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New versions on Nikaidô's coincidence theorem

100%

Chu L.-J., Lin C.-Y.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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2002

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tom 22

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nr 1

79-95

EN

In 1959, Nikaidô established a remarkable coincidence theorem in a compact Hausdorff topological space, to generalize and to give a unified treatment to the results of Gale regarding the existence of economic equilibrium and the theorems in game problems. The main purpose of the present paper is to deduce several generalized key results based on this very powerful result, together with some KKM property. Indeed, we shall simplify and reformulate a few coincidence theorems on acyclic multifunctions, as well as some Górniewicz-type fixed point theorems. Beyond the realm of monotonicity nor metrizability, the results derived here generalize and unify various earlier ones from the classic optimization theory. In the sequel, we shall deduce two versions of Nikaidô's coincidence theorem about Vietoris maps from different approaches.

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Variational inequalities in noncompact nonconvex regions

100%

Lin C.-Y., Chu L.-J.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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2003

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tom 23

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nr 1

5-19

EN

In this paper, a general existence theorem on the generalized variational inequality problem GVI(T,C,ϕ) is derived by using our new versions of Nikaidô's coincidence theorem, for the case where the region C is noncompact and nonconvex, but merely is a nearly convex set. Equipped with a kind of V₀-Karamardian condition, this general existence theorem contains some existing ones as special cases. Based on a Saigal condition, we also modify the main theorem to obtain another existence theorem on GVI(T,C,ϕ), which generalizes a result of Fang and Peterson.

3

Minimax theorems without changeless proportion

100%

Chu L.-J., Tsai C.-N.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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2003

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tom 23

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nr 1

55-92

EN

The so-called minimax theorem means that if X and Y are two sets, and f and g are two real-valued functions defined on X×Y, then under some conditions the following inequality holds: $inf_{y∈Y} sup_{x∈X} f(x,y) ≤ sup_{x∈X} inf_{y∈Y} g(x,y)$. We will extend the two functions version of minimax theorems without the usual condition: f ≤ g. We replace it by a milder condition: $sup_{x∈X} f(x,y) ≤sup_{x∈X}g(x,y)$, ∀y ∈ Y. However, we require some restrictions; such as, the functions f and g are jointly upward, and their upper sets are connected. On the other hand, by using some properties of multifunctions, we define X-quasiconcave sets, so that we can extend the two functions minimax theorem to the graph of the multifunction. In fact, we get the inequality: $inf_{y∈T(X)} sup_{x∈T^{-1}(y)} f(x,y) ≤ sup_{x∈X} inf_{y∈T(x)} g(x,y)$, where T is a multifunction from X to Y.

4

On discontinuous quasi-variational inequalities

100%

Chu L.-J., Lin C.-Y.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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2007

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tom 27

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nr 2

199-212

EN

In this paper, we derive a general theorem concerning the quasi-variational inequality problem: find x̅ ∈ C and y̅ ∈ T(x̅) such that x̅ ∈ S(x̅) and ⟨y̅,z-x̅⟩ ≥ 0, ∀ z ∈ S(x̅), where C,D are two closed convex subsets of a normed linear space X with dual X*, and $T:X → 2^{X*}$ and $S:C → 2^D$ are multifunctions. In fact, we extend the above to an existence result proposed by Ricceri [12] for the case where the multifunction T is required only to satisfy some general assumption without any continuity. Under a kind of Karmardian's condition, we give a partial affirmative answer to an unbounded quasi-variational inequality problem.

Ograniczanie wyników

4 Discussiones Mathematicae, Differential Inclusions, Control and Optimization

4 Chu L.-J.

3 Lin C.-Y.

1 Tsai C.-N.

1 2007

2 2003

1 2002

New versions on Nikaidô's coincidence theorem

Variational inequalities in noncompact nonconvex regions

Minimax theorems without changeless proportion

On discontinuous quasi-variational inequalities