1. Introduction
In contemporary topological theory, iterative techniques are broadly used to find roots of linear and nonlinear systems of equations, differential equations and integral equations. Banach (1922)) introduced a well-liked iterative method. Several authors have extended, improved and generalized Banach's theorem in different ways (Alamgir et al., 2020; Bondar, 2011; Gupta et al., 2015; Gupta et al., 2020; Gupta & Verma, 2020; Jaggi,1976; Samet, & Yazidi, 2011).
Popa (1983) generalized the result of Banach through Hausdorff topological spaces and proved some unique fixed-point theorems.
Theorem 1.1 (Popa, 1983) - Let
where
Remark 1.1: - Any metric space is a Hausdorff-metric space, or easily, Hausdorff spaces in the induced topology.
Jungck (1976) proved a common fixed-point theorem for commuting maps so that one of them is continuous. Sessa (1982) generalized the concept of commuting maps to weakly commutating pairs of self-mappings. Furthermore, Jungck (1986) generalized this idea; first, to compatible mappings and then to weakly compatible mappings (Jungck, 1996).
Results by Banach further extended in several directions for self and pairs of mappings. Some of the latest results on fixed- and common-fixed points can be found in (Gupta & Verma, 2020; Shahi et al., 2014).
Branciari (2002) introduced a new definition for the Lebesgue-integrable function and proved a fixed-point theorem satisfying the contractive condition of an integral type as an analog of the Banach contraction principal.
Definition 1.1 (Branciari, 2002) - A function defined as
Branciari (2002) result was further studied by many other authors and lot of generalizations have been done (Gupta et al., 2012; Gupta & Mani, 2013) and the references there in Samet and Yazidi (2011) gave an extension of Branciari (2002) result by using rational inequality in Hausdorff topological spaces and proved the following theorem:
Theorem 1.2 (Samet & Yazidi, 2011)- Let X be a Hausdorff space and
Let F be self maps of X satisfying the contractive condition so that for each
where
Our main results are the following theorems.
2. Main Results
Theorem 2.1:- Let
Where
Proof:- Let us choose
First, let us suppose that there exists
Second, assume that
where, from Eq. 3:
Let us suppose
Also, if
With the use of Eq. 6 and Eq. 7, and by repeating the above process up to n times, we get:
Thus, we obtain a monotone sequence of non-negative real numbers, which must converge with all its subsequence to some real no
Next, let us claim that z is a fixed point of F.
To prove this, let us suppose z is not a fixed point of F.
The continuity of F and H implies:
On using Eq. 3:
Hence, from Eq. 8, we get :
This is a contradiction to our assumption; thus Fz = z,. That is, 𝑧is a fixed point of F. This completes the proof of Theorem 2.1.
In our next result, we introduce a new contraction to establish a common fixed-point theorem for a pair of self maps in Hausdorff spaces without using the compatibility and commutative property.
Theorem 2. 2:- Let
where
α, β > 0 are constants with α + β > 1. If for some x0 ϵ X, sequence {xn} has a subsequence
Proof:- Le us choose
First, let us suppose that there exists
Second, let us assume that
Where, from Eq. 11, we have:
Let us suppose that
Also, if
Repeating the above process n times, we get
Thus we get a monotone sequence
Now, we show that z is fixed point of F and G.First, we show that z is fixed point of F.
Let us suppose
Let us consider sequence
Where, from Eq. 11:
Thus, from Eq. 15 and Eq. 16:
This is a contradiction. Thus, z is a fixed point of F. Analogously, we can show that z is fixed point of G. This completes the proof of Theorem 2.1.
In order to get the uniqueness of the fixed point for the maps (in Theorem 2.1 and Theorem 2.2), we consider the following assumption:
Theorem 2.3: If we add condition (17) to the hypothesis of Theorem 2.1, we get a unique fixed point of map F.
Proof:- We have proved that
From Eq 2:
Where,
From Eq 18 and Eq 19, we get a contradiction.
Thus z is a unique fixed point of F
Theorem 2.4:- If we add condition (17) to the hypothesis of Theorem 2.2, we get a unique common fixed point for maps F and G.
Proof:- We have proved that
From Eq. 10:
On using (17), we have:
From Eq. 20 and Eq 21, we get a contradiction. Thus, z is a unique common fixed point of 𝐹and𝐺.
Remark 1:- Note that in the above theorems (Theorem 2.1 and Theorem 2.2), the continuity of maps is necessary to get the fixed point; otherwise, the fixed point cannot be guaranteed
Remark 2:- Authors leave here an open problem for further research to get the uniqueness of fixed points in Theorem 2.1 and Theorem 2.2 without assuming condition (17).
3. Applications for the integral type contraction
In this section, we discuss the existence and uniqueness of the fixed point for integral type contractive mappings. Besides being a proper extension, results obtained here are weaker than the result obtained by Samet and Yazidi (2011)), and some other existing results.
Theorem 3.1: Let
where
α, β > 0 are constants with α + β < 1 and
Proof:- By assuming
Theorem 3.2: Let
where
α, β > 0 are constants with α + β < 1 and
Proof:- By taking in Theorem 2.2, we get the result.
4. Conclusions
In this paper, firstly, we derived a fixed-point result (Theorem 2.1) for a self map. In Theorem 2.2, we introduced a contraction to get a common fixed point for a pair of self maps without using the compatibility and commutative property of maps. Theorem 2.3 and Theorem 2.4 proved the uniqueness of the fixed point by assuming an additional assumption on the maps. Some observational remarks and an open problem are given for further research.