A Relation-Theoretic Metrical Fixed Point Theorem for Rational Type Contraction Mapping with an Application

: In this article, we discuss the relation theoretic aspect of rational type contractive mapping to obtain ﬁxed point results in a complete metric space under arbitrary binary relation. Furthermore, we provide an application to ﬁnd a solution to a non-linear integral equation.


Introduction
In 1922, the first prosperous result was postulated by S. Banach [1] in the fixed point theory for contractive mapping.For its modesty, his work functioned as a schematic research tool in a different branch of mathematics.This theorem went in a different direction to verify its effectiveness.Such as (i) Enlarging the ambient space; (ii) Improving the underlying contraction condition; (iii) Weakening the involved metrical notions.
Among the several extensions of the Banach contraction principle to various spaces, some are rectangular metric space, generalized metric space, partial metric space, b-metric space, partial b-metric space, symmetric space and quasi metric space.Partial metric space was introduced by Matthews [2] in 1994.Nowadays, there are many fixed point theories in Partial metric space.
Several researchers stated various contraction conditions [3][4][5][6][7][8] for the fixed point theorem.Inspired by Turinici's [9] work, Ran-Reurings in 2004 formulate the result that there will be a fixed point of self-mappings that is applied only for those points which are comparable to each other by an order relation in partial metric space.Later, the work was extended by J. J. Nieto and R. Rodríguez-López [10].In 1975, Dass and Gupta [11], came up with a new contractive condition termed as a rational type contraction.Later, Canbrera et al. [12] used the result of Dass and Gupta [11] in 2013 to obtain the fixed point results in partial ordered metric space.
Alternatively, Alam and Imdad [13] established a profound generalization of the Banach contraction principle with an amorphous binary relation.With this structure, various relation-theoretic results were proposed in different aspects of the binary relation or contractive condition.
There are too many applications of fixed point theory in the field of ordinary differential equations, systems of matrix equation, integral equations, game theory, economics, optimization models and numerical models in statistics.Moreover, for the multivalued maps in the equilibrium in the duopoly markets and in aquatic ecosystem there are also too many applications.In an ordinary differential equation, the application provided by J. J. Nieto and R. Rodríguez-López [10] and the system of matrix equations by Ran and Reurings [14], the fixed point for iteration to find optimal solution in statistics [15], for the stability problem in Intuitionistics Fuzzy Banach Space [16], and many more such as [17].
This article intends to establish some fixed point theorems under contractive mapping over a complete metric space.Ultimately, an example is provided to establish the result for our assumptions.Furthermore, we provide an application [18] in a non-linear integral equation to obtain a fixed point.

Preliminaries
In this section, we present some basic definitions which will be required in proving our main results.We denote N ∪ {0} as N 0 throughout the paper.Definition 1 ([11]).Let (W, d) be a complete metric space and T a self-mapping on W.Then, T is said to be a rational type contraction if there exist Definition 5 ([19]).Let W be a nonempty set and a binary relation on W.
(i) The dual relation, transpose or inverse of , signified by −1 is interpreted by, (ii) Symmetric closure s of , is defined to be the set ∪ −1 (i.e., s = ∪ −1 ).

Proposition 1 ([13]
).For a binary relation defined on a nonempty set W, Definition 6 ([13]).Consider a nonempty set W and let be a binary relation on W. A sequence Definition 7 ([13]).For a nonempty set W with a self-mapping T on it.Any binary relation on

Definition 8 ([25]
).Let (W, d) be a metric space and a binary relation on W.Then, (W, d) is -complete if every -preserving Cauchy sequence in W converges.
It is obvious that every complete metric space is -complete with respect to a binary relation but not conversely.For instance, Suppose W = (−2, 2] together with the usual metric d.Notice that (W, d) is not complete.Now endow W with the following relation: Then, (W, d) is a -complete metric space.Definition 9 ([22]).Let W be a nonempty set endowed with a binary relation .A subset D of W is called -directed if for each µ, ν ∈ D, there exists κ ∈ W such that (µ, κ) ∈ and (ν, κ) ∈ .
Definition 11 ([13]).Let (W, d) be a metric space.A binary relation on W is termed as d-self-closed if whenever {µ n } is an -preserving sequence and The proof is followed by the symmetrycity of the metric d.

Main Result
In this fragment, we will introduce the fixed point theorem under rational contraction in the relation theoretic sense.Theorem 1.Consider (W, d) as a metric space together with a binary relation and a self-mapping T on it.Assume that the following conditions hold: Then T has a fixed point.
If Tµ 0 = µ 0 then by condition is T-closed we obtain Then, by an induction process, we will obtain Denote γ = δ 2 1−δ 1 < 1, then Equation (1) can be rewritten as Next, for {µ n } to be a Cauchy sequence, let m > n then For m, n → ∞ and as γ < 1, then we obtain lim n,m→∞ d(µ n , µ m ) = 0. So, we have proved that {µ n } is a Cauchy sequence.Since the space (W, d) is a -complete metric space, then there always exists µ ∈ W such that µ n → µ.
Then by continuity of T, we have So, µ is a fixed point of T.
Theorem 2. If in addition to Theorem 1 we have the condition: (vi) T(W) is s -directed.Then T has a unique fixed point.
Proof.Let us suppose that µ, ν are two fixed points, i.e., Tµ = µ and Tν = ν then we have the two cases, Case I: if (µ, ν) ∈ then Case II: if (µ, ν) / ∈ then by T(W) is s -directed then there exists κ ∈ W such that (µ, κ) ∈ and (κ, ν) ∈ .Since is T-closed T n κ will be related to T n µ, i.e., (T n κ, T n µ = µ) ∈ for any n ∈ N 0 .Then, by contractive condition (v) of Theorem 1, for any n ∈ N 0 , we have Then by mathematical induction, we obtain Since δ 2 < 1 then lim n→∞ d(T n κ, µ) = 0, which provides us lim n→∞ T n κ = µ.In a similar fashion we also obtain lim n→∞ T n κ = ν.Then, by the unity of limits we obtain µ = ν.So our supposition that µ and ν are two different fixed points is wrong.Hence, the mapping T has a unique fixed point.
Corollary 1.If we substitute δ 2 = 0 into Theorems 1 and 2, we have the following fixed point theorem.Consider (W, d) as a metric space together with a binary relation and a self mapping T on it.Assume that the following conditions holds: Then, T has a unique fixed point.
Example 1.Consider the metric space W = (−1, 1] with the usual metric d and a binary relation = {(µ, ν) ∈ W 2 : ν > µ ≥ 0} together with a mapping T : W → W defined by It is clear that is T-closed and T is not a continuous function.Now, for (µ, ν) ∈ Then T has fixed point µ = 0. Notice that condition (v) of Theorem 1 does not hold for the whole space (for example, take µ = 1 and ν = 0).Therefore, this example cannot be solved by the existing results, which establishes the importance of our result.

Proof. For the proof, let us define a binary relation on
By assumption (iv) we have µ So, the contractive condition also satisfied.

Conclusions
In this article, we have established the relation theoretical fixed point results for the rational type contraction.One may observe that, for the uniqueness of the fixed point, the s -directed condition can be replaced by other conditions.Here, we also included some contractions that can be obtained on restriction to the rational contraction.Our results deduce some well known fixed point results if the binary relation is universal.The example we provided is unique in that it will satisfy all the relational elements but fails for many elements outside of the relation.Moreover, we provide an abstract version of an application to a non-linear integral equation.Lastly, we include an example that guarantees the existence of such a non-linear integral equation.
Definition 2([19]).Let W be a nonempty set.A subset of W 2 is called a binary relation on W. The subsets W 2 and ∅ of W 2 are in trivial relation.Consider a binary relation on a nonempty set W. For µ, ν ∈ W, one may say that µ and ν are -comparative if either (µ, ν) ∈ or (ν, µ) ∈ .We symbolize it with [µ, ν] ∈ .