Rendiconti del Circolo Matematico di Palermo 57, 433 441 (2008) DOI: 10.1007/s12215-008-0032-5 Akbar Azam Muhammad Arshad Ismat Beg Common fixed points of two maps in cone metric spaces Received: July 3, 2008/ Accepted: November 7, 2008 c Springer-Verlag 2008 Abstract. We prove the existence of points of coincidence and common fixed points of a pair of self mappings satisfying a generalized contractive condition in cone metric spaces. Our results generalize several well-known recent and classical results. Keywords Point of coincidence Common fixed point Contractive type mapping Commuting mapping Compatible mapping Cone metric space. Mathematics Subject Classification (2000) 47H10 54H25 1 Introduction Sessa [12] generalized the concept of commuting mappings [5] by calling self mappings f, g on a metric space X, weakly commuting if and only if d( fgx,gfx) d( fx,gx) for all x X. A. Azam Department of Mathematics, Faculty of Basic and Applied Sciences, International Islamic University, H-10, Islamabad, Pakistan, and Department of Mathematics, F.G. Postgraduate College, H-8, Islamabad, Pakistan E-mail: akbarazam@yahoo.com M. Arshad Department of Mathematics, Faculty of Basic and Applied Sciences, International Islamic University, H-10, Islamabad, Pakistan E-mail: marshad zia@yahoo.com I. Beg (B) Centre for Advanced Studies in Mathematics, Lahore University of Management Sciences, 54792-Lahore, Pakistan E-mail: ibeg@lums.edu.pk
434 A. Azam et al. Commuting mappings are weakly commuting but converse is not true in general (see [12]). Afterwards, Jungck [6] and Pant [8] introduced some less restrictive concepts of compatible mappings and R-weakly commuting mappings in order to improve the concept of weakly commuting mappings and generalized some common fixed point theorems respectively. Later on, it has been noticed that compatible mappings and R-weakly commuting mappings commute at their coincidence points. Jungck and Rhoades [7], then defined a pair of self-mappings to be weakly compatible if they commute at their coincidence points. Recently, Huang and Zhang [3] generalized the notion of metric spaces by replacing the real numbers by ordered Banach space and define cone metric spaces. They have proved Banach contraction mapping theorem and some other fixed point theorems of contractive type mappings in cone metric spaces. Subsequently, Abbas and Jungck [1], Ilic and Rakocevic [4], Rezapour and Hamlbarani [9] and Vetro [13] studied fixed point theorems for contractive type mappings in cone metric spaces. The aim of this paper is to obtain points of coincidence and common fixed points of a pair of self mappings satisfying a generalized contractive type conditionin a cone metric space. Our results generalized several existing fixed point theorems including [1 3,5,9,10]. A subset P of a real Banach space Z is called a cone if it has following properties: (i) P is nonempty closed and P {0}; (ii) 0 a,b R and x, y P>ax+by P; (iii) P ( P)={0}. For a given cone P Z, we can define a partial ordering on Z with respect to P by x y if and only if y x P. We shall write x < y if x y and x y, while x y will stands for y x intp, where intp denotes the interior of P. A cone P is called normal if there is a number κ > 0 such that for all x, y Z, 0 x y > x κ y. (1) The least positive number κ satisfying (1) is called the normal constant of P. There are no normal cones with normal constant κ < 1 [9]. Also [9, example 2.3] shows that there are non-normal cones. In the following we always suppose that Z is a real Banach space and P is a cone in Z with intp φ and is a partial ordering with respect to P. Definition 1 Let X be a nonempty set. Suppose the mapping d : X X Z, satisfies 1. 0 d(x,y), for all x, y X and d(x,y)=0 if and only if x = y;
Common fixed points of two maps in cone metric spaces 435 2. d(x,y)=d(y,x) for all x,y X; 3. d(x,y) d(x,z)+d(z,y) for all x,y,z X. Then d is called a cone metric on X, and(x,d) is called a cone metric space. Let x X, and {x n } be a sequence in X. Ifforeveryc Z, with 0 c there is n 0 N such that for all n n 0, d(x n,x) c, then {x n } is said to be convergent, {x n } converges to x and x is the limit of {x n }. We denote this by lim n x n = x, or x n x, as n. Ifforeveryc X with 0 c there is n 0 N such that for all n,m n 0, d(x n,x m ) c, then {x n } is called a Cauchy sequence in X. If every Cauchy sequence is convergent in X, thenx is called a complete cone metric space. Let us recall [3] that if P is a normal cone, then x n X converges to x X if and only if d(x n,x) 0 as n. Further, x n X is a Cauchy sequence if and only if d(x n,x m ) 0 as n,m. A point x X is called coincidence point of two mappings T, f : X X if fx= Tx. Definition 2 A point y X is called point of coincidence of two mappings T, f : X X if there exists a point x X such that y = fx= Tx. Let (X,d) be a complete cone metric space, P be a normal cone with normal constant κ. Suppose that the mappings T, f : X X satisfy: d(tx,ty) Ad( fx, fy)+bd( fx,tx)+cd( fy,ty) +D d( fx,ty)+ed( fy,tx), (2) for all x, y X where A,B,C,D,E are non-negative real numbers. Huang and Zhang [3] proved that T has a unique fixed point if (a) f = I, wherei is the identity mapping on X (see [2,10,11]) and (b) one of the following is satisfied: (i) B = C = D = E = 0 with A < 1 ([3, theorem 1] ), (ii) A = D = E = 0 with B = C < 1 2 ([3, theorem 3]), (iii) A = B = C = 0 with D = E < 1 2 ([3, theorem 4]). Abbas and Jungck [1] proved that f and T have a unique point of coincidence and unique common fixed point if: (a) the mappings f and T are weakly compatible and (b) one of the following is satisfied: (i) B = C = D = E = 0 with A < 1 ([1, theorem 2.1] ), (ii) A = D = E = 0 with B = C < 1 2 ([1,theorem2.3]), (iii) A = B = C = 0 with D = E < 1 2 ([1, theorem 2.4]). Rezapour and Hamlbarani [9] generalized some results of [3] by omitting the assumption of normality on X.
436 A. Azam et al. 2Mainresults Theorem 1 Let (X, d) be a cone metric space. Suppose the mappings T, f : X X satisfy d(tx,ty) Ad( fx, fy)+b [d( fx,tx)+d( fy,ty)] +C [d( fx,ty)+d( fy,tx)], (3) for all x,y X where A,B,C are non-negative real numbers with A+2B+2C < 1. If T (X) f (X) and f (X) or T(X) is a complete subspace of X, then T and f have a unique point of coincidence. Proof Let x 0 be an arbitrary point in X. Choose a point x 1 in X such that fx 1 = Tx 0. This can be done since T (X) f (X). Similarly, choose a point x 2 in X, such that fx 2 = Tx 1. Continuing this process and having chosen x n in X, we obtain x n+1 in X such that Then fx k+1 = Tx k, k = 0,1,2,... d( fx k+1, fx k+2 )=d(tx k,tx k+1 ) Ad( fx k, fx k+1 )+B [d( fx k,tx k )+d( fx k+1,tx k+1 )] +C [d( fx k,tx k+1 )+d( fx k+1,tx k )] [A +B] d( fx k, fx k+1 )+Bd( fx k+1, fx k+2 ) +C d( fx k, fx k+2 ) [A +B +C] d( fx k, fx k+1 )+[B +C] d( fx k+1, fx k+2 ). It implies that []d( fx k+1, fx k+2 ) [A +B +C] d( fx k, fx k+1 ). That is [ ] A +B +C d( fx k+1, fx k+2 ) d( fx k, fx k+1 ). Moreover, [ ] A +B +C 2 d( fx k+1, fx k+2 ) d( fx k 1, fx k ) [ ] A +B +C k+1 d( fx 0, fx 1 ). Putting, [ ] A +B +C y n = fx n and λ =.
Common fixed points of two maps in cone metric spaces 437 We have, For n > m d(y n,y n+1 ) λ n d(y 0,y 1 ). d(y n,y m ) d(y n,y n 1 )+d(y n 1,y n 2 )+ +d(y m+1,y m ) ( λ n 1 +λ n 2 + +λ m) d(y 0,y 1 ) λ m 1 λ d(y 0,y 1 ). Let 0 c be given. Choose δ > 0 such that c +{x Z : x < δ} P. Also choose a natural number N 1 such that λ m 1 λ d(y 0,y 1 ) {x Z : x < δ}, for all m N 1. Then λ m 1 λ d(y 0,y 1 ) c, for all m N 1. Thus, n > m>d(y n,y m ) λ m 1 λ d(y 0,y 1 ) c, which implies that {y n } is a Cauchy sequence. We assume that f (X) is complete, then there exists u,v X such that y n v = fu. and Choose a natural number N 2 such that for all n N 2 [ ] c() d(y n 1,y n ),d(y n 1,v) 3B d(y n,v) Now, inequality (3) implies that d( fu,tu) d( fu,y n )+d(y n,tu) d(v,y n )+d(tx n 1,Tu) [ ] c(). 3(1 +C) [ c() 3(A +C) d(v,y n )+Ad( fu, fx n 1 )+B[d( fu,tu)+d( fx n 1,Tx n 1 )] +C[d( fu,tx n 1 )+d( fx n 1,Tu)] d(v,y n )+Ad(v,y n 1 )+B[d( fu,tu)+d(y n 1, y n )] +C[d(v,y n )+d(y n 1,v)+ d( fu,tu)] (1 +C)d(v,y n )+(A +C)d(v,y n 1 )+Bd(y n 1,y n ) +(B +C) d( fu,tu). ]
438 A. Azam et al. Consequently, [ ] 1 +C d( fu,tu) [ B + It further implies that [ A +C d(v,y n )+ ] d(y n 1, y n ). d( fu,tu) c 3 + c 3 + c 3 = c. ] d(v,y n 1 ) Thus, d( fu,tu) c, for all m 1. m c So, m d( fu,tu) P, for all m 1. Since m c 0 (as m ) and P is closed, d( fu,tu) P. But P ( P)={0}. Therefore, d( fu,tu)=0. Hence v = fu= Tu. Next we show that f and T have a unique point of coincidence. For this, assume that there exists another point v in X such that v = fu = Tu for some u in X. Now d(v,v )=d(tu,tu ) Ad( fu, fu )+B[d( fu,tu)+d( fu,tu )] +C[d( fu,tu )+d( fu,tu)] Ad(v,v )+C[d(v,v )+d(v,v)] (A +2C) d(v,v ), hence v = v. On the other hand, if we assume that T (X) is complete, then the Cauchy sequence y n = fx n = Tx n 1 converges to v TX. But TX fx which allows us to obtain u fx such that v = fu. The rest of the proof is similar to the previous case. Theorem 2 If in addition to the hypotheses of Theorem 1 the mappings T, f : X X are weakly compatible, then T and f have a unique common fixed point. Proof As in the proof of Theorem 1, there is a unique point of coincidence v of f and T. NowT, f are weakly compatible, therefore Tv= Tfu= ftu= fv. It implies that Tv= fv= w (say). Then w is a point of coincidence of T and f, therefore v = w by uniqueness. Thus v is a unique common fixed point of T and f.
Common fixed points of two maps in cone metric spaces 439 Theorem 3 Let (X,d) be a cone metric space. Suppose that the mappings T, f : X X satisfy (2), for all x,y XwhereA,B,C,D and E are nonnegative real numbers with A + B + C + D + E < 1. If T (X) f (X) and f (X) or T(X) is a complete subspace of X, then T and f have a unique point of coincidence. Moreover, if T, f are weakly compatible, then T and f have a unique common fixed point. Proof By hypothesis for all x,y X, we get, d(ty,tx) Ad( fy, fx)+bd( fy,ty)+cd( fx,tx) +Dd( fy,tx)+ed( fx,ty). It follows that, ( ) B +C d(tx,ty) Ad( fx, fy)+ [d( fx,tx)+d( fy,ty)] 2 ( ) D +E + [d( fx,ty)+ed( fy,tx)]. 2 The required result follows from Theorems 1 and 2. Example 1 Let X = R, Z = R 2, d(x,y) = ( x y,β x y ),β > 0, P = {(x,y) : x,y 0}, T (x)=2x 2 +4x +3 andf (x)=3x 2 +6x +4. Then TX = fx =[1, ) and all the conditions of Theorem 1 are satisfied for [ ) 2 A 3,1, B = C = 0 as we obtain 1 X as a unique point of coincidence Remark 1 (i) Note that in Example 1 1 = f ( 1)=T ( 1). Tf( 1)=T (1)=9and ft( 1)= f (1)=13. Thus T and f are not weakly compatible. It follows that except the weak compatibility of T and f all other hypotheses of Theorem 2 are satisfied but 1 f (1) T (1). It shows that the weak compatibility for T and f in Theorem 2 is an essential condition.
440 A. Azam et al. (ii) In Example 1 if we assume T (x)=2x 2 +4x +1 and f (x)=3x 2 +6x +2 then T and f become weakly compatible and all conditions of Theorems 1, 2 and 3 are satisfied to obtain a unique point of coincidence and a unique common fixed point 1 = f ( 1)=T ( 1). Our next example demonstrates the crucial role of the condition T (X) f (X) in our results. Example 2 Let X = R + (the set of all non-negative real numbers), Z = R 2, d(x,y)=( x y,e x y ), P = {(x,y) : x,y 0}, T (x)=e x and fx= e x+1. Then TX =(0, ) [e, )= fx, d(tx,ty)=( e x e y, e x+1 e y+1 ) = 1 e ( e x+1 e y+1, e x+2 e y+2 ). = 1 d( fx, fy). e It follows that all the assumptions of Theorem 1 except TX fx are satisfied for A = 1 e, B = C = 0. But T and f do not have a point of coincidence in X. Remark 2 Our results generalized several known results included among them are [1, theorems 2.1, 2.3, 2.4], [3, theorems 1, 3, 4] and [9, theorems 2.3, 2.6, 2.7, 2.8]. Acknowledgements The present version of the paper owes much to the precise and kind remarks of the learned referee. References 1. Abbas, M., Jungck, G.: Common fixed point results for non commuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl., 341 (2008), 416 420 2. Hardy, G.E., Roggers, T.D.: A generalization of a fixed point theorem of Reich, Canad. Math. Bull., 16(2) (1973),201 206 3. Huang, L.G., Zhang, X.: Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007), 1468 1476 4. Ilic, D., Rakocevic, V.: Common fixed points for maps on cone metric space, J.Math. Anal. Appl., 341 (2008), 876 882 5. Jungck, G.: Commuting maps and fixed points, Amer. Math. Monthly, 83 (1976), 261 263 6. Jungck, G.: Common fixed points for commuting and compatible maps on compacta,proc. Amer. Math. Soc., 103 (1988), 977 983 7. Jungck, G., Rhoades, B.E.: Fixed points for set valued functions without continuity, Indian J. Pure Appl. Math., 29(3) (1998), 227 238 8. Pant, R.P.: Common fixed points of noncommuting mappings, J. Math. Anal. Appl., 188 (1994), 436 440
Common fixed points of two maps in cone metric spaces 441 9. Rezapour, S., Hamlbarani, R.: Some notes on paper Cone metric spaces and fixed point theorems of contractive mappings., J. Math. Anal. Appl., 345 (2008), 719 724 10. Reich, S.: Some remarks concerning contraction mappings, Canad. Math. Bull., 14(2) (1971), 121 124 11. Rhoads, B.E.: A comparison of various definitions of contractive mappings,trans.aner. Math. Soc., 26 (1977), 257 290 12. Sessa, S.: On a weak commutativity condition of mappings in fixed point considerations, Publ. Inst. Math., 32 (1982), 149 153 13. Vetro, P.: Common fixed points in cone metric spaces, Rend. Circ. Mat. Palermo, 56 (2007), 464 468