1. Introduction
The theory of fixed point is a very active area of research despite having a history of more than hundred years. The strength of fixed point theory lies in its application, which is spread throughout the existing literature fixed point theory. In the field of metric fixed point theory, the first important and significant result was proved by Banach [
1] in 1922. The celebrated Banach contraction principle has been extended and generalized in numerous different directions (see [
2,
3,
4,
5,
6,
7,
8,
9,
10,
11]). To enhance the domain of applicability, I.A. Bakhtin [
2], S. Czerwik [
5], introduced the concept of
b-metric space as a noted improvement of metric spaces and proved fixed point results as an analogue of Banach contraction principle. In the recent past, several research articles dealing with the fixed point theory for single-valued and multivalued mappings in
b-metric spaces and by now there exists a considerable literature in such spaces (see [
12,
13,
14]). On the other hand, with a similar quest, Matthews [
3] employed another way to enlarge the class of metric spaces by introducing the notion of partial metric spaces and established an analogue of Banach contraction principle in such spaces. Thereafter, several metrical fixed point results were extended to partial metric spaces that were essentially inspired by Matthews (see [
15,
16,
17]). Motivated by these two ideas of
b-metric spaces and partial metric spaces, Shukla [
18] introduced the notion of partial
b-metric spaces that is a genuinely sharper version of both
b-metric spaces and partial metric spaces and utilize the same to prove fixed point results in such spaces. Later on, many researchers proved some existence and uniqueness results on a fixed point in partial
b-metric spaces (see [
19,
20,
21]).
In 2014, Ma et al. [
22] established the notion of
-algebra valued metric spaces (in short
-avMS) by replacing the range set
with an unital
-algebra, which is more general class than the class of metric spaces and utilized the same to prove some fixed point results is such spaces. In 2015, Ma et al. [
23] introduce the notion of
-algebra valued
b-metric spaces as a generalization of
-avMS and proved some fixed point results also used their results as an application for an integral type operator. Very soon, Chandok [
24], generalized the class of
-avMS by introducing the class of
-algebra valued partial metric spaces and utilize the same to prove some fixed point theorems.
Inspired by foregoing observations, we enlarge the class of -avbMS and -avPMS by introducing the class of -avPbMS and utilize the same to prove fixed point result. We also furnish some examples which demonstrate the utility of our results. Moreover, we apply our main result to examine the existence and uniqueness of a solution for the system of integral type operators.
This paper consists of five sections, wherein
Section 1 begins with an introduction. In
Section 2, we first recall some related definitions and remarks thereafter we introduce the notion of
-algebra valued partial
b-metric space and discuss its related properties. In
Section 3, we define the contraction condition in the setting of
-algebra valued partial
b-metric space thereafter we prove fixed point result besides giving an example in support of our main result and give two corollaries. In
Section 4, we apply our main result to examine the existence and uniqueness of a solution for the system of Fredholm integral equation and in the last section, we accomplish the conclusion part.
2. Preliminaries
Throughout the paper, we denote by an unital ( unity element I) -algebra with linear involution ∗, such that, for all , , and . A positive element is denoted by , where is a zero element in . If and . The partial ordering on can be defined as follows: if and only if . The pair is said to be an unital ∗-algebra, if it contains the unity element I. A unital ∗-algebra is called a Banach ∗-algebra, if it satisfies along with a complete sub-multiplicative norm. A Banach ∗-algebra satisfying , for all is called a -algebra.
The following definition was introduced by Ma et al. [
22]:
Definition 1. Let. A mappingis called a-av metric on A, if it satisfies the following for all:
- (i)
andiff;
- (ii)
;
- (iii)
.
The tripletis called a-avMS.
In 2015, again Ma et al. [
23] introduced the notion of
-av
b-metric space, as follows:
Definition 2. Letandsuch that. A mappingis called a-av b-metric on A, if it satisfies the following for all:
- (i)
andiff;
- (ii)
; and,
- (iii)
.
The tripletis called a-avbMS.
Remark 1. Clearly, if, then a-avbMS reduces to a-avMS.
Now, we recall the definition of
-algebra valued partial metric space introduced by Chandok et al. [
24].
Definition 3. Let. A mappingis called a-av partial metric on A, if it satisfies the following for all:
- (i)
and;
- (ii)
;
- (iii)
; and,
- (iv)
.
The tripletis called a-avPMS.
Remark 2. Obviously, iffor allthenis a-avMS.
Now, we define -algebra valued partial b-metric space (in short -avPbMS), as follows:
Definition 4. Letandsuch that. A mappingis called a-av partial b-metric on A, if it satisfies the following for all:
- (i)
and;
- (ii)
;
- (iii)
;
- (iv)
.
The tripletis called a-avPbMS.
Observe that, a -avPbMS is a generalization of both -avbMS as well as -avPMS. Obviously, every -avbMS is a -avPbMS with zero self distance and every -avPMS is a -avPbMS with , but converse is not true in general.
Example 1. Letand, the class of bounded and linear operators on a Hilbert space. Defineby (for all):whereand. Then,is a-avPbMS with coefficient. However, it is easy to see thatis neither a-avbMS nor-avPMS. To substantiate the claim, for any non-zero element, we have Therefore,is not a-avbMS. Furthermore, forand, we obtain Therefore, d is not-avPMS on A.
Example 2. Letand. Defineby (for alland)
:where,and. Observe that, d is-avPbM andis a-avPbMS with coefficient. Example 3. Letbe a-avPMS anda-avbMS with coefficienton A. Define a mappingby (for all): Subsequently, d is a-avPbM andis a-avPbMS.
Proof. It is easy to verify that the conditions
of Definition 4 are satisfied. To verify condition
of Definition 4, we have (for all
)
Therefore, d satisfies all the conditions of Definition 4. Hence, is a -avPbMS. □
Let
be a
-avPbMS. Afterwards, open ball of center
and radius
is defined by:
Similarly, the closed ball with center
and radius
is defined by:
The family of open balls (for all
and
)
forms a basis of some topology
on
A.
Lemma 1. Letbe a topological space and. If f is continuous, then every sequence, such thatimplies. The converse holds if A is metrizable.
Definition 5. A sequenceinis called convergent (with respect to) to a point, if for given, there existssuch that, for all. We denote it by Definition 6. A sequenceinis called Cauchy (with respect to), ifexists and it is finite.
Definition 7. The tripletis called complete-avPbMS if every Cauchy sequence in A is convergent to some point a in A such that The following example shows that the limit of convergence in -avPbMS may or may not be unique.
Example 4. Letand. Defineby (for alland):where,and. Then d is-avPbM andis a-avPbMS with coefficient. Now, we construct a constant sequencein A by. Choose,, such that. Subsequently, we have Therefore,, for all. Hence, the limit of convergence in-avPbMS may not be unique.
3. Fixed Point Results
The following definition is utilized in our results:
Definition 8. Letbe a-avPbMS. A mappingis said to be-contraction if there existswithsuch that Our main result runs, as follows:
Theorem 1. Letbe a complete-avPbMS andbe a-contraction. Then f has a unique fixed pointsuch that.
Proof. Choose
for constructing an iterative sequence
by:
We denote
. Now, we assert that
. On setting
and
in (1), we get
Because
, we have
For any
, we have
Thus,
is a Cauchy sequence in
A. Now, by the completeness of
A, there exists
such that
By employing (
3), we have
Now, we will show that
a is a fixed point
f. For any
, we have
Therefore,
a is a fixed point of
f. To show the uniqueness of the fixed point, suppose
, such that
Then, by the definition of
-contraction, we have
so that
a contradiction. Hence,
that is,
f has a unique fixed point. Now, to show that
. Suppose on contrary that
. Subsequently, we have
a contradiction. Therefore,
. This completes the proof. □
To exhibit the utility of Theorem 1, we give the following example.
Example 5. Let, and. Defineby:where. Then,is a complete-avPbMS. Define a mapby: Observe that,, (for all) satisfies Thus, all of the hypothesis of Theorem 1 are satisfied andis unique fixed point of f.
In Theorem 1, by setting
with zero self distance, which is,
for all
, we obtain the result due to Ma et al. [
22].
Corollary 1. Letbe a complete-avMS andbe a-contraction. Afterwards, f has a unique fixed point.
In Theorem 1, by setting
for all
, we obtain the result due to Ma et al. [
23].
Corollary 2. Letbe a complete-avPbMS andbe a-contraction. Afterwards, f has a unique fixed point, such that.