Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 580297, 5 pages http://dx.doi.org/10.1155/2014/580297

Research Article

A Generalization of a Gregus Fixed Point Theorem in Metric Spaces

Marwan A. Kutbi,1 A. Amini-Harandi,2,3 and N. Hussain1

1 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

2 Department of Pure Mathematics, University of Shahrekord, Shahrekord 88186-34141, Iran

3 School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5746, Iran

Correspondence should be addressed to N. Hussain; nhusain@kau.edu.sa Received 30 January 2014; Accepted 22 February 2014; Published 25 March 2014 Academic Editor: Giuseppe Marino

Copyright © 2014 Marwan A. Kutbi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We first introduce a new class of contractive mappings in the setting of metric spaces and then we present certain Gregus type fixed point theorems for such mappings. As an application, we derive certain Gregus type common fixed theorems. Our results extend Gregus fixed point theorem in metric spaces and generalize and unify some related results in the literature. An example is also given to support our main result.

1. Introduction and Preliminaries

Let X be a Banach space and let C be a closed convex subset of X. In 1980 Gregus [1] proved the following result.

Theorem 1. Let T : C ^ C be a mapping satisfying the inequality

\\Tx - Ty\\ <a\\x- y\\ + b\\x- Tx\\ + c\\y- Ty\\, (1)

for all x,y e C, where 0 < a < 1, b,c > 0, and a + b + c = 1. Then T has a unique fixed point.

Fisher and Sessa [2], Jungck [3], and Hussain et al. [4] obtained common fixed point generalizations of Theorem 1. In recent years, many theorems which are closely related to Gregus's Theorem have appeared (see [1-21]). Very recently, Moradi and Farajzadeh [21] extended Gregus fixed point theorem in complete metric spaces.

Theorem 2 ([21, Theorem 2.4]). Let (X,d) be a complete metric space and let T : X ^ X be a mapping such that, for all x,y e X,

d (Tx, Ty) < ad (x, y) + bd (x, Tx)

+ cd (y, Ty) + ed (y, Tx) + fd (x, Ty),

where 0 < a < 1, b,c,e,f>0, b + c > 0, e + f > 0, and a + b + c + e + f= 1. Then T has a unique fixed point.

Let I and T be self-maps of X. A point x e X is a coincidence point (resp., common fixed point) of I and T if Ix = Tx (resp., x = Ix = Tx). The pair {I, T] is called (1) commuting if Tlx = ITx for all x e X;(2)weaklycommuting

[2] if, for all x e X, d(ITx, Tlx) < d(Ix, Tx); (3) compatible

[3] if limn d(TIxn, ITxn) = 0 whenever {xn] is a sequence such that limn Txn = limn Ixn = t for some t in X; (4) weakly compatible if they commute at their coincidence points, that is, if ITx = Tlx whenever Ix = Tx. Clearly, commuting maps are weakly commuting, and weakly commuting maps are compatible. References [2, 3] give examples which show that neither implication is reversible.

The purpose of this paper is to define and to investigate a class of new generalized contractive mappings (not necessarily continuous) on metric spaces. We will prove certain fixed point and common fixed results which are generalizations of the above mentioned theorems.

2. Fixed Point Results

We denote by R+ the set of all nonnegative real numbers and by U the set of all functions u : R+ ^ R+ satisfying

the following conditions: (Q) u is continuous,

(C2) u(t1, t2, t3, t4, t5) is nondecreasing in t1,t2, t3, and t5, (C3) s < t ^ u(s, s, t,0,s + t) < t, for each s,t > 0, (C4) u(t, t, t, 0, u(2t, t, t, t, 3t)) < t, for each t > 0, (C5) u(t, 0, 0, t,t) <t for each t > 0.

Example 3. If u(t1, t2, t3, t4, t5) = at1 + bt2 + bt3 + et4 + et5 for tt e R+, where 0 < a < 1, b,e > 0, and a + 2b + 2e = 1, then ue U.

Example 4. If u(t1,t2,t3,t4,t5) =kmax{i1;i2,i3,(t4 + t5)/2} for t{ e R+, where k e [0,1), then u e U.

Example 5. Let u e U. Then it is easy tosee that we U where

wih'h'h'U'h)

= u(t 1, t2, t3, t4, t5) + L min {i 1, t2, t3, t4, t5 }

for each L> 0.

Now we are ready to state our main result.

Theorem 6. Let (X, d) be a complete metric space and let T : X ^ X be a mapping satisfying

d (Tx, Ty)

<u(d (x, y), d (x, Tx), d (y, Ty), d (y, Tx), d (x, Ty)),

for each x,y e X, where u e U. Then T has a unique fixed point.

Proof. We first show that a = infx£X d(x,Tx) = 0. If d(x, Tx) = 0 for some x e X, then x is a fixed point of T and we are done. So, we may assume that d(x, Tx) > 0 for each x e X. From (4), (C2), and (C3), we have

d(Tz, T2z)

<u(d (z, Tz), d (z, Tz), d (Tz, T2z), 0, d (z, T2z)) <u(d (z, Tz), d (z, Tz), d (Tz, T2z) , 0, d(z,Tz)+d(Tz, T2z)),

and so

d(T z, T2z) < d (z, Tz), for each z e X. (6) Now let {xn} be a sequence such that

a = inf d (x, Tx) = lim d (xn, Txn). (7)

xsX n^rn v n nJ VJ

From (4), (6), and (C2), we get d (Tx^ T3Xn)

<u(d [x^ T2Xn), d (xn, Txn), d (T2x^ T3xn),

d(T2Xn ,TXn),d(xn,T3Xn))

<u(yd (xn, Txn) + d (Txn, T2xn), d (xn, Txn), d (T2x^ T3Xn), d (T2x^ Txn), d Txn) + d (Txn, T2Xn) +d (T2Xn, T3Xn)) < u (2d (xn, Txn), d (xn, Txn), d (xn, Txn), d (xn, Txn), 3d Txn)), for each n e N. From (4), (6), (8), and (C2), we have d(T2x^ T3Xn)

<u(d (Txn, T2xn), d (Txn, T2xn),

d (T2x^ T3Xn), 0 d (Tx^ T3Xn)) <u(d (xn, Txn), d (xn, Txn), d (T2xn, T3xn), 0, u (2.d Txn), d Txn), d Txn),

d(Xn,TXn),3d (Xn TXn^)^)^),

for each n e N. From (6) and (7), we get

a<d (T2xn, T3xn) < d (xn, Txn), (10)

and so by (7)

\imad(T2Xn,T3Xn) = a. (11)

From (7), (9), (11), and (C1), we obtain

a < u(a, a, a, 0, u (2a, a, a, a, 3a)). (12)

Hence by(C4)

a = inf d (x, Tx) = 0. (13)

xeX v '

Now, let

Cn = {xeX:d(x,Tx)<l-}, for each ne N. (14)

Notice that, by (13), Cn = 0 for each ne N.We show that

nlim diam (C~n) = 0. (15)

On the contrary, assume that there are sequences {xn} and {yj with Xn, yn e Cn satisfying

p = Jim, d(xn>yn) > 0.

From (4), (14), and (C2), we have

d (xn> yn) < d (xn> Txn) + d (Txn, Tyn) + d {yn, Tyn) 2

<- + u(d (xn, yn), d {xn, Txn), d (yw Tyn), n

d{y„,Txn),d{xn,Tyn))

<- + u(d {xn, yn), d {xn, Txn), d (yw Tyn), n

d{xn,yn)+d{xn,Txn), d{xn,yn) + d{yn,Tyn)),

for each ne N. From (14), (16), (17), and (C1), we get

p<u(p, 0, 0, p, p), (18)

which contradicts (C5). Thus (15) holds. Hence {Cn} is a decreasing sequence of closed nonempty sets with diam(Cn) ^ 0 and so, by Cantor's intersection theorem,

= [x] for some x e X.

We show that x is a fixed point of T. Since x e Cn, there exists un e Cn such that d(x, un) < 1/n for all n e N. Now for each n e N,we have

d (x, Tx) < d (x, un) + d (un, Tun) + d (Tun, Tx)

< - +u(d (un, x), d (un, Tun), d (x, Tx), n

d (x, Tun), d (un, Tx))

< - + u(d (un, x), d (un, Tun), d (x, Tx), n

d (x, un) + d (un, Tun), d (un, x) +d(x,Tx)). Since u is continuous, from (20),

d (x, Tx)<u (0,0, d (x, Tx),0,d (x, Tx)), (21)

and hence, by (C3), d(x, Tx) = 0; that is, Tx = x. To prove the uniqueness, note that if y is a fixed point of T, then y e f|™i Cn c Cn = {x} and hence x = y. □

Remark 7. Note that, to prove Theorem 2, we may assume that b = c and f = e (see the proof of Theorem 2.4 in [21]). Thus, by Example 3, Theorem 6 is a generalization of the above mentioned Theorem 2 of Moradi and Farajzadeh.

If we take u as in Example 4, from Theorem 6 we get the main result of Ciric [10].

The following corollary improves Theorem 2.4 in [6].

Corollary 8. Let (X, d) be a complete metric space and let T : X ^ X be a mapping satisfying

d (Tx, Ty)

< k max -d(x,y),d (x, Tx), d (y, Ty),

d (y, Tx) + d (x, Ty)} (22)

+ L min [d (x, y), d (x, Tx), d (y, Ty),

d (y, Tx), d (x, Ty)}

for each x,yeX, where k e [0,1) and L > 0. Then T has a unique fixed point.

As an easy consequence of the axiom of choice, [13, page 5], AC5: for every function f: X ^ X, there is a function g such that D(g) = R(f) and for every x e D(g), f(gx) = x], we obtain the following lemma (see also [22]).

Lemma 9. Let X be a nonempty set and let g : X ^ X be a mapping. Then, there exists a subset E c X such that g(E) = g(X) and g : E ^ X is one-to-one.

As an application of Theorem 6, we now establish a common fixed point result.

Theorem 10. Let (X, d) be a metric space and let g,T : X ^ X be mappings satisfying

d (Tx, Ty) < u(d (gx, gy), d (gx, Tx),

d (gy, Ty),d (gy,Tx),d (gx, Ty)),

for each x,yeX, where u e U. Suppose that TX ç gX and gX is complete subspace of X. Then g and T have a unique coincidence point. Further, if g and T are weakly compatible, then they have a unique common fixed point.

Proof. By Lemma 9, there exists E ç X such that g(E) = g(X) and g : E ^ X is one-to-one. We define a mapping G : g(E) ^ g(E) by

G (gx) = T(x),

for all gx e g(E). As g is one-to-one on g(E) and T(X) ç g(X), G is well defined. Thus, it follows from (23) and (24) that

d (G (gx), G (gy)) = d (Tx, Ty)

<u(d (gx, gy), d (gx, Tx), d (gy, Ty), d (gy, Tx), d (gx, Ty)),

for all gx, gy e g(X) = g(E). Thus the function G : g(E) ^ g(E) satisfies all conditions of Theorem 6, so G has a unique fixed point z e g(X). As z e g(X), there exists w e X such

that z = g(w). Thus T(w) = G(gw) = G(z) = z = g(w) which implies that g and T have a unique coincidence point. Further if g and T are weakly compatible, then they have a unique common fixed point. □

If we take u as in Example 5, then from Theorem 10 we obtain the following result which extends many related results in the literature (see [16,17]).

Theorem 11. Let (X, d) be a metric space and let g,T : X ^ X be mappings satisfying

d (Tx, Ty)

< k max -

d (gx, gy), d (gx, Tx), d (gy, Ty), d (gy, Tx) + d (gx, Ty)) (26)

+ L min [d (gx, gy), d (gx, Tx), d (gy, Ty),

d (gy, Tx), d (gx, Ty)}

for each x,y e X, where k e [0,1) and L > 0. Suppose that TX ç gX and gX is a complete subspace of X. Then g and T have a unique coincidence point. Further, if g and T are weakly compatible, then they have a unique common fixed point.

Corollary 12. Let (X, d) be a metric space and let g,T : X ^ X be mappings satisfying

d (Tx, Ty) < k max

d (gx, gy), d (gx, Tx), d (gy, Ty),

d (gy, Tx) + d (gx, Ty)

foreach x,y e X, wherek e [0,1). SupposethatTX c gX and gX is a complete subspace of X. Then g and T have a unique coincidence point. Further if g and T are weakly compatible, then they have a unique common fixed point.

As a linear continuous operator defined on a closed subset of a normed space is closed operator, we obtain the following new common fixed point results as corollaries to Theorem 11.

Corollary 13 (see Fisher and Sessa [2]). Let T and g be two weakly commuting mappings on a closed convex subset C of a Banach space X into itself satisfying the inequality

| | Tx - Ty | | < a max { 11 Tx - gx \ |, 11 Ty - gy \ \ }

+ Lmin {Wgx - gyW, ¡Tx - gx\^, ¡Ty - gyW,

Hgy-TxlUgx-TyH},

for all x,y e C, where a e (0,1) and L > O.Ifg is linear and nonexpansive on C and T(C) c g(C), then T and g have a unique common fixed point in C.

Corollary 14 (see Jungck [3]). Let T and g be compatible self-maps of a closed convex subset C of a Banach space X. Suppose that g is continuous and linear and that T(C) c g(C). IfT and g satisfy inequality (28), then T and g have a unique common fixed point in C.

Now, we illustrate our main result by the following example.

Example 15. Let X= {1,2,3,4} and let d(1,2) = 5/4, d(1,3) = 1, d(1,4) = 7/4, and d(2,3) = d(2,4) = d(3,4) = 2. Then (X, d) is a complete metric space. Let T : X ^ X be given by T1 = 1, T2 = 4, T3 = 4,and T4 = 1. Then it is straightforward to show that

d (Tx, Ty) < - max -

d (x, y), d (x, Tx), d (y, Ty),

d (x, Ty) + d (y, Tx)

for each x,y e X. Then by Corollary 8, T has a unique fixed point (x = 1 is the unique fixed point of T).

Now, we show that T does not satisfy the condition of Theorem 2 of Moradi and Farajzadeh. On the contrary, assume that there exist nonnegative numbers 0 < a < 1, b,c,e,f>0, b + c> 0, e + f > 0 such that

d (Tx, Ty) < ad (x, y) + bd (x, Tx) + cd (y, Ty) + ed (y, Tx) + fd (x, Ty)

for all x,y e X. Let (xx, y^ = (1,2) and (x2,y2) = (2,1). Then from (30), we have

15 5 7

- < -a + 2c + -e + - t, 4 4 4 4

7 5 7 5

- < -a + 2b + -e + - f, 4 4 4 4

which yield

7 < 5a + 4b + 4c + 6e + 6f < 6 (a + b + c + e + f) < 6,

a contradiction. Thus we cannot invoke the above mentioned theorem of Moradi and Farajzadeh (Theorem 2), to show the existence of a fixed point of T.

Remark 16. The technique of proof of Theorem 6 is in line with the proof of Theorem 2.4 in [23]. Therefore the reader interested in fixed point results for generalized contractions/nonexpansive mappings in the general setup of uniformly convex metric spaces is referred to [23].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, Mar-wan A. Kutbi and N. Hussain acknowledge with thanks DSR, KAU, for financial support. A. Amini-Harandi was partially supported by the Center of Excellence for Mathematics, University of Shahrekord, Iran, and by a Grant from IPM (no. 92470412).

References

[1] M. Gregus, "A fixed point theorem in Banach space," Bollettino della Unione Matematica Italiana A, vol. 5, pp. 193-198,1980.

[2] B. Fisher and S. Sessa, "On a fixed point theorem of Gregus," International Journal of Mathematics and Mathematical Sciences, vol. 9, pp. 23-28,1986.

[3] G. Jungck, "On a fixed point theorem of Fisher and Sessa," International Journal of Mathematics and Mathematical Sciences, vol. 13, pp. 497-500,1990.

[4] N. Hussain, B. E. Rhoades, and G. Jungck, "Common fixed point and invariant approximation results for Gregus type I-contractions," Numerical Functional Analysis and Optimization, vol. 28, no. 9-10, pp. 1139-1151, 2007.

[5] A. Aliouche, "Common fixed point theorems of Gregss type for weakly compatible mappings satisfying generalized contractive conditions," Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 707-719, 2008.

[6] V. Berinde, "Some remarks on a fixed point theorem for Ciric-type almost contractions," Carpathian Journal of Mathematics, vol. 25, no. 2, pp. 157-162, 2009.

[7] L. B. Ciric, "On some discontinuous fixed point mappings in convex metric spaces," Czechoslovak Mathematical Journal, vol. 43, no. 118, pp. 319-326,1993.

[8] L. B. Ciric, "On a generalization of a Gregus fixed point theorem," Czechoslovak Mathematical Journal, vol. 50, no. 125, pp. 449-458, 2000.

[9] L. B. Ciric, "On a common fixed point theorem of a Gregus type," Publications de l'Institut Mathematique, vol. 49, no. 63, pp. 174-178, 1991.

[10] L. B. Ciric, "Generalized contractions and fixed-point theorems," Publications de l'Institut Mathematique, vol. 12, pp. 19-26, 1971.

[11] M. L. Diviccaro, B. Fisher, and S. Sessa, "A common fixed point theorem of Gregus type," Publicationes Mathematicae Debrecen, vol. 34, pp. 83-89,1987.

[12] B. Fisher, "Common fixed points on a Banach space," Chung Yuan, vol. 11, pp. 19-26,1982.

[13] R. Herman and E. R. Jean, Equivalents of the Axiom of Choice, North-Holland, Amsterdam, The Netherlands, 1970.

[14] N. J. Huang and Y. J. Cho, "Common fixed point theorems of Gregus type in convex metric spaces," Japanese Mathematics, vol. 48, pp. 83-89, 1998.

[15] N. Hussain, "Common fixed points in best approximation for Banach operator pairs with (Ciric type I-contractions," Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1351-1363, 2008.

[16] Y. J. Cho and N. Hussain, "Weak contractions, common fixed points, and invariant approximations," Journal of Inequalities and Applications, vol. 2009, Article ID 390634,10 pages, 2009.

[17] G. Jungck and N. Hussain, "Compatible maps and invariant approximations," Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1003-1012, 2007.

[18] L. Bing-you, "Fixed point theorem of nonexpansive mappings in convex metric spaces," Applied Mathematics and Mechanics, vol. 10, no. 2, pp. 183-188,1989.

[19] R. N. Mukherjea and V. Verma, "A note on a fixed point theorem of Gregus," Japanese Journal of Mathematics, vol. 33, pp. 745749,1988.

[20] P. P. Murthy, Y. J. Cho, and B. Fisher, "Common fixed points of Gregus type mappings," Glasnik Matematicki, no. 50, pp. 335341,1995.

[21] S. Moradi and A. Farajzadeh, "On Olaleru's open problem on Gregus fixed point theorem," Journal of Global Optimization, vol. 56, no. 4, pp. 1689-1697, 2013.

[22] R. H. Haghi, S. Rezapour, and N. Shahzad, "Some fixed point generalizations are not real generalizations," Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 5, pp. 1799-1803, 2011.

[23] H. Fukhar-ud-din, A. R. Khan, and Z. Akhtar, "Fixed point results for a generalized nonexpansive map in uniformly convex metric spaces," Nonlinear Analysis, Theory, Methods and Applications, vol. 75, pp. 4747-4760, 2012.

Copyright of Journal of Applied Mathematics is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.