Cyclic contractions and fixed point theorems on various generating spaces

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1 Kumari and Panthi Fixed Point Theory and Applications 015) 015:153 DOI /s R E S E A R C H Open Access Cyclic contractions and fixed point theorems on various generating spaces Panda Sumati Kumari 1* and Dinesh Panthi * Correspondence: mumy @gmail.com 1 Department of Mathematics, K.L. University, Vaddeswaram, Andhra Pradesh, India Full list of author information is available at the end of the article Abstract In this paper, we prove some fixed point theorems using various cyclic contractions in weaker forms of generating spaces. MSC: 47H10; 54H5 Keywords: generating space of b-dislocated quasi-metric family; generating space of b-quasi-metric family; generating space of quasi-metric family; generating space of b-dislocated metric family; generating space of dislocated quasi-metric family; generating space of dislocated metric family; cyclic contraction; G bdq -cyclic-banach contraction; G bd -cyclic-kannan contraction; G bdq -cyclic-ciric contraction 1 Introduction In 19, Banach [1] established a remarkable fixed point theorem known as the Banach contraction principle. Theorem 1.1 Banach contraction principle [1]) Let X, d) be a complete metric space and T : X X be a mapping, there exists a number t,0 t <1,such that, for each x, y X, dtx, Ty) tdx, y). Then T has a fixed point. This theorem assures the existence and uniqueness of fixed points of certain self-maps of metric spaces, and it gives a constructive method to find those fixed points. In recent years, a number of generalizations of the above Banach contraction principle have appeared. Of all these, the following generalization of Kannan []andciric[3]standsatthetop. In 1969, Kannan [] initiated a new type of contraction mapping, which is called the Kannan-contraction type. He also established a fixed point result for such a type. Theorem 1. Kannan fixed point theorem []) Let X, d) be a complete metric space and T : X X be a mapping; there exists a number t,0<t < 1, such that, for each x, y X, dtx, Ty) t [ dx, Tx)+dy, Ty) ]. Then T has a fixed point. 015 Kumari and Panthi. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Kumari and Panthi Fixed Point Theory and Applications 015) 015:153 Page of 17 It is interesting that Kannan s fixed point theorem is very predominant because Subrahmanyam [4] proved that Kannan s theorem describes the completeness of the metric. In other words, a metric space X is complete if and only if every Kannan mapping on X has a fixed point. Recently, many authors generalized the Kannan fixed point theorem in some class of spaces see [5 9]). In 1974, Ciric [3] established a fixed point theorem known as the Ciric quasi-contraction principle. Theorem 1.3 Ciric quasi-contraction principle [3]) Let X, d) be a complete metric space. Assume there is a map T : X X such that there exists a constant h,0 h <1,and for each x, y X, dtx, Ty) h max { dx, y), dx, Tx), dy, Ty), dx, Ty), dy, Tx) }. Then T has a fixed point. The above-mentioned theorems play a very important role not only in analysis but also in other areas of science involving mathematics, especially in graph theory, dynamical systems, and programming languages. These renowned theorems have several mathematical and real world illustrations. Since then, many authors proved and generalized the above important theorems in various generalized metric spaces. In 1989, Bakhtin [10] introduced the very interesting concept of a b-metricspace asan analog of metric space. He proved the contraction mapping principle in b-metric space that generalizes the famous Banach contraction principle in metric spaces. Since then many mathematicians have done work involving fixed points for single-valued and multivalued operators in b-metric spaces see for example [11 1]). In 1997, Chang et al. [] introduced the definition of a generating space of a quasimetric family and established some interesting fixed point theorems and coincidence point theorems in the generating space of a quasi-metric family. In 1999, Gue Myung Lee et al. [3] defined a family of weak quasi-metrics in a generating space of quasi-metric family. He proved a Takahashi-type minimization theorem, a generalized Ekeland variational principle, and a general Caristi-type fixed point theorem for set-valued maps in complete generating spaces of a quasi-metric family by using a family of weak quasi-metrics. Also without considering lower semi-continuity, he proved fixed point theorem for set-valued maps in complete generating spaces of a quasi-metric family. In 003, Kirk et al. [4] introduced cyclic contractions in metric spaces and investigated the existence of proximity points and fixed points for cyclic contraction mappings. Since then many results appeared in this field see [ 33]). Sankar Raj and Veeramani [6] proved the existence of best proximity points for relatively non-expansive maps. Later, Karapinar and Erhan [31] proved the existence of fixed points for various types of cyclic contractions in a metric space. In this paper, motivated by the above facts, we first introduce the notions of generating space of b-dislocated quasi-metric family abbreviated G bdq -family ), generating space of b-dislocated metric family abbreviated G bd -family ), generating space of b-quasi-metric

3 Kumari and Panthi Fixed Point Theory and Applications 015) 015:153 Page 3 of 17 family abbreviated G bq -family), generating space of dislocated quasi-metric family abbreviated G dq -family) and generating space of dislocated metric family abbreviated G d - family). By following the approaches of Kirk et al. [4], we introduce the G bdq -cyclic- Banach contraction,theg bd -cyclic-kannan contraction,andtheg bq -cyclic-ciric contraction. Then we prove the existence of fixed point theorems in weaker forms of generating spaces by using the above mentioned contractive conditions. From our results in various generating spaces, we obtain the corresponding theorems for cyclic maps in weaker forms of metric spaces. Moreover, some examples are provided to illustrate the usability of the obtained results. Preliminaries Throughout this paper, we assume that R + =[0, ), N denotes the set of all positive integers. Definition.1 [10]) Let X be a non-empty set, let d : X X [0, ) andlets R +. Then X, d) is said to be b-metric space if the following conditions are satisfied. i) dx, y)=0if and only if x = y for all x, y X. ii) dx, y)=dy, x) for all x, y X. iii) There exists a real number s 1 such that dx, y) s[dx, z) +dz, y)] for all x, y, z X. Example. Let X = R +,defined : X X R + by dx, y) =x y).thenx, d) isa b-metric space with coefficient s =. The concept of a b-metricspaceisveryimportantastheclassofb-metricspacesislarger than that of metric spaces, since a b-metric space is a metric space when s =1. In [34], it was proved that each b-metric on X generates a topology I whose base is the family of open balls B d x, ɛ)={y X/dx, y)<ɛ}. Recently, Kumari [35] discussed some topological aspects of b-dislocated quasi-metric spaces which is more general than of b-metric spaces and also derived some fixed point theorems in such a space. Definition.3 [35]) A b-dislocated quasisimply b dq ) metric on a non-empty set X is a function b dq : X X R + such that for all x, y, z in X and s R +, the following conditions hold. i) b dq x, y) 0. ii) b dq x, y)=0=b dq y, x) x = y. iii) There exists a real number s 1 such that b dq x, y) s[b dq x, z)+b dq z, y)]. Then the pair X, b dq ) is called a b dq -metric space. In [36], we note that if b dq satisfies an extra condition i.e. b dq x, y)=b dq y, x)thenx, b dq ) is called a b-dislocated metric space or simply a b d -metric space). Example.4 Let X = {0, 1, },andletb dq 0, 0) = 1, b dq0, 1) = 0, b dq 0, ) = 1, b dq 1, 0) = 1, b dq 1, 1) = 0, b dq 1, ) =, b dq, 0) = 1, b dq, 1) = 1 and b dq, ) = then X, b dq )isab dq metric space with coefficient s =,butsinceb dq 0, 1) b dq 1, 0), it is not a b d metric space. Obviously X, b dq ) is not a dislocated quasi-metric space as the triangle inequality does not hold; b dq, 1) b dq, 0) + b dq 0, 1).

4 Kumari and Panthi Fixed Point Theory and Applications 015) 015:153 Page 4 of 17 Definition.5 []) Let X be a non-empty set and { : α 0, 1]} a family of mappings of X X into R +.ThenX, ) is called a generating space of a quasi-metric family if it satisfies the following conditions for all x, y, z X: i) x, y)=0if and only if x = y. ii) x, y)= y, x). iii) For any α 0, 1] there exists β 0, α] such that x, z) d β x, y)+d β y, z). iv) For any x, y X, x, y) is non-increasing and left continuous in α. In [37], it was proved that each generating space of a quasi-metric family generates a topology I dα whosebaseisthefamilyofopenballs. Example.6 Let X, d) be a metric space. If we take x, y)=dx, y) for all α 0, 1] and x, y X,thenX, d) is a generating space of a quasi-metric family. For more examples, the reader can refer to []. Now, we first introducing following concepts: Definition.7 Let X be a non-empty set and { : α 0, 1]} a family of mapping of X X into R +.ThenX, ) is called a generating space of a b-quasi-metric family if it satisfies the following conditions for any x, y, z X and s 1. i) x, y)=0if and only if x = y. ii) x, y)= y, x). iii) For any α 0, 1] there exists β 0, α] such that x, z) s[d β x, y)+d β y, z)]. iv) For any x, y X, x, y) is non-increasing and left continuous in α. Definition.8 1. Let X, ) be a G bq -family and {x n } beasequenceinx.wesaythat{x n } G bq -converges to x in X, ) if lim n x n, x)=0for all α 0, 1]. In this case we write x n x.. Let X, ) be a G bq -family and let A X, x X.Wesaythatx is a G bq -limit point of A if there exists a sequence {x n } in A {x} such that lim n x n = x. 3. A sequence {x n } in a G bq -family is called a G bq -Cauchy sequence if, given ɛ >0,there exists n 0 N such that for all n, m n 0,wehave x n, x m )<ɛ or lim n,m x n, x m )=0for all α 0, 1]. 4. A G bq -family X, ) is called complete if every G bq -Cauchy sequence in X is G bq -convergent. Remark.9 Every G bq -convergent sequence in a G bq -family is G bq -Cauchy. Now we introducing the concepts of a generating space of a b-dislocated quasi-metric family simply G bdq -family ) and a generating space of a b-dislocated metric family simply G bd -family ). Definition.10 Let X be a non-empty set and { : α 0, 1]} a family of mapping of X X into R +.ThenX, ) is called a generating space of b-dislocatedmetricfamilyifit satisfied the following conditions for any x, y, z X and s 1. i) x, y)=0implies x = y.

5 Kumari and Panthi Fixed Point Theory and Applications 015) 015:153 Page 5 of 17 ii) x, y)= y, x). iii) For any α 0, 1] there exists β 0, α] such that x, z) s[d β x, y)+d β y, z)]. iv) For any x, y X, x, y) is non-increasing and left continuous in α. Generating space of b-dislocated metric family is a generalization of b-dislocated metric space and generating space of quasi-metric family. Now we introduce the following according to Kumari [38 41]. Definition Let X, ) be a G bd -family and {x n } be a sequence in X.Wesaythat{x n } G bd -converges to x in X, ) if lim n x n, x)=0for all α 0, 1]. In this case we write x n x.. Let X, ) be a G bd -family and let A X, x X.Wesaythatx is a G bd -limit point of A if there exists a sequence {x n } in A {x} such that lim n x n = x. 3. A sequence {x n } in a G bd -family is called a G bd -Cauchy sequence if, given ɛ >0,there exists n 0 N such that for all n, m n 0,wehave x n, x m )<ɛ or lim n,m x n, x m )=0for all α 0, 1]. 4. A G bd -family X, ) is called complete if every G bd -Cauchy sequence in X is G bd -convergent. Remark.1 Every G bd -convergent sequence in a G bd -family is G bd -Cauchy. The pair X, ) is called generating space of dislocated metric family if we take s =1in Definition.10. Definition.13 Let X be a non-empty set and { : α 0, 1] : X X R + }.X, )isa generating space of b-dislocated quasi-metric family simply G bdq -family ) if the following conditions hold for any x, y, z X and s 1. i) x, y)= y, x)=0 x = y. ii) For any α 0, 1] there exists β 0, α] such that x, z) s[d β x, y)+d β y, z)]. iii) For any x, y X, x, y) is non-increasing and left continuous in α. The generating space of a b-dislocated quasi-metric family is a generalization of a b- dislocated quasi-metric space and the generating space of a quasi-metric family. We introducing the concepts of a G bdq -convergent sequence, a G bdq -Cauchy sequence, and a G bdq -complete space according to Kumari [4]. Definition A sequence {x n } in a G bdq -family X, ) is called G bdq -converges to x X if lim n x n, x)=0= lim n x, x n ) for all α 0, 1]. In this case x is called a G bdq -limit of {x n },andwewritex n x.. A sequence {x n } in a G bdq -family X, ) is called G bdq -Cauchy if lim x n, x m )=0= lim x m, x n ) for all α 0, 1]. n,m n,m

6 Kumari and Panthi Fixed Point Theory and Applications 015) 015:153 Page 6 of A G bdq -family X, ) is complete if every Cauchy sequence in it is G bdq -convergent in X. The pair X, ) is called generating space of dislocated quasi-metric family if we take s = 1 in Definition.13. Now, we recall the definition of cyclic map in a metric space. Definition.15 [4]) Let A and B be non-empty subsets of a metric space X, d) and T : A B A B. T is called a cyclic map iff TA) B and TB) A. Definition.16 [4]) Let A and B be non-empty subsets of a metric space X, d). A cyclic map T : A B A B is said to be a cyclic contraction if there exists k [0, 1) such that dtx, Ty) kdx, y)) for all x A and y B. 3 Fixed point theorems on various generating spaces Definition 3.1 Let A and B be non-empty subsets of a G bdq -family X, ). A cyclic map T : A B A B is said to be a G bdq -cyclic-banach contraction if for all α 0, 1] such that Tx, Ty) k x, y) for all x A, y B and k [0, 1 s ). Remark Cyclic contraction is continuous in a G bdq -family. Theorem 3. Let A and B be non-empty closed subsets of a complete G bdq -family X, ). Let T be a cyclic mapping that satisfies the condition of a G bdq -cyclic-banach contraction. Then T has a unique fixed point in A B. Proof Let x A fix) and using the contractive condition of the theorem, we have T x, Tx ) = TTx), Tx ) kdα Tx, x) and Tx, T x ) = Tx, TTx) ) kdα x, Tx). So, T x, Tx ) kδ 1) and Tx, T x ) kδ, ) where δ = max{ Tx, x), x, Tx)}. From 1)and), we have T 3 x, T x) k δ and T x, T 3 x) k δ. For all n N,weget T n+1 x, T n x ) k n δ

7 Kumari and Panthi Fixed Point Theory and Applications 015) 015:153 Page 7 of 17 and T n x, T n+1 x ) k n δ. Let n, m N with m > n, by using the definition of G bdq -family, we have T m x, T n x ) s [ d β T m x, T m 1 x ) + d β T m 1 x, T n x )] sd β T m x, T m 1 x ) + sd β T m 1 x, T n x ) sd β T m x, T m 1 x ) + s d β T m 1 x, T m x ) + s d β T m x, T n x ) sd β T m x, T m 1 x ) + s d β T m 1 x, T m x ) + s 3 d β T m x, T m 3 x ) + s 3 d β T m 3 x, T n x ). By repeating this process, we get T m x, T n x ) sd β T m x, T m 1 x ) + s d β T m 1 x, T m x ) + s 3 d β T m x, T m 3 x ) +. sk m 1 δ + s k m δ + s 3 k m 3 δ + ) = sk m 1 sk s δ 1+ + k + sk m 1 ) δ. 1 s k Taking the limits as m,weget T m x, T n x) 0 for all α [0, 1) as sk <1. Similarly, let n, m N with m < n,wehave T n x, T m x ) sd β T n x, T n+1 x ) + s d β T n+1 x, T n+ x ) + s 3 d β T n+ x, T n+3 x ) + sk n δ + s k n+1 δ + s 3 k n+ δ + = sk n δ 1+sk +sk) + ) sk n ) δ. 1 sk Letting n,weget T n x, T m x) 0 for all α [0, 1) as sk <1. Thus {T n x} is a G bdq -Cauchy sequence. Since X, )isg bdq -complete, we see that {T n x} G bdq -converges to some u X for all α 0, 1]. We note that {T n x} is a sequence in A and {T n 1 x} is a sequence in B in such a way that both sequences tend to the same limit u. Since A and B are closed, we have u A B,andthenA B. Now, we will show that Tu = u. Consider T n x, Tu ) s [ d β T n x, T n+1 x ) + d β T n+1 x, Tu )]. 3)

8 Kumari and Panthi Fixed Point Theory and Applications 015) 015:153 Page 8 of 17 Since T is continuous, lim n d β T n+1 x, Tu)=0andd β T n x, T n+1 x) k n δ. This implies that sd β T n x, T n+1 x) sk n δ sk) n δ. Letting n in the above inequality 3), we get u, Tu)=0, 4) since sk < 1. Similarly, Tu, u)=0. 5) From 4)and5), it follows that u = Tu. Hence u is a fixed point of T. Finally, to obtain the uniqueness of a fixed point, let v X be another fixed point of T such that Tv = v. Then we have u, v)= Tu, Tv) k u, v). 6) Similarly, v, u) k v, u). 7) From 6)and7), we obtain v, u)= u, v) = 0, this implies that u = v. Thus u is a unique fixed point of T. This completes the proof. If we take s = 1 in the above theorem, we obtain the following theorem in generating space of dislocated quasi-metric family. Theorem 3.3 Let A and B be nonempty closed subsets of a complete G dq -family X, ). Let T be a cyclic mapping that satisfies the condition of a G dq -cyclic-banach contraction. Then T has a unique fixed point in A B. If we take = d in Theorem 3., we obtain the following corollary in b-dislocated quasimetric space. Corollary 3.4 Let A and B be non-empty closed subsets of a complete b-dislocated quasimetric space X, d). Let T : A B A B be a cyclic mapping that satisfies the condition dtx, Ty) kdx, y), where k [0, 1 ) for all x A, y Bands 1then T has a unique fixed s point in A B. Example 3.5 Let X = { 1,0,1} and T : A B A B defined by Tx = x. Supposethat A = { 1, 0}, B = {0, 1}. Definethefunctiond : X X [0, ) byd0, 0) = 0, d0, 1) =, d0, 1) = 0, d1, 1) =, d1, 0) = 0, d1, 1) = 0, d 1, 1) = 1, d 1, 1) = 1 and d 1, 0) =. Clearly X, d) isacompleteb-dislocated quasi-metric with the coefficient s =butnot a dislocated quasi-metric, since the triangle inequality does not hold, that is, d 1, 0) d 1, 1) + d1, 0).

9 Kumari and Panthi Fixed Point Theory and Applications 015) 015:153 Page 9 of 17 Now let x A then Tx B.Furtherletx B then Tx A.HencethemapT is cyclic on X because TA) B and TB) A. Now, we have the following cases: Case 1Forx =0 A, y =0 B. Then dtx, Ty)=dT0, T0) = d0, 0) = 0; kdx, y)=kd0, 0) = 0. Thus dtx, Ty) kdx, y). Case Forx =0 A, y =1 B. Then dtx, Ty) = dt0, T1) = d0, 1) = 0; kdx, y) = kd0, 1) = k. Thus dtx, Ty) kdx, y),fork [0, 1 ). Case 3Forx = 1 A, y =0 B. Then dtx, Ty)=dT 1), T0) = d1, 0) = 0; kdx, y)=kd 1, 0) = k. Thus dtx, Ty) kdx, y),fork [0, 1 ). Case 4Forx = 1 A, y =1 B. Then dtx, Ty) = dt 1),T1) = d1, 1) = 0; kdx, y) = kd 1, 1) = k. Thus dtx, Ty) kdx, y),fork [0, 1 ). Thus for all x A, y B and k [0, 1 ), T satisfies all the conditions of the above corollary. Hence T has a unique fixed point in A B.Infact0 A B istheuniquefixedpoint. If we take the parameter s =1and = d in Theorem 3., we obtain the following corollary in dislocated quasi-metric space. Corollary 3.6 Let A and B be non-empty closed subsets of a complete dislocated quasimetric space X, d). Let T : A B A B be a cyclic mapping that satisfies the condition dtx, Ty) kdx, y), where k [0, 1) for all x A, y B. Then T has a unique fixed point in A B. We now give an example to illustrate the above corollary. Example 3.7 Let X =[ 1,1]andT : X X defined by Tx = x.supposethata =[ 1,0] and B = [0, 1]. Define the function d : X X [0, )bydx, y)= x. Hence d is complete dislocated quasi-metric on X. FurtherTA) B and TB) A. Thus T is cyclic on X. Now consider dtx, Ty) =d x ), y = x = x kdx, y) for 1 k <1. 8) Thus T satisfies all the conditions of above corollary and 0 is a unique fixed point.

10 Kumari and Panthi Fixed Point Theory and Applications 015) 015:153 Page 10 of 17 Example 3.8 Let X =[ 1,1]andT : X X defined by Tx = x.supposethata =[ 1,0] and B = [0, 1]. Define the function d : X X [0, )bydx, y)= x + y ; dtx, Ty) =d x ), y = x + y 4 = x + y 4 = 1 ) x + y ) x + y 1 kdx, y), where 1 k <1. 9) Thus T satisfies all the conditions of above corollary and 0 is a unique fixed point of T. Definition 3.9 []) Let X, d) beacompletemetricspace.t : X X is called a Kannan contraction if there exists a number t,0<t < 1,suchthat,foreachx, y X, dtx, Ty) t [ dx, Tx)+dy, Ty) ]. Inspired by the above definition, we introduce the G bd -cyclic-kannan contraction in the setting of generating space of a b-dislocatedmetric family. Definition 3.10 Let A and B be non-empty subsets of a G bd -family X, ). A cyclic map T : A B A B is said to be a G bd -cyclic-kannan contraction if there exists t [0, 1 4s ] where s 1andforanyα 0, 1] such that Tx, Ty) t [ x, Tx)+ y, Ty) ] 10) for all x A, y B and st < 1 4. Now we state and prove the fixed point theorem for a cyclic-kannan contraction in a generating space of b-dislocatedmetric family. Theorem 3.11 Let A and B be non-empty closed subsets of a complete generating space of b-dislocated metric family X, ). Let T be a continuous cyclic mapping that satisfies the condition of a G bd -cyclic-kannan contraction. Then T has a unique fixed point in A B. Proof Let x A fix) and, using the definition of a G bd -cyclic-kannan contraction,wehave Tx, T x ) ) = Tx, TTx) t x, Tx)+t Tx, T x ),

11 Kumari and Panthi Fixed Point Theory and Applications 015) 015:153 Page 11 of 17 so Tx, T x ) ) t x, Tx). 11) By using 10), we have T x, T 3 x ) TTx), T T x )) For all n N,wehave t [ Tx, T x ) + T x, T 3 x )] ) t Tx, T x ) ) t x, Tx). T n x, T n+1 x ) ) t n x, Tx) for all α [0, 1). Let n, m N with m < n, by using the definition of G bd -family, we get T m x, T n x ) s [ T m x, T m+1 x ) + T m+1 x, T n x )] s T m x, T m+1 x ) + s T m+1 x, T m+ x ) + s 3 T m+ x, T m+3 x ) + ) t m ) t m+1 ) t m+ s x, Tx)+s x, Tx)+s 3 x, Tx)+ ) t m ) ) t t = s x, Tx) 1+s + s + ) st m ) x, Tx) 1 1 st ) st m st ) x, Tx). Since st <1, T m x, T n x) 0asm, n. Hence {T n x} is a G bd -Cauchy sequence. Since X, ) is complete generating space of b-dislocated metric family, {T n x} G bd - converges to some u X,i.e.x n u. We note that {T n x} is a sequence in A and {T n 1 x} is a sequence in B in a way that both sequences tend to the same limit u. Since A and B are closed, we have u A B,andthenA B. Now, we prove that Tu = u. Consider Tu = Tlim n x n )=lim n Tx n )=lim n x n+1 = u. Now we prove that u is the unique fixed point. Suppose u is another fixed point of T such that Tu = u. Then from the contractive condition, we get u, u ) = Tu, Tu )

12 Kumari and Panthi Fixed Point Theory and Applications 015) 015:153 Page 1 of 17 t [ u, Tu)+ u, Tu )] t [ u, u)+ u, u )] t [ s [ d β u, u ) + d β u, u )] + s [ d β u, u ) + d β u, u )]] 4ts u, u ), which yields 1 4ts) u, u ) 0. Since 1 4ts >0, u, u )=0. Thus u = u. This completes the proof. If we take s = 1 in the above theorem, we obtain the following theorem in generating space of dislocated metric family. Theorem 3.1 Let A and B be nonempty closed subsets of a complete generating space of dislocated metric family X, ). Let T be a continuous cyclic mapping that satisfies the condition of a G d -cyclic-kannan contraction. Then T has a unique fixed point in A B. If we take = d in Theorem 3.11, we obtain the following corollary in b-dislocated metric space. Corollary 3.13 Let A and B be non-empty closed subsets of a complete b-dislocated metric space X, d). Let T be a continuous cyclic mapping that satisfies the condition dtx, Ty) t [ dx, Tx)+dy, Ty) ], where t 0, 1 ). Then T has a unique fixed point in A B. 4s If we take the parameter s =1and = d in Theorem 3.11, we obtain the following corollary in complete dislocated metric space. Corollary 3.14 Let A and B be non-empty closed subsets of a complete dislocated metric space X, d). Let T be a continuous cyclic mapping that satisfies the condition dtx, Ty) t [ dx, Tx)+dy, Ty) ], where t 0, 1 ). Then T has a unique fixed point in A B. 4 Example 3.15 Let X = [0, 1] and T : A B A B defined by Tx = x 9.SupposethatA, B = [0, 1]. Define the function d : X X [0, )bydx, y)=max{x, y}. Clearly d is a complete dislocated metric space. Case i) If x > y. x dtx, Ty) =d 9, y ) { x = max 9 9, y } 9 = x 9 = 1 9 x) 1 x + y) 9

13 Kumari and Panthi Fixed Point Theory and Applications 015) 015:153 Page 13 of 17 = 1 { max x, x } { + max y, y }) = 1 d x, x ) + d y, y )) t [ dx, Tx)+dy, Ty) ] so 1 9 t < 1 4. Case ii) If x < y, similar proof as above. Hence omitted. Thus T satisfies all the conditions of the above corollary. Hence T has a unique fixed point. In fact 0 is the unique fixed point of T. In 1974, Ciric [3] introduced the concept of a quasi-contraction and extended some results concerning generalized contractions of [43] and[44] to quasi-contractions. He proved fixed point theorems for single-valued quasi-contractions and provided an illustrative example for his extensions. Some generalizations on quasi-contractions can be found in [45 47]. Definition 3.16 [5]) Let X, d) beacompletemetricspace.amapt : X X such that there exists a constant h,0 h <1,andforeachx, y X, dtx, Ty) h max { dx, y), dx, Tx), dy, Ty), dx, Ty), dy, Tx) } is called a Ciric quasi-contraction. Based on above definition, we introduce the following definition in the setting of generating space of b-quasi-metricfamily. Definition 3.17 Let A and B be non-empty subsets of a G bq -family X, ). A cyclic map T : A B A B is said to be a G bq -cyclic-ciric contraction if there exists h [0, s )where s 1andforanyα 0, 1] such that s Tx, Ty) h max { x, y), x, Tx), y, Ty), x, Ty), y, Tx) } for all x A, y B. Now, we derive the existence of fixed point theorems for G bq -cyclic-ciric contraction in G bq -family which is larger than the generating space of a quasi-metric family and metric space. Theorem 3.18 Let A and B be non-empty closed subsets of a complete G bq -family X, ). Let T be a continuous cyclic mapping that satisfies the condition of a G bq -cyclic-ciric contraction. Then T has a unique fixed point in A B. Proof Let x AFix). By using G bq -cyclic-ciric contraction,wehave s Tx, Ty) h max { x, y), x, Tx), y, Ty), x, Ty), y, Tx) }.

14 Kumari and Panthi Fixed Point Theory and Applications 015) 015:153 Page 14 of 17 Consider s Tx, T x ) h max { x, Tx), x, Tx), Tx, T x ), x, T x ), Tx, Tx) } h max { x, Tx), Tx, T x ), s [ d β x, Tx)+d β Tx, T x )]} hsd β x, Tx)+hsd β Tx, T x ), which yields Thus Tx, T x ) h s h x, Tx). Tx, T x ) k x, Tx), where k = h, which implies 0 k <1. s h Choose sk 1. Repeating the same process for all n N,weget T n x, T n+1 x ) k n x, Tx). Now we shall show that {T n x} is a G bq -Cauchy sequence. Let n, m N with m > n,wehave T n x, T m x ) s [ d β T n x, T n+1 x ) + d β T n+1 x, T m x )] sd β T n x, T n+1 x ) + s d β T n+1 x, T n+ x ) + s 3 d β T n+ x, T n+3 x ) +. sk n d β x, Tx)+s k n+1 d β x, Tx)+s 3 k n+ d β x, Tx)+ sk n[ 1+sk +sk) + ]d α x, Tx) skn 1 sk x, Tx). Letting n,weget T n x, T m x) 0 for all α 0, 1] as sk 1. Hence {T n x} is a G bq -Cauchy sequence. Since X is G bq -complete, the sequence {T n x} converges to some point z X. Wenote that {T n x} is a sequence in A and {T n 1 x} is a sequence in B in such a way that both sequences tend to the same limit z. Since A and B are closed, we have z A B,andthenA B. Now, we will show that Tz = z. Consider z, Tz) s [ d β z, T n+1 x ) + d β T n+1 x, Tz )].

15 Kumari and Panthi Fixed Point Theory and Applications 015) 015:153 Page 15 of 17 Letting n,weget z, Tz)=0. This implies z = Tz. Now we prove that z istheuniquefixedpoint. Inordertodothis,letz X be another fixed point of T such that Tz = z. Then we have s z, z ) = s Tz, Tz ) h max { z, z ), z, Tz), z, Tz ), z, Tz ), z, Tz )}, which implies that z, z ) h s ) z, z ) since h s <1, z, z )=0.Thusz = z. This completes the proof. If we take s = 1 in the above theorem, we obtain the following theorem in generating space of quasi-metric family. Theorem 3.19 Let A and B be nonempty closed subsets of a complete generating space of quasi-metric family X, ). Let T be a continuous cyclic mapping that satisfies the condition of a G q -cyclic-kannan contraction. Then T has a unique fixed point in A B. If we take = d in Theorem 3.18, we obtain the following corollary in b-metric space. Corollary 3.0 Let A and B be non-empty closed subsets of a complete b-metric space. Let T : A B A B be a continuous cyclic mapping that satisfies the condition s dtx, Ty) h max{dx, y), dx, Tx), dy, Ty), dx, Ty), dy, Tx)}, where h [0, s ) for all x A, y Band s 1. Then T has a unique fixed point in A B. Example 3.1 Let X = [0, 1] and T : A B A B defined by Tx = x 6.SupposethatA, B = [0, 1]. Define the function d : X X [0, )bydx, y)= x y and s =. Clearly d is complete b-metric on X. FurtherTA) B and TB) A. HenceT is continuous cyclic map on X. Now, we consider x s dtx, Ty) =4d 6, y ) 6 =4 x y 6 = 4 x y 36 = 1 dx, y) 9 h max { dx, y), dx, Tx), dy, Ty), dx, Ty), dy, Tx) }, where 1 h <1.ThusT satisfies all the conditions of the above corollary and 0 is the 9 unique fixed point in A B.

16 Kumari and Panthi Fixed Point Theory and Applications 015) 015:153 Page 16 of 17 If we take the parameter s =1and = d in Theorem 3.18, we obtain the following corollary in a complete metric space. Corollary 3. Let A and B be non-empty closed subsets of a complete metric space. Let T : A B A B be a continuous cyclic mapping that satisfies the condition dtx, Ty) h max{dx, y), dx, Tx), dy, Ty), dx, Ty), dy, Tx)}, where h [0, 1 ) for all x A, y Band s 1. Then T has a unique fixed point in A B. Example 3.3 Let X =[ 1,1]andT : A B A B defined by Tx = x 5.Supposethat A =[ 1,0]andB = [0, 1]. Define the function d : X X [0, )bydx, y)= x y. Clearly d is complete metric on X.FurtherTA) B and TB) A.HenceT is continuouscyclic map on X. Now, we consider dtx, Ty) =d x ) 5, y 5 = x 5 + y 5 = x 5 y 5 = 1 x y 5 = 1 dx, y) 5 h max { dx, y), dx, Tx), dy, Ty), dx, Ty), dy, Tx) }, where 1 h <1.ThusT satisfies all the conditions of the above corollary and 0 is the 5 unique fixed point in A B. Competing interests The authors declare that there is no conflict of interests regarding the publication of this paper. Authors contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details 1 Department of Mathematics, K.L. University, Vaddeswaram, Andhra Pradesh, India. Department of Mathematics, Nepal Sanskrit University, Valmeeki campus, Exhibition road, Kathmandu, Nepal. Acknowledgements The first author would like to express her sincere gratitude to Prof. I. Ramabhadra Sarma for his invaluable support and motivation. The authors would like to express their thanks to the referees for their helpful comments and suggestions. Received: 16 June 015 Accepted: 1 August 015 References 1. Banach, S: Sur les operations dans les ensembles abstraits et leur applications aux equations integrales. Fundam. Math. 3, ). Kannan, R: Some results on fixed points. II. Am. Math. Mon. 76, ) 3. Ciric, LB: A generalization of Banach s contraction principle. Proc. Am. Math. Soc. 45, ) 4. Subrahmanyam, PV: Completeness and fixed-points. Monatshefte Math. 80, ) 5. Ume, JS: Fixed point theorems for Kannan-type maps. Fixed Point Theory Appl. 015, ). doi: /s Enjouji, Y, Nakanishi, M, Suzuki, T: A generalization of Kannan s fixed point theorem. Fixed Point Theory Appl. 009, ). doi: /009/ De la Sen, M: Linking contractive self-mappings and cyclic Meir-Keeler contractions with Kannan self-mappings. Fixed Point Theory Appl. 010, ).doi: /010/57057

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