Some Coincidence and Common Fixed Point Theorems for Two Self Mappings under Generalized Contractive Condition in Cone-b- Metric Space Tamara Sh. Ahmed Dept. of Mathematics/College of Education for pure Science (Ibn-Al-Haitham)/ University of Baghdad Received in : 17 October 2012 , Accepted in : 16 April 2013 Abstract In this paper, we prove some coincidence and common fixed point theorems for a pair of discontinuous weakly compatible self mappings satisfying generalized contractive condition in the setting of Cone-b- metric space under assumption that the Cone which is used is non- normal. Our results are generalizations of some recent results. Key Words: Coincidence and common fixed point, pair of weakly compatible mappings, generalized contractive self mapping, Cone-b- metric space, normal Cone, non-normal Cone. 340 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Introduction Metric fixed point theory is a branch of fixed point theory which finds its primary application in functional analysis. It is a sub-branch of the functional analytic theory in which geometric conditions on the mapping and / or underlying space play a crucial role. Although it has a purely metric facet, it is also a major branch of nonlinear functional analysis with close ties to Banach space geometry, [1]. Historically; the basic idea of metric fixed point principle firstly appeared in explicit from Banach’s thesis 1922 [2,p.5], where it was used to establish the existence of solution to an integral equation. This principle Banach contraction mapping is remarkable in its simplicity contraction; it is perhaps the most widely applied fixed point theorem in all of analysis. This is because the contractive condition on the mapping is simple and easy to test because: (i) IT requires only complete metric space for its setting. (ii) IT provides a contractive algorithm (iterative method). (iii) IT finds almost conociale applications in the theory of differential and integral equations specially the existence solution, uniqueness solution. All these properties motivate authors to study this principle and there appeared many types of contraction mapping on metric space. Recently, Bakhtin [3] introduced b-metric space as a generalization of metric spaces. He proved the Contraction mapping principle in b-metric spaces that generalized the famous Banach Contraction principle in metric spaces. Since then, several papers have dealt with fixed point theory or the variation principle for single-valued and multi-valued operators in b-metric spaces (as shown in [4 ]and [5]). in [6] Haung and Zhang introduced Cone metric spaces as a generalization of metric spaces by replacing the set of real numbers by an ordered Banach space and they proved some fixed point theorems for contractive mappings by using the normality of a Cone in results which expanded certain results of fixed points in metric spaces, and other authors who worked in the same way like [7] and [8]. In [9], Hussain and Shah introduced Cone b-metric spaces as a generalization of b-metric spaces and Cone metric spaces and they established some topological properties in such space and improved some recent results about KKM mappings in the setting of a Cone b-metric space, as well as in [10] they generalized the results of [9] and obtained some fixed point theorems of contractive mappings without the assumption of normality of the Cone. In this paper, we generalized the results of [9] and [10] and prove some coincidence and common fixed point theorems for a pair of discontinuous weakly compatible self mappings satisfying generalized contractive condition by using a certain vector valued altering function satisfying some properties in the setting of Cone-b- metric space where the normality of the Cone is omitted, we shall call this altering function by Cone-b-altering function. Preliminaries Consistent with Haung and Zhang [6], the following definitions: Let E be a normed space and P be a subset of E, P is called a Cone if: (i) P is closed, non empty and P ≠ {0}. (ii) ax + by ∈ P for all x, y ∈ P and non-negative real numbers a, b. (iii) P ∩ (– P) = {0}. Given a Cone P ⊂ E, we define a partial ordering “≤” with respect to P by x ≤ y if and only if y – x ∈ P, we write x < y to indicate that x ≤ y but x ≠ y, while x ≪ y stand for y – x ∈ int(P), where int(P) is the interior of P. 341 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 The Cone P is called normal if there is a number k > 0 such that for all x, y ∈E, 0 ≤ x ≤ y implies ∥x∥ ≤ k∥y∥, the least positive number satisfying the above inequality is called the normal constant of P. Example (2.1): [7] Let E =CR([0,1]) with supremum norm and P ={ f∈ E :f ≥ 0 }where || f|| =sup {|f(xi)| , xi ∈[0,1] } for all f, g ∈ P , put f(x) =x , g(x) =2x , then 0≤f ≤ g , ||f|| =1 , ||g|| =2 . So ||f|| ≤||g|| and K=1 . Therefore P is normal cone with normal constant K=1. Remark (2.2):[7] There are cones are not normal ,the following example show that : Example (2.3):[7] Let E =CR2([0,1]) with the norm ||f|| =||f||∞ +||f΄||∞ and consider the cone P ={ f∈ E :f ≥ 0 } , where ||f||∞ =max {|f(x1)|, |f(x2)|,…. |f(xn)|, xi ∈[0,1] ∀I =1,2,…..,n } ||f΄||∞ =max {|f΄(x1), |f΄(x2),…., |f΄(xn)|, xi ∈[0,1] ∀I =1,2,…..,n } For each k≥1 ,put f(x)=x and g(x) =x2k . Then 0≤g≤f , ||f|| =2 and ||g|| =2k+1, since k||f||<||g||, k is not a normal constant of P. Therefore , P is non –normal cone . In the following we always suppose that E is a normed space , P is a cone in E with int(p)≠∅ and ≤ is a partial ordering with respect to P . Definition 2.4: [6] Let X be non-empty set, a mapping d: X×X → E is called a Cone metric space on X if the following conditions are satisfied: (i) 0 ≤ d(x,y) for all x, y ∈ X with x ≠ y and d(x,y) = 0 if and only if x = y. (ii) d(x,y) = d(y,x) for all x, y ∈ X. (iii) d(x,y) ≤ d(x,z) + d(z,y) for all x, y, z ∈ X. Then the ordered pair (X,d) is called a Cone metric space. Example 2.5 :[6] Let E =R2 with usual norm on R2 defined by ||x|| =max{|x1|,|x2 |} for all x∈R2 , x=(x1,x2), xi∈R,i=1,2, P={(x,y) ∈ E: x,y≥0} ⊂ R2 , X=R and d:X×X→E such that :d(x,y)=(|x-y| ,α|x-y|), where α≥0 is a constsnt .Then (X,d) is a cone metric space . Definition 2.6: [9] Let X be a non empty set and S ≥ 1 be a given real number. A mapping d: X×X → E is said to be Cone b-metric if and only if, for all x, y, z ∈ X, the following conditions are satisfied: (i) 0 < d(x,y) with x ≠ y and d(x,y) = 0 if and only if x = y. (ii) d(x,y) = d(y,x). (iii) d(x,y) ≤ S[d(x,z) + d(z,y)]. The pair (X,d) is called a Cone-b-metric space. Example 2.7:[10] Let X={1,2,3,4}, E=R2, P={(x,y)∈ E :x≥0, y≥0 }. Define d:X×X→E by d(x,y) = ( ) 1 1 x y , x y if x y if x=yθ − − − − ≠       342 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Then (X,d) is a cone b-metric space with the coefficient S = 6 5 . Definition (2.8): [9] Let (X,d) be a Cone b-metric space, x ∈ X and {xn} be a sequence in X. Then (i) {xn} converge to x whenever for every c ∈ E with 0 ≪ c , there is a natural number N such that d(xn,x) ≪ c for all n > N. We denote this by n lim →∞ xn = x or xn → x (n → ∞). (ii) {xn} is Cauchy sequence whenever for every c ∈ E with 0 ≪ c, there is a natural number N such that d(xn,xm) ≪ c for all n, m > N. (iii) (X,d) is a complete Cone-b-metric space if every Cauchy sequence is convergent. Definition (2.9): [11] If Y be any partially ordered set with relation “≤” and f : Y → Y, we say that f is non- deceasing if, x, y ∈ Y, x ≤ y ⇒ f (x) ≤ f (y). Definition (2.10): [11] A function f: P → P is called subadditive if for all x, y ∈ P, f (x + y) ≤ f (x) + f (y). Seong-Hoon Cho [12] defined the ≪-increasing function by following: A function F:P → P is called ≪-increasing if for each x, y ∈ P, x ≪ y if and only if F(x) ≪ F(y). In the following we shall introduce Cone-b-altering function. Definition (2.11): Let (X,d) be a Cone-b-metric space, let F:P → P be a vector valued function, F is called a Cone-b-altering function if: (i) F is non-decreasing, subadditive, ≪-increasing and surjective. (ii) If,for {tn} , ( ) 0lim n n p F t →∞ ⊂ = ↔ 0lim n nt →∞ = (iii) F(αkt) = αkF(t) for α ≥ 1, k=1,2,……. Example (2.12): Let F(t) = t for all t ∈ P then F is Cone-b-altering function. The following lemmas which are necessary through our work in this sequel are often used in Cone metric spaces in which the Cone need not be normal. Lemma (2.13): [8] Let P be a Cone and {an} be a sequence in E. If c ∈ int(P) and 0 ≤ an → 0 (as n → ∞), then there exists N such that for all n > N, we have an ≪ c. Lemma (2.14): [8] Let x, y,z ∈ E, if x ≪ y and y≪ z then x ≪ z. Lemma (2.15): [9] Let P be a Cone and 0 ≤ u ≪ c for each c ∈ int(P), then u = 0. Lemma (2.16): [13] Let P be a Cone. If u ∈ P and u ≤ ku for some 0 ≤ k < 1, then u = 0. The following definitions and proposition are necessary in this sequel. 343 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Definition (2.17): [11] Let X be any non-empty set, f, g:X → X be mappings, a point w ∈ X is called point of coincidence of f and g if there is x ∈ X such that fx = gx = w. Definition (2.18): [11] Let X be any non-empty set, f, g:X → X be mappings, the pair (f, g) is called weakly compatible if x ∈ X, fx = gx ⇒ fgx = gfx. Proposition (2.19): [11] Let X be any non-empty set and f, g:X → X be mappings. If (f,g) is weakly compatible pair and have a unique point of coincidence then it is unique common fixed point of f and g. Main Result Theorem (3.1): Let (X,d) be a Cone metric space with the coefficient S ≥ 1, suppose the mappings f, g:X → X satisfying for all x, y ∈ X: F[d(fx,fy)] ≤ a1F[d(gx,gy)] + a2F[d(fx,gx)] + a3F[d(fy,gy)]+ a4F[d(fx,gy)] + a5F[d(fy,gx)] …(3.1.1) where this constant ai ∈ [0,1) and a1 + a2 + a3 + S(a4 + a5) < 1, i = 1,2,3,4,5 and F be Cone-b- altering function. If f (X) ⊂ g(X) and f (X) is complete, then f and g have a unique point of coincidence. Furthermore if the pair (f,g) is weakly compatible pair then f , g have a unique common fixed point. Proof: Let x0 ∈ X be arbitrary point in X. Since f (X) ⊂ g(X), we can choose a point x1 in X such that f x0 = g x1, if we continue in same way we can choose xn +1 in X for xn in X such that gxn + 1 = fxn for all n ≥ 0. If x = xn + 1, y = xn in (3.1.1), we have: F[d(fxn + 1,fxn)] ≤ a1F[d(gxn + 1,gxn)] + a2F[d(fxn + 1,gxn + 1)] +a3F[d(fxn,gxn)]+a4F[d(fxn + 1,gxn)]+ a5F[d(fxn,gxn + 1)] F[d(fxn + 1, fxn)] ≤ a1F[d(fxn, fx n – 1)] + a2F[d(fxn + 1, fx n)] + a3F[d(fxn, fx n – 1)]+ a4F[d(fxn + 1, fx n – 1)] + a5F[d(fxn, fxn)] ≤ a1F[d(fxn, fx n – 1)] + a2F[d(fxn + 1, fx n)] + a3F[d(fxn, fx n – 1)]+ Sa4F[d(fxn + 1, fx n)] + Sa4F[d(fxn, fxn – 1)] (1 – a2 – a4) F[d(fxn + 1, fx n)] ≤ (a1 + a3 + Sa4) F[d(fxn, fxn – 1)] ...(3.1.2) Using symmetry of (3.1.2) in x, y we have: (1 – a3 – Sa5) F[d(fxn + 1, fx n)] ≤ (a1 + a2 + Sa5) F[d(fxn, fxn – 1)] ...(3.1.3) Now combine (3.1.2) and (3.1.3) we have: F[d(fxn + 1, fx n)] ≤ 1 2 3 4 5 2 3 4 5 2a a a S(a a ) 2 a a S(a a ) + + + + − − − + F[d(fxn, fxn – 1)] Put 1 2 3 4 5 2 3 4 5 2a a a S(a a ) 2 a a S(a a ) λ + + + + = − − − + , We must prove that λ < 1. Since a1 + a2 + a3 + S(a4 + a5) < 1 344 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 ⇒ a2 + a3 + S(a4 + a5) < 1 – a1 ⇒ – a2 – a3 – S(a4 + a5) > a1 – 1 ⇒ 2 – a2 – a3 – S(a4 + a5) > a1 + 1 ⇒ 2 3 4 5 1 1 1 2 a a S(a a ) a 1 < − − − + + ⇒ 1 2 3 4 5 1 2 3 4 5 2 3 4 5 1 2a a a S(a a ) 2a a a S(a a ) 1 2 a a S(a a ) a 1 + + + + + + + + < < − − − + + Therefore 0 ≤ λ < 1, so we have F[d(fxn + 1, fx n)] ≤ λ F[d(fxn, fxn – 1)] ≤…..≤ λn F[d(fx1, f x0)] Now, for any m≥1 ,p≥1, it follows that d(fxm+p, fxm) ≤ S[d(fxm+p, fxm+p - 1) + d(fxm+p - 1, fxm) = Sd(fxm+p, fxm+p- 1) + Sd(fxm+p-1, fxm) ≤ S d(fxm+p, fxm+p - 1)] +s2[ d(fxm+p - 1, fxm+p-2) + d(fxm +p-2, fxm)] = Sd(fxm+p, f xm+p-1) + s2 d(fxm+p-1, f xm+p-2)] + s2d(fxm+p-2, f xm) ≤ Sd(fxm+p, f xm+p-1) + s2 d(fxm+p-1, f xm+p-2)] + s2d(fxm+p-2, f xm)+ s3 d(fxm+p-2, f xm+p-3)+....+sp-1 d(fxm+2, fxm+1)+ sp-1 d(fxm+1, fxm). But by (i) of definition (2.11)of F; F is non –decreasing and sub additive function we have : F[d(fxm+p, fxm)]]≤ F[Sd(fxm+p, f xm+p-1)] +F[ s2 d(fxm+p-1, f xm+p-2)] +F[ s3 d(fxm+p-2, f xm+p- 3)]+....+F[sp-1 d(fxm+2, fxm+1)]+ F[sp-1 d(fxm+1, fxm)] Also by (iii) of above definition we have : F[d(fxm+p, fxm)]]≤ SF[d(fxm+p, f xm+p-1)] + s2F[d(fxm+p-1, f xm+p-2)] + s3 F[d(fxm+p-2, f xm+p- 3)]+....+ sp-1F[ d(fxm+2, fxm+1)]+ sp-1F[ d(fxm+1, fxm)] ≤s 1m pλ + − F[d{fx1,fx0)]+s2 2m pλ + − F[d{fx1,fx0)]+s3 3m pλ + − F[d{fx1,fx0)]+.......+ sp-1 1mλ + F[d{fx1,fx0)]+ sp-1 mλ F[d{fx1,fx0)]. = 1 1[( ) 1]m p ps s s λ λ λ + − − − − F[d{fx1,fx0)]+ sp-1 mλ F[d{fx1,fx0)] ≤ 1p ms s λ λ + − F[d{fx1,fx0)]+ sp-1 mλ F[d{fx1,fx0)]→0 as m→∞ Hence, n ,m lim →∞ d(fxm+p, fxm) = 0 by (ii) of definition (2.11) of F. So by lemma (2.13), there exists k ∈ N such that d(fxm+p, fxm) ≪ c for each c ∈ int(P) and for all m > k.If n=m+p ,so for all m,n >k , {fxn} is a Cauchy sequence in f (X), but f (X) is complete, so the sequence {fxn} must be convergent in f (X), so there exists u ∈ f (X) and fxn → u. Now, since u ∈ f (X) ⊂ g(X), let u = g(v) for some v ∈ X. We show that gv = fv. d(fv,u) ≤ S[d(fv,fxn)] + d(fxn,u)] d(fv,u) ≤ Sd(fv,fxn) + Sd(fxn,u) So by properties of F we have: F[d(fv,u)] ≤ SF[d(fv,fxn)] + SF[d(fxn,u)] By (3.1.1), we have: F[d(fv,u)] ≤ S[a1F[d(gv,gxn)]+a2F[d(fv,gv)]+a3F[d(fxn,gxn)]+a4F[d(fv,gxn)] + a5F[d(fxn,gv)]] + SF[d(fxn,u)] ≤ S[a1F[d(u, fxn – 1)] + a2F[d(fv,u)] + a3F[d(fxn, fxn – 1)] + a4F[d(fv, fxn – 1)] + a5F[d(fxn,u)] ] + SF[d(fxn,u)] = Sa1F[d(u, fxn – 1)] + Sa2F[d(fv,u)] + Sa3F[d(fxn, fxn – 1)] + Sa4F[d(fv, fxn – 1)] + Sa5F[d(fxn,u)] + SF[d(fxn,u)] 345 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 ≤ Sa1F[d(u, fxn – 1)] + Sa2F[d(fv,u)] + S2a3F[d(fxn,u)] + S2a3F[d(u, fxn – 1)] + S2a4F[d(fv,u)] + S2a4F[d(u, fxn – 1)] + Sa5F[d(fxn,u)] + SF[d(fxn,u)] That implies: (1 – Sa2 – S2a4) ≤ 2 2 1 3 4 2 2 4 Sa S a S a 1 Sa S a + + − − F[d(u, fxn – 1)] + 2 3 5 2 2 4 S a Sa S 1 Sa S a + + − − F[d(fxn,u)] Now, let c ∈ int(P) be given. We can choose n0 ∈ N such that F[d(fxn – 1,u)] ≪ F – 1 2 2 4 2 2 1 3 4 1 Sa S a c Sa S a S a 2  − − ⋅  + +  and F[d(fxn,u)] ≪ F – 1 2 2 4 2 3 5 1 Sa S a c S a Sa S 2  − − ⋅  + +  for all n > n0. So by (i) of definition (2.11) of F; F is ≪-increasing and surjective, we have: F[d(fxn – 1,u)] ≪ 2 2 4 2 2 1 3 4 1 Sa S a c Sa S a S a 2  − − ⋅  + +  and F[d(fxn,u)] ≪ F – 1 2 2 4 2 3 5 1 Sa S a c S a Sa S 2  − − ⋅  + +  So that implies; F[d(fv,u)] ≪ c, thus by lemma (2.15) we have F[d(fv,u)] = 0 and so by (ii) of definition (2.11) of F; we obtain that d(fv,u) = 0 and so by fv = u = g(v), thus u is a point of coincidence of f and g. To prove u is unique, suppose u′ is another point of coincidence then there is v′ ∈ X such that u′ = f v′= g v, so by (3.1.1) we have: F[d(fv, fv′)] ≤ a1F[d(gv,gv′)] + a2F[d(fv,gv)] + a3F[d(fv′,gv′)] + a4F[d(fv,gv′)] + a5F[d(fv′,gv)] F[d(u,u′)] ≤ a1F[d(u,u′)] + a2F[d(u,u)] + a3F[d(u′,u′)] + a4F[d(u,u′)] + a5F[d(u′,u)] = (a1 + a4 + a5) F[d(u,u′)] ≤ (a1 + a2 + a3 + a4 + a5) F[d(u,u′)] But S ≥ 1, so we have: F[d(u,u′)] ≤ (a1 + a2 + a3 + Sa4 + Sa5) F[d(u,u′)] F[d(u,u′)] ≤ (a1 + a2 + a3 + S(a4 + a5)) F[d(u,u′)] Since a1 + a2 + a3 + S(a4 + a5) < 1, so by lemma (2.16) we have F[d(u,u′)] = 0 and so d(u,u′) = 0 (i.e.), u = u′. Therefore, f and g have a unique point of coincidence. Moreover, if the pair (f ,g) is weakly compatible then by proposition (2.19), u is unique common fixed point of f and g. Now we have the following corollaries: Corollary (3.2): Let (X,d) be a Cone-b-metric space with the coefficient S ≥ 1. Suppose the mappings f, g:X → X satisfy for all x, y ∈ X: d(fx,fy) ≤ a1d(gx,gy) + a2d(fx,gx) + a3d(fy,gy) + a4d(fx,gy) + a5d(fy,gx) …(3.1.4) where the constant ai ∈ [0,1) and a1 + a2 + a3 + S(a4 + a5) < 1, i = 1,2,3,4,5. If f (X) ⊂ g(X) and f (X) is complete, then f and g have a unique point of coincidence. Furthermore if the pair (f,g) is weakly compatible pair then f , g have a unique common fixed point. Proof: By taking F(t) = t for all t ∈ P, we obtain the required result. 346 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Corollary (3.3): Let (X,d) be a complete Cone-b-metric space with the coefficient S ≥ 1. Suppose the mappings f, g:X → X satisfy for all x, y ∈ X: d(fx,fy) ≤ a1d(x,y) + a2d(fx,x) + a3d(fy,y) + a4d(fx,y) + a5d(fy,x) ...(3.1.5) where the constant ai ∈ [0,1) and a1 + a2 + a3 + S(a4 + a5) < 1, i = 1,2,3,4,5. Then f has a unique fixed point in X. Proof: By taking F(t) = t for all t ∈ P and taking g(x) = x for all x ∈ X, we obtain the required result. The following corollary is theorem (2.3) of [10]. Corollary (3.4): Let (X,d) be a complete Cone-b-metric space with the coefficient S ≥ 1. Suppose f:X → X be a mapping satisfy for all x, y ∈ X: d(fx, fy) ≤ a2d(fx,x) + a3d(fy,y) + a4d(fx,y) + a5d(fy,x) ...(3.1.6) where the constant ai ∈ [0,1) and a2 + a3+ S(a4 + a5) < min{1, 2 5 }, i =2,3,4,5. Then f has a unique fixed point in X. Proof: By taking F(t) = t for all t ∈ P and g(x) = x for all x ∈ X, also by taking a1 = 0 in theorem (3.1) we obtain the required result. The following corollary is theorem (2.1) in [10] . Corollary (3.5): Let (X,d) be a complete Cone-b-metric space with the coefficient S ≥ 1. Suppose f: X → X be a mapping satisfy for all x, y ∈ X: d(fx, fy) ≤ a1d(x,y) ...(3.1.7) where the constant ai ∈ [0,1). Then f has a unique fixed point in X. Proof: By taking F(t) = t for all t∈P and g(x) = x for all x ∈ X, also by taking a2=a3=a4=a5=0 in theorem (3.1) we obtain the required result. Corollary (3.6): Let (X,d) be a Cone-b-metric space with the coefficient S ≥ 1. Suppose the mappings f, g::X → X satisfy for all x, y ∈ X: F[d(fx, fy)] ≤ a1 F[d(gx,gy)]+λ{F[d(fx,gx)]+F[d(fy,gy)]}+β{F[d(fx,gy)+F[d(fy,gx)]} …(3.1.8) where the constants a1, λ, β ∈ [0,1) with a1 + 2λ + 2Sβ < 1 and F be altering function. If f (X) ⊂ g(X) and f (X) is complete, then f and g have a unique point of coincidence. Furthermore if the pair (f,g) is weakly compatible pair then f , g have a unique common fixed point. Proof: By taking λ = a2 = a3 and β = a4 = a5 in theorem (3.1) we obtain the required result. Corollary (3.7): Let (X,d) be a complete Cone-b-metric space with the coefficient S ≥ 1. Suppose the mappings f, g::X → X satisfy for all x, y ∈ X: F[d(fx, fy)] ≤ a4 F[d(fx,gy)] + a5F[d(fy,gy)] … (3.1.11) 347 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 where the constants ai ∈ [0,1) and S(a4 + a5) < 1, i = 4, 5 and F be Cone-b-altering function. If f (X) ⊂ g(X) and f (X) is complete, then f and g have a unique point of coincidence. Furthermore if the pair (f,g) is weakly compatible pair then f , g have a unique common fixed point. Proof: By taking a1 = a2 = a3 = 0 in theorem (3.1) we obtain the required result. References 1. Sami;A. (2005) on Some Results on Multivalued Mappings Concerning Fixed Point Theorems and Ishikawa Iteration, M.Sc Thesis, College of Education Ibn-Al-Haitham, University of Baghdad. 2. Smart, D.R. (1974), Theorems of Fixed Point, Cambridge University, New York. 3. Bakhtin,I.A. (1989), The Contraction Mapping Principle in Almost Metric Spaces, Funct.Anal.,Gos.Ped.Inst.Unianowsk, Vol.30, .26-37. 4. Zerwik,S.C. (1998) Nonlinear Set-Valued Contraction Mappings in b-Metric Spaces, Attisem.Mat.Univ.Modena, Vol.46, .263-276. 5. Boriceanu,M., Bota;M. and Petrusel; A. (2010), Mutivalued Fractals in b-Metric Spaces, Cen.Eur.J.Math., Vol. 8, (.2), .367-377. 6. Haung,L.G. and Zhang,X. (2007) Cone Metric Space and Fixed Point Theorems of Contractive Mappings, J. Math. Anal. and Appl., Vol.332, (2) ,.1468-1476. 7. Rezapour, SH. and Hamlbarani; R. (2008), Some Notes on the Paper Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings, Journal of Mathematical Analysis and Applications, Vol.345,.719-724. 8. Jankovic,S., Kadelburg, Z. and Radenovic; S. (2011), On Cone Metric Space, Asurvey, Nonlinear Analysis, Vol.74,.2591-2601. 9. Hussian,N.and Shah; M.H. (2011), KKM Mappings in Cone b-Metric Spaces, Computes and Mathematics with Applications, Vol.62, 1677-1684. 10. Huang,H. and Xu, SH. (2012) Fixed Point Theorems of Contractive Mappings in Cone b- Metric Spaces and Applications,.1113-13113. 11. Malhotra, S.K., Shukla, S. and Sen, R., (2011), Fixed Point Theorems in Cone Metric Spaces by Altering Distances, International Mathematical Forum, Vol.54, (6) , .2665- 2671. 12. Cho,.S.H. (2012) Fixed Point Theorems for Generalized Contractive Mappings on Cone Metric Spaces, International Journal of Mathematics Analysis, Vol.50, (.6), .2473- 2481. 13. Cho,.S.H.and Bae,J.S. (2011) Common Fixed Point Theorems for Mappings Satisfying Property (E.A) on Cone Metric Space, Mathematics and Computer Modeling, Vol.53,.945- 951. 348 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 بعض مبرھنات نقاط التطابق و النقاط الصامدة المشتركة لتطبیقین ذاتیین (b)یحققان شرطا ً منكمشا ً معمما ً في فضاء قرصي متري من النوع تماره شھاب أحمد جامعة بغداد)/ابن الھیثم(كلیة التربیة للعلوم الصرفة /قسم الریاضیات 2013نیسان 16في ، قبل 2012تشرین األول 17أستلم البحث في : المستخلص في ھذا البحث، تم اثبات بعض مبرھنات نقاط التطابق و النقاط الصامدة المشتركة لزوج من تطبیقات ذاتیة غیر مع افتراض ان (b)مستمرة متبادلة تبادال ً ضعیفا ً یحققان شرطا ً منكمشا ً معمماً في فضاء قرصي متري من النوع ة في ھذا الفضاء غیر طبیعیة. نتائجنا ھي تعمیم لبعض النتائج الحالیة.القرص المستخدم مفتاحیة : نقاط التطابق والنقاط الصامدة المشتركة ، زوج من التطبیقات المتبادلة الضعیفة، تطبیق ذاتي الكلمات ال ، قرص طبیعي، قرص غیر طبیعي .bمنكمش معمم ، فضاء قرص المتري من الصنف 349 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Abstract Definition 2.4: [6] Definition 2.6: [9]