About Semicontinuity of Set-valued Maps and Stability of Quasivariational Inclusions

Anh, Lâm Quốc; Khánh, Phan Quốc; Quy, Dinh Ngoc

doi:10.1007/s11228-014-0276-5

Cited by 6 publications

(1 citation statement)

References 24 publications

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“…For

α \in {α_{1}, α_{2}}

, consider the problem

\begin{matrix} true \\ right ({VR}_{α}^{'}) ({, \begin{matrix} find \bar{normala} \in A such that \bar{normala} \in {normalS}_{1} (\overset{a}{true}), and, \\ for all x \in S_{2} (\overset{a}{true}), α (y, T (\bar{a}, x)), ρ (\overset{a}{true}, x, y) holds . \end{matrix}) \end{matrix}

To see this equivalence, simply take

S = S_{2}

and define a relation R linking

a \in A

and

x \in X

as follows:

R (a, x) holds iff a \in S_{1} (a), and α (y, T (a, x)), ρ (a, x, y) holds .

Note that if

α = α_{1}

, then (VR' α ) is the variational relation problem considered in , . In , it was proved that, if A , X are linear spaces, then by setting

M : = {(a, x) \in A \times X | R (a, x)

…”

Section: Variational Relation Problemsmentioning

confidence: 99%

Topologically-based characterizations of the existence of solutions of optimization-related problems

Khánh

Quan

2015

Math. Nachr.

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Necessary and sufficient conditions for the existence of solutions of optimization‐related problems, defined on general sets, are established by using topologically‐based structures, without the usual linear structure. First, we prove that the existence of solutions to a variational relation problem is equivalent to the existence of either a KKM‐structure or a connectedness structure, satisfying additionally some verifiable conditions. Then, applying these results, we obtain such full (two‐way) characterizations for the existence of invariant points, solutions of quasiequilibrium problems of the Stampacchia‐Minty type, saddle points, and Nash equilibria for noncooperative games. The main advantages of our scheme with the mentioned two structures are that full characterizations for existence are obtained in a unified way, with no convexity assumptions, and also that one has flexibility to choose a suitable structure in investigations. (This flexibility is decisive when the natural underlying structure of the given problem is scarcely employed.)

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“…For