Common Fixed Point Theorem for Compatible. Mapping on Cone Banach Space

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1 International Journal of Mathematical Analysis Vol. 8, 2014, no. 35, HIKARI Ltd, Common Fixed Point Theorem for Compatible Mapping on Cone Banach Space 1 R. K. Gujetiya, 2 Dheeraj Kumari Mali * and 3 Mala Hakwadiya 1 Department of Mathematics, Govt. P. G. College, Neemuch, India 2, 3 Pacific Academy of Higher Education and Research University Udaipur, Rajasthan, India *Corresponding author Copyright 2014 R. K. Gujetiya, Dheeraj Kumari Mali and Mala Hakwadiya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The aim of this paper is to prove a common fixed point theorem for eight self mappings under weakly condition in Cone Banach space. Our results modify the result of P.G. Varghese [7] Mathematics Subject Classification : 47H10, 54H25 Keywords: Cone Banach space, Common Fixed points, Compatible mapping, weakly compatible 1. Introduction In 2007, Huang and Zhang [4] introduced the notion of cone metric space, replacing the set of real numbers by ordered Banach space and proved some fixed point theorems for functions satisfying contractive conditions in these spaces. The

2 1698 R. K. Gujetiya, Dheeraj Kumari Mali and Mala Hakwadiya results in [4] were generalized by Sh. Rezapour and R. Hamlbarani [9] by omitting the normality condition, which is a mile stone in developing fixed point theory in cone metric space. After that several articles on fixed point theorems in cone metric space were obtained by different mathematicians such as M.Abbas, G. Junck [5], D.Ilic [1] etc. Very recently some results on fixed point theorems have been extended to Cone Banach space. E. Karapinar [2] proved some fixed point theorems for self mappings satisfying some contractive condition on a Cone Banach space. Thabet Abdeljawad, E. Karapinar and Kenan Tas [2] have given some generalizations to this theorems. Neeraj Malviya and Sarala Chouhan [6] extended some fixed point theorems to Cone Banach space. We will generalized the result of P.G. Varghese [7]. 2. Preliminaries Definition 2.1. [4] Let (E,. ) be a real Banach space. A subset P E is said to be a cone if and only if (1) P is closed, nonempty and P {0} (2) a, b ϵ R, a, b 0, x, y ϵ P implies ax, by ϵ P (3) P (-P) = {0} For a given cone P subset of E, we define a partial ordering with respect to P by x y if and only if y x ϵ P. We shall write x < y to indicate that x y but x y while x < < y will stand for y - x ϵ int P where int P denotes interior of P and is assumed to be nonempty. Definition 2.2. [4] Let X be a nonempty set. Suppose that the mapping d : X X E satisfies (1) 0 d (x, y) for every x, y ϵ X, d(x, y) = 0 if and only if x = y. (2) d(x, y) = d(y, x) for every x, y ϵ X. (3) d(x, y) d(x, z) + d(z, y) for every x, y, z ϵ X. Then d is a cone metric on X and (X, d) is a cone metric space.

3 Common fixed point theorem 1699 Definition 2.3. [9] Two self maps A and B on a cone normed space (X,. ) are said to be weak-compatible if they commute at their coincidence points, i.e. Ax = Bx implies ABx = BAx. Definition 2.4. [9] Let X be a vector space over R. Suppose the mapping. : X E satisfies (1) x > 0 for all x ϵ X (2) x = 0 if and only if x = 0 (3) x + y x + y for all x, y ϵ X (4) kx = k x for all k ϵ R. Then. is called a norm on X, and (X,. ) is called a cone normed space. Clearly each cone normed space is a cone metric space with metric defined by d (x, y) =. x y. Definition 2.5. [9] Let (X,. ) be a cone normed space, x ϵ X and {x n } a sequence in X. Then (1) {x n } converges to x if for every c ϵ E with o << c there is a natural number N such that. x n x c for all n N. We shall denote it by lim x n = x or x n x. (2) {x n } is a Cauchy sequence, if for every c ϵ E with o << c there is a natural number N such that x n x m c for all n, m N. (3) (X,. ) is a complete cone normed space if every Cauchy sequence is convergent. A complete cone normed space is called a Cone Banach space. Definition 2.6. [7] Let f and g be self mappings on a cone normed space (X,. ), they are said to be compatible if lim fgx n gfx m = 0 for every sequence {x n } in X with lim fx n = lim gx n = y for some point y in X. Proposition 2.7. [9] Let (X,. ) be a cone normed space. P be a normal cone with constant K. Let {x n } be a sequence in X. Then (1) {x n } converges to x if and only if x n x 0 as n.

4 1700 R. K. Gujetiya, Dheeraj Kumari Mali and Mala Hakwadiya (2) {x n } is a Cauchy sequence if and only x n x m 0 as n, m. (3) if the {x n }converges to x and {y n }converges to y then x n y n x y Proposition 2.8. [7] Let f and g be compatible mappings on a cone normed space (X,. ) such that lim fx n = lim gx n for some point y in X and for every sequence {x n } in X. Then lim gfx n = fy if f is continuous. Theorem 2.9. [7] Let f, g, h, I be mappings on Cone Banach space into (X,. ) itself, with x = d(x, 0) satisfying the conditions. (1) hx ly a fx hx + b fx ly + c gy ly for all x, y ϵ X, a, b, c 0, a+2b+c <1. (2) f and g are onto mapping, (3) f is continuous, (4) f and h; g and l commute Then f, g, h and l have a unique common fixed point. 3. Main Result Theorem : Let A, B, C, D, K, M, P and V be mappings on Cone Banach space (X,. ) into itself with X = d(x, 0) satisfying the conditions: (1) V(X) ABC(X) and P(X) DKM(X) (2) Px Vy a ABCx Px + b ABCx Vy + c DKMx Vy For all x, y X, a, b, c 0, a+2b+c < 1. (3) [P, ABC] and [V, DKM] are weakly compatible (4) If one of P(X), ABC(X), V(X), DKM(X) is a complete subspace of X then (4.1) P and ABC have a coincidence point and (4.2) V and DKM have a coincidence point. (5) A,B,C and P ; D,K,M and V commute i.e. PA=AP, BP=PB, PC=CP, BC = CB, AC = CA, AB = BA, DV=VD, KV=VK, VM=MV, DK = KD, KM = MK, DM = MD. Then A, B, C, D, K, M, P and V have a unique common fixed point. Proof: Let x 0 X be arbitrary, then V(x 0 ) X.Since V(X) ABC(X) there exists

5 Common fixed point theorem 1701 x 1 X such that ABC(x 1 ) = V(x 0 ) and for x 1 there exists x 2 X such that DKM(x 2 ) = P(x 1 ) and so on. continuing this process we can define a sequence {y n } in X such that y 2n = ABCx 2n+1 = Vx 2n and y 2n+1 = DKMx 2n+2 = Px 2n+1. We have y 2n 1 y 2n = Px 2n 1 Vx 2n a ABCx 2n 1 Px 2n 1 + b ABCx 2n 1 Vx 2n + c DKMx 2n Vx 2n a y 2n 2 y 2n 1 + b y 2n 2 y 2n + c y 2n 1 y 2n a y 2n 2 y 2n 1 + b[ y 2n 2 y 2n 1 + y 2n 1 y 2n ] + c y 2n 1 y 2n (a+b) y 2n 2 y 2n 1 + (c+b) y 2n 1 y 2n (1-b-c) y 2n 1 y 2n (a+b) y 2n 2 y 2n 1 Let 1 = Now, y 2n 1 y 2n a+b 1 b c, 1 < 1.Therefore a+b 1 b c y 2n 2 y 2n 1 y 2n 1 y 2n 1 y 2n 2 y 2n 1 y 2n 1 y 2n = Px 2n 1 Vx 2n (i) a ABCx 2n+1 Px 2n+1 + b ABCx 2n+1 Vx 2n + c DKMx 2n Vx 2n a y 2n y 2n+1 + b y 2n y 2n + c y 2n 1 y 2n a y 2n y 2n+1 + c y 2n 1 y 2n (1-a) y 2n y 2n+1 c y 2n 1 y 2n y 2n y 2n+1 Let 2 = c 1 a y 2n 1 y 2n c 1 a, 2 < 1.Therefore y 2n y 2n+1 2 y 2n 1 y 2n (ii) Let = max ( 1, 2 ) then < 1.Therefore by (i) and (ii) becomes y 2n 1 y 2n y 2n 2 y 2n 1 and y 2n y 2n+1 y 2n 1 y 2n From (iii) and (iv) we get (iii) (iv)

6 1702 R. K. Gujetiya, Dheeraj Kumari Mali and Mala Hakwadiya y n y n+1 m y 1 y 0 for every n. Let n > m, y n y m y m y m+1 + y m+1 y m y n 1 y n y n y m m + m n 1 y 1 y 0 y n y m αm 1 α y 1 y 0 Let c>0,then there is a δ>0 such that c+n δ (0) H where N δ (0) = {y : y δ}.since < 1 there exists a positive integer N such that αm 1 α y 1 y 0 δ for every m N.Hence αm 1 α y 1 y 0 N δ (0), which implies αm 1 α y 1 y 0 N δ (0).Therefore c αm 1 α y 1 y 0 c + N δ (0) H implies αm 1 α y 1 y 0 c for n, m N.So by definition {y n } is a Cauchy sequence in X. Since X is complete there is an z X such that y n z. Therefore {ABCx 2n+1 }, {DKMx 2n }, {Px 2n+1 } and {Vx 2n } converges to z. Now suppose ABC(X) is complete.then the sequence {y 2n+1 } is contained in ABC(X) and has a limit in ABC(X), call it z. Let w ABC 1 z. Then ABCw = z. We shall use the fact that subsequence {y 2n } also converges to z. Then by (2), putting x = w, y = y 2n, we have Pw Vy 2n a ABCw Pw +b ABCw Vy 2n +c DKMy 2n Vy 2n Letting, we get Pw z a z Pw + b z z + c z z (1-a) Pw z 0 But 1-a > 0. Hence Pw = z. Thus we have ABCw = z = Pw. That is w is coincidence point of P and ABC. Since P(X) DKM(X), Pw = z implies z ϵ DKM(X). Let u ϵ DKM 1 z.then DKMu = z. By (2) putting x = x 2n+1, y = u, we have Px 2n+2 Vu a ABCx 2n+2 Px 2n+2 +b ABCx 2n+2 Vu +c DKMu Vu y 2n+2 Vu a y 2n+2 y 2n+2 + b y 2n+1 Vu + c DKMu Vu Letting, we get z Vu a z z + b z Vu + c z Vu (1-b-c) z Vu 0

7 Common fixed point theorem 1703 But (1-b-c) > 0, hence z = Vu. Since DKMu = z, we have Vu = z = DKMu. That is u is Coincidence point of V and DKM. Since the pair [P, ABC] is weakly compatible therefore P and ABC commute at their Coincidence point that is P(ABCw) = (ABC)Pw or Pz = ABCz. Since the pair [V, DKM] is weakly compatible therefore V and DKM commute at their Coincidence point that is V(DKMu) = (DKM)Vu or Vz = DKMz. Now we prove that Pz = z, by (2), putting x = z, y = x 2n+1 we have, Pz Vx 2n+1 a ABCz Pz +b ABCz Vx 2n+1 + c DKMx 2n+1 Vx 2n+1 Letting, we get Pz z a Pz Pz + b Pz z + c z z (1-b) Pz z 0 But 1-b > 0, hence Pz = z. so Pz = z = ABCz. Now by (2), putting x = x 2n+1 and y = z, we have Px 2n+1 Vz a ABCx 2n+1 Px 2n+1 + b ABCx 2n+1 Vz + c DKMz Vz Letting, we get, z Vz a z z + b z Vz + c Vz Vz (1-b) z Vu 0, But (1-b-c) > 0, hence z = Vz. Thus Vz = DKMz = z. Now by (2), putting x = z, and y = Mz, we have Pz V(Mz) a ABCz Pz + b ABCz V(Mz) + c DKM(Mz) V(Mz) z Mz a z z + b z Mz + c Mz Mz (1-b) z Mz 0 But 1-b > 0, hence Mz = z. since DKMz = z therefore DKz = z. Again by (2), putting x = z, and y = Kz, we have Pz V(Kz) a ABCz Pz + b ABCz V(Kz) + c DKM(Kz) V(Kz) z Kz a z z + b z Kz + c Kz Kz (1-b) z Kz 0 But 1-b > 0, hence Kz = z.since DKz = z,therefore Dz = z. Now by (2), putting x = Cz, and y = z, we have P(Cz) Vz a ABC(Cz) P(Cz) +b ABC(Cz) Vz +c DKMz Vz

8 1704 R. K. Gujetiya, Dheeraj Kumari Mali and Mala Hakwadiya Cz z a Cz Cz + b Cz z + c z z (1-b) Cz z 0 But 1-b > 0, So Cz = z. since ABCz = z.therefore ABz = z. Now by (2), putting x = Bz, and y = z, we have P(Bz) Vz a ABC(Bz) P(Bz) + b ABC(Bz) Vz + c DKMz Vz Bz z a Bz Bz + b Bz z + c z z (1-b) Bz z 0 But 1-b > 0, So Bz = z. since ABz = z. Therefore Az = z. by combinig the above results we have Az = Bz = Cz = Dz = Kz = Mz = Pz = Vz = z. That is z is a common fixed point of A, B, C, D, K, M, P and V. Uniqueness: Suppose g be an another common fixed point of A, B, C, D, K, M, P and V. Then we have, P(z) V(g) a ABC(z) P(z) + b ABC(z) V(g) + c DKM(g) V(g) z g a z z + b z g + c g g (1-b) z g 0 But 1-b > 0, hence z = g. Thus z is a unique common fixed point of A, B, C, D, K, M, P and V. Corollary: Let A, B, D, K, P, and V be mappings on Cone Banach space (X,. ) into itself with X = d(x, 0) satisfying the conditions: (1) V(X) AB(X) and P(X) DK(X) (2) Px Vy a ABx Px + b ABx Vy + c DKx Vy For all x, y X, a, b 0, a+2b < 1. (3) [P, AB] and [V, DK] are weakly compatible (4) If one of P(X), AB(X), V(X), DK(X) is a complete subspace of X then (4.1) P and AB have a coincidence point and (4.2) V and DK have a coincidence point. (5) A,B and P ; D,K and V commute i.e. PA=AP, BP=PB, AB = BA, DV=VD, KV=VK, DK = KD,. Then A, B, D, K, P and V have a unique common fixed point.

9 Common fixed point theorem 1705 Acknowledegment: The Authors are thankful to the anonymous referees for his valuable suggestions for the improvement of this paper. References [1] D. Ilic and V. Rakoevic, Common fixed points for maps on cone metric space, J.Math. Anal. Appl. 341 (2) (2008) 876. [2] Erdal Karapinar, Fixed point Theorems in Cone Banach Space, Hindawi Publishing Corporation, Fixed point Theory and Applications, Article ID , (2009) 9. [3] K. Deimling, Non linear functional analysis, Springer- Veriag, (1985). [4] L G. Huang, X. Zhang, Cone metric space and fixed point theorems, Math. Anal. Appl. 332 (2) (2007) Anal. Appl. 341 (2) (2008) 876. [5] M. Abbas, G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric space, J. Math. Anal. Appl. 341 (2008)416. [6] Neeraj Malviya and Sarla Chouhan, Proving Fixed point Theorems Using General Principles in Cone Banach Space, International Mathematical Forum, 6(3)(2011) 115. [7] P.G. Varghese, K.S. Dersanambika, Common fixed point theorem on cone Banach space, Kathmandu University Journal of Science, Engineering and Technology Vol. 9, No. I, July, 2013, [8] Sh. Rezapour, R Hamlbarani, Some notes on the paper Cone Metric space and Fixed point Theorems of contractive mappings, J. Math, Anal. Appl. 345 (2008) 719. [9] S M Kang and B. Rhoades, Fixed points for four mappings, Math. Japonica, 37(6) (1992), 1053.

10 1706 R. K. Gujetiya, Dheeraj Kumari Mali and Mala Hakwadiya [10] Thabet Abdeljawad, Erdal Karapinar and Kenan Tas, Common fixed point Theorems in Cone Banach spaces, Hacettpe Journal of Mathematics and Statistics, 40(2) (2011) 211. Received: June 11, 2014