Fixed Point Theorems for Contractions and Generalized Contractions in Compact G-Metric Spaces

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1 Journal of Interpolation and Approximation in Scientific Computing 2015 No.1 (2015) Available online at Volume 2015, Issue 1, Year 2015 Article ID jiasc-00071, 8 Pages doi: /2015/jiasc Research Article Fixed Point Theorems for Contractions and Generalized Contractions in Compact G-Metric Spaces G. M. Abd-Elhamed Department of Mathematics College of Science and Humanities studies, Salman Bin Abdul-Aziz University, Saudi Arabia Copyright 2015 c G. M. Abd-Elhamed. 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 research paper is to prove some fixed point theorems for contractions and generalized contractions in compact G-metric spaces. This work considered as a generalization of Edelstein 2] and Suzuki 3]. Keywords: Compact G-Metric Spaces, Fixed Point, Contractions Mapping. Mathematics Subject Classification. 33C05, 33C20, 33C70 1 Introduction The study of fixed points of mappings satisfying certain contractive conditions has been at the center of vigorous research activity, because it has a wide range of applications in different areas such as differential equations, operation research, mathematical economics, biology, chemistry, games theory, and physics. The contraction mapping principle proved in 1] is one of the important theorems in fixed point theory. The following fixed point theorem was proved by Edelstein 2] in Theorem 1.1. Let (X, d ) be a compact metric space and T be a mapping on X. Assume d(t x,ty) < d(x, y) for all x,y X with x y. Then T has a unique fixed point. In 2009, Suzuki 3] improved the results of Banach and Edelstein in the following theorem. Theorem 1.2. Let (X, d ) be a compact metric space and T be a mapping on X. Assume that 1 2 d(x, T x) < d(x, y) implies d(t x, Ty) < d(x, y) for x, y X. Then T has a unique fixed point. In 2006, Mustafa Z. and Sims B. introduced a new class of generalized metric spaces (see 4]) which are called G-metric spaces. Recently, Mustafa Z et. al. studied many fixed point theorems for mapping various contractive conditions on complete G-Metric spaces (see 5] -8]). In this paper, firstly, we prove some fixed point theorems for contractions and generalized contractions in the compact G-metric spaces (Theorem 3.1, Theorem 3.4). Our results generalize Theorem (1.1), Theorem (1.2), in compact G- Metric spaces. Finally, we prove a fixed point theorem for self- mapping satisfying a new contractive condition in compact G-metric spaces which generalize Theorem (3.1). Corresponding author. address: gehadw2010@hotmail.com 20

2 Definitions and Preliminaries We begin by briefly recalling some basic definitions and results for G-Metric spaces (see 4]) that will be needed in the sequel. Definition 2.1. Let X be a non empty set, and 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); (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. Example 2.1. Let R be the set of all real numbers. Define G : R R R R + by G(x,y,z) = x y + y z + z x, for all x,y,z X. Then it is clear that (R,G) is a G-metric space. Proposition 2.1. Let (X,G)be a G-metric space. Then for any x,y,z and a X, it follows that: (1) If G(x,y,z) = 0 then x = y = z, (2) G(x,y,z) G(x,x,y) + G(x,x,z), (3) G(x,y,y) 2G(y,x,x), (4) G(x,y,z) G(x,a,z) + G(a,y,z), (5) G(x,y,z) 2 3 (G(x,y,a) + G(x,a,z) + G(a,y,z)), (6) G(x,y,z) G(x,a,a) + G(y,a,a) + G(z,a,a). Definition 2.2. Let (X,G) be a G-metric space, and (x n ) be a sequence of points of X, we say that (x n ) is G-convergent to x if for any ε > 0, there exists n 0 N such that G(x,x n,x m ) < ε, for all n,m n 0. Proposition 2.2. Let (X,G) be a G-metric space. Then the following are equivalent: (1) (x n ), is G-convergent to x, (2) G(x n,x n,x) 0, as n, (3) G(x n,x,x) 0, as n, (4) G(x m,x n,x) 0, as m,n. Definition 2.3. Let (X,G) be a G-metric space, a sequence (x n ) is called G-Cauchy if given ε > 0, there is n 0 N such that G(x n,x m,x l ) < ε, f or all n,m,l n 0. Definition 2.4. Let (X,G) and (X,G ) be G-metric spaces and let f : (X,G) (X,G ) be a function, then f is said to be G-continuous at a point a X, if given ε > 0, there exists δ > 0 such that x, y X;G(a,x,y) < δ implies G ( f (a), f (x), f (y)) < ε. A function f is G-continuous on X if, and only if, it is G-continuous at all a X. Proposition 2.3. Let (X, G) be a G-metric space. Then the function G(x,y,z) is continuous in all variables. Definition 2.5. 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,G). Definition 2.6. A G-metric space (X,G) is said to be a compact G-metric space if it is G-complete and G-totally bounded.

3 Main results Firstly, we prove a fixed point result of Edelstein in the setting of compact G-metric spaces: Theorem 3.1. Let X be a compact G-metric space. If T : X X satisfies G(T (x 1 ),T (x 2 ),T (x 3 )) < G(x 1,x 2,x 3 ) when x 1 x 2 x 3 in X, then T has a unique fixed point in X and the fixed point can be found as the limit of T n (x 0 ) as n for any x 0 X. Proof. Let given by ϕ : X 0;1) ϕ(x) = G(x,T (x),t 2 (x)) Since X is compact, the function G(x,T (x),t 2 (x)) takes on its minimum value, there is an a X such that G(a,T (a),t 2 (a)) < G(x,T (x),t 2 (x)) f or all x X. We ll show by contradiction that a is a fixed point for T. If T (a) a then the hypothesis about T in the theorem (taking x 1 = a and x 2 = T (a), x 3 = T (T (a))) says G(T (a),t (T (a)),t (T 2 (a))) G(a,T (a),t 2 (a)). which contradicts the minimality of G(a,T (a),t 2 (a)) among all numbers G(x,T (x),t 2 (x)). So T (a) = a. To show T has at most one fixed point in X, suppose T has three fixed points a 1 a 2 a 3. Then G(a 1,a 2,a 3 ) = G(T (a 1 ),T (a 2 ),T (a 3 ))) < G(a 1,a 2,a 3 ). This is impossible, so a 1 = a 2 = a 3, Finally, we show for any x X that the sequence x n = T n (x 0 ) G-converges to a as n : If for some k 0 we have x k = a, then x k+1 = T (x k ) = T (a) = a, and more generally x n = a, for all n k, so (x n ) G-converges to a since the terms of the sequence equal a for all large n. Now we may assume instead that x n a for all n. Let T be a contractive map of a compact G-metric space X into itself and x 0 X. Then show that T x 0,T 2 x 0,T 3 x 0,... G-converge to the unique fixed point of T. Let a denote the unique fixed point of T. Then G(T n+1 x 0,a,a) = G(T (T n x 0 ),Ta,Ta) < G(T n x 0,a,a),n = 1,2,... Thus, ( G(T n+1 x 0,a,a) ) n 1 is a decreasing sequence of nonnegative numbers and hence converges to λ, say, lim n G(T n+1 x 0,a,a) = λ. The sequence (T n x 0 ), being in the compact G-metric space has a G-convergent subsequence (T n kx 0 ) k 1. Let limt n kx 0 = y. Then by continuity of G-metric space we get G(y,a,a) = limg(t n kx 0,a,a) = λ. If λ 0, then y a k k and so λ = G(y,a,a) > G(Ty,Ta,Ta) = lim k G(T (T n k x 0 ),Ta,Ta) lim k supg(t 2 (T n k x 0 ),T 2 a,t 2 a) lim k supg(t n k+1 x 0,T n k+1 n k a,t n k+1 n k a) = lim k supg(t n+1 x 0,a,a) = λ This is a contradiction. So λ =0 and hence y = a. Thus a is the only limit of (T n x 0 ). Example 3.1. Let X = 0,1] and G(x,y,z) = x y + y z + x z be a G-metric on X. Define T : X X by T (x) = 1+x 1. Then T satisfies the condition of Theorem (3.1 ) and so T has a unique fixed point.

4 23 Proof. For all x, y X, we have T x Ty = x y (1 + x)(1 + y), Ty T z = y z (1 + y)(1 + z), T z T x = z x (1 + z)(1 + x) when x y, T x Ty x y = 1 (1 + x)(1 + y) < 1, G(T x,ty,t z) = T x Ty + Ty T z + T z T x < x y + y z + z x = G(x,y,z), Hence T has a unique fixed point in 0,1] at x = Proposition 3.1. A G-continuous, iteratively contractive self-map T on a compact G-metric space (X, G) has a unique fixed point a and liminfg(t n x,t n y,t n z) = 0 f or all x, y, z X, n (3.1) So a is a limit point of every orbit. Proof. Since T is iteratively contractive, i.e., given x y z, there exists r in N with G(T r x,t r y,t r z)<g(x,y,z). (3.2) The existence of a fixed point a follows from Theorem 3.1 with T = T r and uniqueness follows from (3.2). Suppose liminfg(t n x,t n y,t n z) = ε > 0, for some x,y,z X. Since X is compact there exists a limit point (u,v,w) of ((T n x,t n y,t n z)) with G(u,v,w) = ε. Apply (3.2) to u,v,w to get r with G(T r u,t r v,t r w)< ε. Since (u,v,w) is a limit point of ((T n x,t n y,t n z),) so is (T r u,t r v,t r w). Hence, by the definition of ε, ε G(T r u,t r v,t r w),a contradiction. So ε = 0 which gives (3.1). Apply (3.1) with y = z = a to conclude that a is a limit point of every orbit (T n x). Proposition 3.2. A continuous self-map T on a compact G-metric space (X,G) has a fixed point if, and only if, given x y z, there exists h : X X such that T h = ht and G(hx,hy,hz) < G(x,y,z). Proof. Given Tw = w, take hx = w for all x, T h(x) = Tw = w = ht (x),and G(hx,hy,hz) = G(w,w,w) = 0 < G(x,y,z). For the converse take w with G(w,Tw,T 2 w) = ming(x,t x,t 2 x). Suppose w Tw. Get h with T h = ht and x G(h(w),h(Tw),h(T 2 w)) < G(w,Tw,T 2 w). For x = hw, this inequality becomes G(x,T x,t 2 x) < G(w,Tw,T 2 w), contradicting our choice of w. Secondly, we prove a fixed point result of Suzuki type in compact G-metric spaces: Theorem 3.2. Let (X,G) be a compact G- metric space and let T be a mapping on X. Assume that αg(x,t x,t x) < G(x,y,z),implies G(T x,ty,t z) < G(x,y,z), (3.3) for α (0, 2 1 ], x, y, z X. Then T has a unique fixed point.

5 24 Proof. If we consider then there exists a sequence (x n ) in X such that β = inf{g(x,t x,t x) : x X} lim G(x n,t x n,t x n ) = β. Since X is compact G- metric space, there exists v,w X such that a sequence (x n ) is G- converges to v X, and (T x n ) G- converge to w X. We assume β > 0. Hence, by the continuity of the function G, we have We can choose k N such that for n N with n k. Thus, β = lim G(x n,t x n,t x n ) = G(v,w,w) = lim G(x n,w,w) G(x n,w,w) > 2 3 β,and G(x n,t x n,t x n ) < 2 3α β G(x n,t x n,t x n ) < 2 3α β < 1 α G(x n,w,w) αg(x n,t x n,t x n ) < G(x n,w,w) f or n k. By hypotheses (3.3), G(T x n,tw,tw) < G(x n,w,w) holds for n k. This implies G(w,Tw,Tw) = lim G(x n,tw,tw) lim G(x n,w,w) = β From the definition of β, we obtain G(w, Tw, Tw) = β. Since αg(w, Tw, Tw) < G(w, Tw, Tw), we have G(Tw,T 2 w,t 2 w) < G(w,Tw,Tw) = β which contradicts the definition of β. Therefore we obtain β = 0. We next show that T has a fixed point. We assume that T does not have a fixed point. Then G(T x n,t 2 x n,t 2 x n ) < G(x n,t x n,t x n ) (3.4) holds for every n N because 0 < αg(x n,t x n,t x n ) < G(x n,t x n,t x n ). We have lim G(v,T x n,t x n ) = G(v,w,w) = lim G(x n,t x n,t x n ) = β = 0; which implies that (T x n ) also G-converges to v. Using the rectangle inequality ( We put x = v, y = T 2 x n, a = T 2 x n ), we have lim G(v,T 2 x n,t 2 x n ) lim G(v,T x n,t x n ) + lim G(T x n,t 2 x n,t 2 x n ) using (3.4), we get Thus, (T 2 x n ) G-converges to v. If then we have lim G(v,T 2 x n,t 2 x n ) lim G(v,T x n,t x n ) + lim G(x n,t x n,t x n ) = 0. αg(x n,t x n,t x n ) G(x n,v,v) and αg(t x n,t 2 x n,t 2 x n ) G(v,T x n,t x n ) G(x n,t x n,t x n ) G(x n,v,v) + G(T x n,t x n,v) αg(x n,t x n,t x n ) + αg(t x n,t 2 x n,t 2 x n ) < αg(x n,t x n,t x n ) + αg(x n,t x n,t x n ) = 2αG(x n,t x n,t x n ) (2α 1)G(x n,t x n,t x n ) > 0,hence (2α 1) > 0 = α > 1 2

6 25 which is a contradiction. Hence for every n N, either αg(x n,t x n,t x n ) < G(x n,v,v) or αg(t x n,t 2 x n,t 2 x n ) < G(T x n,t x n,v) holds. So by hypotheses (3.3), we conclude that one of the following inequalities holds for all n in an infinite subset of N: G(T x n,t v,t v) < G(x n,v,v) or G(T 2 x n,t 2 x n,t v) < G(T x n,t x n,v) holds. Hence one of the following holds: In the first case, we obtain which implies T v = v. Also, in the second case, we obtain G(v,T v,t v) = lim G(T x n,t v,t v) < lim G(x n,v,v) = 0 G(v,v,T v) = lim G(T 2 x n,t 2 x n,t v) < lim G(T x n,t x n,v) = 0. Hence, we have v is a fixed point of T in both cases, this is a contradiction, therefore T has a fixed point.to prove the uniqueness, fix y, z, x X with y z x. Then since αg(x,t x,t x) = 0 < G(x,y,z), we have G(T x,ty,t z) = G(x,y,z) < G(x,y,z), this is a contradiction, and hence T has a unique fixed point. Corollary 3.1. Let (X,G) be a compact G- metric space and T be a mapping on X. Assume that αg(x,t x,x) < G(x,y,z),implies G(T x,ty,t z) < G(x,y,z) for α (0, 2 1 ], x, y, z X. Then T has a unique fixed point. Example 3.2. Let X = 0,1] and G(x,y,z) = x y + y z + x z be a G-metric on X. Define T : X X by T x = x 4. Then T satisfies the condition of Theorem 2.4 and so T has a unique fixed point. Proof. For all x, y, z X, we have G(T x,ty,t z) = 1 ( x y + y z + x z ) < x y + y z + x z = G(x,y,z) 4 and hence G(T x,ty,t z) < G(x,y,z). Now for x = 0, we have αg(x,t x,t x) = G(0,0,0) = 0 < G(x,y,z),Then for all x,y,z X. αg(x,t x,t x) < G(x,y,z) = G(T x,ty,t z) < G(x,y,z), Finally, We prove a fixed point theorems for self- mapping satisfying a new contractive condition in compact G-metric spaces which generalized the Theorem(3.1): Theorem 3.3. Let T be a continuous mapping of a compact G- metric space X into itself satisfying the condition G(T x,ty,t z) ] G(x,Ty,T z)g(y,t x,t z)g(z,ty,t x)g(x,t x,t x) + G(y,Ty,T z) + G(x,y,z) < α 1 + G(x,Ty,T z)g(y,t x,t z)g(x,t x,t z)g(y,ty,ty)g(z,t z,t z) + G(y,T x,t z)g(x,y,z) + γg(y,t x,t z) + δg(x,y,z), for all x, y, z X,where α, γ, δ are non negative reals such that 2α + γ + δ < 1.Then T has unique fixed point.

7 26 Proof. We define a function f on X, such that f (x) = G(x, T x, x).since T is G-continuous, therefore f is also. From compactness of X, there exist p X, such that f (p) = inf{ f (x) : x X}, G(p,T p, p) < G(x,T x,x) (3.5) if f (p) 0, it follows that T (p) p, then f (T (p)) = G(T p,t (T (p)), p) = G(T p,t 2 p,t p) < G(T p,t 2 p,t p)g(t p,t p,t p)g(p,t 2 p,t p)g(p,t p,t p) + G(T p,t 2 ] p,t p) + G(p,T p, p) α 1 + G(p,T 2 p,t p)g(t p,t p,t p)g(p,t p,t p)g(t p,t 2 p,t 2 p)g(p,t p,t p) + G(T p,t p,t p)g(p,t p, p) (1 α)g(t p,t 2 p,t p) < (α + δ)g(p,t p, p) G(T p,t 2 p,t p) < + γg(t p,t p,t p) + δg(p,t p, p) < αg(p,t p, p) + δg(p,t p, p) + αg(t p,t 2 p,t p), (α + δ) G(p,T p, p) < G(p,T p, p), (1 α) because 2α +γ +δ < 1,which is a contradiction of condition (3.5), and so p = T (p).to prove the uniqueness, suppose that T (p 1 ) = p 1,T (p 2 ) = p 2,then G(T p 1,T p 2,T p 1 ) = G(p 1, p 2, p 1 ) ] G(p2, p 2, p 1 ) + G(p 1, p 2, p 1 ) < α + γg(p 2, p 1, p 1 ) + δg(p 1, p 2, p 1 ) 1 + G(p 2, p 1, p 1 )G(p 1, p 2, p 1 ) by use proposition(2.2), we get ] G(p 1, p 2, p 1 ) G(T p 1,T p 2,T p 1 ) = G(p 1, p 2, p 1 ) < 2α 1 + G(p 1, p 2, p 1 )G(p 1, p 2, p 1 ) where This is a contradiction, hence p is a unique fixed point. + 2γG(p 2, p 1, p 1 ) + δg(p 1, p 2, p 1 ) G(p 1, p 2, p 1 ) < (2α + γ + δ)g(p 1, p 2, p 1 ) < G(p 1, p 2, p 1 ). Theorem 3.4. Let T be a continuous mapping of a compact G- metric space X into itself satisfying the conditions G(T x,ty,t z) ] G(x,Ty,T z)g(y,t x,t z)g(z,ty,t x)g(y,ty,ty) + G(y,Ty,T z) + G(x,y,z) < α 1 + G(x,Ty,T z)g(y,t x,t z)g(x,t x,t z)g(y,ty,ty)g(z,t z,t z) + G(y,T x,t z)g(x,y,z) + δg(x,y,z), f or all x,y,z X, where α,δ are non negative reals such that 2α + δ < 1.Then T has unique fixed point. Proof. We define a function f on X, such that f (x) = G(x,T x,x). From compactness of X, there exist p X, such that f (p) = inf{ f (x) : x X}, if f (p) 0, it follows that T (p) p, then f (T (p)) = G(T p,t 2 p,t p) < G(T p,t 2 p,t p)g(t p,t p,t p)g(p,t 2 p,t p)g(t p,t 2 p,t 2 p) + G(T p,t 2 ] p,t p) + G(p,T p, p) α 1 + G(p,T 2 p,t p)g(t p,t p,t p)g(p,t p,t p)g(t p,t 2 p,t 2 p)g(p,t p,t p) + G(T p,t p,t p)g(p,t p, p) + δg(p,t p, p), G(T p,t 2 p,t p) < αg(p,t p, p) + δg(p,t p, p) + αg(t p,t 2 p,t p), (1 α)g(t p,t 2 p,t p) < (α + δ)g(p,t p, p) G(T p,t 2 p,t p) < (α + δ) G(p,T p, p) < G(p,T p, p), (1 α) because 2α + δ < 1, which is a contradiction, and so p = T (p). Using the same method of Theorem (3.6), we can prove the uniqueness.

8 27 Theorem 3.5. Let T be a continuous mapping of a compact G- metric space X into itself satisfying the conditions G(T x,ty,t z) ] G(x,Ty,T z)g(y,t x,t z)g(z,ty,t x)g(z,t z,t z) + G(y,Ty,T z) + G(x,y,z) < α 1 + G(x,Ty,T z)g(y,t x,t z)g(x,t x,t z)g(y,ty,ty)g(z,t z,t z) + G(y,T x,t z)g(x,y,z) + δg(x,y,z), for all x,y,z X,where α,δ are non negative reals such that 2α + δ < 1.Then T has unique fixed point. Proof. The proof follows from the argument similar to that used in Theorem (3.6). Acknowledgment 1. The author would like to express her sincere gratitude to the referee for the valuable comments and suggestions. 2. This project was supported by the Deanship of Scientific Research at Salman Bin Abdulaziz University, KSA, under the research project No. 2014/01/ The author did this work while she was on leave from Department of Mathematics, College of Girls, Ain Shams University, Cairo - Egypt. References 1] S. Banach, Sur les operations dans les ensembles abstraits et leur application auxéquations integrales, Fun Math, 3 (1922) ] M. Edelstein, On fixed and periodic points under contractive mappings, J London Math Soc, 37 (1962) ] T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal, 71 (2009) ] Z. Mustafa, Sims B. A new approach to generalized metric spaces, J Nonlinear Convex Anal, 7 (2006) ] M. Abbas, T. Nazir, S. Radenović, Some periodic point results in generalized metric spaces, Appl Math Comput, 217 (2010) ] Z. Mustafa, H. Obiedat, F. Awawdeh, Some of fixed point theorem for mapping on complete G-metric spaces, Fixed Point Theory Appl, Article ID , (2008) 12 pages. 7] Z. Mustafa, W. Shatanawi, M. Bataineh, Existence of fixed point result in G-metric spaces, Int J Math Math Sci, Article ID (2009) 10 pages. 8] Z. Mustafa, B. Sims, Fixed point theorems for contractive mappings in complete G-metric space, Fixed Point Theory Appl, Article ID (2009) 10 pages.