The metric function generalizes the concept of distance between two points and hence includes the symmetric property. The aim of this article is to introduce a new and proper extension of Kannan’s fixed point theorem to the case of multivalued maps using Wardowski’s F-contraction. We show that our result is applicable to a class of mappings where neither the multivalued version of Kannan’s theorem nor that of Wardowski’s can be applied to determine the existence of fixed points. Application of our result to the solution of integral equations has been provided. A multivalued Reich type generalized version of the result is also established.
Kannan  generalized the Banach contraction principle in the following manner which assured that even certain discontinuous functions might possess fixed points.
 Let be a complete metric space. The self-map is called a Kannan map if there is a constant such that
for all . Then Υ has a unique fixed point, where the element satisfying is called a fixed point of Υ.
Subrahmanyam  showed that Kannan’s theorem could be used to characterize metric completeness. Reich  further generalized Banach’s Contraction Principle and observed that Kannan’s theorem is a particular case of it with suitable choice of the constants.
 Consider the complete metric space . Suppose the self-map satisfies the following:
where satisfy . Then Υ admits a unique fixed point.
provides Banach contraction principle while produces Kannan’s theorem.
Wardowski  defined the concept of F-contraction as given next.
Let denote the class of all such functions satisfying the following assumptions:
(F1)F is strictly increasing, i.e., for all implies
(F2)For each sequence , if and only if
(F3)There exists such that
If is a metric space, then a mapping is said to be an contraction if there exist , , such that for all ,
Nadler  started the research on fixed points for multivalued maps with the help of Hausdorff concept, i.e., by considering the distance between two arbitrary sets in the following manner.
Let be a complete metric space (in short, MS) and let denote the class of all nonempty closed and bounded subsets of the nonempty set ℑ. Then for , define the map by
where . is called the Pompeiu-Hausdorff metric space generated by the metric .
 is said to be a fixed point of the multivalued map if . The set of all fixed points of Γ is denoted by .
In the MS , is a fixed point of Υ if and only if .
The metric function is continuous in the sense that if are two sequences in ℑ with for some , as , then as . Similarly, the function Δ is continuous because if as , then as for any .
We list the following results to be used in the sequel.
 Suppose that and . If , then there exists satisfying
But there may not exist a point satisfying
However, if is compact, then a point ξ exists satisfying
Reich provided a multivalued version of his famous result as follows.
 A multivalued map (where is the family of nonempty closed subsets of ℑ) is called a Reich-type multivalued -contraction if there are constants satisfying such that
for each .
It was proved in  that a Reich-type multivalued -contraction in a complete MS possesses a fixed point. When and , the above definition reduces to the multivalued version of Kannan-type contraction.
Multivalued version of Wardowski’s theorem was given by Altun et al.  as follows.
 Let be a MS. A multivalued map is called a multivalued F-contraction (MVFC, in short) if there is a constant and such that
for all with .
In a complete MS, an MVFC possesses a fixed point.
Recently, Kannan’s and Reich’s fixed point theorems have been studied and extended in several directions. Particularly we refer to the research of Aydi et al. [10,11], Bojor [12,13], Choudhury and Kundu , Debnath and de La Sen [15,16], Debnath et al. [17,18], Gornicki , Karapinar et al. , Mohammadi et al. . Some important work on the application of multivalued F-contractions were recently carried out by Sgroi and Vetro  and Ali and Kamran .
In this article, first we introduce a proper generalization of Kannan’s theorem for multivalued maps via F-contraction and further introduce a Reich-type generalization of the same. We present an application of our multivalued Kannan-type F-contraction to the solution of integral equations.
2. Multivalued Kannan Type F-contraction
In this section, we provide a proper extension of Kannan’s theorem for multivalued maps using Wardowski’s technique.
Let be a MS. The map is called a generalized multivalued Kannan-type F-contraction (GMKFC, in short) if there are constants satisfying , and such that
for all with , where is the collection of all fixed points of Γ.
Let be a complete MS. A GMKFC, such that is compact for each possesses a fixed point.
Fix and choose . Since is compact, by Lemma 2, we can select such that . Similarly we may consider such that and so on. Continuing this way we generate a sequence satisfying such that .
Assume that for all , because otherwise we obtain a fixed point. Thus , for all .
Taking limit in (6) as and using , we have . Thus there exists such that for all , i.e., for all .
Let with . Then
Since the series is convergent for , we have as . Hence is Cauchy and being complete, we have for some .
We claim that is a fixed point of . We consider the two cases.
Case I: There exists a subsequence of such that for all .
Case II: There exists such that for all . Then
Taking limit in (7) as , we have . Hence . Thus . □
In , Mohammadi et al. studied interpolative multivalued Ćirić-Reich-Rus type F-contraction which is an extension of Reich’s  theorem. It is to be noted that our new result, i.e, Theorem 3 is not a particular case of Theorem 2.7 in . Because in , the condition is not permissible.
Next, we provide an example which shows that Theorem 3 can be used to prove existence of fixed point results for such mappings where neither Kannan’s nor Wardowski’s theorem is applicable.
Consider with the metric
Clearly is a complete MS. Define the multivalued map by
Let . Then and . Thus in this case we can not find any such that , i.e., the multivalued version of Wardowski’s theorem (see Remark 3) is not applicable.
Further, with , if the condition is to be satisfied, then we should have , i.e., , i.e., , which is not satisfied by any . Hence multivalued Kannan’s theorem (see, Remark 2) is also not applicable either.
Finally, if we assume that with , then it is easy to see that the condition is trivially satisfied for any with , and . We observe that .
We present another example to illustrate Theorem 3 as follows.
Consider the set endowed with the usual metric for all . Define the multivalued map by
Let , then clearly . In that case, . Thus, we observe that Γ is a GMKFC with , and any with . Therefore, all conditions of Theorem 3 are satisfied and Γ has a fixed point. In fact, Γ has infinitely many fixed points.
3. Multivalued Reich Type F-Contraction
Here we introduce generalized multivalued Reich-type F-contraction (GMRFC, in short) by increasing the degrees of freedom of the constants in GMKFC. We show that GMKFC introduced in the previous section is a particular case of GMRFC for suitable choice of the constants.
Let be a complete MS. A map is said to be a GMRFC if there exist with , and such that
for all with .
Let be a complete MS. A GMRFC, such that is compact for each admits a fixed point.
Similar to the proof of Theorem 3, we construct a sequence .
Rest of the proof may be obtained in a similar manner as the proof of Theorem 3 and hence omitted. □
In Theorem 4, if we take , then Theorem 3 is obtained. Thus GMKFC introduced in this paper is a particular case of GMRFC when .
Theorem 4 is more general than Theorem 2.7 in  in terms of relaxation of degrees of freedom of the constants involved.
4. An Application to Integral Equations
In this section we present an application of Theorem 3 to the solution of a particular Volterra type integral equation.
Let be the space of all real valued continuous functions defined on . For any and fixed arbitrary , define . It is easy to see that the norm is equivalent to the supremum norm. The metric on is defined by
for all .
Consider the following integral equation
(A) and are continuous;
(B) is increasing for all ;
(C) there is such that for all , the following is true:
Suppose that conditions hold. Further, suppose there exist and with satisfying
for all and . Then the integral Equation (10) has a solution.
Hence Theorem 3 is applicable to and we conclude that has a fixed point. Therefore, the integral Equation (10) has a solution. □
We have introduced new and proper extensions of multivalued Kannan type F-contraction and found its application to the solution of integral equations. It has been shown that our result is applicable to certain class of mappings where neither the multivalued version of Kannnan nor that of Wardowski can be used. Finding metric completeness characterization in terms of GMKFC is a suggested future work.
Author P.D. contributed in Conceptualization, Investigation, Methodology and Writing the original draft; Author H.M.S. contributed in Investigation, Validation, Writing and Editing. All authors have read and agreed to the published version of the manuscript.
This research received no other external funding.
The authors are thankful to the learned referees for careful reading and valuable comments towards improvement of the manuscript. Research of the first author (P. Debnath) is supported by UGC (Ministry of HRD, Govt. of India) through UGC-BSR Start-Up Grant vide letter No. F.30-452/2018(BSR) dated 12 Feb 2019.
Conflicts of Interest
The authors declare no conflict of interest.
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