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 [Formula: see text]-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 .

Background

Banach (1922) established a remarkable fixed point theorem known as the “Banach Contraction Principle.” This renowned principle assures the existence and uniqueness of fixed points of certain self maps of metric spaces and gives a constructive method to find those fixed points.

(Banach1922) Let (X, d) be a complete metric space and be a mapping, there exists a number such that, for each 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.

(Kannan1968) Let (X, d) be a complete metric space and be a mapping, there exists a number such that, for each 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.

(Chatterjea1972) There exists such that, for all Then f has a fixed point.

On the other hand, the traditional theory of a metric space has been generalized in wide directions. Some of such generalizations are dislocated metric spaces (Matthews 1986), dislocated quasi-metric spaces (Zeyada et al. 2006), dislocated symmetric spaces Ramabhadra et al. (2014) and quasi-symmetric spaces (Kumari and Ramana 2014) [for more new spaces and related results can be found in Bakhtin (1989), Branciari (2000), Kumari et al. (2012), Kumari et al. (2015), Kumari et al. (2015)].

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 and Panthi (2015, 2016) introduced the concepts of “generating space of b-dislocated quasi-metric family” (abbreviated “-family”),“generating space of b-dislocated metric family” (abbreviated “-family”) and “generating space of b-quasi-metric family” (abbreviated “-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 denotes the set of all positive integers .

Definition 1

Let X be a non-empty set and a family of mapping of in to Consider the following conditions for any and

The family of self distances are zero:

The family of distances are symmetric:

The family of positive distances between distinct points: implies

For any there exist such that

For any is non-increasing and left continuous in

is called,If then -family becomes generating space of quasi-metric family as defined by Banach (1922).

Generating space of b-quasi-metric family if through

Generating space of b-dislocated metric family if through

Generating space of b-dislocated-quasi metric family if through

Example 2

Let (X, d) be a metric space. If we put instead of d for all and , then 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 whose base is the family of open balls. The “-family” will play a very predominant role in fixed point theory because the class of -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.

Definition 3

Let be a -family and be a net in X. We say that -converges to x in if for all

In this case we write

Definition 4

Let be a -family and let We say that x is a -limit point of A if there exists a net in such that The set of all -limit points of is denoted by D(A).

Definition 5

Let be a -family with The set is called -open ball of radius and center x.

The set is called -closed ball of radius and center x.

Remark 6

In a -familyNow, we state some propositions and corollaries in which can be proved following similar arguments to those given in Kumari (2012), Sarma and Kumari (2012).

if and only if for every there exists such that for all

hold.

-limit point of a net is unique.

Proposition 7

Letthen

if

if

Corollary 8

If we writeforthe operationsatisfies Kuratowski’s closure axioms of Kelley (1960) so that the setandis a topology onX.

Corollary 9

We callthe topological space induced by-family. We callto be closed ifand open if

Proposition 10

Letbe a-family.is a-limit point ofiff for every

Corollary 11

if and only iforfor all

Corollary 12

is open iniff for everythere existssuch that

Proposition 13

Ifand, thenis anopen set in

Proof

Let and Then

Hence is open.

Proposition 14

is a Hausdorff space.

Let , then Choose such that

Then

Corollary 15

Ifthen the collectionisan open base atxinHenceis first countable.

The above corollary yields us to deal with sequences instead of nets.

Definition 16

A sequence in a -family is called a -Cauchy sequence if given there exists such that for all we have or for all

Proposition 17

Every-convergent sequence in a-family is-Cauchy.

Definition 18

A -family is called complete if every -Cauchy sequence in X is -Convergent.

Remark 19

In a -family a subset A of X is said to be closed if for any sequence of points of A such that then

Main results

Definition 20

By we denote the set of all real functions which have the following properties:

is monotone increasing;

lim for any where

Theorem 21

Letbe a complete-family with the co-efficientandsatisfywhereis a continuous monotone increasing mapping such thatfor eachThenThas exactly one fixed point.

First of all, note that, for all and

Let be an arbitrary point in X. Define the iterative sequence as follows:

If we assume that for some ,then we have so is a fixed point of T and the proof is complete. From now on we will assume that for each

Consider

By repeating this procedure, we get,If then which yields is a fixed point of T.

Suppose then

Which implies for each there exists such thatNow our aim is to prove is a -Cauchy sequence.

If we apply induction with respect to n,  to show for all Clearly (3) holds for Let us assume that (3) holds for some

i.e

We have,Now consider,Thus by induction, we get, (3) is satisfied for any

Hence which yields that is a -Cauchy sequence in a complete -family. Thus there exists some u in X such that Also the subsequence -converges to u in X.

For any x, y in X, we havewhich implies T is continuous. Hence

Consider

By taking limits

Hence

Uniqueness: Let u, v be two fixed points of T and

Thus which implies

Consider,A contradiction. Thus Which implies

Hence T has a unique fixed point.

This completes the proof of the theorem.

By taking with we can set the following corollary which generalizes the famous Banach contraction principle in -family.

Corollary 22

Letbe a complete-family with the coefficientand letis a mapping such that

for allwhereThenThas a unique fixed point inX.

By takingin above corollary, we generalize the Banach contraction principle in generating space of quasi-metric family.

Corollary 23

Letbe a complete generating space of quasi-metric family with the coefficientand letbe a mapping such that

for allwhereThenThas a unique fixed point inX.

By takingin 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. Ifis a self mapping andx, ybe any elements ofX. Now consider the following contractive conditions:

(Rhoades1977) There exist nonnegative functionsa, b, csatisfying

such that, for eachCiric.1 (Ciric 1974) There exists nonnegative functionsq, r, s, tsatisfyingsuch that, for eachCiric.2 (Ciric 1971) There exist a constantsuch that for eachNote that above mentioned named contractions, as originally defined by their respective authors. In above contractions, Ciric.2 condition is very significant because, a good number of contractive conditions imply Ciric.2 condition.

Based on the definition of quasi-contraction of Ciric (1971), we introduce the following definition in the setting of-family.

Definition 24

Let be a complete -family with the parameter . If be a self mapping which satisfiesfor all and Then T is called “ s-h generating b-quasi-contraction”.

Definition 25

Let be a complete -family with the parameter . If is a self continuous mapping which satisfies s-h generating b-quasi-contraction, then T has a unique fixed point in X.

Let be an arbitrary point in X. Define the iterative sequence as follows:If we assume that for some ,then we have so is a fixed point of T and the proof is complete. Now we will assume that for each

Now consider,which implies that,where .

Similarly, by the contractive condition of the theorem, we can get below condition:By repeating the same process, we get for all Since and applying limits as we get

Now our aim is to prove is a -Cauchy sequence.

To obtain this, let with

Then we have,By taking the limits as we get as

Hence is a -Cauchy sequence in complete -family Thus there exists some such that -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 and Then by s-h generating b-quasi-contraction, we getSo where and since then we get

Hence which implies

Hence T has a unique fixed point in X.

If we take parameter in the above theorem, we obtain following corollary.

Corollary 26

Letbe a complete generating space of quasi-metric family andis a self continuous mapping which satisfies:for allandThenThas a unique fixed point inX.

If we putin above corollary, we get following corollary.

Corollary 27

Let (X, d) be a complete metric space and ifis a self continuous mapping which satisfies:for allandThenThas a unique fixed point inX.

We now give an example to illustrate the above corollary.

Example 28

Let and

Clearly d is a complete metric on X. Define the self mapping by

For we havefor Clearly 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.

In 1972, Zamfirescu (1972) consolidate the (1,2,3) conditions which is known as Zamfirescu contractive condition and proved a fixed point theorem.

(Zamfirescu1972) There exists real numbers such that for each at least one of the following is true:Then T has a fixed point.

In Rhoades (1977), Rhoades state below conditions,

(Rhoades 19 Rhoades 1977) There exist non-negative functions a, b, c satisfying

such that, for each (Rhoades 19 Rhoades 1977) There exist a constant such that for each Moreover, Rhoades proved that Zamfirescus condition is equivalent to Rhoades 19′&Rhoades 19 conditions. However, recently Berinde (2004) proved that Banach’s, Kannan’s, Chatterjea’s and Zamfirescu’s mappings are weak contractions.

Theorem 29

Let (X, d) is a complete metric space. Ifbe a self continuous mapping satisfying any of the conditions either Rhodes or Zamfirescu or Ciric.1. ThenThas a unique fixed point.

In Rhoades (1977), Rhodes proved below Implications.Hence from Corollary 27, T has a unique fixed point.

Theorem 30

Letbe a complete-family with the parametersand. Ifis a self continuous mapping which satisfiess-h generating b-quasi-contraction, i.e.for allandIf for some positive integeris continuous, thenThas a unique fixed point inX.

We can construct a sequence as in Theorem 30 and conclude that the sequence -converges to some point u in X.

Thus its subsequence -converges to u. Also we have,

Which yields that u is a fixed point of

Now we shall prove that

Let l be a smallest positive integer such that but

If then,which yields,Thus ; where

Similarly,Which implies,Inductively we get,notice that

Thus

Which is a contradiction.

Hence

Which yields u is a fixed point of T.

Now if there exists another point in X such that thenWhich implies A contradiction.

Hence u is a unique fixed point of T in X.

If we take parameter in the above theorem, we obtain following corollary in generating spaces of quasi-metric family.

Corollary 31

Letbe a complete generating space of quasi-metric family andis a continuous mapping which satisfies:for allandIf for some positive integeris continuous, thenThas a unique fixed point inX.

If we takethen we get following corollary.

Corollary 32

Let (X, d) be a complete metric space and ifis a continuous mapping which satisfies:for allandIf for some positive integeris continuous, thenThas a unique fixed point inX.

Based on the definition of Zamfirescu Contraction, we introduce the following definition in the setting of -family.

Definition 33

Let be a complete -family with the parameter . If be a mapping such that for each at least one of the following is true:Then f is called “s-z generating b-quasi-contraction”.

Theorem 34

Letbe a complete-family with the parameters. Ifbe a continuouss-z generating b-quasi-contractionthenfhas a unique fixed point inX.

Put in the above (1), (2) and (3) of Definition 33, gives,Choose and then we getBy repeating this procedure,we obtainSince

Now we prove that is a -Cauchy sequence.

To do this,let m, n are positive integers such that

Now considerApplying we get, as

Thus is a -Cauchy sequence in complete -family.

Which implies there exist some such that

Since f is continuous, we get

Thus u is a fixed point of f. Suppose there exists another point in X such that then

Since Which implies Hence u is a unique fixed point.

We now give an example to illustrate the above theorem.

Example 35

Let Define by for all then is a complete -family. Let be a mapping defined by then for any which implies,

Hence f satisfies the condition(1) of Theorem 34 but f doesn’t satisfy condition(2) of Theorem 34.

SinceHence is the unique fixed point of f in X.

If we take parameter in the above theorem, we obtain following corollary in generating spaces of quasi-metric family.

Corollary 36

Letbe a complete generating spaces of quasi-metric family. Ifbe a continuous mapping such that for eachat least one of the following is true.Thenfhas a unique fixed point.

If we takethen we get following corollary.

Corollary 37

Let (X, d) be a complete metric space. Ifbe a continuous mapping such that for eachat least one of the following is true.Thenfhas a unique fixed point.

Open Question What are the additional conditions as needed in order to establish the existence of a unique fixed point satisfying the condition for any there exists such that in -family

Conclusion In this work, we introduced a new concept of s-h generating b-quasi-contraction., s-z generating b-quasi-contraction.. Also, we derived the existence of fixed point theorems for generating spaces of b-quasi-metric family. Moreover, some examples are provided wherever necessary. Our results may be the motivation to other authors for extending and improving these results to be suitable tools for their applications.