bundles over P3

This paper was begun when the author was graduate student at Scuola Normale Superiore in Pisa     

Nonlinear maps in spaces of distributions

This research was started while the author was visiting University of Pisa, University of Ferrara and I.C.T.P., Trieste in the autumn of 1988