NEW FIXED POINT THEOREMS FOR MODIFIED GENERALIZED RATIONAL − −GERAGHTY CONTRACTION TYPE MAPS

Copyright © 2020 the author(s). This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract: In this paper, we introduce the notion of modified generalized rational α-ψ-Geraghty contraction type maps in the context of metric space and establish some fixed point theorems for such maps. This new contraction map is motivated by the different Geraghty contraction type maps introduced by many authors over the years. Examples are also given to illustrate the validity of our results.


INTRODUCTION
The celebrated Banach Contraction Principle is one of the most important and most used fixed point results in all analysis.Therefore this result has been generalized in different directions by various researchers ever since.In 1973, Geraghty [4] generalized the Banach contraction principle in the setting of a complete metric space by considering an auxiliary function.This remarkable result of Geraghty was further generalized and improved upon by the works of many authors namely Amini-Harandi & Emami [1], Caballero et al. [2] and Gordji et al. [5] etc.In 2012, Samet et al. [16] defined the notion of α-ψ-contractive mappings and obtained remarkable fixed point results.Then, Karapinar & Samet [8] introduced the concept of generalized α-ψ-contractive mappings and obtained fixed point results for such mappings.Salimi et al. [15] again modified the notions of α-ψ-contractive and α-admissible mappings and established fixed point results for such mappings.In 2013, Cho et al. [3] defined the concept of α-Geraghty contraction type maps in the setting of a metric space and proved the existence and uniqueness of a fixed point of such maps.Erdal Karapinar [9] introduced the concept of α-ψ-Geraghty contraction type maps and proved fixed point results generalizing the results obtained by Cho et al. [3].In 2014, Popescu [14] generalized the results of Cho et al. [3] and gave other conditions for the existence and uniqueness of a fixed point of α-Geraghty contraction type maps.Then, K. Anthony Singh [6] introduced extended generalized −− Geraghty contraction type maps and proved some fixed point results generalizing the results of Popescu [14].In 2017, Muhammad Arshad & Aftab Hussain [13] defined generalized rational  − Geraghty contraction type maps and proved some fixed point results.Again very recently, K. Anthony Singh et al. [7] introduced the notion of generalized rational  −− Geraghty contraction type maps and proved some fixed point results.
In this paper, motivated by the different Geraghty contraction type maps introduced by many authors and the works of Popescu [14], Salimi et al. [15], Muhammad Arshad & Aftab Hussain [13], K. Anthony Singh et al. [7], we define modified generalized rational  −− Geraghty contraction type maps in the setting of metric space and obtain the existence and uniqueness of a fixed point of such maps.We also give examples to illustrate the validity of our results.

PRELIMINARIES
In this section, we recall some basic definitions and related results on the topic in the literature.
Let ℱ be the family of all functions By using such a map, Geraghty proved the following interesting result.Theorem 2.1.[4] Let (X,d) be a complete metric space and let T be a mapping on X. Suppose there exists   ℱ such that for all x,y ∈ X, ( Popescu [14] introduced the following two new concepts. Definition 2.2.[14] Let : T X X → be a map and : X X R  → be a function.Then T is said to be α-orbital admissible if ( ) Erdal Karapinar [9] defined the following class of auxiliary functions.
Definition 2.6.[9] Let (X,d) be a metric space and : XX  →be a function.A map : T X X → is called a generalized α-ψ-Geraghty contraction type mapping if there exists  ℱ such that for all , x y X  , (   [6] further introduced the following contraction and proved some fixed point results generalising the results of Popescu [14].Definition 2.8.[6] Let (X,d) be a metric space and : XX  →be a function.A map : T X X → is called an extended generalized α-ψ-Geraghty contraction type map if there exists  ℱ such that for all , Let Ω be the family of all functions  )   : 0, 0,1  → which satisfy the following conditions

x Ty d y Tx M x y d x y d x Tx d y Ty
(1) ( ) Remark 2.9.Here instead of the family ℱ we are introducing a slightly extended family Ω.K. Anthony Singh et al. [7] further introduced the following contraction and proved some fixed point results.

Definition 2.10. [7] Let ( )
, Xd be a metric space and let : XX  → be a function.Then the mapping : T X X → is called a generalized rational  -ψ-Geraghty contraction type mapping if there exists   such that for all , x y X  , If we take ( ) tt  = in Definition 2.10., then T can be called generalized rational  -Geraghty contraction type mapping.

MAIN RESULTS
We now state and prove our main results.
First we introduce some new definitions and concepts and then define modified generalized rational α-ψ-Geraghty contraction type map.These are motivated by the works of Popescu [14], Salimi et al. [15], Muhammad Arshad & Aftab Hussain [13], K. Anthony Singh et al. [7] and the different types of Geraghty contraction type maps introduced by various authors over the years.
Definition 3.1.Let : T X X → be a map and ,: XX  → be two functions.Then T is said to be α-orbital admissible with respect to  if ( ) ( ) , then T becomes an α-orbital admissible mapping and if ( ) T is called an  -orbital subadmissible mapping.Definition 3.2.Let : T X X → be a map and ,: XX  → be two functions.Then T is said to be triangular α-orbital admissible with respect to  if T is α-orbital admissible with respect to  and ( ) ( ) , , , , Suppose that the following conditions are satisfied T is a triangular  -orbital admissible mapping with respect to  , (2) there exists 1 xX  such that ( ) ( ) T is continuous.Then T has a fixed point xX   , and   xX  be such that ( ) ( ) We construct a sequence of points   x is clearly a fixed point of T and the proof is complete.Hence, we suppose that ,, , Here we have ) ) ,, from (1) and the definition of θ, we have , , , which is a contradiction.
Therefore, we have ( ) ( ) = .We show that r = 0.And we suppose on the contrary that r > 0.
Then, we have (2) Now we show that the sequence   By ( 3) and ( 4), we have , Then in view of ( 2) and ( 5), we have Again, we have Taking limit as k → and using ( 2) and ( 6), we obtain ( ) By Lemma 3.3., we get ( ) ( ) . Therefore, we have ) And we see that ( ) and so ** .
x Tx = Hence x * is a fixed point of

T.
For the uniqueness of a fixed point of the mapping T, we consider the following hypothesis: (G) For any two fixed points x and y of T, there exists zX such that ( ) ( ) .
Here we have  Note that condition (ii) can be replaced by the weaker condition ' T is triangular -orbital admissible'.Theorem 3.10.[7] Adding condition (G) to the hypotheses of Theorem 3.9., we obtain that x * is the unique fixed point of T.
Also let a function : XX  → be defined by ( ) Then we show that T is triangular  -orbital subadmissible mapping.


Then T has a unique fixed point xX   and   n Tx converges to x  for each xX  .

1 n
Tx converges to x  .Proof : Let 1 Now by taking limit n →, we have

nxm
is a Cauchy sequence.Let us suppose on the contrary that   n x is not a Cauchy sequence.Then there exists 0   such that, for all positive integers k be the smallest number satisfying the conditions above.Then we have ( ) 6) and taking limit as k → in the above inequality, we obtain 4., we deduce that the sequence   n Tz converges to a fixed point zX   .Then taking limit n → ∞ in the above equality, we get (

1 n
we must have .zx  = Similarly, we get .zy  = Thus we have .yx  = Hence x  is the unique fixed point of T. By taking ( ) ,1 xy  = in Theorem 3.4., we get the following result.Theorem 3.6.Let ( ) , Xd be a complete metric space, : XX  → be a function and let : T X X → be a map.Assume that (iv) T is continuous.Then T has a fixed point xX   and   Tx converges to x  .

7 .,
Example 3.11.Let X = with the metric d defined by ( ) Xd is a complete metric space.And let ( ) Then   .Also let the function  )  )


Then we show that T is triangular α-orbital admissible mapping.Thus T is triangular α-orbital admissible mapping, which is condition (1) of Theorem 3.6.Also, condition (3) of Theorem 3.6.is satisfied because T is continuous.Thus all the conditions of Theorem 3.6.are satisfied and T has a unique fixed point 0.
Now we give our Theorem below.The contraction T defined in the Theorem can be called modified generalized rational α-ψ-Geraghty contraction type map.
 and T is triangular  -orbital admissible mapping with respect to  .
Adding condition (G) to the hypotheses of Theorem 3.4., we obtain that x * is the unique fixed point of T. Due to Theorem 3.4., we obtain that x * X is a fixed point of T. Let y * X be another fixed point of T. Then by hypothesis (G), there exists zX such that ( ) ( )