In this paper, we prove common fixed point theorems for a pair of occasionally weakly compatible maps in Symmetric G-metric space. Our results generalize and extend several relevant common fixed-point theorems from the literature

Manro, Saurabh

English

Kathmandu University Open Journal Systems

KATHMANDU UNIVERSITY JOURNAL OF SCIENCE, ENGINEERING AND TECHNOLOGY   VOL. 9, No. I, July, 2013, pp 175-180 175  FIXED POINT THEOREMS FOR OCCSAIONALLY WEAKLY COMPATIBLE MAPS IN G-METRIC SPACE  Saurabh Manro School of Mathematics and Computer Applications, Thapar University, Patiala (Punjab)  Corresponding address: sauravmanro@hotmail.com  Received 11 May, 2012; Revised 04 February, 2013  ABSTRACT In this paper, we prove common fixed point theorems for a pair of occasionally weakly compatible maps in Symmetric G-metric space. Our results generalize and extend several relevant common fixed point theorems from the literature.    Key words: Symmetric G-metric space, occasionally weakly compatible maps, weakly compatible maps. Subject classification: 2000 AMS:  47H10, 54H25   INTRODUCTION  In 1992, Dhage[1] introduced the concept of D – metric space. Recently, Mustafa and Sims[5] shown that most of the results concerning Dhage’s D – metric spaces are invalid. Therefore, they introduced G – metric space. For more details on G – metric spaces, one can refer to the papers [5]-[8].  In 2006, Mustafa and Sims[6] introduced the concept of G-metric spaces as follows: Definition 1.1.[6] Let X be a nonempty set, and let G: X × X × X   R+ be a function satisfying the following axioms: (G1) G(x, y, z) = 0 if  x = y = z, (G2) 0  <  G(x, x, y), for all x, y   X with x ≠ y,  (G3) G(x, x, y)   ≤  G(x, y, z), for all x, y, z   X with z ≠ y, (G4) G(x, y, z) = G(x, z, y) = G(y, z, x) = … (symmetry in all three variables) and (G5) G(x, y, z) ≤ G(x, a, a) + G(a, y, z) for all x, y, z, a   X, (rectangle inequality) then the function G is called a generalized metric, or, more specifically a G – metric on  X  and the pair (X, G) is called a G – metric space. If condition (G6) also satisfied then (X, G) is called Symmetric G-metric space. (G6)    G(x, y, y) = G(x, x, y) for all x, y   X.  Definition 1.2.[6] Let (X, G) be a G–metric space, and let {xn} a sequence of points in X, a point ‘x’ in X is said to be the limit of the sequence {xn} if,limm nG(x, xn, xm) = 0, and one says that sequence {xn} is G–convergent to x.               Thus, that if xn   x or limn xn = x in a G-metric space (X, G) then for each           > 0,  there exists a positive integer N such that G (x, xn, xm) <   for all m, n ≥ N. KATHMANDU UNIVERSITY JOURNAL OF SCIENCE, ENGINEERING AND TECHNOLOGY   VOL. 9, No. I, July, 2013, pp 175-180 176  Proposition 1.1.[6] Let (X, G) be a G – metric space. Then the following are equivalent:   (1) {xn} is G-convergent to x, (2) G (xn, xn, x)     0 as n    ∞, (3) G(xn, x, x)    0 as n   ∞, (4) G(xm, xn, x)    0 as m, n     ∞.  Definition 1.3.[6] Let (X, G) be a G – metric space. A sequence {xn} is called G – Cauchy if, for each  > 0 there exists a positive integer N such that G (xn, xm, xl) <   for all n, m, l ≥ N; i.e. if G (xn, xm, xl) →0 as n, m, l→∞  Proposition 1.2.[6] If (X,G) is a G – metric space then the following are equivalent: (1) The sequence {xn} is G – Cauchy, (2) for each   > 0 , there exist a positive integer N such that  G(xn, xm, xm) <   for all n, m ≥ N. Proposition 1.3.[6] Let (X, G) be a G – metric space. Then the function G(x, y, z) is jointly continuous in all three of its variables.  Definition 1.4.[6] A G – metric space (X, G) is said to be  G–complete if every G-Cauchy sequence in (X, G) is G-convergent in X.  Proposition 1.4.[6]  A G – metric space (X, G) is G – complete if and only if (X, dG) is a complete metric space. Proposition 1.5.[6]  Let (X, G) be a G – metric space. Then, for any x, y, z, a in X it follows that:  (i) If G(x, y, z) = 0, then x = y = z,   (ii) G(x, y, z) ≤ G(x, x, y) + G(x, x, z),  (iii) G(x, y, y) ≤ 2G(y, x, x),  (iv) G(x, y, z) ≤ G(x, a, z) + G(a, y, z),  (v) G(x, y, z) ≤ 2/3 (G(x, y, a) + G(x, a, z) + G(a, y, z)), (vi)       G(x, y, z) ≤   (G(x, a, a) + G(y, a, a) + G(z, a, a)). In 1996, Jungck [2] introduced the notion of weakly compatible maps as follows:  Definition 1.5.[2] A pair of self mappings ( f , g ) of a metric space is said to be weakly compatible if they commute at the coincidence  points i.e. Tu = Su for some u in X , then TSu = STu.  Definition 1.6. Let (X, G) be a Symmetric G-metric space. f and g be self maps on X. A point x in X is called a coincidence point of f and g iff fx = gx. In this case, w = fx = gx is called a point of coincidence of f and g.  Definition 1.7[3]: A pair of self mappings (f, g ) of a Symmetric G-metric space (X, G) is  said to be weakly compatible if they commute at the coincidence points   i.e., if  fu = gu for some u in X, then fgu = gfu.  It is easy to see that two compatible maps are weakly compatible but converse is not true.  Definition 1.8[3]: Two self mappings f and g of a Symmetric G-metric space (X, G) are  KATHMANDU UNIVERSITY JOURNAL OF SCIENCE, ENGINEERING AND TECHNOLOGY   VOL. 9, No. I, July, 2013, pp 175-180 177  said to be occasionally weakly compatible (owc) iff there is a point x in X which is coincidence point of f and g at which f and g commute. Lemma 1.1[3]: Let (X, G) be a Symmetric G-metric space. f and g be self maps on X and let f and g have a unique point of coincidence, w = fx = gx, then w is the unique common fixed point of f and g.  MAIN RESULTS Following to Matkowski[5], let   be the set of all functions   such that :[0, ) [0, )     be a non-decreasing function with lim ( ) 0nnt  for all [0, )t  . If  , then   is called  - map. If    is  - map, then it is an easy matter to show that  (A)   ( )t t   for all [0, )t  ; (B)    (0) 0.   From now unless otherwise stated, we mean by   the  - map. Now, we introduce and prove our result. Theorem 2.1: Let (X, G) be a Symmetric G-metric space. If f and g are owc self maps on X and  G(fx,fy,fy) ≤  [max{G(gx,gy,gy), G(gx,fy,fy), G(gy,fx,fx), G(gy,fy,fy)}]                (2.1)  for all ,x y X . Then f  and  g  have a unique common fixed point. Proof: Since f and g are owc, there exist a point u X  such that fu = gu and fgu = gfu. We claim that fu is the unique common fixed point of f and g.  We first assert that fu is a fixed point of  f. For, if ffu ≠ fu, then from equation (2.1), we get  G(fu,ffu,ffu) ≤  [max{G(gu,gfu,gfu), G(gu,ffu,ffu), G(gfu,fu,fu), G(gfu,ffu,ffu)}]                       =  [max{G(fu,ffu,ffu), G(fu,ffu,ffu), G(ffu,fu,fu), G(ffu,ffu,ffu)}]                       =  [max{G(fu,ffu,ffu), G(fu,ffu,ffu), G(fu,fu,ffu), 0}]                      =  [max{G(fu,ffu,ffu), G(fu,ffu,ffu), G(fu,ffu,ffu)}]                       =  [G(fu,ffu,ffu)] <  G(fu,ffu,ffu)  a contradiction. So ffu = fu and ffu = fgu = gfu = fu. Hence fu is a common fixed point of f and g.  Now we prove uniqueness. Suppose that ,u v X  such that fu = gu = u and fv = gv = v and u ≠ v. Then from equation (2.1), G(u,v,v) = G(fu,fv,fv) ≤  [max{G(gu,gv,gv), G(gu,fv,fv), G(gv,fu,fu), G(gv,fv,fv)}]                         =   [max{G(u,v,v), G(u,v,v), G(v,u,u), G(v,v,v)}]                                    =   [max{G(u,v,v), G(u,v,v), G(v,v,u), 0}]                                   =   [max{G(u,v,v), G(u,v,v), G(u,v,v), 0}]                                   =  [G(u,v,v)] <  G(u,v,v) a contradiction. So u = v. Therefore, the common fixed point of f and g is unique.  Theorem 2.2: Let (X, G) be a Symmetric G-metric space. Suppose that f, g, S, T are self maps on X and that the pairs {f,  S} and {g, T} are each owc. If  G(fx,gy,gy) < max { G(Sx,Ty,Ty), G(Sx,fx,fx), G(Ty,gy,gy), G(Sx,gy,gy), G(Ty,fx,fx) },                                                                                                                                                (2.2) KATHMANDU UNIVERSITY JOURNAL OF SCIENCE, ENGINEERING AND TECHNOLOGY   VOL. 9, No. I, July, 2013, pp 175-180 178  for all ,x y X . Then f, g, S and T have a unique common fixed point in X. Proof: By hypothesis, there exists points ,x y X such that fx = Sx and gy = Ty. We claim that fx = gy. For, otherwise, by (2.2)  G(fx,gy,gy) < max { G(Sx,Ty,Ty), G(Sx,fx,fx), G(Ty,gy,gy), G(Sx,gy,gy), G(Ty,fx,fx) }                     = max { G(fx,gy,gy), G(fx,fx,fx), G(gy,gy,gy), G(fx,gy,gy), G(gy,fx,fx) }         = max { G(fx,gy,gy), 0, 0, G(fx,gy,gy), G(gy,gy,fx) }                    = max { G(fx,gy,gy), G(fx,gy,gy), G(fx,gy,gy) } =  G(fx,gy,gy) a contradiction. This implies that fx = gy. So fx = Sx = gy = Ty. Moreover, if there is another point z such that fz = Sz, then, using (2.2) it follows that fz = Sz = gy = Ty or fx = fz and w = fx = Sx is the unique point of coincidence of f and S. Then by Lemma 1.1, it follows that w is the unique common fixed point of f and S. By symmetry, there is a unique common fixed point z X  such that z = gz = Tz.  Now, we claim that w = z. Suppose that w ≠ z. Using (2.2),  G(w,z,z) = G(fw,gz,gz)                < max { G(Sw,Tz,Tz), G(Sw,fw,fw), G(Tz,gz,gz), G(Sw,gz,gz), G(Tz,fw,fw) }  G(w,z,z) < max { G(w,z,z), G(w,w,w), G(z,z,z), G(w,z,z), G(z,w,w) }                  = max { G(w,z,z), 0 , 0 , G(w,z,z), G(z,z,w) }                 = max { G(w,z,z), G(w,z,z), G(w,z,z) } = G(w,z,z) This is a contradiction. Therefore w = z and w is a unique point of coincidence of f, g, S and T.  By Lemma 1.1, w is the unique common fixed point of f, g, S and  T.  Corollary 2.1: Let (X, G) be a Symmetric G-metric space. Suppose that f, g, S and T are self maps on X and that the pairs {f , S} and {g , T} are each owc. If  G(fx,gy,gy) ≤  h m(x,y,y)  where          m(x,y,y) = max{G(Sx,Ty,Ty),G(Sx,fx,fx),G(Ty,gy,gy),[G(Sx,gy,gy),G(Ty,fx,fx)]/2},     (2.3)                                                                                                                                             for all ,x y X  and 0 ≤ h < 1, then  f, g, S and T have a unique common fixed point in X. Proof: Since (2.3) is a special case of (2.2), the result follows immediately from Theorem 2.2.  Theorem 2.3. Let A, B, S and T be self maps of Symmetric G-metric space (X, G), satisfying the following conditions: (2.4)   A(X)   T(X), B(X)    S(X),       (2.5)   pairs (A, S) or (B, T) satisfies property E.A.,       (2.6)  for all ,x y X ,  G(Ax, By, By) <  [ max {G(Sx, Ty, Ty), G(Sx, By, By), G(Ty, By, By)}] where  . If one of A(X), B(X), S(X) or T(X) is complete subsets of X then pairs       (A, S) and (B, T) have coincidence point. Further, if (A, S) and (B, T) are weakly compatible then A, B, S and T have unique common fixed point in X. Proof:  Suppose the pair (B, T) satisfies the property (E.A.). Then there exists a sequence {xn} in X such that  limn→∞Bxn = limn→∞Txn = p for some .p X    Since B(X)    S(X), there exists a sequence {yn} in X such that  Bxn = Syn = p. Hence limn→∞Syn  =  p.     We shall show that limn→∞Ayn  =  p. From (2.6), we have G(Ayn, Bxn, Bxn) <  [ max {G(Syn, Txn, Txn), G(Syn, Bxn, Bxn), G(Txn, Bxn, Bxn)}] KATHMANDU UNIVERSITY JOURNAL OF SCIENCE, ENGINEERING AND TECHNOLOGY   VOL. 9, No. I, July, 2013, pp 175-180 179  Taking limit as n , we get  limn→∞G(Ayn, p , p) < [max {G(p, p, p), G(p, p, p), G(p, p, p)}]                                  =  [max { 0, 0, 0}] = (0) = 0. This implies, limn→∞Ayn  =  p. Thus we have,  limn→∞Ayn  = limn→∞Syn  = limn→∞Bxn = limn→∞Txn = p.  Suppose that S(X) is a complete subspace of X. Then p = Su for some .u X  Subsequently, we have    limn→∞Ayn  = limn→∞Sxn =  limn→∞Bxn = limn→∞Txn = p  = Su     Now, we shall show that Au = Su. From (2.6), we have  G(Au, Bxn, Bxn) <  [max{G(Su, Txn, Txn), G(Su, Bxn, Bxn), G(Txn, Bxn, Bxn)}] Taking limit as n  we get  G(Au, Su, Su) <   [max{G(p, p, p), G(p, p, p), G(p, p, p)}]                        =  [max { 0, 0, 0}] =  (0) = 0. Thus, we have Au = Su. Therefore (A, S) have coincidence point.  The weak compatibility of A and S implies that ASu = SAu and thus AAu = ASu = SAu = SSu.             As A(X)   T(X), there exists v X  such that Au = Tv. We claim that Tv = Bv. Suppose not, from (2.6) , we have  G(Au, Bv, Bv) <  [max{G(Su, Tv, Tv), G(Su, Bv, Bv), G(Tv, Bv, Bv)}]                                      = [max{0, G(Au, Bv, Bv), G(Au, Bv, Bv)}]                                       =  [G(Au, Bv, Bv)] < G(Au, Bv, Bv), this implies, Au = Bv. Hence,   Tv = Bv. Therefore (B, T) have coincidence point Thus we have Au = Su = Tv = Bv. The weak compatibility of B and T implies that BTv = TBv = TTv = BBv.    Finally, we show that Au is the common fixed point of A, B, S and T. From (2.6), suppose Au ≠ AAu,  we have  G(Au, AAu, AAu) = G(Au, Au, AAu)                  { by definition of symmetric space} = G(AAu, Bv, Bv) <  [max{G(SAu, Tv, Tv), G(SAu, Bv, Bv), G(Tv, Bv, Bv)}]                                = [max{G(AAu, Bv, Bv), G(AAu, Bv, Bv), G(Bv, Bv, Bv)}]                                =  [max{G(AAu, Bv, Bv), G(AAu, Bv, Bv), 0}]                                 =  [G(AAu, Bv, Bv)] < G(AAu, Bv, Bv), This gives, AAu = Bv = Au and thus AAu = Au. Therefore, Au = AAu = SAu is the common fixed point of A and S. Similarly, we prove that Bv is the common fixed point of B and T. Since Au = Bv, Au is common fixed point of A, B, S and T. The proof is similar when T(X) is assumed to be a complete subspace of X. The cases in which A(X) or B(X) is a complete subspace of X are similar to the cases in which T(X) or S(X), respectively is complete subspace of X as   A(X)   T(X) and B(X)    S(X). Finally now we show that the common fixed point is unique. If possible, let x0 and y0 be two common fixed points of A, B, S and T. Suppose x0 ≠ y0, then by condition (2.6), we have  G(x0, y0, y0) = G(Ax0, By0, By0)                      <  [max {G(Sx0, Ty0, Ty0), G(Sx0, By0, By0), G(Ty0, By0, By0)}] KATHMANDU UNIVERSITY JOURNAL OF SCIENCE, ENGINEERING AND TECHNOLOGY   VOL. 9, No. I, July, 2013, pp 175-180 180                       =  [max {G(x0, y0, y0), G(x0, y0, y0), G(y0, y0, y0)}]                       =  [G(x0, y0, y0)] < G(x0, y0, y0), this implies x0 = y0.  Therefore, the mappings A, B, S and T  have a unique common fixed point. Corollary 2.2. Let A, B and S be self maps of Symmetric G-metric space (X, G), satisfying the following conditions: (2.7)   A(X)   S(X), B(X)    S(X),       (2.8)   pairs (A, S) or (B, S) satisfies property E.A.,       (2.9)  for all ,x y X ,  G(Ax, By, By) <  [max {G(Sx, Sy, Sy), G(Sx, By, By), G(Sy, By, By)}] where  . If one of A(X), B(X) or S(X) is complete subsets of X then pairs (A, S) and (B, S) have coincidence point. Further, if (A, S) and (B, S) are weakly compatible then A, B and S have unique common fixed point in X. Proof:  Take T = S in Theorem 2.3.  ACKNOWLEDGEMENT The author is grateful to an anonymous referee for his fruitful suggestions.  REFERENCES [1] Dhage B C, Generalized metric spaces and mappings with fixed point, Bull.Calcutta Math. Soc. 84 (1992) 329. [2] Jungck G, Commuting mappings and fixed point, Amer. Math. Monthly, 83(1976) 261.  [3] Jungck G & Rhoades B E, Fixed point Theorems for occasionally weakly      compatible mappings, Fixed point theory, 7(2006) 286-296 [4] Matkowski J, “Fixed point theorems for mappings with a contractive iterate at a point,” Proceedings of the American Mathematical Society, 62(2) (1977) 344. [5] Mustafa Z &  Sims B, Some remarks concerning D-metric spaces, Proceedings of International Conference on Fixed Point Theory and Applications, Yokoham  Publishers, Valencia Spain, July 13-19(2004) 189.  [6] Mustafa Z & Sims B, A new approach to a generalized metric spaces, J. Nonlinear Convex Anal., 7(2006) 289. [7] Mustafa Z , Obiedat H & Awawdeh F, Some fixed point theorems for mappings on complete G-metric spaces, Fixed point Theory and Applications, Article  ID 18970, (2008) 12. [8] Mustafa Z, Shatanawi W & Bataineh M, Existence of fixed points results in G-metric spaces, International Journal of Mathematics and Mathematical Sciences, (2009) Article ID. 283028, 10 pages. [9] Shatanawi W, Fixed point theory for contractive mappings satisfying  -maps in G-metric spaces, Fixed Point Theory and Applications, (2010) Article ID 181650, 9 pages doi:10.1155/2010/181650.   

Fixed Point Theorems for Occasionally Weakly Compatible Maps in G-Metric Space

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Fixed Point Theorems for Occasionally Weakly Compatible Maps in G-Metric Space

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