Fixed Point Results via Real-Valued Function Satisfying Integral Type Rational Contraction

In this article, we mainly discuss the existence and uniqueness of ﬁ xed point satisfying integral type contractions in complete metric spaces via rational expression using real-valued functions. We improve and unify many widely known results from the literature. Among these, the work of Rakotch (1962), Branciari (2002), and Liu et al. (2013) is extended. Finally, we conclude with an example presented graphically in favour of our work.


Introduction
We start this section by recalling the definition of Lebesgueintegrable function.Notify L as a function defined as which is nonnegative, summable on each compact subset of R + , and such that for each ε > 0, Branciari [1] in 2002 independently and essentially deduced the following result, as an extension of most famous problem of Banach in 1922.
Theorem 1 [1].Let ðP, dÞ be a complete metric space, 0 < s < 1, and U : P ⟶ P is a map.If for each g, h ∈ P where l ∈ L.Then, z ∈ P is a unique fixed point of U.
Rhoades [2] made a major extension in 2003 of Branciari [1] by proving a more genral result.His proof introduced a number of interesting ideas for other reseachers to study on integral type of contracions.Prior to Branciari [1] works, Kumar et al. [3] had been able to derive Jungck's [4] fixed point result in sense of integral type contractions.Mocanu and Popa [5] proposed following lemmas that are useful for deriving our main theorem.
Lemma 1 [5].Let l ∈ L and ðr s Þ s∈N be a nonnegative sequence with lim s⟶∞ r s = c then Lemma 2 [5].Let l ∈ L and ðr s Þ s∈N be a nonnegative sequence with lim s⟶∞ r s = c then Further, Liu et al. [6] extended Branciari's work by including real-valued function and improved the result of Rakotch [7].
and l ∈ L. Then Pz = z for all z ∈ U.
In addition to previous findings, Gupta et al. [8] in 2012 proposed a work for 2 compatible self-maps and derived a result satisfying integral type contraction.In contrast to Rakotch's result, further in 2013, Gupta and Mani [9] placed a rational contraction using real-valued function and established their theorem.
Theorem 3 [9].Let U be a self-map on a complete metric space ðP, dÞ.If for each g, h ∈ P where l ∈ L and a function γ : ð0,∞Þ ⟶ ½0, 1Þ with lim s⟶n sup γðsÞ < 1 for all n > 0: Then, U has a unique fixed point in P.
About the same time, Liu et al. [10] come with different approach and set up three distinct results for integral type contractions.These studies further give other aspects of integral contractions for researchers, in particular related problems on real-valued functions.Some motivated results on integral type contractions and in metric spaces are refer to see [11][12][13][14][15][16][17].
This article is devoted to state the theorem containing real-valued function and to prove the theorem satisfying integral type rational contraction.Our finding extends and generalized some renowned result.An example with graphical representation has been given in favour of our work.

Fixed Point via Rational Contraction and by
Using Real-Valued Function Geraghty [18] defined the following class of test function which is more general than the Rakotch [7].
Definition 1 [18].Define S = fγjγ: ½0,∞Þ ⟶ ½0, 1Þg satisfies the condition Example 1. Define the function Clearly, Let U be a self-mapping on a complete metric space ðP, dÞ and are such that for each g, h ∈ P where l ∈ L and γ ∈ S. Then U has a unique fixed point.
Proof.Set initial approximation g 0 ∈ P as an any arbitrary point in P. In general, construct fg i g in P such that First, we assert that lim i⟶∞ dðg i , g i+1 Þ = 0. From Equation (12), ∀i ≥ 0, we have where Abstract and Applied Analysis Hence, from Equation ( 15) and using the fact that γ ∈ S, we arrived at a contradiction.Therefore, dðg i , Thus, Equation ( 15) implies that Since γ ∈ S, we have Similarly, Thus, a monotone decreasing sequence f Ð dðg i ,g i+1 Þ 0 lðmÞ dmg of nonnegative reals has obtained, and so there exists s ≥ 0 such that Assume that s > 0: Letting i ⟶ ∞ in Equation ( 15) and using Equation (20), we get s ≤ s, as γ ∈ S, a contradiction, implies s = 0 and hence Next, we assert that sequence fg i g is Cauchy.
Assume that for an ϵ > 0, there exists subsequences fg i s g and fg w s g of fg i g with w s > i s ≥ s, s > 0 satisfying where Triangle inequality implies that From Equation (23), On taking limits and using Equations ( 22), we get Again, we know Therefore, on taking lim s⟶∞ and using ( 23) and ( 22), we get Hence, from Equation (25), on taking lim s⟶∞ and using Equations ( 23), (27), and (29), we get Thus, on letting lim s⟶∞ , Equation (24) implies that where we arrived at a contradiction as γ ∈ S. Therefore, sequence fg i g is Cauchy.Call a limit v such that from ( 12) Now, assert that v is a fixed point of U. Indeed, continuity of U implies that 3 Abstract and Applied Analysis Secondly, assume U is not continuous and let Uv ≠ v.Then, clearly dðUv, vÞ ≠ 0. Assume that dðUv, vÞ > 0. Therefore Take lim i⟶∞ , we obtain Hence from (34), Therefore, Uv = v.For uniqueness, assume there exist a point s ≠ v other than v s.t dðUs, sÞ = 0. Consider, where Since dðs, vÞ/1 + dðs, vÞ < 1, therefore Using the fact that γ ∈ S and from (38), we have This implies s = v, and hence, fixed point of U is unique.This accomplished our proof.
Theorem 5. Let a self-map U on a complete metric space ðP, dÞ such that for each g, h ∈ P l ∈ L and δ, γ ∈ S with δðmÞ + γðmÞ < 1.Then, U has a unique fixed point.
Proof.Since δ, γ ∈ S with δðmÞ + γðmÞ < 1.Let γðmÞ = max fδðmÞ, γðmÞg: Then from Equation (42), we have where Rest of the proof is on the same line of Theorem 4.
If we take lðtÞ = 1, then we have the following two consequence results from our main theorem Corollary 1.Let a self-map U on a complete metric space ðP, dÞ such that for each g, h where and γ ∈ S.Then, U has a unique fixed point.
Example 2. Let P = N. Define the metric dðg, hÞ = max fg, hg, for all g, h ∈ P. Clearly, ðP, dÞ is a metric space.Define a function U : P ⟶ P as UðgÞ = ffiffiffi g p , ∀g ∈ P. Also defined l ∈ L as lðmÞ = 2m, ∀m ∈ R + and δ : R + ⟶ ½0, 1Þ is defined by δðmÞ = 9/10 Figure 1 showing the plot of inequality (12) satisfying the Example 2. Thus, all the conditions of Theorem 4 are satisfied.Clearly, 1 ∈ P is a fixed-point of U. Remark 1. Theorem 4 and Theorem 5 are unified and extended results of Liu et al. [10] and Branciari [1].Remark 2. Corollary 2 is an extension of the result of Rakotch [7] with more general test functions.Remark 3. In Theorem 5, on letting γðtÞ = 0, we obtain the result of 1.2 (result of Liu et al. [10]).

Conclusion
We conclude this note by mentioning that our proved result is a further extension of Branciari result into other settings.Some remarks and an example are given to justify that our results are extension and generalized version of some known results of literature.

Figure 1 :
Figure 1: Graphical representation of inequality satisfying Example 2. Also showing the graph of curves y = ffiffi ffi x p and y = x with fixed point x = 1.