Resultate der Mathematik (3) 1-6 Birkhauser Verlag, Basel

Fixed point theorems in complete metric spaces

J. ACHARI

o. Introdudion.

In recent years many extensions and generalizations of Banach fixed point theorem had been done by many authors. But in all the cases the mappings under consideration involve only two points of the space. Until, recently Pittnauer [6] and also Rhoades [8] studied fixed point theorems for contractive type mappings involving three points of the space. Pittnauer [7] also studied mappings involving four points of the space.

In this paper we have studied a fixed point theorem for a pair of contractive type mappings involving four points of the space. We then extend this result to family of mappings. Finally we have shown that our result contains as special cases that of Wong [10] and also Hardy and Rogers [2], Reich [9], Kannan [3] and Maiti et al. [4,5].

Let (X, d) be a complete metric space. Let cf>i : P ~ [0, (0) [P is the range of d and P is the closure of P] be upper semicontinuous function from the right on P and satisfies the condition

cf>i(t)<tj5 for t>O and cf>i(0)=0,i=1,2,3,4,5. (0.1)

Also let f and g be mappings of X into itself such that

1. Fixed point theorems.

To establish the following theorem we use the method of Boyd and Wong [1] with necessary modifications as required for the more general settings.

2 J. ACHARI

THEOREM 1.1. If f and g be mappings of X into itself satisfying (0.2), then f and g have a common unique fixed point.

Proof. Let x, y E X and we define

u1 = gy, u2 = fx,

Then (0.2) takes the form

d(fgy, gfX)~<Pl[d(fx, gy)]+<pJd(fx. gy)]+<p:\[d(fx, gy)]. (1.1)

Let Xo E X be arbitrary and we construct a sequence {x,,} define by

fx,,-l = x", n = 1,2, . . ..

Let us put x = x,, - l, Y = x" in (1.1), then we have

or

Let n be even and set Cn = d(x,,-b x,,). Then

From (1.3) it is obvious that Cn decreases with n and hence Cn ~ C say, as n ~ 00. If possible, let C > O. Then since <Pi is upper semicontinuous, we obtain in the limit as n ~ 00

C ~ <Pl( C) + <Pi C) + <P3( C)

<3/5(C)

which is impossible unless C = O. Next, we show that the sequence {xn } is Cauchy. Suppose that it is not so. Then

Fixed point theorems in complete metric spaces 3

there exists an e > 0 and sequences of integers {m(k)}, {n(k)} with m(k) > n(k) ~ k

such that

k = 1,2,3 ..... (1.4)

If m(k) is the smallest integer exceeding n(k) for which (1.4) holds, then from the well ordering principle, we have

(1.5)

Then

< C",(k) + e < C k + e,

which implies that dk ~ e as k ~ 00 . Now the following cases are to be considered

a) m is even and n is odd, b) m and n are both odd, c) m is odd and n is even, d) m and n are both even.

Case (a)

:0;;: Cm + 1 + Cn +! + d(gx"" fXn)

letting k ~ 00 we have

e $.3/5(e)

4 J.ACHARI

This is a contradiction if e > O. In the case (b) we have

dk = d(x"" xn)::;; d(x"" xm+1) + d(x",+2, Xm+l) + d(x",+2, Xn+l) + d(Xn, Xn+l)

::;; Cm+2 + Cm+1 + Cn+! + d(gx",+I, fXn)

::;; Cm +2 + Cm +1 + Cn +1 + <Pl[d(x",+b Xn)]+ <P2[d(Xn, fXn-l)]

+ <P3[d(x",+I, gx",)] + <P4[d(Xn, gXm)] + <Ps[d(x"" fXn-l)]

[by putting Ul = Xn, U2 = x",+b U3 = Xn-l U4 = x",]

~ Cm+2 + Cm+1 + Cn+ 1 + <PI (dk + Cm + 1) + <P4( dk + Cm+1) + <Ps(dk + Cn+1).

Letting k ~ 00 in the above inequality we obtain e ::;; ~e, which is a contradiction if e > o. Similarly, the cases (c) and (d) may be disposed of. This leads us to conclude that the sequence {Xn} is Cauchy and since X is complete so there exists a point z E X such that Xn ~ z as n ~ 00. We shall now show that gz = z = fz. Putting U1 = Xn-b U2 = Z, U3 = Xn+), U4 = Xn in (0.2) we get

d(fx..-b gz)::;; <Pl[d(x..+b z )]+ <P2[d(x..-t. ix..+l)] + <P3[d(z, gXn)]

+ <P4[d(Xn-b gXn)] + <Ps[d(z, fXn+l)].

letting n ~ 00 we get d(z, gz) ::;;0, which is a contradiction and hence z = gz. In the same way it is possible to show that fz = z. Thus z is a common fixed point of f and g. If possible let there be another point w( =!= z) such that fw = w = gw. Then putting Ul = U4 = z and U2 = U3 = w in (0.2) we have

d(z, w)::;;<Pl(d(z, w))+<P2(d(z, w))+<P3(d(z, w))

<~ d(z, w)

which is a contradiction. Hence z = w. Finally we note that f and g can not have any other fixed point apart from the common fixed point z. If possible, there exists Wl(=!=Z) such that gW1 =Wl. Then putting U1 =U3 =Z, U2=U4=W1 in (0.2) we get

d(z, WI) = d (fz, gw1)::;; <PI (d (z, WI) + <P4(d (z, WI)) + <Ps(d (z, WI))

<~ d(z, WI)

which is impossible, implying z = WI. Similarly it can be shown that f can not have any other fixed point besides the common fixed point. This completes the proof of the theorem.

Fixed point theorems in complete metric spaces 5

Theorem 1.2. Let fk (k = 1, 2, .... n) be a family of mappings of X into itself. If fk satisfy the conditions

(i) fd2 ... fn commutes with every fk' (ii) d(fd2 ... fnUb fJn~l . .. fl U2)

~ <Pl[d(u1, u2)] + <P2[d(ub fd2 ... fnU3)] + <P3[d(ub fJn~1 ... fl u4)] + <P4[d(ub fJn ~I.·. fl U4)] + <PS[d(Ub fd2· · · fn U3)]

for UbU2,U3,U4EX and <pi(t)(i=1,2 ... 5) satisfies condition (0.1). Then {A} have a unique common fixed point.

Proof. Let f = fd2 .. . fn and g = fJn~l .. . fb then (ii) Takes the form

(iii) d(fu b gU2)~<Pl[d(Ub U2)] + <P2[d(ub fu3)] + <P3[d(U2, gu4)]

+ <P4[d(u1, gU4)] + <PS[d(U2, fu3)].

By Theorem 1.1, f and g have a unique common fixed point z. Then fz = gz = z. For any fb fk (fz) = fkZ. By the assumption, f(fkZ) = fkZ. So fkZ is a fixed point of f and Z is a fixed point of g. By putting U1 = U3 = fkZ and U2 = U4 = Z in (iii) we have

d(fkZ, gz) = d(ffkZ, gz) ~ <Pl[d(fkZ, z)] + <P2[d(fkZ, ffkZ)]

+ <P3[d(z, gz )]+ <P4[d(fkZ, gz)] + <Ps[d(z, ffkZ)]

~ <Pl[d(fkZ, z)] + <P4[d(fkZ, Z )]+ <Ps[d(fkZ, z)]

<~ d(Az, z)

which is a contradiction. Hence fkZ = z(k = 1, 2, ... n). This means that Z is the common fixed point of the family {fd. It can easily be shown that Z is the unique common fixed point of {fk}.

2. Some special cases.

In this section we shall show that our result contains some well known results as special cases. a) If in Theorem 1.1 we put Ul = U3 and U2 = U4, then we get the result of Maiti et

al. [4]. If we define the function <Pi (t) by <PI (t) = a l t, <P2(t) = a2t, <P3(t) = a3t, <P4(t) = a4t, <Ps(t) = ast, then we get the following results as special cases.

b) If we put U1 = U3 and U2 = U4, then we have generalized contraction mapping of Wong [10].

6 J.ACHARI

c) If we put a l = a2 = a3 = 0, a 4 = as = a and U l = U3, U2 = U4, then we get the result of Maiti et al. [5].

d) Putting f = g in Theorem 1.1 and Ul = U3, U2 = U4, we have the theorem of Hardy and Rogers [2].

e) Putting f= g in Theorem 1.1 and Ul = U3, U2 = U4 and a l = a, a 2 = b, a 3 = c, a 4 = as = 0, we have the results of Reich [9].

f) Putting f = g in Theorem 1.1 and a l = a 4 = as = 0, a 2 = a3 = a and U l U3, U2 = U4

we have Kannan [3].

Acknowledgement. The author gratefully acknowledges the support of a fellow­ship from the C.N.R. (Italy) and also expresses his sincere thanks to Prof. Dr. F. Pittnauer for his kind help.

REFERENCES

[1] D. W. BoYD and J. S. W. WONG, On non-linear contractions, Proc. Amer. Math. Soc., 20 (1969), 458-464.

[2] G. E. HARDY and T. D. ROGERS, A generalization of a fixed point theorem of Reich, Canad. Math. Bull., 16 (1973), 201-206.

[3] R. KANNAN, Some results on fixed points, Bull. Cal. Math. Soc., 60 (1968),71-76. [4] M. MAm, J. ACHARI and T. K. PAL, Mappings having common fixed points, Tunghai J., 18

(1977), 271-274. [5] --, Mappings having common fixed points, Pure and Appl. Math. Sci., 3 (1976), 101-104. [6] F. PfITNAUER, Ein fixpunksatz in metrischen Raumen, Archiv der Math., 26 (1975), 421-426. [7] --, A fixed point theorem in complete metric spaces, to appear in Periodica Math. Hungarica. [8] B. E. RHOADES, A fixed point theorem in metric spaces, to appear. [9] S. REICH, Kannan's fixed point theorems, Boll. U.M.I., 4 (1971), 1-11.

[10] C. S. WONG, Common fixed points of two mappings, Pacific J. Math., 48 (1973), 299-312.

Munschifdanga, P. O. Raghunathpur Dist. Purulia (WB.), India

Eingegangen am 2. Mai 1978