Volume 2010, Article ID 814970,10pages doi:10.1155/2010/814970
Research Article
Fixed Point Theorem of Half-Continuous Mappings
on Topological Vector Spaces
Imchit Termwuttipong and Thanatkrit Kaewtem
Department of Mathematics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
Correspondence should be addressed to Imchit Termwuttipong,[email protected]
Received 11 December 2009; Accepted 10 January 2010
Academic Editor: Anthony To Ming Lau
Copyrightq2010 I. Termwuttipong and T. Kaewtem. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Some fixed point theorems of half-continuous mappings which are possibly discontinuous defined on topological vector spaces are presented. The results generalize the work of Philippe Bich2006 and several well-known results.
1. Introduction
Almost a century ago, L. E. J. Brouwer proved a famous theorem in fixed point theory, that any continuous mapping from the closed unit ball of the Euclidean spaceRnto itself has a fixed point. Later in 1930, J. Schauder extended Brouwer’s theorem to Banach spacessee
1.
In 2008, Herings et al.see2proposed a new type of mapping which is possibly discontinuous. They called such mappings locally gross direction preservingand proved that every locally gross direction preserving mapping defined on a nonempty polytope the convex hull of a finite subset ofRnhas a fixed point. Their work both allows discontinuities of mappings and generalizes Brouwer’s theorem.
Later, Bichsee 3 extended the work of Herings et al. to an arbitrary nonempty compact convex subset of Rn.Moreover, in4, Bich established a new class of mappings which contains the class of locally gross direction preserving mappings. He called the mappings in that class half-continuousand proved that if Cis a nonempty compact convex subset of a Banach space and f : C → C is half-continuous, then f has a fixed point. Furthermore, in the same work, Bich extended the notion of half-continuity to multivalued mappings and proved fixed point theorems which generalize several well-known results.
and also show that several well-known theorems can be obtained from our results. The paper is organized as follows. InSection 2, some notations, terminologies, and fundamental facts are reviewed. Sections3and4, the fixed point theorems are proved. Finally, inSection 5, we give some consequent results on inward and outward mappings.
2. Preliminaries
A mapping F from a set X into 2Y the set of nonempty subsets of a set Y is called a
multivalued mappingfrom X into Y, and thefibersof F aty ∈ Y are the setF−y {x ∈
X:y∈Fx}.A mappingf:X → Yis called aselectionofFiffx∈Fxfor allx∈X. LetX, Y be topological spaces. A mappingF :X → 2Y is calledupper semicontinuous
u.s.c.if for eachx0 ∈X and neighborhoodV ofFx0inY, there exists a neighborhoodU
ofx0inXsuch thatFx⊆Vfor allx∈U. By aneighborhoodof a pointxinX, we mean any
open subset ofXthat containsx.
Let E be a topological vector space t.v.s., not necessarily Hausdorff and E∗ the topological dual ofE. In this paper, we considerE∗equipped with the topology of compact convergence. ThenE∗ is a t.v.s. We say thatE∗separates pointsofE, if wheneverxandyare distinct points of E,then px/py for somep ∈ E∗. If E∗ separates points of E, then a topology onEis Hausdorff. By Hahn-Banach theorem, ifEis locally convex Hausdorff, then
E∗separates points ofE, but the converse is not true, for an example, see5,6.
LetC ⊆ E andF : C → 2E. A mapping F is calledupper demicontinuous u.d.cif for each x0 ∈ Cand any open half-spacethe set of the form{x ∈ E : px > α}, where p ∈E∗\ {0}andα∈ RHinEcontainingFx
0, there exists a neighborhoodUofx0inC
such thatFx⊆Hfor allx∈U. It is clear that a u.s.c. multivalued mapping is u.d.c. but the converse is not truesee7. It is convenient to writep, xinstead ofpxforp ∈E∗and
x∈E.The reason for this is that often the vectorxand/or the continuous linear functionalp may be given in a notation already containing parentheses or other complicated form.
The following useful results are recalled to be referred.
Theorem 2.1 Browder 8. Let C be a nonempty compact convex subset of a locally convex Hausdorff t.v.s.E.If ϕ : C → E∗ is a continuous mapping, then there exists u0 ∈ Csuch that
ϕu0, v−u0 ≤0for allv∈C.
Theorem 2.2 Ben-El-Mechaiekh et al.1. Let X be a paracompact Hausdorff space andY a convex subset of a t.v.s. SupposeΦ: X → 2Y is a multivalued mapping having nonempty convex values and open fibers, thenΦhas a continuous selection.
Theorem 2.3see6. LetA, Bbe disjoint nonempty convex subsets of a locally convex Hausdorff t.v.s.E. IfAis compact andBis closed, then there existsp ∈E∗andα1, α2 ∈Rsuch thatp, x < α1< α2<p, yfor allx∈Aandy∈B.
Theorem 2.4see6. LetEbe a t.v.s. whoseE∗separates points. Suppose thatAandBare disjoint nonempty compact convex sets inE.Then there existsp ∈ E∗ such thatsup{p, x : x ∈ A} < inf{p, y:y∈B}.
Theorem 2.5see9. LetXbe a topological space,Ya compact Hausdorffspace, andF :X → 2Y
a multivalued mapping with nonempty closed values. ThenFis u.s.c. if and only if the graph{x, y:
3. Half-Continuous Mappings
Now, we introduce the notion of half-continuity on t.v.s., and investigate some of their properties.
Definition 3.1. LetCbe a subset of a t.v.s.E.A mappingf :C → Eis said to behalf-continuous
if for eachx∈Cwithx /fxthere existp∈E∗and a neighborhoodWofxinCsuch that
p, fy−y>0 3.1
for ally∈Wwithy /fy.
By the name “half-continuous,” it induces us to think that continuous mappings should be half-continuous. The following theorem tells us that ifE∗ separates points ofE, then the statement is affirmative.
Proposition 3.2. LetEbe a t.v.s. whoseE∗ separates points andCa nonempty subset ofE.Then every continuous mappingf :C → Eis half-continuous.
Proof. Letx ∈Cbe such thatx /fx.SinceE∗separates points onE, we may assume that, p, fx−x >0 for somep∈E∗.Since the mappingz → p, fz−zis continuous, there exists a neighborhoodW ofxinCsuch thatp, fy−y> 0 for ally∈ W.Therefore,f is half-continuous.
The hypothesis thatE∗separates points ofEcannot be relaxed as will be shown in the following examples.
Example 3.3. Let E be a nontrivial vector space. Then the topology {∅, E} makes E into a locally convex t.v.s. that is not Hausdorffand E∗ {0} see10. SoE∗ does not separate points onE. Consequently, every continuous self-mapping onEwhich is not the identity, is not half-continuous.
Example 3.4. For 0< p <1,Lp0,1is a Hausdorfft.v.s. withLp0,1∗{0}see6.
Remark 3.5. There are some half-continuous mappings which are not continuous. For example
4, letf:R → Rbe defined by
fx ⎧ ⎨ ⎩
3 ifx∈0,1,
2 otherwise.
3.2
It is clear thatfis half-continuous but not continuous.
Moreover, half-continuity is not closed under the composition, the addition, and the scalar multiplication. To see this consider a half-continuous mapping g on R defined by
Proposition 3.6. LetCbe a nonempty subset of a t.v.s.Eandf :C → E. Thenfis half-continuous if and only if for anyβ∈R, the mappingx →1−βxβfxis half-continuous.
Proof. The sufficiency is clear. To prove the necessity, letβ∈Rand letg :C → Ebe defined bygx 1−βxβfxfor allx∈C.Letx∈Cbe such thatx /gx.Thenx /fxand hence there existp∈E∗ and a neighborhoodW ofxinCsuch thatp, fy−y >0 for all
y∈Wwithy /fy. Then for eachy∈Wwithy /gy,
p, gy−yp,1−βyβfy−yβp, fy−y. 3.3
Ifβ >0, then done. Otherwise, consider−pinstead ofp.
Next, we give a sufficient condition for mappings on t.v.s. to be half-continuous.
Proposition 3.7. LetCbe a nonempty subset of a t.v.s.Eand f : C → E. Suppose that for each x∈Cwithx /fx, there existp∈E∗such thatp, fx−x>0p, fx−x<0andp◦fis loweruppersemicontinuous atx. Thenfis half-continuous.
Proof. Letx∈Cbe such thatx /fx.Then there existsp∈E∗such thatp, fx−x>0 and
p◦f is lower semicontinuous atx.Letα ∈Rbe such thatp, fx−x > α > 0. Sincepis continuous atx, there exists a neighborhoodV ofxinEsuch that|p, x−z|< αfor allz∈V. This implies that
β: inf
z∈V
p, x−zp, fx−x>inf
z∈V
p, x−zα≥0. 3.4
By lower semicontinuity ofp◦f, there exists a neighborhoodUofxinCsuch that
p, fy>p, fx−β 3.5
for ally∈U.Then, for eachy∈U∩V withy /fy, we have from3.4and3.5that
p, fy−y>p, fx−βp,−y≥p, fx−x−βinf
z∈V
p, x−z0. 3.6
Therefore,fis half-continuous.
The latter case follows from the fact thatf is upper semicontinuous if and only if−f is lower semicontinuous.
Remark 3.8. If E is a Banach space, then Proposition 3.7 is Proposition 2.4 in 4. By considering the mapping f in Remark 3.5, we note that the converse ofProposition 3.7 is not truesee4.
LetXandY be sets. Letfandgbe mappings fromX toY. The setCf, g {x∈X :
Theorem 3.9. LetCbe a nonempty compact convex subset of a locally convex Hausdorfft.v.s.Eand f, g:C → C. Suppose thatg:C → Cis bijective continuous and for eachx∈Cwithgx/fx
there existp∈E∗and a neighborhoodWofg−1xinCsuch thatp, fy−gy>0for ally∈W withgy/fy. ThenCf, gis nonempty.
Proof. Suppose thatCf, g ∅. DefineΦ:C → 2E∗by
Φx p∈E∗: there exists a neighborhoodW ofg−1xinCsuch that
p, fy−gy>0∀y∈W withgy/fy
3.7
for allx ∈C.Clearly,Φxis nonempty for allx ∈C.Letx ∈C, p, q ∈Φxandλ ∈ 0,1. There are neighborhoodsW1andW2ofg−1xinCsuch that
∀y∈W1, g
y/fy⇒p, fy−gy>0,
∀y∈W2, g
y/fy⇒q, fy−gy>0. 3.8
Clearly,λp 1−λq∈E∗andW W1∩W2is a neighborhood ofg−1xinC. For eachy∈W
withgy/fy,
λp 1−λq, fy−gyλp, fy−gy 1−λq, fy−gy>0. 3.9
Hence,λp 1−λq∈Φx.This implies thatΦxis convex.
Next, letp ∈ E∗ andx ∈ Φ−p.There exists a neighborhoodW ofg−1xinCsuch thatp, fy−gy > 0 for ally ∈ W withgy/fy. Thenx ∈ gW ⊆ Φ−p.Sinceg is open,Φ−pis open inC.From Theorems2.1and2.2, there exists a continuous selection
ϕ:C → E∗ofΦandx0∈Csuch that for everyy∈C,
ϕx0, y−x0
≤0. 3.10
Sinceg is surjective,x0 gz0for somez0 ∈ C, and henceϕgz0, fz0−gz0 ≤ 0.
Also, sinceϕgz0∈Φgz0,ϕgz0, fz0−gz0>0, which is a contradiction.
IfginTheorem 3.9is the identity mapping, then the following result is immediate.
Corollary 3.10. LetCbe a nonempty compact convex subset of a locally convex Hausdorfft.v.s.E. If f:C → Cis half-continuous, thenfhas a fixed point.
Remark 3.11. IfEis a Banach space, then the previous corollary is the Theorem 3.1 in4.
The following result is obtained fromProposition 3.2andCorollary 3.10.
4. Half-Continuous Multivalued Mappings
Now, we consider half-continuity of multivalued mappings and prove that under a certain assumption they have fixed point.
Definition 4.1. LetCbe a subset of a t.v.s.E.A mappingF:C → 2Eis said to behalf-continuous
if for eachx∈Cwithx /∈Fxthere existsp∈E∗and a neighborhoodWofxinCsuch that
∀y∈W, y /∈Fy⇒ ∀z∈Fy, p, z−y>0. 4.1
The following proposition gives a sufficient condition for a multivalued mapping to be half-continuous.
Proposition 4.2. LetCbe a nonempty subset of a locally convex Hausdorfft.v.s.E. IfF:C → 2Eis a u.d.c. mapping with nonempty closed convex values, thenFis half-continuous.
Proof. Assume thatF :C → 2E is u.d.c. with nonempty closed convex values. Letx∈Cbe such thatx /∈Fx. Suppose thatF fails to be half-continuous. ByTheorem 2.3, there exists
p∈E∗andα∈Rsuch that
p, x< α <p, y 4.2
for ally ∈ Fx. This implies that Fx ⊆ H : p−1α,∞. Since F is u.d.c., there exists a neighborhood Uof xin Csuch thatFy ⊆ H for all y ∈ U. Set V U\H. Then V is a neighborhood ofxinC. Since F is not half-continuous, there existsxV ∈ V \FxVand zV ∈FxVsuch that
p, zV−xV ≤0. 4.3
SincexV ∈U,FxV⊆H, sozV ∈H. Then, by4.3,α <p, zV ≤ p, xV. This means that xV ∈H, which is a contradiction. Therefore,Fis half-continuous.
Remark 4.3. However, there are some half-continuous mappings which are not u.d.c.. To see this, consider the mappingF:R → 2Rdefined by
Fx ⎧ ⎨ ⎩
−1,1 ifx /0,
{0} ifx0.
4.4
ThenFis half-continuous but not u.d.c. at 0.
In case that E is a t.v.s. whoseE∗ separates points, we need more assumptions on the mapping as the following result. The proof is analogous to that of Proposition 4.2, by applyingTheorem 2.4.
Proposition 4.4. LetEbe a t.v.s. whoseE∗separates points andCa nonempty subset ofE.IfF :
Next, we will prove the main result which guarantees the possessing of fixed points if the multivalued mapping is half-continuous. To do this, we need the following lemma.
Lemma 4.5. LetCbe a nonempty subset of a t.v.s.EandF :C → 2E. IfFis half-continuous, then Fhas a half-continuous selection.
Proof. Assume thatFis half-continuous. Letfbe any selection ofF. Definef:C → Eby
fx
⎧ ⎨ ⎩
x ifx∈Fx,
fx ifx /∈Fx.
4.5
Clearly,fis a selection ofF. To show thatfis half-continuous, letx∈Cbe such thatx /fx. Thenx /∈Fxand hence there existsp∈E∗and a neighborhoodWofxinCsuch that
∀y∈W, y /∈Fy⇒ ∀z∈Fy, p, z−y>0. 4.6
It follows thatp,fy−yp, fy−y>0 for everyy∈Wwithy /fy.
Corollary 3.10andLemma 4.5yield the following main result.
Theorem 4.6. LetCbe a nonempty compact subset of a locally convex Hausdorfft.v.s.E. IfF:C → 2C is half-continuous, thenFhas a fixed point.
The following result is immediately obtained fromTheorem 4.6andProposition 4.2.
Corollary 4.7. LetCbe a nonempty compact convex subset of a locally convex Hausdorfft.v.s.E. If F:C → 2Cis u.d.c. with nonempty closed convex values, thenFhas a fixed point.
It is well known that ifCis a subset of a topological spaceXandF :C → 2Xhas closed graph, then the set of fixed points ofF is closed inC. FromCorollary 4.7andTheorem 2.5, we have the following corollary.
Corollary 4.8 Kakutani-Fan-Glicksberg, see 11, 12. Let C be a nonempty compact convex subset of a locally convex Hausdorfft.v.s.E. IfF :C → 2C is u.s.c. with nonempty closed convex values, then the set of fixed points ofFis nonempty and compact.
5. Inward and Outward Mappings
In case that the half-continuous mappingf is a nonself-mapping onCbutfhas some nice property, thenfstill possesses a fixed point inC. We state the results in the following theorem.
Proof. Suppose thatf has no fixed point. For eachx∈C,letΛx {λ ∈R:λ <1 andλx
1−λfx∈C}.DefineF:C → 2Cby
Fx λx 1−λfx:λ∈Λx 5.1
for allx∈C.ThenFx/∅for everyx∈C.It is not difficult to see thatFis half-continuous. ByTheorem 4.6, there existsx0 ∈Fx0∩Candα∈Λx0such thatx0αx0 1−αfx0.
It follows thatx0fx0, which is a contradiction.
Remark 5.2. FromTheorem 5.1, forx ∈Cwithx /fx, if there isλ < 0 such thatz :λx
1−λfx∈C, thenfx, in fact, is the element inC. Indeed, by settingμλ/λ−1, then 0< μ <1 and so, by convexity ofC,fx μx 1−μz∈C.
Recall that theline segmentjoining vectorsxandyinEis the setx, y {λx1−λy: 0≤λ≤1}. As a special case ofTheorem 5.1we obtain the following corollary.
Corollary 5.3Fan-Kaczynski, see1. LetCbe a nonempty compact convex subset of a locally convex Hausdorfft.v.s.E. Suppose thatf :C → Eis continuous and for eachx∈Cwithx /fx the line segmentx, fxcontains at least two points ofC,thenfhas a fixed point.
Next, we derive a generalization of a fixed point theorem due to F. E. Browder and B. R. Halpern. To do this, let us recall the definition of inward and outward mappings.
Definition 5.4 see1. LetC be a subset of a vector space E. A mappingf : C → E is calledinward resp.,outward if for eachx ∈ C there existsλ > 0 resp.,λ < 0 satisfying
xλfx−x∈C.
Theorem 5.5. LetCbe a nonempty compact convex subset of a locally convex Hausdorfft.v.s.E. Then every half-continuous inward (or outward) mappingf:C → Ehas a fixed point.
Proof. Suppose thatf :C → Eis a half-continuous inward mapping. Letx∈Cbe such that
x /fx.There existsλ > 0 such that xλfx−x ∈ C. By lettingβ 1−λand apply
Theorem 5.1,fhas a fixed point.
Next, assume thatfis outward. Defineg :C → Ebygx 2x−fxfor allx∈C. Theng is inward and, byProposition 3.6,g is half-continuous. Hence, there isx0 ∈Csuch
thatx0gx0 2x0−fx0. That isx0fx0.
Remark 5.6. In Theorem 5.5, if f is a continuous inward or outward mapping, then
Theorem 5.5is the theorem proved by F. E. Browder1967and B. R. Halpern1968 see
1.
In the final part, we prove the fixed points theorem for half-continuous inward and outward multivalued mappings.
Definition 5.7see 7. LetCbe a subset of a vector space E. A mappingF : C → 2E is calledinwardresp.,outwardif for eachx∈Cthere existsy ∈Fxandλ >0resp.,λ <0 satisfyingxλy−x∈C.
Proof. LetF :C → 2Ebe a half-continuous mapping. Suppose thatFis inward but it has no fixed point. DefineG:C → 2Cby
Gx {u∈C: there existsv∈Fxandλ >0 such thatuxλv−x} 5.2
for allx ∈ C.We can see thatGxis nonempty for allx ∈ CandGis half-continuous. By
Theorem 4.6, there existsx0 ∈C∩Gx0, v∈Fx0,andα >0 such thatx0 x0αv−x0.
That isx0∈Fx0, which is a contradiction.
Next, assume thatF is outward. DefineH : C → 2E byHx 2x−Fxfor all
x∈C. It is easy to see thatHis half-continuous. Letx∈Cbe arbitrary. There existsy∈Fx andλ < 0 satisfyingxλy−x∈ C. Thenx −λ2x−y−x xλy−x∈ C. Since 2x−y∈Hx,His inward. Thusx0 2x0−vfor somex0 ∈Hx0∩Candv∈Fx0. That
isx0∈Fx0.
Any selection of half-continuous inward multivalued mappings may not be inward as shown in the following example. LetF:0,1 → 2Rbe defined by
Fx ⎧ ⎨ ⎩
x1,∞ if x∈0,1,
{0,1,2} if x1. 5.3
Clearly,Fis inward half-continuous but a selectionf:0,1 → RofFdefined byfx x2 if 0≤x <1 andfx 2 ifx1 is not inward.
Remark 5.9. If the half-continuity ofFis replaced by upper semicontinuity, thenTheorem 5.8
is the result of Halpern-Bergman1968 see7and Fan1969 see13.
As an interesting special case ofTheorem 5.8, we obtain the following corollary.
Corollary 5.10. LetCbe a nonempty compact convex subset of a locally convex Hausdorfft.v.s.E. Suppose thatF:C → 2Eis half-continuous and for eachx∈C,Fx∩Cis nonempty, thenFhas a fixed point.
6. Discussion
It is worth to notice that there exists a multivalued mapping which is not half-continuous but some of its selection is half-continuous. For example, letF:0,1 → 20,1be defined by
Fx ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
3 4,1
∪ {0} ifx∈
0,1 2
,
3 4
ifx∈
1 2,1
.
Then F is not half-continuous since 4.1 fails for x 1/2. Nevertheless, a mapping f :
0,1 → 0,1defined by
fx ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
1 ifx∈
0,1 2
,
3
4 ifx∈
1 2,1
6.2
is a half-continuous selection ofF.
From Theorem 4.6 we see that if a multivalued mapping F has a half-continuous selection, then F has a fixed point. It is interesting to investigate the conditions for a multivalued mapping to induce a half-continuous selection.
Acknowledgments
The second author is financially supported by Mahidol Wittayanusorn School. This work is dedicated to Professor Wataru Takahashi on his retirement.
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