SOME APPROXIMATE FIXED POINT RESULTS FOR VARIOUS CONTRACTION TYPE MAPPINGS

: In this text, we investigate approximate ﬁxed point results for various contraction mappings in a metric space. This manuscript’s intention is to demonstrate ε -ﬁxed point results on metric spaces (not necessarily complete) by using contraction mappings such as B -contraction, convex contraction, and so on. The ﬁndings are extensions of several others, including the Kannan-type mapping, the Chatterjea-type mapping, and the S. A. M. Mohsenalhosseini-type mapping, etc. A few examples are included to illustrate the results. Finally, we discuss some applications of approximate ﬁxed point results in the ﬁeld of applied mathematics rigorously.


INTRODUCTION
Researchers from various fields have contributed to the development of science and technology by using fixed point theory. It is common knowledge that large scale problems involving fixed point theory can be solved quickly. As a result, many researchers have focused on creating fixed point theory approaches in recent years and have presented numerous effective techniques for locating fixed points in complex issues. It also allows for nonlinear analysis. Furthermore, fixed point theory is used to address a variety of problems in applied mathematics. These are now crucial in many mathematics related fields and their applications, notably in economics, astronomy, dynamical systems, decision theory, and parameter estimation. In the early 1900s, the mathematician Brouwer [6], called the father of fixed point theory, proved the fixed point theorem for continuous mapping on finite dimensional spaces. In 1922, Banach [1] proved the famous Banach contraction principle. Then, using contraction mappings on metric spaces, numerous experts expanded the Banach principle and provided numerous theorems. A fixed point theorem for operators that are not required to be continuous was established by Kannan (see [17], [18]). Chatterjea [7] has researched a similar kind of contraction condition and proposed his fixed point theorem. Zamfirescu [30] combined the above two operators and found a new contraction operator for proving the fixed point theorem.Ćirić [8] invented generalised contractions and found a fixed point theorem by using them. Following that, researchers Hardy and Rogers [12] proved a new fixed point theorem using a new contraction operator. In [26], Reich introduced his contraction operator and proposed fixed point findings. Similarly, Bianchini [5] proved fixed point theorem by using another contraction mapping. The unique fixed point theorem for weakly B-contraction mapping was proved by Marudai et al. in [21].
Let us consider a selfmap M : T → T . A fixed point is a point (say, t 0 ) which is equal to Mt 0 .
That is, d(Mt 0 ,t 0 ) = 0. Assume that a mapping has a fixed point, t 0 . In which case the point (t 0 ,t 0 ) is located on its diagram. The conditions for fixed points existence are very strict. As a result, there is no assurance that fixed points will always exist. In the absence of exact fixed points, approximate fixed points may be used because the fixed point methods have overly strict limitations. This is the main reason to find an approximate fixed point (ε-fixed point). One can see, the point Mt 0 is "very near" to the point t 0 . An approximate fixed point is a point that is nearly located at its respective fixed point. Here, the distance is less than ε, i.e., d(Mt 0 ,t 0 ) < ε.
Initially, In 2003, Tijs at el. [29] proved the existence of approximate fixed points turns out to be still guaranteed under various weakened versions of the well-known fixed point theorems of Brouwer, Kakutani, and Banach (refer, Theorems 2.1 and 2.2). Moreover, he proved approximate fixed point results for contraction maps and nonexpansive maps in Theorems 3.1 and 4.1, respectively. After that, the author Berinde [2] proved approximate fixed point results (Qualitative theorems) by using various operators (Kannan, Chatterjea, Zamfirescu, and weak contractions) in metric spaces (not necessarily complete). Further, he found the diameter of the approximate fixed points (Quantitative theorems) by using two main lemmas (see also [3], [4]).
Dey and Saha [11] extended these results, and they found the diameter of the approximate fixed point for the Reich operator tends to zero when ε approaches zero. In the same manner, S. A. M.
Mohsenialhosseini (see, [23], [24], and [25]) derived some new approximate fixed point results for cyclical contraction mappings. Also, he extended these results to a family of contraction mappings and found a common fixed point for the Mohseni-Saheli contraction mapping.
The first scholars to investigate a generalisation of the Banach fixed point theorem while simultaneously using a contraction condition of the rational type were Dass and Gupta [9]. Jaggi [16], used a contraction condition of the rational type to prove a fixed point theorem in complete metric spaces. Later, Harjani et al. [13] extended Jaggi's findings to partially ordered metric spaces. Rational contraction conditions have been heavily employed in both the fixed point and common fixed point locations. Also, the authours Tijani et al. [28] proved approximate fixed point results using rational operators: Let (T, d) be a metric space and M : T → T be a selfmap.
Moreover, the author Istratescu (refer [14], [15], and [22]) proved fixed point theorems by using various convex contraction mappings. The scope of this paper is to establish approximate fixed point results in a metric space (not necessarily complete) by using contraction mappings such as B-contraction [21], convex contraction [14], etc. Furthermore, many articles provided (see [10], [19], and [27]) some definitions, which helped more to find approximate fixed point This manuscripts remaining portions are displayed as follows. In Section 2, we recall the notations, basic notions, and essential definitions needed throughout the paper. In Section 3, we prove the main concept related to approximate fixed point results using various contraction, rational contraction, and convex contraction mappings. In Section 4, we go one step further and find the applications of approximate fixed point results in a wide range of applied mathematical topics. Finally, In Section 5, we reach a conclusion.

PRELIMINARIES
In this section, some notations, basic notions, essential definitions and lemmas from earlier works are recalled. These are then employed throughout the remainder of the main results of the manuscript.  Remark 2.6. [2] In the following, by D(K) for a set K = / 0 we will understand the diameter of the set K, i.e., Definition 2.7.
[2] Let (T, d) be a metric space, M : T → T a operator and ε > 0. We define the diameter of the set F ε (M), i.e., Let (T, d) be a metric space, M : T → T an operator and ε > 0. We assume that: Then: , for all t, r ∈ T.
Definition 2.11. [14] A continuous mapping M : T → T is said to be a convex contraction of order 2 if there exists constants l 0 , l 1 ∈ [0, 1) such that the following conditions hold: Definition 2.12. [14] Let M : T → T be a continuous map. Then M is said to be n-convex contraction if there exists l 0 , l 1 , ..., l n−1 ∈ (0, 1) such that the following conditions hold:

MAIN RESULTS
In this section, we prove some approximate fixed point theorems for various contraction mappings on metric spaces, including the B-contraction, the Bianchini contraction, and the convex contraction mappings and their related consequences. Proof. Fix t 0 ∈ T and a sequence {t n } is defined by t n+1 = Mt n , for all n ≥ 0. Which implies that {t n } is a Cauchy sequence. That is, for every ε > 0, there exists k 0 ∈ N such that for every Therefore, t n ∈ F ε (M) = / 0, for all ε > 0. Hence, M has an approximate fixed point (ε-fixed point). Hence, M has an ε-fixed point. Then M has an ε-fixed point and  Consider, So, for all θ > 0, there exists φ (θ ) = γ > 0, such that    , for all t, r ∈ T.
Prove that M has an ε-fixed point and Proof. Let ε > 0, t 0 ∈ M and a sequence {t n } is defined by t n+1 = Mt n , for all n ≥ 0.
Remark 3.11. We have proved many approximate fixed point results by using various operators on metric spaces (not necessarily complete). In the following table, one can see the diameters of various contraction operators and the diameters of a few rational contraction operators.

APPLICATIONS
Approximate fixed point theory covers a wide range of applications in applied mathematics, particularly differential geometry, numerical analysis, and so on. By reading [20] and its references therein, one can find a variety of applications involving approximate fixed point results in the field of applied mathematics. The two examples below demonstrate how to apply approximate fixed point findings in differential equations.

CONCLUSION
This work provides a series of contraction and rational type contraction mappings to demonstrate several approximate fixed point theorems on metric spaces (not necessarily complete). It is essential to note that all of the conclusions made in the current paper generate better constrained approximations of fixed points, mostly in minimising condition ε −→ 0. In order to confirm the presence of an approximate fixed points, alternative discoveries presented in the later can be demonstrated in a lower environment. Thus, the concept of an approximate fixed points (ε-fixed points) is just as significant as the concept of fixed points.