A Unique Common Fixed Point Theorem in Cone Metric Type Spaces

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1 Universal Journal of Applied Mathematis (): 33-38, 03 DOI: 0.389/ujam A Unique Common Fixed Point Theorem in Cone Metri Type Spaes K. P. R. Rao, G.N.V.Kishore,, P.R.Sobhana Babu 3 Department of Mathematis, Aharya Nagarjuna University, Nagarjuna Nagar, Guntur , Andhra Pradesh, India Department of Mathematis, Baba Institute of Tehnology and Sienes, P.M.Palem, Madhurawada Visahapatnam , Andhra Pradesh, India 3 Department of Mathematis, Ramahandra College of Engineering, Vatluru(V), Eluru , West Godavari Dist., Andhra Pradesh, India Corresponding Author: ishore.apr@gmail.om Copyright 03 Horizon Researh Publishing All rights reserved. Abstrat In this paper, we have given a unique ommon fixed point theorem for four mappings in a one metri type spae whih is a slight variant of theorems of [, 0]. We also gave an example to illustrate our main theorem. Finally we have given an example in whih theorem of [0] is not appliable whereas our orollary is appliable. Keywords Cone Metri Type Spae, Wealy Compatible Maps, Common Fixed Points 000 Mathematis Subjet Classifiation: 54H5, 47H0, 54E50 Introdution Replaing the real numbers, as the o-domain of a metri, by an ordered Banah spae, the authors Huang and Zhang [8] introdued the onept of one metri spaes. They desribed the onvergene in one metri spaes, introdued their ompleteness and proved some fixed point theorems for ontrative mappings. Cones and ordered normed spaes have some appliations in optimization theory. The initial wor of Huang and Zhang [8] inspired many authors to prove fixed point theorems and ommon fixed point theorems for two or more mappings on one metri spaes, for example [, 3, 4, 5, 6, 7,,, 3] et. First we state some nown definitions and lemmas. Definition. ([8]) Let E be a real Banah spae and P be a subset of E. P is alled a one if and only if : (i) P is losed, non - empty and P 0; (ii) a, b R, a, b 0, x, y P ax + by P (iii) P ( P ) = 0. For a given one P, we define a partial ordering with respet to P by x y if and only if x y P. We write x < y to indiate that x y but x y, while x y will stand for y x Int P, Int P denotes the interior of P. The one P E is said to be solid if Int P. Analogously with the definition of one metri spaes in [8], we onsider the one metri type spaes given by [9]. Definition. ([9]) Let X be a nonempty set and E a real Banah spae with one P. A vetor valued funtion d : X X E is said to be a one metri type funtion on X with onstant if the following onditions are satisfied (d ) 0 d(x, y) for all x, y X and d(x, y) = 0 if and only if x = y; (d ) d(x, y) = d(x, y) for all x, y X (d 3 ) d(x, y) [d(x, z) + d(z, y)] for all x, y, z X.

2 34 A Unique Common Fixed Point Theorem in Cone Metri Type Spaes The pair (X, d) is alled a one metri type spae (shortly CMTS). For = in Definition., we obtain a one metri spae introdued in [8]. Definition.3 ([9]): Let (X, d) be a CMTS and x n a sequene in X (i) x n onverges to x X if for every E with 0, there exists n 0 N suh that d(x n, x) for all n > n 0. We write lim n x n = x or x n x as n. (ii) If for every E with 0, there exists n 0 N suh that d(x n, x m ) for all n > n 0, m > n 0, then x n is alled a Cauhy sequene in X. (iii) If every Cauhy sequene is onvergent in X, then X is alled a omplete CMTS. The following properties hold in the ase of CMTS. Lemma.4 ([9]): Let (X, d) be a CMTS over ordered real Banah spae E with a one P. The following properties hold (a, b, E). (i) If a b and b then a (ii) If 0 a for all Int P, then a = 0. (iii) If a λa, where a P and 0 λ < then a = 0. (iv) Let x n 0 in E and let 0. Then there exists n 0 N suh that x n for eah n > n 0 Alesandar et al. [] proved the following theorem. Theorem.5 ([]): Let (X, d) be a CMTS with onstant and P a solid one. Suppose that F, G, S, T : X X are suh that (i) S(X) G(X), T (X) F (X), (ii) for some onstant λ ( ) 0, for all x, y X there exists [d(f x, T y) + d(gy, T y)] u(x, y) d(f x, Gy), d(f x, Sx), d(gy, T y), suh that d(sx, T y) λ u(x, y) holds (iii) one of S(X), T (X), F (X) and G(X) is a omplete sub spae of X (iv) the pairs (S, F ) and (T, G) are wealy ompatible. Then F, G, S and T have a unique ommon fixed point. Reeently Marija P. Stani et al [0] proved ommon fixed point theorem under ontrative ondition of Ćirić s type on one metri type spaes. Theorem.6 ([0]): Let (X, d) be a CMTS with onstant [, ] relative to a solid one P. Let S, T : X X be suh that for some onstant λ ( 0, ) for all x, y X there exists suh that d(sx, T y) λu(x, y) holds. Then S and T have a unique ommon fixed point. u(x, y) d(x, y), d(x, Sx), d(y, T y), d(x, T y), d(y, Sx) In this paper we prove a unique ommon fixed point theorem for four mappings whih is a slight variant of Theorem.5 [] and Theorem.6 [0] when F = G = I. (Identity map) Results Theorem. Let (X, d) be a one metri type spae with onstant [, ] relative to a solid one P. S, T, F, G : X X be satisfying (..) d (Sx, T y) λu(x, y), where u (x, y) d(f x, Gy), d(f x, Sx), d(gy, T y), d(f x, T y), d(gy, Sx) for all x, y X, where 0 < λ < and λ <, (..) S(X) G(X), T (X) F (X), (..3) one of F (X), G(X), S(X) and T (X) is a omplete sub spae of X, (..4) the pairs (S, F ) and (T, G) are wealy ompatible. Then S, T, F and G have a unique ommon fixed point in X. Let

3 Universal Journal of Applied Mathematis (): 33-38, Let x 0 X. Define y n = Sx n = Gx n+, y n+ = T x n+ = F x n+, n = 0,,,. where u (x n+, x n+ ) we have the following ases d(y n+, y n+ ) λd(y n+, y n ) or d(y n+, y n+ ) = d(sx n+, T x n+ ) λu (x n+, x n+ ), d(y n+, y n ), d(y n+, y n+ ), d(y n, y n+ ), 0, d(y n, y n+ ) d(y n+, y n+ ) λd(y n+, y n+ ) whih implies that d(y n+, y n+ ) = 0 or d(y n+, y n+ ) λ d(y n, y n+ ) λ [d(y n, y n+ ) + d(y n+, y n+ )]. That is d(y n+, y n+ ) λ λ d(y n, y n+ ) λd(y n, y n+ ). In all ases, d(y n+, y n+ ) λ d(y n+, y n ), Similarly we an show that d(y n+, y n ) λ d(y n, y n ). Thus d(y n+, y n ) λ d(y n, y n ), so that For m > n, we have d(y n+, y n ) λ n d(y 0, y ). d(y n, y m ) d(y n, y n+ ) + d(y n+, y n+ ) + + m n d(y m, y m ) ( λ n + λ n+ + + m n λ m ) d(y 0, y ) λ n λ d(y 0, y ), sine λ <. Now by (iv) and (i) of Lemma.4, it follows that for every Int P, there exists n 0 N suh that d(y n, y m ) for every m > n > n 0. Thus y n is a Cauhy sequene. Suppose that F (X) is a omplete sub spae of X. Sine y n+ = F x n+ F (X), there exist z and p in X suh that y n+ z = F (p). Sine y n is Cauhy, we have y n z. d(sp, F p) [d(sp, y n+ ) + d(y n+, F p)] = d(sp, T x n+ ) + d(y n+, z) λ u(p, x n+ ) + d(y n+, z), where u(p, x n+ ) d(f p, y n ), d(f p, Sp), d(y n, y n+ ), d(f p, y n+), d(y n, Sp). For any 0 and suffiiently large n, at least one of the following ases hold: d(sp, F p) λ d(z, y n ) + d(y n+, z) λ λ + = ; d(sp, F p) λ d(f p, Sp) + d(y n+, z) Thus d(sp, F p) λ d(y n+, z) ( λ) = ; λ d(sp, F p) λd(y n, y n+ ) + d(y n+, z) λ [d(y n, z) + d(z, y n+ )] + d(y n+, z) λ λ + (λ + ) (λ + ) = ; d(sp, F p) λ d(z, y n+) + d(y n+, z) ( λ + ) +) = ; d(sp, F p) λ d(y n, Sp) + d(y n+, z) ( λ λ [d(y n, z) + d(f p, Sp)] + d(y n+, z)

4 36 A Unique Common Fixed Point Theorem in Cone Metri Type Spaes i.e. d(sp, F p) λ ( ) d(y λ n, z) + ( )d(y λ n+, z) λ ( λ) ( λ) λ + ( λ) ( λ) 4 =. Thus in all ases, we have d(sp, F p) for every Int P. Hene Sp = F p = z. Sine the pair (F, S) is wealy ompatible, we have F z = Sz. d(sz, z) d(sz, y n+ ) + d(y n+, z) = d(sz, T x n+ ) + d(y n+, z) λu(z, x n+ ) + d(y n+, z), where u(z, x n+ ) d(f z, y n ), 0, d(y n, y n+ ), d(f z, y n+), d(y n, Sz) = d(sz, y n ), 0, d(y n, y n+ ), d(sz, y n+), d(y n, Sz). We have the following ases: d(sz, z) λd(sz, y n ) + d(y n+, z) λ d(sz, z) + λ d(z, y n ) + d(y n+, z) i. e. d(sz, z) λ ( λ ) d(y n, z) + ( λ ) d(y n+, z) d(sz, z) λd(y n, y n+ ) + d(y n+, z) d(sz, z) λ d(sz, y n+) + d(y n+, z) d(sz, z) λ d(y n, Sz) + (y n+, z) Thus in all ases, we have i.e. d(sz, z) λ ( λ ) ( λ ) λ + ( λ ) λ d(y n, z) + λ d(z, y n+ ) + d(y n+, z) λ λ + (λ + ) (λ + ) = ; λ λ d(sz, z) + d(z, y n+) + d(y n+, z) ( λ i. e. d(sz, z) ( + ) ) d(z, y λ n+ ) ( λ ( + ) ) λ ( ) λ ( λ + ) = ; λ d(y n, z) + λ d(z, Sz) + d(y n+, z) ( ) λ λ λ ( λ) d(y n, z) + ( )d(y λ n+, z) ( λ) λ + ( λ) d(sz, z) for every Int P. Hene Sz = z. Thus F z = Sz = z. Sine S(X) G(X), there exists v X suh that z = Sz = Gv. d(z, T v) = d(sz, T v) λu(z, v), where u(z, v) 0, 0, d(z, T v), ( λ) d(z, T v), 0. ( λ ) 4 =. = ;

5 Universal Journal of Applied Mathematis (): 33-38, Sine and λ <, it follows that T v = z. Thus Gv = z = T v. Sine the pair (T, G) is a wealy ompatible, we have Gz = T z. d(z, T z) = d(sz, T z) λu(z, z), where u(z, z) d(z, T z), 0, 0, Sine and λ <, it follows that T z = z. Thus Gz = T z = z. Hene z is a ommon fixed point of S, T, F and G. Uniqueness of ommon fixed point follows easily from (..). Now we give an example whih illustrates our main Theorem. d(z, T z), d(z, T z). Example. Let X = [0, ), E = R and P = [0, ). Define d(x, y) = x y, x, y X, then d(x, y) = x y [ x z + z y ] = x z + z y + x z z y [ x z + z y ] = [d(x, z) + d(z, y)] Thus (X, d) is not a one metri spae and it is a one metri type spae with onstant =. Let S, T, F and G be self maps on X defined by S(x) = 3 sinx, T (x) = 3 sin3x, F (x) = x and G(x) = 3x. Then d(sx, T y) = sinx sin3y 3 3 = ( ) ( ) x + 3y x 3y 3 os sin x 3y 3 = 3 d(f x, Gy). Clearly (..), (..3) and (..4) are satisfied and 0 is the unique ommon fixed point of S, G, F and G. Corollary.3 Let (X, d) be a one metri type spae with onstant [, ] relative to a solid one P. S, T : X X be satisfying d (Sx, T y) λu(x, y), where u (x, y) d(x, y), d(x, Sx), d(y, T y), for all x, y X, where 0 < λ < and λ <. Then S and T have a unique ommon fixed point in X. d(x, T y), d(y, Sx) Let Now we give an example in whih Corollary.3 is appliable whereas Theorem.6 of [0] is not appliable. Example.4 Let (X, d), E and P be as in Example. Let Sx = 3 sinx x X Then d(sx, Sy) = sinx siny 3 3 = ( ) ( ) x + y x y 3 os sin x y 3 = d(x, y). 3 Here λ = 3 / ( 0, 4) = ( 0, ). Hene Theorem.6 [0] is not appliable with T = S. Clearly Corollary.3 is appliable with T = S.

6 38 A Unique Common Fixed Point Theorem in Cone Metri Type Spaes 3 Disussion and Conlusion Our main Theorem. is a slight variant of Theorem.5. We have given an Example. to illustate our main Theorem.. We also show that our Corollary.3 is appliable whereas Theorem.6 is not appliable by means Example.4 REFERENCES [] Alesandar S. Cvetović, Marija P. Stanić, Sladjana Dimitrijević and Suzana Simić, Common Fixed Point Theorems for Four Mappings on Cone Metri Type Spae, Fixed Point Theory and Appliations, Volume 0, Artile ID 58975, 5 pages, doi:0.55/0/ [] D. Turoglu, M. Abuloha and T. Abdeljawad, KKM mappings in one metri spaes and some fixed point theorems, Nonlinear Analysis: Theory, Methods and Appliations, 7(), (00), [3] D. Turoglu, M. Abuloha and T. Abdeljawad, Fixed points of generalized ontration mappings in one metri spaes, Mathematial Communiations, 6, (0), [4] E. Karapínar, Fixed point theorems in one Banah spaes, Fixed Point Theory Appl., Volume 009, (009), 9 pages, Artile ID [5] E. Karapínar, Some non - unique fixed point theorems of Ciri type on one metri spaes, Abstr. Appl. Anal., Volume 00, (00), 4 pages, Artile ID 3094, doi:0.55/00/3094. [6] E. Karapínar and D. A. Turoglu, Best approximations theorem for a ouple in one Banah Spae, Fixed Point Theory Appl., Volume 00, (00), 9 pages, Artile ID [7] I. Altun, B. Damjanovi and D. Djori, Fixed point and ommon fixed point theorems on ordered one metri spaes, Applied Mathematis Letters, 3(3), (00), [8] L. G. Huang and X. Zhang, Cone metri spaes and fixed point theorems of Contrative mappings, J. Math.Anal.Appl.,33(), (007), [9] M. A. Khamsi and N. Hussain, KKM mappings in metri type spaes, Nonlinear Analysis : Theory, Methods and Appliations, 73(9), (00), [0] Marija P. Stani, Alesandar S. Cvetovi, Suzana Simi, Sladjana Dimitrijevi, Common fixed point under ontrative ondition of Ciri s type on one metri type spaes, Fixed Point Theory and Appliations 0, 0:35 doi:0.86/ [] Sh. Rezapour, R. Hamlbarani, Some Notes on the Paper Cone Metri Spaes and Fixed Point Theorems of Contrative Mappings, J. Math. Anal. Appl., 347(008), [] T. Abdeljawad and E. Karapínar, Quasi - one metri spaes and generalizations of Caristi Kir s theorem, Fixed Point Theory Appl., Volume 009 (009), Artile ID , 9 pages. [3] T. Abdeljawad, E. Karapínar and K.Tas, Common fixed point theorems in one Banah spaes, Haet.J.Math.Stat., 40(), (0), - 7.