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Internat. J. M,th. & M,th. Sci.VOI. [8 NO. 3 (1995) 613-616

BEST APPROXIMATION AND FIXED POINTS INSTRONG M-STARSHAPED METRIC SPACES

M. A. AL-THAGAFI

D(’l)aztment of 5IathematicsKing Abdul Aziz University

P. O. Box 30608, Jeddah 21487, Saudi Azabia

(Received January 11, 1994)

613

ABSTRACT. We introduced strong M-starsh,qed ,netric spaces. For these spaces, we obtainedtwo fixed-point theorems generalizing a result of W. G. Dotson, aud two theorems extending and

subsuming several known results on the existence of fixed points of best approximation.

KEY WORDS AND PHRASES. Best aI)poximation, strong M-starshaped netric spaces, fixed

i)oints.1991 AMS SUBJECT CLASSIFICATION CODES. 41A65, 54H25, 47H10.

1. INTRODUCTION.Let X be a metric space, T: X---X. K and D subsets of X, and p E X. T is "),-nonexpansive

on D if d(Tx, Ty)<_d(.r,!,) for every x,y D with d(x,y)<_ "v. T is "y-contraction on D if

d(Tx, Ty) <_ Ad(x,y) for some A [0,1) and for every x,y D with d(.r,y) _< 7. X is a 7-chainablemetric space if for any pair x,y

_X, there exists a finite chain of points x0. x, -,x,_ ,x,, in X

with x()=x and x,,=.q such that d(x,_ ..r,)_< "), for = 1,2,.- .,. The set of best K-approximations to p, denoted by BK(p), is the set of all x G K such that d(x, p) 6(p,K), where6(p,K) inf.. e Kd(z,p).

Brosowski [1] prowd that if X is a nomed linear space, T:X--,X is nonexpansive with a

fixed point p, K C X with T(K) C K, then T has a fixed point in B,<(p) provided that BK(p) is

nonempty, compact and convex. Singh ([6], Theorem 1) relaxed the convexity of BK(p) be

starshapedness. However, Hicks and Humphries ([4], p. 221) showed that the conclusion ofSingh’s result still holds whenever T(K)C I( is replaced by T(OK)C If. Subrahmanyam ([81,Theorem 3) proved that if X is a normed linear space, T:X--X is 6(p,K)-nonexpansive with a

fixed point p, K C X with T(K) C K. then T has a fixed point in BK(p) provided that K is a

finite-dimensional subspace of X. However, Singh ([7], Theorem 1) showed that the conclusion of

Subrahmanyam’s result still holds whenever the finite-dinensionality of K is replaced by thefollowing conditions:

(i) BK(P) is nonempty, compact and starshaped, and

(ii) T is continuous on B:(p).Recently, Sahab and Khan ([5J. Theorem 3.1) showed that Singh’s second result still holdswhenever X is a strong convex metric space (se" Definition 3.1 below).

Our aim, in this pal,er, is to establish results extending and subsumang the above bestapproximation results. To do this. we introdt(e 3l-starshaped and strong 3l-starshaped metric

614 M. A. AL-THA(;AFI

spaces. Convex and starshaped m’t,ric spa’. of Takahashi [9] are examples of /-starshapedmetric spaces. Then w’ obtained two fixed-p,int th,orems generalizing a result of W. G. Dotson.We prove the first using a r,sult of M. Edelstein and the second using Banach’s contraction

principle. Using the first theorem, we estallislwd our results on the existence of fixed points of

best approximation.

For later us’, we state the following result of M. Edelstein ([3], Theorem 5.2).THBOM 1.1. Let D be a complete and -chainable metric space. If S:DD is 2-

contraction, then S has a mfique fixed point in D.2. NBD POINTS IN STRONG M-STARSHAPBD MBTC SPACES.

DEFINITION 2.1. Let X be a metric bp,,, M C X and I [0,1].(a) X is M-starshaped if there exists a mapping W:X x 3I x I, satisfying

d(x, W(y,q,1)) ,d(=’,y) + (1 1)d(x,q)

for every x, g X, all q M and all , I.

(b) X is strong M-starshaped if it is M-starshaped and V satisfies

d(iV(x,q,A),W(y,q,$)) $d(x,y)

for every x, y X, all q M and all k I.

(c) X is (strong) convex if it is (strong) X-starshaped. X is starshaped if it is {q}-stshaped for sone q X.

Convex and starshaped metric spaces were introduced by Takahashi [9]. Each normed linem"

space X is a strong convex metric space with W defined by W(x,q,$)= Xx + (1- X)q for every

x,q X and all $ I.DEFITION 2.2. Let X be a M-starshaped metric space. A subset D of X is q-starshaped

if there exists q DM with W(Dx {q} x I)C D. A q-starshaped subset of a convex metric

space is called starshaped.THEOM 2.3. Let X be a strong M-starshaped metric space and D C X. If D is

compact, -chMnable and q-starshaped, and T:DD is ?-nonexpansive, then T has a fixed point

in D.PROOF. For each positive integer n, let .- n and T.x W(Tx, q,$.) for all x D.

By the -nonexpansiveness of T on D, each T,, satisfies

d(T.x,T.y) d(W(Tx, q,$.),W(Ty, q,$.)) $.d(Tx, Ty) $.d(x,y)

for every x,y D with d(x,y) 7. Note that D is q-starshaped and T:DD. So each T. is a 7-contraction selfmap of D. Since D is 7-chainable, Theorem 1.1 shows that each T. has a unique

fixed point x. D. By the compactness of D, there exists a subsequence {x.,} of {x.} with

lim,_x., x0 D. Since

d(Tx.,,x.,) d(Tz,,,,W(Tx.,,q,2.,)) (1 $.,)d(Tx.,,q)for all i, then lim,_d(Tx.,,x.,)= O. Now, the -nonexpansiveness of T on D implies its

continuity, and hence x0 is a fixed point of T.The following is a result of Dotson ([2], Theorem 1).COROLLARY 2.4. Let X be a normcd linear space and D C X. If D is compact and

starshaped and if T:DD is nonexpansive, then T has a fixed point in D.

BEST ,\PPROXIYb\TION AND F1XEI) POINT THEOREMS 615

PROOF. For ev’rv ";, > 0, D is -chainalle and T is ,-nonexpansiv’. [-1

THEOREM 2.5. Let X be a strong M-st,ushaped metric space and D C X. If D is compact

and q-starslaped and if T: D+D is nonexpansiw,, then T has a fixed point in D.

The proof of Th’orem 2.5 is simil,r to the one giw’n for Theorem 2.a; however, we use

Banach’s contra tion principle insteal of Theorem 1.1. Note that Corollary 2.4 follows also from

Theorem 2.5.

3. BEST APPROXIMATION IN STRONG M-STAHAPED METRIC SPACES.LEMMA a.1. Let X be a strong M-starshapcd metric space, K C X and p X. If BK(P) is

q-starshaped, then Ba-(p) is (p,K)-chainable.

PROOF. For z,y B,-(p), let

y, q, .ra =y.

Since B(p)is q-starshaped, then x0, a’,.q,.r belong to Bh.(p). Now, the strong M-starshapedness of X implies that

Therefore BK(p)is a(p,K)-chainable.

LEMMA 3.2. Let X be a M-starshaped metric space, K C X and p X. Then

BK(p) C OK K.PROOF. Let y B(p} and let 2, Rr each positive integer Then

d(p,W(y,v,.k,)) .k,,a(p,y) < a(v,K)

which implies that }V(y, p, 2,} K for every n. Since

d(y, tV(y,p,L,)) 5 ( ,,)d(y, p)= (

for all n, then lim,W(y, p,,} y. So each neighborhood of y contains at least one W(y, p,2,),hence y OK.THEOM 3.3. Let X be a strong M-sarshaped metric space, T:XX, K C X, d

p 6 X a fixed point of T. If Bg(p) is compact and q-starshaped, T(K)C K, ad T is 6(p,K)-nonexpansive on BK(P)O {p}, then T has a fixed point in Bu(p).

PROOF. Let y B,;(p). Then Ty G K and, by the 3(p.K)-nonexpansivencss of T on

BK(p)O {p}, d(Ty, p) 5 d(y,p). Thus Ty Be(p) and so T:Bu(p)Bh.(p ). Now, Theorem 2.3,

with D Bg(p), and Lemma 3.1 show that T has a fixed point in BK(p).THEOM 3.4. Let X be a strong M-starshaped metric space, T:XX,K C X, d

p M a fixed point of T. If Bu(p) is compact and q-starshaped, T(OKK)C K, and T is

6(p,K)-nonexpansive on Bu(p)U {p}, then T has a fixed point in

PROOF. Let yGBu(p). Then yGOKK by Lemma 3.2. Since T(OKK) CK, then

Ty 6 K. Now, the a(p,K)-nonexpansiveness on Bu(p)U {p} implies that Ty 6 Bg(p}. Therefore

T:Bg(p)BK(p). Now, Theorem 2.3. with D Bu(p), and Lenana 3.1 sh,,w that T has a fixed

point in BK(p).

616 M.A. AL-THACAFI

COROLLARY 3.5. Let X 1)e a strong conv,x netric space, T:X--,X, K C X, and p E X a

fixed point of T. If B(p) is compact and tarhai),(l, T(OKK)C K. anal T is (p,K)-nonexpansive on BK(I, U {p}, th(’n T has a f’ix,’d point in B, (p).

All best approximation results mentiom.d in section hllow from Corollary 3.5. Moreover,if T(OK K) C K is r,lla,-(1 by T(K) C K il Corollary 3.5. we will have Sahab and Khan result

([5], Theorem 3.1) and it will follow from Tlwoem 3.3. Note that their assumption for the

contimdtv of T on B-(p) is implied by th 6(p, K)-nonexpansiveness of T on B((p)U {p}.

REFERENCES

BROSOWSKI, B., Fixpunkts/itze in der approximations-theorie, Mathem.at,ca (Cluj) 11(I 969), 195- 220.

DOTSON, JR., W.G., On fixed pointb oi nonexpansive mappings in nonconvex sets, Proc.Amer. Math. Soc. 38 (1973), 155-156.

EDELSTEIN, M., An extension of Banach’s contraction principle, Proc. Amer. Math. Soc.12 (1961), 7-10.

HICKS, T.L. and HUMPHRIES, M.D., A note on fixed-point theorens, J. Approx. Theory34 (1982), 221-225.

5. SAHAB, S.A. and KHAN, M.S., Best approximation in spaces with convex structure, BMl.Inst. Math. A cad. S’tnica 17 (1989), 59-63.

6. SINGH, S.P., An application of a fixed point theorem to approximation theory, J. Approx.Theory 25 (1979), 89-90.

7. SINGH, S.P., Applications of fixed point theorems in approximation theory, AppliedNonlinear Analys,s (Ed., V. Lakshmikantham), Academic Press, New York (1979),389-397.

8. SUBRAHMANYAM, P.V., An application of a fixed point theorem to best approximation,J. Approx. Theory 20 (1977), 165-172.

9. TAKAHASHI, W., A convexity in metric spaces and nonexpasive mappings, Kodai Math.Sere. Rep. 22 (1970), 142-149.

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