JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 580297 10.1155/2014/580297 580297 Research Article A Generalization of a Greguš Fixed Point Theorem in Metric Spaces Kutbi Marwan A. 1 Amini-Harandi A. 2, 3 Hussain N. 1 Marino Giuseppe 1 Department of Mathematics King Abdulaziz University P.O. Box 80203 Jeddah 21589 Saudi Arabia kau.edu.sa 2 Department of Pure Mathematics University of Shahrekord Shahrekord 88186-34141 Iran sku.ac.ir 3 School of Mathematics Institute for Research in Fundamental Sciences (IPM) Tehran 19395-5746 Iran ipm.ac.ir 2014 2532014 2014 30 01 2014 22 02 2014 25 3 2014 2014 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 Greguš type fixed point theorems for such mappings. As an application, we derive certain Greguš type common fixed theorems. Our results extend Greguš 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 Greguš  proved the following result.

Theorem 1.

Let T : C C be a mapping satisfying the inequality (1) T x - T y a x - y + b x - T x + c y - T y , for all x , y C , where 0 < a < 1 , b , c 0 , and a + b + c = 1 . Then T has a unique fixed point.

Fisher and Sessa , Jungck , and Hussain et al.  obtained common fixed point generalizations of Theorem 1. In recent years, many theorems which are closely related to Greguš’s Theorem have appeared (see ). Very recently, Moradi and Farajzadeh  extended Greguš fixed point theorem in complete metric spaces.

Theorem 2 ([<xref ref-type="bibr" rid="B21">21</xref>, 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 X , (2) d ( T x , T y ) a d ( x , y ) + b d ( x , T x ) + c d ( y , T y ) + e d ( y , T x ) + f d ( x , T y ) , 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 X is a coincidence point (resp., common fixed point) of I and T if I x = T x (resp., x = I x = T x ). The pair { I , T } is called (1) commuting if T I x = I T x for all x X ; (2) weakly commuting  if, for all x X , d ( I T x , T I x ) d ( I x , T x ) ; (3) compatible  if lim n d ( T I x n , I T x n ) = 0 whenever { x n } is a sequence such that lim n T x n = lim n I x n = t for some t in X ; (4) weakly compatible if they commute at their coincidence points, that is, if I T x = T I x whenever I x = T x . 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 + the set of all nonnegative real numbers and by 𝒰 the set of all functions u : + 5 + satisfying the following conditions:

u is continuous,

u ( t 1 , t 2 , t 3 , t 4 , t 5 ) is nondecreasing in t 1 , t 2 , t 3 , and t 5 ,

s < t u ( s , s , t , 0 , s + t ) < t , for each s , t > 0 ,

u ( t , t , t , 0 , u ( 2 t , t , t , t , 3 t ) ) < t , for each t > 0 ,

u ( t , 0,0 , t , t ) < t for each t > 0 .

Example 3.

If u ( t 1 , t 2 , t 3 , t 4 , t 5 ) = a t 1 + b t 2 + b t 3 + e t 4 + e t 5 for t i + , where 0 < a < 1 , b , e > 0 , and a + 2 b + 2 e = 1 , then u 𝒰 .

Example 4.

If u ( t 1 , t 2 , t 3 , t 4 , t 5 ) = k max { t 1 , t 2 , t 3 , ( t 4 + t 5 ) / 2 } for t i + , where k [ 0,1 ) , then u 𝒰 .

Example 5.

Let u 𝒰 . Then it is easy to see that w 𝒰 where (3) w ( t 1 , t 2 , t 3 , t 4 , t 5 ) = u ( t 1 , t 2 , t 3 , t 4 , t 5 ) + L min { t 1 , t 2 , t 3 , t 4 , t 5 } , 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 (4) d ( T x , T y ) u ( d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( y , T x ) , d ( x , T y ) ) , for each x , y X , where u 𝒰 . Then T has a unique fixed point.

Proof.

We first show that α = inf x X d ( x , T x ) = 0 . If d ( x , T x ) = 0 for some x X , then x is a fixed point of T and we are done. So, we may assume that d ( x , T x ) > 0 for each x X . From (4), ( C 2 ), and ( C 3 ), we have (5) d ( T z , T 2 z ) u ( d ( z , T z ) , d ( z , T z ) , d ( T z , T 2 z ) , 0 , d ( z , T 2 z ) ) u ( d ( z , T z ) , d ( z , T z ) , d ( T z , T 2 z ) , 0 , m m m m l d ( z , T z ) + d ( T z , T 2 z ) ) , and so (6) d ( T z , T 2 z ) d ( z , T z ) , for    each    z X . Now let { x n } be a sequence such that (7) α = inf x X d ( x , T x ) = lim n d ( x n , T x n ) . From (4), (6), and ( C 2 ), we get (8) d ( T x n , T 3 x n ) m m u ( d ( x n , T 2 x n ) , d ( x n , T x n ) , d ( T 2 x n , T 3 x n ) , m m m m d ( T 2 x n , T x n ) , d ( x n , T 3 x n ) ) m m u ( d ( x n , T x n ) + d ( T x n , T 2 x n ) , d ( x n , T x n ) , m m m m m d ( T 2 x n , T 3 x n ) , d ( T 2 x n , T x n ) , d ( x n , T x n ) m m m m m + d ( T x n , T 2 x n ) + d ( T 2 x n , T 3 x n ) ) m m u ( 2 d ( x n , T x n ) , d ( x n , T x n ) , d ( x n , T x n ) , m m m m m d ( x n , T x n ) , 3 d ( x n , T x n ) ) , for each n . From (4), (6), (8), and ( C 2 ), we have (9) d ( T 2 x n , T 3 x n ) u ( d ( T x n , T 2 x n ) , d ( T x n , T 2 x n ) , d ( T 2 x n , T 3 x n ) , 0 , d ( T x n , T 3 x n ) ) u ( d ( x n , T x n ) , d ( x n , T x n ) , d ( T 2 x n , T 3 x n ) , 0 , u ( 2 d ( x n , T x n ) , d ( x n , T x n ) , d ( x n , T x n ) , d ( x n , T x n ) , 3 d ( x n , T x n ) ) ( T 2 x n , T 3 x n ) ) , for each n . From (6) and (7), we get (10) α d ( T 2 x n , T 3 x n ) d ( x n , T x n ) , and so by (7) (11) lim n d ( T 2 x n , T 3 x n ) = α . From (7), (9), (11), and ( C 1 ), we obtain (12) α u ( α , α , α , 0 , u ( 2 α , α , α , α , 3 α ) ) . Hence by ( C 4 ) (13) α = inf x X d ( x , T x ) = 0 . Now, let (14) C n = { x X : d ( x , T x ) 1 n } , for    each    n . Notice that, by (13), C n for each n . We show that (15) lim n diam ( C n ¯ ) = 0 . On the contrary, assume that there are sequences { x n } and { y n } with x n , y n C n satisfying (16) ρ = lim n d ( x n , y n ) > 0 . From (4), (14), and ( C 2 ), we have (17) d ( x n , y n ) d ( x n , T x n ) + d ( T x n , T y n ) + d ( y n , T y n ) 2 n + u ( d ( x n , y n ) , d ( x n , T x n ) , d ( y n , T y n ) , d ( y n , T x n ) , d ( x n , T y n ) ) 2 n + u ( d ( x n , y n ) , d ( x n , T x n ) , d ( y n , T y n ) , d ( x n , y n ) + d ( x n , T x n ) , d ( x n , y n ) + d ( y n , T y n ) ) , for each n . From (14), (16), (17), and ( C 1 ) , we get (18) ρ u ( ρ , 0,0 , ρ , ρ ) , which contradicts ( C 5 ). Thus (15) holds. Hence { C n ¯ } is a decreasing sequence of closed nonempty sets with diam ( C n ¯ ) 0 and so, by Cantor’s intersection theorem, (19) n = 1 C n ¯ = { x ¯ } for    some    x ¯ X . We show that x ¯ is a fixed point of T . Since x ¯ C n ¯ , there exists u n C n such that d ( x ¯ , u n ) < 1 / n for all n . Now for each n , we have (20) d ( x ¯ , T x ¯ ) d ( x ¯ , u n ) + d ( u n , T u n ) + d ( T u n , T x ¯ ) 2 n + u ( d ( u n , x ¯ ) , d ( u n , T u n ) , d ( x ¯ , T x ¯ ) , d ( x ¯ , T u n ) , d ( u n , T x ¯ ) ) 2 n + u ( d ( u n , x ¯ ) , d ( u n , T u n ) , d ( x ¯ , T x ¯ ) , d ( x ¯ , u n ) + d ( u n , T u n ) , d ( u n , x ¯ ) + d ( x ¯ , T x ¯ ) ) . Since u is continuous, from (20), (21) d ( x ¯ , T x ¯ ) u ( 0,0 , d ( x ¯ , T x ¯ ) , 0 , d ( x ¯ , T x ¯ ) ) , and hence, by ( C 3 ) , d ( x ¯ , T x ¯ ) = 0 ; that is, T x ¯ = x ¯ . To prove the uniqueness, note that if y ¯ is a fixed point of T , then y ¯ n = 1 C n n = 1 C n ¯ = { 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 ). 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 Ćirić .

The following corollary improves Theorem 2.4 in .

Corollary 8.

Let ( X , d ) be a complete metric space and let T : X X be a mapping satisfying (22) d ( T x , T y ) k max { d ( y , T x ) + d ( x , T y ) 2 d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( y , T x ) + d ( x , T y ) 2 } + L min { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( y , T x ) , d ( x , T y ) } for each x , y X , where k [ 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 D ( g ) , f ( g x ) = x ], we obtain the following lemma (see also ).

Lemma 9.

Let X be a nonempty set and let g : X X be a mapping. Then, there exists a subset E 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 (23) d ( T x , T y ) u ( d ( g x , g y ) , d ( g x , T x ) , d ( g y , T y ) , d ( g y , T x ) , d ( g x , T y ) ) , for each x , y X , where u 𝒰 . Suppose that T X g X and g X 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 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 (24) G ( g x ) = T ( x ) , for all g x g ( E ) . As g is one-to-one on g ( E ) and T ( X ) g ( X ) , G is well defined. Thus, it follows from (23) and (24) that (25) d ( G ( g x ) , G ( g y ) ) = d ( T x , T y ) u ( d ( g x , g y ) , d ( g x , T x ) , d ( g y , T y ) , d ( g y , T x ) , d ( g x , T y ) ) , for all g x , g y 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 g ( X ) . As z g ( X ) , there exists w X such that z = g ( w ) . Thus T ( w ) = G ( g w ) = 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 (26) d ( T x , T y ) m k max { d ( g y , T x ) + d ( g x , T y ) 2 d ( g x , g y ) , d ( g x , T x ) , d ( g y , T y ) , m m m m m m l l d ( g y , T x ) + d ( g x , T y ) 2 } m m + L min { d ( g x , g y ) , d ( g x , T x ) , d ( g y , T y ) , m m m m m m l l d ( g y , T x ) , d ( g x , T y ) } for each x , y X , where k [ 0,1 ) and L 0 . Suppose that T X g X and g X 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 (27) d ( T x , T y ) k max { d ( g y , T x ) + d ( g x , T y ) 2 d ( g x , g y ) , d ( g x , T x ) , d ( g y , T y ) , d ( g y , T x ) + d ( g x , T y ) 2 } , for each x , y X , where k [ 0,1 ) . Suppose that T X g X and g X 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 [<xref ref-type="bibr" rid="B9">2</xref>]).

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 (28) T x - T y a max { T x - g x , T y - g y } + L min { g x - g y , T x - g x , T y - g y , g y - T x , g x - T y } , for all x , y C , where a ( 0,1 ) and L 0 . If g is linear and nonexpansive on C and T ( C ) g ( C ) , then T and g have a unique common fixed point in C .

Corollary 14 (see Jungck [<xref ref-type="bibr" rid="B17">3</xref>]).

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 ) g ( C ) . If T 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 T 1 = 1 , T 2 = 4 , T 3 = 4 , and T 4 = 1 . Then it is straightforward to show that (29) d ( T x , T y ) 7 8 max { d ( x , T y ) + d ( y , T x ) 2 d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) + d ( y , T x ) 2 } for each x , y 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 (30) d ( T x , T y ) a d ( x , y ) + b d ( x , T x ) + c d ( y , T y ) + e d ( y , T x ) + f d ( x , T y ) for all x , y X . Let ( x 1 , y 1 ) = ( 1,2 ) and ( x 2 , y 2 ) = ( 2,1 ) . Then from (30), we have (31) 7 4 5 4 a + 2 c + 5 4 e + 7 4 f , 7 4 5 4 a + 2 b + 7 4 e + 5 4 f , which yield (32) 7 5 a + 4 b + 4 c + 6 e + 6 f 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 . 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 .

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, Marwan 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).

Greguš M. A fixed point theorem in Banach space Bollettino della Unione Matematica Italiana A 1980 5 193 198 Fisher B. Sessa S. On a fixed point theorem of Greguš International Journal of Mathematics and Mathematical Sciences 1986 9 23 28 Jungck G. On a fixed point theorem of Fisher and Sessa International Journal of Mathematics and Mathematical Sciences 1990 13 497 500 Hussain N. Rhoades B. E. Jungck G. Common fixed point and invariant approximation results for Greguš type I-contractions Numerical Functional Analysis and Optimization 2007 28 9-10 1139 1151 2-s2.0-35148863642 10.1080/01630560701563842 Aliouche A. Common fixed point theorems of Gregšs type for weakly compatible mappings satisfying generalized contractive conditions Journal of Mathematical Analysis and Applications 2008 341 1 707 719 2-s2.0-38849090073 10.1016/j.jmaa.2007.10.054 ZBL1138.54031 Berinde V. Some remarks on a fixed point theorem for Ćirić-type almost contractions Carpathian Journal of Mathematics 2009 25 2 157 162 2-s2.0-77954452480 ZBL1249.54078 Ćirić L. B. On some discontinuous fixed point mappings in convex metric spaces Czechoslovak Mathematical Journal 1993 43 118 319 326 ZBL0814.47065 Ćirić L. B. On a generalization of a Greguš fixed point theorem Czechoslovak Mathematical Journal 2000 50 125 449 458 Ćirić L. B. On a common fixed point theorem of a Greguš type Publications de l'Institut Mathématique 1991 49 63 174 178 Ćirić L. B. Generalized contractions and fixed-point theorems Publications de l'Institut Mathématique 1971 12 19 26 ZBL0234.54029 Diviccaro M. L. Fisher B. Sessa S. A common fixed point theorem of Greguš type Publicationes Mathematicae Debrecen 1987 34 83 89 Fisher B. Common fixed points on a Banach space Chung Yuan 1982 11 19 26 Herman R. Jean E. R. Equivalents of the Axiom of Choice 1970 Amsterdam, The Netherlands North-Holland Huang N. J. Cho Y. J. Common fixed point theorems of Greguš type in convex metric spaces Japanese Mathematics 1998 48 83 89 Hussain N. Common fixed points in best approximation for Banach operator pairs with Ćirić type I-contractions Journal of Mathematical Analysis and Applications 2008 338 2 1351 1363 2-s2.0-34848917578 10.1016/j.jmaa.2007.06.008 ZBL1134.47039 Cho Y. J. Hussain N. Weak contractions, common fixed points, and invariant approximations Journal of Inequalities and Applications 2009 2009 10 2-s2.0-76649144207 10.1155/2009/390634 390634 Jungck G. Hussain N. Compatible maps and invariant approximations Journal of Mathematical Analysis and Applications 2007 325 2 1003 1012 2-s2.0-33750327353 10.1016/j.jmaa.2006.02.058 Bing-you L. Fixed point theorem of nonexpansive mappings in convex metric spaces Applied Mathematics and Mechanics 1989 10 2 183 188 2-s2.0-34249976795 10.1007/BF02014826 Mukherjea R. N. Verma V. A note on a fixed point theorem of Greguš Japanese Journal of Mathematics 1988 33 745 749 Murthy P. P. Cho Y. J. Fisher B. Common fixed points of Greguš type mappings Glasnik Matematicki 1995 50 335 341 Moradi S. Farajzadeh A. On Olaleru's open problem on Greguš fixed point theorem Journal of Global Optimization 2013 56 4 1689 1697 Haghi R. H. Rezapour S. Shahzad N. Some fixed point generalizations are not real generalizations Nonlinear Analysis, Theory, Methods and Applications 2011 74 5 1799 1803 2-s2.0-78651357943 10.1016/j.na.2010.10.052 Fukhar-ud-din H. Khan A. R. Akhtar Z. Fixed point results for a generalized nonexpansive map in uniformly convex metric spaces Nonlinear Analysis, Theory, Methods and Applications 2012 75 4747 4760 2-s2.0-84859768254 10.1016/j.na.2012.03.025