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Necessary and sufficient condition for common fixed point theorems.

1. INTRODUCTION AND PRELIMINARIES

Prior to 1968 all work involving fixed points used the Banach contraction principle. In 1968 Kannan [9] proved a fixed point theorem for a map satisfying a contractive condition that did not require continuity at each point. This paper was a genesis for a multitude of fixed point papers over the next two decades (see for example, [10] for a listing and comparison of many of these definitions). A number of these papers dealt with fixed points for more than one map. In some cases commutativity between the maps was required in order to obtain a common fixed point. Sessa [12] coined the term weakly commuting. Jungck [4] generalized the notion of weak commutativity by introducing the concept of compatible maps and then weakly compatible maps [6]. There are examples that show that each of these generalizations of commutativity is a proper extension of the previous definition. Also, Jungck established necessary and sufficient conditions for the existence of common fixed points for commuting mappings. In [3], Jungck proved the following theorem.

Theorem 1.1. Let (X, d) be a complete metric space. A continuous mapping f : X [right arrow] X has a fixed point in X if and only if there exists a mapping g: X [right arrow] f(X) which commutes with f and satisfies the following inequality

d(gx, gy) [less than or equal to] [alpha] d(fx, fy)

for each x, y [member of] X, where [alpha] [member of] (0,1). In fact, f and g have a unique common fixed point.

Fisher [2] extended Theorem 1.1 to three self maps as follows:

Theorem 1.2. Let (X,d) be a complete metric space. Continuous mappings f,g: X [right arrow] X have a common fixed point in X if and only if there exists a continuous mapping T: X [right arrow] f (X) [intersection] g(X) which commutes with f and g and satisfies the following inequality

d(Tx, Ty) [less than or equal to] [alpha]d(fx, gy)

for each x,y [member of] X, where [alpha] [member of] (0,1).In fact, f, g and T have a unique common fixed point in X.

In 1995, Rhoades et al. [11] proved the following theorem in a more general setting.

Theorem 1.3. Let (X,d) be a complete metric space. Continuous mappings f,g: X [right arrow] X have a common fixed point in X if and only if there exists a continuous mapping T: X [right arrow] f (X) [intersection] g(X) which is compatible with f and g and satisfies the following inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for each x,y [member of] X, where [alpha] [member of] (0,1) and w: [R.sup.+] [right arrow] [R.sup.+] is a continuous function such that 0 < w(r) < r for all r > 0 with r - w(r) is an increasing function for each r > 0. In fact, f, g and T have a unique common fixed point in X.

Remark 1.1. Let [phi]: [R.sup.+] - [R.sup.+] be a continuous, non decreasing function such that [phi](r) < r for all r > 0. Define, [phi]: [R.sup.+] [right arrow] [R.sup.+] by [phi](t)= t - w(t), where w: [R.sup.+] [right arrow] [R.sup.+] is a function considered in Theorem 1.3, then p is a continuous, non decreasing function such that [phi](r) < r for all r > 0. Conversely, if we define, w: [R.sup.+] [right arrow] [R.sup.+] by w(t)= t - [phi](t), where [phi]: [R.sup.+] [right arrow] [R.sup.+] is continuous, non decreasing function such that [phi](r) < r for all r > 0, then w is a continuous function such that 0 < w(r) < r, for all r > 0 with r - w(r) is an increasing function for each r > 0. Hence the contractive condition of Theorem 1. 3 can be replaced by the following condition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

for all x, y [member of] X.

In [6], Jungck improved Theorem 1.1, by eliminating continuity and self map requirements, weakening the commutativity condition and, by generalizing the underlying space to a semi metric space.

Recently, Al-Thagafi and Shahzad [1] (see also, [8]) gave the following definition which is a proper generalization of nontrivial weakly compatible maps which do have coincidence points.

Definition 1.1. Two selfmaps f and g of a set X are said to be occasionally weakly compatible (owc) if there is a point x in X which is a coincidence point of f and g at which f and g commute.

Definition 1.2. Let X be a set. A symmetric on X is a mapping d: X x X [right arrow] [0, [infinity]) such that

d(x,y )= 0 if x = y, and d(x,y)= d(y,x) for x,y [member of] X.

In the sequel, we shall also need the following lemma from [8].

Lemma 1.1. Let X be a set, f, g owc selfmaps of X. If f and g have a unique point of coincidence, w := fx = gx, then w is the unique common fixed point of f and g.

In this paper, we provide necessary and sufficient conditions for the existence of common fixed points for three self maps in a symmetric space which is more general than a metric space, dropping the condition of continuity on any of the map involved therein. Our results improves and generalize the results of [2], [3], [6], [11] and many other comparable results in the literature.

2. COMMON FIXED POINT THEOREMS

The following result generalizes Theorem 2.1 of [6] and the main result of [3].

Theorem 2.1. Let (X, d) be a symmetric space with symmetric d and D [subset or equal to] X. Mapping T : D [right arrow] X has a fixed point if and only if there exists mapping f : D [right arrow] D [intersection] TD satisfying the inequality

d(fx,fy) [less than or equal to ] ad(Tx,Ty) + b max{d(fx,Tx),d(fy,Ty)} + c max{d(Tx, Ty), d(Tx, fx), d(Ty, fy)} (2.1)

for all x, y [member of] D, and the pair {f, T} is occasionally weakly compatible, where a, b, c [less than or equal to] 0, a + b + c = 1 and a + c < [square root of a]. Indeed, f and T have a unique common fixed point.

Proof. Suppose that T has a fixed point a in D. Define f: D [right arrow] X by fx = a for x [member of] D. Thus, f(D) = {a} = {Ta} [subset or equal to] T(D). This implies that there exists an x [member of] X such that fx = Tx and that fTx = Tfx. Hence the pair {f,T} is occasionally weakly compatible. Moreover (2.1) holds trivially, and the given conditions are necessary. To prove sufficiency, there exists a point u [member of] D such that fu = Tu. Suppose that there exists another point v [member of] D for which fv = Tv. Then, from (2.1),

d(fu, fv) [less than or equal to] [less than or equal to] ad(fu, fv) + b max{0,0} + c max{d(fu, fv), 0,0} = (a + c)d(fu, fv).

Since a + c < 1, the above inequality implies that d(fu, fv) = 0, which, in turn implies that fu = fv. Therefore fu is unique. From Lemma 1.1, f and T have a unique common fixed point.

Corollary 2.1. Let X be a set with a symmetric d, and let D [subset or equal to] X. A mapping f: D [right arrow] X has a fixed point if and only if there exists a mapping g: D [right arrow] D [intersection] f(D) such that g(D) is closed, the pair {f,g} is occasionally weakly compatible and there exists an [alpha] [member of] (0, 1) such that

d(gx, gy) [less than or equal to ] [alpha]d(fx, fy)

for each x,y [member of] D. In fact, f and g have a unique common fixed point.

Theorem 2.2. Let X be a set with a symmetric d. Mappings f,g: X [right arrow] X have a common fixed point in X if and only if there exists a mapping T: X [right arrow] fX [intersection] gX satisfying (1.1) and such that the pairs {f,T} and {g,T} are occasionally weakly compatible. In fact, f, g and T have a unique common fixed point in X.

Proof. First we show that the condition is necessary. For this, let fz = gz = z for some z in X. Define, T: X [right arrow] f X [intersection] gX by Tx = z. Then the pairs {f,T} and {g,T} are occasionally weakly compatible and inequality (1.1) holds for all x [member of] X. Conversely, suppose that there exists a map T: X [right arrow] f X [intersection] gX satisfying (1.1) such that the pairs {f, T} and {g, T} are occasionally weakly compatible. Suppose there exist points x, y in X such that Tx = fx and Ty = gy. We claim that Tx = Ty. By (1.1),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

that is, Tx = fx = Ty = gy. Moreover, if there is another point z such that Tz = fz, then, using (1.1) it follows that Tz = fz = Ty = gy, or Tx = Tz and w = Tx = fx is the unique point of coincidence of T and f. By Lemma 1.1, w is the only common fixed point of T and f. By symmetry there is a unique point z [member of] X such that z = Tz = gz. Suppose that w [not equal t] z. Using (1.1),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

a contradiction. Therefore w = z and w is a common fixed point. By the preceding argument it is clear that w is unique.

Theorem 2.3 of [11] is also a special case of Theorem 2.2.

Corollary 2.2. Let X be a set with a symmetric d. Mappings f, g: X [right arrow] X have a common fixed point in X if and only if there exists a mapping T: X [right arrow] fX [intersection] gX satisfying the inequality

d(Tx,Ty) [less than or equal to] [alpha] max {d(Tx, fx),d(Ty,gy),d(fx,gy), 1/2[d(fx,Ty) + d(fy,Tx)]},

and the pairs {f, T} and {g, T} are occasionally weakly compatible. In fact, f, g and T have a unique common fixed point in X.

Proof. Follows from Theorem 2.2, by defining [phi](t) = [alpha]t, t [greater than or equal to] 0, and [alpha] [member of] (0,1).

Corollary 2.3. Let X be a set with a symmetric d. Mapping T: X [right arrow] X has a fixed point in X if and only if there exists a mapping f: X [right arrow] TX satisfying the inequality

d(fx,fy) < a max j d(fx,Tx),d(fy,Ty),d(Tx,Ty),

1/2[d(fx,Ty) + d(fy,Tx)}},

and the pair {f, T} is occasionally weakly compatible. In fact, f and T have a unique common fixed point in X.

Corollary 2.4. Let X be a set with a symmetric d. Mappings f,g: X [right arrow] X have a common fixed point in X if and only if there exists a mapping T: X [right arrow] fX [intersection] gX satisfying the inequality

d(Tx,Ty) [less than or equal to] [phi](max/ d(Tx, fx),d(Ty,gy),d(fx,gy), 1/2[d(Tx,gy) + d(Ty,fx)]^ (2.2)

where [phi]: [R.sup.+] [right arrow] [R.sup.+], [phi] upper semi continuous with [phi](t) < t for each t > 0, and the pairs {f, T} and {g, T} are occasionally weakly compatible. In fact, f, g and T have a unique common fixed point in X.

Received: May 07, 2009.

REFERENCES

[1] M. A. Al-Thagafi and Naseer Shahzad: Generalized I-nonexpansive self-maps and invariant approximations, Acta Math. Sinica, 24(2008), No. 5, 867-876.

[2] B. Fisher: Mappings 'with a common fixed point, Math. Sem. Notes, 7(1979), 81-84.

[3] G. Jungck: Commuting mappings and fixed points, Amer. Math. Monthly, 83(1976), 261-263.

[4] G. Jungck: Compatible mappings and common fixed points, Internat. J. Math. Math. Sci., 9(1986), No. 4, 771-779.

[5] G. Jungck: Common fixed points for commuting and compatible maps on compacta,Proc. Amer. Soc., 103(1988), 977-983.

[6] G. Jungck: Common fixed points for noncontinuous nonself maps on nonmetric spaces, Far East J. Math. Sci., 4(1996), 199-215.

[7] G. Jungck and B. E. Rhoades: Fixed point for set valued functions without continuity, Indian J. Pure Appl. Math., 29(1998), No. 3, 227-238.

[8] G. Jungck and B. E. Rhoades: Fixed point theorems for occasionally weakly compatible mappings, Fixed Point Theory., 7(2006), 287-296.

[9] R. Kannan: Some results on fixed points, Bull. Calcutta Math. Soc., 60(1968), 71-76.

[10] B. E. Rhoades: A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc., 26(1977), 257-290.

[11] B. E. Rhoades, K. Tiwary and G. N. Singh: A common fixed point theorem for compatible mappings, Indian J. Pure and Appl. Math., 26(1995), No. 5, 403-409.

[12] S. Sessa: On a weak commutativity condition of mappings in fixed point consideration, Publ. Inst. Math. Soc., 32(1982), 149-153.

[13] S. L. Singh and S. N. Mishra: Coincidence and fixed points of nonself hybrid contractions, J. Math. Anal. Appl., 256(2001), 486-497.

Indiana University

Department of Mathematics

Bloomington, IN 47405-7106

E-mail address: rhoades@indiana.edu

Indiana University

Department of Mathematics

Bloomington, IN 47405-7106

E-mail address: abbas@indiana.edu
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Author:Rhoades, B.E.; Abbas, Mujahid
Publication:Journal of Advanced Mathematical Studies
Article Type:Report
Geographic Code:1USA
Date:Jul 1, 2009
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