Some fixed point theorems in generating space of b-quasi-metric family

The purpose of this work is to study some properties of “Generating space of b-quasi-metric family”(simply \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{bq}$$\end{document}Gbq-family) and derive some fixed point theorems using some standard contractions. Presented theorems extend and generalize many well-known results in the literature of fixed point theory .

Then f has a fixed point.
It is well known that Banach's contraction principle is one of the decisive result of functional analysis. A huge number of generalizations of the Banach contraction principle have appeared. Of all these, the following generalization of Kannan (1968) and Chatterjea (1972) stands at the top.
(Kannan 1968) Let (X, d) be a complete metric space and f : X → X be a mapping, there exists a number t, 0 < t < 1 2 , such that, for each x, y ∈ X, Then T has a fixed point. It is interesting that Kannan's fixed point theorem is very predominant because Subrahmanyam (1975) 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. (Chatterjea 1972) There exists α ∈ [0, 1) such that, for all x, y ∈ X, Then f has a fixed point.
In 1997, Chang et al. (1997) introduced a definition of "generating space of quasi-metric family" which is a generalization of quasi-metric space. He proved some interesting fixed point theorems and coincidence point theorems in generating space of quasi-metric family.
Later, Lee et al. (1999) define a family of weak quasi-metrics in a generating space of quasi-metric family. He proved 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 quasi-metric family by using a family of weak quasi-metrics. He also proved fixed point theorem for set-valued maps in complete generating spaces of quasi-metric family without considering of lower semi-continuity.
Very recently, Kumari andPanthi (2015, 2016) introduced the concepts of "generating space of b-dislocated quasi-metric family" (abbreviated "G bdq -family"), "generating space of b-dislocated metric family" (abbreviated "G bd -family") and "generating space of b-quasi-metric family" (abbreviated "G bq -family"). Also she proved the existence of unique fixed point theorems in weaker forms of generating spaces by using various cyclic contractive conditions. Through out this paper, we assume that R + = [0, ∞), N denotes the set of all positive integers.
Definition 1 Let X be a non-empty set and {d α : α ∈ (0, 1]} a family of mapping d α of X × X in to R + . Consider the following conditions for any x, y, z ∈ X and s ≥ 1.
The family of self distances are zero: The family of distances are symmetric: The family of positive distances between distinct points: For any x, y ∈ X, d α (x, y) is non-increasing and left continuous in α.
Generating space of b-dislocated metric family if d 2 through d 5 .
Generating space of b-dislocated-quasi metric family if d 3 through d 5 . If s = 1 then G bq -family becomes generating space of quasi-metric family as defined by Banach (1922).
Example 2 Let (X, d) be a metric space. If we put d α instead of d for all α ∈ (0, 1] and x, y ∈ X, then (X, d α ) is a generating space of quasi-metric family.
In Fan (1993), it was proved that each generating space of quasi-metric family generates a topology I d α whose base is the family of open balls. The "G bq -family" will play a very predominant role in fixed point theory because the class of G bq -family is larger than generating space of quasi-metric family.
Motivated by above, In this paper, we establish the existence of a topology induced by a Generating space of b-quasi-metric family. Moreover, we derive some unique fixed point theorems.

Some properties of generating space of b-quasi-metric family
In 1880s, A French mathematician H Poincare introduced topological methods in studying nonlinear problems of mathematical analysis. One of main ideas was to utilize fixed point theorems. Together with the study under the topological structure derived from Poincares analysis motivation, L E J Brouwers fixed point theorem came into the world. Since then, the fixed point theory became a major branch of topology and afterwards it consistently became a major theme of the research.
Due to importance of topology in fixed point theory, we discuss some topological structures in b-quasi-metric family as below.
In this case we write lim n→∞ x n = x. Definition 4 Let (X, d α ) be a G bq -family and let A ⊆ X, x ∈ X. We say that x is a G bq -limit point of A if there exists a net {x n } in A − {x n } such that lim n→∞ x n = x. The set of all G bq -limit points of A ⊆ X is denoted by D(A).
Remark 6 In a G bq -family 3. G bq -limit point of a net is unique. Now, we state some propositions and corollaries in (X, d α , I) which can be proved following similar arguments to those given in Kumari (2012), Sarma and Kumari (2012). Kelley (1960) so that the set I = {A/A ⊂ X and A c = A c } is a topology on X.

Proposition 7 Let
The above corollary yields us to deal with sequences instead of nets.
Proposition 17 Every G bq -convergent sequence in a G bq -family is G bq -Cauchy.

Remark 19
In a G bq -family (X, d α ), a subset A of X is said to be closed if for any sequence {x n } of points of A such that lim n→∞ x n = x then x ∈ A.

Main results
Definition 20 By we denote the set of all real functions ϒ : [0, ∞) → [0, ∞) which have the following properties: Theorem 21 Let (X, d α ) be a complete G bq -family with the co-efficient s ≥ 1, p > 1 and T : X → X satisfy s p d α (Tx, Ty) ≤ ϒ(d α (x, y)) ; ∀x, y ∈ X. where : R + → R + is a continuous monotone increasing mapping such that lim n→∞ ϒ n (t) = 0 for each t > 0. Then T has exactly one fixed point.
Let x 0 be an arbitrary point in X. Define the iterative sequence {x n } as follows: If we assume that x n+1 = x n for some n ∈ N,then we have x n = x n+1 = T (x n ), so x n is a fixed point of T and the proof is complete. From now on we will assume that for each n ∈ N, x n+1 � = x n .
If we apply induction with respect to n, to show for all n ∈ N, Clearly (3) holds for n = 1. Let us assume that (3) holds for some n ∈ N.

Now consider,
Thus by induction, we get, (3) is satisfied for any n ≥ 1.
Hence d α (T k (x), T k+n (x)) < η. which yields that {x n } is a G bq -Cauchy sequence in a complete G bq -family. Thus there exists some u in X such that lim n→∞ x n = u. Also the subsequence {x n+1 } G bq -converges to u in X.
For any x, y in X, we have which implies T is continuous. Hence lim n→∞ Tx n = Tu.
By taking limits n → ∞, d α (u, Tu) = 0. Hence u = Tu. Uniqueness: Let u, v be two fixed points of T and u � = v.
Hence T has a unique fixed point. This completes the proof of the theorem. By taking ϒ(t) = t with 0 ≤ < 1 s p , we can set the following corollary which generalizes the famous Banach contraction principle in G bq -family.

Corollary 22
Let (X, d α ) be a complete G bq -family with the coefficient s ≥ 1, p > 1 and let T : X → X is a mapping such that d α (Tx, Ty) ≤ d α (x, y) for all x, y ∈ X, where 0 ≤ < 1 s p . Then T has a unique fixed point in X.
By taking s = 1 in above corollary, we generalize the Banach contraction principle in generating space of quasi-metric family. (10) Corollary 23 Let (X, d α ) be a complete generating space of quasi-metric family with the coefficient s > 1, and let T : X → X be a mapping such that d α (Tx, Ty) ≤ d α (x, y) for all x, y ∈ X, where 0 ≤ < 1. Then T has a unique fixed point in X.
By taking d α = d in above corollary, we get Banach contraction principle in complete metric space. Rhoades (1977) collected some contractive conditions considered by various authors and established implications and non-implications between them. We noted some contractive conditions as mentioned below.
Let (X, d) be a metric space. If T : X → X is a self mapping and x, y be any elements of X. Now consider the following contractive conditions: (Rhoades 1977) (1971), we introduce the following definition in the setting of G bq -family.
Definition 24 Let (X, d α ) be a complete G bq -family with the parameter s ≥ 1, p > 1. If T : X → X be a self mapping which satisfies for all x, y ∈ X and h ∈ [0, s p−1 s+1 ). Then T is called " s-h generating b-quasi-contraction".
Definition 25 Let (X, d α ) be a complete G bq -family with the parameter s ≥ 1, p > 1. If T : X → X is a self continuous mapping which satisfies s-h generating b-quasi-contraction, then T has a unique fixed point in X.
Proof Let x 0 be an arbitrary point in X. Define the iterative sequence {x n } as follows: (11) s p d α (Tx, Ty) ≤ h.max d α (x, y), d α (x, Tx), d α (y, Ty), d α (x, Ty), d α (y, Tx) If we assume that x n+1 = x n for some n ∈ N,then we have x n = x n+1 = T (x n ), so x n is a fixed point of T and the proof is complete. Now we will assume that for each n ∈ N, x n+1 � = x n . Now consider, which implies that, where c = h s p−1 −h . Similarly, by the contractive condition of the theorem, we can get below condition: By repeating the same process, we get for all n ≥ 2.
Since 0 ≤ c < 1 and applying limits as n → ∞, we get d α (x n , x n+1 ) → 0. Now our aim is to prove {x n } is a G bq -Cauchy sequence.
To obtain this, let m, n > 0 with m > n. Then we have, By taking the limits as n, m → ∞, we get d α (x n , x m ) → 0 as cs < 1.
Hence {x n } is a G bq -Cauchy sequence in complete G bq -family (X, d α ). Thus there exists some u ∈ X such that {x n } G bq -converges to u.
Since T is a continuous mapping, Hence u is a fixed point of T.
Uniqueness Let us suppose that u and v are two fixed points of T where Tu = u and Tv = v. Then by s-h generating b-quasi-contraction, we get Hence T has a unique fixed point in X. If we take parameter s = 1 in the above theorem, we obtain following corollary.

Corollary 26
Let (X, d α ) be a complete generating space of quasi-metric family and T : X → X is a self continuous mapping which satisfies: for all x, y ∈ X and h ∈ [0, 1 2 ). Then T has a unique fixed point in X. If we put d α = d in above corollary, we get following corollary.

Corollary 27
Let (X, d) be a complete metric space and if T : X → X is a self continuous mapping which satisfies: for all x, y ∈ X and h ∈ [0, 1 2 ). Then T has a unique fixed point in X. We now give an example to illustrate the above corollary.
Clearly d is a complete metric on X. Define the self mapping T : X → X by T (x) = x 3 . For x, y ∈ [0, 1], we have for 1 3 ≤ h < 1 2 . Clearly x = 0 is the unique fixed point of T.
Three eminent conditions (1), (2) and (3) are made significant contribution in the area of fixed point theory and applications. After these three results, a huge number of papers have been written by several authors to those results either improve or generalize some of the conditions (1), (2) or (3), or even the three conditions simultaneously.
(18) Proof We can construct a sequence {x n } as in Theorem 30 and conclude that the sequence {x n } G bq -converges to some point u in X.
Thus its subsequence {x n k }(n k = k) G bq -converges to u. Also we have, T q (u) = T q ( lim k→∞ x n k ) = lim k→∞ (T q (x n k )) = lim k→∞ (x k+q ) = u Which yields that u is a fixed point of T q . Now we shall prove that Tu = u. Let l be a smallest positive integer such that T l u = u but T m u � = u.(m = 1, 2, . . . l − 1). If l > 1 then, which yields, Thus d α (u, Tu) < k.d α (T l−1 u, u); where k = h s p−1 −h . Similarly, Which implies, Inductively we get, notice that k < 1.