Fixed Point Theory and the Ulam Stability

Brzdęk, Janusz; Cădariu, Liviu; Ciepliński, Krzysztof

doi:https://doi.org/10.1155/2014/829419

Journal of Function Spaces

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Ulam’s Type Stability and Fixed Points Methods

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Volume 2014 | Article ID 829419 | https://doi.org/10.1155/2014/829419

Fixed Point Theory and the Ulam Stability

Janusz Brzdęk,¹Liviu Cădariu,²and Krzysztof Ciepliński^3,4

Academic Editor: Zbigniew Leśniak

Received26 May 2014

Accepted09 Jul 2014

Published13 Aug 2014

Abstract

The fixed point method has been applied for the first time, in proving the stability results for functional equations, by Baker (1991); he used a variant of Banach's fixed point theorem to obtain the stability of a functional equation in a single variable. However, most authors follow the approaches involving a theorem of Diaz and Margolis. The main aim of this survey is to present applications of different fixed point theorems to the theory of stability of functional equations, motivated by a problem raised by Ulam in 1940.

1. Introduction

Speaking of the stability of a functional equation we follow a question raised in 1940 by Ulam, concerning approximate homomorphisms of groups (see [1]). The first partial answer (in the case of Cauchy’s functional equation in Banach spaces) to Ulam’s question was given by Hyers (see [2]). After his result a great number of papers (see for instance monographs [3–5], survey articles [6–14], and the references given there) on the subject have been published, generalizing Ulam’s problem and Hyers’s theorem in various directions and to other (not necessarily functional) equations.

The method used by Hyers in [2] (quite often called the direct method) has been successfully applied for study of the stability of large variety of equations, but unfortunately, as it was shown in [15], it does not work in numerous significant cases. Apart from it, there are also several other efficient approaches to the Hyers-Ulam stability, using different tools, for example, the method of invariant means (introduced in [16]), the method based on sandwich theorems (see [17]), and the method using the concept of shadowing (see [18]).

It this paper we discuss the fixed point method, which is the second most popular technique of proving the stability of functional equations. It was used for the first time by Baker (see [19]) who applied a variant of Banach’s fixed point theorem to obtain the Hyers-Ulam stability of a functional equation in a single variable. At present, numerous authors follow Radu’s approach (see [20]) and make use of a theorem of Diaz and Margolis. Our aim is to show connections between different fixed point theorems and the theory of stability, inspired by the problem of Ulam (see [5, 7, 9]).

The paper contains both classical and more recent results. In Section 2 we present applications of some classical fixed point theorems. Section 3 shows a somewhat different (but still fixed point) approach, when the results on the stability are simple consequences of the proved (new) fixed point theorems. In Section 4 we deal with the stability of the fixed point equation and its generalization. Section 5 contains final remarks.

In the paper denotes the set of positive integers and we put , .

2. Applications of Known Fixed Point Theorems

2.1. (Some Variants of) Banach’s Theorem

The fixed point method was used for the investigation of the Hyers-Ulam stability of functional equations for the first time by Baker in [19], where actually he applied the following variant of Banach’s fixed point theorem.

Theorem 1 ([19], Theorem 1). Assume that is a complete metric space and is a contraction (i.e., there is a such that for all ). Then has a unique fixed point . Moreover,

He obtained in this way the subsequent result concerning the stability of a quite general functional equation in a single variable.

Theorem 2 ([19], Theorem 2). Let be a nonempty set, be a complete metric space, , , and
If , and then there is a unique function such that

Theorem 2 with gives the following.

Corollary 3 ([19], Theorem 3). Let be a nonempty set, a real (or complex) Banach space, , , (or ), and If , and then there exists a unique function such that

The following stability result for a more general functional equation has been deduced in [21] from Theorem 1.

Theorem 4 ([21], Theorem 2.2). Let be a nonempty set, be a complete metric space, , , and Assume also that , are such that and there exists an with Then there is a unique function such that

In some recent papers the authors applied the weighted space method to prove the generalized Hyers-Ulam stability properties of several nonlinear functional equations. We recall that the weighted space method uses the classical mathematical results in spaces endowed with weighted distances. In those papers, the classical mathematical result is just the Banach fixed point theorem. This new method is used to prove a stability result for (4), described in the following theorem.

Theorem 5 ([22], Theorem 2.1). Let be a nonempty set, a complete metric space, and the functions , , satisfy for any , and .
If satisfies the inequality then there exists a solution of (4) such that

The results in Theorems 2, 4, and 5 have been extended in [23], where a result on the generalized Hyers-Ulam stability of the nonlinear equation has been obtained, also by the weighted space method. Here, is a nonempty set, is a complete metric space, and are given mappings (the unknown function in (18) is ).

Theorem 6 ([23], Theorem 2). Suppose that and satisfy for some given function . Suppose also that fulfils the inequality for all and for all .
If is a mapping with the property then there exists a unique such that

For the proof it is enough to show that the set is a complete metric space with the weighted metric moreover, it can be proved that the nonlinear operator given by is a strictly contractive self-mapping of , with the Lipschitz constant .

On the other hand, if then (18) becomes where , , are given mappings and is the unknown function. The above equation is called a linear functional equation and was intensively investigated by a lot of authors (e.g., Kuczma et al. in [24] obtained some results concerning monotonic, regular, and convex solutions of (27)). The following theorem contains a generalized Hyers-Ulam stability result for the above linear functional equation, obtained as a particular case of Theorem 6.

Let us consider a nonempty set , a real (or complex) Banach space , endowed with the norm and the given functions , (or ) and .

Theorem 7 ([23], Theorem 5). Let and satisfy for some fixed mapping . Let fulfil for all and for all . If has the property then there exists a unique mapping , defined by for , such that

The following outcome proved by the weighted space method concerns a generalized Hyers-Ulam stability for a general class of the Volterra nonlinear integral equations, in Banach spaces.

Let us consider a Banach space over the (real or complex) field , an interval () and the continuous given functions and . We write and denote by the norm in .

The result on stability of the nonlinear Volterra integral equation ( is an unknown function, and are continuous given mappings and is a fixed nonzero scalar), reads as follows.

Theorem 8 ([23], Theorem 8). Suppose that there exists a positive constant such that Suppose also that is a continuous function, which satisfies
If is continuous and has the property and if then there exists a unique such that

Note that, if we replace in Theorem 8 the functions and by and , respectively, and write , then the theorem takes the subsequent equivalent, and a bit simpler, form.

Theorem 9. Suppose that there is a positive constant such that Suppose also that is a continuous function, which satisfies
If is continuous and has the property then there exists a unique such that

Next, following [25], we recall some notations.

Let be a nonempty set, a complete metric space, and . Then is a complete metric space and , where denotes the set of all such that there is a real constant with

Let , , and . Put

In [25], the authors used the contraction principle to get the following fixed point result.

Theorem 10 ([25], Proposition 1.1). Let be a nonempty set, be a complete metric space, , , and . If , , and , then and has a unique fixed point in . Moreover,

Next, applying Theorem 10, they have proved the following theorem.

Theorem 11 ([25], Theorem 2.1). Let be a nonempty set, , , and an automorphism of . Assume also that is a complete metric space, is continuous and is an endomorphism of . If , , , and mappings satisfy then there exists a unique mapping such that

Theorem 11 with and gives the following corollary, which corresponds to the results in [26–31].

Corollary 12 ([25], Corollary 2.1). Let be a nonempty set, , and be an automorphism of . Assume also that is a complete metric space, is continuous and is an endomorphism of . If , , and a mapping satisfies then there exists a unique mapping fulfilling (50) and

Another consequence of Theorem 11 is the following.

Theorem 13 ([25], Theorem 3.1). Let be a nonempty set, and be a bijection. Assume also that is a complete metric space and is continuous. If , , , and mappings satisfy (49) and then there exists a unique mapping such that

Theorem 13 with and implies the following.

Corollary 14 ([25], Corollary 3.1). Let be a nonempty set, and a bijection. Assume also that is a complete metric space and is continuous. If , , and a mapping satisfies the inequality then there exists a unique mapping fulfilling (55) and

Theorem 15 ([25], Theorem 2.2). Let be a nonempty set, , , and an endomorphism of . Assume also that is a complete metric space, is continuous and is an automorphism of . If , , , and mappings satisfy inequalities (48) and (49), then there exists a unique mapping such that (50) holds,

Theorem 15 with and yields the next corollary.

Corollary 16 ([25], Corollary 2.3). Let be a nonempty set, , and an endomorphism of . Assume also that is a complete metric space, is continuous and is an automorphism of . If , , and a mapping satisfies inequality (52), then there exists a unique mapping such that (50) holds and

Another consequence of Theorem 15 is the following.

Theorem 17 ([25], Theorem 3.2). Let be a nonempty set, and . Assume also that is a complete metric space and is a continuous bijection. If , , , and mappings satisfy (49) and (48), then there exists a unique mapping such that (55) holds,

Theorem 17 with and implies the following.

Corollary 18 ([25], Corollary 3.2). Let be a nonempty set, and . Assume also that is a complete metric space and is a continuous bijection. If , , and a mapping satisfies (57), then there exists a unique mapping such that (55) holds and

Let us finally mention that it is also shown in [25] that the above results imply some classical outcomes on the generalized stability of the Cauchy functional equation.

2.2. Other Classical Theorems

In this section, we present applications of three other fixed point theorems. To formulate the first of them we need two more definitions.

A nondecreasing function is called a comparison function [32, 33] or Matkowski gauge function [34, 35] if

Given such a mapping and a metric space , we say that a function is a Matkowski -contraction if

We can now recall Matkowski’s fixed point theorem from [36].

Theorem 19. If is a complete metric space and is a Matkowski contraction, then has a unique fixed point and the sequence converges to for every .

In [37], this theorem was applied to prove the following generalization of Theorem 2.

Theorem 20 ([37], Theorem 2.2). Let be a nonempty set, be a complete metric space, , . Assume also that where is a comparison function, and let , be such that (3) holds. Then there is a unique function satisfying (4) and
Moreover,

Theorem 20 with gives the subsequent result.

Corollary 21 ([37], Corollary 2.3). Let be a nonempty set, a complete metric space, . Assume also that is a Matkowski -contraction and let , be such that Then there is a unique function satisfying the equation and condition (66). The function is given by

On the other hand, in [38], the following variant of Ćirić’s fixed point theorem was proved.

Theorem 22 ([38], Theorem 2.1). Assume that is a complete metric space and is a mapping such that where satisfy for all and some fixed . Then has a unique fixed point and

Next, Baker’s idea and Theorem 22 were used to obtain the following result concerning the stability of (4).

Theorem 23 ([38], Theorem 2.2). Let be a nonempty set, a complete metric space, , and where satisfy (73) for all and some . If , and (3) holds, then there is a unique function satisfying (4) and

A consequence of Theorem 23 is the following.

Corollary 24 ([38], Theorem 2.3). Let be a nonempty set, a real or complex Banach space, , , (here denotes the Banach algebra of all bounded linear operators on ), and If , and then there exists a unique function satisfying the equation and condition (10).

Now, let us recall the Markov-Kakutani theorem (see [39, 40]).

Theorem 25. Let be a linear topological space and let be a nonempty convex compact subset of . Let be a family of affine continuous self-mappings of such that Then there is a common fixed point of family ; that is,

Theorem 25 has been applied in [41] to provide an alternative (quite involved) proof of the following classical stability result due to Hyers [2].

Theorem 26. Let be an abelian semigroup, , , and Then there exists an additive function such that

2.3. Fixed Point Alternatives Theorems on Generalized Metric Space

In this part of the paper, we show how several fixed points alternatives can be used to get some Hyers-Ulam stability results.

In order to do this let us first recall (see [42, 43]) that is said to be a generalized metric on a nonempty set if and only if for any we have

In 2002, at the 14th European Conference on Iteration Theory (ECIT 2002 - Evora, Portugal), L. Cădariu and V. Radu delivered a lecture titled “On the stability of the Cauchy functional equation: a fixed points approach.” They presented a generalized Hyers-Ulam stability result for the Cauchy functional equation, in the case when the equation perturbation is controlled by a given mapping , with a simple property of contractive type. Their idea was to obtain the existence of the exact solution and the error estimations by using the following fixed point alternative theorem of Diaz and Margolis [44].

Theorem 27. Let be a complete generalized metric space. Assume that is a strictly contractive operator with the Lipschitz constant . Then, for each given element , either(), for all , or()there exists such that . Actually, if holds, then the following three conditions are valid.()The sequence converges to a fixed point of .() is the unique fixed point of in the set ()If , then

Remark 28. If the fixed point exists, it is not necessarily unique in the whole space ; this may depend on the starting approximation. It is worth noting that, in case , the pair is a complete metric space and . Therefore, properties ()–() follow from Banach’s Contraction Principle.
This method has been next used in [20] (for the additive Cauchy equation) and in [45] (for Jensen’s equation).
Now, let us remind one of the most classical results, which was first proved by the direct method: for in [46] (see also [47], where a similar result has been obtained under some regularity assumptions), and for in [48] (see also for instance [5] and [20, Theorem]; information and recent results on the case can be found in [49, 50]).

Theorem 29. Let be a real normed space, a real Banach space, , and be such that Then there exists a unique additive mapping such that

The lecture from ECIT 2002 was materialized in [51] in the following extension of Theorem 29.

Theorem 30 ([51], Theorem 2.5). Let be two linear spaces over the same (real or complex) field, a complete -normed space, , , and . Assume that with satisfies If there exist an and a positive constant such that the mapping has the property and satisfies the condition then there exists a unique additive mapping such that

Let be a linear space over the field . Recall that a mapping is called a -norm provided it has the following properties: if and only if ; , ; , .

The idea emphasized in [20, 45, 51] has been subsequently used for the quadratic equation in [52], the cubic equation in [53], the quartic equation in [54], equations and in [55], and the monomial equation in [56]. We present that last result below. To this end, let us recall that a function (mapping an abelian group into a real vector space ) is called a monomial function of degree ( is a fixed positive integer) if it is a solution of the monomial functional equation where the difference operator is defined in the following manner: and, inductively,

Theorem 31 ([56], Theorem 2.1). Let be a group that is uniquely divisible by (i.e., for any there exists a unique with ), a (real or complex) complete -normed space, and fulfil the following property: Suppose , with , satisfies the condition If there exists a positive constant such that the mapping satisfies the inequality then there exists a unique monomial mapping of degree with

In [43], Mihet has given one more generalization of Theorem 2; he obtained it by proving another fixed point alternative.

Recall that a mapping is called a generalized strict comparison function if it is nondecreasing, , and . Given such a mapping and a generalized metric space , we say that a function is a strict -contraction if it satisfies inequality (64).

Now, we can formulate the fixed point result from [43].

Theorem 32 ([43], Theorem 2.2). Let be a complete generalized metric space and a strict -contraction such that for an . Then has a unique fixed point in the set and the sequence converges to for every . Moreover,

Using this theorem we can get the following generalization of Theorem 2.

Theorem 33 ([43], Theorem 3.1). Let be a nonempty set, a complete metric space, , . Assume also that where is a generalized strict comparison function, and let , be such that (3) holds. Then there is a unique function satisfying (4) and

We also have the following.

Theorem 34 ([43], Theorem 4.1). Let be a nonempty set, a complete metric space, , . Assume also that and are such that (69) holds. If is a generalized strict comparison function satisfying inequality (64), then there is a unique mapping such that (70) and (106) hold. The function is given by formula (71).

We end this section with some applications of the fixed point alternatives of the Bianchini-Grandolfi and Matkowski types. To this end, we introduce some notations and definitions.

A nondecreasing function is said to be a c-comparison function [32] or Bianchini-Grandolfi gauge function [35, 57] if for each the series is convergent (here denotes the th iteration of ).

A self-mapping of the metric space for which there exists a -comparison function such that is called a Bianchini-Grandolfi contraction [34] (see also the notion of a Matkowski contraction in Section 2.2).

The following result is the fixed point alternative of Bianchini and Grandolfi [57] (see [58, Lemma 2.1]).

Theorem 35. Let be a complete generalized metric space, that is, one for which may assume infinite values, and a Bianchini-Grandolfi contraction. Then, for each , either (), for all , or()there exists such that for . If holds, then the sequence converges to a fixed point of , is the unique fixed point of in the set

Next, we introduce the notion of an admissible pair. Let us consider a 2-divisible group and denote by the family of all mappings with the properties:(i) is continuous at , for each ;(ii) is superadditive; that is where for each .

Let us consider a comparison function , an element and numbers and .

Given and , we say that is a -admissible pair of order if

Now, we are in a position to present a stability theorem for monomial functional equation (94) of degree .

Theorem 36 ([58], Theorem 3.1). Let be a 2-divisible group, a (real or complex) complete -normed space, be a c-comparison function, and be a -admissible pair of order , with a . Suppose that and with satisfy the inequality If there exist and such that then there exists a unique monomial function of degree such that

Remark 37. If (in Theorem 36) , with and having suitable properties, then we obtain [58, Corollary 4.1]; if , and with an , then we get Theorem 31.

The fixed point alternative of Bianchini and Grandolfi has been used in [59] to prove a generalized Hyers-Ulam stability result for the additive Cauchy functional equation. In what follows, we yet recall the fixed point alternative of Matkowski and the corresponding outcome on stability of the Cauchy equation.

Theorem 38 ([59], Lemma 3.2). Let be a complete generalized metric space and a -Matkowski contraction. Then, for each , either (), for all , or()there exists an such that , for all . If holds, then the sequence converges to a fixed point of and is the unique fixed point of in the set moreover, if additionally the mapping is a bijection, then

The above fixed point result has been used to prove the following stability result for the Cauchy equation.

Theorem 39 ([59], Theorem 2.5). Let us consider a real linear space , a complete -normed space , a comparison function , and a -admissible pair of order with a . Let us suppose that the mapping is an increasing bijection and that a mapping with satisfies the inequality where is a given function such that If there exists a such that then there exists a unique additive mapping such that

3. New Fixed Point Theorems and Their Applications

In this section, we present a somewhat different fixed point approach to the stability of functional equations, in which the stability results are simple consequences of some new fixed point theorems.

Given a nonempty set and a metric space , we define a mapping ( denotes the family of all functions mapping a set into a set ) by

With this notation, we have the following.

Theorem 40 ([60], Theorem 1). Let be a nonempty set, a complete metric space, , , and be given by If is an operator satisfying the inequality and functions and are such that then for every the limit exists and the function , defined in this way, is a unique fixed point of with

A consequence of Theorem 40 is the following result on the stability of a quite wide class of functional equations in a single variable.

Corollary 41 ([60], Corollary 3). Let be a nonempty set, be a complete metric space, , , , a function satisfy the inequality for any and , and be an operator defined by Assume also that is given by (120) and functions and are such that and (123) holds. Then for every limit (124) exists and the function is a unique solution of the functional equation satisfying inequality (125).

Another application of Theorem 40 has been given in [61]; it concerns stability of the polynomial equation (for a survey on related results see [62]).

Next, following [63], we consider the case of non-Archimedean metric spaces (let us mention here that the first paper dealing with the Hyers-Ulam stability of functional equations in non-Archimedean normed spaces was [64], whereas [65] seems to be the first one in which the stability problem in a particular type of these spaces was considered). In order to do this, we introduce some notations and definitions.

Let be a nonempty set. For any we write provided and we say that an operator is nondecreasing if it satisfies the condition

Moreover, given a sequence in , we write provided

We will also use the following hypothesis concerning operators : for every sequence in with .

Finally, let us recall that a metric on a nonempty set is called non-Archimedean (or an ultrametric) provided

We can now formulate the following fixed point theorem.

Theorem 42 ([63], Theorem 1). Let be a nonempty set, a complete non-Archimedean metric space and a nondecreasing operator satisfying hypothesis . If is an operator satisfying inequality (121) and functions and are such that and (122) holds, then for each limit (124) exists and the function , defined in this way, is a fixed point of with If, moreover, then is the unique fixed point of satisfying (135).

An immediate consequence of Theorem 42 is the following result on the stability of (129) in the complete non-Archimedean metric spaces.

Corollary 43 (see [63]). Let be a nonempty set, a complete non-Archimedean metric space, , , , a function satisfy the inequality for any and , and an operator defined by (127). Assume also that is given by and functions and are such that (128) and (134) hold. Then for every limit (124) exists and the function is a solution of functional equation (129) satisfying inequality (135).

A variant of Theorem 42 in arbitrary complete metric spaces was also proved in [63, Theorem 2]. A slightly improved version of this outcome was obtained in [66] in response to an open problem concerning the uniqueness of the mapping , defined below. Namely, we have the following result.

Theorem 44 ([66], Corollary 2.2). Let be a nonempty set, a complete metric space and a nondecreasing operator satisfying the hypothesis for every sequence of elements of and every . If is an operator satisfying the inequality and the functions , are such that then, for every , the limit exists and the function , defined in this way, is a fixed point of with Moreover, if the condition holds, then the mapping is the unique fixed point of with the property

Theorem 44 is a consequence of the following fixed point theorem for a class of operators satisfying some very general conditions (recall that given , we say that is -contractive if for any and with for , it follows that for ).

Theorem 45 ([66], Theorem 2.1). Let be a nonempty set, a complete metric space and . Suppose that is -contractive, hypotheses and hold, and fufils Then, for every , the limit exists and the mapping is the unique fixed point of with the property Moreover, if we have then is the unique fixed point of with the property

It is also worth noting that Theorem 40 can be directly obtained from Theorem 45 (see [66] for details).

Given nonempty sets and functions , , we define an operator by and we say that is an operator of substitution provided with some and . Moreover, if is continuous for each (with respect to a topology in ), then we say that is continuous.

Theorem 46 ([67], Theorem 2.1). Let be a nonempty set, a complete metric space, , , and Assume also that for every , is nondecreasing, , , and (122) holds. Then for every limit (124) exists and inequality (125) is satisfied. Moreover, the following two statements are true. (i)If is a continuous operator of substitution or is continuous at for each , then is a fixed point of .(ii)If is subadditive; that is, for all , , then has at most one fixed point such that there exists with

Actually, it can be deduced from the proof of [67, Theorem 2.1] that Theorem 46 can be derived from Theorem 44.

Theorem 46 with immediately gives the following generalization of Theorem 2.

Corollary 47 ([67], Corollary 2.1). Let be a nonempty set, a complete metric space, , and Assume also that , , (151) holds, , for every , is nondecreasing, is continuous and Then for every the limit exists, (125) holds and is a solution of (4). Moreover, if is subadditive for every and , then is the unique solution of (4) fulfilling (153).

Some results related to those presented above, proved for functions taking values in Riesz spaces, can be found in [68].

Further applications of Theorem 40 have been proposed in [49, 50] (in solving a problem of Th. M. Rassias concerning optimality of estimations in Theorem 29) and in [69] (in proving stability of the equation of -Wright affine functions); in particular, it has been discovered in this way in [49] that the property of hyperstability for the additive Cauchy equation appears quite often in a natural way (see [8, 70] for more information on this issue and related results); generalizations of that approach have been presented in [71, 72]. Similar methods (also involving Theorem 42) have been applied for some other equations in [73–78].

4. Stability of the Fixed Point Equation and Its Generalization

In [79] one can find the following definition (as well as some related notions concerning the generalized Ulam-Hyers stability).

Let be a metric space and . We say that the fixed point equation is Ulam-Hyers stable if there is a such that for any and with there exists an satisfying (157) and

Let us recall (see [79, 80]) that is called a weakly Picard operator if for every the sequence is convergent and its limit, denoted by , is a fixed point of . Given such an operator and a , we say that is a -weakly Picard operator if

The following two results comes from [79].

Theorem 48 ([79], Remark 2.1). Let be a metric space and . If is a -weakly Picard operator, then (157) is Ulam-Hyers stable.

Theorem 49 ([79], Remark 2.2). Let be a metric space, a set and . Assume also that satisfies for . If for every , the equation is Ulam-Hyers stable, then (157) is also Ulam-Hyers stable.

Let us mention here that in [79] some applications of these outcomes (e.g., to the stability of an integral equation) are also presented.

Now, let and be metric spaces, and . The coincidence equation is called Ulam-Hyers stable if there is a such that for any and with there exists an satisfying (162) and (159).

Let and . We say that is a -weakly Picard pair if there exists a weakly Picard operator such that

Theorem 50 ([79], Remark 6.1). Let , be metric spaces, and . If is a -weakly Picard pair, then (162) is Ulam-Hyers stable.

More results on these as well as related problems can be found in [81–95].

5. Final Remarks

Applications of different fixed point theorems to the Ulam type stability have been presented in this survey. On the other hand, some fixed point theorems can be derived from such stability results; we refer to [96–98] for suitable examples. Below we show how to get another such an example.

Let be a metric space. We denote by the family of all nonempty subsets of . The convergence of subsets of is with respect to the Hausdorff metric derived from the metric . The number is said to be the diameter of . The next theorem has been obtained in [99, Theorem 2] (cf. [100, Theorem 1]; we refer the reader to [99–102] for further similar results and to [103] for a survey on the subject).

Theorem 51 ([99], Theorem 2). Let be a nonempty set, , , and satisfy Then one of the following two statements is valid. (i)If is complete and then, for each , the limit exists, the function is single-valued and it is the unique function (which maps into ) such that and for .(ii)If then F is single-valued and for .

It is easily seen that Theorem 51 yields the following fixed point result.

Corollary 52. Let be a nonempty set, , , , and satisfy (165). Write for and . Then one of the following two statements is valid. (a)If is complete and then, for each , the limit exists, is single-valued and it is the unique fixed point of such that for .(b)If then F is a single-valued fixed point of .

Analogously, a fixed point result can be deduced from the main outcome in [104], concerning stability of a generalization of the Volterra integral equation.

We end the paper with an example of stability result for the Cauchy additive equation, proved in [26, Corollary 1] through a modified fixed point approach (somewhat analogous to that in [105]); it corresponds to Theorem 29.

Theorem 53 ([26], Corollary 1). Let be a normed space, with a nonempty such that , and satisfy the inequality Then there exists a unique function such that

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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