About the Local Stability of the Four Cauchy Equations Restricted on a Bounded Domain and of Their Pexiderized Forms

Abstract

The Ulam-Hyers stability of functional equations is widely studied from various points of view by many authors. The present paper is concerned with local stability of the four Cauchy equations restricted on a bounded domain. These results can be easily adapted to the corresponding Pexiderized equations.

1. Introduction

After this introduction, in Section 2 the local stability of the additive equation 𝜑𝑥+𝑦=𝜑(𝑥)+𝜑𝑦(1) and, as a consequence, of the logarithmic equation 𝜑𝑥𝑦=𝜑(𝑥)+𝜑𝑦(2) both restricted on a bounded domain in 𝑅² is studied.

It is well known that the problem of stability was posed, for the additive equation, by S. Ulam and was solved by Hyers [1] in 1941, with reference to the equation valid on the whole space. Afterwards, stability was widely studied by many authors, from various points of view, considering further equations on the whole space or putting them in very general settings (see, for instance, [2, 3]).

As for the “local” stability of equations on a restricted domain, first results can be found in [4, 5] (see also [6]) and they concern substantially the set of functions 𝑓∶𝐷_𝑓⊂𝑅→(𝑆,‖⋅‖), satisfying the condition of 𝛿-additivity 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦<𝛿(3) restricted either on the triangular domain 𝐸₀=𝐸(0,0;𝑟)=𝑥,𝑦∈𝑅²∶𝑥≥0,𝑦≥0,𝑥+𝑦<𝑟(4) for some given 𝑟>0, or the unbounded domain 𝐾_𝑎=𝑥,𝑦∈𝑅²∶|𝑥|+𝑦>𝑎(5) for some given 𝑎>0.

In the present paper (Section 2) the bounded restricted domain of inequality (3) will be assumed to be the triangle ()=𝐸=𝐸𝑎,𝑏;𝑟𝑥,𝑦∈𝑅²∶𝑥≥𝑎,𝑦≥𝑏,𝑥+𝑦<𝑎+𝑏+𝑟(6) for some given (𝑎,𝑏)∈𝑅² and 𝑟>0.

It has to be remarked that in the classical paper [1] by Hyers as well as in case of restricted domains studied in [4, 5], the solutions of the “equation” correlated to the inequality (3) on the given domain are either additive functions on the whole space (in cases of the general result by Hyers and of the domain 𝐾_𝑎 defined in (5)) or the restrictions to 𝐷_𝑓=[0,𝑟) of functions 𝐹 additive on 𝑅² (in case of a domain like 𝐸₀ defined in (4)).

On the contrary, when a restricted domain like that in (6) is assumed, the local solution of the corresponding additive equation, restricted to the same set, is different from the restriction on the domain of 𝑓 of some function, which is additive in the whole space 𝑅² (see [7]).

Therefore, in order to adhere to the sense of Ulam’s question in case of a restricted domain like 𝐸(𝑎,𝑏;𝑟) too, the locally 𝛿-additive function 𝑓 has to be compared with the local solution of the corresponding exact equation restricted to the same set 𝐸(𝑎,𝑏;𝑟).

In this frame, in Section 2, first the local stability of the additive Cauchy equation restricted to 𝐸(𝑎,𝑏;𝑟) will be proved (Theorem 1); then, as a consequence of this result, the local stability of the logarithmic Cauchy equation (2) will be proved (Theorem 6).

These results can be easily extended to the Pexiderized forms of the same equations.

Notice that the problem of the “local” stability for the remaining two Cauchy equations, 𝜑(𝑥)𝜑𝑦,𝜑(𝑥)𝜑𝑦,𝑥+𝑦=𝜑(7)𝑥𝑦=𝜑(8) restricted to bounded domains, requires a suitable slightly different approach because of the peculiar properties of the local solutions of such equations when they are restricted on bounded domains (see [8, 9]).

This problem will be the object of Section 3 of the present paper, where results of local stability of (7) and (8) will be proved (Theorems 15 and 16, resp.).

2. About the Additive and the Logarithmic Cauchy’s Equations

2.1. A Result on Local Stability of the Additive Equation

In the set of functions 𝑓 of a real variable with values in a normed space 𝑆=(𝑆,‖⋅‖), let us consider the inequality 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦<𝛿(9) for some 𝛿>0 and (𝑥,𝑦)∈𝐸(𝑎,𝑏;𝑟), defined in (6), for given (𝑎,𝑏)∈𝑅² and 𝑟>0.

As usual, the projections of 𝐸 will be denoted by 𝐸_𝑥,𝐸∶=𝑥∈𝑅∶∃𝑦∈𝑅suchthat𝑥,𝑦∈𝐸_𝑦,𝐸∶=𝑦∈𝑅∶∃𝑥∈𝑅suchthat𝑥,𝑦∈𝐸_𝑥+𝑦∶=𝑥+𝑦∈𝑅∶𝑥,𝑦∈𝐸.(10)

For 𝐸(𝑎,𝑏;𝑟) as defined in (6), the projections are 𝐸_𝑥=[𝑎,𝑎+𝑟), 𝐸_𝑦=[𝑏,𝑏+𝑟), 𝐸_𝑥+𝑦=[𝑎+𝑏,𝑎+𝑏+𝑟), and the domain 𝐷_𝑓 of functions 𝑓 satisfying (9) is 𝐷_𝑓=𝐸_𝑥∪𝐸_𝑦∪𝐸_𝑥+𝑦.

We purpose to check whether each 𝑓∶𝐷_𝑓→(𝑆,‖⋅‖) satisfying (9) for (𝑥,𝑦)∈𝐸(𝑎,𝑏;𝑟) is uniformly approached on 𝐷_𝑓 by some 𝜑∶𝐷_𝑓→𝑆 satisfying the additive equation restricted to 𝐸(𝑎,𝑏;𝑟), namely, by a function 𝜑 of the following form (see [7]): 𝜑(𝑡)=ℎ₀(𝑡)+𝛼,𝑡∈𝐸_𝑥,ℎ₀(𝑡)+𝛽,𝑡∈𝐸_𝑦,ℎ₀(𝑡)+𝛼+𝛽,𝑡∈𝐸_𝑥+𝑦,(11) where ℎ₀∶𝑅→𝑆 is additive on the whole space 𝑅² and 𝛼,𝛽∈𝑆 are constant.

A positive answer is given by the following.

Theorem 1. Let 𝑓∶𝐷_𝑓→(𝑆,‖⋅‖), (𝑆,‖⋅‖) being a Banach space, satisfy (9) for some 𝛿>0 and every (𝑥,𝑦)∈𝐸(𝑎,𝑏;𝑟), defined in (6), for given (𝑎,𝑏)∈𝑅² and 𝑟>0; 𝐷_𝑓=𝐸_𝑥∪𝐸_𝑦∪𝐸_𝑥+𝑦.
Then there exists (at least) a function 𝐻∶𝑅→𝑆, additive on 𝑅², such that the function 𝐹∶𝐷_𝑓→𝑆 defined by 𝐹(𝑡)=𝐻(𝑡)+𝑓(𝑎)−𝐻(𝑎),𝑡∈𝐸_𝑥,𝐻(𝑡)+𝑓(𝑏)−𝐻(𝑏),𝑡∈𝐸_𝑦,𝐻(𝑡)+𝑓(𝑎)+𝑓(𝑏)−𝐻(𝑎)−𝐻(𝑏),𝑡∈𝐸_𝑥+𝑦(12) has both the following properties:(i)𝐹is a (local) solution of the additive equation restricted on 𝐸(𝑎,𝑏;𝑟);(ii)𝐹 approaches uniformly 𝑓 on 𝐷_𝑓, and 𝑓(𝑡)−𝐹(𝑡)<15𝛿(13) holds for every 𝑡∈𝐷_𝑓.

In order to prove Theorem 1, let us premise two lemmas.

Lemma 2. Let 𝑓∶𝐷_𝑓=𝐸_𝑥∪𝐸_𝑦∪𝐸_𝑥+𝑦→(𝑆,‖⋅‖) satisfy the condition (9) restricted to the set 𝐸=𝐸(𝑎,𝑏;𝑟) defined in (6). Then the functions 𝛾_𝑖∶[0,𝑟)→𝑆, 𝑖=1,2,3, defined by 𝛾₁(𝑡)∶=𝑓(𝑎+𝑡)−𝑓(𝑎),𝛾₂(𝑡)()(𝑏),𝛾∶=𝑓𝑏+𝑡−𝑓₃(𝑡)∶=𝑓(𝑎+𝑏+𝑡)−𝑓(𝑎+𝑏)(14) for 𝑡∈[0,𝑟) satisfy both the following inequalities: 𝛾_𝑖(𝑡)−𝛾_𝑗(𝑡𝛾)<2𝛿𝑓𝑜𝑟𝑖,𝑗=1,2,3,(15)_𝑖𝜉+𝜂−𝛾_𝑖(𝜉)−𝛾_𝑖𝜂<4𝛿𝑓𝑜𝑟𝑖=1,2,3,𝜉,𝜂∈𝐸₀=𝐸(0,0;𝑟).(16)

Proof of Lemma 2. Let us prove (15) firstly for 𝑖=1, 𝑗=2. For 𝑡∈[0,𝑟) the points (𝑎+𝑡,𝑏) and (𝑎,𝑏+𝑡) belong to 𝐸(𝑎,𝑏;𝑟); hence, from (9) both the following inequalities hold 𝑓()−𝑓()(𝑏𝑓()−𝑓()(𝑎𝑎+𝑏+𝑡𝑎+𝑡+𝑓)<𝛿,𝑎+𝑏+𝑡𝑏+𝑡+𝑓)<𝛿,(17) whence, 𝑓(𝑎+𝑡)+𝑓(𝑏)−𝑓(𝑏+𝑡)−𝑓(𝑎)<2𝛿,(18) namely, (15) for 𝑖=1, 𝑗=2.
Similarly we prove (15), for 𝑖=1, 𝑗=3, assuming the pairs (𝑎,𝑏) and (𝑎+𝑡,𝑏) with 𝑡∈[0,𝑟), from the formulas 𝑓()((𝑏𝑓()(𝑎(𝑏𝑎+𝑏+𝑡−𝑓𝑎+𝑡)−𝑓)<𝛿,𝑎+𝑏−𝑓)−𝑓)<𝛿,(19) and for 𝑖=2,𝑗=3, assuming the pairs (𝑎,𝑏), (𝑎,𝑏+𝑡), and 𝑡∈[0,𝑟).
In order to prove (16) let us assume (𝜉,𝜂)∈𝐸₀=𝐸(0,0;𝑟).
For 𝑖=1, from (15) and (9) we get Φ₁∶=𝛾₁𝜉+𝜂−𝛾₁(𝜉)−𝛾₁𝜂=𝑓(𝑎−𝑓()(𝑎−𝑓(𝑎()−𝑓(𝑏)=𝑓(𝑏−𝑓()‖𝜎‖𝑎+𝜉+𝜂−𝑓)𝑎+𝜉−𝑓)𝑎+𝜂−𝑓)=𝑓𝑎+𝜉+𝜂−𝑓𝑎+𝜉𝑏+𝜂−𝑓+𝜎𝑎+𝜉+𝜂+𝑓)𝑎+𝜉+𝑓𝑏+𝜂−𝜎<2𝛿;(20) since (𝑎+𝜉+𝜂,𝑏)∈𝐸(𝑎,𝑏;𝑟) and (𝑎+𝜉,𝑏+𝜂)∈𝐸(𝑎,𝑏;𝑟), it follows from (9) that Φ₁=𝑓𝑎+𝑏+𝜉+𝜂+𝜌₁−𝑓𝑎+𝑏+𝜉+𝜂+𝜌₂−𝜎,(21) with ‖𝜎‖<2𝛿, ‖𝜌_𝑘‖<𝛿 for 𝑘=1,2, whence (16) for 𝑖=1.
Similarly, for 𝑖=2, 𝑎 and 𝑏 interchanged.
As for 𝛾₃ with (𝜉,𝜂)∈𝐸₀=𝐸(0,0;𝑟), from (9) Φ₃∶=𝛾₃𝜉+𝜂−𝛾₃(𝜉)−𝛾₃𝜂=𝑓(−𝑓()(−𝑓(=𝑓()𝑎+𝑏+𝜉+𝜂−𝑓𝑎+𝑏)𝑎+𝑏+𝜉−𝑓𝑎+𝑏)𝑎+𝑏+𝜂−𝑓𝑎+𝑏)𝑎+𝜉+𝑓𝑏+𝜂+𝜏₁−𝑓(𝑎+𝜉)+𝑓(𝑏)+𝜏₂−𝑓(𝑎)+𝑓𝑏+𝜂+𝜏₃+𝑓(𝑎)+𝑓(𝑏)+𝜏₄,(22) where ‖𝜏_𝑘‖<𝛿, 𝑘=1,2,3,4, whence ‖Φ₃‖<4𝛿.
Lemma 2 is proved.

Lemma 3. If 𝑓∶𝐷_𝑓=𝐸_𝑥∪𝐸_𝑦∪𝐸_𝑥+𝑦→(𝑆,‖⋅‖), 𝑆 being a Banach space, satisfies (9) on 𝐸=𝐸(𝑎,𝑏;𝑟), then each of the functions 𝛾_𝑖(𝑡), 𝑖=1,2,3, defined in (14) for 𝑡∈[0,𝑟), is uniformly approached on [0,𝑟) by the restriction of a function 𝐻_𝑖∶𝑅→𝑆 additive on 𝑅².
The following inequalities hold: 𝛾_𝑖(𝑡)−𝐻_𝑖(𝑡[)𝛾)<12𝛿,𝑡∈0,𝑟,𝑖=1,2,3,(23)_𝑖(𝑡)−𝐻_𝑗(𝑡)<14𝛿,𝑡∈[0,𝑟),𝑖,𝑗=1,2,3.(24)

Proof of Lemma 3. Since each function 𝛾₁, 𝛾₂, 𝛾₃ is 4𝛿-additive on 𝐸(0,0;𝑟) (from Lemma 2), formula (23) follows immediately from a known result [4, Lemma 2]:
“Let 𝑓∶[0,𝑑)→(𝑋,‖⋅‖), 𝑋a Banach space, be 𝛿-additive on 𝐸_𝑑 = {(𝑥,𝑦)∈𝑅²∶0≤𝑥<𝑑,0≤𝑦<𝑑,𝑥+𝑦<𝑑}.
Then there exists at least one additive function 𝐿∶𝑅→𝑋such that𝑓(𝑥)−𝐿(𝑥)<3𝛿𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦𝑥∈[0,𝑑).”(25) Moreover, for every 𝑖,𝑗=1,2,3 and 𝑡∈[0,𝑟), formula (24) follows from (15) and (23); in fact 𝛾_𝑖(𝑡)−𝐻_𝑗(𝑡)=𝛾_𝑖(𝑡)−𝛾_𝑗(𝑡)+𝛾_𝑗(𝑡)−𝐻_𝑗(𝑡)<14𝛿.(26) This means that the restriction to [0,𝑟) of each additive 𝐻_𝑗, 𝑗=1,2,3, approaches uniformly on [0,𝑟) each function 𝛾₁, 𝛾₂, 𝛾₃.
Lemma 3 is proved.

Proof of Theorem 1. According to (24) in Lemma 3, each function 𝛾_𝑖(𝑡), 𝑖=1,2,3, defined in (14) for 𝑡∈[0,𝑟), namely, 𝛾₁(𝑡)∶=𝑓(𝑎+𝑡)−𝑓(𝑎),𝛾₂(𝑡)()(𝑏),𝛾∶=𝑓𝑏+𝑡−𝑓₃(𝑡)∶=𝑓(𝑎+𝑏+𝑡)−𝑓(𝑎+𝑏),(27) is uniformly approached on [0,𝑟) by each of the additive functions 𝐻_𝑗∶𝑅→𝑆, 𝑗=1,2,3.
Let us define 𝐹_𝑖∶𝐷_𝑓→𝑆, 𝑖=1,2,3, as follows: 𝐹_𝑖(𝑡)=𝐻_𝑖(𝑡)−𝐻_𝑖(𝑎)+𝑓(𝑎),𝑡∈𝐸_𝑥,𝐻_𝑖(𝑡)−𝐻_𝑖(𝑏)+𝑓(𝑏),𝑡∈𝐸_𝑦,𝐻_𝑖(𝑡)−𝐻_𝑖(𝑎+𝑏)+𝑓(𝑎)+𝑓(𝑏),𝑡∈𝐸_𝑥+𝑦.(28) Such functions 𝐹_𝑖, 𝑖=1,2,3, satisfy obviously the additive equation restricted to 𝐸(𝑎,𝑏;𝑟).
Moreover, thanks to Lemmas 2 and 3, each function 𝐹_𝑖 approaches uniformly 𝑓 on 𝐷_𝑓 as in formula (13); in fact, for arbitrary (𝑥,𝑦)∈𝐸(𝑎,𝑏;𝑟), 𝑥=𝑎+𝑡∈𝐸_𝑥,𝛾₁(𝑡)=𝛾₁()(𝑥)(𝑎),𝐹𝑥−𝑎=𝑓−𝑓_𝑖(𝑥)(𝑥=𝐻−𝑓)_𝑖(𝑥)−𝐻_𝑖(𝑎)+𝑓(𝑎)−𝛾₁()(𝑎)=𝐻𝑥−𝑎+𝑓_𝑖(𝑥−𝑎)−𝛾₁(𝑥−𝑎)<14𝛿;(29) similarly for 𝑦∈𝐸_𝑦, 𝛾₂(𝑦−𝑏)=𝑓(𝑦)−𝑓(𝑏).
On the projection 𝐸_𝑥+𝑦, where 𝑥+𝑦=𝑎+𝑏+𝑡 and 𝛾₃(𝑥+𝑦−𝑎−𝑏)=𝑓(𝑥+𝑦)−𝑓(𝑎+𝑏), we get from Lemma 3 and formula (9) 𝐹_𝑖=𝐻𝑥+𝑦−𝑓𝑥+𝑦_𝑖(𝑎)(𝑏)−𝛾𝑥+𝑦−𝑎−𝑏+𝑓+𝑓₃()≤𝐻𝑥+𝑦−𝑎−𝑏+𝑓𝑎+𝑏_𝑖𝑥+𝑦−𝑎−𝑏−𝛾₃+𝑓()(𝑎)(𝑏𝑥+𝑦−𝑎−𝑏𝑎+𝑏−𝑓−𝑓)<15𝛿.(30) Therefore, each function 𝐹_𝑖∶𝐷_𝑓→𝑆, 𝑖=1,2,3, satisfies (13), and Theorem 1 is proved.

Remark 4. The foregoing study was developed as though the projections 𝐸_𝑥, 𝐸_𝑦, and 𝐸_𝑥+𝑦 were pairwise disjoint.
If two of them overlap, for instance 𝐸_𝑥∩𝐸_𝑦 is nonempty, in every common point the values given by the different parts of formulas of approximating function 𝐹 have to be the same.
More in particular, if the set 𝐷_𝑓=𝐸_𝑥∪𝐸_𝑦∪𝐸_𝑥+𝑦 is connected, in (28) the equations 𝑓(𝑎)−𝐻_𝑖(𝑎)=𝑓(𝑏)−𝐻_𝑖(𝑏)=𝑓(𝑎)+𝑓(𝑏)−𝐻_𝑖(𝑎+𝑏)(31) hold, whence 𝑓(𝑎)−𝐻_𝑖(𝑎)=𝑓(𝑏)−𝐻_𝑖(𝑏)=0. In this case, the locally 𝛿-additive 𝑓 is uniformly approached on the whole 𝐷_𝑓 by the restriction to 𝐷_𝑓 of a function 𝐻∶𝑅→𝑆, additive on 𝑅² (𝐻=𝐻_𝑖, 𝑖 either =1, or =2, or =3).

2.2. A Result on Local Stability of the Logarithmic Equation

On the ground of the results in Section 2.1 it is easy to prove the local stability of the logarithmic Cauchy equation (2) restricted to the bounded domain 𝐽=𝐽(𝑎,𝑏;𝑟)∶=𝑥,𝑦∈𝑅²∶𝑒^𝑎≤𝑥<𝑒^𝑎+𝑟,𝑒^𝑏≤𝑦<𝑒^𝑏+𝑟,𝑒^𝑎+𝑏≤𝑥𝑦<𝑒^{𝑎+𝑏+𝑟}(32) for given (𝑎,𝑏)∈𝑅² and 𝑟>0.

The projections 𝐽_𝑥, 𝐽_𝑦, 𝐽_𝑥𝑦 of 𝐽 are given by 𝐽_𝑥∶=𝑥∈𝑅⁺∶∃𝑦∈𝑅⁺suchthat𝑥,𝑦∈𝐽=𝑒^𝑎,𝑒^𝑎+𝑟,𝐽_𝑦∶=𝑦∈𝑅⁺∶∃𝑥∈𝑅⁺suchthat𝑥,𝑦∈𝐽=𝑒^𝑏,𝑒^𝑏+𝑟,𝐽_𝑥𝑦∶=𝑥𝑦∈𝑅⁺∶𝑥,𝑦∈𝐽=𝑒^𝑎+𝑏,𝑒^{𝑎+𝑏+𝑟}.(33)

Since the local stability of (2) depends on the comparison of every 𝑓 satisfying 𝑓𝑥𝑦−𝑓(𝑥)−𝑓𝑦<𝛿(34) for (𝑥,𝑦)∈𝐽(𝑎,𝑏;𝑟) with some solution 𝜑 of the corresponding equation (2) restricted to 𝐽(𝑎,𝑏;𝑟), let us premise (Lemma 5) the local solution of (2).

Lemma 5. Let 𝑆 be a real linear space; if 𝜑∶𝐷_𝑓=𝐽_𝑥∪𝐽_𝑦∪𝐽_𝑥𝑦→𝑆 satisfies (2) on the bounded domain 𝐽=𝐽(𝑎,𝑏;𝑟) defined in (32), then there exists a function ℎ₀∶𝑅→𝑆, additive on 𝑅², such that 𝜑(𝑡)=ℎ₀(ln𝑡)−ℎ₀(𝑎)+𝜑𝑒^𝑎,𝑡∈𝐽_𝑥,ℎ₀(ln𝑡)−ℎ₀(𝑏)+𝜑𝑒^𝑏,𝑡∈𝐽_𝑦,ℎ₀(ln𝑡)−ℎ₀(𝑎)−ℎ₀(𝑏)+𝜑𝑒^𝑎+𝑏,𝑡∈𝐽_𝑥𝑦.(35)

Proof of Lemma 5. By the usual substitutions 𝑥=𝑒^𝑢,𝑢=ln𝑥,𝜑(𝑥)=𝜑𝑒^𝑢=∶𝑔(𝑢)=𝑔(ln𝑥),𝑦=𝑒^𝑣,𝑣=ln𝑦,𝜑𝑦=𝜑𝑒^𝑣(𝑣),,𝜑𝑒=∶𝑔=𝑔ln𝑦𝑢+𝑣=ln𝑥𝑦𝑥𝑦=𝜑^𝑢+𝑣=∶𝑔(𝑢+𝑣)=𝑔ln𝑥𝑦,(36) the domain 𝐽(𝑎,𝑏;𝑟) is transformed into a set 𝐸(𝑎,𝑏;𝑟) like the one defined in (6). Let us consider 𝐸=𝐸(𝑎,𝑏;𝑟)=(𝑢,𝑣)∈𝑅².∶𝑢≥𝑎,𝑣≥𝑏,𝑢+𝑣<𝑎+𝑏+𝑟(37) Equation (2) is transformed into 𝑔(𝑢+𝑣)=𝑔(𝑢)+𝑔(𝑣),(𝑢,𝑣)∈𝐸(𝑎,𝑏;𝑟).(38) Therefore, using Theorem 1, we obtain (35) with additive ℎ₀.

The local stability of the logarithmic equation (2) is stated by the following.

Theorem 6. Let (𝑆,‖⋅‖) be a Banach space; if 𝑓∶𝐷_𝑓=𝐽_𝑥∪𝐽_𝑦∪𝐽_𝑥𝑦→(𝑆,‖⋅‖) satisfies 𝑓𝑥𝑦−𝑓(𝑥)−𝑓𝑦<𝛿(39) for some 𝛿>0 and every (𝑥,𝑦)∈𝐽(𝑎,𝑏;𝑟), defined in (32), for given (𝑎,𝑏)∈𝑅² and 𝑟>0, then there exists (at least) a function 𝐻∶𝑅→𝑆, additive on 𝑅², such that the function 𝐿∶𝐷_𝑓→𝑆 defined by 𝐿(𝑡)=𝐻(ln𝑡)−𝐻(𝑎)+𝑓𝑒^𝑎,𝑡∈𝐽_𝑥=𝑒^𝑎,𝑒^𝑎+𝑟,𝐻(ln𝑡)−𝐻(𝑏)+𝑓𝑒^𝑏,𝑡∈𝐽_𝑦=𝑒^𝑏,𝑒^𝑏+𝑟,𝐻()(𝑎)(𝑏)𝑒ln𝑡−𝐻−𝐻+𝑓^𝑎+𝑓𝑒^𝑏,𝑡∈𝐽_𝑥𝑦=𝑒^𝑎+𝑏,𝑒^{𝑎+𝑏+𝑟},(40) satisfies both the following properties:(i)𝐿 is a local solution of the logarithmic equation on the restricted domain 𝐽(𝑎,𝑏;𝑟);(ii)𝐿 approaches uniformly 𝑓on 𝐷_𝑓, and 𝑓(𝑡)−𝐿(𝑡)<15𝛿(41) holds for every 𝑡∈𝐷_𝑓.

Proof of Theorem 6. The usual substitutions 𝑥=𝑒^𝑢, 𝑦=𝑒^𝑣 (like in proof of Lemma 5) transform the inequality (39) restricted to the set 𝐽(𝑎,𝑏;𝑟) into 𝑔(𝑢+𝑣)−𝑔(𝑢)−𝑔(𝑣)<𝛿(42) restricted to the set 𝐸=𝐸(𝑎,𝑏;𝑟), defined in (6).
Now, we can follow the same line of proof as in Section 2.1 by defining the functions 𝛾_𝑖∶[0,𝑟)→𝑆, 𝑖=1,2,3, related to 𝑔; namely, 𝛾₁(𝑡)∶=𝑔(𝑎+𝑡)−𝑔(𝑎),𝛾₂(𝑡)()(𝑏),𝛾∶=𝑔𝑏+𝑡−𝑔₃(𝑡)∶=𝑔(𝑎+𝑏+𝑡)−𝑔(𝑎+𝑏);(43) then there exist functions 𝐻_𝑖∶𝑅→𝑆, 𝑖=1,2,3, additive on 𝑅², such that each of the functions 𝐺_𝑖, 𝑖=1,2,3, 𝐺_𝑖(𝑡)=𝐻_𝑖(𝑡)−𝐻_𝑖(𝑎)+𝑔(𝑎),𝑡∈𝐸_𝑢=[),𝐻𝑎,𝑎+𝑟_𝑖(𝑡)−𝐻_𝑖(𝑏)+𝑔(𝑏),𝑡∈𝐸_𝑣=[),𝐻𝑏,𝑏+𝑟_𝑖(𝑡)−𝐻_𝑖(𝑎)−𝐻_𝑖(𝑏)+𝑔(𝑎)+𝑔(𝑏),𝑡∈𝐸_𝑢+𝑣=[𝑎+𝑏,𝑎+𝑏+𝑟),(44)is a local solution of the equation 𝑔(𝑢+𝑣)=𝑔(𝑢)+𝑔(𝑣) restricted to 𝐸(𝑎,𝑏;𝑟), and 𝑔(𝑡)−𝐺_𝑖(𝑡)<15𝛿,𝑡∈𝐷_𝑔=𝐸_𝑢∪𝐸_𝑣∪𝐸_𝑢+𝑣(45) holds for 𝑖=1,2,3.
Now let us come back to 𝑓, by the substitutions which transformed (39) into (42), beginning by the transformation of functions 𝐺_𝑖 defined in (44); on 𝐽_𝑥 (from 𝐸_𝑢) 𝐺_𝑖(ln𝑥)=𝐻_𝑖(ln𝑥)−𝐻_𝑖(𝑎)+𝑓𝑒^𝑎,𝑥=𝑒^𝑢∈𝑒^𝑎,𝑒^𝑎+𝑟,(46) similarly, for 𝐺_𝑖(ln𝑦) on 𝐽_𝑦 (from 𝐸_𝑣) and for 𝐺_𝑖(ln𝑥𝑦) on 𝐽_𝑥𝑦 (from 𝐸_𝑢+𝑣).
By the definition 𝐿_𝑖(𝜏)∶=𝐺_𝑖(ln𝜏),𝑖=1,2,3,(47) formula (44) changes into𝐿_𝑖(𝜏)=𝐻_𝑖(ln𝜏)−𝐻_𝑖(𝑎)+𝑓𝑒^𝑎,𝜏∈𝐽_𝑥=𝑒^𝑎,𝑒^𝑎+𝑟,𝐻_𝑖(ln𝜏)−𝐻_𝑖(𝑏)+𝑓𝑒^𝑏,𝜏∈𝐽_𝑦=𝑒^𝑏,𝑒^𝑏+𝑟,𝐻_𝑖(ln𝜏)−𝐻_𝑖(𝑎)−𝐻_𝑖(𝑏)+𝑓𝑒^𝑎+𝑓𝑒^𝑏,𝜏∈𝐽_𝑥𝑦=𝑒^𝑎+𝑏,𝑒^{𝑎+𝑏+𝑟}.(48)Obviously each 𝐿_𝑖(𝜏), 𝑖=1,2,3, satisfies the logarithmic equation restricted to 𝐽(𝑎,𝑏;𝑟).
In order to prove the approximation stated in (41), let us begin by the projection 𝐽_𝑥: for 𝑥∈𝐽_𝑥; then 𝑢=ln𝑥, 𝑔(𝑢)=𝑓(𝑥), and 𝑢∈[𝑎,𝑎+𝑟); hence, from (44) 𝐺_𝑖(𝑢)=𝐺_𝑖(ln𝑥)=𝐿_𝑖(𝑥)=𝐻_𝑖(ln𝑥)−𝐻_𝑖(𝑎)+𝑓𝑒^𝑎(49) and from (45) 𝑓(𝑥)−𝐿_𝑖(𝑥)<15𝛿for𝑥∈𝐽_𝑥.(50) Similarly for 𝑓 on 𝐽_𝑦 and on 𝐽_𝑥𝑦.
Therefore, (41) is true with 𝐿=𝐿₁ or 𝐿=𝐿₂ or 𝐿=𝐿₃, and Theorem 6 is proved.

Remark 7. Remarks about the consequence of a possible overlapping of the projections of the given restricted domain, like those in Remark 4, could be repeated here.

2.3. About the Pexiderized Forms of the Foregoing Equations

Stability results for the Pexiderized forms of the additive and the logarithmic equations, namely, 𝐴(𝑥)𝑦,(),𝐴(𝑥)𝑦,(),𝑥+𝑦=𝐵+𝐶𝑥,𝑦∈𝐸𝑎,𝑏;𝑟𝑥𝑦=𝐵+𝐶𝑥,𝑦∈𝐽𝑎,𝑏;𝑟(51) can be easily stated on the ground of the foregoing Theorems 1 and 6.

In fact, when the inequality 𝜓𝑥+𝑦−𝜆(𝑥)−𝜇𝑦<𝛿(52) is satisfied for every (𝑥,𝑦)∈𝐸(𝑎,𝑏;𝑟), the statement of Theorem 1 can be easily adapted to the condition (52), because the functions 𝜆(𝑥), 𝜇(𝑦) and 𝜓(𝑥+𝑦) play a role like that of the restrictions of 𝑓 to the projections 𝐸_𝑥, 𝐸_𝑦, 𝐸_𝑥+𝑦 in case of a unique function 𝑓.

Similarly for 𝜌𝑥𝑦−𝜏(𝑥)−𝜎𝑦<𝛿(53) restricted to 𝐽(𝑎,𝑏;𝑟), by use of Theorem 6.

Remark 8. In case of a Pexiderized equation on restricted domain, overlapping of the projections of the given bounded domain obviously produces no changes in the result.

3. About the Remaining Two Cauchy Equations (7) 𝜑(𝑥+𝑦)=𝜑(𝑥)𝜑(𝑦) and (8) 𝜑(𝑥𝑦)=𝜑(𝑥)𝜑(𝑦) on a Bounded Restricted Domain

3.1. Preliminaries

As for (7) 𝜑(𝑥+𝑦)=𝜑(𝑥)𝜑(𝑦) the restricted domain is assumed to be 𝐸(𝑎,𝑏;𝑟) defined in (6); the domain of (8) 𝜑(𝑥𝑦)=𝜑(𝑥)𝜑(𝑦) is 𝐽(𝑎,𝑏;𝑟) defined in (32) for fixed real 𝑎, 𝑏 and 𝑟>0.

Let us premise the local solutions of the above equations (see papers [8, 9] and [10], resp.).

Lemma 9. Let 𝜑∶𝐷_𝜑=𝐸_𝑥∪𝐸_𝑦∪𝐸_𝑥+𝑦→𝑅 satisfy (7) restricted to 𝐸(𝑎,𝑏;𝑟) defined in (6).
If and only if there exists some (𝑥^′,𝑦^′)∈𝐸(𝑎,𝑏;𝑟), (𝑥^′∈𝐸_𝑥,𝑦^′∈𝐸_𝑦) such that 𝜑(𝑥^′)≠0 and 𝜑(𝑦^′)≠0, the following properties (P₁), (P₂), (P₃) hold: (P₁)𝜑(𝑡)≠0 for every 𝑡∈𝐸_𝑥+𝑦 (hence for every 𝑡∈𝐷_𝜑); (P₂)sgn𝜑(𝑡) is constant on each projection 𝐸_𝑥,𝐸_𝑦,𝐸_𝑥+𝑦 (not necessarily the same in different projections); (P₃)𝜑(𝑡) is given on 𝐷_𝜑 by the following formulas: (i)if 𝑡∈𝐸_𝑥, 𝜑(𝑡)=𝐴𝑒^{𝐺(𝑡)−𝐺(𝑎)}, (ii)if 𝑡∈𝐸_𝑦, 𝜑(𝑡)=𝐵𝑒^{𝐺(𝑡)−𝐺(𝑏)}, (iii)if 𝑡∈𝐸_𝑥+𝑦, 𝜑(𝑡)=𝐴𝐵𝑒^{𝐺(𝑡)−𝐺(𝑎+𝑏)}, where 𝐺∶𝑅→𝑅 is additive on 𝑅²; 𝐴≠0, 𝐵≠0 are constant.

Remark 10. Notice that 𝜑 restricted to each of the projections 𝐸_𝑥, 𝐸_𝑦, 𝐸_𝑥+𝑦 is the restriction of a solution Φ∶𝑅→𝑅 of the equation Φ𝑥+𝑦=𝐾Φ(𝑥)Φ𝑦(54) valid on the whole 𝑅², for suitable 𝐾≠0.
Since this equation can be written as 𝐾Φ𝑥+𝑦=𝐾Φ(𝑥)𝐾Φ𝑦,(55) we get 𝐾Φ(𝑡)=𝑒^ℎ0(𝑡), for some additive 𝐺(𝑡), whence formulas in (P₃) of Lemma 9 for 𝑒𝐾=^𝐺(𝑎)𝐴𝑒,𝐾=^𝐺(𝑏)𝐵𝑒,𝐾=^{𝐺(𝑎+𝑏)},𝐴𝐵(56) respectively, in 𝐸_𝑥, 𝐸_𝑦, 𝐸_𝑥+𝑦.

Lemma 11 (see [10]). The general nowhere vanishing solution 𝜑∶𝐷_𝜑=𝐽_𝑥∪𝐽_𝑦∪𝐽_𝑥𝑦→𝑅 of (8) restricted to the set 𝐽(𝑎,𝑏;𝑟) defined in (32) is given by the following formulas: 𝜑(𝑥)=𝑒^ℎ(ln𝑥)⋅𝑒^−ℎ(𝑎)⋅𝜑𝑒^𝑎,𝑥∈𝐽_𝑥=𝑒^𝑎,𝑒^𝑎+𝑟,𝜑𝑦=𝑒^ℎ(ln𝑦)⋅𝑒^−ℎ(𝑏)⋅𝜑𝑒^𝑏,𝑦∈𝐽_𝑦=𝑒^𝑏,𝑒^𝑏+𝑟,𝜑𝑥𝑦=𝑒^{ℎ(ln𝑥𝑦)}⋅𝑒^{−ℎ(𝑎+𝑏)}⋅𝜑𝑒^𝑎+𝑏,𝑥𝑦∈𝐽_𝑥𝑦=𝑒^𝑎+𝑏,𝑒^{𝑎+𝑏+𝑟},(57) where ℎ∶𝑅→𝑅 is additive on 𝑅².

Remark 12. As in Remark 10, we can see that the local solution 𝜑 of (8), restricted to each of the projections 𝐽_𝑥, 𝐽_𝑦, 𝐽_𝑥𝑦, is the restriction of a solution Ψ∶𝑅⁺→𝑅 of a more general equation Ψ𝑥𝑦=𝐾Ψ(𝑥)Ψ𝑦,𝑥,𝑦∈𝑅⁺×𝑅⁺(58) for suitable 𝐾≠0.
From 𝐾Ψ(𝑥𝑦)=𝐾Ψ(𝑥)𝐾Ψ(𝑦), it follows that Ψ(𝑡)=(1/𝐾)𝑒^ℎ(ln𝑡), with 𝑒𝐾=^ℎ(𝑎)𝜑(𝑒^𝑎)ontheprojection𝐽_𝑥,𝑒𝐾=^ℎ(𝑏)𝜑𝑒^𝑏on𝐽_𝑦,𝑒𝐾=^{ℎ(𝑎+𝑏)}𝜑𝑒^𝑎+𝑏on𝐽_𝑥𝑦.(59)

3.2. How the Question of Local Stability of (7) or (8) Has to Be Properly Formulated?

The foregoing Remarks 10 and 12, which point out a connection of the restricted equation under consideration with more general equations, namely, Ψ𝑥+𝑦=𝐾Ψ(𝑥)Ψ𝑦,Ψ𝑥𝑦=𝐾Ψ(𝑥)Ψ𝑦,(60) suggest the following forms of perturbation of such equations: 𝑓𝑥+𝑦=𝑒^{𝛿⋅𝜃(𝑥,𝑦)}𝑓(𝑥)𝑓𝑦,(7)_𝛿𝑓𝑥𝑦=𝑒^{𝛿⋅𝜃(𝑥,𝑦)}𝑓(𝑥)𝑓𝑦,(8)_𝛿 for −1<𝜃(𝑥,𝑦)<1 and some fixed 𝛿>0.

Moreover, it is known (see [8, 9]) that the local solutions of the restricted equations (7) or (8), which vanish somewhere, are expressed by formulas containing arbitrary functions; therefore, the problem of the local stability seems to be significant in the set of nowhere vanishing functions 𝑓.

In this frame, the perturbed forms (7)_𝛿 and (8)_𝛿 can be written equivalently as 𝑒^−𝛿<𝑓𝑥+𝑦𝑓(𝑥)𝑓𝑦<𝑒^𝛿,(7)^′_𝛿𝑒^−𝛿<𝑓𝑥𝑦𝑓(𝑥)𝑓𝑦<𝑒^𝛿.(8)^′_𝛿

The stability results which follow are framed in this context.

3.3. A Sign Property concerning the Perturbed Forms of the Exponential Equation and the Power Equation

Here, we will be concerned with the condition (7)^′_𝛿, (𝑥,𝑦)∈𝐸(𝑎,𝑏;𝑟) defined in (6) for some fixed 𝛿>0, in the set of functions 𝑓∶𝐷_𝑓=𝐸_𝑥∪𝐸_𝑦∪𝐸_𝑥+𝑦→𝑅, such that 𝑓(𝑡)≠0 for every 𝑡∈𝐷_𝑓.

Let us premise a remark about signs of nowhere vanishing functions 𝑓 satisfying (7)^′_𝛿 on 𝐸(𝑎,𝑏;𝑟). From Lemma 9, Property (P₂), it is known that every nowhere vanishing solution of the exponential Cauchy equation restricted to 𝐸(𝑎,𝑏;𝑟) keeps a constant sign in each of the projections 𝐸_𝑥, 𝐸_𝑦, 𝐸_𝑥+𝑦 of 𝐸(𝑎,𝑏;𝑟).

We will see that a similar property is true also for every solution of the restricted condition (7)^′_𝛿, which is rewritten here as follows: 𝑓𝑥+𝑦=𝑒^{𝛿⋅𝜃(𝑥,𝑦)}𝑓(𝑥)𝑓𝑦,−1<𝜃𝑥,𝑦<1.(7)^′_𝛿𝜃 From (7)^′_𝛿𝜃, assuming (𝑥,𝑦)=(𝑎,𝑏+𝜉) and (𝑥,𝑦)=(𝑎+𝜉,𝑏) with 0≤𝜉<𝑟, we get 𝑓(𝑎+𝑏+𝜉)=𝑒^𝜃′𝛿𝑓(𝑎)𝑓(𝑏+𝜉),𝜃^′(),𝑓()=𝜃𝑎,𝑏+𝜉(61)𝑎+𝑏+𝜉=𝑒^{𝜃′′𝛿}𝑓(𝑎+𝜉)𝑓(𝑏),𝜃^′′=𝜃(𝑎+𝜉,𝑏),(62) whence 𝑓(𝑎)𝑓(𝑏+𝜉)=𝑒^{(𝜃′′−𝜃′)𝛿}𝑓()𝑓(𝑏)[),𝑓()𝑎+𝜉,𝜉∈0,𝑟(63)𝑏+𝜉=𝑒^{(𝜃′′−𝜃′)𝛿}𝑓(𝑏)𝑓(𝑎)𝑓(𝑎+𝜉).(64)

Moreover, from (7)^′_𝛿𝜃 for (𝑥,𝑦) = (𝑎+𝜉,𝑏+𝜉), 0≤𝜉<𝑟/2, 𝜃^′′′ = 𝜃(𝑎+𝜉,𝑏+𝜉), 𝑓(𝑎+𝑏+2𝜉) = 𝑒^{𝜃′′′𝛿}𝑓(𝑎+𝜉)𝑓(𝑏+𝜉)=𝑒^{𝜃′′′𝛿}𝑓(𝑎+𝜉)²𝑒^{(𝜃′′−𝜃′)𝛿}𝑓(𝑏)/𝑓(𝑎); hence, 𝑓(𝑡) has constant sign in 𝐸_𝑥+𝑦.

As a consequence, from (62), (64) it follows that 𝑓 has constant signs also in 𝐸_𝑥 and in 𝐸_𝑦 (the signs of 𝑓(𝑎) and of 𝑓(𝑏), resp.).

This proves the following.

Lemma 13. Every nowhere vanishing function 𝑓∶𝐷_𝑓=𝐸_𝑥∪𝐸_𝑦∪𝐸_𝑥+𝑦→𝑅 satisfying (7)_𝛿 in 𝐸(𝑎,𝑏;𝑟) keeps constant sign in each of the projections 𝐸_𝑥, 𝐸_𝑦, 𝐸_𝑥+𝑦 of 𝐸(𝑎,𝑏;𝑟).

Similarly, we can prove a sign property concerning the perturbed form of the power equation.

Let us consider now the condition (8)^′_𝛿; namely, 𝑒^−𝛿<𝑓𝑥𝑦𝑓(𝑥)𝑓𝑦<𝑒^𝛿,𝑥,𝑦∈𝐽(𝑎,𝑏;𝑟)(65) for some fixed 𝛿>0, assuming 𝑓(𝑡)≠0 for every 𝑡∈𝐷_𝑓=𝐽_𝑥∪𝐽_𝑦∪𝐽_𝑥𝑦.

The usual substitutions of variables 𝑥,𝑦 allow us to use the foregoing results about the exponential equation. Put𝑥=𝑒^𝑢,𝑢=ln𝑥,𝑓(𝑥)=𝑓𝑒^𝑢=∶𝑔(𝑢),𝑎≤𝑢<𝑎+𝑟𝑦=𝑒^𝑣,𝑣=ln𝑦,𝑓𝑦=𝑓𝑒^𝑣=∶𝑔(𝑣),𝑏≤𝑣<𝑏+𝑟,(66)whence 𝑓(𝑥𝑦)=∶𝑔(𝑢+𝑣), 𝑎+𝑏≤𝑢+𝑣<𝑎+𝑏+𝑟.

Then (65) is transformed into 𝑒^−𝛿<𝑔(𝑢+𝑣)𝑔(𝑢)𝑔(𝑣)<𝑒^𝛿,(𝑢,𝑣)∈𝐸(𝑎,𝑏;𝑟);(67) namely 𝑔(𝑢+𝑣)=𝑒^{𝛿⋅𝜃(𝑢,𝑣)}𝑔(𝑢)𝑔(𝑣),−1<𝜃(𝑢,𝑣)<1.(65)_𝜃

Therefore, from Lemma 13, it follows that 𝑔(𝑢) has constant sign (=sgn𝑔(𝑎)) in [𝑎,𝑎+𝑟), whence 𝑓(𝑥) has constant sign (=sgn𝑓(𝑒^𝑎)) in [𝑒^𝑎,𝑒^𝑎+𝑟); similarly for 𝑔(𝑣) in [𝑏,𝑏+𝑟), namely for 𝑓(𝑦) in [𝑒^𝑏,𝑒^𝑏+𝑟) and for 𝑔(𝑢+𝑣) in [𝑎+𝑏,𝑎+𝑏+𝑟), namely 𝑓(𝑥𝑦) in [𝑒^𝑎+𝑏,𝑒^{𝑎+𝑏+𝑟}).

Hence, the following result is proved.

Lemma 14. Every nowhere vanishing function 𝑓∶𝐷_𝑓=𝐽_𝑥∪𝐽_𝑦∪𝐽_𝑥𝑦→𝑅 satisfying (65) restricted to 𝐽(𝑎,𝑏;𝑟) has constant signs in each of the projections 𝐽_𝑥, 𝐽_𝑦, 𝐽_𝑥𝑦 of 𝐽(𝑎,𝑏;𝑟).

3.4. A Result of Local Stability for the Exponential Cauchy Equation

In the set of functions 𝑓∶𝐷_𝑓=𝐸_𝑥∪𝐸_𝑦∪𝐸_𝑥+𝑦→𝑅 such that 𝑓(𝑡)≠0 for every 𝑡∈𝐷_𝑓, let us consider the inequality  (7)^′_𝛿, with (𝑥,𝑦)∈𝐸(𝑎,𝑏;𝑟) for some fixed 𝛿>0.

From (7)^′_𝛿𝑓−𝛿<ln𝑥+𝑦𝑓(𝑥)𝑓𝑦<𝛿;(68) since 𝑓0<𝑥+𝑦𝑓(𝑥)𝑓𝑦=𝑓𝑥+𝑦𝑓(𝑥)𝑓𝑦,(69)𝑓 satisfies ln𝑓𝑥+𝑦−ln𝑓(𝑥)−ln𝑓𝑦<𝛿;(70) namely for 𝜆(𝑡)∶=ln|𝑓(𝑡)|, 𝜆𝑥+𝑦−𝜆(𝑥)−𝜆𝑦<𝛿,𝑥,𝑦∈𝐸(𝑎,𝑏;𝑟).(71)

On the ground of Theorem 1, there exists (at least) one additive function 𝐻∶𝑅→𝑅, such that the function 𝐿∶𝐷_𝑓→𝑆 defined by 𝐿(𝑥)=𝐻(𝑥)+𝜆(𝑎)−𝐻(𝑎),𝑥∈𝐸_𝑥,𝐿𝑦=𝐻𝑦+𝜆(𝑏)−𝐻(𝑏),𝑦∈𝐸_𝑦,𝐿𝑥+𝑦=𝐻𝑥+𝑦+𝜆(𝑎)+𝜆(𝑏)−𝐻(𝑎)−𝐻(𝑏),𝑥+𝑦∈𝐸_𝑥+𝑦,(72) is a local solution of the additive Cauchy equation restricted to 𝐸(𝑎,𝑏;𝑟), such that |𝜆(𝑡)−𝐿(𝑡)|<15𝛿forevery𝑡∈𝐷_𝑓.(73)

Since |𝑓(𝑡)|=𝑒^𝜆(𝑡), whence, 𝑓(𝑡)=sgn𝑓(𝑡)⋅𝑒^𝜆(𝑡),𝑡∈𝐷_𝑓,(74) we get 𝑓(𝑡)=sgn𝑓(𝑡)⋅𝑒^{𝐿(𝑡)+15𝛿⋅𝜃(𝑡)},−1<𝜃(𝑡)<1,𝑡∈𝐷_𝑓,(75) substitution of 𝐿(𝑡) by its explicit formulas gives the following:for𝑡∈𝐸_𝑥∶𝑒^{−15𝛿⋅𝜃(𝑡)}𝑓(𝑡)=sgn𝑓(𝑎)⋅𝑒^{𝐻(𝑡)+ln|𝑓(𝑎)|−𝐻(𝑎)}=sgn𝑓(𝑎)⋅𝑓(𝑎)⋅𝑒^{−𝐻(𝑎)}⋅𝑒^𝐻(𝑡)=𝑓(𝑎)⋅𝑒^{−𝐻(𝑎)}⋅𝑒^𝐻(𝑡),(76)and similarly for𝑡∈𝐸_𝑦,𝑒^{−15𝛿⋅𝜃(𝑡)}𝑓(𝑡)=𝑓(𝑏)⋅𝑒^{−𝐻(𝑏)}⋅𝑒^𝐻(𝑡),for𝑡∈𝐸_𝑥+𝑦,𝑒^{−15𝛿⋅𝜃(𝑡)}𝑓(𝑡)=𝑓(𝑎)⋅𝑓(𝑏)⋅𝑒^{−𝐻(𝑎)−𝐻(𝑏)}⋅𝑒^𝐻(𝑡).(77)

By defining 𝐹(𝑡)∶=𝑒^{−15𝛿⋅𝜃(𝑡)}𝑓(𝑡),𝑡∈𝐸_𝑥∪𝐸_𝑦∪𝐸_𝑥+𝑦,−1<𝜃(𝑡)<1,(78) it is easily proved that 𝐹𝑥+𝑦=𝐹(𝑥)𝐹𝑦for𝑥,𝑦∈𝐸(𝑎,𝑏;𝑟).(79)

Moreover, from (78), 𝑒^−15𝛿⋅𝐹(𝑡)<𝑓(𝑡)<𝑒^15𝛿⋅𝐹(𝑡),𝑡∈𝐷_𝑓.(80) This means that the values of 𝑓(𝑡) in 𝐷_𝑓 are “near” (in dependence on 𝛿) the values of a local solution 𝐹(𝑡) of the corresponding equation restricted to the same domain 𝐸(𝑎,𝑏;𝑟) and give the following theorem of local stability.

Theorem 15. If the function 𝑓∶𝐷_𝑓=𝐸_𝑥∪𝐸_𝑦∪𝐸_𝑥+𝑦→𝑅, is nowhere vanishing in its domain 𝐷_𝑓 and satisfies  (7)^′_𝛿, for some given 𝛿>0 and every (𝑥,𝑦)∈𝐸(𝑎,𝑏;𝑟) defined in (6) for given (𝑎,𝑏)∈𝑅² and 𝑟>0, then there exists (at least) an additive function 𝐻∶𝑅→𝑅 such that the function 𝐹∶𝐷_𝑓→𝑅𝐹(𝑡)=𝑓(𝑎)𝑒^{−𝐻(𝑎)}⋅𝑒^𝐻(𝑡),𝑡∈𝐸_𝑥,𝑓(𝑏)𝑒^{−𝐻(𝑏)}⋅𝑒^𝐻(𝑡),𝑡∈𝐸_𝑦,𝑓(𝑎)𝑓(𝑏)𝑒^{−𝐻(𝑎)−𝐻(𝑏)}⋅𝑒^𝐻(𝑡),𝑡∈𝐸_𝑥+𝑦(81) has both the properties:(i)𝐹 is a nowhere vanishing local solution of the exponential Cauchy equation 𝐹(𝑥+𝑦)=𝐹(𝑥)𝐹(𝑦) restricted to 𝐸(𝑎,𝑏;𝑟);(ii)the values 𝐹(𝑡) are near the values 𝑓(𝑡) on 𝐷_𝑓; more exactly 𝑒^−15𝛿⋅𝐹(𝑡)<𝑓(𝑡)<𝑒^15𝛿⋅𝐹(𝑡),𝑡∈𝐷_𝑓.(82)

3.5. A Result on Local Stability of the Power Cauchy Equation

In the set of nowhere vanishing functions 𝑓∶𝐷_𝑓=𝐽_𝑥∪𝐽_𝑦∪𝐽_𝑥𝑦→𝑅, let us consider the inequality (65), (𝑥,𝑦)∈𝐽(𝑎,𝑏;𝑟)⊂𝑅² defined in (32), for some given 𝛿>0.

The usual substitutions 𝑥=𝑒^𝑢,𝑢=ln𝑥,𝑓(𝑥)=𝑓𝑒^𝑢=∶𝑔(𝑢),𝑦=𝑒^𝑣,𝑣=ln𝑦,𝑓𝑦=𝑓𝑒^𝑣=∶𝑔(𝑣),𝑥𝑦=𝑒^𝑢+𝑣,𝑓𝑥𝑦=∶𝑔(𝑢+𝑣),(83) transform the condition (65) into 𝑒^−𝛿<𝑔(𝑢+𝑣)𝑔(𝑢)𝑔(𝑣)<𝑒^𝛿,(𝑢,𝑣)∈𝐸(𝑎,𝑏;𝑟).(65)_𝑔

Hence, thanks to Theorem 15 (referred to 𝑔(𝑡) instead of 𝑓(𝑡)), there exists (at least) one additive function 𝐻∶𝑅→𝑅 such that the function 𝐺∶𝐷_𝑔→𝑅 defined by 𝐺(𝑢)=𝑔(𝑎)⋅𝑒^{−𝐻(𝑎)}⋅𝑒^𝐻(𝑢),𝑢∈𝐸_𝑢,𝐺(𝑣)=𝑔(𝑏)⋅𝑒^{−𝐻(𝑏)}⋅𝑒^𝐻(𝑣),𝑣∈𝐸_𝑣,𝐺(𝑢+𝑣)=𝑔(𝑎)𝑔(𝑏)⋅𝑒^{−𝐻(𝑎)−𝐻(𝑏)}⋅𝑒^{𝐻(𝑢+𝑣)},𝑢+𝑣∈𝐸_𝑢+𝑣,(84) satisfies the exponential equation 𝐺(𝑢+𝑣)=𝐺(𝑢)𝐺(𝑣) restricted to 𝐸(𝑎,𝑏;𝑟) and approaches 𝑔 on 𝐷_𝑔=𝐸_𝑢∪𝐸_𝑣∪𝐸_𝑢+𝑣 as follows: 𝑒^−15𝛿𝐺(𝑡)<𝑔(𝑡)<𝑒^15𝛿𝐺(𝑡),𝑡∈𝐷_𝑔.(82)_𝑔

Formula (82)_𝑔 can be rewritten as 𝑒^{15𝛿⋅𝜃(𝑡)}𝑔(𝑡)=𝐺(𝑡),−1<𝜃(𝑡)<1,(82)^′_𝑔 for 𝑡∈𝐸_𝑢∪𝐸_𝑣∪𝐸_𝑢+𝑣.

From the definition of 𝐺, for 𝑢∈𝐸_𝑢=[𝑎,𝑎+𝑟) then 𝑔(𝑢)=𝑓(𝑥), 𝐺(𝑢)=𝐺(ln𝑥)=𝑓(𝑒^𝑎)⋅𝑒^{−𝐻(𝑎)}⋅𝑒^𝐻(ln𝑥), similarly for 𝑣∈𝐸_𝑣=[𝑏,𝑏+𝑟) then 𝑔(𝑣)=𝑓(𝑦), 𝐺(ln𝑦)=𝑓(𝑒^𝑏)⋅𝑒^{−𝐻(𝑏)}⋅𝑒^𝐻(ln𝑦), and for 𝑢+𝑣∈𝐸_𝑢+𝑣=[𝑎+𝑏,𝑎+𝑏+𝑟): 𝐺(ln(𝑥𝑦))=𝑓(𝑒^𝑎)𝑓(𝑒^𝑏)⋅𝑒^{−𝐻(𝑎)−𝐻(𝑏)}⋅𝑒^{𝐻(ln𝑥𝑦)}.

Hence, by defining Φ∶𝐷_𝑓→𝑅 as follows: Φ(𝑥)∶=𝑒^{15𝛿⋅𝜃(𝑢)}𝑓(𝑥)=𝐺(ln𝑥)=𝑓𝑒^𝑎𝑒^{−𝐻(𝑎)}𝑒^𝐻(ln𝑥),Φ𝑦∶=𝑒^{15𝛿⋅𝜃(𝑣)}𝑓𝑦=𝐺ln𝑦=𝑓𝑒^𝑏𝑒^{−𝐻(𝑏)}𝑒^𝐻(ln𝑦),Φ𝑥𝑦∶=𝑒^{15𝛿⋅𝜃(𝑢+𝑣)}𝑓𝑒𝑥𝑦=𝐺ln𝑥𝑦=𝑓^𝑎𝑓𝑒^𝑏𝑒^{−𝐻(𝑎)−𝐻(𝑏)}𝑒^{𝐻(ln(𝑥𝑦))},(85) we get 𝑒^{15𝛿⋅𝜃(𝑡)}𝑓(𝑡)=Φ(𝑡),−1<𝜃(𝑡)<1,𝑡∈𝐽_𝑥∪𝐽_𝑦∪𝐽_𝑥𝑦;(86) namely, 𝑒^−15𝛿Φ(𝑡)<𝑓(𝑡)<𝑒^15𝛿Φ(𝑡),𝑡∈𝐷_𝑓=𝐽_𝑥∪𝐽_𝑦∪𝐽_𝑥𝑦.(87)

This proves the following property of local stability of the “power” Cauchy equation.

Theorem 16. Let the nowhere vanishing function 𝑓∶𝐷_𝑓⊂𝑅⁺→𝑅 satisfy the condition (65) for some given 𝛿>0 and every (𝑥,𝑦)∈𝐽(𝑎,𝑏;𝑟) defined in (32), for given (𝑎,𝑏)∈𝑅² and 𝑟>0; 𝐷_𝑓=𝐽_𝑥∪𝐽_𝑦∪𝐽_𝑥𝑦; then there exists (at least) an additive function 𝐻∶𝑅→𝑅 such that the function Φ∶𝐷_𝑓→𝑅 defined by Φ(𝑡)=𝑓𝑒^𝑎𝑒^{−𝐻(𝑎)}⋅𝑒^𝐻(ln𝑡),𝑡∈𝐽_𝑥,𝑓𝑒^𝑏𝑒^{−𝐻(𝑏)}⋅𝑒^𝐻(ln𝑡),𝑡∈𝐽_𝑦,𝑓𝑒^𝑎𝑓𝑒^𝑏𝑒^{−𝐻(𝑎)−𝐻(𝑏)}⋅𝑒^𝐻(ln𝑡),𝑡∈𝐽_𝑥𝑦(88) has both the following properties:(i)Φ is a local solution of the Cauchy equation Φ(𝑥𝑦)=Φ(𝑥)Φ(𝑦) restricted to 𝐽(𝑎,𝑏;𝑟);(ii)the values of Φ(𝑡) are near the values 𝑓(𝑡) in 𝐷_𝑓; more exactly 𝑒^−15𝛿⋅Φ(𝑡)<𝑓(𝑡)<𝑒^15𝛿⋅Φ(𝑡),𝑡∈𝐷_𝑓=𝐽_𝑥∪𝐽_𝑦∪𝐽_𝑥𝑦.(89)

3.6. Remark about the Pexiderized Forms of the Foregoing Equations

According to the remarks at the end of Section 2, the stability results given by Theorems 15 and 16 can be easily adapted to the Pexiderized forms of the corresponding equations, namely, to 𝛼(𝑥)𝛾𝑦(),𝜌(𝑥)𝜏𝑦(),𝑥+𝑦=𝛽restrictedto𝐸𝑎,𝑏;𝑟𝑥𝑦=𝜎restrictedto𝐽𝑎,𝑏;𝑟(90) for nowhere vanishing functions 𝛽∶𝐸_𝑥⟶𝑅,𝛾∶𝐸_𝑦⟶𝑅,𝛼∶𝐸_𝑥+𝑦⟶𝑅,𝜎∶𝐽_𝑥⟶𝑅,𝜏∶𝐽_𝑦⟶𝑅,𝜌∶𝐽_𝑥𝑦⟶𝑅.(91)