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Internat. J. Math. & Math. Sci.VOL. 21 NO. 3 (1998) 467-h70

467

AN APPLICATION OF FIXED POINT THEOREMSIN BEST APPROXIMATION THEORY

H.K. PATHAKDepartment of Mathematics

Kalyan MahavidyalayaBhilai Nagar (M.P.) 490 006, INDIA

Y.J. CliODepartment of Mathematics

Gyeongsang National UniversityChinju 660-701, KOREA

S.M. KANGDepartment of Mathematics

Gyeongsang National UniversityChinju 660-701, KOREA

(Received February 7, 1996 and in revised form June 18, 1996)

ABSTRACT. In this paper, we give an application of Jungck’s fixed point theorem to best ap-

proximation theory, which extends the results of Singh and Sahab et al.

KEY WORDS AND PHRASES: Contractive operator, best approximant, compatible map-

pings, fixed point.

1991 AMS SUBJECT CLASSIFICATION CODES: 54H25, 47H10.

Let X be a normed linear space. A mapping T X X is said to be contractwe on X (resp.,on a subset C of X) if IITx- Tyll <_ IIx Yll for all x, y in X (resp., C). The set of fixed points of

T on X is denoted by F(T). If is a point of X, then for 0 < a _< 1, we define the set Da of best

(C, a)-approximants to consists of the points y in C such that

Let D denote the set of best C-approximants to . For a 1, our definition reduces to the set D of

best C-approximants to . A subset C of X is said to be starshaped with, respect to a point q E C

if, for all x in C and all A 5 [0,1], Az + (1 A)q C. The point p is called the star-centre of C. Aconvex set is starshaped with respect to each of its points, but not conversely. For an example, the

set C {0} [0,1] LI [1, 0] {0} is starshaped with respect to (0, 0) e C as the star-centre of C, but

it is not convex.

In this paper, we give an application of Jungck’s fixed point theorem to best approximation

theory, which extends the results of Sahab et al. [9] and Singh [10].By relaxing the linearity of the operator T and the convexity of D in the original statement of

Brosowski [1], Singh [10] proved the following:

Theorem 1. Let C be a T-invariant subset of a normed linear space X. Let T C C be a

contractive operator on C and let F(T). If D c_ X is nonempty, compact and starshaped, then

D f F(T) 0.In the subsequent paper [11], Singh observed that only the nonexpansiveness of T on D’ DU{}

is necessary. Further, Hicks and Humphries [4] have shown that the assumption T" C C can be

weakened to the condition T" OC C if y C, i.e., y E D is not necessarily in the interior of C,where OC denotes the boundary of C.

Recently, Sahab, Khan and Sessa [9] generalized Theorem as in the following:

468 H. K. PATHAK, Y. J. CHO AND S. M. KANG

Theorem 2. Let X be a Banach space. Let T, I X X be operators and C be a subset of Xsuch that T" OC C and 5: F(T)f3 F(I). Further, suppose that T and I satisfy

(1)

for all x, y in D’, I is linear, continuous on D and ITx TIx for all x in D. If D is nonempty,

compact and starshaped with respect to a point q F(I) and I(D) D, then Df3F(T)f3F(I) .Recall that two self-maps I and T of a metric space (X, d) with d(x, y) IIx- Yll for all x, y X

are said to be compatible on X if

.h_m d(ITx., TIx.)(= .h_rn IlITz. Tlz.ll) 0

whenever there is a sequence {x. } in X such that Tx., Ix. t, as n oo, for some tin X ([6]-[8]).We shall use N to denote the set of positive integers and CI(S) to denote the closure of a set S.

For our main theorem, we need the following:

Proposition 3. [8] Let T and I be compatible self-maps of a metric space (X, d) with I beingcontinuous. Suppose that there exist real numbers r > 0 and a (0,1) such that for all x, y X,

d(Tx, Ty) <_ rd(Ix, Iy) + a max(d(Tx, Ix), d(Ty, Iy)}.

,Then Tw Iw for some w X if and only if A f3{Cl(T(Ko)) n N} # $, where for each

go {x X" d(Tx, Ix) <_ - }.On the other hand, using this proposition, Jungck [8] proved the following:Theorem 4. Let I and T be compatible self-maps of a closed convex subset C of a Banach spaceX. Suppose that I is continuous and linear with T(C) c_ I(C). If there exists an a e (0,1) such

that for all x, y C,

IITx T,II _< alllx lull + (1 a) max{llTx lxll, IITy- IulI}, (2)

then I and T have a unique common fixed point in C.By using this theorem, we extend Theorem 2 as in the following:

Theorem 5. Let X be a Banach space. Let T, I X X be operators and C be a subset of Xsuch that T" c9C C and 5: F(T)N F(1). Further, suppose that Tand I satisfy (2) for all x, yin D’ Do t3 {5:} U E, where E {q X Ix=,Tx. q, {xo} C Do}, 0 < a < 1, I is linear,continuous on Do and T, I are compatible in D. If D is nonempty, cbmpact and convex, and

I(D) Do, then D f F(T) f F(I) .Proof. Let y D and hence Iy is in D since I(D) D. lurther, if y OC, then Ty is in Csince T(OC) c C. From (2), it follows that

[ITy- 5:[] ][Ty-

< allly I5:11 + (1 a) max{llTy I11, IITS:

which implies a]]Ty 5:1] <IIIy- 11 and so T is in D. Thus T maps Do into itself.

By hypothesis, we have 5: TS: I5:. Then Proposition 3 implies that

A {CI(T(K.))’n N} # 0.

Suppose that w 6 A. Then for each n N, there exists y. T(Ko) such that d(w, I/o) < 1/n.Consequently, for such n, we can and do choose x. K. such that d(w, Tx.) < 1In and so Tx. w.

But since x. 6 K., d(Tx., Ix.) < 1/n and therefore Ixo w. Thus we have

lira Iz, lira Tx, w. (3)

FIXED POINT THEOREMS 69

Therefore, for a sequence {z} in D= the existence of (3) is guaranteed whenever D= C K,. Moreover,w 6 E. Since I and T are compatible and I is continuous, we have lirn_am TIz,, Iw and

lim._am I:x,., Iw. By (2), we have

IITIx. Y:ll []TIx. Vl] < al]Ix. I]1 + (1 a)max{I]Tlx.

which implies, as n

IIIw ll -< alllw- ll.Hence Iw . By (2) again, we have

wgi IIT- 11-< (- )IIT- 11, and T .Next, we consider

IlTw T.II < allIw Ix-II + (1 -a)max{llVw -/toll, IITx.which gives II wll < all wll as n oo, and so w, i.e., w Iw Tw. By Theorem 4, w

must be unique. Hence E (w}. Then D; D U (w} D’Let {k. } be a monotonically non-decreasing sequence of real numbers such that 0 _< k. < 1 and

li".am k. 1. Let {x,} be a sequence in D’ satisfying (3). For each n e N, define a mapping

T,’D’,, D’,, byT,,x, k.Tx, + (1- k.)p. (4)

It is possible to define such a mapping T, for each n N since D’ is starshaped with respect to

p6F(I).Since I is linear, we have

T.Ix, k.TIx, + (1 k.)p, IT,x, k.ITx, + (1 k,.,)p.

By compatibility of I and T, we have for each n q N,

0 <_ lim liT.Ix, IT.x,< k. lira IIrlz, ITxjI + lira (1 -/)llp-

=0

and so

lira lIT.Iz, IT.z, o

whenever lim,._.am Ixj lim3_am T.x w since we have

lira T,.,x, k, lira Tx, + (1 k,.,)w

Thus, I and T, are compatible on D’ for each n and T,,(D’) C D’,, I(D).On the other hand, by (2), for all x, y 6 D’, we have, for all j > n and n fixed,

liT: T.II<_ alllz- I11 + (1 -a)max{llT- I11, IIT- I11}

<_

(X k.)llT- Pll / IIT.- I11}-

470 H.K. PATHAK, Y. J. CHO AND S. M. KANG

Hence for all j

_n, we have

liT,a:- T.Yll < alllx- Iyll ++ IIT,z lall, (1 k,)llTy Pll + IIT.,y IylI}

Thus, since lim3_00 k 1, from (5), for every n E N, we have

<3--*00

+which implies

()

liT,a: T.,yll- allIx -/Yll / (- a) max{llT.,x Xxll, IIT. I11}

for all x, y E D. Therefore, by Theorem 4, for every n N, T, and I have a unique common fixed

point x, in D’, i.e., for every n N, we have

F(T,) N F(I) {x, }.

Now, the compactness of D= ensures that {x} has a convergent subsequence {x,,, } which converges

to a point z in D=. Since

x,,, T,,x, k,,Tx,,, + (1 k,,)q (6)

and T is continuous, we have, as x in (6), z Tz, i.e., z D= f3 F(T).Further, the continuity of I implies that

Iz I(,li_m z,.,,)= ,lim Ix,.,, ,li.m_ z,.,, z,

i.e., z F(I). Therefore, we have z D a F(T) N F(I) and so

D F(T) F(I) # .This completes the proof.

ACKNOWLEDGEMENT. The first author was supported in part by U.G.C., New Delhi, India,

and the second and third authors were supported in part by the Basic Research Institute Program,Ministry of Education, Korea, 1996, Project No. BSRI-96-1405.

REFERENCES

1. B. Brosowski, Fixpunktsatze in der Approximation-theorie, Mathematica (Cluj) 11 (1969), 195-220.

2. B. Fisher and S. Sessa, On a fixed point theorem o.f GreguJ, Internat. J. Math. & Math. Sci. 9(1986), 23-28.

3. M. Gregu, A fixed point theorem in Banach spaces, Boll. Un. Mat. Ital. (5)17-A (1980), 193-198.

4. T. L. Hicks and M. D. Humphries, A note on fixed point theorems, J. Approx. Theroy 34 (1982),221-225.

5. G. Jungck, An ifffixed point criterion, Math. Mag. 49(1) (1976), 32-34.6. G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. & Math. Sci. 9

(1986), 771-779.7. G. Jtmgck, Common fix points .or commuting and compatible maps on compacta, Proc. Amer.

Math. So. 103 (1988), 977-983.8. G. Jungck, On a fixed point theorem of Fisher and Sessa, Internat. J. Math. & Math. Sci. 13

(1990), 497-500.9. S. A. Sahab, M. S. Khan and S. Sessa, A result in best approximation theory, J. Approx. TheoryS (19),

10. S. P. Singh, An application of a fixed point theorem to approximation theory, J. Approx. Theory25 (1979), 89-90.

11. S. P. Singh, Application of fixed point theorems in approximation theory, "Applied NonlinearAnalysis" (Edited by V. Lakshmikantham), Academic Press, New York, 1979.

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