AbstractThis note gives some fixed point theorems for lower and upper semi-continuous mappings and mappings with open lower sections defined on non-compact and non-convex sets. It will be noted that the conditions of our theorems are not only sufficient but also necessary. Also our theorems generalize some well-known fixed point theorems such as the Kakutani fixed point theorem and the Brouwer-Schauder fixed point theorem by relaxing the compactness and convexity conditions

Tian, Guoqiang

Elsevier - Publisher Connector 

Fixed points theorems for mappings with non-compact and non-convex domains 

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 158, 161-167 (1991) 
Fixed Points Theorems for Mappings with 
Non-compact and Non-convex Domains 
GUOQIANG TIAN 
Department of Economics, Texas A & M University, 
College Station, Texas 77843 
Submitted by E. Stanley Lee 
Received November 27, 1989 
This note gives some fixed point theorems for lower and upper semi-continuous 
mappings and mappings with open lower sections defined on non-compact and 
non-convex sets. It will be noted that the conditions of our theorems are not only 
sufftcient but also necessary. Also our theorems generalize some well-known 
fixed point theorems such as the Kakutani fixed point theorem and the 
Brouwer-Schauder fixed point theorem by relaxing the compactness and convexity 
conditions. f? 1991 Academx Press, Inc. 
1. INTRODUCTION 
Fixed point theorems have wide applications to problems in optimiza- 
tion, game theory, and economics (say, e.g., Arrow and Debreu [l], 
Border [4], Debreu [6], Nash [16], Shafer and Sonnenschein [18], and 
others). In particular, it is a key mathematical tool in proving the existence 
of equilibrium. However, most of the fixed point theorems in the literature 
(such as those in Border [4], IstraIescu [ll], Joshi and Bose [12], 
Smart [ 191, and the references cited therein) are proved upon compact 
convex sets and only give sufficient conditions for the existence of fixed 
points of mappings. Also, the fixed point theorems obtained by Halpern 
[S, 91, Browder [S], Fan [7], Petryshyn and Fitzpatrick [17], and 
Lim [:14] among others assumed that the (weakly) inward mappings are 
upper semi-continuous or upper demi-continuous. To my best knowledge, 
no results are variable for lower semi-continuous inward mappings. The 
purpose of this note is to present some necessary and sufficient conditions 
for the existence of fixed points of lower semi-continuous and upper semi- 
continuous correspondences and correspondences with open lower sections 
defined on non-compact and non-convex sets in the Frtchet topological 
vector space and locally convex Hausdorff topological vector space. It will 
be noted that these results generalize the Kakutani’s fixed point theorem 
161 
0022-247X/91 $3.00 
Capynght q~> 1991 by Academic Press, Inc. 
All rights of reproduction in any form reserved. 
162 WJoQIANG I-IAN 
and the Brouwer-Schauder fixed point theorem by relaxing the compact- 
ness and convexity conditions. As an application, Tian 120, 211 recently 
used the results obtained in this note to prove the existence theorems on 
abstract economies with a non-compact strategy space. Though this note 
only considers the existence of fixed points of the lower and upper semi- 
continuous mappings and mappings with lower open sections, we think 
some of the techniques developed in the note can be applied to extend 
other fixed point theorems for, say, nonexpansive and weakly inward 
mappings with non-closed and non-convex domains. 
2. NOTION AND DEFINITIONS 
Let X and Y be two topological spaces, and let 2’ be the set of all sub- 
sets of Y. A correspondence G: X -+ 2’ is said to be upper semi-continuous 
(in short, u.s.c.) if the set {XE X: G(x) c V} is open in X for every open 
subset V of Y. A correspondence G: X --) 2 ’ is said to be lower semi- 
continuous (in short, 1,s.~) if the set {x E X: G(x) n V # @} is open in X for 
every open subset V of Y. A correspondence G : X + 2 ’ is said to have open 
upper sections if, for every x E X, G(X) is open in Y. A correspondence 
G: X-+2’ is said to have open lower sections if the set G-‘(y)= 
(xEX:~EG(X)} . IS o p en in X for every y E Y. Let D be a non-empty 
convex subset of the locally convex Hausdorff topological vector space E 
andletZ,(x)={~~~E:~x+(1-~~)y~Dandi.~[O,1)}.Acorrespondence 
G: D + 2E is said to be inward on D if G(x) n I,(s) # $3 for all x E D, or 
equivalently, for every x E D, there exists a y E G(x) and j* E [0, 1) such that 
Ax + (1 - A) y ED. A correspondence G: D + 2E is said to be weakly inward 
on D if G(x) n cl In(x) # @. 
3. THE EXISTENCE OF FIXED POINTS OF MAPPINGS 
Before proceeding to the main theorems, we state some technical lemmas 
which were due to Michael [lS, Proposition 2.51 and Yannelis and 
Prabhakar [22, Fact 6.11. 
LEMMA 1. Let X and Y be two topological spaces and let 4 : X + 2 ‘, and 
II/ : X --* 2’ be correspondences uch that 
(i) d is I.s.c. and has open upper sections, 
(ii) $ is I.s.c., 
(iii) for all x E X, b(x) n $(x) # 0. 
Then the correspondence 0 : X -+ 2 ’ defined by O(x) = b(x) n $(x) is 1.s.c. 
FIXED POINT THEOREMS 163 
LEMMA 2. Let X and Y be two topological spaces and I$: X -+ 2’, and 
I+!I: X--t 2’ be correspondences having open lower sections. Then the corre- 
spondence 8: X + 2’ defined by %(x) = d(x) n It/(x) has open lower sections. 
Our main results in the following give necessary and sufficient conditions 
for the existence of fixed points of mappings. 
THEOREM 1. Let X be a non-empty subset in a Frechet topological vector 
space E. Suppose that F: X -+ 2E is a lower semi-continuous correspondence 
with non-empty closed convex values. Then the necessary and sufficient 
condition for the existence of a fixed point x* E F(x*) is that there exists 
a non-empty compact convex subset Cc X such that 
F(x)nC#1ZI VXEC. (1) 
Proof Necessity. Suppose F has a fixed point x* E F(x*). Let 
C = (.Y* }. Then the singleton set C is clearly compact and convex and 
F(x) n C # @ for all x E C. 
Sufficiency. Suppose there exists a non-empty compact convex set C 
such that F(x) n C # @ for all x E C. 
Define a correspondence K: C --, 2c by, for each x E C, 
K(x) = F(x) n C. (2) 
Since the correspondence G : C -+ 2’ defined by G(x) = C for all x E C is 
clearly 1.s.c. and has open upper sections in C and F is i.s.c., K is 1.s.c. by 
Lemma 1. Note that also K(x) is non-empty, compact, and convex for 
all XE C. Therefore, by the Michael selection theorem (cf. Aubin [2, 
Theorem 15.3.5]), there exists a continuous function .f: C + C such that 
f(x) E K(x) for all XE C. Now, by applying the Brouwer-Schauder fixed 
point theorem (cf. Aubin [2, p. 284]), there exists a point x* E C such that 
x* =f (x*) E K(x*) c F(x*) and thus F has a fixed point x* E F(x*). 1 
Remark 1. Even though F is 1.s.c. and C is a non-empty compact and 
convex subset, the correspondence K: C --, 2’ defined by K(x) = F(x) n C 
for each x E C need not be 1.s.c. on C if it is not true K(x) # @ for all x E C. 
The following simple example illustrates this. 
EXAMPLE 1. Let X=[O,3] and Y=[1,5]. Let F:X-t2Y defined by 
F(x) ‘= [ 1 +x, 2 +x]. If we let C= [0, 21, then K is not 1.~. on C since 
(xEC:F(x)nC#@}=[O,l]isnotopeninC. 
Remark 2. Note that the condition (1) implies that the correspondence 
F is inward on C since Cc Z,(x) for all x E C. 
164 GUOQIANG TIAN 
Theorem 1 requires that X be a non-empty subset in a Frechet topologi- 
cal vector space. We can extend the above theorem to a non-empty 
paracompact subset in a locally convex Hausdorff topological space if F is 
strengthened to have open lower sections (cf. Yannelis and Prabhakar [22. 
p. 2371). 
THEOREM 2. Let X be a non-empty paracompact subset in a locally con- 
vex Hausdorff topological vector space E. Suppose that F: X + 2” has open 
lower sections such that F(x) is non-empty, closed. and convex ,for all x E X. 
Then the necessary and sufficient condition for the existence of a ,fixed point 
x* E F(x*) is that there exists a non-empty compact convex subset Cc X 
such that 
F(x)nC#@’ QxEC. (3) 
Proof The proof of necessity is the same. The proof of sufficiency is 
similar to the above. Suppose there exists a non-empty compact convex set 
C such that F(x) n C # @ for all x E C. 
Define a correspondence K: C -+ 2c by, for each x E C, 
K(x) = F(x) n C. (4) 
Then, by Lemma 2, K has open lower sections by noting that the corre- 
spondence G: C + 2c defined by G(x) = C for all x E C has open lower 
sections. And K(x) is non-empty, compact, and convex for all XE C. 
Therefore, by the Theorem 3.1 in Yannelis and Prabhakar [22], there 
exists a continuous function f: C -+ C such that f(x) E K(x) for all XE C. 
Hence, by the Brouwer-Schauder fixed point theorem, there exists a point 
x* E C such that x* =,f(x*) E K(x*) c F(x*) and thus F has a fixed point 
x* E F(x*). 1 
We now give a similar theorem for the upper semi-continuous corre- 
spondence. 
THEOREM 3. Let X be a non-empty subset in a locally convex Hausdorff 
topological vector space E. Suppose that F: X -+ 2E is an upper semi- 
continuous correspondence with non-empty closed convex values. Then the 
necessary and sufficient condition for the existence of a fixed point 
x* E F(x*) is that there exists a non-empty compact convex subset Cc X 
such that 
F(x)nC#O Qx E C. (5) 
Proof: This theorem can be proved by applying Theorem 2 in 
Halpern [9] by noting Cc Z,(x) for all x E C. Here we give a direct proof. 
We first show that the correspondence K: C -+ 2c by K(x) = F(x) I-I C for 
FIXED POINT THEOREMS 165 
each x E C is U.S.C. In fact, since F is an upper semi-continuous corre- 
spondence with non-empty closed convex values, it is closed (i.e., its graph 
is closed) by Proposition 3.7 of Aubin and Ekeland [3]. Thus, by 
Theorem 8 of Aubin and Ekeland [3], K is an upper semi-continuous 
correspondence with non-empty compact values. Therefore, by Kakutani’s 
fixed point theorem (cf. Aubin [2, p. 284]), there exists x* EC such that 
x* E K(x*) and thus F has a fixed point x* E F(x*). 1 
Theorem 3 above can be extended to the following theorem which 
generalizes the fixed point theorems of Browder [S], Halpern [S, 91, and 
Halpern and Bergman [lo] by relaxing the compactness and convexity 
sets. 
THEOREM 4. Let X be a non-empty subset in a locally convex Hausdorff 
topological vector space E. Suppose that F: X + 2” is an upper semi- 
continuous correspondence with non-empty closed convex values. Then the 
necessary and sufficient condition for the existence of a fixed point 
x* E F(x*) is that there exists a non-empty compact convex subset Cc X 
such that F is weakly inward on C. 
Proof We only need to show the sufficiency. Since F is weakly inward 
on the non-empty compact convex set C and F: C + 2” is an upper semi- 
continuous correspondence with closed convex values, by Theorem 2 in 
Halpern [9], there exists a points x* E C such that x* E F(x*). 1 
Remark 3. Observe that in case that X is a non-empty compact convex 
subset and F is a mapping from X into X, the sufficiency conditions of 
Theorems l-3 are satisfied by C= X. So Theorems 2 and 3 generalize 
Theorem 3.2 in Yannelis and Prabhakar [22] and the Kalutani fixed 
theorem to non-compact and non-convex sets. 
Remark 4. From the proofs of Theorems 1 and 3, we can see that there 
still exists a fixed point of F if the condition that F has non-empty closed 
convex values on X is weakened to the condition that F has non-empty 
closed convex values on C. 
When a correspondence becomes a single-valued function, we have the 
following corollary by applying Theorem 3: 
COROLLARY 1. Let X be a non-empty subset in a locally convex 
Hausdorff topological vector space E. Suppose that f: X + E is a continuous 
function. Then the necessary and sufficient condition for the existence of a 
fixed point x* =f (x*) is that there exists a compact convex subset Cc X 
such that 
f(x)cC VXEC. (6) 
166 CXOQIANG TIAI\; 
Thus, the above corollary generalizes the Brouwer-Schauder fixed point 
theorem by relaxing the compactness and convexity conditions. 
Our theorems can prove the existence of a fixed point of a mapping 
which may have empty, non-compact, or non-convex values and whose 
domain may be non-compact and non-convex. The following simple 
example illustrates this. 
EXAMPLE 2. Let X= (-x, u) u (b, + ~TJ) c [w, which is non-compact 
and non-convex. Here 0 <a< h. Define an upper semi-continuous corre- 
spondence F: X-t R by, for all s E X, 
F&x)=(-a,,~-a]u[b, +x), 
which is non-compact and non-convex. So we cannot apply, say, the 
Kakutani fixed point theorem to prove the existence of the fixed point of F. 
However, if we take C= [b, b + 21, then C is a compact and convex 
interval, F is an upper semi-continuous correspondence with non-empty 
compact convex values on C, and 
F(x)nC#@ VXEC. 
Thus, by Theorem 3 and Remark 4, F has a fixed point. 
REFERENCES 
1. K. ARROW AND D. DEBREU, Existence of equilibrium for a competitive economy, 
Econometrica 22 (1954), 265-290. 
2. J. P. AUBIN, “Mathematical Methods of Game and Economic Theory,” North-Holland, 
Amsterdam, 1979. 
3. J. P. AUBIN AND I. EKELAND, “Applied Nonlinear Analysis,” Wiley, New York, 1984. 
4. K. C. BORDER, “Fixed Point Theorems with Applications to Economics and Game 
Theory,” Cambridge Univ. Press, London/New York, 1985. 
5. F. E. BROWDER, Fixed point theorem for multivalued mappings in topological vector 
spaces, Ma& Ann. 177 (1968), 283-301. 
6. G. DEBREU, A social equilibrium existence theorem, Proc. Nat/. Acad. Sci. (1. S. A. 38 
(1952), 386-393. 
7. K. FAN, Extensions of two fixed points theorems of F. E. Browder, Malh. Z. 112 (1969), 
234-240. 
8. B. HALPERN, “Fixed-Point Theorems for Outward Maps,” Doctoral Thesis, U.C.L.A., 
1965. 
9. B. HALPERN, Fixed point theorems for set-valued maps in infinite dimensional spaces, 
Math. Ann. 189 (1970), 87-98. 
10. B. HALPERN AND G. BERGMAN, A fixed-point theorem for inward and outward maps, 
Trans. Amer. Math. Sot. 130, No. 2 (1968), 353-358. 
Il. V. I. ISTRAIESCU, “Fixed Point Theory,” Reidel, Dordrecht, 1981. 
FIXED POINT THEOREMS 167 
12. M. C. JOSHI AND R. K. BOSE, “Some Topics in Nonlinear Functional Analysis,” Wiley, 
New York, 1985. 
13. S. KAKUTANI, A generalization of Brouwer’s fixed point theorem, Duke Math. J. 8 (1941), 
457-459. 
14. T. C. LIM, On asymptotic centres and fixed points of nonexpansive mappings, Cunad. 
J. Math. 32 (1980), 421430. 
15. E. MICHAEL, Continuous selections, I, Ann. qf Math. 63 (1956), 361-382. 
16. J. NASH, Equilibrium points in N-person games, Proc. Null. Acad. Sci. 36 (1950), 48-49. 
17. W. V. PETRYSHYN AND P. M. FITZPATRICK, Fixed-point theorems for multivalued non- 
compact inward maps, J. Math. Anal. Appl. 46 (1974), 756-767. 
18. W. SHAFER AND H. SONNENSCHEIN, Equilibrium in abstract economies without ordered 
preferences, J. Math. Econom. 2 (1975), 345-348. 
19. D. R. SMART. “Fixed Point Theorems,” Cambridge Univ. Press, London/New York, 1974. 
20. G. TIAN, “An Equilibrium Existence Theorem on Abstract Economies,” Working Paper 
No. 88-30. Department of Economics, Texas A&M University, 1988. 
21. G. 'TIAN, Equilibrium in abstract economies with a non-compact infinite dimensional 
strategy space, an infinite number of agents and without ordered preference, Econ. Lelt. 
33 (1990), 203-206. 
22. N. C. YANNELIS AND N. D. PRABHAKAR, Existence of maximal elements and equilibria in 
linear topological spaces, J. Math. Econom. 12 (1983), 223-245. 


file:///data/remote/core/dit/data/elsevier/pdf/267/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS8wMDIyMjQ3eDkxOTAyNzQ0.pdf

Fixed points theorems for mappings with non-compact and non-convex domains

Abstract

Similar works

Full text

Available Versions

Elsevier - Publisher Connector

Elsevier - Publisher Connector