On Some Recent Developments in Ulam's Type Stability

Abstract

We present a survey of some selected recent developments (results and methods) in the theory of Ulam's type stability. In particular we provide some information on hyperstability and the fixed point methods.

1. Introduction

The theory of Ulam's type stability (also quite often connected, e.g., with the names of Bourgin, Găvruţa, Ger, Hyers, and Rassias) is a very popular subject of investigations at the moment. In this expository paper we do not give an introduction to it or an ample historical background; for this we refer to [1–11]. Here we only want to attract the readers attention to some selected topics by presenting some new results and methods in several areas of the theory, which have not been treated at all or only marginally in those publications and which are somehow connected to the research interests of the authors of this paper. Also the number of references is significantly limited (otherwise the list of references would be the major part of the paper) and is only somehow representative (but certainly not fully) to the subjects discussed in this survey.

First we present a brief historical background for the stability of the Cauchy equation. Next we discuss some aspects of stability and nonstability of functional equations in single variable, some methods of proofs applied in that theory (the Forti method and the methods of fixed points), stability in non-Archmedean spaces, selected results on functional congruences, stability of composite type functional equations (in particular of the Gołąb-Schinzel equation and its generalizations), and finally the notion of hyperstability. We end the paper with remarks also on some other miscellaneous issues.

2. Some Classical Results Concerning the Cauchy Equation

Throughout this paper ℕ, ℤ, ℝ, and ℂ denote, as usual, the sets of positive integers, integers, reals, and complex numbers, respectively. Moreover, ℝ₊∶=[0,∞) and ℕ₀∶=ℕ∪{0}.

For the beginning let us mention that the first known result on stability of functional equations is due to Pólya and Szegő [12] and reads as follows.

For every real sequence (𝑎_𝑛)_𝑛∈ℕ withsup_{𝑛,𝑚∈ℕ}𝑎_𝑛+𝑚−𝑎_𝑛−𝑎_𝑚≤1, (2.1)there is a real number 𝜔 such thatsup_𝑛∈ℕ𝑎_𝑛−𝜔𝑛≤1. (2.2)Moreover, 𝜔=lim_𝑛→∞𝑎_𝑛𝑛. (2.3) But the main motivation for study of that subject is due to Ulam (cf. [13]), who in 1940 in his talk at the University of Wisconsin presented some unsolved problems and among them was the following question.

Let 𝐺₁ be a group and (𝐺₂,𝑑) a metric group. Given 𝜀>0, does there exist 𝛿>0 such that if 𝑓∶𝐺₁→𝐺₂ satisfies𝑑𝑓𝑥𝑦,𝑓(𝑥)𝑓𝑦<𝛿,𝑥,𝑦∈𝐺₁, (2.4)then a homomorphism 𝑇∶𝐺₁→𝐺₂ exists with𝑑𝑓(𝑥),𝑇(𝑥)<𝜀,𝑥∈𝐺₁? (2.5) In 1941 Hyers [14] published the following answer to it.

Let 𝑋 and 𝑌 be Banach spaces and 𝜀>0. Then for every 𝑔∶𝑋→𝑌 with sup_{𝑥,𝑦∈𝑋}𝑔𝑥+𝑦−𝑔(𝑥)−𝑔𝑦≤𝜀, (2.6) there exists a unique function 𝑓∶𝑋→𝑌 such that sup_𝑥∈𝑋𝑔(𝑥)(𝑥𝑓(𝑥)𝑦−𝑓)≤𝜀,𝑥+𝑦=𝑓+𝑓,𝑥,𝑦∈𝑋. (2.7)(2.8) We can describe that latter result saying that the Cauchy functional equation (2.8) is Hyers-Ulam stable (or has the Hyers-Ulam stability) in the class of functions 𝑌^𝑋. For examples of various possible definitions of stability for functional equations and some discussions on them we refer to [9].

The result of Hyers was extended by Aoki [15] (for 0<𝑝<1; see also [16–18]), Gajda [19] (for 𝑝>1), and Rassias [20] (for 𝑝<0; see also [21, p. 326] and [22]), in the following way.

Theorem 2.1. Let 𝐸₁ and 𝐸₂ be two normed spaces, let 𝐸₂ be complete, 𝑐≥0, and let 𝑝≠1 be a real number. Let 𝑓∶𝐸₁→𝐸₂ be an operator such that 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝑐‖𝑥‖^𝑝+𝑦^𝑝,𝑥,𝑦∈𝐸₁⧵{0}. (2.9) Then there exists a unique additive operator 𝑇∶𝐸₁→𝐸₂ with 𝑓(𝑥)(𝑥≤𝑐‖𝑥‖−𝑇)^𝑝1−2^𝑝−1,𝑥∈𝐸₁⧵{0}. (2.10)

A further generalization was suggested by Bourgin [22] (see also [2, 6–8, 23]), without a proof, and next rediscovered and improved many years later by Găvruţa [24]. Below, we present the Găvruţa type result in a bit generalized form (on the restricted domain), which can be easily derived from [25, Theorem 1].

Corollary 2.2. Let 𝑋 be a linear space over a field with 2≠0 and let 𝑌 be a Banach space. Let 𝑉⊂𝑋 be nonempty, 𝜑∶𝑉²→ℝ, and 𝑓∶𝑉→𝑌 satisfy 𝑔𝑥+𝑦−𝑔(𝑥)−𝑔𝑦≤𝜑𝑥,𝑦,𝑥,𝑦∈𝑉,𝑥+𝑦∈𝑉. (2.11) Suppose that there is 𝜀∈{−1,1} such that 2^𝜀𝑉⊂𝑉 and 𝐻(𝑥)∶=^∞_𝑖=02^−𝑖𝜀𝜑2^𝑖𝜀𝑥,2^𝑖𝜀𝑥<∞,𝑥∈𝑉,liminf_𝑛→∞2^−𝑛𝜀𝜑2^𝑛𝜀𝑥,2^𝑛𝜀𝑦=0,𝑥,𝑦∈𝑉. (2.12)(2.13) Then there exists a unique 𝐹∶𝑉→𝑌 such that 𝐹(𝑥)𝑦𝐹(𝑥)(𝑥𝑥+𝑦=𝐹+𝐹,𝑥,𝑦∈𝑉,𝑥+𝑦∈𝑉,−𝑓)≤𝐻₀(𝑥),𝑥∈𝑉, (2.14) where 𝐻₀(𝑥)2∶=⁻¹𝐻(𝑥)𝐻2,𝑖𝑓𝜀=1,₋₁𝑥,𝑖𝑓𝜀=−1. (2.15)

Corollary 2.2 generalizes several already classical results on stability of (2.8). In fact, if we take 𝜀=−1 and 𝜑𝑥,𝑦∶=𝐿₁‖𝑥‖^𝑝+𝐿₂𝑦^𝑞+𝐿₃‖𝑥‖^𝑟𝑦^𝑠,𝑥,𝑦∈𝑉 (2.16) with some 𝐿₁,𝐿₂,𝐿₃∈ℝ₊, 𝑝,𝑞∈(1,∞), and 𝑟,𝑠∈ℝ with 𝑟+𝑠>1, then 𝐻₀ has the form 𝐻₀(𝑥)=𝐿₁‖𝑥‖^𝑝2^𝑝+𝐿−2²𝑦^𝑞2^𝑞+L−2³𝑦^𝑟+𝑠2^𝑟+𝑠−2,𝑥∈𝑉. (2.17) On the other hand, if 𝜀=1, 𝑉⊂𝑋⧵{0} and 𝜑𝑥,𝑦∶=𝛿+𝐿₁‖𝑥‖^𝑝+𝐿₂𝑦^𝑞+𝐿₃‖𝑥‖^𝑟𝑦^𝑠,𝑥,𝑦∈𝑉, (2.18) with some 𝛿,𝐿₁,𝐿₂,𝐿₃∈ℝ₊, 𝑞,𝑟∈(−∞,1), and 𝑟,𝑠∈ℝ with 𝑟+𝑠<1, then 𝐻₀(𝑥)𝐿=𝛿+¹‖𝑥‖^𝑝2−2^𝑝+𝐿₂𝑦^𝑞2−2^𝑞+𝐿₃𝑦^𝑟+𝑠2−2^𝑟+𝑠,𝑥∈𝑉. (2.19) It is easily seen that, in this way, with 𝑉=𝑋 and 𝐿₁=𝐿₂=𝐿₃=0 we get the result of Hyers [14], with 𝑉=𝑋, 𝑝=𝑞, 𝐿₁=𝐿₂ and 𝛿=𝐿₃=0 we obtain Theorem 2.1, with 𝑉=𝑋 and 𝛿=𝐿₁=𝐿₂=0 we have the results of Rassias [26, 27].

Remark 2.3. Actually, as it is easily seen in the proof of [25, Theorem 1], it is enough to assume in Corollary 2.2 that (𝑋,+) is a commutative semigroup that is uniquely divisible by 2 (i.e., for each 𝑥∈𝑋 there exists a unique 𝑦∈𝑋 with 𝑥=𝑦+𝑦.)
For recent results on stability of some conditional versions of the Cauchy functional equation (2.8) we refer to, for example, [28–31].

3. Stability of the Linear Functional Equation in Single Variable

In this section 𝕂∈{ℝ,ℂ}, 𝑋 stands for a Banach space over 𝕂, 𝑆 is a nonempty set, 𝐹∶𝑆→𝑋, 𝑚∈ℕ, 𝑓₁,…,𝑓_𝑚∶𝑆→𝑆, and 𝑎₁,…,𝑎_𝑚∶𝑆→𝕂, unless explicitly stated otherwise.

The functional equation 𝜑(𝑥)=^𝑚_𝑖=1𝑎_𝑖(𝑥)𝜑𝑓_𝑖(𝑥)+𝐹(𝑥), (3.1) for 𝜑∶𝑆→𝑋, is known as the linear functional equation of order 𝑚. For some information on it we refer to [32, 33] and the references therein.

A simply particular case of functional equation (3.1), with 𝑆∈{ℕ₀,ℤ}, is the difference equation: 𝑦_𝑛=^𝑚_𝑖=1𝑎_𝑖(𝑛)𝑦_𝑛+𝑖+𝑏_𝑛,𝑛∈𝑆, (3.2) for sequences (𝑦_𝑛)_𝑛∈𝑆 in 𝑋, where (𝑏_𝑛)_𝑛∈𝑆 is a fixed sequence in 𝑋, namely, (3.1) becomes difference equation (3.2) with 𝑓_𝑖(𝑛)=𝑛+𝑖,𝑦_𝑛∶=𝜑(𝑛)=𝜑𝑓₁(0),𝑏_𝑛∶=𝐹(𝑛),𝑛∈𝑆. (3.3) There are only few results on stability of (3.1), and actually only of some particular cases of it. For example, [34, Corollary 4] (cf. [34, Remark 5]) yields the following stability result.

Corollary 3.1. Assume that 𝑞(𝑥)∶=^𝑚_𝑖=1𝑎_𝑖(𝑥)<1,𝑥∈𝑆, (3.4) and 𝜀∶𝑆→ℝ₊ are such that 𝜀𝑓_𝑖(𝑥)≤𝜀(𝑥),𝑞𝑓_𝑖(𝑥)≤𝑞(𝑥),𝑥∈𝑆,𝑖=1,…,𝑚, (3.5) (e.g., 𝜀 and 𝑞 are constant). If a function 𝜑∶𝑆→𝑋 satisfies the inequality 𝜑(𝑥)−^𝑚_𝑖=1𝑎_𝑖(𝑥)𝜑𝑓_𝑖(𝑥)−𝐹(𝑥)≤𝜀(𝑥),𝑥∈𝑆, (3.6) then there exists a unique solution 𝜓∶𝑆→𝑋 to (3.1) with 𝜑(𝑥)(𝑥≤𝜀(𝑥)−𝜓)1−𝑞(𝑥),𝑥∈𝑆. (3.7)

The assumption (3.4) seems to be quite restrictive. So far we only know that it can be avoided for some special cases of (3.1). For instance, this is the case when each function 𝑎_𝑖 is constant, 𝑎_𝑚 is nonzero, and 𝑓_𝑖=𝑓^𝑖 for 𝑖=1,…,𝑚 (with some function 𝑓∶𝑆→𝑆), where as usual, for each 𝑝∈ℕ₀, 𝑓^𝑝 denotes the 𝑝th iterate of 𝑓, that is, 𝑓⁰(𝑥)=𝑥,𝑓^𝑝+1(𝑥)=𝑓𝑓^𝑝(𝑥),𝑝∈ℕ₀,𝑥∈𝑆. (3.8) Then (3.1) can be written in the following form 𝜑𝑓^𝑚(𝑥)=^𝑚−1_𝑖=0𝑑_𝑖𝜑𝑓^𝑖(𝑥)+𝐹(𝑥), (3.9) with some 𝑑₀,…,𝑑_𝑚−1∈𝕂, and [35, Theorem 2] implies the following stability result.

Theorem 3.2. Let 𝛿∈ℝ₊, 𝑑₀,…,𝑑_𝑚−1∈𝕂, 𝜑_𝑠∶𝑆→𝑋 satisfy 𝜑_𝑠𝑓^𝑚(𝑥)−^𝑚−1_𝑗=0𝑑_𝑗𝜑_𝑠𝑓^𝑗(𝑥)−𝐹(𝑥)≤𝛿,𝑥∈𝑆, (3.10) and 𝑟₁,…,𝑟_𝑚∈ℂ denote the roots of the characteristic equation 𝑟^𝑚−^𝑚−1_𝑗=0𝑑_𝑗𝑟^𝑗=0. (3.11) Assume that one of the following three conditions is valid.  1^∘|𝑟_𝑗|>1 for 𝑗=1,…,𝑚.  2^∘|𝑟_𝑗|∈(1,∞)∪{0} for 𝑗=1,…,𝑚 and 𝑓 is injective.  3^∘|𝑟_𝑗|≠1 for 𝑗=1,…,𝑚 and 𝑓 is bijective. Then there is a solution 𝜑∶𝑆→𝑋 of (3.9) with 𝜑_𝑠(𝑥)(𝑥≤𝛿−𝜑)1−𝑟₁⋅…⋅1−𝑟_𝑚,𝑥∈𝑆. (3.12) Moreover, in the case where 1^∘ or 3^∘ holds, 𝜑 is the unique solution of (3.9) such that sup_𝑥∈𝑆𝜑_𝑠(𝑥)−𝜑(𝑥)<∞. (3.13)

The following example (see [35, Example 1]) shows that the statement of Theorem 3.2 need not to be valid in the general situation if |𝑟_𝑗|=1 for some 𝑗∈{1,…,𝑚}.

Example 3.3. Fix 𝛿>0. Let 𝑆=𝑋=𝕂 and let the functions 𝑓 and 𝜑_𝑠 be given by 𝑓(𝑥)=𝑥+1,𝜑_𝑠(𝑥)𝛿∶=2𝑥²,𝑥∈𝕂. (3.14) Then it is easily seen that 𝜑_𝑠𝑓²(𝑥)−2𝜑_𝑠𝑓(𝑥)+𝜑_𝑠(𝑥=𝛿)2(𝑥+2)²−𝛿(𝑥+1)²+𝛿2𝑥²=𝛿,𝑥∈𝕂. (3.15) Suppose that 𝜑∶𝕂→𝕂 is a solution of 𝜑𝑓²(𝑥)=2𝜑𝑓(𝑥)−𝜑(𝑥). (3.16) Clearly, 𝑟²−2𝑟+1=0 (3.17) is the characteristic equation of (3.16) with the roots 𝑟₁=𝑟₂=1. Let 𝜓(𝑥)∶=𝜑(𝑥+1)−𝜑(𝑥),𝑥∈𝕂. (3.18) Then it is easily seen that 𝜓(𝑥+1)=𝜓(𝑥) for 𝑥∈𝕂, whence by a simple induction on 𝑛∈ℕ we get 𝜑(𝑛)=𝜑(0)+𝑛𝜓(0),𝑛∈ℕ. (3.19) Consequently lim_𝑛→∞𝜑_𝑠(𝑛)−𝜑(𝑛)=lim_𝑛→∞𝛿2𝑛²−𝜑(0)−𝑛𝜓(0)=∞, (3.20) which means that sup_𝑥∈𝕂𝜑_𝑠(𝑥)−𝜑(𝑥)=∞. (3.21) Thus we have shown that the statement of Theorem 3.2 is not valid in this case.

Estimation (3.12) is not optimal at least in some cases; for details we refer to [36, Remark 1.5, and Theorem 3.1] (see also [37]).

For some investigations of stability of the functional equation 𝜑𝑓^𝑚(𝑥)=^𝑚_𝑗=1𝑎_𝑗(𝑥)𝜑𝑓^𝑚−𝑗(𝑥)+𝐹(𝑥), (3.22) with 𝑚>1, we refer to [38] (note that the equation is a special case of (3.1) and a generalization of (3.9)). Here we only present one simplified result from there.

To this end we need a hypothesis concerning the roots of the equations 𝑧^𝑚−^𝑚_𝑗=1𝑎_𝑗(𝑥)𝑧^𝑚−𝑗=0, (3.23) with 𝑥∈𝑆, which reads as follows.

(ℋ) Functions 𝑟₁,…,𝑟_𝑚∶𝑆→ℂ satisfy the condition ^𝑚_𝑖=1𝑧−𝑟_𝑖(𝑥)=𝑧^𝑚−^𝑚_𝑗=1𝑎_𝑗(𝑥)𝑧^𝑚−𝑗,𝑥∈𝑆,𝑧∈ℂ. (3.24) Hypothesis (ℋ) means that, for every 𝑥∈𝑆, 𝑟₁(𝑥),…,𝑟_𝑚(𝑥)∈ℂ are the complex roots of (3.23). Clearly, the functions 𝑟₁,…,𝑟_𝑚 are not unique, but for every 𝑥∈𝑆 the sequence 𝑟₁(𝑥),…,𝑟_𝑚(𝑥) (3.25) is uniquely determined up to a permutation. Moreover, 0∉𝑎_𝑚(𝑆) if and only if 0∉𝑟_𝑗(𝑆) for each 𝑗=1,…,𝑚 (see [38, Remark 1]).

We say that a function 𝑔∶𝑆→𝑋 is 𝑓-invariant provided 𝑔𝑓(𝑥)=𝑔(𝑥),𝑥∈𝑆. (3.26) Now we are in a position to present a result that can de deduced from [38, Theorem 1].

Theorem 3.4. Let 𝜀₀∶𝑆→ℝ₊, let (ℋ) be valid, and let 𝑟_𝑗 be 𝑓-invariant for 𝑗>1.
Assume that 0∉𝑎_𝑚(𝑆) and 𝜑_𝑠∶𝑆→𝑋 fulfills the inequality 𝜑_𝑠𝑓^𝑚(𝑥)−^𝑚_𝑗=1𝑎_𝑗(𝑥)𝜑_𝑠𝑓^𝑚−𝑗(𝑥)−𝐹(𝑥)≤𝜀₀(𝑥),𝑥∈𝑆. (3.27) Further, suppose that 𝜀₁(𝑥)∶=^∞_𝑘=0𝜀₀𝑓^𝑘(𝑥)∏^𝑘_𝑝=0𝑟₁𝑓^𝑝(𝑥)𝜀<∞,𝑥∈𝑆,_𝑗(x)∶=^∞_𝑘=0𝜀_𝑗−1𝑓^𝑘(𝑥)𝑟_𝑗(𝑥)^𝑘+1<∞,𝑥∈𝑆,𝑗>1. (3.28) Then (3.22) has a solution 𝜑∶𝑆→𝑋 with 𝜑_𝑠(𝑥)−𝜑(𝑥)≤𝜀_𝑚(𝑥),𝑥∈𝑆. (3.29)

As it folows from [38, Remark 8], the form of 𝜑 in Theorem 3.4 can be explicitly described in some recurrent way.

Some further results on stability of (3.9), particular cases of it and some other similar equations in single variable can be found in [1, 35, 39–51]. For instance, it has been shown in [34, 52, 53] that stability of numerous functional equations of this kind is a direct consequence of some fixed point results. We deal with that issue in the section on the fixed point methods.

At the end of this part we would like to suggest some terminology that might be useful in the investigation of stability also for some other equations (as before, 𝐵^𝐷 denotes the class of functions mapping a nonempty set 𝐷 into a nonempty set 𝐵). Moreover, that terminology could be somehow helpful in clarification of the notion of nonstability, which is very briefly discussed in the next section.

Definition 3.5. Let 𝒞⊂ℝ^𝑆₊ be nonempty and let 𝒯 be an operator mapping 𝒞 into ℝ^𝑆₊. We say that (3.1) is 𝒯-stable (with uniqueness, resp.) provided for every 𝜀∈𝒞 and 𝜑∶𝑆→𝑋 with 𝜑_𝑠(𝑥)−^𝑚_𝑖=1𝑎_𝑖(𝑥)𝜑_𝑠𝑓_𝑖(𝑥)−𝐹(𝑥)≤𝜀(𝑥),𝑥∈𝑆 (3.30) there exists a (unique, resp.) solution 𝜑∶𝑆→𝑋 of (3.1) such that 𝜑(𝑥)−𝜑(𝑥)≤𝒯𝜀(𝑥),𝑥∈𝑆. (3.31)
In connection with the original statement of Ulam's problem we might think of yet another definition that seems to be quite natural and useful sometimes.

Definition 3.6. Let 𝜀∶𝑆→ℝ₊ and 𝐿∈ℝ₊. We say that functional equation (3.1) is (𝜀,𝐿)-stable (with uniqueness, resp.,) provided for every function 𝜑∶𝑆→𝑋 satisfying (3.30), there exists a (unique, resp.,) solution 𝜑∶𝑆→𝑋 to (3.1) such that 𝜑(𝑥)−𝜑(𝑥)≤𝐿𝜀(𝑥),𝑥∈𝑆. (3.32) Given 𝑎∶𝑆→ℝ₊⧵{0}, for each 𝜙∶𝑆→ℝ₊ we write 𝒜^𝑓_𝑎𝜙(𝑥)∶=^∞_𝑗=0𝜙𝑓^𝑗(𝑥)∏^𝑗_𝑘=0𝑎𝑓^𝑘(𝑥),𝑥∈𝑆,𝒟∶=𝜀∶𝑆→ℝ⁰₊∶𝒜^𝑓_𝑎𝜀(𝑥)<∞,𝑥∈𝑆. (3.33) Then 𝒜^𝑓_𝑎 is an operator mapping 𝒟 into ℝ^𝑆 and, according to Theorem 3.4, the functional equation 𝜓𝑓(𝑥)=𝑎(𝑥)𝜓(𝑥)+𝐹(𝑥),𝑥∈𝑆 (3.34) (i.e., (3.22) with 𝑚=1) is 𝒜^𝑓_𝑎-stable with uniqueness (cf. [48, Theorem 2.1]).
Further, note that for every 𝜀∈ℝ^𝑆₊ with 𝜀𝑓(𝑡)≤𝜀(𝑡),𝑡∈𝑆,𝑠∶=inf_𝑡∈𝑆|𝑎(𝑡)|>1, (3.35) we have 𝒜^𝑓_𝑎𝜀(𝑥)≤^∞_𝑗=0𝜀(𝑥)s^𝑘=𝜀(𝑥)𝑠−1,𝑥∈𝑆, (3.36) and consequently (3.34) is (𝜀,𝐿)-stable with 1𝐿∶=.𝑠−1 (3.37)

4. Nonstability

There are only few outcomes of which we could say that they concern nonstability of functional equation. The first well-known one is due to Gajda [19] and answers a question raised by Rassias [54]. Namely, he gave an example of a function showing that a result analogous to that described in Theorem 2.1 cannot be obtained for 𝑝=1 (for further such examples see [21]; cf. also, e.g., [55, 56]).

In general it is not easy to define the notion of nonstability precisely, mostly because at the moment there are several notions of stability in use (see [9, 57]). For instance, we could understand nonstability as in Example 3.3. The other possibility is to refer to Definitions 3.5 and 3.6 and define 𝒯-nonstability and (𝜀,𝐿)-nonstability, respectively. Finally, if there does not exist an 𝐿∈ℝ₊ such that the equation is (𝜀,𝐿)-stable, then we could say that it is 𝜀-nonstable.

For some further propositions of such definitions and preliminary results on nonstability we refer to [58–62]. As an example we present below the result from [60, Theorem 1] concerning nonstability of the difference equation 𝑥_𝑛+1=𝑎_𝑛𝑥_𝑛+𝑏_𝑛,𝑛∈ℕ₀, (4.1) where (𝑥_𝑛)_𝑛≥0 and (𝑏_𝑛)_𝑛≥0 are sequences in 𝑋 and (𝑎_𝑛)_𝑛≥0 is a sequence in 𝕂.

Theorem 4.1. Let (𝜀_𝑛)_𝑛≥0 be a sequence of positive real numbers, (𝑏_𝑛)_𝑛≥0 a sequence in 𝑋, and (𝑎_𝑛)_𝑛≥0 a sequence in 𝕂 with the property lim_𝑛→∞𝜀_𝑛𝑎_𝑛+1𝜀_𝑛+1=1. (4.2) Then there exists a sequence (𝑦_𝑛)_𝑛≥0 in 𝑋 with 𝑦_𝑛+1−𝑎_𝑛𝑦_𝑛−𝑏_𝑛≤𝜀_𝑛,𝑛∈ℕ₀, (4.3) such that, for every sequence (𝑥_𝑛)_𝑛≥0 in 𝑋 satisfying recurrence (4.1), sup_𝑛∈ℕ𝑥_𝑛−𝑦_𝑛𝜀_𝑛−1=∞. (4.4)

The issue of nonstability seems to be a new promising area for research.

5. Stability and Completeness

It is well known that the completeness of the target space is of great importance in the theory of Hyers-Ulam stability of functional equations; we could observe this fact for the stability of the Cauchy equation in the second section.

In [63], Forti and Schwaiger proved that if 𝑋 is a commutative group containing an element of infinite order, 𝑌 is a normed space, and the Cauchy functional equation is Hyers-Ulam stable in the class 𝑌^𝑋, then the space 𝑌 has to be complete (let us also mention here that Moszner [64] showed that all four assumptions are essential to get the completeness of 𝑌).

The above-described effect, stability implies completeness, was recently proved for some other equations (see [65–68]). Here we present only one result of this kind. It concerns the quadratic equation 𝑓𝑥+𝑦+𝑓𝑥−𝑦=2𝑓(𝑥)+2𝑓𝑦 (5.1) and comes from [67].

Theorem 5.1. Let 𝑋 be a finitely generated free commutative group and 𝑌 be a normed space. If (5.1) is Hyers-Ulam stable in the class 𝑌^𝑋, then the space 𝑌 is complete.

6. The Method of Forti

As Forti [43] (see also, e.g., [69]) has clearly demonstrated, the stability of functional equations in single variable, in particular of the form: Ψ∘𝐹∘𝑎=𝐹 (6.1) plays a basic role in many investigations of the stability of functional equations in several variables. Some examples presenting that method can be found in [25, 70, 71] (see also [72]). Here we give only one such example that corrects [70, Corollary 3.2], which unfortunately has been published with some details confused. The main tool is the following theorem (see [70, Theorem 2.1]; cf. [43]).

Theorem 6.1. Assume that (𝑌,𝑑) is a complete metric space, 𝐾 is a nonempty set, 𝑓∶𝐾→𝑌, Ψ∶𝑌→𝑌, 𝑎∶𝐾→𝐾, ℎ∶𝐾→ℝ₊, 𝜆∈ℝ₊, 𝑑(𝑥)(𝑥)(𝑥)𝑑Ψ(𝑥)𝑦𝐻(𝑥)Ψ∘𝑓∘𝑎,𝑓≤ℎ,𝑥∈𝐾,,Ψ≤𝜆𝑑𝑥,𝑦,𝑥,𝑦∈𝑌,∶=^∞_𝑖=0𝜆^𝑖ℎ𝑎^𝑖(𝑥)<∞,𝑥∈𝐾. (6.2) Then, for every 𝑥∈𝐾, the limit 𝐹(𝑥)∶=lim_𝑛→∞Ψ^𝑛∘𝑓∘𝑎^𝑛(𝑥) (6.3) exists and 𝐹∶𝐾→𝑌 is the unique function such that (6.1) holds and 𝑑𝑓(𝑥),𝐹(𝑥)≤𝐻(𝑥),𝑥∈𝐾. (6.4)

The next corollary presents the corrected version of [70, Corollary 3.2] and its proof. Let us make some preparations for it.

First, let us recall that a groupoid (𝐺,+) (i.e., a nonempty set 𝐺 endowed with a binary operation +∶𝐺²→𝐺) is uniquely divisible by 2 provided, for each 𝑥∈𝑋, there is a unique 𝑦∈𝑋 with 𝑥=2𝑦∶=𝑦+𝑦; such 𝑦 we denote by (1/2)𝑥. Next, we use the notion: 2⁰𝑥∶=𝑥,2^𝑛𝑥=22^𝑛−1𝑥, (6.5) and (only if the groupoid is uniquely divisible by 2) 2^−𝑛1𝑥=22^−𝑛+1𝑥, (6.6) for every 𝑥∈𝐺, 𝑛∈ℕ.

A groupoid (𝐺,+) is square symmetric provided the operation + is square symmetric, that is, 2(𝑥+𝑦)=2𝑥+2𝑦 for 𝑥,𝑦∈𝐺; it is easy to show by induction that, for each 𝑛∈ℕ (for all 𝑛∈ℤ, if the groupoid is uniquely divisible by 2), we have 2^𝑛𝑥+𝑦=2^𝑛𝑥+2^𝑛𝑦,𝑥,𝑦∈𝐺. (6.7)

Clearly every commutative semigroup is a square symmetric groupoid. Next, let 𝑋 be a linear space over a field 𝕂, 𝑎,𝑏∈𝕂, 𝑧∈𝑋, and define a binary operation ∗∶𝑋²→𝑋 by 𝑥∗𝑦∶=𝑎𝑥+𝑏𝑦+𝑧,𝑥,𝑦∈𝑋. (6.8) Then it is easy to check that (𝑋,∗) provides a simple example of a square symmetric groupoid.

The square symmetric groupoids have been already considered in several papers investigating the stability of some functional equations (see, e.g., [73–79]). For a description of square symmetric operations we refer to [80].

Finally, we say that (𝐺,+,𝑑) is a complete metric groupoid provided (𝐺,+) is a groupoid, (𝐺,𝑑) is a complete metric space, and the operation +∶𝐺²→𝐺 is continuous, in both variables simultaneously, with respect to the metric 𝑑.

Now we are in a position to present the mentioned above corrected version of [70, Corollary 3.2].

Corollary 6.2. Let (𝑋,+) and (𝑌,+) be square symmetric groupoids, (𝑌,+) be uniquely divisible by 2, (𝑌,+,𝑑) be a complete metric groupoid, 𝐾⊂𝑋, 2𝐾⊂𝐾 (i.e., 2𝑎∈𝐾 for 𝑎∈𝐾), and 𝜒∶𝑋²→ℝ₊. Suppose that there exist 𝜉,𝜂∈ℝ₊ such that 𝜉𝜂<1, 𝑑121𝑥,2𝑦𝜒≤𝜉𝑑𝑥,𝑦,x,𝑦∈𝑌,2𝑥,2𝑦≤𝜂𝜒𝑥,𝑦,𝑥,𝑦∈𝐾, (6.9) and 𝜑∶𝐾→𝑌 satisfies 𝑑𝜑𝑥+𝑦,𝜑(𝑥)+𝜑𝑦≤𝜒𝑥,𝑦,𝑥,𝑦∈𝐾,𝑥+𝑦∈𝐾. (6.10) Then there is a unique function 𝐹∶𝐾→𝑌 with 𝐹(𝑥)𝑦𝑑𝜑(𝑥)(𝑥)≤𝜉𝑥+𝑦=𝐹+𝐹,𝑥,𝑦∈𝐾,𝑥+𝑦∈𝐾,,𝐹𝜒()1−𝜉𝜂𝑥,𝑥,𝑥∈𝐾. (6.11)(6.12)

Proof. From (6.10), with 𝑥=𝑦, we obtain 𝑑(𝜑(2𝑥),2𝜑(𝑥))≤𝜒(𝑥,𝑥) for 𝑥∈𝐾, which yields 𝑑12𝜑(2𝑥),𝜑(𝑥)≤𝜉𝑑𝜑(2𝑥),2𝜑(𝑥)≤𝜉𝜒(𝑥,𝑥),𝑥∈𝐾. (6.13) Hence, by Theorem 6.1 (with 𝜆=𝜉, 𝑓=𝜑, Ψ(𝑧)=(1/2)𝑧, ℎ(𝑥)=𝜉𝜒(𝑥,𝑥), and 𝑎(𝑥)=2𝑥) the limit 𝐹(𝑥)∶=lim_𝑛→∞2^−𝑛𝜑2^𝑛𝑥 (6.14) exists for every 𝑥∈𝐾 and 𝑑𝜑(𝑥),𝐹(𝑥)≤𝜉𝜒(𝑥,𝑥)^∞_𝑖=0𝜉𝜂^𝑖≤𝜉𝜒()1−𝜉𝜂𝑥,𝑥,𝑥∈𝐾. (6.15) Next, by (6.7) and (6.10), for every 𝑥,𝑦∈𝐾 with 𝑥+𝑦∈𝐾, we have 𝑑2^−𝑛𝜑2^𝑛𝑥+𝑦,2^−𝑛𝜑2^𝑛𝑥+2^−𝑛𝜑2^𝑛𝑦≤𝜉𝜂^𝑛𝜒𝑥,𝑦, (6.16) for 𝑛∈ℕ, whence letting 𝑛→∞ we deduce that 𝐹 is a solution of (6.11).
It remains to show the uniqueness of 𝐹. So suppose that 𝐺∶𝐾→𝑌, 𝑑𝜑(𝑥)(𝑥)≤𝜉,𝐺𝜒()𝐺(𝑥)𝑦1−𝜉𝜂𝑥,𝑥,𝑥∈𝐾,𝑥+𝑦=𝐺+𝐺,𝑥,𝑦∈𝐾,𝑥+𝑦∈𝐾. (6.17)(6.18) Then 𝑑(𝐹(𝑥),𝐺(𝑥))≤𝑑𝐹(𝑥),𝜑(𝑥)+𝑑𝜑(𝑥),𝐺(𝑥)≤2𝜉𝜒()1−𝜉𝜂𝑥,𝑥,𝑥∈𝐾, (6.19) and by induction it is easy to show that (6.11) and (6.18) yield 𝐹(2^𝑛𝑥)=2^𝑛𝐹(𝑥) and 𝐺(2^𝑛𝑥)=2^𝑛𝐺(𝑥) for every 𝑥∈𝐾 and 𝑛∈ℕ. Hence, for each 𝑥∈𝐾, 𝑑(𝐹(𝑥),𝐺(𝑥))=𝑑2^−𝑛𝐹2^𝑛𝑥,2^−𝑛𝐺2^𝑛𝑥≤𝜉^𝑛𝜒2^𝑛𝑥,2^𝑛𝑥≤𝜉𝜂^𝑛𝜒(𝑥,𝑥). (6.20) Since 𝜉𝜂<1, letting 𝑛→∞ we get 𝐹=𝐺.

7. The Fixed Point Methods

Apart from the classical method applied by Hyers and its modification proposed by Forti (see also [72]), the fixed point methods seem to be the most popular at the moment in the investigations of the stability of functional equations, both in single and several variables. Although the fixed point method was used for the first time by Baker [39] who applied a variant of Banach's fixed point theorem to obtain the Hyers-Ulam stability of the functional equation 𝑓(𝑡)=𝐹𝑡,𝑓𝜑(𝑡), (7.1) most authors follow Radu's approach (see [81], where a new proof of Theorem 2.1 for 𝑝∈ℝ₊⧵{1} was given) and make use of a theorem of Diaz and Margolis. Here we only present one of the recent results obtained in this way.

Let us recall that a mapping 𝑓∶𝑉^𝑛→𝑊, where 𝑉 is a commutative group, 𝑊 is a linear space, and 𝑛 is a positive integer, is called multiquadratic if it is quadratic in each variable. Similarly we define multiadditive and multi-Jensen mappings. Some basic facts on multiadditive functions can be found for instance in [82] (where their application to the representation of polynomial functions is also presented), whereas for the general form of multi-Jensen mappings and their connection with generalized polynomials we refer to [83].

The stability of multiadditive, multi-Jensen, and multiquadratic mappings was recently investigated in [68, 84–93]. In particular, in [88] Radu's approach was applied to the proof of the following theorem.

Theorem 7.1. Let 𝑊 be a Banach space and for every 𝑖∈{1,…,𝑛}, let 𝜑_𝑖∶𝑉^𝑛+1→ℝ₊ be a mapping such thatlim_𝑗→∞14^𝑗𝜑_𝑖2^𝑗𝑥₁,𝑥₂,…,𝑥_𝑛+1=⋯=lim_𝑗→∞14^𝑗𝜑_𝑖𝑥₁,…,𝑥_𝑖−2,2^𝑗𝑥_𝑖−1,𝑥_𝑖,…,𝑥_𝑛+1=lim_𝑗→∞14^𝑗𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,2^𝑗x_𝑖,2^𝑗𝑥_𝑖+1,𝑥_𝑖+2,…,𝑥_𝑛+1=lim_𝑗→∞14^𝑗𝜑_𝑖𝑥₁,…,𝑥_𝑖+1,2^𝑗𝑥_𝑖+2,𝑥_𝑖+3,…,𝑥_𝑛+1=⋯=lim_𝑗→∞14^𝑗𝜑_𝑖𝑥₁,…,𝑥_𝑛,2^𝑗𝑥_𝑛+1=0,𝑥₁,…,𝑥_𝑛+1∈𝑉^𝑛+1, (7.2)𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,2𝑥_𝑖,2𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛≤4𝐿_𝑖𝜑_𝑖𝑥₁,…,𝑥_𝑖,𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛,𝑥₁,…,𝑥_𝑛∈𝑉^𝑛, (7.3) for an 𝐿_𝑖∈(0,1). If 𝑓∶𝑉^𝑛→𝑊 is a mapping satisfying, for any 𝑖∈{1,…,𝑛}, 𝑓𝑥₁,…,𝑥_𝑖−1,0,𝑥_𝑖+1,…,𝑥_𝑛=0,𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖+1,…,𝑥_𝑛∈𝑉^𝑛−1, (7.4)𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖+𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑛+𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖−𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑛−2𝑓𝑥₁,…,𝑥_𝑛−2𝑓𝑥₁,…,𝑥_𝑖−1,𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑛≤𝜑_𝑖𝑥₁,…,𝑥_𝑖,𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑛,𝑥₁,…,𝑥_𝑖,𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑛∈𝑉^𝑛+1, (7.5) then for every 𝑖∈{1,…,𝑛} there exists a unique multiquadratic mapping 𝐹_𝑖∶𝑉^𝑛→𝑊 such that 𝑓𝑥₁,…,𝑥_𝑛−𝐹_𝑖𝑥₁,…,𝑥_𝑛≤14−4𝐿_𝑖𝜑_𝑖𝑥₁,…,𝑥_𝑖,𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛,𝑥₁,…,𝑥_𝑛∈𝑉^𝑛. (7.6)

Baker's idea (to prove his result it is enough to define suitable (complete) metric space and (contractive) operator, which form follows from the considered equation (in this case 𝑇(𝑎)(𝑡)∶=𝐹(𝑡,𝑎(𝜑(𝑡)))), and apply the (Banach) fixed point theorem) was used by several mathematicians, who applied other fixed point theorems to extend and generalize Baker's result. Now, we present some of these recent outcomes.

To formulate the first of them, let us recall that a mapping 𝛾∶ℝ₊→ℝ₊ is called a comparison function if it is nondecreasing and lim_𝑛→∞𝛾^𝑛(𝑡)=0,𝑡∈(0,∞). (7.7) In [94], Matkowski's fixed point theorem was applied to the proof of the following generalization of Baker's result.

Theorem 7.2. Let 𝑆 be a nonempty set, let (𝑋,𝑑) be a complete metric space, 𝜑∶𝑆→𝑆, and 𝐹∶𝑆×𝑋→𝑋. Assume also that 𝑑(𝐹(𝑡,𝑢),𝐹(𝑡,𝑣))≤𝛾(𝑑(𝑢,𝑣)),𝑡∈𝑆,𝑢,𝑣∈𝑋, (7.8) where 𝛾∶ℝ₊→ℝ₊ is a comparison function, and let 𝑔∶𝑆→𝑋, 𝛿>0 be such that 𝑑𝑔(𝑡),𝐹𝑡,𝑔𝜑(𝑡)≤𝛿,𝑡∈𝑆. (7.9) Then there is a unique function 𝑓∶𝑆→𝑋 satisfying (7.1) and 𝜌𝑓,𝑔∶=sup𝑑𝑓(𝑡),𝑔(𝑡),𝑡∈𝑆<∞. (7.10) Moreover, 𝜌(𝑓,𝑔)−𝛾(𝜌(𝑓,𝑔))≤𝛿.

On the other hand, in [95], Baker's idea and a variant of Ćirić's fixed point theorem were used to obtain the following result concerning the stability of (7.1).

Theorem 7.3. Let 𝑆 be a nonempty set, let (𝑋,𝑑) be a complete metric space, 𝜑∶𝑆→𝑆, and 𝐹∶𝑆×𝑋→𝑋 and 𝑑𝐹(𝑡,𝑥),𝐹𝑡,𝑦≤𝛼₁𝑥,𝑦𝑑𝑥,𝑦+𝛼₂𝑥,𝑦𝑑(𝑥,𝐹(𝑡,𝑥))+𝛼₃𝑥,𝑦𝑑𝑦,𝐹𝑡,𝑦+𝛼₄𝑥,𝑦𝑑𝑥,𝐹𝑡,𝑦+𝛼₅𝑥,𝑦𝑑𝑦,𝐹(𝑡,𝑥),𝑡∈𝑆,𝑥,𝑦∈𝑋, (7.11) where 𝛼₁,…,𝛼₅∶𝑋×𝑋→ℝ₊ satisfy ⁵_𝑖=1𝛼_𝑖𝑥,𝑦≤𝜆, (7.12) for all 𝑥,𝑦∈𝑋 and a 𝜆∈[0,1). If 𝑔∶𝑆→𝑋, 𝛿>0, and (7.9) holds, then there is a unique function 𝑓∶𝑆→𝑋 satisfying (7.1) and 𝑑𝑓(𝑡)(𝑡)≤()𝛿,𝑔2+𝜆2(1−𝜆),𝑡∈𝑆. (7.13)

A consequence of Theorem 7.3 is the following result on the stability of the linear functional equation of order 1.

Corollary 7.4. Let 𝑆 be a nonempty set, let 𝐸 be a real or complex Banach space, 𝜑∶𝑆→𝑆, 𝛼∶𝑆→𝐸, 𝐵∶𝑆→𝔏(𝐸) (here 𝔏(𝐸) denotes the Banach algebra of all bounded linear operators on 𝐸), 𝜆∈[0,1), and ‖𝐵(𝑡)‖≤𝜆,𝑡∈𝑆. (7.14) If 𝑔∶𝑆→𝐸, 𝛿>0, and 𝑔(𝑡)−𝛼(𝑡)+𝐵(𝑡)𝑔𝜑(𝑡)≤𝛿,𝑡∈𝑆, (7.15) then there exists a unique function 𝑓∶𝑆→𝐸 satisfying 𝑓(𝑡)=𝛼(𝑡)+𝐵(𝑡)𝑓𝜑(𝑡),𝑡∈𝑆, (7.16) and the condition 𝑓(𝑡)(𝑡≤𝛿−𝑔)1−𝜆,𝑡∈𝑆. (7.17)

In [96], Miheţ gave one more generalization of Baker's result. In order to do this he proved a fixed point alternative and used it in the proof of this generalization. To formulate Miheţ's theorem, let us recall that a mapping 𝛾∶[0,∞]→[0,∞] is called a generalized strict comparison function if it is nondecreasing, 𝛾(∞)=∞, lim_𝑛→∞𝛾^𝑛(𝑡)=0,𝑡∈(0,∞),lim_𝑡→∞𝑡−𝛾(𝑡)=∞. (7.18)

Theorem 7.5. Let 𝑆 be a nonempty set, let (𝑋,𝑑) be a complete metric space, 𝜑∶𝑆→𝑆, and 𝐹∶𝑆×𝑋→𝑋. Assume also that 𝑑(𝐹(𝑡,𝑢),𝐹(𝑡,𝑣))≤𝛾(𝑑(𝑢,𝑣)),𝑡∈𝑆,𝑢,𝑣∈𝑋, (7.19) where 𝛾∶[0,∞]→[0,∞] is a generalized strict comparison function and let 𝑔∶𝑆→𝑋, 𝛿>0 be such that (7.9) holds. Then there is a unique function 𝑓∶𝑆→𝑋 satisfying (7.1) and 𝑑𝑓(𝑡),𝑔(𝑡)≤sup𝑠>0∶𝑠−𝛾(𝑠)≤𝛿,𝑡∈𝑆. (7.20)

A somewhat different fixed point approach to the Hyers-Ulam stability of functional equations, in which the stability results are simple consequences of some new fixed point theorems, can be found in [34, 52, 53, 97].

Given a nonempty set 𝑆 and a metric space (𝑋,𝑑), we define Δ∶(𝑋^𝑆)²→ℝ₊^𝑆 by Δ𝜉,𝜇(𝑡)∶=𝑑𝜉(𝑡),𝜇(𝑡),𝜉,𝜇∈𝑋^𝑆,𝑡∈𝑆. (7.21) Now, we are in a position to present the following fixed point theorem from [34].

Theorem 7.6. Let 𝑆 be a nonempty set, let (𝑋,𝑑) be a complete metric space, 𝑘∈ℕ, 𝑓₁,…,𝑓_𝑘∶𝑆→𝑆, 𝐿₁,…,𝐿_𝑘∶𝑆→ℝ₊, and let Λ∶ℝ^𝑆₊→ℝ^𝑆₊ be given by (Λ𝛿)(𝑡)∶=^𝑘_𝑖=1𝐿_𝑖(𝑡)𝛿𝑓_𝑖(𝑡),𝛿∈ℝ^𝑆₊,𝑡∈𝑆. (7.22) If 𝒯∶𝑋^𝑆→𝑋^𝑆 is an operator satisfying the inequality Δ𝒯𝜉,𝒯𝜇(𝑡)≤ΛΔ𝜉,𝜇(𝑡),𝜉,𝜇∈𝑋^𝑆,𝑡∈𝑆 (7.23) and functions 𝜀∶𝑆→ℝ₊ and 𝑔∶𝑆→𝑋 are such that Δ𝒯𝑔,𝑔(𝑡)≤𝜀(𝑡),𝑡∈𝑆,_∞𝑛=0Λ^𝑛𝜀(𝑡)=∶𝜎(𝑡)<∞,𝑡∈𝑆, (7.24)(7.25) then for every 𝑡∈𝑆 the limit lim_𝑛→∞𝒯^𝑛𝑔(𝑡)=∶𝑓(𝑡) (7.26) exists and the function 𝑓∶𝑆→𝑋, defined in this way, is a unique fixed point of 𝒯 with Δ𝑔,𝑓(𝑡)≤𝜎(𝑡),𝑡∈𝑆. (7.27)

A consequence of Theorem 7.6 is the following result on the stability of a quite wide class of functional equations in a single variable.

Corollary 7.7. Let 𝑆 be a nonempty set, let (𝑋,𝑑) be a complete metric space, 𝑘∈ℕ, 𝑓₁,…,𝑓_𝑘∶𝑆→𝑆, 𝐿₁,…,𝐿_𝑘∶𝑆→ℝ₊, and let a function Φ∶𝑆×𝑋^𝑘→𝑋 satisfy the inequality 𝑑Φ𝑡,𝑦₁,…,𝑦_𝑘,Φ𝑡,𝑧₁,…,𝑧_𝑘≤^𝑘_𝑖=1𝐿_𝑖(𝑡)𝑑𝑦_𝑖,𝑧_𝑖, (7.28) for any (𝑦₁,…,𝑦_𝑘),(𝑧₁,…,𝑧_𝑘)∈𝑋^𝑘 and 𝑡∈𝑆, and 𝒯∶𝑋^𝑆→𝑋^𝑆 be an operator defined by 𝒯𝜑(𝑡)∶=Φ𝑡,𝜑𝑓₁(𝑡),…,𝜑𝑓_𝑘(𝑡),𝜑∈𝑋^𝑆,𝑡∈𝑆. (7.29) Assume also that Λ is given by (7.22) and functions 𝑔∶𝑆→𝑋 and 𝜀∶𝑆→ℝ₊ are such that 𝑑𝑔(𝑡),Φ𝑡,𝑔𝑓₁(𝑡),…,𝑔𝑓_𝑘(𝑡)≤𝜀(𝑡),𝑡∈𝑆 (7.30) and (7.25) holds. Then for every 𝑡∈𝑆 limit (7.26) exists and the function 𝑓∶𝑆→𝑋 is a unique solution of the functional equation Φ𝑡,𝑓𝑓₁(𝑡),…,𝑓𝑓_𝑘(𝑡)=𝑓(𝑡),𝑡∈𝑆 (7.31) satisfying inequality (7.27).

Let us also mention here that very recently Cădariu et al. [97] improved the above two outcomes considering, instead of that given by (7.22), a more general operator Λ.

Next, following [53], we deal with the case of non-Archimedean metric spaces. In order to do this, we introduce some notations and definitions.

Let 𝑆 be a nonempty set. For any 𝛿₁,𝛿₂∈ℝ^𝑆₊ we write 𝛿₁≤𝛿₂ provided 𝛿₁(𝑡)≤𝛿₂(𝑡),𝑡∈𝑆, (7.32) and we say that an operator Λ∶ℝ^𝑆₊→ℝ^𝑆₊ is nondecreasing if it satisfies the condition Λ𝛿₁≤Λ𝛿₂,𝛿₁,𝛿₂∈ℝ^𝑆₊,𝛿₁≤𝛿₂. (7.33) Moreover, given a sequence (𝑔_𝑛)_𝑛∈ℕ in ℝ^𝑆₊, we write lim_𝑛→∞𝑔_𝑛=0 provided lim_𝑛→∞𝑔_𝑛(𝑡)=0,𝑡∈𝑆. (7.34) We will also use the following hypothesis concerning operators Λ∶ℝ^𝑆₊→ℝ^𝑆₊:

(𝒞) lim_𝑛→∞Λ𝛿_𝑛=0 for every sequence (𝛿_𝑛)_𝑛∈ℕ in ℝ^𝑆₊ with lim_𝑛→∞𝛿_𝑛=0.

Finally, recall that a metric 𝑑 on a nonempty set 𝑋 is called non-Archimedean (or an ultrametric) provided 𝑑(𝑥,𝑧)≤max𝑑𝑥,𝑦,𝑑𝑦,𝑧,𝑥,𝑦,𝑧∈𝑋. (7.35) We can now formulate the following fixed point theorem.

Theorem 7.8. Let 𝑆 be a nonempty set, let (𝑋,𝑑) be a complete non-Archimedean metric space, and let Λ∶ℝ^𝑆₊→ℝ^𝑆₊ be a nondecreasing operator satisfying hypothesis (𝒞). If 𝒯∶𝑋^𝑆→𝑋^𝑆 is an operator satisfying inequality (7.23) and functions 𝜀∶𝑆→ℝ₊ and 𝑔∶𝑆→𝑋 are such that lim_𝑛→∞Λ^𝑛𝜀=0, (7.36) and (7.24) holds, then for every 𝑡∈𝑆 limit (7.26) exists and the function 𝑓∶𝑆→𝑋, defined in this way, is a fixed point of 𝒯 with Δ𝑔,𝑓(𝑡)≤sup_𝑛∈ℕ0Λ^𝑛𝜀(𝑡)=∶𝜎(𝑡),𝑡∈𝑆. (7.37) If, moreover, (Λ𝜎)(𝑡)≤sup_𝑛∈ℕ0Λ^𝑛+1𝜀(𝑡),𝑡∈𝑆, (7.38) then 𝑓 is the unique fixed point of 𝒯 satisfying (7.37).

An immediate consequence of Theorem 7.8 is the following result on the stability of (7.31) in complete non-Archimedean metric spaces.

Corollary 7.9. Let 𝑆 be a nonempty set, (𝑋,𝑑) be a complete non-Archimedean metric space, 𝑘∈ℕ, 𝑓₁,…,𝑓_𝑘∶𝑆→𝑆, 𝐿₁,…,𝐿_𝑘∶𝑆→ℝ₊, and a function Φ∶𝑆×𝑋^𝑘→𝑋 satisfy the inequality 𝑑Φ𝑡,𝑦₁,…,𝑦_𝑘,Φ𝑡,𝑧₁,…,𝑧_𝑘≤max_{{}𝑖∈1,…,𝑘}𝐿_𝑖(𝑡)𝑑𝑦_𝑖,𝑧_𝑖, (7.39) for any (𝑦₁,…,𝑦_𝑘),(𝑧₁,…,𝑧_𝑘)∈𝑋^𝑘 and 𝑡∈𝑆, and 𝒯∶𝑋^𝑆→𝑋^𝑆 be an operator defined by (7.29). Assume also that Λ is given by (Λ𝛿)(𝑡)∶=max_{{}𝑖∈1,…,𝑘}𝐿_𝑖(𝑡)𝛿𝑓_𝑖(𝑡),𝛿∈ℝ^𝑆₊,𝑡∈𝑆, (7.40) and functions 𝑔∶𝑆→𝑋 and 𝜀∶𝑆→ℝ₊ are such that (7.30) and (7.36) hold. Then for every 𝑡∈𝑆 limit (7.26) exists and the function 𝑓∶𝑆→𝑋 is a solution of functional equation (7.31) satisfying inequality (7.37).

Given nonempty sets 𝑆,𝑍 and functions 𝜑∶𝑆→𝑆, 𝐹∶𝑆×𝑍→𝑍, we define an operator ℒ^𝐹_𝜑∶𝑍^𝑆→𝑍^𝑆 by ℒ^𝐹_𝜑𝑔(𝑡)∶=𝐹𝑡,𝑔𝜑(𝑡),𝑔∈𝑍^𝑆,𝑡∈𝑆, (7.41) and we say that 𝒰∶𝑍^𝑆→𝑍^𝑆 is an operator of substitution provided 𝒰=ℒ^𝐺_𝜓 with some 𝜓∶𝑆→𝑆 and 𝐺∶𝑆×𝑍→𝑍. Moreover, if 𝐺(𝑡,⋅) is continuous for each 𝑡∈𝑆 (with respect to a topology in 𝑍), then we say that 𝒰 is continuous.

The following fixed point theorem was proved in [52].

Theorem 7.10. Let 𝑆 be a nonempty set, let (𝑋,𝑑) be a complete metric space, Λ∶𝑆×ℝ₊→ℝ₊, 𝒯∶𝑋^𝑆→𝑋^𝑆, 𝜑∶𝑆→𝑆, and Δ𝒯𝛼,𝒯𝛽(𝑡)≤Λ𝑡,Δ𝛼∘𝜑,𝛽∘𝜑(𝑡),𝛼,𝛽∈𝑋^𝑆,𝑡∈𝑆. (7.42) Assume also that for every 𝑡∈𝑆, Λ_𝑡∶=Λ(𝑡,⋅) is nondecreasing, 𝜀∶𝑆→ℝ₊, 𝑔∶𝑆→𝑋, ^∞_𝑛=0ℒ^Λ_𝜑^𝑛𝜀(𝑡)=∶𝜎(𝑡)<∞,𝑡∈𝑆, (7.43) and (7.24) holds. Then for every 𝑡∈𝑆 limit (7.26) exists and inequality (7.27) is satisfied. Moreover, the following two statements are true. (i)If 𝒯 is a continuous operator of substitution or Λ_𝑡 is continuous at 0 for each 𝑡∈𝑆, then 𝑓 is a fixed point of 𝒯. (ii)If Λ_𝑡 is subadditive (that is, Λ_𝑡(𝑎+𝑏)≤Λ_𝑡(𝑎)+Λ_𝑡(𝑏), (7.44) for all 𝑎,𝑏∈ℝ₊) for each 𝑡∈𝑆, then 𝒯 has at most one fixed point 𝑓∈𝑋^𝑆 such that Δ𝑔,𝑓(𝑡)≤𝑀𝜎(𝑡),𝑡∈𝑆, (7.45)for a positive integer 𝑀.

Theorem 7.10 with 𝒯=ℒ^𝐹_𝜑 immediately gives the following generalization of Baker's result.

Corollary 7.11. Let 𝑆 be a nonempty set, let (𝑋,𝑑) be a complete metric space, 𝐹∶𝑆×𝑋→𝑋, Λ∶𝑆×ℝ₊→ℝ₊, and 𝑑𝐹(𝑡,𝑥),𝐹𝑡,𝑦≤Λ𝑡,𝑑𝑥,𝑦,𝑡∈𝑆,𝑥,𝑦∈𝑋. (7.46) Assume also that 𝜑∶𝑆→𝑆, 𝜀∶𝑆→ℝ₊, (7.43) holds, 𝑔∶𝑆→𝑋, for every 𝑡∈𝑆, Λ_𝑡∶=Λ(𝑡,⋅) is nondecreasing, 𝐹(𝑡,⋅) is continuous, and 𝑑𝑔(𝑡),𝐹𝑡,𝑔𝜑(𝑡)≤𝜀(𝑡),𝑡∈𝑆. (7.47) Then for every 𝑡∈𝑆 the limit 𝑓(𝑡)∶=lim_𝑛→∞ℒ^𝐹_𝜑^𝑛𝑔(𝑡) (7.48) exists, (7.27) holds and 𝑓 is a solution of (7.1). Moreover, if for every 𝑡∈𝑆, Λ_𝑡 is subadditive and 𝑀∈ℕ, then 𝑓∶𝑆→𝑋 is the unique solution of (7.1) fulfilling (7.45).

Let us finally mention that the fixed point method is also a useful tool for proving the Hyers-Ulam stability of differential (see [98, 99]) and integral equations (see for instance [100–102]). Some further details and information on the connections between the fixed point theory and the Hyers-Ulam stability can be found in [103].

8. Stability in Non-Archimedean Spaces

Let us recall that a non-Archimedean valuation in a field 𝕂 is a function |⋅|∶𝕂→ℝ₊ with |𝑟|||=|𝑟𝑠|||𝑟|,|𝑠=0,iff𝑟=0,𝑟𝑠||,𝑟,𝑠∈𝕂,𝑟+𝑠≤max{||},𝑟,𝑠∈𝕂. (8.1) A field endowed with a non-Archimedean valuation is said to be non-Archimedean. Let 𝑋 be a linear space over a field 𝕂 with a non-Archimedean valuation that is nontrivial (i.e., we additionally assume that there is an 𝑟₀∈𝕂 such that 0≠|𝑟₀|≠1). A function ‖⋅‖∶𝑋→ℝ₊ is said to be a non-Archimedean norm if it satisfies the following conditions: ‖𝑥‖‖‖=|𝑟𝑥‖𝑥‖,𝑦=0,iff𝑥=0,𝑟𝑥|‖,𝑟∈𝕂,𝑥∈𝑋,𝑥+𝑦≤max‖,𝑥,𝑦∈𝑋. (8.2) If ‖⋅‖∶𝑋→ℝ₊ is a non-Archimedean norm in 𝑋, then the pair (𝑋,‖⋅‖) is called a non-Archimedean normed space.

If (𝑋,‖⋅‖) is a non-Archimedean normed space, then it is easily seen that the function 𝑑_𝑋∶𝑋²→ℝ₊, given by 𝑑_𝑋(𝑥,𝑦)∶=‖𝑥−𝑦‖, is a non-Archimedean metric on 𝑋. Therefore non-Archimedean normed spaces are special cases of metric spaces. The most important examples of non-Archimedean normed spaces are the 𝑝-adic numbers ℚ_𝑝 (here 𝑝 is any prime number), which have gained the interest of physicists because of their connections with some problems coming from quantum physics, 𝑝-adic strings, and superstrings (see, for instance, [104]).

In [105], correcting the mistakes in the proof given by the second author in 1968, Arriola and Beyer showed that the Cauchy functional equation is Hyers-Ulam stable in ℝ^Q𝑝. Schwaiger [106] did the same in the class of functions from a commutative group which is uniquely divisible by 𝑝 to a Banach space over ℚ_𝑝. In 2007, Moslehian and Rassias [107] proved the generalized Hyers-Ulam stability of the Cauchy equation in a more general setting, namely, in complete non-Archimedean normed spaces. After their results a lot of papers (see, for instance, [87–89, 93] and the references given there) on the stability of other equations in such spaces have been published. Here we present only one example of these outcomes which is a generalization of the result of Moslehian and Rassias and was obtained in [87] (cf. also Theorem 7 in [106]).

Theorem 8.1. Let 𝑉 be a commutative semigroup and 𝑊 be a complete non-Archimedean normed space over a non-Archimedean field of characteristic different from 2. Assume also that 𝑛∈ℕ and for every 𝑖∈{1,…,𝑛}, 𝜑_𝑖∶𝑉^𝑛+1→ℝ₊ is a mapping such that for each (𝑥₁,…,𝑥_𝑛+1)∈𝑉^𝑛+1, lim_𝑗→∞1|2|^𝑗𝜑_𝑖2^𝑗𝑥₁,𝑥₂,…,𝑥_𝑛+1=…=lim_𝑗→∞1|2|^𝑗𝜑_𝑖𝑥₁,…,𝑥_𝑖−2,2^𝑗𝑥_𝑖−1,𝑥_𝑖,…,𝑥_𝑛+1=lim_𝑗→∞1|2|^𝑗𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,2^𝑗𝑥_𝑖,2^𝑗𝑥_𝑖+1,𝑥_𝑖+2,…,𝑥_𝑛+1=lim_𝑗→∞1|2|^𝑗𝜑_𝑖𝑥₁,…,𝑥_𝑖+1,2^𝑗𝑥_𝑖+2,𝑥_𝑖+3,…,𝑥_𝑛+1=…=lim_𝑗→∞1|2|^𝑗𝜑_𝑖𝑥₁,…,𝑥_𝑛,2^𝑗𝑥_𝑛+1=0, (8.3) and the limit lim_𝑘→∞1max|2|^𝑗𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,2^𝑗𝑥_𝑖,2^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛∶0≤𝑗<𝑘, (8.4) denoted by 𝜑_𝑖(𝑥₁,…,𝑥_𝑛), exists. If 𝑓∶𝑉^𝑛→𝑊 is a function satisfying 𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖+𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑛−𝑓𝑥₁,…,𝑥_𝑛