Abstract
In this article, we mainly discuss the existence and uniqueness of fixed 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.
1. Introduction
We start this section by recalling the definition of Lebesgue-integrable function. Notify as a function defined as which is nonnegative, summable on each compact subset of , and such that for each
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 be a complete metric space, , and is a map. If for each where . Then, is a unique fixed point of .
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 and be a nonnegative sequence with then
Lemma 2 [5]. Let and be a nonnegative sequence with then
Further, Liu et al. [6] extended Branciari’s work by including real-valued function and improved the result of Rakotch [7].
Theorem 2 [6]. Let be a self-mapping on a complete metric space satisfying for each , s.t. and . Then for all .
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 be a self-map on a complete metric space . If for each where and a function with for all Then, has a unique fixed point in .
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 [11–17].
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.
2. 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 satisfies the condition
Example 1. Define the function Clearly, and
Theorem 4. Let be a self-mapping on a complete metric space and are such that for each where and . Then has a unique fixed point.
Proof. Set initial approximation as an any arbitrary point in . In general, construct in such that
First, we assert that .
From Equation (12), , we have
where
Now, if , then Hence, from Equation (15) and using the fact that , we arrived at a contradiction. Therefore, and so
Thus, Equation (15) implies that
Since , we have
Similarly,
Thus, a monotone decreasing sequence of nonnegative reals has obtained, and so there exists such that
Assume that Letting in Equation (15) and using Equation (20), we get , as , a contradiction, implies and hence
Lemma 1 implies
Next, we assert that sequence is Cauchy.
Assume that for an there exists subsequences and of with satisfying
, consider
where
Triangle inequality implies that
From Equation (23), .
On taking limits and using Equations (22), we get
Again, we know
Therefore, on taking and using (23) and (22), we get
Hence, from Equation (25), on taking and using Equations (23), (27), and (29), we get
Thus, on letting , Equation (24) implies that
where we arrived at a contradiction as . Therefore, sequence is Cauchy. Call a limit such that from (12)
Now, assert that is a fixed point of .
Indeed, continuity of implies that
Secondly, assume is not continuous and also let . Then, clearly . Assume that . Therefore
where
Take , we obtain
Hence from (34),
Therefore, .
For uniqueness, assume there exist a point other than s.t . Consider,
where
Since , therefore
Using the fact that and from (38), we have
This implies , and hence, fixed point of is unique. This accomplished our proof.
Theorem 5. Let a self-map on a complete metric space such that for each where and with . Then, has a unique fixed point.
Proof. Since with . Let Then from Equation (42), we have where Rest of the proof is on the same line of Theorem 4.
If we take , then we have the following two consequence results from our main theorem
Corollary 1. Let a self-map on a complete metric space such that for each where and . Then, has a unique fixed point.
Corollary 2. Let a self-map on a complete metric space such that for each where and with . Then has a unique fixed point.
Example 2. Let . Define the metric , for all . Clearly, is a metric space. Define a function as . Also defined as and is defined by
Figure 1 showing the plot of inequality (12) satisfying the Example 2. Thus, all the conditions of Theorem 4 are satisfied. Clearly, is a fixed-point of .
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 , we obtain the result of 1.2 (result of Liu et al. [10]).
Remark 4. If we take in Corollary 2, we deduc the result of Geraghty [18].
3. 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.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflicts of interest.