• Title, Summary, Keyword: lower semicontinuity

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ON THE LOWER SEMICONTINUITY OF THE SOLUTION SETS FOR PARAMETRIC GENERALIZED VECTOR MIXED QUASIVARIATIONAL INEQUALITY PROBLEMS

  • HUNG, NGUYEN VAN
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.6
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    • pp.1777-1795
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    • 2015
  • In this paper, we establish sufficient conditions for the solution set of parametric generalized vector mixed quasivariational inequality problem to have the semicontinuities such as the inner-openness, lower semicontinuity and Hausdorff lower semicontinuity. Moreover, a key assumption is introduced by virtue of a parametric gap function by using a nonlinear scalarization function. Then, by using the key assumption, we establish condition ($H_h$(${\gamma}_0$, ${\lambda}_0$, ${\mu}_0$)) is a sufficient and necessary condition for the Hausdorff lower semicontinuity, continuity and Hausdorff continuity of the solution set for this problem in Hausdorff topological vector spaces with the objective space being infinite dimensional. The results presented in this paper are different and extend from some main results in the literature.

SADDLE POINTS OF VECTOR-vALUED FUNCTIONS IN TOPOLOGICAL VECTOR SPACES

  • Kim, In-Sook
    • Journal of the Korean Mathematical Society
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    • v.37 no.5
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    • pp.849-856
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    • 2000
  • We give a new saddle point theorem for vector-valued functions on an admissible compact convex set in a topological vector space under weak condition that is the semicontinuity of two function scalarization and acyclicty of the involved sets. As application, we obtain the minimax theorem.

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Weak semicontinuity for unbounded operators

  • Kim, Hyoungsoon
    • Bulletin of the Korean Mathematical Society
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    • v.34 no.3
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    • pp.447-457
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    • 1997
  • Let A be a $C^*$-algebra and $A^**$ its enveloping von Neumann algebra. Pedersen and Akemann developed four concepts of lower semicontinuity for elements of $A^**$. Later, Brown suggested using only three classes: strongly lsc, middle lsc, and weakly lsc. In this paper, we generalize the concept of weak semicontinuity [1, 3] to the case of unbounded operators affiliated with $A^**$. Also we consider the generalized version of the conditions of the Brown's theorem [3, Proposition 2.2 & 3.27] for unbounded operators.

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MIXED VECTOR FQ-IMPLICIT VARIATIONAL INEQUALITY WITH LOCAL NON-POSITIVITY

  • Lee, Byung-Soo
    • Communications of the Korean Mathematical Society
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    • v.24 no.3
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    • pp.425-432
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    • 2009
  • This paper introduces a local non-positivity of two set-valued mappings (F,Q) and considers the existences and properties of solutions for set-valued mixed vector FQ-implicit variational inequality problems and set-valued mixed vector FQ-complementarity problems in the neighborhood of a point belonging to an underlined domain K of the set-valued mappings, where the neighborhood is contained in K. This paper generalizes and extends many results in [1, 3-7].

MIXED VECTOR FQ-IMPLICIT VARIATIONAL INEQUALITIES WITH FQ-COMPLEMENTARITY PROBLEMS

  • Lee, Byung-Soo
    • Honam Mathematical Journal
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    • v.31 no.2
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    • pp.247-258
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    • 2009
  • This paper introduces new mixed vector FQ-implicit variational inequality problems and corresponding mixed vector FQ-implicit complementarity problems for set-valued mappings, and studies the equivalence between them under certain assumptions in Banach spaces. It also derives some new existence theorems of solutions for them with examples under suitable assumptions without monotonicity. This paper generalizes and extends many results in [8, 10, 19-22].