Contraction Mapping in Cone b-hexagonal Metric Spaces

Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 1651 1662 Research India Publications http://www.ripublication.com/gjpam.htm Contraction Mapping in Cone b-hexagonal Metric Spaces Abba Auwalu 1 Department of Mathematics, Near East University, Nicosia-TRNC, Mersin 10, Turkey. Ali Denker Department of Mathematics, Near East University, Nicosia-TRNC, Mersin 10, Turkey. Abstract In this paper, we introduce the notion of coneb-hexagonal metric space and prove Banach contraction mapping principle in cone b-hexagonal metric space setting without assuming the normality condition. Our results extend and improve some recent results in the literature. AMS subject classification: 47H10, 54H25. Keywords: Cone b-hexagonal metric space; cone metric space; fixed point; Banach Contraction mapping principle. 1. Introduction The study of fixed point theory plays an important role in applications of many branches of mathematics. In 1922, Banach [1] introduced the concept of Banach contraction mapping principle. Due to wide applications of this concept, the study of existence and uniqueness of fixed points of a mapping and common fixed points of one, two or more mappings has become a subject of great interest. Many authors proved the Banach contraction principle in various generalized metric spaces. 1 Corresponding author.

1652 Abba Auwalu and Ali Denker In 2007, Huang and Zhang [2] introduced the concept of a cone metric space, they replaced the set of real numbers by an ordered Banach space and proved some fixed point theorems for contractive type conditions in cone metric spaces. Later on many authors have proved some fixed point theorems for different contractive types conditions in cone metric spaces (for e.g., [3, 4, 5, 6]). Recently, Hussain and Shah [7] introduced the concept of cone b - metric space as a generalization of cone metric space. They also improved some recent results about KKM mappings in cone b - metric spaces. Very recently, Garg and Agarwal [8] introduced the notion of cone hexagonal metric space and proved Banach contraction mapping principle in a normal cone hexagonal metric space setting. Motivated and inspired by the results of [7, 8], it is our purpose in this paper to continue the study of fixed point of mapping in non-normal cone b-hexagonal metric space setting. Our results extend and improve the results of [4, 5, 8], and many others in the literature. 2. Preliminaries We present some definitions and Lemmas, which will be needed in the sequel. Definition 2.1. [2] Let E be a real Banach space and P subset of E. P is called a cone if and only if: (1) P is closed, nonempty, and P {0}; (2) a,b R, a,b 0 and x,y P ax + by P ; (3) x P and x P x = 0. Given a cone P E, we defined a partial ordering with respect to P by x y if and only if y x P. We shall write x<yto indicate that x y but x =y, while x y will stand for y x int(p), where int(p) denotes the interior of P. Definition 2.2. [2] A cone P is called normal if there is a number k>0 such that for all x,y E, the inequality 0 x y x k y. (2.1) The least positive number k satisfying (2.1) is called the normal constant of P. In this paper, we always suppose that E is a real Banach space and P is a cone in E with int(p) and is a partial ordering with respect to P. Definition 2.3. [2] Let X be a nonempty set. Suppose the mapping ρ : X X E satisfies:

Fixed Point Theorem in Cone b-hexagonal Metric Spaces 1653 (1) 0 < ρ(x, y) for all x,y X and ρ(x,y) = 0 if and only if x = y; (2) ρ(x,y) = ρ(y,x) for all x,y X; (3) ρ(x,y) ρ(x,z) + ρ(z,y) for all x,y,z X. Then ρ is called a cone metric on X, and (X, ρ) is called a cone metric space. Remark 2.4. The concept of a cone metric space is more general than that of a metric space, because each metric space is a cone metric space where E = R and P =[0, ) (e.g., see [2]). Definition 2.5. [7] Let X be a nonempty set and s 1 be a given real number. Suppose the mapping ρ : X X E satisfies: (1) 0 < ρ(x, y) for all x,y X and ρ(x,y) = 0 if and only if x = y; (2) ρ(x,y) = ρ(y,x) for all x,y X; (3) ρ(x,y) s[ρ(x,z) + ρ(z,y)] for all x,y,z X. Then ρ is called a cone b - metric on X, and (X, ρ) is called a cone b - metric space. Remark 2.6. Every cone metric space is cone b - metric space. The converse is not necessarily true (e.g., see [7]). Definition 2.7. [8] Let X be a nonempty set. Suppose the mapping d : X X E satisfies: (1) 0 < d(x, y) for all x,y X and d(x,y) = 0 if and only if x = y; (2) d(x,y) = d(y,x) for x,y X; (3) d(x,y) d(x,z)+d(z,w)+d(w,u)+d(u, v)+d(v,y)for all x,y,z,w,u,v X and for all distinct points z, w, u, v X {x,y} [hexagonal property]. Then d is called a cone hexagonal metric on X, and (X, d) is called a cone hexagonal metric space. Definition 2.8. Let X be a nonempty set and s 1 be a given real number. Suppose the mapping d : X X E satisfies: (1) 0 < d(x, y) for all x,y X and d(x,y) = 0 if and only if x = y; (2) d(x,y) = d(y,x) for x,y X; (3) d(x,y) s[d(x,z)+d(z,w)+d(w,u)+d(u, v)+d(v,y)] for all x,y,z,w,u,v X and for all distinct points z, w, u, v X {x,y} [b-hexagonal property].

1654 Abba Auwalu and Ali Denker Then d is called a cone b-hexagonal metric on X, and (X, d) is called a cone b - hexagonal metric space. Observe that if s = 1, then the hexagonal property in a cone hexagonal metric space is satisfied, however it does not hold true when s>1. Thus the class of coneb-hexagonal metric spaces is effectively larger than that of the cone hexagonal metric spaces. Also, every cone metric space is a coneb-hexagonal metric space, but the converse need not be true. The following examples illustrate the above remarks. Example 2.9. Let X =[0, 5], E= R 2 and P ={(x, y) : x,y 0} is a cone in E. Define d : X X E as follows: d(x,y) = ( x y 4, x y 4 ). Then (X, d) isaconeb-hexagonal metric space. Indeed d(x,y) 2[d(x,z)+d(z,w)+d(w,u)+d(u, v)+d(v,y)] for all x,y,z,w,u,v X. However, (X, d) is not a cone b - metric space because it lacks the triangular property: d(0, 5) = (625, 625) >(514, 514) = 2[d(0, 1) + d(1, 5)]. Definition 2.10. Let (X, d) beaconeb-hexagonal metric space. Let {x n } be a sequence in X and x X. If for every c E with 0 c there exist n 0 N and that for all n>n 0, d(x n,x) c, then {x n } is said to be convergent and {x n } converges to x, and x is the limit of {x n }. We denote this by lim n x n = x or x n x as n. Definition 2.11. Let (X, d) beaconeb-hexagonal metric space. Let {x n } be a sequence in X and x X. If for every c E, with 0 c there exist n 0 N such that for all n,m>n 0,d(x n,x m ) c, then {x n } is called Cauchy sequence in X. Definition 2.12. Let (X, d) be a cone b-hexagonal metric space. If every Cauchy sequence is convergent in (X, d), then X is called a complete cone b-hexagonal metric space. Definition 2.13. Let P be a cone defined as above and let be the set of non decreasing continuous functions ϕ : P P satisfying: 1. 0 <ϕ(t)<tfor all t P \{0}, 2. the series n 0 ϕ n (t) converge for all t P \{0}. From (1), we have ϕ(0) = 0, and from (2), we have lim n 0 ϕ n (t) = 0 for all t P \{0}.

Fixed Point Theorem in Cone b-hexagonal Metric Spaces 1655 3. Main Results In this section, we derive the main result of our work, which is an extension of Banach contraction principle in coneb-hexagonal metric space. Theorem 3.1. Let (X, d) be a complete cone b-hexagonal metric space with s 1. Suppose the mapping S : X X satisfy the contractive condition: d(sx,sy) ( d(x,y) ), (3.1) for all x,y X, where ϕ. Then S has a unique fixed point in X. Proof. Let x 0 be an arbitrary point in X. Define a sequence {x n } in X such that x n+1 = Sx n, for all n = 0, 1, 2,... We assume that x n =x n+1, for all n N. Then, from (3.1), it follows that d(x n,x n+1 ) = d(sx n 1,Sx n ) ( d(x n 1,x n ) ) = ϕ ( d(sx n 2,Sx n 1 ) ) 2( d(x n 2,x n 1 ) ) It again follows that It further follows that. n( d(x 0,x 1 ) ). (3.2) d(x n,x n+2 ) = d(sx n 1,Sx n+1 ) ( d(x n 1,x n+1 ) ) = ϕ ( d(sx n 2,Sx n ) ) 2( d(x n 2,x n ) ). n( d(x 0,x 2 ) ). (3.3) d(x n,x n+3 ) = d(sx n 1,Sx n+2 ) ( d(x n 1,x n+2 ) ) = ϕ ( d(sx n 2,Sx n+1 ) ) 2( d(x n 2,x n+1 ) ). n( d(x 0,x 3 ) ), and (3.4)

1656 Abba Auwalu and Ali Denker d(x n,x n+4 ) = d(sx n 1,Sx n+3 ) ( d(x n 1,x n+3 ) ) In similar way, for k = 1, 2, 3,...,we get = ϕ ( d(sx n 2,Sx n+2 ) ) 2( d(x n 2,x n+2 ) ). n( d(x 0,x 4 ) ). (3.5) d(x n,x n+4k+1 ) n( d(x 0,x 4k+1 ) ), (3.6) d(x n,x n+4k+2 ) n( d(x 0,x 4k+2 ) ), (3.7) d(x n,x n+4k+3 ) n( d(x 0,x 4k+3 ) ), (3.8) d(x n,x n+4k+4 ) n( d(x 0,x 4k+4 ) ). (3.9) By using (3.2) andb-hexagonal property, we have Similarly, d(x 0,x 5 ) s [ d(x 0,x 1 ) + d(x 1,x 2 ) + d(x 2,x 3 ) + d(x 3,x 4 ) + d(x 4,x 5 ) ] + ϕ 4( d(x 0,x 1 ) )] [ 4 s ϕ i( d(x 0,x 1 ) )]. d(x 0,x 9 ) s [ d(x 0,x 1 ) + d(x 1,x 2 ) + d(x 2,x 3 ) + d(x 3,x 4 ) + d(x 4,x 5 ) ] + d(x 5,x 6 ) + d(x 6,x 7 ) + d(x 7,x 8 ) + d(x 8,x 9 ) ] + ϕ 4( d(x 0,x 1 ) )] + ϕ 5( d(x 0,x 1 ) ) + ϕ 6( d(x 0,x 1 ) ) + ϕ 7( d(x 0,x 1 ) ) + ϕ 8( d(x 0,x 1 ) )] [ 8 s ϕ i( d(x 0,x 1 ) )]. Now by induction, we obtain for each k = 0, 1, 2, 3,... [ 4k d(x 0,x 4k+1 ) s ϕ i( d(x 0,x 1 ) )]. (3.10)

Fixed Point Theorem in Cone b-hexagonal Metric Spaces 1657 Also by (3.2), (3.3) andb-hexagonal property, we have d(x 0,x 6 ) s [ d(x 0,x 1 ) + d(x 1,x 2 ) + d(x 2,x 3 ) + d(x 3,x 4 ) + d(x 4,x 6 ) ] Similarly, + ϕ 4( d(x 0,x 2 ) )] [ 3 s ϕ i( d(x 0,x 1 ) ) + ϕ 4( d(x 0,x 2 ) )]. d(x 0,x 10 ) s [ d(x 0,x 1 ) + d(x 1,x 2 ) + d(x 2,x 3 ) + d(x 3,x 4 ) + d(x 4,x 5 ) + d(x 5,x 6 ) + d(x 6,x 7 ) + d(x 7,x 8 ) + d(x 8,x 10 ) ] + ϕ 4( d(x 0,x 1 ) )] + ϕ 5( d(x 0,x 1 ) ) + ϕ 6( d(x 0,x 1 ) ) + ϕ 7( d(x 0,x 1 ) ) + ϕ 8( d(x 0,x 2 ) )] [ 7 s ϕ i( d(x 0,x 1 ) ) + ϕ 8( d(x 0,x 2 ) )]. By induction, we obtain for each k = 1, 2, 3,... d(x 0,x 4k+2 ) s ϕ i( d(x 0,x 1 ) ) + ϕ 4k( d(x 0,x 2 ) )]. (3.11) Again by (3.2), (3.4) andb-hexagonal property, we have d(x 0,x 7 ) s [ d(x 0,x 1 ) + d(x 1,x 2 ) + d(x 2,x 3 ) + d(x 3,x 4 ) + d(x 4,x 7 ) ] Similarly, + ϕ 4( d(x 0,x 3 ) )] [ 3 s ϕ i( d(x 0,x 1 ) ) + ϕ 4( d(x 0,x 3 ) )]. d(x 0,x 11 ) s [ d(x 0,x 1 ) + d(x 1,x 2 ) + d(x 2,x 3 ) + d(x 3,x 4 ) + d(x 4,x 5 ) + d(x 5,x 6 ) + d(x 6,x 7 ) + d(x 7,x 8 ) + d(x 8,x 11 ) ] + ϕ 4( d(x 0,x 1 ) )] + ϕ 5( d(x 0,x 1 ) ) + ϕ 6( d(x 0,x 1 ) ) + ϕ 7( d(x 0,x 1 ) ) + ϕ 8( d(x 0,x 3 ) ) [ 7 s ϕ i( d(x 0,x 1 ) ) + ϕ 8( d(x 0,x 3 ) )].

1658 Abba Auwalu and Ali Denker So by induction, we obtain for each k = 1, 2, 3,... d(x 0,x 4k+3 ) s ϕ i( d(x 0,x 1 ) ) + ϕ 4k( d(x 0,x 3 ) )]. (3.12) In fact, by (3.2), (3.5) andb-hexagonal property, we have d(x 0,x 8 ) s [ d(x 0,x 1 ) + d(x 1,x 2 ) + d(x 2,x 3 ) + d(x 3,x 4 ) + d(x 4,x 8 ) ] Similarly, + ϕ 4( d(x 0,x 4 ) )] [ 3 s ϕ i( d(x 0,x 1 ) ) + ϕ 4( d(x 0,x 4 ) )]. d(x 0,x 12 ) s [ d(x 0,x 1 ) + d(x 1,x 2 ) + d(x 2,x 3 ) + d(x 3,x 4 ) + d(x 4,x 5 ) + d(x 5,x 6 ) + d(x 6,x 7 ) + d(x 7,x 8 ) + d(x 8,x 12 ) ] + ϕ 4( d(x 0,x 1 ) )] + ϕ 5( d(x 0,x 1 ) ) + ϕ 6( d(x 0,x 1 ) ) + ϕ 7( d(x 0,x 1 ) ) + ϕ 8( d(x 0,x 4 ) ) [ 7 s ϕ i( d(x 0,x 1 ) ) + ϕ 8( d(x 0,x 4 ) )]. By induction, we obtain for each k = 1, 2, 3,... d(x 0,x 4k+4 ) s ϕ i( d(x 0,x 1 ) ) + ϕ 4k( d(x 0,x 4 ) )]. (3.13) Using inequality (3.6) and (3.10) for k = 1, 2, 3,...,we have d(x n,x n+4k+1 ) n( d(x 0,x 4k+1 ) ) [ 4k n ϕ i( d(x 0,x 1 ) )]) n [ 4k ϕ i( d(x 0,x 1 ) + d(x 0,x 2 ) + d(x 0,x 3 ) + d(x 0,x 4 ) )]) [ n ϕ i( d(x 0,x 1 ) + d(x 0,x 2 ) + d(x 0,x 3 ) + d(x 0,x 4 ) )]). (3.14)

Fixed Point Theorem in Cone b-hexagonal Metric Spaces 1659 Similarly for k = 1, 2, 3,...,inequalities (3.7) and (3.11) implies that d(x n,x n+4k+2 ) n( d(x 0,x 4k+2 ) ) n ϕ i( d(x 0,x 1 ) ) + ϕ 4k( d(x 0,x 2 ) )]) n ϕ i( d(x 0,x 1 ) + d(x 0,x 2 ) + d(x 0,x 3 ) + d(x 0,x 4 ) ) + ϕ 4k( d(x 0,x 1 ) + d(x 0,x 2 ) + d(x 0,x 3 ) + d(x 0,x 4 ) )]) [ 4k n ϕ i( d(x 0,x 1 ) + d(x 0,x 2 ) + d(x 0,x 3 ) + d(x 0,x 4 ) )]) [ n ϕ i( d(x 0,x 1 ) + d(x 0,x 2 ) + d(x 0,x 3 ) + d(x 0,x 4 ) )]). (3.15) Again for k = 1, 2, 3,...,inequalities (3.8) and (3.12) implies that d(x n,x n+4k+3 ) n( d(x 0,x 4k+3 ) ) n ϕ i( d(x 0,x 1 ) ) + ϕ 4k( d(x 0,x 3 ) )]) n ϕ i( d(x 0,x 1 ) + d(x 0,x 2 ) + d(x 0,x 3 ) + d(x 0,x 4 ) ) + ϕ 4k( d(x 0,x 1 ) + d(x 0,x 2 ) + d(x 0,x 3 ) + d(x 0,x 4 ) )]) [ 4k n ϕ i( d(x 0,x 1 ) + d(x 0,x 2 ) + d(x 0,x 3 ) + d(x 0,x 4 ) )]) [ n ϕ i( d(x 0,x 1 ) + d(x 0,x 2 ) + d(x 0,x 3 ) + d(x 0,x 4 ) )]). (3.16)

1660 Abba Auwalu and Ali Denker Again for k = 1, 2, 3,...,inequalities (3.9) and (3.13) implies that d(x n,x n+4k+4 ) n( d(x 0,x 4k+4 ) ) n ϕ i( d(x 0,x 1 ) ) + ϕ 4k( d(x 0,x 4 ) )]) n ϕ i( d(x 0,x 1 ) + d(x 0,x 2 ) + d(x 0,x 3 ) + d(x 0,x 4 ) ) + ϕ 4k( d(x 0,x 1 ) + d(x 0,x 2 ) + d(x 0,x 3 ) + d(x 0,x 4 ) )]) [ 4k n ϕ i( d(x 0,x 1 ) + d(x 0,x 2 ) + d(x 0,x 3 ) + d(x 0,x 4 ) )]) [ n ϕ i( d(x 0,x 1 ) + d(x 0,x 2 ) + d(x 0,x 3 ) + d(x 0,x 4 ) )]). Thus, by inequalities (3.14), (3.15), (3.16) and (3.17) we have, for each m, (3.17) [ d(x n,x n+m ) n ϕ i( d(x 0,x 1 )+d(x 0,x 2 )+d(x 0,x 3 )+d(x 0,x 4 ) )]). (3.18) ( Since s [ ϕ i( d(x 0,x 1 ) + d(x 0,x 2 ) + d(x 0,x 3 ) + d(x 0,x 4 ) )]) converges (by definition 2.13), where d(x 0,x 1 )+d(x 0,x 2 )+d(x 0,x 3 )+d(x 0,x 4 ) P \{0} and P is closed, ( then s [ ϕ i( d(x 0,x 1 ) + d(x 0,x 2 ) + d(x 0,x 3 ) + d(x 0,x 4 ) )]) P \{0}. Hence ( [ lim n ϕn s ϕ i( d(x 0,x 1 ) + d(x 0,x 2 ) + d(x 0,x 3 ) + d(x 0,x 4 ) )]) = 0. Then for given c 0, there is a natural number N 1 such that [ ϕ n ϕ i( d(x 0,x 1 )+d(x 0,x 2 )+d(x 0,x 3 )+d(x 0,x 4 ) )]) c, n N 1. (3.19) Thus from (3.18) and (3.19), we have d(x n,x n+m ) c, for all n N 1.

Fixed Point Theorem in Cone b-hexagonal Metric Spaces 1661 Therefore, {x n } is a Cauchy sequence in (X, d). Since X is complete, then there exists a point z X such that lim x n = lim Sx n 1 = z. n n Now, we will show that z is a fixed point of S, i.e. Sz = z. Given c 0, we choose N 2,N 3 N such that d(z,x n ) c 5s, n N 2 and d(x n,x n+1 ) c 5s, n N 3. Since x n =x m for n =m, therefore by b-hexagonal property, we have d(sz,z) s [ d(sz,sx n ) + d(sx n,sx n+1 ) + d(sx n+1,sx n+2 ) + d(sx n+2,sx n+3 ) + d(sx n+3,z) ] s [ ϕ ( d(z,x n ) ) + d(x n+1,x n+2 ) + d(x n+2,x n+3 ) + d(x n+3,x n+4 ) + d(x n+4,z) ] <s [ d(z,x n ) + d(x n+1,x n+2 ) + d(x n+2,x n+3 ) + d(x n+3,x n+4 ) + d(x n+4,z) ] [ c s 5s + c 5s + c 5s + c 5s + c ] = c, for all n N, 5s where N = max{n 2,N 3 }. Since c is arbitrary we have d(sz,z) c, m N. Since m c m 0asm, we conclude c d(sz,z) d(sz,z) as m. Since P is m closed, d(sz,z) P. Hence d(sz,z) P P. By definition of cone we get that d(sz,z) = 0, and so Sz = z. Therefore, S has a fixed point that is z in X. Next we show that z is unique. For suppose z be another fixed point of S such that Sz = z. Therefore, d(z,z ) = d(sz,sz ) ( d(z,z ) ) <d(z,z ). Hence, z = z. This completes the proof of the theorem. Corollary 3.2. Let (X, d) be a complete coneb-hexagonal metric space. Suppose the mapping S : X X satisfy the following: d(s m x,s m y) ( d(x,y) ), (3.20) for all x,y X, where ϕ. Then S has a unique fixed point in X. Proof. From Theorem 3.1, we conclude that S m has a fixed point say z. Hence Sz = S(S m z) = S m+1 z = S m (Sz). (3.21) Then Sz is also a fixed point to S m. By uniqueness of z, we have Sz = z. Corollary 3.3. ee [8]) Let (X, d) be a cone hexagonal metric space, P be a normal cone, and the mapping S : X X satisfy the following: d(sx,sy) λd(x, y),

1662 Abba Auwalu and Ali Denker for all x,y X, where λ [0, 1). Then S has a unique fixed point in X. Proof. Define ϕ : P P by ϕ(t) = λt, and s = 1. Then it is clear that ϕ satisfies the conditions in definition 2.13. Hence the results follows from Theorem 3.1. Corollary 3.4. ee [4]) Let (X, d) be a cone rectangular metric space, P be a normal cone, and the mapping S : X X satisfy the following: d(sx,sy) λd(x, y), for all x,y X, where λ [0, 1). Then S has a unique fixed point in X. Proof. The results follows from the fact that every cone rectangular metric space is cone hexagonal metric space, and Corollary 3.3 above. References [1] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundamenta Mathematicae, 3 (1922), 133 181. [2] L. G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, Journal of Mathematical Analysis and Applications, 332 (2007), no. 2, 1468 1476. [3] D. Hie and V. Rakocevic, Common fixed points for maps on cone metric space, Journal of Mathematical Analysis and Applications, 341 (2008), 876 882. [4] A. Azam, M. Arshad, and I. Beg, Banach contraction principle on cone rectangular metric spaces, Applicable Analysis and Discrete Mathematics, 3 (2009), no. 2, 236 241. [5] A. Auwalu and E. Hincal, A Note on Banach Contraction Mapping principle in Cone Hexagonal Metric Spaces, British Journal of Mathematics & Computer Science, 16 (2016), no. 1, 1 12. [6] A. Auwalu and A. Denker, Banach Contraction Mapping principle in Cone Heptagonal Metric Spaces, Global Journal of Pure and Applied Mathematics, 13 (2017), no. 2, xxx-xxx. [7] N. Hussain, MH. Shah, KKM mapping in cone b-metric spaces, Computers & Mathematics with Applications, 62 (2011), no. 4, 1677 1684. [8] M. Garg, Banach Contraction Principle on Cone hexagonal Metric Space, Ultra Scientist, 26 (2014), no. 1, 97 103.