Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 394216, 7 pages

http://dx.doi.org/10.1155/2013/394216

Research Article

On the Aleksandrov-Rassias Problems on Linear -Normed Spaces

Yumei Ma

Department of Mathematics, Dalian Nationality University, Dalian 116600, China

Received 11 May 2013; Accepted 17 July 2013

Academic Editor: Ji Gao

Copyright © 2013 Yumei Ma. 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.

This paper generalizes T. M. Rassias' results in 1993 to -normed spaces. If and are two real -normed spaces and is -strictly convex, a surjective mapping preserving unit distance in both directions and preserving any integer distance is an -isometry.

1. Introduction

Let and be two metric spaces. A mapping is called an isometry if satisfies for all , where and denote the metrics in the spaces and , respectively. For some fixed number , suppose that preserves distance , that is, for all with , we have , then is called a conservative (or preserved) distance for the mapping . In particular, we denote DOPP as preserving the one distance property and SDOPP as preserving the strong one distance property and also for .

In 1970 [1], Aleksandrov posed the following problem. Examine whether the existence of a single conservative distance for some mapping implies that is an isometry. This question is of great significance for the Mazur-Ulam Theorem [2].

In 1993, T. M. Rassias and P. Šemrl proved the following.

Theorem 1 (see [3]). Let and be two real normed linear spaces such that one of them has a dimension greater than one. Assume also that one of them is strictly convex. Suppose that is a surjective mapping that satisfies SDOPP. Then, is an affine isometry (a linear isometry up to translation).

Theorem 2 (see [3]). Let and be two real normed linear spaces such that one of them has a dimension greater than one. Suppose that is a Lipschitz mapping. Assume also that is a surjective mapping satisfying (SDOPP). Then, is an isometry.

Since 2004, the Aleksandrov problem in -normed spaces has been discussed, and some results are obtained [4–8].

Definition 3 (see [7]). Let be a real linear space with and , a function, then is called a linear -normed space if for any and all

 : are linearly dependent, : for every permutation of , : , : . The function is called the -norm on .

Definition 4 (see [8]). Let and be two real linear -normed spaces.

(i)A mapping is defined to be an -isometry if for all ,

(ii)A mapping is called the -distance one preserving property (-DOPP) if for , , it follows that .(iii)A mapping is called the -strong distance one preserving property (-SDOPP) if for , , it follows that and conversely.(iv)A mapping is called an -Lipschitz if for all ,

Definition 5 (see [7]). The points of are called -collinear if for every , is linearly dependent.

Definition 6. is said to be -strictly convex normed spaces if for any , and imply that and are linearly dependent.

C. Park and T. M. Rassias obtained the following.

Theorem 7 (see [8]). Let and be real linear -normed spaces. If a mapping satisfies the following conditions:

(i) has the -DOPP,(ii) is -Lipschitz,(iii) preserves the -collinearity,(iv) preserves the -collinearity,

then is an -isometry.

In 2009, Gao [6] researched another -isometry and gave the 2-strictly convex concept [6].

In this paper, we generalize T. M. Rassias Theorems 1 and 7 on -strictly convex normed spaces .

2. Main Results

The proof of the following lemma was presented in [9], to be published; the proof is given again for the convenience of readers.

Lemma 8. Let be an -normed space such that has dimension greater than and . Suppose that for . Then, there exists such that

Proof. Since are linearly independent and , then there exists with .

Set . For any , we have

Let us define by

then, we obtain

Set

Clearly, . And we have

On the other hand,

Thus,

Define by

It follows that

Thus, .

Obviously, is continuous on . Using the mean value theorem, there exists such that .

Set , , we have

And from , we have

Lemma 9. Let and be two real linear -normed spaces whose dimensions are greater than , and let be -strictly convex normed space. Suppose that is a surjective mapping satisfying (-SDOPP) with preserving distance for any . Then, preserves distance for any .

Proof. Firstly, is injective. Suppose, on the contrary, that there are , , such that . As , it follows that there exist vectors such that are linearly independent. Then, .

Set

Clearly,

Then

This implies that , which is a contradiction. Therefore, is a bijective mapping.

Let and satisfying

By Lemma 8, we can find with

Set

Clearly, we have

It follows from the hypothesis of preserving any integer ; then,

Clearly, we have

We conclude that

Otherwise, if for some , we have with or such that

Suppose that . Then,

Assume that

Set

Then, for ,

Since is bijective and preserves -SDOPP on both directions. Then, there exists with which satisfies that

However, by (20), , and thus , are linear dependent. Then,

This contradiction implies that

This also contradicts with (26). Since is -strictly convex, then there exists such that

Then,

Since

then . Thus,

Similarly,

Hence,

Lemma 10. Let and be real -normed spaces such that . If a mapping preserves the distance for each , then preserves the distance zero.

Proof. Choose such that ; that is, are linearly dependent. Assume that is a maximum linearly independent group of (. As , we can find a finite sequence of vectors such that are linearly independent. Hence, it holds that

We will prove that

for every . Let . We can find a vector such that are linearly independent. Set

for arbitrarily fixed . Then,

Since , we get

Since preserves the distance , we see that

For , we set

for any . Then, we have

for each . Since are linearly dependent, we get

and hence,

which together with (48) implies that

for all . By a similar argument, we further obtain that

In view of (45), (50), and (51), we conclude that

where denotes either or for .

Since preserves the distance for any , it follows from (52) that

where is an arbitrary positive integer. Hence, we conclude that

which implies that preserves the distance zero.

Remark 11. In Lemma 2.2 ([9] to be published), we give the same method under the condition of preserving 2-colinear.

Theorem 12. Let and be real -normed spaces such that and is -strictly convex. If a surjective mapping has the -SDOPP and preserves the distance for any , then is an affine -isometry.

Proof. Assume that for .

Take positive integers such that

Set

for , and

Clearly, for ,

According to Lemma 8, there exists such that

It follows from Lemma 9 that we have

for .

On the other hand,

Hence

Suppose that

For any , with

find a positive integer satisfying .

Set . Clearly, , and .

It follows that and

Then (63) is not valid. Hence,

Corollary 13. Let and be two real linear -normed spaces. Suppose that mapping preserves any positive integer -distance and Lipschitz condition. Then, is an -isometry.

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities in China and Education Department of Liaoning province in china.

References

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