DEMONSTRATIO MATHEMATICA Vol. XLVI No 1 2013 Luljeta Kikina, Kristaq Kikina A FIXED POINT THEOREM IN GENERALIZED METRIC SPACES Abstract. A generalized metric space has been defined by Branciari as a metric space in which the triangle inequality is replaced by a more general inequality. Subsequently, some classical metric fixed point theorems have been transferred to such a space. In this paper, we continue in this direction and prove a version of Fisher s fixed point theorem in generalized metric spaces. 1. Introduction and preliminaries In 2000, Branciari [3] introduced the concept of generalized metric space where the triangular inequality of a metric space has been replaced by an tetrahedral inequality. It is easy to see that, every metric space is a generalized metric space, but the converse need not be true [3]. He presented the well-known Banach s fixed point theorem in such a space and the following were taken for granted: a) A g.m.s is a topological space with neighborhood basis given by B = {B(x,r) : x X,r > 0}, where B(x,r) = {y X : d(x,y) < r} is the open ball with center x and radius r. b) A generalized metric d is continuous in each of coordinates and c) A g.m.s is a Hausdorff space. These were taken also for granted in ([1], [4], [5]). In 2009, Sarma et al [10] showed that the above propositions are not true, and in the same time they present a version of Banach s contraction principle in a Hausdorff g.m.s. When transferring a theorem from metric spaces to g.m.s, some more conditions should perhaps be added, for example: the continuity of generalized distance d, the Hausdorffity of (X,d), etc. 2000 Mathematics Subject Classification: 47H10, 54H25. Key words and phrases: Cauchy sequence, generalized metric spaces, fixed point, implicit relation.
182 L. Kikina, K. Kikina We conclude that the Hausdorff condition of g.m.s in Theorem 1.3 [10] is not necessary, because uniqueness of the limit of a Cauchy sequence (a n ) follows without the Hausdorff condition, in the same way as in our main Theorem 2.1. In this paper, we prove a version of Fisher s fixed point theorem [6] in generalized metric spaces. Corollaries of our main result are the theorems Rhoades [8], Kanan [7], Bianchini [2], Reich [9] extended in g.m.s. Firstly, we will give some known definitions and notations. Definition 1.1. [3] Let X be a set and d : X 2 R + a mapping such that for all x,y X and for all distinct points z,w X, each of them different from x and y, one has (i) d(x,y) = 0 if and only if x = y, (ii) d(x,y) = d(y,x), (iii) d(x,y) d(x,z)+d(z,w)+d(w,y) (rectangular property). Then d is called a generalized metric and (X,d) is a generalized metric space (or shortly g.m.s.). Let (x n ) be a sequence in X and x X. If for every ε > 0 there is an n 0 N such that d(x n,x) < ε, for all n > n 0, 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. If for every ε > 0 there is an n 0 N such that d(x n,x n+m ) < ε, for all n > n 0, then (x n ) is called a Cauchy sequence in X. If every Cauchy sequence is convergent in X, then X is called a complete generalized metric space. Any metric space is a g.m.s but the converse is not true: Example 1.2. [10] Let A = {0,2}, B = { 1 n : n N},X = A B. Define d on X X as follows: d(x,y) = 0 if x = y,d(x,y) = 1 if x y and {x,y} A or {x,y} B, d(x,y) = d(y,x) = y if x A and y B. Then it is easy to see that (X,d) is a generalized metric space but (X,d) is not a standard metric space because it lacks the triangular property: ( 1 1 = d 2, 1 ) ( ) ( 1 > d 3 2,0 +d 0, 1 ) = 1 3 2 + 1 3 = 5 6. In this g.m.s the sequence ( 1 n ) converges to both 0 and 2 and it is not a Cauchy sequence. (X,d) is not a Hausdorff space and d is not continuous distance in a sense presented in [3], lim n d( 1 n, 1 2 ) d(0, 1 2 ). Let T : X X be a mapping where X is a g.m.s. For each x X, let O(x) = {x,tx,t 2 x,...}
A fixed point theorem in generalized metric spaces 183 which will be called the orbit of T at x. O(x) is called T-orbitally complete if and only if every Cauchy sequence which is contained in O(x) converges to a point in X. First, we introduce and consider a class of implicit relations which will give a general character to the main Theorem 2.1. Definition 1.3. The set of all upper semi-continuous functions with 4 variables f : R 4 + R satisfying the properties: (a) f is non decreasing in respect with each variable, (b) f(t,t,t,t) t, t R + will be noted F 4 and every such function will be called a F 4 -function. Some examples of F 4 -function are as follows: 1. f(t 1,t 2,t 3,t 4 ) = max{t 1,t 2,t 3,t 4 }. 2. f(t 1,t 2,t 3,t 4 ) = [max{t 1 t 2,t 2 t 3,t 3 t 4,t 4 t 1 }] 1 2. 3. f(t 1,t 2,t 3,t 4 ) = [max{t p 1,tp 2,tp 3,tp 4 }]1 p. 4. f(t 1,t 2,t 3,t 4 )=(a 1 t p 1 +a 2t p 2 +a 3t p 3 +a 4t p 4 p, )1 wherep>0, 0 a i, 4 i=1 a i 1. 5. f(t 1,t 2,t 3,t 4 ) = t 1+t 2 +t 3 3 or f(t 1,t 2,t 3,t 4 ) = t 1+t 2 2 etc. 2. Main results We now prove the following fixed point theorem that generalizes and extends a theorem of Fisher [6] from metric spaces to g.m.s. Theorem 2.1. Let (X, d),(y, ρ) be two generalized metric spaces. Let T : X Y, S : Y X be two mappings, at least one of them being continuous and the following inequalities are satisfied: (1) (2) d(stx,stx ) cf 1 (d(x,x ),d(x,stx),d(x,stx ),ρ(tx,tx )), ρ(tsy,tsy ) cf 2 (ρ(y,y ),ρ(y,tsy),ρ(y,tsy ),d(sy,sy )) for all x,x X and y,y Y, where 0 c < 1 and f 1,f 2 F 4. If there exists x 0 X such that O(x 0 ) is ST-orbitally complete in X and O(Tx 0 ) is TS-orbitally complete in Y, then ST has a unique fixed point α X and TS has a unique fixed point β Y. Further, Tα = β and Sβ = α. Proof. Let x 0 be an arbitrary point in X. Define the sequences (x n ) and (y n ) inductively as follows: x n = Sy n = (ST) n x 0 and y 1 = Tx 0, y n+1 = Tx n = (TS) n y 1, n 1. Denote d n = d(x n,x n+1 ) and ρ n = ρ(y n,y n+1 ), n = 1,2,...
184 L. Kikina, K. Kikina By the inequality (2), for y = y n 1 and y = y n, we get: ρ n = ρ(y n,y n+1 ) = ρ(tsy n 1,TSy n ) cf 2 (ρ(y n 1,y n ),ρ(y n 1,TSy n 1 ),ρ(y n,tsy n ),d(sy n 1,Sy n )) = cf 2 (ρ n 1,ρ n 1,ρ n,d n 1 ). By this inequality and properties of f 2, it follows: (3) ρ n cmax{d n 1,ρ n 1 }. By the inequality (1), for x = x n 1 and x = x n, we get: and so d n = d(x n,x n+1 ) = d(stx n 1,STx n ) cf 1 (d(x n 1,x n ),d(x n 1,STx n 1 ),d(x n,stx n ),ρ(tx n 1,Tx n )) = cf 1 (d n 1,d n 1,d n,ρ n ) By this inequality and (3), we obtain d n cmax{d n 1,ρ n }. (4) d n cmax{d n 1,ρ n 1 }. Using the mathematical induction, by the inequalities (3) and (4), we get: (5) d n c n 1 max{d 1,ρ 1 } c n 1 max{d 0,ρ 1 } = c n 1 l, ρ n c n 1 max{d 1,ρ 1 } c n 1 max{d 0,ρ 1 } = c n 1 l, where l = max{d 0,ρ 1 }. So lim n d(x n,x n+1 ) = lim n ρ(y n,y n+1 ) = 0. Applying the inequality (2), we get ρ(y n,y n+2 ) = ρ(tsy n 1,TSy n+1 ) By (5) and properties of f 2, we get cf 2 (ρ(y n 1,y n+1 ),ρ(y n 1,TSy n 1 ), ρ(y n+1,tsy n+1 ),d(sy n 1,Sy n+1 )) = cf 2 (ρ(y n 1,y n+1 ),ρ n 1,ρ n+1,d(x n 1,x n+1 )). (6) ρ(y n,y n+2 ) cmax{ρ(y n 1,y n+1 ),c n 2 l,d(x n 1,x n+1 )}. Similarly, using (1), we obtain d(x n,x n+2 ) = d(stx n 1,STx n+1 ) cf 1 (d(x n 1,x n+1 ),d(x n 1,STx n 1 ), d(x n+1,stx n+1 ),ρ(tx n 1,Tx n+1 )) = cf 1 (d(x n 1,x n+1 ),d n 1,d n+1,ρ(y n,y n+2 ))
A fixed point theorem in generalized metric spaces 185 and so (7) d(x n,x n+2 ) cmax{ρ(y n 1,y n+1 ),c n 2 l,d(x n 1,x n+1 )}. Using the mathematical induction, we get: d(x n,x n+2 ) cmax{ρ(y n 1,y n+1 ),c n 2 l,d(x n 1,x n+1 )} cmax{cmax{ρ(y n 2,y n ),c n 3 l,d(x n 2,x n )},c n 2 l } = c 2 max{ρ(y n 2,y n ),c n 3 l,d(x n 2,x n )} c n 1 max{ρ(y 1,y 3 ),l,d(x 1,x 3 )} = c n 1 l or for all n N. Similarly d(x n,x n+2 ) c n 1 l ρ(y n,y n+2 ) c n 1 l where l = max{ρ(y 1,y 3 ),l,d(x 1,x 3 )}. We divide the proof into two cases. Case I: Suppose x n = x m for some n,m N,n m. Let n > m. Then (ST) n x 0 = (ST) n m (ST) m x 0 = (ST) m x 0 i.e. (ST) k α = α where k = n m and (ST) m x 0 = α. Now if k > 1, by (5), where as x 0 is considered α, we have: d(α,stα) = d[(st) k α,(st) k+1 α] c k 1 max{d(α,stα),ρ(tα,tstα)} and so (8) d(α,stα) c k 1 ρ(tα,tstα). By the equality x n = x m, it follows that y n+1 = y m+1. Then (TS) n Tx 0 = (TS) n m (TS) m Tx 0 = (TS) m Tx 0 i.e. (TS) k β = β where k = n m and (TS) m Tx 0 = β. Similarly, by (5), we have: (9) ρ(tα,tstα) = ρ(β,tsβ) = ρ[(ts) k Tα,(TS) k+1 Tα] c k max{d(α,stα),ρ(tα,tstα)} c k d(α,stα). By (8) and (9) it follows d(α,stα) = ρ(β,tsβ) = 0. Thus, α is a fixed point for ST and β is a fixed point for TS. Case II: Assume that x n x m for all n m. Then (x n ) = ((ST) n x 0 ) is a sequence of distinct point and for m > n+1, we have
186 L. Kikina, K. Kikina ( ) If m > 2 is odd then writing m = 2k + 1,k 1 (by rectangular property) we can easily show that d(x n,x n+m ) [d(x n,x n+1 )+d(x n+1,x n+2 )+d(x n+2,x n+3 )+ + +d(x n+2k,x n+2k+1 )] c n 1 l+c n l+c n+1 l+ +c n+2k 1 l = c n 1 l 1 c2k+1 1 c c n 1 l 1 c. ( ) If m > 2 is even then writing m = 2k,k 2 and using the same arguments as before we can get d(x n,x n+m ) [d(x n,x n+2 )+d(x n+2,x n+3 )+d(x n+3,x n+4 )+ + +d(x n+2k 1,x n+2k )] c n 1 l+c n l+c n+1 l+...+c n+2k 1 l c n 1 l 1 c2k+1 1 c c n 1 l 1 c. Thus combining all the cases we have d(x n,x n+m ) c n 1 l 1 c for all n,m N. Therefore, lim n d(x n,x n+m ) = 0. It implies that (x n ) is a Cauchy sequence in X. Since O(x 0 ) is ST-orbitally complete, there exists an α X such that lim n x n = α. In the same way, we show that the sequence (y n ) is a Cauchy sequence and exists a β Y such that lim n y n = β. The limits α and β are unique. Assume that α α is also lim n x n. Since x n x m for all n m, there exists a subsequence (x nk ) of (x n ) such that x nk α and x nk α for all k N. Without loss of generality, assume that (x n ) is this subsequence. Then, by rectangular property of Definition 1.1 we obtain d(α,α ) d(α,x n )+d(x n,x n+1 )+d(x n+1,α ). Letting n tend to infinity we get d(α,α ) = 0 and so α = α. In the same way for β. Now, suppose that T is continuous, we have: ( ) lim y n = lim Tx n 1 β = Tα. n n Later we will show that Sβ = α. Let us prove that STα = α. In contrary, if STα α, the sequence (x n ) does not converge to STα and there exists a subsequence (x nk ) of (x n ) such that x nk STα for all k N. Then by rectangular property of Definition 1.1 we obtain d(stα,α) d(stα,x nk 1 )+d(x nk 1,x nk )+d(x nk,α). Then if k, we get (10) d(stα,α) lim k d(stα,x n k 1 ).
A fixed point theorem in generalized metric spaces 187 Using (1) for x = α and x = x n 1 we obtain d(stα,x n ) cf 1 (d(α,x n 1 ),d(α,stα),d(x n 1,STx n 1 ),ρ(tα,tx n 1 )) Then if n, we get = cf 1 (d(α,x n 1 ),d(α,stα),d(x n 1,x n ),ρ(β,y n )). (11) lim n d(stα,x n) cf 1 (0,d(α,STα),0,0) cd(α,stα). By (10) and (11), since lim k d(stα,x nk 1 ) lim n d(stα,x n ), we have d(stα,α) = 0 and (12) STα = α. By ( ) and (12) we obtain: STα = Sβ = α and TSβ = Tα = β. Thus, we proved that the points α and β are fixed points of ST and TS respectively. Let us prove the uniqueness (for case I and II in the same time). Assume that α α is also a fixed point of ST. By (1) for x = α and x = α we get: and so, we have d(α,α ) = d(stα,stα ) cf 1 (d(α,α ),0,0,ρ(Tα,Tα )) (13) d(α,α ) cρ(tα,tα ). If Tα Tα, in a similar way by (2) for y = Tα and y = Tα, we have: (14) ρ(tα,tα ) cd(α,α ). By (13) and (14) we get: d(α,α ) = 0. Thus, we have again α = α. The uniqueness of β follows similarly. This completes the proof of the theorem. 3. Corollaries For different expressions of f 1 and f 2 we get different theorems. In the special case f 1 = f 2 = f, where f (t 1,t 2,t 3,t 4 ) = max{t 1,t 2,t 3,t 4 } we have the extension of Fisher theorem [6] to g.m.s: Corollary 3.1. Let (X, d),(y, ρ) be two generalized metric spaces. Let T : X Y, S : Y X be two mappings, at least one of them being continuous and the following inequalities are satisfied: d(stx,stx ) cmax{d(x,x ),d(x,stx),d(x,stx ),ρ(tx,tx )}, ρ(tsy,tsy ) cmax{ρ(y,y ),ρ(y,tsy),ρ(y,tsy ),d(sy,sy )}
188 L. Kikina, K. Kikina for all x,x X and y,y Y, where 0 c < 1. If there exists x 0 X such that O(x 0 ) is ST-orbitally complete in X and O(Tx 0 ) is TS-orbitally complete in Y, then ST has a unique fixed point α in X and TS has a unique fixed point β in Y. Further, Tα = β and Sβ = α. The next corollary follows from Theorem 2.1 in case f 1 (t 1,t 2,t 3,t 4 ) = a 1t 1 +a 2 t 2 +a 3 t 3 +a 4 t 4 a 1 +a 2 +a 3 +a 4, f 2 (t 1,t 2,t 3,t 4 ) = b 1t 1 +b 2 t 2 +b 3 t 3 +b 4 t 4 b 1 +b 2 +b 3 +b 4, c = max{a 1 +a 2 +a 3 +a 4,b 1 +b 2 +b 3 +b 4 } and extends a theorem of Reich [9] from one metric space to two g.m.s: Corollary 3.2. Let (X, d),(y, ρ) be two generalized metric spaces. Let T : X Y, S : Y X be two mappings, at least one of them being continuous and the following inequalities are satisfied: d(rsx,rsx ) a 1 d(x,x )+a 2 d(x,rsx)+a 3 d(x,rsx )+a 4 ρ(sx,sx ), ρ(sry,sry ) b 1 ρ(y,y )+b 2 ρ(y,sry)+b 3 ρ(y,sry )+b 4 d(ry,ry ) for all x,x X;y,y Y, where a 1,a 2,a 3,a 4,b 1,b 2,b 3,b 4 are nonnegative numbers such that 0 a 1 +a 2 +a 3 +a 4 < 1, 0 b 1 +b 2 +b 3 +b 4 < 1. If there exists x 0 X such that O(x 0 ) is ST-orbitally complete in X and O(Tx 0 ) is TS-orbitally complete in Y, then ST has a unique fixed point α X and TS has a unique fixed point β Y. Moreover, Tα = β and Sβ = α. The next corollary follows from Theorem 2.1 for f 1 = f 2 = f, where f (t 1,t 2,t 3,t 4 ) = t 2 +t 3 +t 4 3 and extends a theorem of Kannan [7] from one metric space to two generalized metric spaces: Corollary 3.3. Let (X, d),(y, ρ) be two generalized metric spaces. Let T : X Y, S : Y X be two mappings, at least one of them being continuous and the following inequalities are satisfied: d ( RSx,RSx ) c[d(x,rsx)+d ( x,rsx ) +ρ ( Sx,Sx ) ], ρ ( SRy,SRy ) c[ρ(y,sry)+ρ ( y,sry ) +d ( Ry,Ry ) ] for all x,x X; y,y Y, where c (0, 1 3 ). If there exists x 0 X such that O(x 0 ) is ST-orbitally complete in X and O(Tx 0 ) is TS-orbitally complete in Y, then ST has a unique fixed point α in X and TS has a unique fixed point β in Y. Moreover, Tα = β and Sβ = α.
A fixed point theorem in generalized metric spaces 189 Corollary 3.4. Let (X,d) be generalized metric space, T : X X a mapping of X into itself and the following inequality is satisfied: (1 ) d(tx,ty) cf(d(x,y),d(x,tx),d(y,ty)) for all x,y X, where 0 c < 1. If there exists x 0 X such that O(x 0 ) is T-orbitally complete in X, then T has a unique fixed point α in X. Proof. By the Theorem 2.1, if we take: Y = X,ρ = d,x = y, the mapping S is the identity mapping in X, f 1 = f 2 = f, f (t 1,t 2,t 3,t 4 ) = f (t 1,t 2,t 3 ), then the inequalities (1) and (2) are reduced in inequality (1 ). This corollary is a generalization and extension of Rhoades theorem [8] in g.m.s. For different expressions of f, in the Corollary 3.4, we get different theorems. For example: For f (t 1,t 2,t 3 ) = max{t 1,t 2,t 3 }, we have an extension of the Rhoades theorem [8] in a g.m.s. For f (t 1,t 2,t 3 ) = t 1, we have the Banach s contraction principle in a g.m.s [10]. For f (t 1,t 2,t 3 ) = t 2+t 3 2, we have the Kannan s fixed point theorem in a g.m.s [1]. For f (t 1,t 2,t 3 ) = max{t 2,t 3 }, we have an extension of the Bianchini theorem [2] in a g.m.s. For f (t 1,t 2,t 3 ) = at 1+bt 2 +ct 3 a+b+c, where a, b, c are nonnegative numbers such that, a+b+c < 1, we have an extension of the Reich theorem [9] in a g.m.s. Remark 3.5. We can obtain many other similar results for different f. References [1] A. Azam, M. Arshad, Kannan fixed point theorem on generalized metric spaces, J. Nonlinear Sci. Appl. 1 (2008), 45 48. [2] R. M. T. Bianchini, Su un problema di S. Reich riguardante la teoria dei punti fissi, Boll. Un. Mat. Ital. 5 (1972), 103 108. [3] A. Branciari, A fixed point theorem of Banach-Caccippoli type on a class of generalized metric spaces, Publ. Math. Debrecen 57 (2000), 31 37. [4] P. Das, A fixed point theorem on a class of generalized metric spaces, Korean J. Math. Sciences 1 (2002), 29 33. [5] P. Das, L. K. Dey, A fixed point theorem in a generalized metric space, Soochow J. Math. 33 (2007), 33 39. [6] B. Fisher, Related fixed point on two metric spaces, Math. Sem. Notes, Kobe Univ. 10 (1982), 17 26. [7] R. Kannan, Some results on fixed points II, Amer. Math. Monthly 76 (1969), 405 408. [8] B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977), 256 290.
190 L. Kikina, K. Kikina [9] S. Reich, Some remarks concerning contraction mappings, Canad. Math. Bull. 14 (1971), 121 124. [10] I. R. Sarma, J. M. Rao, S. S. Rao, Contractions over generalized metric spaces, J. Nonlinear Sci. Appl. 2 (2009), 108 182. DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE FACULTY OF NATURAL SCIENCES UNIVERSITY OF GJIROKASTRA ALBANIA E-mails: kristaqkikina@yahoo.com gjonileta@yahoo.com Received March 25, 2011; revised version July 18, 2011.