(C α,c β )-Admissible Functions in Quasi-Pseudometric Type Spaces

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1 Sohag J. Math. 2, No. 2, (2015) 51 Sohag Journal of Mathematics An International Journal (,C β )-Admissible Functions in Quasi-Pseudometric Type Spaces Yaé Ulrich Gaba epartment of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa Received: 27 Jul. 2014, Revised: 24 Jan. 2015, Accepted: 28 Jan Published online: 1 May 2015 Abstract: In this article, we give some fixed point results in left K-complete quasi-pseudometric type spaces for self-mappings that are (,C β )-admissible. Keywords: quasi-pseudometric, left(right)k-completeness, (,C β )-admissible 1 Introduction and Preliminaries In the last few years the theory of quasi-metric spaces and other related structures such as quasi-normed cones and asymmetric normed linear spaces (see for instance [3]) has been at the center of rigorous research activity, because such a theory provides a convenient framework in the study of several problems in theoretical computer science, approximation theory and convex analysis. The existence of fixed points and common fixed points of mappings satisfying certain contractive conditions in this setting, has also been discussed. Recently, in [1], Eniola et al. discussed the newly introduced notion of quasi-pseudometric type spaces as a logical equivalent to metric type spaces- introduced by M. A. Khamsi[4]-when the initial distance-like function is not symmetric. The aim of this paper is to analyze the existence of fixed points for mapping defined on a left K-complete quasi-pseudometric type space (X,, K). The present results generalize another one already obtained by Gaba in [2], where the author considered(α, γ)-contractions. efinition 11 Let X be a nonempty set, and let the function : X X [0, ) satisfy the following properties: (1) (x,x)=0 for any x X; (2) (x,y) K ( (x,z 1 )+(z 1,z 2 )+ +(,y) ) for any points x,y,z i X, i=1,2,...,n and some constant K > 0. Then (X,,K) is called a quasi-pseudometric type space. Moreover, if (x,y) = 0 = (y,x) = x = y, then is said to be a T 0 -quasi-pseudometric type space. The latter condition is referred to as the T 0 -condition. Remark 12 Let be a quasi-pseudometric type on X, then the map 1 defined by 1 (x,y) = (y,x) whenever x, y X is also a quasi-pseudometric type on X, called the conjugate of. We shall also denote 1 by t or. It is easy to verify that the function s defined by s := 1, i.e. s (x,y)=max{(x,y),(y,x)} defines a metric-type (see [4]) on X whenever is a T 0 -quasipseudometric type. If we substitute the property (1) by the following property (3) : (x,y)=0 x=y, we obtain a T 0 -quasi-pseudometric type space directly. Moreover, for K = 1, we recover the classical pseudometric, hence quasi-pseudmetric type spaces generalize quasi-pseudometrics. It is worth mentioning that if (X,, L) is a pseudometric type space, then for any L K, (X,, L) is also a pseudometric type space. We give the following example to illustrate the above comment. Example 13 Let X = {a, b, c} and the mapping : X X [0, ) defined by (a,b) = (c,b) = 1/5, (b,c) = (b,a) = Corresponding author gabayae2@gmail.com

2 52 Y. U. Gaba: (,C β )-Admissible Functions in... (c,a) = 1/4, (a,c) = 1/2, (x,x) = 0 for any x X. Since 1 2 = (a,c)>(a,b)+(b,c)= 9 20, then we conclude that X is not a quasi-pseudometric space. Nevertheless, with K= 2, it is very easy to check that(x,,2) is a quasi-pseudometric type space. efinition 14 Let (X,, K) be a quasi-pseudometric type space. The convergence of a sequence ( ) to x with respect to, called -convergence or left-convergence and denoted by x, is defined in the following way x (x,xn ) 0. (1) Similarly, the convergence of a sequence ( ) to x with respect to 1, called 1 -convergence or right-convergence and denoted by x 1 n x, is defined in the following way 1 x (,x) 0. (2) Finally, in a quasi-pseudometric type space (X,, K), we shall say that a sequence( ) s -converges to x if it is both left and right convergent to x, and we denote it as x s n x or x when there is no confusion. Hence s x x and x 1 n x. efinition 15 A sequence ( ) in a quasi-pseudometric type space(x,,k) is called (a) left K-Cauchy with respect to if for every ε > 0, there exists n 0 N such that n,k : n 0 k n (x k, )<ε; (b) right K-Cauchy with respect to if for every ε > 0, there exists n 0 N such that n,k : n 0 k n (,x k )<ε; (c) s -Cauchy if for every ε > 0, there exists n 0 N such that Remark 16 n,k n 0 (,x k )<ε. A sequence is left K-Cauchy with respect to d if and only if it is right K-Cauchy with respect to 1. A sequence is s -Cauchy if and only if it is both left and right K-Cauchy. efinition 17 A quasi-pseudometric type space (X,, K) is called left-complete provided that any left K-Cauchy sequence is -convergent. efinition 18 A quasi-pseudometric type space (X,, K) is called right-complete provided that any right K-Cauchy sequence is -convergent. efinition 19 A T 0 -quasi-pseudometric type space (X,, K) is called bicomplete provided that the metric type s on X is complete. 2 Main Results We begin by recalling the following. efinition 21(Compare [2]) Let (X,, K) be a quasipseudometric type space. A function T : X X is called - sequentially continuous or left-sequentially continuous if for any -convergent sequence ( ) with x, the sequence(t ) -converges to T x, i.e. T Tx. We then introduce the following definition. efinition 22 Let (X,, K) be a quasi-pseudometric type space, f : X X and α,β : X X [0, ) be mappings and > 0,C β 0. We say that T is(,c β )-admissible with respect to K if the following conditions hold: (C1) α(x,y) = α( f x, f y), whenever x,y X; (C2) β(x,y) C β = β( f x, f y) C β, whenever x,y X; (C3) 0 C β / < 1/K. We begin by the following lemma. Lemma 23 Let (X,, K) be a quasi-pseudometric type space and let ( ) be a sequence in X. Then (x 0, ) K(x 0,x 1 )+K 2 (x 1,x 2 )+ + + K n 1 ( 2, 1 )+K n 1 ( 1, ). From Lemma 23, we deduce the following lemma. Lemma 24 (Compare [1, Lemma 38]) Let (X,, K) be a quasi-pseudometric type space and let ( ) be a sequence in X such that (,+1 ) λ ( 1, ) for all n 0, for some 0 < λ < 1/K. Then ( ) is a left K-Cauchy sequence. Similarly, Lemma 25 Let (X,, α) be a quasi-pseudometric type space and let ( ) be a sequence in X such that (+1, ) λ (, 1 ) for all n 0, for some 0 < λ < 1/K. Then ( ) is a right K-Cauchy sequence.

3 Sohag J. Math. 2, No. 2, (2015) / 53 We now state our main fixed point theorem. Theorem 26 Let(X,,K) be a Hausdorff left K-complete f : X X is(,c β )-admissible with respect to K. α(x,y)( f x, f y) β(x,y)(x,y) (3) (i) f is -sequentially continuous; (ii) there exists x 0 X such that α(x 0, f x 0 ) and β(x 0, f x 0 ) C β. Proof. 27 Let x 0 X such that α(x 0, f x 0 ) and β(x 0, f x 0 ) C β. efine the sequence ( ) by = f n x 0 = f 1. Without loss of generality, we can always assume that +1 for all n N, since if 0 = 0 +1 for some n 0 N, the proof is complete. Since f is (,C β )-admissible with respect to K and α(x 0, f x 0 ) = (x 0,x 1 ), we deduce that α(x 1,x 2 ) = ( f x 0, f x 1 ). By continuing this process, we get that α(,+1 ), for all n 0. Similarly, we establish that β(,+1 ) C β, for all n 0. Using (3), we get and hence (,+1 ) α( 1, )(,+1 ) β( 1, )( 1, ) C β ( 1, ), (,+1 ) C β ( 1, ) for all n 1. By Lemma 24, since 0 C β < 1/K, we derive that ( ) is a left K-Cauchy sequence. Since (X, d) is left K-complete and T -sequentially continuous, there exists x such that x and +1 f x. Since X is Hausdorff, x = f x. Corollary 28 Let (X,, K) be a Hausdorff left that f : X X is (,C β )-admissible with respect to K. α(x,y)( f x, f y) β(x,y)(x,y) (4) (i) f is 1 -sequentially continuous; (ii) there exists x 0 X such that α( f x 0,x 0 ) and β( f x 0,x 0 ) C β. Corollary 29 Let (X,, K) be a bicomplete f : X X is (,C β )-admissible with respect to K. α(x,y)( f x, f y) β(x,y)(x,y) (5) (i) f is s -sequentially continuous; (ii) there exists x 0 X such that α(x 0, f x 0 ) and (iii) the functions α and β are symmetric, i.e. α(a,b) = α(b,a) for any a,b X and β(a,b)=β(b,a) for any a,b X. Proof. 210 Following the proof of Theorem 26, it is clear that the sequence ( ) in X defined by +1 = f for all n = 0,1,2, is s -Cauchy. Since (X, s ) is complete and f s -sequentially continuous, there exists x such that x s n x and x s n+1 f x. Since (X, s ) is Hausdorff, x = f x. Remark 211 In fact, we do not need α and β to be symmetric. It is enough, for the result to be true, to have a point x 0 X for which α(x 0, f x 0 ), α( f x 0,x 0 ), β(x 0, f x 0 ) C β and β( f x 0,x 0 ) C β. Otherwise stated, it is enough to have a point x 0 X for which min{α(x 0, f x 0 ),α( f x 0,x 0 )} and min{β(x 0, f x 0 ),β( f x 0,x 0 } C β. We give the following results which are in fact consequences of the Theorem 26. Theorem 212 Let (X,, K) be a Hausdorff left that f : X X is (,C β )-admissible with respect to K. α(x,y)( f x, f y) β(x,y)(x,y) (6) (i) there exists x 0 X such that α(x 0, f x 0 ) and (ii) if ( ) is a sequence in X such that α(,+1 ) and β(,+1 ) C β for all n=1,2, and x, then there exists a subsequence((k) ) of( ) such that α(x,(k) ) and β(x,(k) ) C β for all k. Proof. 213 Following the proof of Theorem 26, we know that the sequence ( ) defined by +1 = f for all n = 0,1,2, -converges to some x and satisfies α(,+1 ) and β(,+1 ) C β for n 1. From

4 54 Y. U. Gaba: (,C β )-Admissible Functions in... the condition (ii), we know there exists a subsequence ((k) ) of ( ) such that α(x,(k) ) and β(x (k) ) C β for all k. Since f is satisfies (6), we get ( f x,(k)+1 )= ( f x, f (k) ) α(x,(k) )( f x, f (k) ) β(x,(k) )(x,(k) ) C β (x,(k) ). Letting k, we obtain ( f x,(k)+1 ) 0. Since X is Hausdorff, we have that f x = x. This completes the proof. Corollary 214 Let (X,, K) be a Hausdorff right that f : X X is (,C β )-admissible with respect to K. α(x,y)( f x, f y) β(x,y)(x,y) (7) (i) there exists x 0 X such that α( f x 0,x 0 ) and β( f x 0,x 0 ) C β ; (ii) if ( ) is a sequence in X such that α(+1, ) and β(+1, ) C β for all n=1,2, and x 1 n x, then there exists a subsequence((k) ) of( ) such that α((k),x) and β((k),x) C β for all k. Corollary 215 Let (X,, K) be a bicomplete f : X X is (,C β )-admissible with respect to K. α(x,y)( f x, f y) β(x,y)(x,y) (8) (i) there exists x 0 X such that α(x 0, f x 0 ) and (ii) if ( ) is a sequence in X such that α(x m, ) and β(x m, ) C β for all n,m N and x s n x, then there exists a subsequence ((k) ) of ( ) such that α(x,(k) ) and β(x,(k) ) C β for all k; (iii) the functions α, β are symmetric. In regard of Remark (211), another variant of the above corollary can be stated as follows: Corollary 216 Let (X,, K) be a bicomplete f : X X is (,C β )-admissible with respect to K. α(x,y)( f x, f y) β(x,y)(x,y) (9) (i) there exists x 0 X such that min{α( f x 0,x 0 ),α(x 0, f x 0 )} and min{β( f x 0,x 0 ),β(x 0, f x 0 )} C β ; (ii) if ( ) is a sequence in X such that min{α(x m, ),α(,x m )} and min{β(x m, ),β(,x m )} C β for all n,m N and x s n x, then there exists a subsequence ((k) ) of ( ) such that min{α(x,(k) ),α((k),x)} and min{β(x,(k) ),β((k),x)} C β for all k. We conclude this paper by the following theorem. Theorem 217 Let (X,, K) be a bicomplete T 0 -quasi-pseudometric space. Suppose that f : X X is (,C β )-admissible with respect to K. α(x,y) s ( f x, f y) β(x,y) s (x,y) (10) (i) there exists x 0 X such that α(x 0, f x 0 ) and (ii) if ( ) is a sequence in X such that α(+1, ) and β(+1, ) C β for all n=0,1,2, and x s n x, then α(,x) and β(,x) C β for all n N. Proof. 218 Let x 0 X such that α(x 0, f x 0 ) and ; β(x 0, f x 0 ) C β. efine a sequence ( ) by = f n x 0 = f 1. Without loss of generality, we can always assume that +1 for all n N, since if 0 = 0 +1 for some n 0 N, the proof is complete. Following the proof of Theorem 26, we see that ( ) is a s -Cauchy sequence α(+1, ) and β(+1, ) C β for all n = 0,1,2,. Since (X, s ) is complete, there exists x such that x s n x and. Now, using the contractive condition (10) and condition(ii), we deduce that [ s (x, f x ) K s (x,+1 )+ C ] α s (+1, f x )) K s (x,+1 )+ K α(,x ) s ( f, f x ) K s (x,+1 )+ K β(,x ) s (,x )) K s (x,+1 )+ Kβ s (,x ). Letting n, we obtain that s (x, f x ) 0, that is x = f x. This complete the proof.

5 Sohag J. Math. 2, No. 2, (2015) / 55 References [1] E. F. Kazeem, C. A. Agyingi and Y. U. Gaba; On Quasi-Pseudometric Type Spaces, Chinese Journal of Mathematics, vol. 2014, Article I , 7 pages, doi: /2014/ [2] Y. U. Gaba Startpoints and (α,γ)-contractions in quasi-pseudometric spaces, Journal of Mathematics, Volume 2014 (2014), Article I , 8 pages, [3] K. Włodarczyk, R. Plebaniak; Asymmetric structures, discontinuous contractions and iterative approximation of fixed and periodic points, Fixed Point Theory and Applications 2013 (2013) 128, [4] M. A. Khamsi; Remarks on Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings, Fixed Point Theory and Application, volume 2010(2010), 7 pages, doi: /2010/ Yaé U. Gaba completed his honours degree in Pure Mathematics at University of Abomey-Calavi (Bénin). He obtained a MSc from the African University of Science and Technology (Abuja- Nigeria) and he is currently enrolled as a Ph.. student at the University of Cape Town (South- Africa).