Research Article Common Fixed Point Theorems for Four Mappings on Cone Metric Type Space

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1 Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 011, Article ID 58975, 15 pages doi: /011/58975 Research Article Common Fixed Point Theorems for Four Mappings on Cone Metric Type Space Aleksandar S. Cvetković, 1 Marija P. Stanić, Sladjana Dimitrijević, and Suzana Simić 1 Department of Mathematics, Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 1110 Belgrade, Serbia Department of Mathematics and Informatics, Faculty of Science, University of Kragujevac, Radoja Domanovića 1, Kragujevac, Serbia Correspondence should be addressed to Aleksandar S. Cvetković, acvetkovic@mas.bg.ac.rs Received 9 December 010; Revised 6 January 011; Accepted 3 February 011 Academic Editor: Fabio Zanolin Copyright q 011 Aleksandar S. Cvetković 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. In this paper we consider the so called a cone metric type space, which is a generalization of a cone metric space. We prove some common fixed point theorems for four mappings in those spaces. Obtained results extend and generalize well-known comparable results in the literature. All results are proved in the settings of a solid cone, without the assumption of continuity of mappings. 1. Introduction Replacing the real numbers, as the codomain of a metric, by an ordered Banach space we obtain a generalization of metric space. Such a generalized space, called a cone metric space, was introduced by Huang and Zhang in 1. They described the convergence in cone metric space, introduced their completeness, and proved some fixed point theorems for contractive mappings on cone metric space. Cones and ordered normed spaces have some applications in optimization theory see. The initial work of Huang and Zhang 1 inspired many authors to prove fixed point theorems, as well as common fixed point theorems for two or more mappings on cone metric space, for example, In this paper we consider the so-called a cone metric type space, which is a generalization of a cone metric space and prove some common fixed point theorems for four mappings in those spaces. Obtained results are generalization of theorems proved in 13.For some special choices of mappings we obtain theorems which generalize results from 1, 8, 15.

2 Fixed Point Theory and Applications All results are proved in the settings of a solid cone, without the assumption of continuity of mappings. The paper is organized as follows. In Section we repeat some definitions and wellknown results which will be needed in the sequel. In Section 3 we prove common fixed point theorems. Also, we presented some corollaries which show that our results are generalization of some existing results in the literature.. Definitions and Notation Let E be a real Banach space and P asubsetofe. Byθ we denote zero element of E and by int P the interior of P. The subset P is called aconeif and only if i P is closed, nonempty and P / θ}; ii a, b R, a, b 0, and x, y P imply ax by P; iii P P θ}. For a given cone P, a partial ordering with respect to P is introduced in the following way: x y if and only if y x P. Onewritesx y to indicate that x y, butx / y. If y x int P, onewritesx y. If int P /, the cone P is called solid. InthesequelwealwayssupposethatE is a real Banach space, P is a solid cone in E, and is partial ordering with respect to P. Analogously with definition of metric type space, given in 16, we consider cone metric type space. Definition.1. Let X be a nonempty set and E a real Banach space with cone P. Avectorvalued function d : X X E is said to be a cone metric type function on X with constant K 1 if the following conditions are satisfied: d 1 θ d x, y for all x, y X and d x, y θ if and only if x y; d d x, y d y, x for all x, y X; d 3 d x, y K d x, z d z, y for all x, y, z X. The pair X, d is called a cone metric type space in brief CMTS. Remark.. For K 1inDefinition.1 we obtain a cone metric space introduced in 1. Definition.3. Let X, d be a CMTS and x n } asequenceinx. c 1 x n } converges to x X if for every c E with θ c there exists n 0 N such that d x n,x c for all n>n 0. We write lim n x n x, orx n x, n. c If for every c E with θ c there exists n 0 N such that d x n,x m c for all n, m > n 0,thenx n } is called a Cauchy sequence in X. If every Cauchy sequence is convergent in X, thenx is called a complete CMTS.

3 Fixed Point Theory and Applications 3 Example.4. Let B e i i 1,...,n} be orthonormal basis of R n with inner product,.let p>0, and define X p x x : 0, 1 R n, 1 0 x t,e k p dt R, k 1,...,n },.1 where x represents class of element x with respect to equivalence relation of functions equal almost everywhere. We choose E R n and P B y R n ( y, e i ) 0, i 1,...,n }.. We show that P B is a solid cone. Let y k P B, k N, with property lim k y k y. Since scalar product is continuous, we get lim k y k,e i lim k y k,e i y, e i, i 1,...,n. Clearly, it must be y, e i 0, i 1,...,n,and,hence,y P B,thatis,P B is closed. It is obvious that θ / e 1 P B / θ},andfora, b 0, and all z, y P B,wehave az by, e i a z, e i b y, e i 0, i 1,...,n. Finally, if z P B P B we have z, e i 0and z, e i 0, i 1,...,n,andit follows that z, e i 0, i 1,...,n,and,sinceB is complete, we get z 0. Let us choose z n i 1 e i. It is obvious that z int P B, since if not, for every ε>0thereexistsy/ P B such that 1 y, e i n i 1 1 y, e i 1/ z y <ε. If we choose ε 1/4, we conclude that it must be y, e i > 1 1/4 > 0, hence y P B, which is contradiction. Finally, define d : X p X p P B by d ( f, g ) n i 1 e i 1 0 (( f g ) t,e i ) p dt, f, g Xp..3 Then it is obvious that X p,d is CMTS with K p 1.Letf, g, h be functions such that f, e 1 1, g,e 1, h, e 1 0, and f, e i g,e i h, e i 0, i,...,n,withp give d f, g 9e 1, d f, h e 1,andd h, g 4e 1,whichproves5e 1 d f, h d h, g d f, g 9e 1,but9e 1 d f, g d f, h d h, g 10e 1. The following properties are well known in the case of a cone metric space, and it is easy to see that they hold also in the case of a CMTS. Lemma.5. Let X, d be a CMTS over-ordered real Banach space E with a cone P. The following properties hold a, b, c E. p 1 If a b and b c, thena c. p If θ a c for all c int P,thena θ. p 3 If a λa,wherea P and 0 λ<1,thena θ. p 4 Let x n θ in E and let θ c. Then there exists positive integer n 0 such that x n c for each n>n 0. Definition.6 see 17. LetF, G : X X be mappings of a set X. Ify Fx Gx for some x X, thenx is called a coincidence point of F and G, andy is called a point of coincidence of F and G.

4 4 Fixed Point Theory and Applications Definition.7 see 17. LetF and G be self-mappings of set X and C F, G x X : Fx Gx}. ThepairF, G} is called weakly compatible if mappings F and G commute at all their coincidence points, that is, if FGx GFx for all x C F, G. Lemma.8 see 5. Let F and G be weakly compatible self-mappings of a set X. IfF and G have a unique point of coincidence y Fx Gx, theny is the unique common fixed point of F and G. 3. Main Results Theorem 3.1. Let X, d be a CMTS with constant 1 K and P a solid cone. Suppose that self-mappings F, G, S, T : X X are such that SX GX, TX FX and that for some constant λ 0, 1/K for all x, y X there exists u ( x, y ) Kd ( Fx,Gy ),Kd Fx,Sx, Kd ( Gy, Ty ),K d( Fx,Ty ) d ( Gy, Sx ) }, 3.1 such that the following inequality d ( Sx, Ty ) λ K u( x, y ), 3. holds. If one of SX, TX, FX,orGX is complete subspace of X, thens, F} and T, G} have a unique point of coincidence in X. Moreover,ifS, F} and T, G} are weakly compatible pairs, then F, G, S, and T have a unique common fixed point. Proof. Let us choose x 0 X arbitrary. Since SX GX, thereexistsx 1 X such that Gx 1 Sx 0 z 0.SinceTX FX,thereexistsx X such that Fx Tx 1 z 1.Wecontinueinthis manner. In general, x n 1 X is chosen such that Gx n 1 Sx n z n,andx n X is chosen such that Fx n Tx n 1 z n 1. First we prove that d z n,z n 1 αd z n 1,z n, n 1, 3.3 where α maxλ, λk/ λk }, which will lead us to the conclusion that z n } is a Cauchy sequence, since α 0, 1 it is easy to see that 0 <λk/ λk < 1. Toprovethis,itis necessary to consider the cases of an odd integer n and of an even n. For n l 1, l N0, wehaved z l 1,z l d Sx l,tx l 1,andfrom 3. there exists u x l,x l 1 Kd Fx l,gx l 1,Kd Fx l,sx l, Kd Gx l 1,Tx l 1,K d Fx } l,tx l 1 d Gx l 1,Sx l Kd z l 1,z l,kd z l 1,z l, Kd z } l,z l, 3.4

5 Fixed Point Theory and Applications 5 such that d z l 1,z l λ/k u x l,x l 1. Thus we have the following three cases: i d z l 1,z l λd z l 1,z l ; ii d z l 1,z l λd z l 1,z l, which, because of property p 3, implies d z l 1,z l θ; iii d z l 1,z l λ/ d z l,z l,thatis,byusing d 3, d z l 1,z l λk d z l,z l 1 λk d z l 1,z l, 3.5 which implies d z l 1,z l λk/ λk d z l,z l 1. Thus, inequality 3.3 holds in this case. For n l, l N0, wehave d z l,z l 1 d Sx l,tx l 1 λ K u x l,x l 1, 3.6 where u x l,x l 1 Kd Fx l,gx l 1,Kd Fx l,sx l, Kd Gx l 1,Tx l 1,K d Fx } l,t l 1 d Gx l 1,Sx l Kd z l 1,z l,kd z l,z l 1, Kd z } l 1,z l Thus we have the following three cases: i d z l,z l 1 λd z l 1,z l ; ii d z l,z l 1 λd z l,z l 1, which implies d z l,z l 1 θ; iii d z l,z l 1 λ/ d z l 1,z l 1 λk/ d z l 1,z l λk/ d z l,z l 1, which implies d z l,z l 1 λk/ λk d z l,z l 1. So, inequality 3.3 is satisfied in this case, too. Therefore, 3.3 is satisfied for all n N0, and by iterating we get d z n,z n 1 α n d z 0,z Since K 1, for m>nwe have d z n,z m Kd z n,z n 1 K d z n 1,z n K m n 1 d z m 1,z m ( Kα n K α n 1 K m n α m 1) d z 0,z Kαn 1 Kα d z 0,z 1 θ, as n.

6 6 Fixed Point Theory and Applications Now, by p 4 and p 1, it follows that for every c int P there exists positive integer n 0 such that d z n,z m c for every m>n>n 0,soz n } is a Cauchy sequence. Let us suppose that SX is complete subspace of X. Completeness of SX implies existence of z SX such that lim n z n lim n Sx n z. Then, we have lim Gx n 1 lim Sx n lim Fx n lim Tx n 1 z, n n n n 3.10 that is, for any θ c, forsufficiently large n we have d z n,z c. Sincez SX GX, there exists y X such that z Gy. Letusprovethatz Ty.From d 3 and 3.,wehave d ( Ty,z ) Kd ( Ty,Sx n ) Kd Sxn,z λu ( x n,y ) Kd z n,z, 3.11 where u ( x n,y ) Kd ( Fx n,gy ),Kd Fx n,sx n,kd ( Gy, Ty ),K d( Fx n,ty ) d ( )} Gy, Sx n Kd z n 1,z,Kd z n 1,z n,kd ( z, Ty ),K d( z n 1,Ty ) } d z, z n. 3.1 Therefore we have the following four cases: i d Ty,z Kλd z n 1,z Kd z n,z Kλ c/ Kλ K c/ K c,asn ; ii d Ty,z Kλd z n 1,z n Kd z n,z Kλ c/ Kλ K c/ K c,asn ; iii d Ty,z Kλd z, Ty Kd z n,z, thatis, d ( Ty,z ) K 1 Kλ d z n,z K 1 Kλ 1 Kλ c c, as n ; 3.13 K iv d Ty,z Kλ/ d z n 1,Ty d z, z n Kd z n,z,thatis,becauseof d 3, d ( Ty,z ) Kλ ( Kd zn 1,z Kd ( z, Ty ) d z, z n ) Kd z n,z, 3.14 which implies d ( Ty,z ) [ 1 K ( ) ] λ Kλ 1 K λ/ d z n 1,z K d z n,z K λ K λ K λ K λ c K λ K λ K λ c c, as n, K λ 3.15 since from 1 K andλ 0, 1/K we have λ<1/k /K, and therefore 1 K λ/ > 0.

7 Fixed Point Theory and Applications 7 Therefore, d Ty,z c for each c int P. So,by p we have d Ty,z θ, thatis, Ty Gy z, y is a coincidence point, and z is a point of coincidence of T and G. Since TX FX,thereexistsv X such that z Fv.LetusprovethatSv z. From d 3 and 3., wehave d Sv, z Kd Sv, Tx n 1 Kd Tx n 1,z λu v, x n 1 Kd z n 1,z, 3.16 where u v, x n 1 Kd Fv,Gx n 1,Kd Fv,Sv,Kd Gx n 1,Tx n 1,K d Fv,Tx } n 1 d Gx n 1,Sv Kd z, z n,kd z, Sv,Kd z n,z n 1,K d z, z } n 1 d z n,sv Therefore we have the following four cases: i d Sv, z Kλd z, z n Kd z n 1,z ; ii d Sv, z Kλd z, Sv Kd z n 1,z ; iii d Sv, z Kλd z n,z n 1 Kd z n 1,z ; iv d Sv, z Kλ/ d z, z n 1 d z n,sv Kd z n 1,z. By the same arguments as above, we conclude that d Sv, z θ, thatis,sv Fv z. So, z is a point of coincidence of S and F, too. Now we prove that z is unique point of coincidence of pairs S, F} and T, G}. Suppose that there exists z which is also a point of coincidence of these four mappings, that is, Fv Gy Sv Ty z.from 3., d z, z d ( Sv, Ty ) λ K u( v, y ), 3.18 where u ( v, y ) Kd ( Fv,Gy ),Kd Fv,Sv,d ( Gy,Ty ),K d( Fv,Ty ) d ( Gy,Sv ) } 3.19 Kd z, z,θ}. Using p 3 we get d z, z θ, thatis,z z. Therefore, z is the unique point of coincidence of pairs S, F} and T, G}.Ifthesepairsareweakly compatible, then z is the unique common fixed point of S, F, T, andg, bylemma.8. Similarly, we can prove the statement in the cases when FX, GX, ortx is complete.

8 8 Fixed Point Theory and Applications We give one simple, but illustrative, example. Example 3.. Let X R, E R, andp 0,.Letusdefined x, y x y for all x, y X. Then X, d is a CMTS, but it is not a cone metric space since the triangle inequality is not satisfied. Starting with Minkowski inequality see 18 for p, by using the inequality of arithmetic and geometric means, we get x z x y y z x y x z ( x y y z ). 3.0 Here, K. Let us define four mappings S, F, T, G : X X as follows: Sx M ax b, Fx ax b, Tx M cx d, Gx cx d, 3.1 where x X, a / 0, c / 0, and M < 1/. Since SX FX TX GX X we have trivially SX GX and TX FX. Also,X is a complete space. Further, d Sx, Ty M ax b M cy d M d Fx,Gy, that is, there exists λ M < 1/ 1/K such that 3. is satisfied. According to Theorem 3.1, S, F} and T, G} have a unique point of coincidence in X, that is, there exists unique z X and there exist x, y X such that z Sx Fx Ty Gy. It is easy to see that x b/a, y d/c, andz 0. If S, F} is weakly compatible pair, we have SFx FSx, which implies Mb b, that is, b 0. Similarly, if T, G} is weakly compatible pair, we have TGy GTy, which implies Md d, thatis,d 0. Then x y 0, and z 0 is the unique common fixed point of these four mappings. The following two theorems can be proved in the same way as Theorem 3.1, sowe omit the proofs. Theorem 3.3. Let X, d be a CMTS with constant K and P a solid cone. Suppose that selfmappings F, G, S, T : X X are such that SX GX, TX FX and that for some constant λ 0, /K for all x, y X there exists u ( x, y ) Kd ( Fx,Gy ),Kd Fx,Sx,Kd ( Gy, Ty ),K d( Fx,Ty ) d ( Gy, Sx ) }, 3. such that the following inequality d ( Sx, Ty ) λ K u( x, y ), 3.3 holds. If one of SX, TX, FX,orGX is complete subspace of X, thens, F} and T, G} have a unique point of coincidence in X. Moreover,ifS, F} and T, G} are weakly compatible pairs, then F, G, S, and T have a unique common fixed point.

9 Fixed Point Theory and Applications 9 Theorem 3.4. Let X, d be a CMTS with constant K 1 and P a solid cone. Suppose that selfmappings F, G, S, T : X X are such that SX GX, TX FX and that for some constant λ 0, 1/K for all x, y X there exists u ( x, y ) Kd ( Fx,Gy ),Kd Fx,Sx,Kd ( Gy, Ty ), d( Fx,Ty ) d ( Gy, Sx ) }, 3.4 such that the following inequality d ( Sx, Ty ) λ K u( x, y ), 3.5 holds. If one of SX, TX, FX,orGX is complete subspace of X, thens, F} and T, G} have a unique point of coincidence in X. Moreover,ifS, F} and T, G} are weakly compatible pairs, then F, G, S, and T have a unique common fixed point. Theorems 3.1 and 3.4 are generalizations of 13, Theorem.. As a matter of fact, for K 1, from Theorems 3.1 and 3.4,weget 13, Theorem.. If we choose T S and G F, fromtheorems3.1, 3.3, and3.4 we get the following resultsfortwomappingsoncmts. Corollary 3.5. Let X, d be a CMTS with constant 1 K and P a solid cone. Suppose that self-mappings F, S : X X are such that SX FX and that for some constant λ 0, 1/K for all x, y X there exists u ( x, y ) Kd ( Fx,Fy ),Kd Fx,Sx,Kd ( Fy,Sy ),K d( Fx,Sy ) d ( Fy,Sx ) }, 3.6 such that the following inequality d ( Sx, Sy ) λ K u( x, y ), 3.7 holds. If FX or SX is complete subspace of X, thenf and S have a unique point of coincidence in X. Moreover, if F, S} is a weakly compatible pair, then F and S have a unique common fixed point. Corollary 3.6. Let X, d be a CMTS with constant K and P a solid cone. Suppose that selfmappings F, S : X X are such that SX FX and that for some constant λ 0, /K for all x, y X there exists u ( x, y ) Kd ( Fx,Fy ),Kd Fx,Sx,Kd ( Fy,Sy ),K d( Fx,Sy ) d ( Fy,Sx ) }, 3.8 such that the following inequality d ( Sx, Sy ) λ K u( x, y ), 3.9

10 10 Fixed Point Theory and Applications holds. If FX or SX is complete subspace of X, thenf and S have a unique point of coincidence in X. Moreover, if F, S} is a weakly compatible pair, then F and S have a unique common fixed point. Corollary 3.7. Let X, d be a CMTS with constant K 1 and P a solid cone. Suppose that selfmappings F, S : X X are such that SX FX and that for some constant λ 0, 1/K for all x, y X there exists u ( x, y ) Kd ( Fx,Fy ),Kd Fx,Sx,Kd ( Fy,Sy ), d( Fx,Sy ) d ( Fy,Sx ) }, 3.30 such that the following inequality d ( Sx, Sy ) λ K u( x, y ), 3.31 holds. If FX or SX is complete subspace of X, thenf and S have a unique point of coincidence in X. Moreover, if F, S} is a weakly compatible pair, then F and S have a unique common fixed point. Theorem 3.8. Let X, d be a CMTS with constant K 1 and P a solid cone. Suppose that selfmappings F, G, S, T : X X are such that SX GX, TX FX and that there exist nonnegative constants a i, i 1,...,5, satisfying a 1 a a 3 K maxa 4,a 5 } < 1, a 3 K a 4 K < 1, a K a 5 K < 1, 3.3 such that for all x, y X inequality d ( Sx, Ty ) a 1 d ( Fx,Gy ) a d Fx,Sx a 3 d ( Gy, Ty ) a 4 d ( Fx,Ty ) a 5 d ( Gy, Sx ), 3.33 holds. If one of SX, TX, FX,orGX is complete subspace of X, thens, F} and T, G} have a unique point of coincidence in X. Moreover,ifS, F} and T, G} are weakly compatible pairs, then F, G, S, and T have a unique common fixed point. Proof. We define sequences x n } and z n } as in the proof of Theorem 3.1. First we prove that d z n,z n 1 αd z n 1,z n, n 1, 3.34 where a1 a 3 a 5 K α max 1 a a 5 K, a } 1 a a 4 K, a 3 a 4 K which implies that z n } is a Cauchy sequence, since, because of 3.3, it is easy to check that α 0, 1. To prove this, it is necessary to consider the cases of an odd and of an even integer n.

11 Fixed Point Theory and Applications 11 have For n l 1, l N0, wehaved z l 1,z l d Sx l,tx l 1,andfrom 3.33 we d Sx l,tx l 1 a 1 d Fx l,gx l 1 a d Fx l,sx l a 3 d Gx l 1,Tx l 1 a 4 d Fx l,tx l 1 a 5 d Gx l 1,Sx l, 3.36 that is, d z l 1,z l a 1 d z l 1,z l a d z l 1,z l a 3 d z l,z l 1 a 4 d z l 1,z l 1 a 5 d z l,z l a 1 a 3 d z l,z l 1 a d z l 1,z l a 5 d z l,z l a 1 a 3 d z l,z l 1 a d z l 1,z l a 5 Kd z l,z l a 5 Kd z l 1,z l a 1 a 3 a 5 K d z l,z l 1 a a 5 K d z l 1,z l. Therefore, d z l 1,z l a 1 a 3 a 5 K 1 a a 5 K d z l,z l 1, 3.38 that is, inequality 3.34 holds in this case. Similarly, for n l, l N0, wehaved z l,z l 1 d Sx l,tx l 1,andfrom 3.33 we have d Sx l,tx l 1 a 1 d Fx l,gx l 1 a d Fx l,sx l a 3 d Gx l 1,Tx l 1 a 4 d Fx l,tx l a 5 d Gx l 1,Sx l, that is, d z l,z l 1 a 1 d z l 1,z l a d z l 1,z l a 3 d z l,z l 1 a 4 d z l 1,z l 1 a 5 d z l,z l a 1 a d z l 1,z l a 3 d z l,z l 1 a 4 d z l 1,z l 1 a 1 a d z l 1,z l a 3 d z l,z l 1 a 4 Kd z l 1,z l a 4 Kd z l,z l 1 a 1 a a 4 K d z l 1,z l a 3 a 4 K d z l,z l

12 1 Fixed Point Theory and Applications Thus, d z l,z l 1 a 1 a a 4 K 1 a 3 a 4 K d z l 1,z l, 3.41 and inequality 3.34 holds in this case, too. BythesameargumentsasinTheorem 3.1 we conclude that z n } is a Cauchy sequence. Let us suppose that SX is complete subspace of X. Completeness of SX implies existence of z SX such that lim n z n lim n Sx n z. Then, we have lim n Gx n 1 lim n Sx n lim n Fx n lim n Tx n 1 z, 3.4 that is, for any θ c, forsufficiently large n we have d z n,z c. Sincez SX GX, there exists y X such that z Gy. Letusprovethatz Ty.From d 3 and 3.33, wehave d ( Ty,z ) Kd ( ) Ty,Sx n Kd Sxn,z a 1 Kd ( Fx n,gy ) a Kd Fx n,sx n a 3 Kd ( Gy, Ty ) a 4 Kd ( Fx n,ty ) a 5 Kd ( Gy, Sx n ) Kd Sxn,z a 1 Kd z n 1,z a Kd z n 1,z n a 3 Kd ( z, Ty ) 3.43 a 4 Kd ( z n 1,Ty ) a 5 Kd z, z n Kd z n,z a 1 Kd z n 1,z a Kd z n 1,z n a 3 Kd ( z, Ty ) a 4 K d z n 1,z a 4 K d ( z, Ty ) a 5 Kd z, z n Kd z n,z. The sequence z n } converges to z, soforeachc int P there exists n 0 N such that for every n>n 0 d ( Ty,z ) 1 1 a 3 K a 4 K ( ) a 1 Kd z n 1,z a Kd z n 1,z n a 4 K d z n 1,z a 5 Kd z, z n Kd z n,z a 1 K 1 a 3 K a 4 K 1 a 3K a 4 K c a 1 K 5 a K 1 a 3 K a 4 K 1 a 3K a 4 K c a K 5 c, a 4 K 1 a 3 K a 4 K 1 a 3K a 4 K c a 4 K 5 a 5 K 1 a 3 K a 4 K 1 a 3K a 4 K c a 5 K 5 K 1 a 3 K a 4 K 1 a 3K a 4 K c K

13 Fixed Point Theory and Applications 13 because of 3.3. Now,by p it follows that d Ty,z θ, thatis,ty z. So,wehave Ty Gy z, thatis,y is a coincidence point, and z is a point of coincidence of mappings T and G. Since TX FX, thereexistsv X such that z Fv. LetusprovethatSv z, too. From d 3 and 3.33,wehave d Sv, z Kd Sv, Tx n 1 Kd Tx n 1,z a 1 Kd Fv,Gx n 1 a Kd Fv,Sv a 3 Kd Gx n 1,Tx n 1 a 4 Kd Fv,Tx n 1 a 5 Kd Gx n 1,Sv Kd Tx n 1,z a 1 Kd z, z n a Kd z, Sv a 3 Kd z n,z n a 4 Kd z, z n 1 a 5 Kd z n,sv Kd Tx n 1,z a 1 Kd z, z n a Kd z, Sv a 3 Kd z n,z n 1 a 4 Kd z, z n 1 a 5 K d z n,z a 5 K d Sv, z Kd Tx n 1,z, and by the same arguments as above, we conclude that d Sv, z θ, thatis,sv Fv z. Thus, z is a point of coincidence of mappings S and F, too. Suppose that there exists z which is also a point of coincidence of these four mappings, that is, Fv Gy Sv Ty z.from 3.33 we have d z, z d ( Sv, Ty ) a 1 Kd ( Fv,Gy ) a Kd Fv,Sv a 3 Kd ( Gy,Ty ) a 4 Kd ( Fv,Ty ) a 4 Kd ( Gy,Sv ) 3.46 a 1 Kd z, z a Kd z, z a 3 Kd z,z a 4 Kd z, z a 5 Kd z,z a 1 a 4 a 5 Kd z, z, and because of p 3 it follows that z z. Therefore, z is the unique point of coincidence of pairs S, F} and T, G}, and we have z Sv Fv Gy Ty.IfS, F} and T, G} are weakly compatible pairs, then z is the unique common fixed point of S, F, T, andg, bylemma.8. The proofs for the cases in which FX, GX,orTX is complete are similar. Theorem 3.8 is a generalization of 13, Theorem.8. Choosing K 1 from Theorem 3.8 we get the following corollary. Corollary 3.9. Let X, d be cone metric space and P a solid cone. Suppose that self-mappings F, G, S, T : X X are such that SX GX, TX FX and that there exist nonnegative constants a i, i 1,...,5, satisfying a 1 a a 3 maxa 4,a 5 } < 1, such that for all x, y X inequality d ( Sx, Ty ) a 1 d ( Fx,Gy ) a d Fx,Sx a 3 d ( Gy, Ty ) a 4 d ( Fx,Ty ) a 5 d ( Gy, Sx ), 3.47

14 14 Fixed Point Theory and Applications holds. If one of SX, TX, FX,orGX is complete subspace of X, thens, F} and T, G} have a unique point of coincidence in X. Moreover,ifS, F} and T, G} are weakly compatible pairs, then F, G, S, and T have a unique common fixed point. If we choose T S and G F, fromtheorem 3.8, we get the following result for two mappings on CMTS. Corollary Let X, d be a CMTS with constant K 1 and P a solid cone. Suppose that selfmappings F, S : X X are such that SX FX and that there exist nonnegative constants a i, i 1,...,5, satisfying a 1 a a 3 K maxa 4,a 5 } < 1, a 3 K a 4 K < 1, a K a 5 K < 1, 3.48 such that for all x, y X inequality d ( Sx, Sy ) a 1 d ( Fx,Fy ) a d Fx,Sx a 3 d ( Fy,Sy ) a 4 d ( Fx,Sy ) a 5 d ( Fy,Sx ), 3.49 holds. If one of SX or FX is complete subspace of X, thens and F have a unique point of coincidence in X.Moreover,ifF, S} is a weakly compatible pair, then F and S have a unique common fixed point. Acknowledgments The authors are indebted to the referees for their valuable suggestions, which have contributed to improve the presentation of the paper. The first two authors were supported in part by the Serbian Ministry of Science and Technological Developments Grant no References 1 L.-G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, Journal of Mathematical Analysis and Applications, vol. 33, no., pp , 007. K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, P. Vetro, Common fixed points in cone metric spaces, Rendiconti del Circolo Matematico di Palermo, vol. 56, no. 3, pp , D. Ilić and V. Rakočević, Common fixed points for maps on cone metric space, Journal of Mathematical Analysis and Applications, vol. 341, no., pp , M. Abbas and G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp , C.DiBariandP.Vetro, ϕ-pairs and common fixed points in cone metric spaces, Rendiconti del Circolo Matematico di Palermo, vol. 57, no., pp , S. Radenović, Common fixed points under contractive conditions in cone metric spaces, Computers & Mathematics with Applications, vol. 58, no. 6, pp , G. Jungck, S. Radenović, S. Radojević, and V. Rakočević, Common fixed point theorems for weakly compatible pairs on cone metric spaces, Fixed Point Theory and Applications, vol. 009, Article ID , 13 pages, B. S. Choudhury and N. Metiya, The point of coincidence and common fixed point for a pair of mappings in cone metric spaces, Computers & Mathematics with Applications, vol. 60, no. 6, pp , 010.

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