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Generalized Vector Quasivariational Inclusion Problems with Moving Cones

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Abstract

This paper deals with the generalized vector quasivariational inclusion Problem (P1) (resp. Problem (P2)) of finding a point (z 0,x 0) of a set E×K such that (z 0,x 0)∈B(z 0,x 0)×A(z 0,x 0) and, for all η∈A(z 0,x 0),

$$\begin{array}{l}F(z_0,x_0,\eta)\subset G(z_0,x_0,x_0)+C(z_0,x_0)\cr \mathrm{[resp.}F(z_0,x_0,x_0)\subset G(z_0,x_0,\eta)+C(z_0,x_0)],\end{array}$$

where A:E×K→2K, B:E×K→2E, C:E×K→2Y, F,G:E×K×K→2Y are some set-valued maps and Y is a topological vector space. The nonemptiness and compactness of the solution sets of Problems (P1) and (P2) are established under the verifiable assumption that the graph of the moving cone C is closed and that the set-valued maps F and G are C-semicontinuous in a new sense (weaker than the usual sense of semicontinuity).

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Authors and Affiliations

  1. Institute of Mathematics, Hanoi, Vietnam

    P. H. Sach

  2. Department of Mathematics, National Changhua University of Education, Changhua, 50058, Taiwan

    L. J. Lin

  3. Ninh Thuan College of Pedagogy, Ninh Thuan, Vietnam

    L. A. Tuan

Authors
  1. P. H. Sach
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  2. L. J. Lin
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  3. L. A. Tuan
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Corresponding author

Correspondence to L. J. Lin.

Additional information

Communicated by P.L. Yu.

The authors thank the referees whose remarks helped improving the paper. This work was completed during a visit of the first author to the National Changhua University of Education, Changhua, Taiwan. The financial support of the National Science Council of the Republic of China is gratefully acknowledged. The third author is indebted to the National Foundation for Science and Technology Development, Hanoi, Vietnam, for partial support.

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Sach, P.H., Lin, L.J. & Tuan, L.A. Generalized Vector Quasivariational Inclusion Problems with Moving Cones. J Optim Theory Appl 147, 607–620 (2010). https://doi.org/10.1007/s10957-010-9670-9

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