d-Neighborhood system and generalized F-contraction in dislocated metric space.

This paper, gives an answer for the Question 1.1 posed by Hitzler (Generalized metrics and topology in logic programming semantics, 2001) by means of "Topological aspects of d-metric space with d-neighborhood system". We have investigated the topological aspects of a d-neighborhood system obtained from dislocated metric space (simply d-metric space) which has got useful applications in the semantic analysis of logic programming. Further more we have generalized the notion of F-contraction in the view of d-metric spaces and investigated the uniqueness of fixed point and coincidence point of such mappings.

Background

Metrics appear everywhere in Mathematics: Geometry, Probability, statistics, coding theory, graph theory, pattern recognition, networks, computer graphics, molecular biology, theory of information and computer semantics are some of the fields in which metrics and/or their cousins play a significant role. The notion of metric spaces introduced by Frechet (1906), is one of the helpful topic in Analysis. Banach (1922) proved a fixed point theorem for contraction mapping in a complete metric space. The Banach contraction theorem is one of the primary result of functional analysis. After Banach contraction theorem, huge number of fixed point theorems have been established by various authors and they made different generalizations of this theorem.

Matthews (1985) generalized Banach contraction mapping theorem in dislocated metric space. Hitzler (2001) introduce the notion of dislocated metric (d-metric) space and presented variants of Banach contraction principle for various modified forms of a metric space including dislocated metric space and applied them to semantic analysis of logic programs. Hitzler (2001) has applied fixed point theorems for self maps on dislocated metric spaces, quasi dislocated metric spaces, generalized ultra metric spaces in his thesis “Generalized Metrics and Topology in Logic Programming Semantics”. In this context, Hitzler raised some related questions on the topological aspects of dislocated metrics.

Recently, Sarma and Kumari (2012) initiated the concept of d-balls and established topological properties on d-metric space. In the context of d-metric space, many papers have been published concerning fixed point, coincidence point and common fixed point theorems satisfying certain contractive conditions in dislocated metric space (see Karapinar and Salimi 2013; Kumari et al. 2012a, b; Zoto et al. 2014; Ahamad et al. 2013; Ren et al. 2013) which become an interesting topic in nowadays.

Of late several weaker forms of metric are extensively used in various fields such as programming languages, qualitative domain theory and so on.

Motivated by above, we give an answer for the Question 1.1 posed by Hitzler, further more we discuss some topological properties in d-neighborhood system obtained from dislocated metric space. Moreover, we generalize the notion of F-contraction initiated by Wardowski (2012) and we prove fixed point theorem. Our established results generalize similar results in the framework of dislocated metric space. Further more, we provide coincidence theorem in the setting of d-neighborhood systems.

Preliminaries and notations

First, we collect some fundamental definitions, notions and basic results which are used throughout this section. For more details, the reader can refer to Hitzler (2001).

Definition 2.1

Let X be a set. A relation is called a d-membership relation (on X) if it satisfies the following property for all and and implies .

Definition 2.2

Let X be a set, let be a d-membership relation on X and let be a collection of subsets of X for each . We call a d- for x if it satisfies the following conditions.

If then

If then

If then there is a with such that for all we have

If and then

Each is called a d-neighborhood of x. Finally, let X be a set and be a d-membership relation on X and, for each , let be a d-neighborhood system for x. Then is called a d-topological space, where

Proposition 2.3

Let X be a nonempty set. A distance on X is a mapA pair (X, d) is known asdislocated metric space (Simply d-metric space) ifdsatisfies the following conditions

() = 0

()

() for allx, y, zinX

If and the set and is called the ball with center at x and radius

Proposition 2.4

Letbe ad-metric space. Define thed-membership relationas the relationthere existsfor whichFor eachletbe the collection of all subsetsA of Xsuch that. Then is ad-neighborhood system forx; for each

Definition 2.5

Let be a d-topological spaces and let A net d-converges to if for each d-neighborhood U of x we have that is eventually in U, that is, there exists some such that for each

Definition 2.6

Let be a d-metric space and let be a d-topological spaces as in Proposition 2.4. Let be a sequence in X. Then converges in if and only if d-converges in

Definition 2.7

Let X and Y be d-topological spaces and let be a function. Then f is d-continuous at if for each d-neighborhood V of in Y there is a d-neighborhood U of in X such that We Say f is d-continuous on X if f is d-continuous at each

Theorem 2.8

LetXandYbed-topological spaces and letbe a function. Thenfis ad-continuous if and only if for each netinXwhichd-converges to someis a net inYwhichd-converges to

Proposition 2.9

Letandbed-metric spaces, letbe a function and letandbe thed-topological spaces obtained from, respectivelyas in Proposition 2.4. Thenf is d-continuous atif and only if for eachthere exists asuch that

Definition 2.10

Let be a d-metric space, let be a contraction with contractivity factor and let be the d- topological space obtained from d-metric as in Proposition 2.4. Then f is d-continuous.

Topological aspects of d-metric space with d-neighborhood system

The following question was put forth in Hitzler Thesis.

(Question 1.1). Question: Is there a reasonable notion of d-open set corresponding to the notions of d-neighborhood, d-convergence and d-continuity.

We provided an answer for the above open question by constructing below theorems.

Theorem 3.1

Letbe ad-topological space. Definefor each. Thenis a topology onX.

Proof

Clearly contains X and

Let be an indexed family of non-empty elements of .

Let which implies that for some

Thus there exists such that . Which implies that

Let be any finite intersection of elements of

We have to prove that To obtain this, first we prove that if then Let .

Which implies that and then there exists and there exists

Which implies and .

Thus Hence by induction, we get

Definition 3.2

Let be a d-topological space and be a d-open if for every there exists

Definition 3.3

Let be a d-topological space and is d-open then is d-closed.

Definition 3.4

Let be a d-topological space and A point x in A is called an interior point of A if

Remark

Interior point of A is an open set.

Definition 3.5

Let be a d-topological space and A point x in X is said to be limit point of A if for every there exist in A such that

Definition 3.6

Let (X, d) be a d-metric space and . If there is a number such that then f is called a contraction.

Definition 3.7

( Sarma and Kumari 2012) Let (X, d) be a d-metric space and be a mapping. Write and . We call points of Z(f) as coincidence point of f. Clearly every point of Z(f) is a fixed point of f but the converse is not necessarily true.

Theorem 3.8

A subsetis said to bed-closediff a netinFd-convergestoxthen

Suppose is d-closed.

Let be a net in F such that lim

We shall prove that

Let us suppose which implies that which is open.

Thus there exists such that

As there exists such that .

Since lim there exists such that for .

Hence A contradiction.

It follows that

Conversely, assume that if a net in Fd-converges to x then

We shall prove that is d-closed.

is d-open.

For this we have to prove that for every there exists such that

Suppose for some there exists such that

Let

As is a direct set under set inclusion

Thus is a net.

Let .

If

Thus implies that .

It follows that lim.

Which implies that A Contradiction.

So for all there exists such that

Which completes the proof.

For each is a d-neighborhood of x.

Theorem 3.9

Letbe ad-topological space and letbe the collection of all subsetsUofXsuch thatThenis said to be a basis for a topology onXif

For eachthere existssuch that

Ifthere existsand

(i) is clear.

Since implies

So there exists such that

Since balls are d-neighborhood, choose

Then and

Lemma 3.10

LetXbe any set andbe basis for the topologiesandrespectively. Then the following are equivalent.

finer than

and each basis elementwiththere exists a basis elementsuch thatand

Theorem 3.11

Letbe the topology induced from thed-topological spaceobtained fromd-metric as in Proposition 2.4,be the topology induced by thed-metric then

Let Then the collection is a basis for and is a basis for Clearly since is a d-neighborhood.

Let and such that

Since there exists such that

Which implies .

So Hence

Theorem 3.12

Letbe ad-topological space andandthe following are equivalent, assumefor every

There existssuch that lim

For everythere existsinAsuch that

Let there exists such that

Since (1) holds, lim.

Which implies that, there exists N such that

Let and then .

It follows that

So Hence (2) holds.

Assume that (2) holds. Let there exists in A such that

i.e there exists such that

Let .

Which implies that and .

Hence .

Which yields lim Hence (1) holds.

Theorem 3.13

Letbe thed-topological space obtained fromd-metricas in Proposition 2.4 .Then balls ared-open.

Let be a ball with center at x and radius .

It sufficies to prove that is d-open.

i.e we shall prove for every there exists such that

Since implies .

Choose .

As is a d-neighborhood, now let .

So it is sufficient to prove that .

Let .

This implies that .

Then .

It follows that .

Hence

Theorem 3.14

Letbe ad-topological space obtained fromd-metricas in Proposition 2.4. Thenis a Haussdorff space.

Suppose .

Let us choose

Let be the d-neighborhoods of x and y respectively.

It sufficies to prove .

Let .

Which implies that and .

So and .

It follows .

Which is a contradiction.

Theorem 3.15

Letbe ad-topological space obtained formd-metricas in Proposition 2.4. Then singleton sets ared-closed in

Let , we have to prove that is d-closed or it is sufficies to prove is d-open.

i.e for each there exists such that

Since implies .

Which yields .

Thus, there is a d-neighborhood, such that .

Hence is d-closed.

Corollary 3.16

Letbe ad-topological space obtained formd-metric. Thenis a-space.

Corollary 3.17

Letbe ad-topological space. Then the collectionis an open base atxforX.

Main theorems

Wardowski (2012) introduced a new type of contraction called F-contraction and proved a new fixed point theorem concerning F-contraction and supported by computational data illustrate the nature of F-contractions. In this section, we present a theorem which generalizes the Wardowski’s theorem.

Definition 4.1

(Wardowski 2012) Let be a mapping satisfying,

F is strictly increasing, i.e for all such that

For each sequence of positive numbers iff

There exists such that

A mapping is said to be an F-contraction if there exists such that for all

Theorem 4.2

(Sgroi and Vetro 2013) Let (X, d) be a complete metric space and letbe an F-contraction thenThas a unique fixed pointand for everya sequenceis convergent to

In the literature one can find some interesting papers concerning F-contractions; (see for example Cosentino and Vetro 2014; Sgroi and Vetro 2013; Secelean 2013; Paesano and Vetro 2014; Hussain and Salimi 2014).

Definition 4.3

By we denote the set of all monotone decreasing real functions such that iff and

Lemma 4.4

Letand, then fromit follows that

Routine.

Theorem 4.5

Let (X, d) be ad-metric space,andsatisfying subadditive property. Definebyfor anyThen

is ad-metric space.

limiff lim

(X, d) is complete iffis complete.

Let .

Which yields implies

So follows from

Now consider

since g is subadditive.

.

It follows that .

Hence is a d-metric. This completes the proof of (1).

Let lim. It follows that lim

Which implies lim

Suppose lim By above lemma, it follows that lim Which completes the proof of (2).

Let us suppose that (X, d) is complete. Thus for every there exist such that for all .

Which yields lim.

Which implies lim; because g is continuous at 0. So is a Cauchy sequence in By using (2), we get is complete.

Conversely suppose that is complete.

Let is a Cauchy sequence in

Then for every there exist such that for all

Thus lim.

It follows that lim.

By above Lemma, lim.

Which implies that is a Cauchy sequence in (X, d). By using (2) we conclude that (X, d) is complete.

Definition 4.6

Let be a mapping satisfying,

F is strictly increasing, i.e for all such that

For each sequence of positive numbers iff

There exists such that . A mapping is said to be an -contraction if there exists such that for all ,

Theorem 4.7

Let (X, d) be a completed-metric spaceand letbe an-contraction. ThenThas a unique fixed point.

Define by for any and By lemma 4.4 and theorem 4.5 it follows that is a d-metric space.

We have, when implies

Then by using same proof as in Theorem 4.2, we can conclude that T has a unique fixed point.

Theorem 4.8

Letbe a completed-metric space and letbe a contraction andbe thed-topological space obtained from. Thenfhas a unique coincidence point forf.

Let . Choose .

Then is a cauchy sequence and converges in to some point u.

i.eu = lim Since f is a contraction it is also d-continuous, by Proposition 2.10,

Hence  = 0. Since , .

Thus u is a coincidence point of f.

Uniqueness

Let us suppose that v be the another coincidence point such that .

Thus and . By using triangle inequality, which implies .

Hence .