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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 380743, 9 pages
About the Local Stability of the Four Cauchy Equations Restricted on a Bounded Domain and of Their Pexiderized Forms
Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
Received 13 June 2013; Accepted 3 September 2013
Academic Editor: Janusz Brzdęk
Copyright © 2013 Fulvia Skof. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
After this introduction, in Section 2 the local stability of the additive equation and, as a consequence, of the logarithmic equation both restricted on a bounded domain in 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  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 ) and they concern substantially the set of functions , satisfying the condition of -additivity restricted either on the triangular domain for some given , or the unbounded domain for some given .
It has to be remarked that in the classical paper  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 defined in (5)) or the restrictions to of functions additive on (in case of a domain like 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 of some function, which is additive in the whole space (see ).
Therefore, in order to adhere to the sense of Ulam’s question in case of a restricted domain like too, the locally -additive function has to be compared with the local solution of the corresponding exact equation restricted to the same set .
In this frame, in Section 2, first the local stability of the additive Cauchy equation restricted to 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, 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]).
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 of a real variable with values in a normed space , let us consider the inequality for some and , defined in (6), for given and .
As usual, the projections of will be denoted by
We purpose to check whether each satisfying (9) for is uniformly approached on by some satisfying the additive equation restricted to , namely, by a function of the following form (see ): where is additive on the whole space and are constant.
A positive answer is given by the following.
Theorem 1. Let , being a Banach space, satisfy (9) for some and every , defined in (6), for given and ; .
Then there exists (at least) a function , additive on , such that the function defined by has both the following properties:(i)is a (local) solution of the additive equation restricted on ;(ii) approaches uniformly on , and holds for every .
In order to prove Theorem 1, let us premise two lemmas.
Proof of Lemma 2. Let us prove (15) firstly for , . For the points and belong to ; hence, from (9) both the following inequalities hold
namely, (15) for , .
Similarly we prove (15), for , , assuming the pairs and with , from the formulas and for , assuming the pairs , , and .
In order to prove (16) let us assume .
For , from (15) and (9) we get since and , it follows from (9) that with , for , whence (16) for .
Similarly, for , and interchanged.
As for with , from (9) where , , whence .
Lemma 2 is proved.
Lemma 3. If , being a Banach space, satisfies (9) on , then each of the functions , , defined in (14) for , is uniformly approached on by the restriction of a function additive on .
The following inequalities hold:
Proof of Lemma 3. Since each function , , is -additive on (from Lemma 2), formula (23) follows immediately from a known result [4, Lemma 2]:
Let , a Banach space, be -additive on = .
Then there exists at least one additive function such that Moreover, for every and , formula (24) follows from (15) and (23); in fact This means that the restriction to of each additive , , approaches uniformly on each function , , .
Lemma 3 is proved.
Proof of Theorem 1. According to (24) in Lemma 3, each function , , defined in (14) for , namely,
is uniformly approached on by each of the additive functions , .
Let us define , , as follows: Such functions , , satisfy obviously the additive equation restricted to .
Moreover, thanks to Lemmas 2 and 3, each function approaches uniformly on as in formula (13); in fact, for arbitrary , similarly for , .
On the projection , where and , we get from Lemma 3 and formula (9) Therefore, each function , , satisfies (13), and Theorem 1 is proved.
Remark 4. The foregoing study was developed as though the projections , , and were pairwise disjoint.
If two of them overlap, for instance is nonempty, in every common point the values given by the different parts of formulas of approximating function have to be the same.
More in particular, if the set is connected, in (28) the equations hold, whence . In this case, the locally -additive is uniformly approached on the whole by the restriction to of a function , additive on (, either =1, or =2, or =3).
2.2. A Result on Local Stability of the Logarithmic Equation
The projections , , of are given by
Since the local stability of (2) depends on the comparison of every satisfying for with some solution of the corresponding equation (2) restricted to , let us premise (Lemma 5) the local solution of (2).
Proof of Lemma 5. By the usual substitutions the domain is transformed into a set like the one defined in (6). Let us consider Equation (2) is transformed into Therefore, using Theorem 1, we obtain (35) with additive .
The local stability of the logarithmic equation (2) is stated by the following.
Theorem 6. Let be a Banach space; if satisfies for some and every , defined in (32), for given and , then there exists (at least) a function , additive on , such that the function defined by satisfies both the following properties:(i) is a local solution of the logarithmic equation on the restricted domain ;(ii) approaches uniformly on , and holds for every .
Proof of Theorem 6. The usual substitutions , (like in proof of Lemma 5) transform the inequality (39) restricted to the set into
restricted to the set , defined in (6).
Now, we can follow the same line of proof as in Section 2.1 by defining the functions , , related to ; namely, then there exist functions , , additive on , such that each of the functions , , is a local solution of the equation restricted to , and holds for .
Now let us come back to , by the substitutions which transformed (39) into (42), beginning by the transformation of functions defined in (44); on (from ) similarly, for on (from ) and for on (from ).
By the definition formula (44) changes intoObviously each , , satisfies the logarithmic equation restricted to .
In order to prove the approximation stated in (41), let us begin by the projection : for ; then , , and ; hence, from (44) and from (45) Similarly for on and on .
Therefore, (41) is true with or or , 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
In fact, when the inequality is satisfied for every , the statement of Theorem 1 can be easily adapted to the condition (52), because the functions , and play a role like that of the restrictions of to the projections , , in case of a unique function .
Similarly for restricted to , 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.
Lemma 9. Let satisfy (7) restricted to defined in (6).
If and only if there exists some , such that and , the following properties (P1), (P2), (P3) hold: (P1) for every (hence for every ); (P2) is constant on each projection (not necessarily the same in different projections); (P3) is given on by the following formulas: (i)if , , (ii)if , , (iii)if , , where is additive on ; , are constant.
Remark 10. Notice that restricted to each of the projections , , is the restriction of a solution of the equation
valid on the whole , for suitable .
Since this equation can be written as we get , for some additive , whence formulas in (P3) of Lemma 9 for respectively, in , , .
Remark 12. As in Remark 10, we can see that the local solution of (8), restricted to each of the projections , , , is the restriction of a solution of a more general equation
for suitable .
From , it follows that , with
The foregoing Remarks 10 and 12, which point out a connection of the restricted equation under consideration with more general equations, namely, suggest the following forms of perturbation of such equations: for and some fixed .
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 .
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 , defined in (6) for some fixed , in the set of functions , such that for every .
Let us premise a remark about signs of nowhere vanishing functions satisfying on . From Lemma 9, Property (P2), it is known that every nowhere vanishing solution of the exponential Cauchy equation restricted to keeps a constant sign in each of the projections , , of .
This proves the following.
Similarly, we can prove a sign property concerning the perturbed form of the power equation.
The usual substitutions of variables allow us to use the foregoing results about the exponential equation. Putwhence , .
Then (65) is transformed into namely
Therefore, from Lemma 13, it follows that has constant sign (=) in , whence has constant sign (=) in ; similarly for in , namely for in and for in , namely in .
Hence, the following result is proved.
Lemma 14. Every nowhere vanishing function satisfying (65) restricted to has constant signs in each of the projections , , of .
3.4. A Result of Local Stability for the Exponential Cauchy Equation
On the ground of Theorem 1, there exists (at least) one additive function , such that the function defined by is a local solution of the additive Cauchy equation restricted to , such that
Since , whence, we get substitution of by its explicit formulas gives the following:and similarly
By defining it is easily proved that
Moreover, from (78), This means that the values of in are “near” (in dependence on ) the values of a local solution of the corresponding equation restricted to the same domain and give the following theorem of local stability.
Theorem 15. If the function , is nowhere vanishing in its domain and satisfies , for some given and every defined in (6) for given and , then there exists (at least) an additive function such that the function has both the properties:(i) is a nowhere vanishing local solution of the exponential Cauchy equation restricted to ;(ii)the values are near the values on ; more exactly
3.5. A Result on Local Stability of the Power Cauchy Equation
The usual substitutions transform the condition (65) into
Hence, thanks to Theorem 15 (referred to instead of ), there exists (at least) one additive function such that the function defined by satisfies the exponential equation restricted to and approaches on as follows:
From the definition of , for then , , similarly for then , , and for : .
Hence, by defining as follows: we get namely,
This proves the following property of local stability of the “power” Cauchy equation.
Theorem 16. Let the nowhere vanishing function satisfy the condition (65) for some given and every defined in (32), for given and ; ; then there exists (at least) an additive function such that the function defined by has both the following properties:(i) is a local solution of the Cauchy equation restricted to ;(ii)the values of are near the values in ; more exactly
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 for nowhere vanishing functions
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