Abstract

In this paper, we introduce the notion of parametric A-metric space as generalisation of parametric metric space and parametric S-metric space. Further we prove some fixed point theorem of expansive mapping in the setting of parametric A-metric space.

1. Introduction

Fixed point theory and different forms of generalization of metric space is one of the interesting topic for many researchers. This can be witnessed from the vast literature available in this topic. In order to study some forms of generalization of metric space one can see the research papers in ^{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 19, 20, 21, 23, 24, 25, 26} and references there in. As one of the generalization Sedghi et. al. ¹ introduced the concept of S-metric space. The definition of S-metric space is as follows.

Definition 1.1. ¹ Let X be a nonempty set. An S-metric on X is a function that satisfies the following conditions,

1.

2. if and if x=y=z,

3. for each

The pair (X, S) is called S-metric space.

Further the definition of S-metric space is generated by extending to n-tuple by Abbas et.al. ². The extended structure is called A-metric. The definition of A-metric space is given as follows.

Definition 1.2. ² Let X be a nonempty set. A function is called an A-metric on X if for any the following conditions hold:

(A1)

(A2) if and only

(A3)

The pair (X, A) is called an A-metric space.

In another achievement towards the generalization of metric space Hussain et. al. ¹⁵ introduced the concept of parametric metric space. They proved some fixed point theorems in complete parametric metric space. The concept of parametric metric space is further extended to parametric S-metric space by Nihal et. al. ¹⁶. Using some expensive mappings, they proved a fixed point theorem on a parametric S-metric space. They also gave examples of parametric S-metric which is not generated by any parametric metric space. Hussain et. al. ¹⁷ also introduced the concept of parametric b-metric space and investigates the existence of fixed points under various contractive conditions in this space. Reshuni et. al. ¹⁸ further established some fixed point, common fixed and coincidence point theorems for expansive type mappings in parametric metric space and parametric b-metric spaces. We recall following definitions etc.

Definition 1.3. ¹⁵ Let X be a nonempty set and let be a function. P is called a parametric metric on X if,

(P1) P (a, b, t) = 0 if and only if a = b,

(P2) P (a, b, t) = P (b, a, t) ,

(P3) P (a, b, t) ≤ P (a, x, t) + P (x, b, t),

for each a, b, x X and all t > 0.

The pair (X, P) is called a parametric metric space.

Definition 1.4. ¹⁵ Let (X, P) be a parametric metric space and let be a sequence in X:

(1) converges to x if and only if there exists such that for all and t > 0; that is, It is denoted by

(2) is called a Cauchy sequence if, for all t > 0,

(3) (X, P) is called complete if every Cauchy sequence is convergent.

Definition 1.5. ¹⁶ Let X be a nonempty set and let be a function. is called a parametric S-metric on X if,

(PS1) if and only if ,

(PS2) for each and all t > 0. The pair is called a parametric S-metric space.

Lemma 1.1. ¹⁶ Let be a parametric S-metric space. Then we have for each and all t > 0.

Lemma 1.2. ¹⁶ Let (X, P) be a parametric metric space and let the function be defined by for each and all t > 0. Then is a parametric S-metric and the pair is a parametric S-metric space.

Lemma 1.3. ¹⁶ Let be a parametric S-metric space. If converges to x, then x is unique.

Lemma 1.4. ¹⁶ Let be a parametric S-metric space. If converges to x, then is Cauchy.

Definition 1.6. Let be a parametric S-metric space and let be a self-mapping of X. T is said to be a continuous mapping at x in X if for any sequence in X and all t > 0 such that

Motivated by the concepts introduced by Abbas et. al. ², Hussian et. al. ¹⁵ and Nihal et. al. ¹⁶, we further investigate and extend the concept of parametric metric space and parametric S-metric to parametric A-metric space. We also give some properties of parametric A-metric space. Further, we prove some fixed point theorems for various expansive mappings in the setting of parametric A-metric space.

2. Parametric A-Metric Spaces

In this section, we introduce the notion of parametric A-metric space and give some basic properties of this space.

Definition 2.1. Let X be a nonempty set and let be a function. is called a parametric A-metric on X if,

(PA1) if and only if

(PA2)

for each and all t ≥ 0. The pair is called a parametric A-metric space. Now we give the following examples of parametric A-metric spaces.

Example 1. Let and let the function be denoted by

for each and all t > 0, where is a continuous function. Then is a parametric A-metric and the pair is a parametric A-metric space.

Example 2. Let and let the function be defined by

for each and all t > 0, where is a continuous function. Then is a parametric A-metric space.

We prove the following Lemma which can be considered as the symmetry condition in a parametric A-metric space.

Lemma 2.1. Let be a parametric A-metric space. Then we have

(1)

for each and all a > 0.

Proof. Using the condition (PA2), we obtain

(2)

(3)

From (2) and (3) we have

Lemma 2.2. Let be a parametric A-metric space. If converges to x, then x is unique.

Proof. Let and let with . Then there exists such that

for each all t > 0 and If we take then using condition (PA2) and Lemma 2.1, we have

for each Therefore and .

Lemma 2.3. Let be a parametric A-metric space. If converges to x, then is Cauchy.

Proof: By similar argument as in Lemma 2.2. One can easily follow the result.

Definition 2.2. Let be a parametric A-metric space and let be a self-mapping of X. T is said to be continuous mapping at x in X if for any sequence in X and all t > 0 such that

3. Some Fixed Point Results

In this section, we give some fixed-point results for expansive mappings in a complete parametric A-metric space.

Definition 3.1. Let be a parametric A-metric space and let T be a self-mapping of X. (AP1) There exists real numbers and such that

for each and all t > 0.

Theorem 3.1. Let be a complete parametric A-metric space and let T be a surjective self-mapping of X. If T satisfies condition (AP1), then T has a unique fixed point in X.

Proof. Using the hypothesis, it can be easily seen that T is injective. Indeed, if we take Ta = Tb, then, using condition (AP1), we get

for all t > 0 and so that is, we have a = b since. Let us denote the inverse mapping of T by F. Let and define the sequence as follows:

(4)

Suppose that for all n. Using condition (AP1) and Lemma 2.1, we have

which implies that

(5)

Clearly, we have . Hence, we have

(6)

If we put , then we get k > 1, since Repeating this process in condition (6), we find

(7)

for all t > 0.

Let with m > n ≥ 1. Using inequality (7) and condition (PA2), we have

(8)

If we take limit for , we obtain

(9)

Therefore is Cauchy. Then there exists such that

(10)

since is a complete parametric A-metric space. Using the surjectivity hypothesis, there exists a point such that Tx = y. From condition (AP1), we have

(11)

If we take limit for , we obtain

(12)

which implies that y = x and T y = y. Now we show the uniqueness of y. Let z be another fixed point of T with Using condition (AP1) and Lemma 2.1, we get

(13)

which implies that y = z, since

Consequently, T has a unique fixed point y.

We give some examples which satisfy the conditions of Theorem 3.1.

Example 3. Let be a complete A-metric space with the A-metric defined in Example 2. Let us define the self-mapping as

(14)

for all with β > 1, and the function as

(15)

for all Then T satisfies the conditions of Theorem 3.1 with and

Then T has a unique fixed point x = 0 in X.

Example 4. Let be a complete A-metric space with the A-metric defined in Example 2. Let us define the self-mapping as

(16)

for all with β > 1, and the function as

(17)

for all then T satisfies the conditions of Theorem 3.1 with

and Then T has a unique fixed point x = 0 in X.

Corollary 3.1. Let be a complete parametric A-metric space and let T be a surjective self-mapping of X. If there exist real numbers and such that

(18)

for each and all t > 0, then T has a unique fixed point in X.

If we take and and and in Theorem 3.1 and Corollary 3.1, respectively, then we obtain the following corollaries.

Corollary 3.2. Let be a complete parametric A-metric space and let T be a surjective self-mapping of X. If there exists a real number k > 1 such that

(19)

for each and all t > 0, then T has a unique fixed point in X.

Corollary 3.3. Let be a complete parametric A-metric space and let T be a surjective self-mapping of X. If there exist a positive integer m and a real number k > 1 such that

(20)

for each and all t > 0, then T has a unique fixed point in X.

Proof. From Corollary 3.2, by a similar way used in the proof of Theorem 3.1, it can be easily seen that has a unique fixed point a in X. Also we have

(21)

and so we obtain that Ta is a fixed point for . We get, Ta = a, since a is the unique fixed point.

References