Abstract and Applied Analysis

Volume 2013 (2013), Article ID 648709, 12 pages

http://dx.doi.org/10.1155/2013/648709

Research Article

Approximate Multi-Jensen, Multi-Euler-Lagrange Additive and Quadratic Mappings in -Banach Spaces

Tian Zhou Xu

School of Mathematics, Beijing Institute of Technology, Beijing 100081, China

Received 26 April 2013; Accepted 31 July 2013

Academic Editor: Krzysztof Ciepliński

Copyright © 2013 Tian Zhou Xu. 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 prove the generalized Hyers-Ulam stability of multi-Jensen, multi-Euler-Lagrange additive, and quadratic mappings in -Banach spaces, using the socalled direct method. The corollaries from our main results correct some outcomes from Park (2011).

1. Introduction and Preliminaries

In 2005, Prager and Schwaiger (see [1] and also [2]) introduced the notion of multi-Jensen functions with the connection with generalized polynomials and obtained their general form. In 2008, (see [3]) they also proved the Hyers-Ulam stability of multi-Jensen equation, whereas Ciepliński (see [4, 5]) showed its generalized stability: in the spirit of Bourgin (see [6]) and Găvruţa (see [7]), and in the spirit of Aoki (see [8]) and Rassias (see [9]). Recently, some further results on the stability of multi-Jensen mappings were obtained in [10–14]. We refer the reader to [15–19] for more information on different aspects of stability of functional equations.

In this paper, we deal with the generalized Hyers-Ulam stability of multi-Jensen, multi-Euler-Lagrange additive, and quadratic mappings in -Banach spaces. The corollaries from our main results correct some outcomes from [20]. The results of Sections 2 and 4 generalize those from [12].

The concept of 2-normed spaces was initially developed by Gähler [21, 22] in the middle of the 1960s, while that of -normed spaces can be found in [23, 24]. Since then, many others have studied this concept and obtained various results (see [23, 25–27]).

Throughout this paper, stands for the set of all positive integers and represents the set of all real numbers. Moreover, we fix two positive integers and .

We recall some basic facts concerning -normed spaces.

Definition 1. Let and let be a real linear space with , and let be a function satisfying the following properties:

(N1) if and only if are linearly dependent,(N2) is invariant under permutation,(N3),(N4)

for all and . Then the function is called an -norm on , and the pair is called an -normed space.

A sequence in an -normed space is said to a converge to some in the -norm if

for every . Every convergent sequence has exactly one limit. If is the limit of the sequence , then we write . For any convergent sequences and of elements of , the sequence is convergent and

If, moreover, is a convergent sequence of real numbers, then the sequence is also convergent and

A sequence in an -normed space is said to be a Cauchy sequence with respect to the -norm if

for every . A linear -normed space in which every Cauchy sequence is convergent is called an -Banach space.

Example 2. For , the Euclidean -norm is defined by

where for each .

Example 3. The standard -norm on , a real inner product space of dimension dim , is as follows:

where denotes the inner product on . If , then this -norm is exactly the same as the Euclidean -norm mentioned earlier. For , this -norm is the usual norm .

In what follows, we will also use the following lemma from [19].

Lemma 4. Let be an -normed space. Then,

(1)for and , a real number,

for all ,(2) for all ,(3)if for all , then ,(4)for a convergent sequence in ,

for all .

2. Approximate Multi-Jensen Mappings

First, we prove the stability of the system of equations defining multi-Jensen mappings in -Banach spaces. For a given mapping , we define the difference operators

Theorem 5. Let be a commutative group uniquely divisible by , and, be an -Banach space. Assume also that for every , is a mapping such that

If is a function satisfying

then for every , there exists a multi-Jensen mapping for which

For every , the function is given by

Proof. Fix , and . By (12) and (11), we get

Hence,

and consequently for any nonnegative integers and such that , we obtain

Therefore, from (10), it follows that is a Cauchy sequence. Since is an -Banach space, this sequence is convergent and we define by (14). Putting , letting in (17), and using Lemma 4 and (10), we see that (13) holds.

Finally, fix , , and note that according to (12), we have

Next, fix , and assume that (the same arguments apply to the case where ). From (12), it follows that

Letting in the above two inequalities and using (10) and Lemma 4, we see that the mapping is multi-Jensen.

Theorem 6. Let be a real linear space and, be an -Banach space. Assume also that for every , is a mapping such that

If is a function satisfying conditions (11) and (12), then for every there exists a multi-Jensen mapping for which

For every , the function is given by

Proof. Fix , , and . By (12) and (11), we get

and consequently for any non-negative integers and such that , we obtain

Therefore, from (20), it follows that is a Cauchy sequence. Since is an -Banach space, this sequence is convergent and we define by (22). Putting , letting in (24), and using Lemma 4 and (20), we see that (21) holds.

Finally, fix , and note that according to (12), we have

Next, fix , and assume that (the same arguments apply to the case where ). From (12), it follows that

Letting in the previous two inequalities and using (20) and Lemma 4, we see that the mapping is multi-Jensen.

As applications of Theorems 5 and 6 we get the following corollaries.

Corollary 7. Let be a real normed linear space and, be an -Banach space. Assume also that and are such that . If is a function satisfying (11) and

then for every there exists a multi-Jensen mapping for which

for all , .

Corollary 8. Let be a real normed linear space and let be an -Banach space. Assume also that and are such that or . If is a function satisfying (11) and

then for every , there exists a multi-Jensen mapping for which

for all , .

From Corollary 8, we obtain the following corollary which corrects Theorems 3.1 and 3.2 from [20].

Corollary 9. Let be a real normed linear space and be an -Banach space. Assume also that and are such that . If is a function satisfying and

then there exists a Jensen mapping for which

3. Approximate Multi-Euler-Lagrange Additive Mappings

In this section, we prove the stability of the system of equations defining multi-Euler-Lagrange additive mappings.

Throughout this section, let be a real linear space and let be an -Banach space, and are fixed with .

A mapping is called a multiEuler-Lagrange additive mapping as follows if it satisfies the Euler-Lagrange additive equations in each of their arguments as follows:

for all and all . If , then the multi-Euler-Lagrange additive mapping is multiadditive (see [28]). For a given mapping , we define the difference operators

Theorem 10. Assume that for every , is a mapping such that

If is a function satisfying

then for every , there exists a unique multi-Euler-Lagrange additive mapping for which

For every , the function is given by

Proof. Fix , , and . By (36), we get

whence

For any nonnegative integers and with , using (40) we get

which tends to zero as tends to infinity. Therefore, from (35) it follows that is a Cauchy sequence in -Banach space and it thus converges. Hence, we can define by

Putting , letting in (41), and using (35), we see that (37) holds.

Now, fix also , and from (36), we have

Next, fix , , and assume that (the same arguments apply to the case where ). From (36) it follows that

Letting in the above two inequalities and using (35) and Lemma 4, we see that the mapping is multi-Euler-Lagrange additive.

Now, let us finally assume that is another multi-Euler-Lagrange additive mapping satisfying (37). Then we have

and therefore .

Theorem 11. Assume that for every , is a mapping such that

If is a function satisfying (36), then for every , there exists a unique multi-Euler-Lagrange additive mapping for which

For every , the function is given by

Proof. Fix , , and . By (36) we get

For any non-negative integers and with , using (49), we get

which tends to zero as tends to infinity. Therefore from (46), it follows that is a Cauchy sequence in -Banach space and it thus converges. Hence, we can define by

Putting , letting in (50), and using (46), we see that (47) holds. The further part of the proof is similar to the proof of Theorem 10.

As applications of Theorems 10 and 11, we get the following corollaries.

Corollary 12. Let be a real normed linear space and, be an -Banach space. Assume also that and are such that . If is a function satisfying

then for every there exists a unique multi-Euler-Lagrange additive mapping for which

for all , .

Corollary 13. Let be a real normed linear space and let be an -Banach space. Assume also that and are such that or . If is a function satisfying

then for every , there exists a unique multi-Euler-Lagrange additive mapping for which

for all , .

From Corollary 13 we obtain the following corollary which corrects Theorems 2.1 and 2.2 from [20].

Corollary 14. Let be a real normed linear space and be an 2-Banach space. Assume also that and are such that . If is a function satisfying

then there exists a unique additive mapping for which

4. Approximate Multi-Euler-Lagrange Quadratic Mappings

In this section, we prove the stability of the system of equations defining multi-Euler-Lagrange quadratic mappings.

Throughout this section, let be a real linear space and let be an -Banach space, and are fixed with .

Rassias [29] introduced the notion of a generalized Euler-Lagrange-type quadratic mapping, and investigated its generalized stability.

A mapping is called a multi-Euler-Lagrange quadratic mapping, if it satisfies the Euler-Lagrange quadratic equations in each of their arguments:

for all and all .

If , then the multi-Euler-Lagrange quadratic mapping is multiquadratic (see [30]). Letting in (58), we get . Putting in (58), we have

Replacing by and by in (58), respectively, we obtain

From (59) and (60), one gets

for all and all .

For a given mapping , we define the difference operators

Theorem 15. Assume that for every , is a mapping such that

If is a function satisfying condition (11) and

then for every , there exists a unique multi-Euler-Lagrange quadratic mapping for which

For every , the function is given by

Proof. Fix , , and . By (64), we get

From (67), we obtain

and consequently for any non-negative integers and such that , we get

Therefore from (63), it follows that is a Cauchy sequence. Since is an -Banach space, this sequence is convergent and we define by (66). Putting , letting in (69) and using Lemma 4 and (63), we see that (65) holds.

Now, fix also and note that according to (64), we have

Next, fix , , and assume that (the same arguments apply to the case where ). From (64), it follows that

Letting in the above two inequalities and using (63), and Lemma 4 we see that the mapping is multi-Euler-Lagrange quadratic.

Now, let us finally assume that is another multi-Euler-Lagrange quadratic mapping satisfying (65) and note that according to (61) and using Lemma 4, and (63) we have

Therefore, by Lemma 4, we can conclude that .

Similar to Theorem 15, one can get the following.

Theorem 16. Assume that for every , is a mapping such that

If is a function satisfying condition (11) and

then for every , there exists a unique multi-Euler-Lagrange quadratic mapping for which

For every the function is given by

Proof. Fix , , and . By (74), we obtain

and consequently for any non-negative integers and such that , we get

Therefore, from (73), it follows that is a Cauchy sequence. Since is an -Banach space, this sequence is convergent and we define by (76). Putting , letting in (78), and using Lemma 4 and (73) we see that (75) holds. The further part of the proof is similar to the proof of Theorem 15.

As applications of Theorems 15 and 16, we get the following corollaries.

Corollary 17. Let be a real normed linear space and, be an -Banach space. Assume also that and are such that . If is a function satisfying

then for every , there exists a unique multi-Euler-Lagrange quadratic mapping for which

for all , .

Corollary 18. Let be a real normed linear space and let be an -Banach space. Assume also that and are such that or . If is a function satisfying

then for every , there exists a unique multi-Euler-Lagrange quadratic mapping for which

for all , .

For , Corollary 18 yields the following corollary which corrects Theorems 4.1 and 4.2 from [20].

Corollary 19. Let be a real normed linear space and let be an 2-Banach space. Assume also that and are such that . If is a function satisfying

then there exists a unique quadratic mapping for which

Acknowledgments

This project was supported by the National Natural Science Foundation of China (NNSFC) (Grant no. 11171022). The author would like to thank Professor Krzysztof Ciepliński and anonymous referees for their valuable comments and suggestions.

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