Abstract
We show that the Prešić type operators of several variables can be regarded as an operator of a single variable and the fixed point problem of Prešić type can be regarded as a classical fixed point problem. We extend the recent result of Ćirić and Prešić by using the classical approach of Prešić. The key of the proof is based on the mappings introduced by Kada, Suzuki, and Takahashi. We also discuss the convergence problems of recursive real sequences and the Volterra integral equations as an application of our result.
Dedicated to the memory of Professor Wataru Takahashi
1. Introduction
Let be a metric space and be a fixed integer. Suppose that is given. The following two problems were studied by Prešić [1]:(i)Find such that(ii)For any given , if we definethen what can we say about the convergence of the sequence ?
This problem is very interesting and was further studied by many authors, for example, see [2–4].
In 1965, Prešić [1] proved the following interesting theorem which can be regarded as a generalization of the classical fixed point theorem proposed by Banach [5]. Let be a positive integer and let be nonnegative real numbers such that . Let be a family of mappings such that
Theorem P1. Let be a complete metric space. Let be a sequence of positive real numbers such that and . Let be a sequence of operators in such thatLet be a sequence in such that are arbitrary and
Then, the following statements are true.(a)There exists an element such that (b)There exists an operator such that converges to uniformly; that is, (c)The element is the only one satisfying Equation (1)
Theorem P1, where , is nothing but Banach’s fixed point theorem [5].
Remark 1. It was shown in [6] that the condition is superfluous.
As a consequence of Theorem P1, we have the following result.
Theorem P2. Let be a complete metric space and let be a positive integer. Let be nonnegative real numbers such that . Let . Let be a sequence in such that are arbitrary and Equation (2) holds. Then, the following statements are true.(a)There exists an element such that (b)The element is the only one satisfying Equation (1).
Ćirić and Prešić [2] proved the following improvement of Theorem P2.
Theorem CP. Let be a complete metric space. Let be a positive integer and let be a positive real number such that . Let be such thatLet be a sequence in such that are arbitrary and Equation (2) holds. Then, the following statements are true.(a)There exists an element such that and Equation (1) holds.(b)If, in additionthen the element is the only one satisfying Equation (1).
Remark 2. It is clear that if satisfies Expression (3), then it satisfies Expressions (6) and (7).
Proof. Suppose that satisfies Expression (3) with nonnegative constants such that . To show that satisfies Expression (6) with , let . It follows thatTo see that satisfies Equation (7), let . It follows thatThis completes the proof.☐
It is natural to ask the following:
Question: Is it possible to generalize Theorem CP by using the approach of Theorem P1?
In this paper, we answer the question above by considering a wider class of mappings than those satisfying Expression (6). The class of mappings studied in this paper is motivated by the one in work of Kada et al. [7]. Some progress on these mappings can be found in [8].
Theorem KST (see [7]). Let be a complete metric space. Let be given and let be a mapping such that the following conditions hold:(1)(KST1) for all (2)(KST2) If , then
Then, every sequence with arbitrary and for all converges to a fixed point of .
2. Main Results
2.1. On the Fixed Point Problem of Prešić Type
We first show that the fixed point problem of Prešić type is equivalent to the classical fixed point problem. Suppose that is a nonempty set and is an integer. The fixed point problem of Prešić type for a given mapping is to find such that
We denote by the set of all solutions of the problem above. We show that this problem is connected to the classical fixed point problem. To simplify the notation of the following result, we write and where .
Theorem 3. Suppose that is a nonempty set and is an integer. Suppose that is given. Then, there exists a mapping such that the following statements hold.(a)If , then (b)If , then
Proof. Define bywhere .
The statement (a) is trivial. We prove the statement (b). Suppose that . It follows thatThis implies thatThis completes the proof.☐
2.2. On the Class of Mappings
In this subsection, we discuss the following classes of mappings. The first two classes are from Theorem P and Theorem CP and the last one from Theorem KST.
Suppose that is a metric space and is an integer. Suppose that are nonnegative real numbers. Define
Remark 4. Suppose that is a metric space and is an integer. The following statements are true.(a)If are nonnegative real numbers, then , where (b)If is a nonnegative real number and , then , where , , is defined by Equation (11) and
Proof. The statement (a) is trivial. We prove the statement (b). Let . It is clear that is a metric on . We now prove that . To see that satisfies Condition (KST1), let . We write and . It follows from thatNote thatPut where . It follows from Equation (6) thatFinally, we show that satisfies Condition (KST2). To see this, let be a sequence in and let be such that . For each , we write and . It follows from the definition of thatIn particular, we have . This implies that and . We considerHence, . Sincewe have . Hence, . This completes the proof.☐
Remark 5. The classes and are closed under the pointwise convergence, that is, if is a sequence in (, respectively) and there exists a mapping such thatthen (, respectively).
Proof. Suppose that is a sequence in and there exists a mapping such thatWe prove that . To see this, let . Note thatIt follows thatIn particular,This implies that . The case that is a sequence in can be done similarly.☐
2.3. A Generalization of Theorem KST and Its Consequences
The following result is analogous to Theorem P1 with a wider class of mappings.
Theorem 6. Let be a complete metric space and be given. Let be a sequence of positive real numbers such that . Let be a sequence of operators in such thatLet be a sequence in such that is arbitrary and for all . Then, the following statements are true.(a)There exists an element such that (b)There exists an operator such that converges to uniformly(c)If satisfies Condition (KST2), then
Lemma 7. Let and be two sequences of nonnegative real numbers such that . Let be given. Ifthen .
Proof. Note that, for each , we have . In particular, for all . Hence, the conclusion follows.☐
Proof of Theorem 6. First, we note thatIt follows from Lemma 7 that . In particular, is a Cauchy sequence and hence for some by the completeness of . Hence, (a) is proved.
We now prove (b). For each and for each , we note thatIt follows that is a Cauchy sequence and hence exists. We then define byIt follows then thatTo see that , let . Since , there exists an integer such that for all . For each and , we haveThat is, for all . Hence,Finally, we assume that satisfies (KST2). Note that . Moreover,It follows from Condition (KST2) that .
We are now ready to give an affirmative answer of the problem in the introduction. In fact, we can generalize Theorem CP by using the approach of Theorem P1.
Theorem 8. Let be a complete metric space and be a fixed integer. Let be a positive real number such that . Let be a sequence of positive real numbers such that . Let be a sequence of operators in such thatLet be a sequence in such that are arbitrary andThen, the following statements are true.(a)There exists an element such that (b)There exists an operator such that converges uniformly to and (c), that is, is the only solution of the fixed point problem of Prešić type for , provided that
Proof. Suppose that , , and are given as in the statement of the theorem. Let and be defined as in Theorem 3 and Remark 4. For each , we define bywhere . By Remark 4, we have for all . Define be a sequence in byWe prove that for each . To see this, we haveNext, we prove thatIn fact, for each and , we haveIt follows from Theorem 6 that there exists an element such that . In particular, we have . It also follows from Theorem 6 that there exists an operator such that converges uniformly to . Note that, for each , we haveWe now define (from ) byIn particular, we have(i)(ii) for all .It follows from each and Remark 5 that . Hence . In particular, satisfies Condition KST2 and hence by Theorem 6. This implies thatHence,The uniqueness of the fixed point of Prešić type is obvious if the additional hypothesis is assumed.☐
3. Applications
We finally discuss some applications of our result.
3.1. Some Convergence Problem of Recursive Real Sequences
We reconsider the following example studied by Ćirić and Prešić [2, Example 1] and give some remark.
Example 9. Let be a metric space endowed with the usual metric and be defined byIt was proved in [2] thatThe author of [2] claimed that . We note that for all . In fact, let , , and ; it follows thatUsing our approach, let andwhere and . Moreover, let be defined bywhere . We can follow the proof in Remark 4 to show thatNext, we show that satisfies Condition (KST2). Note that so . Let be a sequence in and let be such that . For each , we write and . It follows from the definition of thatIn particular, we have . This implies that and . We consider the following two cases.
Case 1. . In this case, we may assume that and are sequences in and hence . This implies that and hence . Then, .
Case 2. . In this case, we may assume that and are sequences in and hence . This implies that and hence . Then .
Hence, . In particular, we can apply this example to our Theorem 8. Note that, since , the condition (7) cannot be omitted for the uniqueness of the solution as claimed in [2].
We now discuss the following convergence problem of real sequences inspired by [6].
Example 10. Suppose that is a real sequence satisfying the following recursive relation: and for each It is clear that this example is related to the preceding one. Table 1 shows the numerical experiment regarding to the problem with respect to the initial inputs and . For example, if and , then ; if and , then .
3.2. Volterra Type Integral Equations
We discuss a further application of our Theorem 8 in the context of Volterra type integral equations.
Theorem 12. Suppose that is the space of continuous real-valued functions defined on an interval , where , equipped with the supremum metric defined bySuppose that and are continuous functions such that(i) and satisfy the Lipschitz condition for the second argument with constants and , respectively, that is,(ii)Suppose that is a sequence in such that where for all . If and for each then for all and there exists an element such that and is a unique solution of the following Volterra type integral equationwhere .
Proof. Now, for each , we define , by for each ,To apply our Theorem 8, it is sufficient to prove that(a) for all (b) for all and for all We now prove the statements (a) and (b).(a)Let and let . Then,(b)Let and let . Then,Note thatIt follows from our Theorem 8, where , there exists an element such thatMoreover,where is defined byand . Finally, to show that is a unique solution of the Volterra type integral Equation (59), we show that for all . Let . Then,This completes the proof.☐
4. Conclusion
We show that the fixed point problem of Prešić type (with respect to several variables) can be regarded a fixed point problem (of a single variable) by using a product space approach. With an appropriate metric on the product space, the Ćirić-Prešić operator can be regarded a mapping studied by Kada et al. in the product space. In particular, we deduce the fixed point result under a weaker assumption. We apply our result for the convergence problems of recursive real sequences and the Volterra type integral equations.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
The authors contributed equally and significantly in writing this paper. They read and approved the final manuscript.
Acknowledgments
NB was supported by Post-Doctoral Training Program from Khon Kaen University, Thailand (Grant No. PD2563-02-10). KS was supported by Rajamangala University of Technology Rattanakosin (RMUTR) (Grant No. C-38/2562).