Further extension of Fan‑Browder coincidence and fixed point theorems
著者（英） Ken Urai, Kohei Shiozawa journal or
publication title
Doshisha Shogaku (The Doshisha Business Review)
volume 66
number 1
page range 125‑132
year 2014‑07‑25
権利（英） Doshisha Daigaku Shogakkai
The Association of Commerce Doshisha University
URL http://doi.org/10.14988/pa.2017.0000013679
Further Extension of Fan-Browder Coincidence and Fixed Point Theorems
Ken Urai
*Kohei Shiozawa
†Ⅰ Introduction
Ⅱ A Simple Extension of Fan-Browder Coincidence Theorem
Ⅲ Extensions of Fan-Browder Coincidence Theorems
Ⅰ Introduction
The fixed-point and coincidence theorems of F. Browder and K. Fan (see Browder (1968) and Fan (1969)) are well known and frequently used in mathematical economics and game- theoretic equilibrium arguments. A series of their results are based on the next famous fixed- point theorem known asBrowder’s fixed point theorem:
Theorem 1: (Browder 1968, Theorem 1). Let X be a non-empty compact convex subset of a Hausdorff topological vector space E. Let T : X→X be a non-empty and convex valued correspondence. Suppose further that for all y∈X, T−1(y) ＝{x∈Xdef. ｜y∈T(x)} is open in X. ThenT : X→X has a fixed pointx＊∈X.
Note that in the above the vector space,E, is assumed to be Hausdorff, but not necessarily to be locally convex.
In the context of economics, the vector space duality is often interpreted as the duality between commodities and prices. Some natural assumptions on agents’ behaviors in response to prices would therefore enable us to utilize rather weak topological conditions on the commodity space to ensure the existence of an equilibrium. From this motivation, the authors have studied several extensions of Kakutani’s fixed-point theorem and their applications to the economic equilibrium theory (see Urai and Hayashi (2000), Urai (2000), Urai and Yoshimachi
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*Graduate School of Economics, Osaka University, Machikaneyama, Toyonaka, Osaka 560−0043, Japan. E-mail : [email protected]−u.ac.jp
†Graduate School of Economics, Osaka University, Machikaneyama, Toyonaka, Osaka 560−0043, Japan. E-mail : [email protected]−u.ac.jp
（125）１２５
(2004 a), Urai and Yokota (2005), Urai (2010)).
In this paper, we provide several extensions of the coincidence and fixed-point theorems in Browder (1968) and Fan (1969). Our proofs depend merely on Browder’s fixed-point theorem as given above. Moreover, it can be seen that our results directly show one of the most general kinds of Kakutani’s fixed point theorem on Hausdorff topological vector spaces.
Ⅱ A Simple Extension of Fan-Browder Coincidence Theorem
First, we directly show a simple extension of the Fan-Browder coincidence theorem by applying Broder’s fixed-point theorem. This holds in the special case that the range of the correspondencex│!x＋(F(x)−G(x)) happens to be a subset of its domain.
Theorem 2: (A Simple Extension of Fan-Browder Coincidence Theorem). Let X be a non- empty compact convex subset of a Hausdorff topological vector spaceE overR andF : X→E and G : X→E be two non-empty valued correspondences. Assume the following two conditions :
(i) For eachx∈X, x＋(F(x)−G(x))⊂X ;
(ii) For each x such that 0∈／F(x)−G(x), element px in topological dual E′of E and open neighborhood U(x) of x exist such that "px, y#＞0 for all y∈F(z)−G(z) andz∈
Ux.
Then there exists a pointx＊∈X such thatF(x＊)∩G(x＊)≠ .
Proof: Assume that for each x∈X, F(x)∩G(x)＝ , so that 0∈／F(x)−G(x). By condition (i), we can define a non-empty valued correspondence !: X→X as !(x)＝x＋(F(x)−G(x)).
Note that for all x∈X, x∈／!(x). Take any x∈X. By condition (ii), there exist an open neighborhood U(x) of x and an element px∈E′such that "px, z#＞0 holds for all x′∈U(x) andz∈F(x′)−G(x′) (＝!(x′)−x′). Thus {U(x)｜x∈X} is an open covering of X, so it has a finite subcovering {U(x1), . . . ,U(xn)}. Let {αi: X→[0,1]｜i＝1, . . . ,n} be a partition of unity subordinate to {U(x1), . . . ,U(xn)}. Define a correspondenceΨ: X→X as Ψ(x)＝{x＋z
∈X｜!n
i＝1 αi(x)"pxi, z#＞0} for each x∈X. It is clear that !(x)⊂Ψ(x), hence in particular Ψ(x)≠ , and Ψ(x) is convex for all x∈X. Moreover, Ψ−1(y) is open in X for all y∈X. Indeed, for any x∈Ψ−1(y), we have that y＝x＋z and !n
i＝1αi(x)"pxi, z#＞0 or, equivalently,
!n
i＝1αi(x)"pxi, y#−!n
i＝1αi(x)"pxi, x#＞0. Since the left-hand side of this inequality is
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continuous, there exists a neighborhoodV ofx such thatx′∈V‖ ‖
＞!n
i＝1αi(x′)!pxi, y"−!n i＝1
αi(x′)!pxi, x′"＞0, i.e.,V⊂Ψ−1(y). Therefore, by Browder’s fixed point theorem,Ψhas a fixed pointx＊∈X. This means that!n
i＝1αi(x＊)!pxi, x＊−x＊"＞0, a contradiction. ■ The above special case immediately implies the following generalization of Kakutani’s fixed-point theorem in Urai and Hayashi (2000) :
Theorem 3: (Urai and Hayashi 2000, Theorem 4). Let X be a non-empty compact convex subset of a Hausdorff topological vector space E over R and F : X→X be a non-empty valued correspondence. Assume the following condition :
(K) For each x such that x∈／F(x), element px in topological dual E′of E and open neighborhoodU(x) ofx exist such that!px, y"＞0 for ally∈F(z)−z andz∈U(x).
Then fixed-pointx＊∈X ofF, x＊∈F(x＊), exists.
Proof: In Theorem 2, letG : X→E be the identity map. Then condition (i) in Theorem 2 is automatically satisfied. It is also easy to check that condition (K) for F implies condition (ii) in Theorem 2, andx＊∈F(x)∩G(x) is nothing but the fixed-point ofF. ■ In this theorem (as not in the theorem but in the proof of Urai and Hayashi (2000, Theorem 4)), the topological vector spaceE need not be locally convex.
Ⅲ Extensions of Fan-Browder Coincidence Theorems
Next, we show an extension of the Fan-Browder coincidence theorem through the following two theorems, Theorem 4 and Theorem 5. (Theorem 4 is an abstract version of Theorem 2 of Browder(1968) ; Theorems 5 and 6 are extensions of Theorems 5 and 6 of Fan (1969).)
Theorem 4: Let X be a non-empty compact convex subset of a Hausdorff topological vector space E overR, and Ψ be a (single-valued) mapping from X to E′(the topological dual of E). If f(x) ＝!def. Ψ(x), y−x"is a continuous function of x on X for each fixed y in X, then there exists a pointx0∈X such that
!Ψ(x0),y−x0"≦0 for ally∈X.
Proof: Suppose that the assertion is false. Then for each x0∈X there exists an elementy∈X
Further Extension of Fan-Browder Coincidence and Fixed Point Theorems（Urai・Shiozawa）（127）１２７
such that!Ψ(x0),y−x0"＞0. LetT : X→X be a correspondence defined as T(x0)def.＝{y∈X｜!Ψ(x0),y−x0"＞0}, wherex0∈X.
Note that T(x0)≠ for each x0∈X by the preceding remark, and obviously T(x0) is convex for all x0∈X. Moreover, T−1(y)＝{x∈Xdef. ｜y∈T(x)} is open in X for all y∈X from the assumption of this theorem. Indeed, x∈T−1(y) ‖ ‖
＞y∈T(x) ‖ ‖
＞ !Ψ(x), y−x"＞0 ‖ ‖
＞∃
U(x) : a neighborhood ofx s.t.!Ψ(x′),y−x′"＞0 for all x′∈U(x) ‖ ‖
＞y∈T(x′) for allx′
∈U(x) ‖ ‖
＞U(x)⊂T−1(y).
By Browder’s fixed point theorem, there exists an element x＊∈X such that x＊∈T(x＊).
This means that 0＜!Ψ(x＊), x＊−x＊"＝!Ψ(x＊), 0"＝0, a contradiction. ■
Theorem 5: Let X be a non-empty compact convex subset of a Hausdorff topological vector space E over R, and F : X→E and G : X→E be two non-empty valued correspondences.
Assume the following two conditions :
(i) For eachx∈X,∪λ＞0(x＋λ(F(x)−G(x))∩X≠ ;
(ii) For eachx such thatF(x) andG(x) can be strictly separated by a closed hyperplane, there exist an elementpx in topological dualE′ofE and an open neighborhoodU(x) of x such that!px, y"＞0 for ally∈F(z)−G(z) andz∈U(x).
Then there exists a pointx＊∈X for which F(x＊) and G(x＊) cannot be strictly separated by a closed hyperplane.
Proof: Suppose that for all x∈X, F(x) and G(x) are strictly separated by a closed hyperplane. By condition (ii), there exist px∈E′and an open neighborhood U(x) of x such that!px, y"＞0 for all y∈F(z)−G(z) andz∈U(x). Since {U(x)}x∈Xis an open covering of a compact setX it has a finite subcovering {U(xi)}ni＝1. Let {αi: X→[0, 1]｜i＝1, . . . ,n} be a partition of unity subordinate to {U(x1), . . . ,U(xn)} and define a single-valued mappingΨ: X→E′as Ψ(x)＝def.!n
i＝1αi(x)pxifor eachx∈X. Then
!Ψ(x), y"＞0 for allx∈X andy∈F(x)−G(x)， (1)
or equivalently
!Ψ(x), u−v"＞0 for allx∈X, u∈F(x), andv∈G(x)， (2) sinceαi(x)＞0 ‖ ‖
＞x∈U(xi) ‖ ‖
＞ !pxi, y"＞0 fory∈F(x)−G(x).
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Note that !Ψ(x), y−x"＝!n
i＝1αi(x)!pxi, y−x"and hence f(x) ＝!def. Ψ(x), y−x"is a continuous function of x on X for each fixedy inX. By Theorem 4, there exists an element x0∈X such that
!Ψ(x0), y−x0"≦0 for ally∈X. (3)
By condition (i), there exist three pointsu0∈F(x0), v0∈G(x0), y0∈X and a real number λ＞0 such thatx0＋λ (u0−v0)＝y0 or, equivalently,u0−v0＝1
λ(y0−x0). Then by (3) we have
!Ψ(x0),u0−v0"＝1
λ!Ψ(x0),y0−x0"≦0 (4)
wherex0∈X,u0∈F(x0), and v0∈G(x0). This contradicts (2). ■ Now, an extension of the Fan-Browder coincidence theorem follows directly as an application of Theorem 5.
Theorem 6: (The First Extension of the Fan-Browder Coincidence Theorem). Let X be a non-empty compact convex set in a locally convex, Hausdorff topological vector spaceE over R. Let F : X→E and G : X→E be two non-empty valued correspondences. Assume the following conditions :
(i) For each x∈X, there exist three points y∈X, u∈F(x), v∈G(x) and a real number λ＞0 such thaty−x＝λ(u−v) ;
(ii) For eachx such thatF(x) andG(x) can be strictly separated by a closed hyperplane, element px in topological dualE′ofE and open neighborhood U(x) of x exist such that
!px, y"＞0 for ally∈F(z)−G(z) andz∈U(x) ;
(iii) For eachx∈X, F(x) andG(x) are closed convex sets in E and at least one of them is compact.
Then there exists a pointx＊∈X for whichF(x＊) andG(x＊) have a non-empty intersection.
Proof: Note that all the assumptions of the previous theorem are satisfied here ; in particular, condition (i) of both theorems are equivalent. Hence, the theorem follows from the fact that in a locally convex space, two disjoint closed convex sets of which at least one is
compact can be strictly separated by a closed hyperplane. ■
In Theorem 6, as well as in Theorem 5, condition (ii) is weaker than assuming the upper
Further Extension of Fan-Browder Coincidence and Fixed Point Theorems（Urai・Shiozawa）（129）１２９
demi-continuity ofF andG. It would also be possible to directly assume a condition like (ii) in Theorem 6 for the relation between 0 andF(x)−G(x), as in our simple extension given by Theorem 2.
Theorem 7: (The Second Extension of the Fan-Browder Coincidence Theorem). LetX be a non-empty compact convex set in a Hausdorff topological vector spaceE overR. LetF : X→ E and G : X→E be two non-empty valued correspondences. Assume the following two conditions :
(i) For each x∈X, there exist three points y∈X, u∈F(x), v∈G(x) and a real number λ＞0 such thaty−x＝λ(u−v) ;
(ii) For eachx such that 0∈／F(x)−G(x), there exist an elementpxin the topological dual E′ofE and an open neighborhoodU(x) ofx such that!px, y"＞0 for all y∈F(z)−G(z) andz∈U(x).
Then there exists a pointx＊∈X for whichF(x＊) andG(x＊) have a non-empty intersection.
Proof: Suppose that for all x∈X, F(x)∩G(x)＝ , and hence, 0∈／F(x)−G(x). Then by condition (ii) we havepx∈E′and open neighborhood U(x) ofx such that !px, y"＞0 for all y
∈F(z)−G(z) and z∈U(x). Since {U(x)}x∈X is an open covering of X, a finite subcovering {U(xi)}ni＝1exists. Let {αi: X→[0, 1]｜i＝1, . . . , n} be a partition of unity subordinate to {U(x1), . . . ,U(xn)} and define single-valued mappingΨ: X→E′asΨ(x)def.＝!n
i＝1αi(x)pxifor eachx∈X. Then
!Ψ(x), y"＞0 for allx∈X andy∈F(x)−G(x)， (5)
or, equivalently,
!Ψ(x), u−v"＞0 for allx∈X, u∈F(x), andv∈G(x), (6) sinceαi(x)＞0 ‖ ‖
＞x∈U(xi) ‖ ‖
＞!pxi, y"＞0 fory∈F(x)−G(x).
Note that !Ψ(x), y−x"＝!n
i＝1αi(x)!pxi, y−x"and hence f(x) ＝!def. Ψ(x), y−x"is a continuous function of x on X for each fixedy inX. By Theorem 4, there exists an element x0∈X such that
!Ψ(x0),y−x0"≦0 for ally∈X. (7)
By condition (i), there exist three pointsu0∈F(x0), v0∈G(x0), y0∈X and a real number λ＞0
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such thatx0＋λ(u0−v0)＝y0or, equivalently,u0−v0＝1
λ(y0−x0). Then by (7) we have
!Ψ(x0),u0−v0"＝1
λ !Ψ(x0),y0−x0"≦0 (8)
wherex0∈X,u0∈F(x0), andv0∈G(x0). This contradicts (6). ■ As Theorem 2 gives one of the most general extensions of Kakutani’s fixed-point theorem, we could also use Theorem 7 to obtain an extension of the Urai-Hayashi fixed-point theorem.
Theorem 8: (An Extension of the Urai-Hayashi Fixed-Point Theorem). Let X be a non- empty compact convex subset of a Hausdorff topological vector spaceE overR, andF : X→ E be a non-empty valued correspondence. Assume the following conditions :
(i) For eachx such thatx∈／F(x), two pointsy∈X andu∈F(x), and a real number, λ＞ 0, exist such thaty−x＝λ(u−x) ;
(K) For eachx such thatx∈／F(x), there exist an element px in the topological dualE′of E and an open neighborhoodU(x) ofx such that!px, y〉＞0 for all y∈F(z)−z andz∈
U(x).
Then there exists a fixed-pointx＊∈X ofF,x＊∈F(x＊).
Proof: Assume that there is no fixed-point ofF. Then by consideringG : X→E in Theorem 7 to be the identity map, condition (i) in this theorem implies (i) in Theorem 7. It is also easy to see that condition (K) for F implies condition (ii), so that we have x＊∈F(x＊)∩G(x＊) a
fixed-point ofF, a contradiction. ■
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Urai, K. and Yokota, K. (2005) : “Generalized dual system structure and fixed point theorems for multivalued mappings,” in Kokyuroku, Research Institute for Mathematical Sciences, Kyoto University.
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