• Title/Summary/Keyword: generalized variational inequalities

Search Result 59, Processing Time 0.033 seconds

LOCAL CONVERGENCE OF NEWTON'S METHOD FOR PERTURBED GENERALIZED EQUATIONS

  • Argyros Ioannis K.
    • The Pure and Applied Mathematics
    • /
    • v.13 no.4 s.34
    • /
    • pp.261-267
    • /
    • 2006
  • A local convergence analysis of Newton's method for perturbed generalized equations is provided in a Banach space setting. Using center Lipschitzian conditions which are actually needed instead of Lipschitzian hypotheses on the $Fr\'{e}chet$-derivative of the operator involved and more precise estimates under less computational cost we provide a finer convergence analysis of Newton's method than before [5]-[7].

  • PDF

A NOTE ON A REGULARIZED GAP FUNCTION OF QVI IN BANACH SPACES

  • Kum, Sangho
    • Journal of the Chungcheong Mathematical Society
    • /
    • v.27 no.2
    • /
    • pp.271-276
    • /
    • 2014
  • Recently, Taji [7] and Harms et al. [4] studied the regularized gap function of QVI analogous to that of VI by Fukushima [2]. Discussions are made in a finite dimensional Euclidean space. In this note, an infinite dimensional generalization is considered in the framework of a reflexive Banach space. To do so, we introduce an extended quasi-variational inequality problem (in short, EQVI) and a generalized regularized gap function of EQVI. Then we investigate some basic properties of it. Our results may be regarded as an infinite dimensional extension of corresponding results due to Taji [7].

NONLINEAR ALGORITHMS FOR A COMMON SOLUTION OF A SYSTEM OF VARIATIONAL INEQUALITIES, A SPLIT EQUILIBRIUM PROBLEM AND FIXED POINT PROBLEMS

  • Jeong, Jae Ug
    • Korean Journal of Mathematics
    • /
    • v.24 no.3
    • /
    • pp.495-524
    • /
    • 2016
  • In this paper, we propose an iterative algorithm for finding a common solution of a system of generalized equilibrium problems, a split equilibrium problem and a hierarchical fixed point problem over the common fixed points set of a finite family of nonexpansive mappings in Hilbert spaces. Furthermore, we prove that the proposed iterative method has strong convergence under some mild conditions imposed on algorithm parameters. The results presented in this paper improve and extend the corresponding results reported by some authors recently.

ERROR BOUNDS FOR NONLINEAR MIXED VARIATIONAL-HEMIVARIATIONAL INEQUALITY PROBLEMS

  • A. A. H. Ahmadini;Salahuddin;J. K. Kim
    • Nonlinear Functional Analysis and Applications
    • /
    • v.29 no.1
    • /
    • pp.15-33
    • /
    • 2024
  • In this article, we considered a class of nonlinear variational hemivariational inequality problems and investigated a gap function and regularized gap function for the problems. We discussed the global error bounds for such inequalities in terms of gap function and regularized gap functions by utilizing the Clarke generalized gradient, relaxed monotonicity, and relaxed Lipschitz continuous mappings. Finally, as applications, we addressed an application to non-stationary non-smooth semi-permeability problems.

Some existence theorems for generalized vector variational inequalities

  • Lee, Gue-Myung;Kim, Do-Sang;Lee, Byung-Soo
    • Bulletin of the Korean Mathematical Society
    • /
    • v.32 no.2
    • /
    • pp.343-348
    • /
    • 1995
  • Let X and Y be two normed spaces and D a nonempty convex subset of X. Let $T : X \ to L(X,Y)$ be a mapping, where L(X,Y) is the space of all continuous linear mappings from X into Y. And let $C : D \to 2^Y$ be a set-valued map such that for each $x \in D$, C(x) is a convex cone in Y such that Int $C(x) \neq 0 and C(x) \neq Y$, where Int denotes the interior.

  • PDF

ON GAP FUNCTIONS OF VARIATIONAL INEQUALITY IN A BANACH SPACE

  • Kum, Sang-Ho;Lee, Gue-Myung
    • Journal of the Korean Mathematical Society
    • /
    • v.38 no.3
    • /
    • pp.683-695
    • /
    • 2001
  • In this paper we are concerned with theoretical properties of gap functions for the extended variational inequality problem (EVI) in a general Banach space. We will present a correction of a recent result of Chen et. al. without assuming the convexity of the given function Ω. Using this correction, we will discuss the continuity and the differentiability of a gap function, and compute its gradient formula under tow particular situations in a general Banach space. Our results can be regarded as infinite dimensional generalizations of the well-known results of Fukushima, and Zhu and Marcotte with soem modifications.

  • PDF

NONLINEAR VARIATIONAL INEQUALITIES AND FIXED POINT THEOREMS

  • Park, Sehie;Kim, Ilhyung
    • Bulletin of the Korean Mathematical Society
    • /
    • v.26 no.2
    • /
    • pp.139-149
    • /
    • 1989
  • pp.Hartman and G. Stampacchia [6] proved the following theorem in 1966: If f:X.rarw. $R^{n}$ is a continuous map on a compact convex subset X of $R^{n}$ , then there exists $x_{0}$ ..mem.X such that $x_{0}$ , $x_{0}$ -x>.geq.0 for all x.mem.X. This remarkable result has been investigated and generalized by F.E. Browder [1], [2], W. Takahashi [9], S. Park [8] and others. For example, Browder extended this theorem to a map f defined on a compact convex subser X of a topological vector space E into the dual space $E^{*}$; see [2, Theorem 2]. And Takahashi extended Browder's theorem to closed convex sets in topological vector space; see [9, Theorem 3]. In Section 2, we obtain some variational inequalities, especially, generalizations of Browder's and Takahashi's theorems. The generalization of Browder's is an earlier result of the first author [8]. In Section 3, using Theorem 1, we improve and extend some known fixed pint theorems. Theorems 4 and 8 improve Takahashi's results [9, Theorems 5 and 9], respectively. Theorem 4 extends the first author's fixed point theorem [8, Theorem 8] (Theorem 5 in this paper) which is a generalization of Browder [1, Theroem 1]. Theorem 8 extends Theorem 9 which is a generalization of Browder [2, Theorem 3]. Finally, in Section 4, we obtain variational inequalities for multivalued maps by using Theorem 1. We improve Takahashi's results [9, Theorems 21 and 22] which are generalization of Browder [2, Theorem 6] and the Kakutani fixed point theorem [7], respectively.ani fixed point theorem [7], respectively.

  • PDF

AN M/G/1 QUEUE WITH GENERALIZED VACATIONS AND EXHAUSTIVE SERVICE

  • Lim, Jong-Seul;Lee, Sang-Heon
    • Journal of applied mathematics & informatics
    • /
    • v.6 no.1
    • /
    • pp.309-320
    • /
    • 1999
  • Models of single-server queues with vacations have been widely used to study the performance of many computer communi-cation and production systems. In this paper we analyze an M/G/1 queue with generalized vacations and exhaustive service. This sys-tem has been shown to possess a stochastic decomposition property. That is the customer waiting time in this system is distributed as the sum of the waiting time in a regular M/G/1 queue with no va-cations and the additional delay due to vacations. Herein a general formula for the additional delay is derived for a wide class of vacation policies. The formula is also extended to cases with multiple types of vacations. Using these new formulas existing results for certain vacation models are easily re-derived and unified.

ELEMENTS OF THE KKM THEORY ON CONVEX SPACES

  • Park, Se-Hie
    • Journal of the Korean Mathematical Society
    • /
    • v.45 no.1
    • /
    • pp.1-27
    • /
    • 2008
  • We introduce a new concept of convex spaces and a multimap class K having certain KKM property. From a basic KKM type theorem for a K-map defined on an convex space without any topology, we deduce ten equivalent formulations of the theorem. As applications of the equivalents, in the frame of convex topological spaces, we obtain Fan-Browder type fixed point theorems, almost fixed point theorems for multimaps, mutual relations between the map classes K and B, variational inequalities, the von Neumann type minimax theorems, and the Nash equilibrium theorems.