International Journal of Analysis and Applications ISSN 2291-8639 Volume 4, Number 2 (2014), 100-106 http://www.etamaths.com SOME COMMON FIXED POINT THEOREMS IN GENERALIZED VECTOR METRIC SPACES RAJESH SHRIVASTAVA1, RAJENDRA KUMAR DUBEY2, PANKAJ TIWARI1,∗ Abstract. In this paper we give some theorems on point of coincidence and common fixed point for two self mappings satisfying some general contractive conditions in generalized vector spaces. Our results generalize some well-known recent results in this direction. 1. Introduction and preliminaries In 2003, Mustafa and Sims [6] introduced a more appropriate and robust notion of a generalized metric space as follows. Definition 1.1. [6]. Let X be a nonempty set, and let G : X ×X ×X → [0,∞) be a function satisfying the following axioms: (1) G(x,y,z) = 0 if and only if x = y = z; (2) G(x,x,y) > 0, for all x 6= y; (3) G(x,y,z) ≥ G(x,x,y), for all x,y,z ∈ X; (4) G(x,y,z) = G(x,z,y) = G(z,y,x) = · · · (symmetric in all three variables); (5) G(x,y,z) ≤ G(x,w,w) + G(w,y,z), for all x,y,z,w ∈ X. 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. A Riesz space is an ordered vector space and a lattice. Let E be a Riesz space with the positive cone E+ = {x ∈ E : x ≥ 0}. If {an} is a decreasing sequence in E such that inf an = a, write an ↓ a. Definition 1.2. The Riesz space E is said to be Achimedean if 1 n an ↓ 0 holds for every E+. Definition 1.3. A sequence {bn} is said to be order convergent (or o-convergent) to b if there is a sequence {an} in E satisfying an ↓ 0 and |bn − b| ≤ an for all n, and written bn o−→ b or o− lim bn = b, where |a| = sup{a,−a} for any a ∈ E. Definition 1.4. A sequence {bn} is said to be order-Cauchy (or o-Cauchy) if there exists a sequence {an} in E such that an ↓ 0 and |bn − bn+p| ≤ an holds for all n and p. Definition 1.5. The Riesz space E is said to be o − Cauchy complete if every o−Cauchy sequence in o− convergent. For notion and other facts regarding Riesz spaces we refer to [1]. 2010 Mathematics Subject Classification. 47H10. Key words and phrases. Reisz space, Generalized vector space, coincidence point, fixed point. c©2014 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 100 SOME COMMON FIXED POINT THEOREMS 101 2. Vector G-metric spaces In this section we introduce the following concepts and properties of Vector G- metric spaces. Definition 2.1. Let X be a non-empty set and E be a Riesz space. The function G : X ×X ×X → E is said to be vector G−metric if it is satisfying the following properties : (VGM-1) G(x,y,z) = 0 if and only if x = y = z, (VGM-2) G(x,y,z) ≤ G(x,w,w) + G(w,y,z), for all x,y,z,w ∈ X. Also the triple (X,G,E) is said to be vector G−metric space. For arbitrary elements x,y,z,w ∈ X of a vector G−metric space, the following statements are satisfied (1) G(x,x,y) > 0, for all x 6= y; (2) G(x,y,z) ≥ G(x,x,y), for all x,y,z ∈ X; (3) G(x,y,z) = G(x,z,y) = G(z,y,x) = · · · (symmetric in all three variables). Example 2.2. A Riesz space E is a vector G−metric space with G : E×E×E → E defined by G(x,y,z) = |x − y| + |y − z| + |z − x|. This vector G−metric space is called to be absolute valued G−metric space on E. It is well known that R2 is a Riesz space with coordinatwise ordering defined by (x1,y1) ≤ (x2,y2) ⇐⇒ x1 ≤ x2 and y1 ≤ y2 for (x1,y1), (x2,y2) ∈ R2. Again R2 is a Riesz space with lexicographical ordering defined by (x1,y1) ≤ (x2,y2) ⇐⇒ x1 < x2 or x1 = x2 and y1 ≤ y2. Note that R2 is Archimedean with coordinatwise ordering but not with lexico- graphical ordering. Example 2.3. Let G : R2 ×R2 ×R2 → R2 defined by G((x1,y1), (x2,y2), (x3,y3)) = (αx ∗,βy∗) where x∗ = |x1−x2|+|x2−x3|+|x3−x1| and y∗ = |y1−y2|+|y2−y3|+|y3−y1| also α,β are positive real numbers. Then G is a vector G−metric space. Let G : R×R×R → R2 defined by G(x,y,z) = (αw,βw) where w = |x−y|+ |y−z|+ |z−x| , α,β ≥ 0 and α + β > 0. Then G is a vector G−metric space. Definition 2.4. A sequence {xn} in a vector G−metric space (X,G,E) vectorial G−convergence to some x ∈ E, written xn G,E−−→ x, if there is a sequence {an} in E such that an ↓ 0 and satisfying, (1) G(xn,xn,x) ≤ an, (2) G(xn,x,x) ≤ an, 102 SHRIVATAVA, DUBEY AND TIWARI (3) G(xn,xm,x) ≤ an, for all n. Definition 2.5. A sequence {xn} is called GE−Cauchy sequence whenever there exists a sequence {an} in E such that an ↓ 0 and G(xn,xm,xm) ≤ an holds for all n and m. Definition 2.6. A vector G-metric space is said to be complete if each GE − Cauchy sequence in X is E − convergens to a limit in X. Using the above definitions, we have the following properties. If xn G,E−−→ x, then (1) The limit x unique, (2) Every subsequence of {xn} E−converges to x, (3) If also yn G,E−−→ y and zn G,E−−→ z, then G(xn,yn,zn) o−→ G(x,y,z). When E = R then the concepts of vectorial GE −convergence and convergence in G−metric space are the same also the concepts of GE −Cauchy sequence and G−Cauchy sequence are the same. Remark 2.7. It E is a Riesz space and a ≤ ka where a ∈ E+ k ∈ [0, 1), then a = 0. Proof. The condition a ≤ ka means that −(1 −k)a = ka−a ∈ E+. Since a ∈ E+ and 1 −k > 0, then also (1 −k)a ∈ E+. Thus we have (1 −k)a = 0 and a = 0. � 3. Main Results Theorem 3.1. Let X be a vector G−metric space with E is Archimedean. Suppose the mappings S,T : X → X satisfying the following conditions, (i) for all x,y,z ∈ X and α,β,γ,δ ∈ [0, 1) such that 0 ≤ α + β + γ + δ < 1 G(Tx,Ty,Tz) ≤ αG(Sx,Sy,Sz) + βG(Sx,Tx,Tx) + γG(Sy,Ty,Ty) + δG(Sz,Tz,Tz)(3.1) or G(Tx,Ty,Tz) ≤ αG(Sx,Sy,Sz) + βG(Sx,Sx,Tx) + γG(Sy,Sy,Ty) + δG(Sz,Sz,Tz)(3.2) (ii) T(X) ⊆ S(X), (iii) T(X) or S(X) is complete subspace of X. Then S and T have a unique point of coincidence in X. Moreover, if S and T are weakly compatible, then they have a unique common fixed point in X. Proof. Let x0 be an arbitrary point in X, since T(X) ⊆ S(X) so we can choose a point x1 ∈ X such that Sx1 = Tx0. In general we can choose Sxn+1 = Txn = yn for all n. Now, form 3.1 we have G(Sxn,Sxn+1,Sxn+1) = G(Txn−1,Txn,Txn) ≤ (α + β)G(Sxn−1,Sxn,Sxn) + (γ + δ)G(Sxn,xn+1,xn+1) G(Sxn,Sxn+1,Sxn+1) ≤ α + β 1 − (γ + δ) G(Sxn−1,Sxn,Sxn).(3.3) Let q = α+β 1−(γ+δ) , then 0 ≤ q < 1 since 0 ≤ α + β + γ + δ < 1. So G(Sxn,Sxn+1,Sxn+1) ≤ qG(Sxn−1,Sxn,Sxn).(3.4) SOME COMMON FIXED POINT THEOREMS 103 Continuing in the same way, we have G(Sxn,Sxn+1,Sxn+1) ≤ qnG(Sx0,Sx1,Sx1).(3.5) Therefore, for all n,m ∈ N,n < m, we have by (VGM-2) G(yn,ym,ym) ≤ G(yn,yn+1,yn+1) + G(yn+1,yn+2,yn+2) + ... + G(ym−1,ym,yn) ≤ (qn + qn+1 + qn+2 + ... + qm−1)G(y0,y1,y1) ≤ qn 1 −q G(y0,y1,y1). Now, since E is Archimedean then {yn} is an GE−Cauchy sequence in X. Since the range of S contains the range of T and the range of at least one is GE−complete, so there is w ∈ X such that Sxn G,E−−→ w. Hence there exists a sequence {an} ∈ E such that an ↓ 0 and G(Sxn,w,w) ≤ an. On the other hand, we can find u ∈ X such that Sw = u. Let us show that Tw = u, we have G(Tw,u,u) ≤ G(Tw,Txn,Txn) + G(Txn,u,u) ≤ αG(Sw,Sxn,Sxn) + βG(Sw,Tw,Tw) + (γ + δ)G(Sxn,Txn,Txn) + an+1 ≤ (α + β + γ + δ + 1)an+1. Since the infimum of the sequence on the right side of the above inequality are zero, then Tw = u. Therefore, w is a point of coincidence of T and S. If w1 is another point of coincidence then there is w1 ∈ X with w1 = Tw1 = Sw1. Now from 3.1 it follows that G(w,w1,w1) = 0, that is w = w1. If S and T are weakly compatible, then it is obvious that w is unique common fixed point of T and S in X. If S and T satisfies condition 3.2, then the argument is similar to that above. However to show that the sequence {xn} is GE −Cauchy sequence, we start with G(Sxn,Sxn,Sxn+1) = G(Txn−1,Txn−1,Txn) ≤ (α + β + γ)G(Sxn−1,Sxn−1,Sxn) + δG(Sxn,xn+1,xn+1) G(Sxn,Sxn+1,Sxn+1) ≤ α + β + γ 1 − δ G(Sxn−1,Sxn,Sxn).(3.6) Let q = α+β+γ 1−δ , then 0 ≤ q < 1 since 0 ≤ α + β + γ + δ < 1. So G(Sxn,Sxn,Sxn+1) ≤ qG(Sxn−1,Sxn−1,Sxn).(3.7) Continuing in the same way, we have G(Sxn,Sxn,Sxn+1) ≤ qnG(Sx0,Sx0,Sx1).(3.8) Then for all n,m ∈ N,n < m, we have by (VGM-2) we prove the remaining part of the proof. � Corollary 3.2. Let X be a vector G−metric space with E is Archimedean. Suppose the mappings S,T : X → X satisfying the following conditions, 104 SHRIVATAVA, DUBEY AND TIWARI (i) for all x,y,z ∈ X and α,β,γ,δ ∈ [0, 1) such that 0 ≤ α + β + γ + δ < 1 G(Tmx,Tmy,Tmz) ≤ αG(Smx,Smy,Smz) + βG(Smx,Tmx,Tmx) +γG(Smy,Tmy,Tmy) + δG(Smz,Tmz,Tmz) or G(Tmx,Tmy,Tmz) ≤ αG(Smx,Smy,Smz) + βG(Smx,Smx,Tmx) +γG(Smy,Smy,Tmy) + δG(Smz,Smz,Tmz) (ii) T(X) ⊆ S(X), (iii) T(X) or S(X) is complete subspace of X. Then S and T have a unique point of coincidence in X. Moreover, if S and T are weakly compatible, then they have a unique common fixed point in X, also Tm and Sm are GE − continuous at u. . Proof. From Theorem 3.1, we see that Tm and Sm have a unique common fixed point (say u), that is, Tm(u) = u. but T(u) = T(Tm(u)) = Tm+1(u) = Tm(T(u)), so T(u) is another fixed point for Tm and by uniqueness Tu = u.Similarly we can show that Su = u. � Theorem 3.3. Let X be a vector G−metric space with E is Archimedean. Suppose the mappings S,T : X → X satisfying the following conditions, (i) for all x,y,z ∈ X and α ∈ [0, 1) such that, G(Tx,Ty,Tz) ≤ α{G(Sx,Sy,Sz),G(Sx,Tx,Tx),G(Sy,Ty,Ty),G(Sz,Tz,Tz)}(3.9) or G(Tx,Ty,Tz) ≤ α{G(Sx,Sy,Sz),G(Sx,Sx,Tx),G(Sy,Sy,Ty),G(Sz,Sz,Tz)}(3.10) (ii) T(X) ⊆ S(X), (iii) T(X) or S(X) is complete subspace of X. Then S and T have a unique point of coincidence in X. Moreover, if S and T are weakly compatible, then they have a unique common fixed point in X, also T and S are GE − continuous at u. Proof. Let x0 be an arbitrary point in X, since T(X) ⊆ S(X) so we can choose a point x1 ∈ X such that Sx1 = Tx0. In general we can choose Sxn+1 = Txn = yn for all n. Now, form 3.9 we have G(Sxn,Sxn+1,Sxn+1) = G(Txn−1,Txn,Txn) ≤ α{G(Sxn−1,Sxn,Sxn),G(Sxn,Sxn+1,Sxn+1)} G(Sxn,Sxn+1,Sxn+1) ≤ αG(Sxn−1,Sxn,Sxn) G(Sxn,Sxn+1,Sxn+1) ≤ αG(Sxn−1,Sxn,Sxn).(3.11) Continuing in the same way, we have G(Sxn,Sxn+1,Sxn+1) ≤ αnG(Sx0,Sx1,Sx1).(3.12) Therefore, for all n,m ∈ N,n < m, we have by (VGM-2) SOME COMMON FIXED POINT THEOREMS 105 G(yn,ym,ym) ≤ G(yn,yn+1,yn+1) + G(yn+1,yn+2,yn+2) + ... + G(ym−1,ym,yn) ≤ (αn + αn+1 + αn+2 + ... + αm−1)G(y0,y1,y1) ≤ αn 1 −α G(y0,y1,y1). Now, since E is Archimedean then {yn} is an GE−Cauchy sequence in X. Since the range of S contains the range of T and the range of at least one is GE−complete, so there is w ∈ X such that Sxn G,E−−→ w. Hence there exists a sequence {an} ∈ E such that an ↓ 0 and G(Sxn,w,w) ≤ an. On the other hand, we can find u ∈ X such that Sw = u. Let us show that Tw = u, we have G(Tw,u,u) ≤ G(Tw,Txn,Txn) + G(Txn,u,u) ≤ α max{G(Sw,Sxn,Sxn),G(Sw,Tw,Tw), G(Sxn,Sxn+1,Sxn+1),G(Sxn,Sxn+1,Sxn+1)} + an+1 ≤ (α + 1)an Since the infimum of the sequence on the right side of the above inequality are zero, then Tw = u. Therefore, w is a point of coincidence of T and S. If w1 is another point of coincidence then there is w1 ∈ X with w1 = Tw1 = Sw1. Now from 3.9 it follows that G(w,w1,w1) = 0, that is w = w1. If S and T are weakly compatible, then it is obvious that w is unique common fixed point of T and S in X. If S and T satisfies condition 3.10, then the argument is similar to that above. However to show that the sequence {xn} is GE −Cauchy sequence, we start with G(Sxn,Sxn,Sxn+1) = G(Txn−1,Txn−1,Txn) ≤ α max{G(Sxn−1,Sxn−1,Sxn),G(Sxn,xn+1,xn+1)} G(Sxn,Sxn+1,Sxn+1) ≤ αG(Sxn−1,Sxn−1,Sxn). Continuing in the same way, we have G(Sxn,Sxn,Sxn+1) ≤ αnG(Sx0,Sx0,Sx1). 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Aydi: A fixed point result involving a generalized weakly contractive condition in G-metric spaces, Bull. Math. Anal. Appl., 3(2011), No. 4, 180-188. 1Department of Mathematics, Govt. Science & Commerce College, Benazir Bhopal- India 2Department of Mathematics, Govt. Science P.G. College, Reewa -India ∗Corresponding author