Fixed Point Theory and the Ulam Stability

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, ,