@1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 Fixed Point Theorem for Uncommuting Mappings Salwa S. Abd Alaa Abd-ullah Dept. of Mathematics/College of Education for Pure Science(Ibn Al- Haitham) University of Baghdad Received in:19 June 2012 Accepted in:15 October 2012 Abstract In this paper we prove a theorem about the existence and uniqueness common fixed point for two uncommenting self-mappings which defined on orbitally complete G-metric space. Where we use a general contraction condition. Key words : G-metric space, orbitaliy complete, commuting mappings, common fixed point. 312 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 Introduction and preliminaries A number of authors have defined contractive type mappings on a usual complete metric space X which are generalizations of the well known Banach’s contraction principle [1:pp. 175-206], and which have the property that each such mapping has a unique fixed point . The fixed point can always be found by using Picard iteration (i.e. iterative sequence[Zidler: pp.15-30 ] ), beginning with some initial choice x0∈X. And then many authors have extended, generalized and improved Banach’s contraction principle in different ways some these ways are depending on commuting mappings, compatible mappings, weakly commuting mappings, … ets (such as, see [3,4,5,6] ). Recently, Branciari [7] introduced a generalization of metric space and proved a general version of Banach’s contraction principle. And then ,P.Das[8], P.Das and L.Dey [9] , S.Mordi [10] and Akram, Zafar and Siddiqui[11] prove other results about the existence of fixed points and common fixed points for mappings defined on complete G-metric space. Throughout this paper R+ is denoted by non-negative real numbers and N is positive integer numbers. Now we begin with the following definition. Definition 1. 1[11]: Let X be a nonempty set. Suppose that the mapping ρ: X × X →R+ such that for all x , y ∈ X and for all distinct points z ,v ∈ X\ {x, y}, satisfies: 1. ρ (x, y) = 0 if and only if x = y, 2. ρ (x, y) = ρ (y, x), 3. ρ (x, y) ≤ ρ (x, z) + ρ (z,v) +ρ (v, y), (rectangular property), Then the ordered pair (X, ρ) is called a generalized metric space (or shortly G-metric space.). Note that, any metric space is G-metric space but the converse is not true, for examples, Example1.2: Let X={a,b,c,d,}. Define ρ :X × X →R by ρ (a,b)= ρ (b,a)= 3 , ρ (b,c)= ρ (c,b)= ρ (a,c)= ρ (c,a)=1, ρ (a,d)= ρ (d,a)= ρ (b,d)= ρ (d,b)= ρ (c,d)= ρ (d,c)=4. It is easily to show that (X, ρ ) is G-metric space and it is not metric space ,since ρ (a,b) ρ (a,c)+ ρ (c,b) 3 1 + 1 Example1.3: Consider X=R , µ :X × X →R and µ (x,y)= (x-y)2 , Clearly µ is not G-metric space and so is not metric space since, for x=2, y=0, z=1 and w=1/2.We have µ (2,0) > µ (2,1)+ µ (1,1/2) + µ (1/2 ,0) Example1.4: Let ρ: R 2 →R+ be a mapping such that ρ(x,y)=max{µ (x,z), µ (z,w), µ (w,y)}, whereas in example above, then ρ is G-metric space. Therefore, G-metric space is a proper extension of a metric space. Also, one can generate many G-metric spaces by usual sense, such as: Example 1.5: If ρ(x,y) G-metric space ρ1 (x,y)= ρ(x,y) / (1+ ρ(x,y)) also G-metric space. Remark1.6 [7]: the G-metric space is continues function on X × X. Remark1.7 [11]: As in the usual metric space settings, a G-metric space is a topological space with respect to the basis given by B={B(x,r): x ∈X, r∈ R+},where B(x,r)={y ∈X: ρ (x,y) < r} is open ball centered by x and with radius r. Definition1.9[11]: Let (X, ρ ) be a G-metric space. A sequence {xn} in X is said to to be a Cauchy sequence if for any ε > 0 there exists nε in N such that for all m, n ∈ N and m, n > nε , one has ρ(xn,xn+m)<ε. 313 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 The space (X,ρ ) is called complete if every Cauchy sequence in X is convergent. Definition1.10: Let T be a self mapping on X. Let x0∈ X. A sequence {Tnx} in X is said to be an orbit of x by T and denoted by O(x, n)= {x,Tx,T2x,…,Tnx} , for all n ∈ N. Also, O (x, ∞) ={x,Tx,T2x,…}. Definition 1.11[9]: Let T be mapping on a G-metric space (X, ρ ) into itself. (X, ρ ) is said to be T-orbitally complete if and only if every Cauchy sequence in O (x, ∞) converges in X, for some x ∈ X. Now we introduced the following concept Definition 1.12: Let T1S be two self mappings on a G-metric space X. X is called ST-orbitally complete if for x0∈X the sequence {x0,Tx0,STx0,TSTx0,….} converges to a point in X. or the sequence {xn} converges to a point in X where x0∈X, x2n+1=Tx2n, x2n+2=Sx2n+1 … (1) For all n∈N {0}. Definition1.13: A point x in X is a common fixed point of two self-mappings on G-metric space X if Tx = Sx = x. Definition1.14: Let T and S be self mappings on G-metric space X. T and S are commuting mappings if there exists a point x in X such that T x = S x and T S x = S T x.. Main results Let Φ be a family of functions such that ϕ ∈ Φ mean that ϕ :R+→R+ is continuous from the right, non-decreasing and satisfy the condition φ(t) < t for t > 0 and φ(0)=0. It is easy to have the following lemma Lemma2.1: If ϕ1, ϕ2 ∈ Φ, then there is some ϕ3 ∈ Φ such that max{ ϕ1(t), ϕ2(t)} ≤ ϕ3 (t) for all t > 0. Proof: we can see ϕ3 as ϕ1+ ϕ2. Lemma 2.1[12]: Let φ ∈ Φ, then φ n(t) → 0 as n →+∞ for every t >0. Proposition2.3: Let (X,ρ) be a G-metric space. Let S,T : X →X be mappings. If for each x,y in X and T and S satisfy the condition: ρ(STx,TSy)≤ max{ϕ1(1/2[ρ(x,Sy)+ ρ(y,Tx)]), ϕ2(ρ(x,Tx)), ϕ3(ρ(y,Sy)), ϕ4(ρ(x,y))} for all x,y ∈ X, where ϕi ∈ Φ (i = 1,2,3,4) .. (2) Then the sequence {xn} defined by (1) is a Cauchy sequence. Proof: Let x0∈X and {xn} be a sequence as in (1).The proof includes two steps: Step 1: to show that limn→∞ ρ(xn,xn+1)=0, let x1,x2∈{xn} and M = max{ ρ(x0,x1), ρ(x1,x2) }. Since all ϕ i are non-decreasing functions by (2), ρ(x2,x3)=ρ(STx0,TSx1) ≤ max{ϕ1(1/2[ρ(x0,Sx1)+ρ(x1,Tx0)]),ϕ2(ρ(x0,Tx0)),ϕ3(ρ(x1,Sx1)),ϕ4(ρ(x0,x1))} ≤ max{ϕ1(M), ϕ2(M),ϕ3(M),ϕ4(M)} ≤ ϕ (M) … (3) where ϕ ∈ Φ. Therefore, we have ρ(x3,x4)= ρ(STx1,TSx2) ≤ max{ϕ1(1/2[ρ(x1,Sx2)+ρ(x2,Tx1)]),ϕ2(ρ(x1,Tx1)),ϕ3(ρ(x2,Sx2)),ϕ4(ρ(x1,x2))} ≤ max{ ϕ1(M), ϕ2(ϕ (M)), ϕ3(M),ϕ4(M)}≤ ϕ (M), … (4) 314 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 Using (2),(3) and (4),we get ρ(x4,x5)=ρ(STx2,TSx3) ≤ max{ϕ1(1/2[ρ(x3,Sx2)+ρ(x2,Tx3)]),ϕ2(ρ(x2,Tx2),ϕ3(ρ(x3,Sx3),ϕ4(ρ(x2,x3)} ≤ max{ ϕ1(ϕ (M)), ϕ2(ϕ (M)), ϕ3(ϕ (M)), ϕ4(ϕ (M))} ≤ ϕ2(M) …(5) Again from (2),(4) and (5), we get ρ(x5,x6)=ρ(STx3,TSx4) ≤ max{ϕ1(1/2[ρ(x3,Sx4)+ ρ(x4,Tx3)]), ϕ2( ρ(x3,Tx3)), ϕ3( ρ(x4,Sx4)), ϕ4(ρ(x3,x4))} ≤ max{ ϕ1(ϕ (M)), ϕ2(ϕ 2(M)), ϕ3(ϕ (M)), ϕ4(ϕ (M))}≤ ϕ 2(M), ...(6) In general, by induction, we get ρ(xn,xn+1)≤ ϕ[n/2](M) for n ≥ 2, where [n/2] stands for the greatest integer not exceeding n/2. Since ϕ ∈ Φ, by lemma2.2 it follows that ϕ n(M) → 0 as n → +∞ for every M >0. Thus, we obtain ρ(xn,xn+1)→0 as n →∞. … (7) Step2: Suppose that proposition is not true. Then there exists an ε > 0 such that for each i∈ N, there exist positive integers ni, mi, with i ≤ ni 0. Then S and T have a unique common fixed point. Proof: Let x0 ∈ X and {xn} defined by (1). From proposition2.3, {xn} is a Cauchy sequence. Since X is a ST-orbitally complete G-metric space, there exists u ∈ X such that {xn} converges to u. Then we have ρ(x2n+1,x2m+2)=ρ(TSx2n−1,STx2m) ≤ max{ϕ1(1/2[ρ(x2n−1,Sx2m)+ρ(x2m,T2n−1)]),ϕ2(ρ(x2n−1,x2n)), ϕ3(ρ(x2m,x2m+1)),ϕ4( ρ(x2n−1,x2m))} ≤ max{ϕ1(1/2[ρ(x2n−1,x2m+1)+ ρ(x2m, x 2n)]),ϕ2( ρ(x2n−1,x2n)), ϕ3(ρ(x2m,x2m+1)),ϕ4( ρ(x2n−1,x2m))}. Thus, we obtain limn→∞ ρ(x2n+1,u) = 0. Assume that u ≠ Su or u ≠ Tu. Then, by hypothesis, we have 0 < inf {ρ(x,u)+ ρ(x, Sx)+ ρ(y,Ty): x,y ∈ X } = inf {ρ(x2n+1,u)+ ρ(x2n+1,Sx2n+1)+ρ(x2n+2,Tx2n+2): n ∈ N } = inf {ρ(x2n+1,u)+ ρ(x2n+1,x2n+2)+ ρ(x2n+2, x2n+3): n ∈ N } = 0. This is a contradiction. Therefore, we have u = Su = Tu.On the other hand, we can prove the existence of a unique common fixed point of S and T by a method similar to that of Theorem 2.4. We can prove the following corollary taking T= I ,the identity mapping, in Theorem 2.4 . Corollary 2.7:Let (X, ρ) be a ST-orbitally complete G-metric space. Let S and T be self mappings on X satisfying ρ(Sx,Sy)≤ max{ϕ1(1/2[ρ(x,Sy)+ ρ(y,x)]), ϕ3(ρ(y,Sy)), ϕ4(ρ(x,y))} for all x,y ∈ X, where ϕi ∈ Φ (i = 1,3,4). If S is a continuous, then S has a unique fixed point. We can prove the following corollary taking T= I ,the identity mapping, in Corollary 2.5 . Corollary 2.8: Let (X, ρ) be a ST-orbitally complete G-metric space. Let S be self mapping on X satisfying ρ(Sx,Sy)≤ α max{1/2[ρ(x,Sy)+ρ(y,x)],ρ(y,Sy),ρ(x,y)} for all x,y∈X, for all x,y ∈ X. The sequence {xn} is defined by x0 ∈ X, xn+1 = S xn . If S is continuous, then S has a unique fixed point. We can prove the following corollary taking T= I ,the identity mapping, in Corollary 2.6. 317 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 Corollary 2.9 : Let (X, ρ) be a ST-orbitally complete G-metric space. Let S be self mapping on X satisfying (2) of proposition 2.3, and, for each u ∈ X with u ≠ Su , let inf {ρ(x,u)+ ρ(x, Sx) : x,y ∈ X } > 0. Then S have a unique fixed point. References 1. Agarwal R. P.;’Regan D. O and Sahu D.R. , (2009), Fixed point theory for Lipschitzian type mappings with applications, Springer verlag, New York. 2. Zidler E. (1986),”Non linear functional analysis and application, (I-Fixed point theorems), Springer verlag, New York. 3. Jungck N. (1988), Common fixed point for commuting and compatible maps on compacta proc. Amer. Math. Soc., 103:(3), 977-983. 4. Joshi M. L. and Mehta J. G. (2010), Common fixed point for weakly compatible maps in complete metric spaces, Int. J of Computer Appl., 11:(4), 451-459. 5. Singh S. L.; Hematulin A. and R. Pant, (2009), New coincidence common fixed point theorems, Applied General Topology, 10:(1), 121-130. 6. Tiwary K.; Basu T. and Sen S. (1995), Some common fixed point theorems in complete metric space, Soochow Journal of mathematics, 21:(4), 32-34. 7. Branciari A. (2000), A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. Debrecen, 57:(1-2), 31-37. 8. Das P. (2002), A fixed point theorem on a class of generalized metric spaces, Korean J. Math. Sc., 9:(1), 29-33. 9. Das P. and Dey L.K. (2007), A fixed point theorem in generalized metric space, Soochow J. Math., 33:(1), 33-39. 10. Moradi S.,(2000), Kannan fixed-point theorem on complete metric spaces and on generalized metric spaces depended an another function,Mathematics Subject Classification , Primary 46J10, 46J15, 47H10, 1- 6 . 11. Akram M., Zafar A. A.,. Siddiqui A. A ,(2011), Common fixed point theorems for self maps of a generalized metric space satisfying A-contraction type condition,Int. J. Math., 5:(16), 757-763 . 12. A. Razani, Z. Mazlumi Nezhad and M. Boujary, (2009), A fxed point theorem for w- distance , Applied Sciences,( 11): 114-117. 318 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 حول النقطة الصامدة لتطبیقین غیر متبادلین مبرھنة سلوى سلمان عبد الء عبد هللاآ جامعة بغداد / )ابن الھیثم(كلیة التربیة للعلوم الصرفة /الریاضیات علوم قسم 2012تشرین االول 15قبل البحث في: ، 2012حزیران 19استلم البحث في: الخالصة في ھذا البحث برھنا نتیجة حول وجود ووحدانیة نقطة صامدة مشتركة ةلتطبیقین غیر متبادلین معرفین على فضاء G – . متري كامل مساریا ، إذ استخدمنا تطبیق انكماشي معمم متري ، كامل مساریا ، تطبیقات غیر متبادلة، نقطة صامدة مشتركة. – Gفضاء : الكلمات المفتاحیة 319 | Mathematics