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Abstract and Applied Analysis
Volumeย 2011, Article IDย 502903, 15 pages
http://dx.doi.org/10.1155/2011/502903
Research Article

Stability in Generalized Functions

Department of Mathematics, Sogang University, Seoul 121-741, Republic of Korea

Received 31 July 2011; Revised 17 September 2011; Accepted 21 September 2011

Academic Editor: Dumitruย Baleanu

Copyright ยฉ 2011 Young-Su Lee. 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.

Abstract

We consider the following additive functional equation with ๐‘›-independent variables: โˆ‘๐‘“(๐‘›๐‘–=1๐‘ฅ๐‘–โˆ‘)=๐‘›๐‘–=1๐‘“(๐‘ฅ๐‘–โˆ‘)+๐‘›๐‘–=1๐‘“(๐‘ฅ๐‘–โˆ’๐‘ฅ๐‘–โˆ’1) in the spaces of generalized functions. Making use of the heat kernels, we solve the general solutions and the stability problems of the above equation in the spaces of tempered distributions and Fourier hyperfunctions. Moreover, using the mollifiers, we extend these results to the space of distributions.

1. Introduction

The most famous functional equation is the Cauchy equation๐‘“(๐‘ฅ+๐‘ฆ)=๐‘“(๐‘ฅ)+๐‘“(๐‘ฆ),(1.1) any solution of which is called additive. It is well known that every measurable solution of (1.1) is of the form ๐‘“(๐‘ฅ)=๐‘Ž๐‘ฅ for some constant ๐‘Ž. In 1941, Hyers proved the stability theorem for (1.1) as follows.

Theorem 1.1 (see [1]). Let ๐ธ1 be a normed vector space, ๐ธ2 a Banach space. Suppose that ๐‘“โˆถ๐ธ1โ†’๐ธ2 satisfies the inequality โ€–๐‘“(๐‘ฅ+๐‘ฆ)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฆ)โ€–โ‰ค๐œ–,(1.2) for all ๐‘ฅ,๐‘ฆโˆˆ๐ธ1, then there exists the unique additive mapping ๐‘”โˆถE1โ†’๐ธ2 such that โ€–๐‘“(๐‘ฅ)โˆ’๐‘”(๐‘ฅ)โ€–โ‰ค๐œ–,(1.3) for all ๐‘ฅโˆˆ๐ธ1.

The above stability theorem was motivated by Ulam [2]. As noted in the above theorem, the stability problem of the functional equations means how the solution of the inequality differs from the solution of the original equation. Forti [3] noticed that the theorem of Hyers is still true if ๐ธ1 is replaced by an arbitrary semigroup. In 1950 Aoki [4] and in 1978 Rassias [5] generalized Hyersโ€™ result to the unbounded Cauchy difference. Thereafter, many authors studied the stability problems for (1.1) in various settings (see [6, 7]).

During the last decades, stability problems of various functional equations have been extensively studied and generalized by a number of authors (see [8โ€“17]). Among them, the following additive functional equation with ๐‘›-independent variables:๐‘“๎ƒฉ๐‘›๎“๐‘–=1๐‘ฅ๐‘–๎ƒช=๐‘›๎“๐‘–=1๐‘“๎€ท๐‘ฅ๐‘–๎€ธ+๐‘›๎“๐‘–=1๐‘“๎€ท๐‘ฅ๐‘–โˆ’๐‘ฅ๐‘–โˆ’1๎€ธ(1.4) was proposed by Nakmahachalasint [18], where ๐‘› is a positive integer with ๐‘›>1 and ๐‘ฅ0โ‰ก๐‘ฅ๐‘›. He solved the general solutions and the stability problems for the above equation. Actually, he proved that (1.4) is equivalent to (1.1).

In this paper, in a similar manner as in [19โ€“23], we solve the general solutions and the stability problems for (1.4) in the spaces of generalized functions such as the space ๐’ฎ๎…ž(โ„๐‘š) of tempered distributions, the space โ„ฑ๎…ž(โ„๐‘š) of Fourier hyperfunctions, and the space ๐’Ÿ๎…ž(โ„๐‘š) of distributions. Making use of the pullbacks, we first reformulate (1.4) and the related inequality in the spaces of generalized functions as follows:๐‘ขโˆ˜๐ด=๐‘›๎“๐‘–=1๐‘ขโˆ˜๐‘ƒ๐‘–+๐‘›๎“๐‘–=1๐‘ขโˆ˜๐ต๐‘–,โ€–โ€–โ€–โ€–(1.5)๐‘ขโˆ˜๐ดโˆ’๐‘›๎“๐‘–=1๐‘ขโˆ˜๐‘ƒ๐‘–โˆ’๐‘›๎“๐‘–=1๐‘ขโˆ˜๐ต๐‘–โ€–โ€–โ€–โ€–โ‰ค๐œ–,(1.6) where ๐ด, ๐‘ƒ๐‘–, and ๐ต๐‘– are the functions defined by ๐ด๎€ท๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›๎€ธ=๐‘ฅ1+โ‹ฏ+๐‘ฅ๐‘›,๐‘ƒ๐‘–๎€ท๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›๎€ธ=๐‘ฅ๐‘–๐ต,1โ‰ค๐‘–โ‰ค๐‘›,๐‘–๎€ท๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›๎€ธ=๐‘ฅ๐‘–โˆ’๐‘ฅ๐‘–โˆ’1,1โ‰ค๐‘–โ‰ค๐‘›.(1.7) Here โˆ˜ denotes the pullback of generalized functions, and the inequality โ€–๐‘ฃโ€–โ‰ค๐œ– in (1.6) means that |โŸจ๐‘ฃ,๐œ‘โŸฉ|โ‰ค๐œ–โ€–๐œ‘โ€–๐ฟ1 for all test functions ๐œ‘.

In Section 2, we will prove that every solution ๐‘ข in ๐’ฎ๎…ž(โ„๐‘š) or โ„ฑ๎…ž(โ„๐‘š) of (1.5) has the form ๐‘ข=๐‘Žโ‹…๐‘ฅ,(1.8) where ๐‘Žโˆˆโ„‚๐‘š. Also, we shall figure out that every solution ๐‘ข in ๐’ฎ๎…ž(โ„๐‘š) or โ„ฑ๎…ž(โ„๐‘š) of the inequality (1.6) can be written uniquely in the form ๐‘ข=๐‘Žโ‹…๐‘ฅ+๐œ‡(๐‘ฅ),(1.9) where ๐œ‡ is a bounded measurable function such that โ€–๐œ‡โ€–๐ฟโˆžโ‰ค((10๐‘›โˆ’3)/(2(2๐‘›โˆ’1)))๐œ–. Subsequently, in Section 3, these results are extended to the space ๐’Ÿ๎…ž(โ„๐‘š).

2. Stability in โ„ฑ๎…ž(โ„๐‘š)

We first introduce the spaces of tempered distributions and Fourier hyperfunctions. Here, we use the ๐‘š-dimensional notations, |๐›ผ|=๐›ผ1+โ‹ฏ+๐›ผ๐‘š, ๐›ผ!=๐›ผ1!โ‹ฏ๐›ผ๐‘š!, ๐œ๐›ผ=๐œ๐›ผ11โ‹ฏ๐œ๐›ผ๐‘š๐‘š, and ๐œ•๐›ผ=๐œ•๐›ผ11โ‹ฏ๐œ•๐›ผ๐‘š๐‘š, for ๐œ=(๐œ1,โ€ฆ,๐œ๐‘š)โˆˆโ„๐‘š, ๐›ผ=(๐›ผ1,โ€ฆ,๐›ผ๐‘š)โˆˆโ„•๐‘š0, where โ„•0 is the set of nonnegative integers and ๐œ•๐‘—=๐œ•/๐œ•๐œ๐‘—.

Definition 2.1 (see [24, 25]). We denote by ๐’ฎ(โ„๐‘š) the Schwartz space of all infinitely differentiable functions ๐œ‘ in โ„๐‘š satisfying โ€–๐œ‘โ€–๐›ผ,๐›ฝ=sup๐‘ฅโˆˆโ„๐‘š||๐‘ฅ๐›ผ๐œ•๐›ฝ๐œ‘||(๐‘ฅ)<โˆž,(2.1) for all ๐›ผ,๐›ฝโˆˆโ„•๐‘š0. A linear functional ๐‘ข on ๐’ฎ(โ„๐‘š) is said to be tempered distribution if there exist a constant ๐ถโ‰ฅ0 and a nonnegative integer ๐‘ such that ||||๎“โŸจ๐‘ข,๐œ‘โŸฉโ‰ค๐ถ||๐›ฝ|||๐›ผ|,โ‰ค๐‘sup๐‘ฅโˆˆโ„๐‘š||๐‘ฅ๐›ผ๐œ•๐›ฝ๐œ‘||,(2.2) for all ๐œ‘โˆˆ๐’ฎ(โ„๐‘š). The set of all tempered distributions is denoted by ๐’ฎ๎…ž(โ„๐‘š).

Note that tempered distributions are generalizations of ๐ฟ๐‘-functions. These are very useful for the study of Fourier transforms in generality, since all tempered distributions have a Fourier transform. Imposing the growth condition on โ€–โ‹…โ€–๐›ผ,๐›ฝ in (2.1), a new space of test functions has emerged as follows.

Definition 2.2 (see [26]). We denote by โ„ฑ(โ„๐‘š) the set of all infinitely differentiable functions ๐œ‘ in โ„๐‘š such that โ€–๐œ‘โ€–๐ด,๐ต=sup๐‘ฅ,๐›ผ,๐›ฝ||๐‘ฅ๐›ผ๐œ•๐›ฝ๐œ‘||(๐‘ฅ)๐ด|๐›ผ|๐ต|๐›ฝ|๐›ผ!๐›ฝ!<โˆž,(2.3) for some positive constants ๐ด,๐ต depending only on ๐œ‘. The strong dual of โ„ฑ(โ„๐‘š), denoted by โ„ฑ๎…ž(โ„๐‘š), is called the Fourier hyperfunction.

It can be verified that the seminorm (2.3) is equivalent to โ€–๐œ‘โ€–โ„Ž,๐‘˜=sup๐‘ฅ,๐›ผ||๐œ•๐›ผ||๐œ‘(๐‘ฅ)exp๐‘˜|๐‘ฅ|โ„Ž|๐›ผ|๐›ผ!<โˆž(2.4) for some constants โ„Ž,๐‘˜>0. It is easy to see the following topological inclusions:โ„ฑ(โ„)โ†ช๐’ฎ(โ„),๐’ฎ๎…ž(โ„)โ†ชโ„ฑ๎…ž(โ„).(2.5) Taking the inclusions (2.5) into account, it suffices to consider the space โ„ฑ๎…ž(โ„๐‘š). In order to solve the general solutions and the stability problems for (1.4) in the spaces โ„ฑ๎…ž(โ„๐‘š) and ๐’ฎ๎…ž(โ„๐‘š), we employ the ๐‘š-dimensional heat kernel, fundamental solution of the heat equation, ๐ธ๐‘ก(โŽงโŽชโŽจโŽชโŽฉ๐‘ฅ)=๐ธ(๐‘ฅ,๐‘ก)=(4๐œ‹๐‘ก)โˆ’๐‘š/2๎‚ตโˆ’exp|๐‘ฅ|2๎‚ถ4๐‘ก,๐‘ฅโˆˆโ„๐‘š,๐‘ก>0,0,๐‘ฅโˆˆโ„๐‘š,๐‘กโ‰ค0.(2.6) Since for each ๐‘ก>0, ๐ธ(โ‹…,๐‘ก) belongs to the space โ„ฑ(โ„๐‘š), the convolution ๎ซ๐‘ขฬƒ๐‘ข(๐‘ฅ,๐‘ก)=(๐‘ขโˆ—๐ธ)(๐‘ฅ,๐‘ก)=๐‘ฆ,๐ธ๐‘ก๎ฌ(๐‘ฅโˆ’๐‘ฆ),๐‘ฅโˆˆโ„๐‘š,๐‘ก>0(2.7) is well defined for all ๐‘ข in โ„ฑ๎…ž(โ„๐‘š), which is called the Gauss transform of ๐‘ข. Subsequently, the semigroup property ๎€ท๐ธ๐‘กโˆ—๐ธ๐‘ ๎€ธ(๐‘ฅ)=๐ธ๐‘ก+๐‘ (๐‘ฅ),๐‘ฅโˆˆโ„๐‘š,๐‘ก,๐‘ >0(2.8) of the heat kernel is very useful to convert (1.5) into the classical functional equation defined on upper-half plane. We also use the following famous result, so-called heat kernel method, which states as follows.

Theorem 2.3 (see [27]). Let ๐‘ขโˆˆ๐’ฎ๎…ž(โ„๐‘š), then its Gauss transform ฬƒ๐‘ข is a ๐ถโˆž-solution of the heat equation ๎‚€๐œ•๎‚๐œ•๐‘กโˆ’ฮ”ฬƒ๐‘ข(๐‘ฅ,๐‘ก)=0,(2.9) satisfying the following:(i) there exist positive constants ๐ถ, ๐‘€, and ๐‘ such that ||||ฬƒ๐‘ข(๐‘ฅ,๐‘ก)โ‰ค๐ถ๐‘กโˆ’๐‘€(1+|๐‘ฅ|)๐‘inโ„๐‘šร—(0,๐›ฟ),(2.10)(ii)ฬƒ๐‘ข(๐‘ฅ,๐‘ก)โ†’๐‘ข as ๐‘กโ†’0+ in the sense that for every ๐œ‘โˆˆ๐’ฎ(โ„๐‘š), โŸจ๐‘ข,๐œ‘โŸฉ=lim๐‘กโ†’0+๎€œฬƒ๐‘ข(๐‘ฅ,๐‘ก)๐œ‘(๐‘ฅ)๐‘‘๐‘ฅ.(2.11)Conversely, every ๐ถโˆž-solution ๐‘ˆ(๐‘ฅ,๐‘ก) of the heat equation satisfying the growth condition (2.10) can be uniquely expressed as ๐‘ˆ(๐‘ฅ,๐‘ก)=ฬƒ๐‘ข(๐‘ฅ,๐‘ก) for some ๐‘ขโˆˆ๐’ฎโ€ฒ(โ„๐‘š).

Similarly, we can represent Fourier hyperfunctions as a special case of the results as in [28]. In this case, the estimate (2.10) is replaced by the following.

For every ๐œ–>0, there exists a positive constant ๐ถ๐œ– such that

||||ฬƒ๐‘ข(๐‘ฅ,๐‘ก)โ‰ค๐ถ๐œ–๎‚ต๐œ–๎‚ตexp|๐‘ฅ|๐‘ก+1๎‚ถ๎‚ถinโ„ร—(0,๐›ฟ).(2.12)

We need the following lemma in order to solve the general solutions for the additive functional equation in the spaces of โ„ฑ๎…ž(โ„๐‘š) and ๐’ฎ๎…ž(โ„๐‘š). In what follows, we denote ๐‘ฅ0โ‰ก๐‘ฅ๐‘› and ๐‘ก0โ‰ก๐‘ก๐‘›.

Lemma 2.4. Suppose that ๐‘“โˆถโ„๐‘šร—(0,โˆž)โ†’โ„‚ is a continuous function satisfying ๐‘“๎ƒฉ๐‘›๎“๐‘–=1๐‘ฅ๐‘–,๐‘›๎“๐‘–=1๐‘ก๐‘–๎ƒช=๐‘›๎“๐‘–=1๐‘“๎€ท๐‘ฅ๐‘–,๐‘ก๐‘–๎€ธ+๐‘›๎“๐‘–=1๐‘“๎€ท๐‘ฅ๐‘–โˆ’๐‘ฅ๐‘–โˆ’1,๐‘ก๐‘–+๐‘ก๐‘–โˆ’1๎€ธ,(2.13) for all ๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›โˆˆโ„๐‘š, ๐‘ก1,โ€ฆ,๐‘ก๐‘›>0, then the solution ๐‘“ has the form ๐‘“(๐‘ฅ,๐‘ก)=๐‘Žโ‹…๐‘ฅ,(2.14) for some ๐‘Žโˆˆโ„‚๐‘š.

Proof. Putting (๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›)=(0,โ€ฆ,0) in (2.13) yields ๐‘“๎ƒฉ0,๐‘›๎“๐‘–=1๐‘ก๐‘–๎ƒช=๐‘›๎“๐‘–=1๐‘“๎€ท0,๐‘ก๐‘–๎€ธ+๐‘›๎“๐‘–=1๐‘“๎€ท0,๐‘ก๐‘–+๐‘ก๐‘–โˆ’1๎€ธ,(2.15) for all ๐‘ก1,โ€ฆ,๐‘ก๐‘›>0. In view of (2.15), we see that ๐‘โˆถ=lim๐‘กโ†’0+๐‘“(0,๐‘ก)(2.16) exists. Letting ๐‘ก1=โ‹ฏ=๐‘ก๐‘›โ†’0+ in (2.15) gives ๐‘=0. Setting (๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›)=(๐‘ฅ,0,โ€ฆ,0) and letting ๐‘ก1=๐‘ก, ๐‘ก2=โ‹ฏ=๐‘ก๐‘›โ†’0+ in (2.13), we have ๐‘“(โˆ’๐‘ฅ,๐‘ก)=โˆ’๐‘“(๐‘ฅ,๐‘ก),(2.17) for all ๐‘ฅโˆˆโ„๐‘š, ๐‘ก>0. Substituting (๐‘ฅ1,๐‘ฅ2,๐‘ฅ3,โ€ฆ,๐‘ฅ๐‘›) with (๐‘ฅ,๐‘ฆ,0,โ€ฆ,0) and letting ๐‘ก1=๐‘ก, ๐‘ก2=๐‘ , ๐‘ก3=โ‹ฏ=๐‘ก๐‘›โ†’0+ in (2.13), we obtain from (2.17) that ๐‘“(๐‘ฅ+๐‘ฆ,๐‘ก+๐‘ )+๐‘“(๐‘ฅโˆ’๐‘ฆ,๐‘ก+๐‘ )=2๐‘“(๐‘ฅ,๐‘ก),(2.18) for all ๐‘ฅ,๐‘ฆโˆˆโ„๐‘š, ๐‘ก,๐‘ >0. Putting ๐‘ฆ=0 in (2.18) yields ๐‘“(๐‘ฅ,๐‘ก+๐‘ )=๐‘“(๐‘ฅ,๐‘ก),(2.19) which shows that ๐‘“(๐‘ฅ,๐‘ก) is independent with respect to ๐‘ก>0. For that reason, we see from (2.18) that ๐น(๐‘ฅ)โˆถ=๐‘“(๐‘ฅ,๐‘ก) satisfies ๐น(๐‘ฅ+๐‘ฆ)+๐น(๐‘ฅโˆ’๐‘ฆ)=2๐น(๐‘ฅ),(2.20) for all ๐‘ฅ,๐‘ฆโˆˆโ„๐‘š. Replacing ๐‘ฅ by (๐‘ฅ+๐‘ฆ)/2 and ๐‘ฆ by (๐‘ฅโˆ’๐‘ฆ)/2 in (2.20), we have ๐น(๐‘ฅ+๐‘ฆ)=๐น(๐‘ฅ)+๐น(๐‘ฆ),(2.21) for all ๐‘ฅ,๐‘ฆโˆˆโ„๐‘š. Given the continuity, we obtain ๐‘“(๐‘ฅ,๐‘ก)=๐น(๐‘ฅ)=๐‘Žโ‹…๐‘ฅ,(2.22) for some ๐‘Žโˆˆโ„‚๐‘š.

From the above lemma, we can solve the general solutions for the additive functional equation in the spaces of โ„ฑ๎…ž(โ„๐‘š) and ๐’ฎ๎…ž(โ„๐‘š).

Theorem 2.5. Every solution ๐‘ข in โ„ฑโ€ฒ(โ„๐‘š) (or ๐’ฎโ€ฒ(โ„๐‘š), resp.) of (1.5) has the form ๐‘ข=๐‘Žโ‹…๐‘ฅ,(2.23) for some ๐‘Žโˆˆโ„‚๐‘š.

Proof. Convolving the tensor product ๐ธ๐‘ก1(๐‘ฅ1)โ‹ฏ๐ธ๐‘ก๐‘›(๐‘ฅ๐‘›) of the heat kernels on both sides of (1.5), we have ๎€บ๎€ท๐ธ(๐‘ขโˆ˜๐ด)โˆ—๐‘ก1๎€ท๐‘ฅ1๎€ธโ‹ฏ๐ธ๐‘ก๐‘›๎€ท๐‘ฅ๐‘›๐œ‰๎€ธ๎€ธ๎€ป๎€ท1,โ€ฆ,๐œ‰๐‘›๎€ธ=๎ซ๐‘ขโˆ˜๐ด,๐ธ๐‘ก1๎€ท๐œ‰1โˆ’๐‘ฅ1๎€ธโ‹ฏ๐ธ๐‘ก๐‘›๎€ท๐œ‰๐‘›โˆ’๐‘ฅ๐‘›=๎ƒก๎€œโ‹ฏ๎€œ๐ธ๎€ธ๎ฌ๐‘ข,๐‘ก1๎€ท๐œ‰1โˆ’๐‘ฅ1+๐‘ฅ2+โ‹ฏ+๐‘ฅ๐‘›๎€ธ๐ธ๐‘ก2๎€ท๐œ‰2โˆ’๐‘ฅ2๎€ธโ‹ฏ๐ธ๐‘ก๐‘›๎€ท๐œ‰๐‘›โˆ’๐‘ฅ๐‘›๎€ธ๐‘‘๐‘ฅ2โ‹ฏ๐‘‘๐‘ฅ๐‘›๎ƒข=๎ƒก๎€œโ‹ฏ๎€œ๐ธ๐‘ข,๐‘ก1๎€ท๐œ‰1+โ‹ฏ+๐œ‰๐‘›โˆ’๐‘ฅ1โˆ’โ‹ฏโˆ’๐‘ฅ๐‘›๎€ธ๐ธ๐‘ก2๎€ท๐‘ฅ2๎€ธโ‹ฏ๐ธ๐‘ก๐‘›๎€ท๐‘ฅ๐‘›๎€ธ๐‘‘๐‘ฅ2โ‹ฏ๐‘‘๐‘ฅ๐‘›๎ƒข=๎ซ๎€ท๐ธ๐‘ข,๐‘ก1โˆ—โ‹ฏโˆ—๐ธ๐‘ก๐‘›๐œ‰๎€ธ๎€ท1+โ‹ฏ+๐œ‰๐‘›โˆ’๐‘ฅ1=๎ซ๎€ธ๎ฌ๐‘ข,๐ธ๐‘ก1+โ‹ฏ+๐‘ก๐‘›๎€ท๐œ‰1+โ‹ฏ+๐œ‰๐‘›๎€ท๐œ‰๎€ธ๎ฌ=ฬƒ๐‘ข1+โ‹ฏ+๐œ‰๐‘›,๐‘ก1+โ‹ฏ+๐‘ก๐‘›๎€ธ,๎€บ๎€ท๐‘ขโˆ˜๐‘ƒ๐‘–๎€ธโˆ—๎€ท๐ธ๐‘ก1๎€ท๐‘ฅ1๎€ธโ‹ฏ๐ธ๐‘ก๐‘›๎€ท๐‘ฅ๐‘›๐œ‰๎€ธ๎€ธ๎€ป๎€ท1,โ€ฆ,๐œ‰๐‘›๎€ธ๎€ท๐œ‰=ฬƒ๐‘ข๐‘–,๐‘ก๐‘–๎€ธ,๎€บ๎€ท๐‘ขโˆ˜๐ต๐‘–๎€ธโˆ—๎€ท๐ธ๐‘ก1๎€ท๐‘ฅ1๎€ธโ‹ฏ๐ธ๐‘ก๐‘›๎€ท๐‘ฅ๐‘›๐œ‰๎€ธ๎€ธ๎€ป๎€ท1,โ€ฆ,๐œ‰๐‘›๎€ธ๎€ท๐œ‰=ฬƒ๐‘ข๐‘–โˆ’๐œ‰๐‘–โˆ’1,๐‘ก๐‘–+๐‘ก๐‘–โˆ’1๎€ธ,(2.24) where ฬƒ๐‘ข is the Gauss transform of ๐‘ข. Thus, (1.5) is converted into the following classical functional equation: ๎ƒฉฬƒ๐‘ข๐‘›๎“๐‘–=1๐‘ฅ๐‘–,๐‘›๎“๐‘–=1๐‘ก๐‘–๎ƒช=๐‘›๎“๐‘–=1๎€ท๐‘ฅฬƒ๐‘ข๐‘–,๐‘ก๐‘–๎€ธ+๐‘›๎“๐‘–=1๎€ท๐‘ฅฬƒ๐‘ข๐‘–โˆ’๐‘ฅ๐‘–โˆ’1,๐‘ก๐‘–+๐‘ก๐‘–โˆ’1๎€ธ,(2.25) for all ๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›โˆˆโ„๐‘š, ๐‘ก1,โ€ฆ,๐‘ก๐‘›>0. It follows from Lemma 2.4 that the solution ฬƒ๐‘ข of (2.25) has the form ฬƒ๐‘ข(๐‘ฅ,๐‘ก)=๐‘Žโ‹…๐‘ฅ,(2.26) for some ๐‘Žโˆˆโ„‚๐‘š. Letting ๐‘กโ†’0+ in (2.26), we finally obtain the general solution for (1.5).

We are going to solve the stability problems for the additive functional equation in the spaces of โ„ฑ๎…ž(โ„๐‘š) and ๐’ฎ๎…ž(โ„๐‘š).

Lemma 2.6. Suppose that ๐‘“โˆถโ„๐‘šร—(0,โˆž)โ†’โ„‚ is a continuous function satisfying |||||๐‘“๎ƒฉ๐‘›๎“๐‘–=1๐‘ฅ๐‘–,๐‘›๎“๐‘–=1๐‘ก๐‘–๎ƒชโˆ’๐‘›๎“๐‘–=1๐‘“๎€ท๐‘ฅ๐‘–,๐‘ก๐‘–๎€ธโˆ’๐‘›๎“๐‘–=1๐‘“๎€ท๐‘ฅ๐‘–โˆ’๐‘ฅ๐‘–โˆ’1,๐‘ก๐‘–+๐‘ก๐‘–โˆ’1๎€ธ|||||โ‰ค๐œ–,(2.27) for all ๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›โˆˆโ„๐‘š, ๐‘ก1,โ€ฆ,๐‘ก๐‘›>0, then there exists a unique ๐‘Žโˆˆโ„‚๐‘š such that ||||โ‰ค๐‘“(๐‘ฅ,๐‘ก)โˆ’๐‘Žโ‹…๐‘ฅ10๐‘›โˆ’32(2๐‘›โˆ’1)๐œ–,(2.28) for all ๐‘ฅโˆˆโ„๐‘š, ๐‘ก>0.

Proof. Putting (๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›)=(0,โ€ฆ,0) in (2.27) yields |||||๐‘“๎ƒฉ0,๐‘›๎“๐‘–=1๐‘ก๐‘–๎ƒชโˆ’๐‘›๎“๐‘–=1๐‘“๎€ท0,๐‘ก๐‘–๎€ธโˆ’๐‘›๎“๐‘–=1๐‘“๎€ท0,๐‘ก๐‘–+๐‘ก๐‘–โˆ’1๎€ธ|||||โ‰ค๐œ–,(2.29) for all ๐‘ก1,โ€ฆ,๐‘ก๐‘›>0. In view of (2.29), we see that ๐‘โˆถ=limsup๐‘กโ†’0+๐‘“(0,๐‘ก)(2.30) exists. Letting ๐‘ก1=โ‹ฏ=๐‘ก๐‘›โ†’0+ in (2.29) gives ๐œ–|๐‘|โ‰ค.2๐‘›โˆ’1(2.31) Setting (๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›)=(๐‘ฅ,0,โ€ฆ,0) and letting ๐‘ก1=๐‘ก, ๐‘ก2=โ‹ฏ=๐‘ก๐‘›โ†’0+ in (2.27), we have ||||๐‘“(๐‘ฅ,๐‘ก)+๐‘“(โˆ’๐‘ฅ,๐‘ก)+(2๐‘›โˆ’3)๐‘โ‰ค๐œ–,(2.32) for all ๐‘ฅโˆˆโ„๐‘š, ๐‘ก>0. Substituting (๐‘ฅ1,๐‘ฅ2,๐‘ฅ3,โ€ฆ,๐‘ฅ๐‘›)=(๐‘ฅ,๐‘ฅ,0,โ€ฆ,0) and letting ๐‘ก1=๐‘ก2=๐‘ก, ๐‘ก3=โ‹ฏ=๐‘ก๐‘›โ†’0+ in (2.13), we obtain ||||๐‘“(2๐‘ฅ,2๐‘ก)โˆ’3๐‘“(๐‘ฅ,๐‘ก)โˆ’๐‘“(โˆ’๐‘ฅ,๐‘ก)โˆ’๐‘“(0,2๐‘ก)โˆ’(2๐‘›โˆ’5)๐‘โ‰ค๐œ–,(2.33) for all ๐‘ฅโˆˆโ„๐‘š, ๐‘ก>0. Adding (2.33) to (2.32) yields ||||๐‘“(2๐‘ฅ,2๐‘ก)โˆ’2๐‘“(๐‘ฅ,๐‘ก)โˆ’๐‘“(0,2๐‘ก)+2๐‘โ‰ค2๐œ–,(2.34) for all ๐‘ฅโˆˆโ„๐‘š, ๐‘ก>0. Letting ๐‘ก1=๐‘ก2=๐‘ก, ๐‘ก3=โ‹ฏ=๐‘ก๐‘›โ†’0+ in (2.29) gives ||||4๐‘“(0,๐‘ก)+(2๐‘›โˆ’5)๐‘โ‰ค๐œ–,(2.35) for all ๐‘ก>0. Combining (2.31), (2.34), and (2.35), we have |||๐‘“(2๐‘ฅ,2๐‘ก)2|||โ‰คโˆ’๐‘“(๐‘ฅ,๐‘ก)10๐‘›โˆ’3๐œ–4(2๐‘›โˆ’1),(2.36) for all ๐‘ฅโˆˆโ„๐‘š, ๐‘ก>0. Making use of induction argument in (2.36), we obtain ||||๐‘“๎€ท2๐‘˜๐‘ฅ,2๐‘˜๐‘ก๎€ธ2๐‘˜||||โ‰คโˆ’๐‘“(๐‘ฅ,๐‘ก)10๐‘›โˆ’32(2๐‘›โˆ’1)๐œ–,(2.37) for all ๐‘˜โˆˆโ„•, ๐‘ฅโˆˆโ„๐‘š, ๐‘ก>0. Replacing ๐‘ฅ, ๐‘ก by 2๐‘™๐‘ฅ, 2๐‘™๐‘ก in (2.37), respectively, and dividing the result by 2๐‘™, we see that for ๐‘˜โ‰ฅ๐‘™>0, ||||๐‘“๎€ท2๐‘˜+๐‘™๐‘ฅ,2๐‘˜+๐‘™๐‘ก๎€ธ2๐‘˜+๐‘™โˆ’๐‘“๎€ท2๐‘™๐‘ฅ,2๐‘™๐‘ก๎€ธ2๐‘™||||โ‰ค10๐‘›โˆ’32๐‘™+1(2๐‘›โˆ’1)๐œ–.(2.38) Since the right-hand side of (2.38) tends to 0 as ๐‘™โ†’โˆž, the sequence {2โˆ’๐‘˜๐‘“(2๐‘˜๐‘ฅ,2๐‘˜๐‘ก)} is a Cauchy sequence which converges uniformly. Thus, we may define ๐ด(๐‘ฅ,๐‘ก)โˆถ=lim๐‘˜โ†’โˆž๐‘“๎€ท2๐‘˜๐‘ฅ,2๐‘˜๐‘ก๎€ธ2๐‘˜,(2.39) for all ๐‘ฅโˆˆโ„๐‘š, ๐‘ก>0. Now, we verify from (2.27) that the function ๐ด satisfies ๐ด๎ƒฉ๐‘›๎“๐‘–=1๐‘ฅ๐‘–,๐‘›๎“๐‘–=1๐‘ก๐‘–๎ƒช=๐‘›๎“๐‘–=1๐ด๎€ท๐‘ฅ๐‘–,๐‘ก๐‘–๎€ธ+๐‘›๎“๐‘–=1๐ด๎€ท๐‘ฅ๐‘–โˆ’๐‘ฅ๐‘–โˆ’1,๐‘ก๐‘–+๐‘ก๐‘–โˆ’1๎€ธ,(2.40) for all ๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›โˆˆโ„๐‘š, ๐‘ก1,โ€ฆ,๐‘ก๐‘›>0. As observed in Lemma 2.4, the continuous solution of (2.40) has the form ๐ด(๐‘ฅ,๐‘ก)=๐‘Žโ‹…๐‘ฅ,(2.41) for some ๐‘Žโˆˆโ„‚๐‘š. It follows from (2.37) that ๐ด is the unique function in โ„๐‘šร—(0,โˆž) satisfying ||||โ‰ค๐‘“(๐‘ฅ,๐‘ก)โˆ’๐ด(๐‘ฅ,๐‘ก)10๐‘›โˆ’32(2๐‘›โˆ’1)๐œ–,(2.42) for all ๐‘ฅโˆˆโ„๐‘š, ๐‘ก>0.

From the above lemma, we have the following stability theorem for the additive functional equation in the spaces of โ„ฑ๎…ž(โ„๐‘š) and ๐’ฎ๎…ž(โ„๐‘š).

Theorem 2.7. Suppose that ๐‘ข in โ„ฑ๎…ž(โ„๐‘š) (or ๐’ฎ๎…ž(โ„๐‘š), resp.) satisfies the inequality (1.6), then there exists a unique ๐‘Žโˆˆโ„‚๐‘š such that โ€–๐‘ขโˆ’๐‘Žโ‹…๐‘ฅโ€–โ‰ค10๐‘›โˆ’32(2๐‘›โˆ’1)๐œ–.(2.43)

Proof. Convolving the tensor product ๐ธ๐‘ก1(๐‘ฅ1)โ‹ฏ๐ธ๐‘ก๐‘›(๐‘ฅ๐‘›) of the heat kernels on both sides of (1.6), we have |||||๎ƒฉฬƒ๐‘ข๐‘›๎“๐‘–=1๐‘ฅ๐‘–,๐‘›๎“๐‘–=1๐‘ก๐‘–๎ƒชโˆ’๐‘›๎“๐‘–=1๎€ท๐‘ฅฬƒ๐‘ข๐‘–,๐‘ก๐‘–๎€ธโˆ’๐‘›๎“๐‘–=1๎€ท๐‘ฅฬƒ๐‘ข๐‘–โˆ’๐‘ฅ๐‘–โˆ’1,๐‘ก๐‘–+๐‘ก๐‘–โˆ’1๎€ธ|||||โ‰ค๐œ–,(2.44) for all ๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›โˆˆโ„๐‘š, ๐‘ก1,โ€ฆ,๐‘ก๐‘›>0, where ฬƒ๐‘ข is the Gauss transform of ๐‘ข. By Lemma 2.6, we have ||||โ‰คฬƒ๐‘ข(๐‘ฅ,๐‘ก)โˆ’๐‘Žโ‹…๐‘ฅ10๐‘›โˆ’32(2๐‘›โˆ’1)๐œ–,(2.45) for all ๐‘ฅโˆˆโ„๐‘š, ๐‘ก>0. Letting ๐‘กโ†’0+ in (2.45), we obtain the conclusion.

3. Stability in ๐’Ÿ๎…ž(โ„๐‘š)

In this section, we shall extend the previous results to the space of distributions. Recall that a distribution ๐‘ข is a linear functional on ๐ถโˆž๐‘(โ„๐‘š) of infinitely differentiable functions on โ„๐‘š with compact supports such that for every compact set ๐พโŠ‚โ„๐‘š, there exist constants ๐ถ>0 and ๐‘โˆˆโ„•0 satisfying ||||๎“โŸจ๐‘ข,๐œ‘โŸฉโ‰ค๐ถ|๐›ผ|โ‰ค๐‘||๐œ•sup๐›ผ๐œ‘||,(3.1) for all ๐œ‘โˆˆ๐ถโˆž๐‘(โ„๐‘š) with supports contained in ๐พ. The set of all distributions is denoted by ๐’Ÿ๎…ž(โ„๐‘š). It is well known that the following topological inclusions hold: ๐ถโˆž๐‘(โ„๐‘š)โ†ช๐’ฎ(โ„๐‘š),๐’ฎ๎…ž(โ„๐‘š)โ†ช๐’Ÿ๎…ž(โ„๐‘š).(3.2) As we see in [19, 20, 23], by the semigroup property of the heat kernel, (1.5) can be controlled easily in the spaces โ„ฑ๎…ž(โ„๐‘š) and ๐’ฎ๎…ž(โ„๐‘š). But we cannot employ the heat kernel in the space ๐’Ÿ๎…ž(โ„๐‘š). For that reason, instead of the heat kernel, we use the regularizing functions. We denote by ๐œ“ the function on โ„๐‘š satisfying ๎ƒฏ๎‚€โˆ’๎€ท๐œ“(๐‘ฅ)=๐ดexp1โˆ’|๐‘ฅ|2๎€ธโˆ’1๎‚,|๐‘ฅ|<1,0,|๐‘ฅ|โ‰ฅ1,(3.3) where ๎‚ต๎€œ๐ด=|๐‘ฅ|<1๎‚€โˆ’๎€ทexp1โˆ’|๐‘ฅ|2๎€ธโˆ’1๎‚๎‚ถ๐‘‘๐‘ฅโˆ’1.(3.4) It is easy to see that ๐œ“ is an infinitely differentiable function supported in the set {๐‘ฅโˆถ|๐‘ฅ|โ‰ค1} with โˆซ๐œ“(๐‘ฅ)๐‘‘๐‘ฅ=1. For each ๐‘ก>0, we define ๐œ“๐‘ก(๐‘ฅ)โˆถ=๐‘กโˆ’๐‘š๐œ“(๐‘ฅ/๐‘ก), then ๐œ“๐‘ก has all the properties of ๐œ“ except that the support of ๐œ“๐‘ก is contained in the ball of radius ๐‘ก with center at 0. If ๐‘ขโˆˆ๐’Ÿ๎…ž(โ„๐‘š), then for each ๐‘ก>0, (๐‘ขโˆ—๐œ“๐‘ก)(๐‘ฅ)=โŸจ๐‘ข๐‘ฆ,๐œ“๐‘ก(๐‘ฅโˆ’๐‘ฆ)โŸฉ is a smooth function in โ„๐‘š and (๐‘ขโˆ—๐œ“๐‘ก)(๐‘ฅ)โ†’๐‘ขas๐‘กโ†’0+ in the sense of distributions, that is, for every ๐œ‘โˆˆ๐ถโˆž๐‘(โ„๐‘š),โŸจ๐‘ขโˆ—๐œ“๐‘ก๎€œ๎€ท,๐œ‘โŸฉ=๐‘ขโˆ—๐œ“๐‘ก๎€ธ(๐‘ฅ)๐œ‘(๐‘ฅ)๐‘‘๐‘ฅโŸถโŸจ๐‘ข,๐œ‘โŸฉas๐‘กโŸถ0+.(3.5) For each ๐‘ก>0, the function ๐‘ขโˆ—๐œ“๐‘ก is called a regularization of ๐‘ข, and the transform which maps ๐‘ข to ๐‘ขโˆ—๐œ“๐‘ก is called a mollifier. Making use of the mollifiers, we can solve the general solution for the additive functional equation in the space ๐’Ÿ๎…ž(โ„๐‘š) as follows.

Theorem 3.1. Every solution ๐‘ข in ๐’Ÿ๎…ž(โ„๐‘š) of (1.5) has the form ๐‘ข=๐‘Žโ‹…๐‘ฅ,(3.6) for some ๐‘Žโˆˆโ„‚๐‘š.

Proof. Convolving the tensor product ๐œ“๐‘ก1(๐‘ฅ1)โ‹ฏ๐œ“๐‘ก๐‘›(๐‘ฅ๐‘›) of the regularizing functions on both sides of (1.5), we have ๎€บ๎€ท๐œ“(๐‘ขโˆ˜๐ด)โˆ—๐‘ก1๎€ท๐‘ฅ1๎€ธโ‹ฏ๐œ“๐‘ก๐‘›๎€ท๐‘ฅ๐‘›๐œ‰๎€ธ๎€ธ๎€ป๎€ท1,โ€ฆ,๐œ‰๐‘›๎€ธ=โŸจ๐‘ขโˆ˜๐ด,๐œ“๐‘ก1๎€ท๐œ‰1โˆ’๐‘ฅ1๎€ธโ‹ฏ๐œ“๐‘ก๐‘›๎€ท๐œ‰๐‘›โˆ’๐‘ฅ๐‘›๎€ธโŸฉ=๎ƒก๎€œโ‹ฏ๎€œ๐œ“๐‘ข,๐‘ก1๎€ท๐œ‰1โˆ’๐‘ฅ1+๐‘ฅ2+โ‹ฏ+๐‘ฅ๐‘›๎€ธ๐œ“๐‘ก2๎€ท๐œ‰2โˆ’๐‘ฅ2๎€ธโ‹ฏ๐œ“๐‘ก๐‘›๎€ท๐œ‰๐‘›โˆ’๐‘ฅ๐‘›๎€ธ๐‘‘๐‘ฅ2โ‹ฏ๐‘‘๐‘ฅ๐‘›๎ƒข=๎ƒก๎€œโ‹ฏ๎€œ๐œ“๐‘ข,๐‘ก1๎€ท๐œ‰1+โ‹ฏ+๐œ‰๐‘›โˆ’๐‘ฅ1โˆ’โ‹ฏโˆ’๐‘ฅ๐‘›๎€ธ๐œ“๐‘ก2๎€ท๐‘ฅ2๎€ธโ‹ฏ๐œ“๐‘ก๐‘›๎€ท๐‘ฅ๐‘›๎€ธ๐‘‘๐‘ฅ2โ‹ฏ๐‘‘๐‘ฅ๐‘›๎ƒข๎€ท๐œ“=โŸจ๐‘ข,๐‘ก1โˆ—โ‹ฏโˆ—๐œ“๐‘ก๐‘›๐œ‰๎€ธ๎€ท1+โ‹ฏ+๐œ‰๐‘›โˆ’๐‘ฅ1๎€ธโŸฉ=๎€ท๐‘ขโˆ—๐œ“๐‘ก1โˆ—โ‹ฏโˆ—๐œ“๐‘ก๐‘›๐œ‰๎€ธ๎€ท1+โ‹ฏ+๐œ‰๐‘›๎€ธ,๎€บ๎€ท๐‘ขโˆ˜๐‘ƒ๐‘–๎€ธโˆ—๎€ท๐œ“๐‘ก1๎€ท๐‘ฅ1๎€ธโ‹ฏ๐œ“๐‘ก๐‘›๎€ท๐‘ฅ๐‘›๐œ‰๎€ธ๎€ธ๎€ป๎€ท1,โ€ฆ,๐œ‰๐‘›๎€ธ=๎€ท๐‘ขโˆ—๐œ“๐‘ก๐‘–๐œ‰๎€ธ๎€ท๐‘–๎€ธ,๎€บ๎€ท๐‘ขโˆ˜๐ต๐‘–๎€ธโˆ—๎€ท๐œ“๐‘ก1๎€ท๐‘ฅ1๎€ธโ‹ฏ๐œ“๐‘ก๐‘›๎€ท๐‘ฅ๐‘›๐œ‰๎€ธ๎€ธ๎€ป๎€ท1,โ€ฆ,๐œ‰๐‘›๎€ธ=๎€ท๐‘ขโˆ—๐œ“๐‘ก๐‘–โˆ—๐œ“๐‘ก๐‘–โˆ’1๐œ‰๎€ธ๎€ท๐‘–โˆ’๐œ‰๐‘–โˆ’1๎€ธ.(3.7) Thus, (1.5) is converted into the following functional equation: ๎€ท๐‘ขโˆ—๐œ“๐‘ก1โˆ—โ‹ฏโˆ—๐œ“๐‘ก๐‘›๐‘ฅ๎€ธ๎€ท1+โ‹ฏ+๐‘ฅ๐‘›๎€ธ=๐‘›๎“๐‘–=1๎€ท๐‘ขโˆ—๐œ“๐‘ก๐‘–๐‘ฅ๎€ธ๎€ท๐‘–๎€ธ+๐‘›๎“๐‘–=1๎€ท๐‘ขโˆ—๐œ“๐‘ก๐‘–โˆ—๐œ“๐‘ก๐‘–โˆ’1๐‘ฅ๎€ธ๎€ท๐‘–โˆ’๐‘ฅ๐‘–โˆ’1๎€ธ,(3.8) for all ๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›โˆˆโ„๐‘š, ๐‘ก1,โ€ฆ,๐‘ก๐‘›>0. In view of (3.8), it is easy to see that, for each fixed ๐‘ฅโˆˆโ„๐‘š, ๐‘“(๐‘ฅ)โˆถ=lim๐‘กโ†’0+๎€ท๐‘ขโˆ—๐œ“๐‘ก๎€ธ(๐‘ฅ)(3.9) exists. Putting (๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›)=(0,โ€ฆ,0) and letting ๐‘ก1=โ‹ฏ=๐‘ก๐‘›โ†’0+ in (3.8) yield ๐‘“(0)=0. Setting (๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›)=(๐‘ฅ,0,โ€ฆ,0) and letting ๐‘ก1=๐‘ก, ๐‘ก2=โ‹ฏ=๐‘ก๐‘›โ†’0+ in (3.8) give ๎€ท๐‘ขโˆ—๐œ“๐‘ก๎€ธ๎€ท(โˆ’๐‘ฅ)=โˆ’๐‘ขโˆ—๐œ“๐‘ก๎€ธ(๐‘ฅ),(3.10) for all ๐‘ฅโˆˆโ„๐‘š, ๐‘ก>0. Substituting (๐‘ฅ1,๐‘ฅ2,๐‘ฅ3,โ€ฆ,๐‘ฅ๐‘›) with (๐‘ฅ,๐‘ฆ,0,โ€ฆ,0) and letting ๐‘ก1=๐‘ก, ๐‘ก2=๐‘ , ๐‘ก3=โ‹ฏ=๐‘ก๐‘›โ†’0+ in (3.8), we obtain from (3.10) that ๎€ท๐‘ขโˆ—๐œ“๐‘กโˆ—๐œ“๐‘ ๎€ธ๎€ท(๐‘ฅ+๐‘ฆ)+๐‘ขโˆ—๐œ“๐‘กโˆ—๐œ“๐‘ ๎€ธ๎€ท(๐‘ฅโˆ’๐‘ฆ)=2๐‘ขโˆ—๐œ“๐‘ก๎€ธ(๐‘ฅ),(3.11) for all ๐‘ฅ,๐‘ฆโˆˆโ„๐‘š, ๐‘ก,๐‘ >0. Letting ๐‘กโ†’0+ in (3.11) yields ๎€ท๐‘ขโˆ—๐œ“๐‘ ๎€ธ๎€ท(๐‘ฅ+๐‘ฆ)+๐‘ขโˆ—๐œ“๐‘ ๎€ธ(๐‘ฅโˆ’๐‘ฆ)=2๐‘“(๐‘ฅ),(3.12) for all ๐‘ฅ,๐‘ฆโˆˆโ„๐‘š, ๐‘ >0. Putting ๐‘ฆ=0 in (3.12) gives ๐‘“๎€ท(๐‘ฅ)=๐‘ขโˆ—๐œ“๐‘ ๎€ธ(๐‘ฅ),(3.13) for all ๐‘ฅโˆˆโ„๐‘š, ๐‘ >0. Applying (3.13) to (3.11), we see that ๐‘“ satisfies ๐‘“(๐‘ฅ+๐‘ฆ)+๐‘“(๐‘ฅโˆ’๐‘ฆ)=2๐‘“(๐‘ฅ),(3.14) which is equivalent to the Cauchy equation (1.1) for all ๐‘ฅ,๐‘ฆโˆˆโ„๐‘š. Since ๐‘“ is a smooth function in view of (3.13), it follows that ๐‘“(๐‘ฅ)=๐‘Žโ‹…๐‘ฅ for some ๐‘Žโˆˆโ„‚๐‘š. Letting ๐‘ โ†’0+ in (3.13), we finally obtain the general solution for (1.5).

Now, we shall extend the stability theorem for the additive equation mentioned in the previous section to the space ๐’Ÿ๎…ž(โ„๐‘š).

Theorem 3.2. Suppose that ๐‘ข in ๐’Ÿ๎…ž(โ„๐‘š) satisfies the inequality (1.6), then there exists a unique ๐‘Žโˆˆโ„‚๐‘š such that โ€–๐‘ขโˆ’๐‘Žโ‹…๐‘ฅโ€–โ‰ค10๐‘›โˆ’32(2๐‘›โˆ’1)๐œ–.(3.15)

Proof. It suffices to show that every distribution satisfying (1.6) belongs to the space ๐’ฎ๎…ž(โ„๐‘š). Convolving the tensor product ๐œ“๐‘ก1(๐‘ฅ1)โ‹ฏ๐œ“๐‘ก๐‘›(๐‘ฅ๐‘›) on both sides of (1.6), we have |||||๎€ท๐‘ขโˆ—๐œ“๐‘ก1โˆ—โ‹ฏโˆ—๐œ“๐‘ก๐‘›๐‘ฅ๎€ธ๎€ท1+โ‹ฏ+๐‘ฅ๐‘›๎€ธโˆ’๐‘›๎“๐‘–=1๎€ท๐‘ขโˆ—๐œ“๐‘ก๐‘–๐‘ฅ๎€ธ๎€ท๐‘–๎€ธโˆ’๐‘›๎“๐‘–=1๎€ท๐‘ขโˆ—๐œ“๐‘กiโˆ—๐œ“๐‘ก๐‘–โˆ’1๐‘ฅ๎€ธ๎€ท๐‘–โˆ’๐‘ฅ๐‘–โˆ’1๎€ธ|||||โ‰ค๐œ–,(3.16) for all ๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›โˆˆโ„๐‘š, ๐‘ก1,โ€ฆ,๐‘ก๐‘›>0. In view of (3.16), it is easy to see that for each fixed ๐‘ฅ, ๐‘“(๐‘ฅ)โˆถ=limsup๐‘กโ†’0+๎€ท๐‘ขโˆ—๐œ“๐‘ก๎€ธ(๐‘ฅ)(3.17) exists. Putting (๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›)=(0,โ€ฆ,0) and letting ๐‘ก1=โ‹ฏ=๐‘ก๐‘›โ†’0+ in (3.16) yield ||๐‘“||โ‰ค๐œ–(0).2๐‘›โˆ’1(3.18) Setting (๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›)=(๐‘ฅ,0,โ€ฆ,0) and letting ๐‘ก1=๐‘ก, ๐‘ก2=โ‹ฏ=๐‘ก๐‘›โ†’0+ in (3.16), we have ||๎€ท๐‘ขโˆ—๐œ“๐‘ก๎€ธ๎€ท(๐‘ฅ)+๐‘ขโˆ—๐œ“๐‘ก๎€ธ||(โˆ’๐‘ฅ)+(2๐‘›โˆ’3)๐‘“(0)โ‰ค๐œ–,(3.19) for all ๐‘ฅโˆˆโ„๐‘š, ๐‘ก>0. Substituting (๐‘ฅ1,๐‘ฅ2,๐‘ฅ3,โ€ฆ,๐‘ฅ๐‘›) with (๐‘ฅ,๐‘ฆ,0,โ€ฆ,0) and letting ๐‘ก1=๐‘ก, ๐‘ก2=๐‘ , ๐‘ก3=โ‹ฏ=๐‘ก๐‘›โ†’0+ in (3.16), we have ||๎€ท๐‘ขโˆ—๐œ“๐‘กโˆ—๐œ“๐‘ ๎€ธ๎€ท(๐‘ฅ+๐‘ฆ)โˆ’๐‘ขโˆ—๐œ“๐‘กโˆ—๐œ“๐‘ ๎€ธ๎€ท(๐‘ฆโˆ’๐‘ฅ)โˆ’2๐‘ขโˆ—๐œ“๐‘ก๎€ธ(๎€ท๐‘ฅ)โˆ’๐‘ขโˆ—๐œ“๐‘ ๎€ธ(๎€ท๐‘ฆ)โˆ’๐‘ขโˆ—๐œ“๐‘ ๎€ธ(||โˆ’๐‘ฆ)โˆ’(2๐‘›โˆ’5)๐‘“(0)โ‰ค๐œ–,(3.20) for all ๐‘ฅ,๐‘ฆโˆˆโ„๐‘š, ๐‘ก,๐‘ >0. It follows from (3.19) that the inequality (3.20) can be rewritten as ||๎€ท๐‘ขโˆ—๐œ“๐‘กโˆ—๐œ“๐‘ ๎€ธ๎€ท(๐‘ฅ+๐‘ฆ)โˆ’๐‘ขโˆ—๐œ“๐‘กโˆ—๐œ“๐‘ ๎€ธ๎€ท(๐‘ฆโˆ’๐‘ฅ)โˆ’2๐‘ขโˆ—๐œ“๐‘ก๎€ธ||(๐‘ฅ)+2๐‘“(0)โ‰ค2๐œ–,(3.21) for all ๐‘ฅ,๐‘ฆโˆˆโ„๐‘š, ๐‘ก,๐‘ >0. Letting ๐‘กโ†’0+ in (3.21) yields ||๎€ท๐‘ขโˆ—๐œ“๐‘ ๎€ธ๎€ท(๐‘ฅ+๐‘ฆ)โˆ’๐‘ขโˆ—๐œ“๐‘ ๎€ธ||(๐‘ฆโˆ’๐‘ฅ)โˆ’2๐‘“(๐‘ฅ)+2๐‘“(0)โ‰ค2๐œ–,(3.22) for all ๐‘ฅ,๐‘ฆโˆˆโ„๐‘š, ๐‘ >0. Using (3.19) we may write the inequality (3.22) as ||๎€ท๐‘ขโˆ—๐œ“๐‘ ๎€ธ๎€ท(๐‘ฅ+๐‘ฆ)+๐‘ขโˆ—๐œ“๐‘ ๎€ธ||(๐‘ฅโˆ’๐‘ฆ)โˆ’2๐‘“(๐‘ฅ)+(2๐‘›โˆ’1)๐‘“(0)โ‰ค3๐œ–,(3.23) for all ๐‘ฅ,๐‘ฆโˆˆโ„๐‘š, ๐‘ก,๐‘ >0. Putting ๐‘ฆ=0 in (3.23) and dividing the result by 2 give |||๎€ท๐‘ขโˆ—๐œ“๐‘ ๎€ธ(๐‘ฅ)โˆ’๐‘“(๐‘ฅ)+(2๐‘›โˆ’1)2|||โ‰ค3๐‘“(0)2๐œ–,(3.24) for all ๐‘ฅโˆˆโ„๐‘š,๐‘ >0. From (3.23) and (3.24), we have ||||๐‘“(๐‘ฅ+๐‘ฆ)+๐‘“(๐‘ฅโˆ’๐‘ฆ)โˆ’2๐‘“(๐‘ฅ)โ‰ค6๐œ–,(3.25) which is equivalent to |||๎‚€2๐‘“๐‘ฅ+๐‘ฆ2๎‚|||โˆ’๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฆ)โ‰ค6๐œ–,(3.26) for all ๐‘ฅ,๐‘ฆโˆˆโ„๐‘š. Thus, by virtue of the result as in [29], there exists a unique function ๐‘”โˆถโ„๐‘šโ†’โ„‚ satisfying ๐‘”(๐‘ฅ+๐‘ฆ)=๐‘”(๐‘ฅ)+๐‘”(๐‘ฆ)(3.27) such that ||||||||๐‘“(๐‘ฅ)โˆ’๐‘”(๐‘ฅ)โ‰ค6๐œ–+๐‘”(0),(3.28) for all ๐‘ฅโˆˆโ„๐‘š. It follows from (3.18), (3.24), and (3.28) that ||๎€ท๐‘ขโˆ—๐œ“๐‘ ๎€ธ||||๐‘”||(๐‘ฅ)โˆ’๐‘”(๐‘ฅ)โ‰ค8๐œ–+(0),(3.29) for all ๐‘ฅโˆˆโ„๐‘š, ๐‘ >0. Letting ๐‘ โ†’0+ in (3.29), we obtain โ€–||||๐‘ขโˆ’๐‘”(๐‘ฅ)โ€–โ‰ค8๐œ–+๐‘”(0).(3.30) Inequality (3.30) implies that โ„Ž(๐‘ฅ)โˆถ=๐‘ขโˆ’๐‘”(๐‘ฅ) belongs to (๐ฟ1)๎…ž=๐ฟโˆž. Thus, we conclude that ๐‘ข=๐‘”(๐‘ฅ)+โ„Ž(๐‘ฅ)โˆˆ๐’ฎ๎…ž(โ„๐‘š).

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