A NOTE ON THE MINIMAL ESSENTIAL SET OF COINCIDENT POINTS FOR SET-VALUED MAPPINGS

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A NOTE ON THE MINIMAL ESSENTIAL SET OF COINCIDENT POINTS FOR SET-VALUED MAPPINGS

LUO QUN

Received 12 May 2004 and in revised form 23 November 2004

Motivated by the ideas of Kinoshita, we introduce the concept of minimal essential set of the coincident points for set-valued mappings, and we prove that there exists at least one minimal essential set and one essential component of the coincident points for set-valued mappings (satisfying some conditions).

1. Introduction

Kinoshita [3] introduced the notion of essential component to the set of fixed points and proved that for any continuous mapping of the Hilbert cube into itself, there exists at least one essential component of the set of its fixed points. The natural extension of fixed point theory is the study of coincident points. Tan et al. [5] introduced the concept of essential coincident points for multivalued mappings, they also discussed the generic stability of coincident points for multivalued mappings. However, as can be seen inExample 2.9, there exist no essential coincident points.

In this paper, motivated by the ideas of Kinoshita, we introduce the concept of minimal essential set of the coincident points for set-valued mappings, and we prove that there exists at least one minimal essential set of the coincident points for set-valued mappings (satisfying some conditions), and hence there exists at least one essential component of the coincident points.

2. Preliminaries

LetKbe a subset of a metric space (E,d); for anyδ >0, we denote byO(K,δ)= {x∈E: d(x,K)< δ}the open neighborhood ofKwith radiusδinE.

LetXbe a nonempty compact convex subset of a Banach spaceV. Let S=

f :X−→2^Xupper semicontinuous and nonempty closed convex values, (2.1) where 2^Xdenotes the family of all nonempty subsets ofX.

For any f,f∈S, define ρ1

f,f=sup

x∈X

Hf(x),f(x), (2.2)

Journal of Applied Mathematics and Stochastic Analysis 2005:2 (2005) 89–95 DOI:10.1155/JAMSA.2005.89

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whereHis the Hausdorﬀmetric defined onX. Clearly, (S,ρ1) is a complete metric space.

Let Y=

(f,g)∈S×S:f,g∈S, for anyx∈BdX,f(x)−g(x)∩

λ>0

λ(X−x)

= ∅ , (2.3) where BdX denotes the boundary ofX, then (Y,ρ) is a complete metric space, where ρ((f,g), (f,g))=ρ1(f,f) +ρ1(g,g).

Theorem2.1. Y⊂S×Sis a closed subset.

Proof. Letyα=(fα,gα)∈Y withyα→y=(f,g)∈S×S. Since (fα,gα)∈Y, for anyx∈ BdX, one has

fα(x)−gα(x)∩

λ>0

λ(X−x)

= ∅. (2.4)

Then there existu_α∈ f_α(x) andv_α∈g_α(x) such that uα−vα∈

λ>0

λ(X−x). (2.5)

Note that f_α→f,g_α→g,Xis compact,{u_α}has a cluster pointu0∈ f(x), and{v_α}has a cluster pointv0∈g(x). Without loss of generality, we may assume thatuα→u0∈f(x), vα→v0∈g(x).

(1) If there exists infinite α such that u_α=v_α, then u0=v0, and hence u0−v0∈

λ>0λ(X−x).

(2) If there exists infiniteαsuch thatuα=vα, then there existsk >0 such that uα−vα∈

0<λ<k

λ(X−x). (2.6)

Hence there existsλ_αwith 0< λ_α< ksuch thatu_α−v_α∈λ_α(X−x). So there existsz_α∈X such thatuα−vα=λα(zα−x). Note thatXis compact,{zα}has a cluster pointz0∈X, we may assume thatzα→z0. And since 0< λα< k, we may assume thatλα→λ0(≥0), hence

u0−v0=λ0

z0−x. (2.7)

If λ0=0, then u0−v0=0∈

λ>0λ(X−x). If λ0=0, then u0−v0=λ0(z0−x)∈ λ0(X−x)⊂

λ>0λ(X−x).Hence, for anyx∈BdX, f(x)−g(x)∩

λ>0

λ(X−x)

= ∅. (2.8)

ThereforeY⊂S×Sis a closed subset.

For any y=(f,g)∈Y, we denote byCC(y)= {x∈X: f(x)∩g(x)= ∅}the set of coincident points of the set-valued mappings f andg, by [2, Theorem 10],CC(y)= ∅, thusy→CC(y) indeed defines a set-valued mapping of coincident points fromY toX and we have the following theorem.

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Theorem2.2. The mappingCC:Y→2^Xis upper semicontinuous with nonempty compact values.

Proof. For anyy=(f,g)∈Y, we need to prove thatCC(y)⊂X is compact. Let a sequence{xα} ⊂CC(y) andxα→x0∈X. Sincexα∈CC(y), we have f(xα)∩g(xα)= ∅.

Suppose that f(x0)∩g(x0)= ∅, then there existsδ >0 such that Ofx0

,δ∩Ogx0

,δ_{= ∅}. (2.9)

By upper semicontinuities of f andg, and sincex_α→x0, there existsα0such that for any α > α0,f(xα)⊂O(f(x0),δ) andg(xα)⊂O(f(x0),δ), thenf(xα)∩g(xα)= ∅, which contradicts the fact that f(xα)∩g(xα)= ∅, hencex0∈CC(y) and henceCC(y) is compact.

SinceXis compact, we want to prove that the mappingCCis upper semicontinuous, we only need to prove that the GraphCCofCCis closed:

GraphCC=

(y,x)∈Y×X:x∈CC(y), y∈Y. (2.10) Let a sequence{(yα,xα)}⊂GraphCCand (yα,xα)→(y0,x0)∈Y×X. Denoteyα=(fα,gα), y0=(f0,g0), thenxα∈CC(yα) and fα(xα)∩gα(xα)= ∅.

Suppose that f0(x0)∩g0(x0)= ∅, then there existsδ^∗>0 such that Of0

x0

,δ^∗∩Og0

x0

,δ^∗= ∅. (2.11)

Since f_α→f0,g_α→g0,x_α→x0, and f0,g0are upper semicontinuous, there existsα^∗such that

fα

xα

⊂O

f0

xα

,δ^∗ 2

⊂Of0

x0

,δ^∗, ∀α > α^∗,

gα

xα

⊂O

g0

xα

,δ^∗ 2

⊂Og0

x0

,δ^∗), ∀α > α^∗.

(2.12)

Hence f_α(x_α)∩g_α(x_α)= ∅, which contradicts the fact that f_α(x_α)∩g_α(x_α)= ∅. So the mappingCCis upper semicontinuous with nonempty compact values.

For each y∈Y, the component of a pointx∈CC(y) is the union of all connected subsets ofCC(y) which contain the pointx, see [1, page 356], components are connected closed subsets ofCC(y) and are also connected compact. It is easy to see that the components of two distinct points ofCC(y) either coincide or are disjoint, so that all components constitute a decomposition ofCC(y) into connected pairwise disjoint compact subsets, that is,

CC(y)=

α∈Λ

Cα(y), (2.13)

whereΛis an index set, for anyα∈Λ,C_α(y) is a nonempty connected compact subset and for anyα,β∈Λ(α=β),C_α(y)∩C_β(y)= ∅.

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Definition 2.3. Fory∈Y,CC(y)=

α∈ΛC_α(y),C_α(y) is called an essential component if for each open setOcontainingC_α(y), there existsδ >0 such that for anyy∈Y with ρ(y,y)< δ,CC(y)∩O= ∅.

Definition 2.4. Fory∈Y,e(y)⊂CC(y) is a nonempty closed set,e(y) is called an essen- tial set ofCC(y) (with respect toY) if for any open setUwithU⊃e(y), there isδ >0 such that for anyy∈Y withρ(y,y)< δ,CC(y)∩U= ∅.

Definition 2.5. For y∈Y,m(y)⊂CC(y) is an essential set,m(y) is called a minimal essential set ofCC(y) (with respect toY) ifm(y) is a minimal element of the family of essential sets ofCC(y) ordered by set inclusion.

Remark 2.6. Ife1(y)⊂CC(y) is an essential set ofCC(y) (with respect toY),e2(y)⊂ CC(y) is closed, ande1(y)⊂e2(y), thene2(y) is also an essential set ofCC(y).

Remark 2.7. Ifx∈CC(y) is an essential coincident point (see [5]) ofCC(y), then{x}is an essential set ofCC(y);e(y)⊂CC(y) is an essential set ande(y)= {x}, thenx∈CC(y) is an essential coincident point ofCC(y).

Remark 2.8. IfA⊂CC(y) is closed,x∈A⊂CC(y), andxis an essential coincident point ofCC(y), thenAis an essential set and{x}is a minimal essential set ofCC(y).

Example 2.9. LetX=[0, 1], for anyx∈X,f(x)=[0,x],g(x)=[x, 1], theny=(f,g)∈Y and CC(y)= {x∈[0, 1] : f(x)∩g(x)= ∅} =[0, 1]. But x0 is not an essential coincident point for anyx0∈CC(y). Ifx0∈(0, 1), for allε >0, takeδ >0 (δ < ε/2) such that O(x0,δ)=(x0−δ,x0+δ)⊂[0, 1].

Define the set-valued mappingsf^ε,g^ε:X→2^Xby g^ε(x)=g(x),

f^ε(x)=











[0,x], x∈

0,x0−δ,

0,

1− ε 2δ

x+ ε

2δ

x0−δ, x∈

x0−δ,x0

,

0,

1 + ε 2δ

x− ε

2δ

x0+δ, x∈

x0,x0+δ,

[0,x], x∈

x0+δ, 1,

(2.14)

then y^ε=(f^ε,g^ε)∈Y andρ(y,y^ε)< ε, butCC(y^ε)∩O(x0,δ)= ∅, hencex0∈(0, 1) is not an essential coincident point.

Similarly, ifx0=1, for allε: 0< ε <1/2, takeδ >0 (δ < ε/2) such that (1−δ, 1]⊂(0, 1].

Define the set-valued mappingsf^ε,g^ε:X→2^Xby g^ε(x)=g(x), f^ε(x)=









[0,x] ifx∈[0, 1−δ],

0,

1− ε

2δ

x+ ε 2δ(1−δ)

ifx∈(1−δ, 1].

(2.15)

Ifx0=0, for allε >0 (<1/2), takeδ >0 (δ < ε/2) such that [0,δ)⊂[0, 1).

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Define the set-valued mappingsf^ε,g^ε:X→2^Xby f^ε(x)=f(x),

g^ε(x)=









1− ε 2δ

x+ε

2, 1

ifx∈[0,δ],

[x, 1] ifx∈(δ, 1].

(2.16)

Hence, for anyx0∈CC(y)=[0, 1],x0is not an essential coincident point.

3. The minimal essential set of coincident points By Zorn lemma, we obtain the following theorem.

Theorem3.1. For anyy∈Y, there exists at least one minimal essential set ofCC(y).

Proof. ByTheorem 2.2, the map CC:Y→2^X is upper semicontinuous and CC(y) is compact for anyy∈Y, thenCC(y) is an essential set.

LetE(y) denote the family of all essential sets ofCC(y) ordered by set inclusion. Let {eα(y)}α∈Γ be a decreasing chain ofE(y), then limeα(y)=

α∈Γeα(y)= ∅and is compact. Denotinge(y)₌lime_α(y), we need to prove thate(y) is the lower bound of the chain{e_α}α∈Γ, that is,e(y)∈E(y). Sincee_α(y) is compact, by [4, page 43],H(e_α(y),e(y))

→0, whereH is the Hausdorﬀ metric defined onX, hence for any open set Owith O⊃e(y), there isα1∈Γsuch thateα(y)⊂Ofor anyα > α1. Sinceeα(y) is an essential set ofCC(y), there existsδ >0 such thatCC(y)∩O= ∅for anyy∈Y withρ(y,y)< δ, thene(y) is an essential set ofCC(y),e(y) is the lower bound of the chain{eα}α∈Γ. There- fore, by Zorn lemma,E(y) has a minimal element and this minimal element is a minimal

essential set ofCC(y).

Theorem3.2. For anyy∈Y, the minimal essential set ofCC(y)is connected.

Proof. Letm(y) be a minimal essential set ofCC(y). Suppose thatm(y) was not connected, then there exist two nonempty closed setsC1(y),C2(y) and two openU1,U2such thatC1(y)⊂U1,C2(y)⊂U2 andm(y)=C1(y)∪C2(y),U1∩U2= ∅. Becausem(y) is a minimal essential set ofCC(y),C1(y) andC2(y) are not essential sets. SinceC1(y) and C2(y) are compact, there exist two open sets isV1andV2which satisfy

C1(y)⊂V1⊂V¯1⊂U1, C2(y)⊂V2⊂V¯2⊂U2, (3.1) where ¯V_idenotes the closure ofV_i,i=1, 2.

For anyδ >0, there existy1=(f1,g1),y2=(f2,g2)∈Y withρ(y,y1)< δ,ρ(y,y2)< δ such that

CCy1

∩V1= ∅, CCy2

∩V2= ∅. (3.2)

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Define two set-valued mapsf^∗:X→2^Xandg^∗:X→2^Xas follows:

f^∗(x)=











f1(x) ifx∈V¯1, f2(x) ifx∈V¯2,

ξ(x)f1(x) +η(x)f2(x) ifx∈X\V¯1∪V¯2,

g^∗(x)=











g1(x) ifx∈V¯1, g2(x) ifx∈V¯2,

ξ(x)g1(x) +η(x)g2(x) ifx∈X\V¯1∪V¯2,

(3.3)

where

ξ(x)= dx, ¯V2

dx, ¯V2

+dx, ¯V1

, η(x)= dx, ¯V1

dx, ¯V2

+dx, ¯V1

. (3.4)

It is easy to see thaty^∗=(f^∗,g^∗)∈Y, thenCC(y^∗)= ∅andCC(y^∗)∩(V1∪V2)= ∅. Sinceρ(y,y^∗)=ρ1(f,f^∗) +ρ1(g,g^∗), by [6, Lemma 3.1], we haveρ(y,y^∗)< δ, but m(y)⊂C1(y)∪C2(y)⊂V1∪V2, byDefinition 2.4,m(y) is not an essential set ofCC(y), which contradicts the fact thatm(y) is a minimal essential set, hencem(y) is connected

and the proof is complete.

By Theorems3.1and3.2, we have the following corollaries.

Corollary3.3. For anyy∈Y, there exists at least one connected minimal essential set of CC(y).

Corollary3.4. For anyy∈Y, there exists at least one essential component ofCC(y).

Proof. For anyy∈Y, byCorollary 3.3, there exists at least one connected minimal essential setm(y) ofCC(y), sincem(y) is connected, there exists a componentM(y) ofCC(y) such thatm(y)⊂M(y), byDefinition 2.3,M(y) is an essential component ofCC(y).

Remark 3.5. Ifg(x)=xfor anyx∈X, then for any f ∈Sandx∈BdX, f(x)−g(x)∩

λ>0

λ(X−x)

=

f(x)−x∩

λ>0

λ(X−x)

= ∅. (3.5)

Thereforey=(f,g)∈Y andCC(y)=F(f), whereF(f) denotes the set of fixed points of f.

ByCorollary 3.4, we have the following corollary.

Corollary3.6. For any f ∈S, there is at least one essential component ofF(f).

Remark 3.7. Corollary 3.6is a generalization of [3, Theorem 3].

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Example 3.8. LetX=[−1, 1], f(x)=



{0}, −1≤x <0, [0,x], 0≤x≤1, g(x)=





[x,−1], −1≤x <0, [0,x−1], 0≤x≤1.

(3.6)

Theny=(f,g)∈YandCC(y)= {x∈[0, 1] : f(x)∩g(x)= ∅} =[0, 1]⊂[−1, 1].

Suppose that [0, 1] is not an essential set ofCC(y), then there exists an open setUwith U⊃[0, 1] (LetU=(−ε, 1], 0< ε <1), for allδ >0, there existsy^δ∈Y withH(y,y^δ)< δ such thatCC(y^δ)∩U= ∅, that is,CC(y^δ)⊂[−1,−ε].

Takeδ=ε/4, for anyy⁰=(f⁰,g⁰)∈Ywithρ1(f,f⁰)< δ/2 andρ1(g,g⁰)< δ/2, one has ρ(y,y⁰)< δ, and for allx∈[−1,−ε],H(f(x),f⁰(x))< ρ1(f,f⁰)< δ/2,H(g(x),g⁰(x))<

ρ1(g,g⁰)< δ/2, then f⁰(x)⊂(−δ/2,δ/2), g⁰(x)⊂[−ε+δ/2,−1]=[(−7/2)δ,−1], and [−δ/2,δ/2]∩[(−7/2)δ,−1]= ∅, hence f⁰(x)∩g⁰(x)= ∅for anyx∈[−1,−ε],CC(y⁰)

⊂(−ε, 1] which contradicts the fact thatCC(y⁰)⊂[−1,−ε]. Therefore, [0, 1] is an essen- tial set and hence [0, 1] is a minimal essential set.

Acknowledgment

This research was supported by the Natural Science Foundation of Guangdong Province, China.

References

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[3] S. Kinoshita,On essential components of the set of fixed points, Osaka Math. J.4(1952), 19–22.

[4] E. Klein and A. C. Thompson,Theory of Correspondences, Canadian Mathematical Society Se- ries of Monographs and Advanced Texts, John Wiley & Sons, New York, 1984.

[5] K. K. Tan, J. Yu, and X. Z. Yuan,The stability of coincident points for multivalued mappings, Nonlinear Anal. Series A: Theory and Methods25(1995), no. 2, 163–168.

[6] J. Yu and Q. Luo,On essential components of the solution set of generalized games, J. Math. Anal.

Appl.230(1999), no. 2, 303–310.

Luo Qun: Department of Mathematics, Zhaoqing University, Zhaoqing, Guangdong 526061, China

E-mail address:[email protected]