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Stability of general multi-Euler-Lagrange quadratic functional equations in non-Archimedean fuzzy normed spaces
Advances in Difference Equations volume 2012, Article number: 119 (2012)
Abstract
In this paper we prove the generalized Hyers-Ulam stability of the system defining general Euler-Lagrange quadratic mappings in non-Archimedean fuzzy normed spaces over a field with valuation using the direct and the fixed point methods.
MSC:39B82, 39B52, 46H25.
1 Introduction
Let be a field. A valuation mapping on is a function such that for any the following conditions are satisfied: (i) and equality holds if and only if ; (ii) ; (iii) .
A field endowed with a valuation mapping will be called a valued field. The usual absolute values of and are examples of valuations. A trivial example of a non-Archimedean valuation is the function taking everything except for 0 into 1 and . In the following we will assume that is non-trivial, i.e., there is an such that .
If the condition (iii) in the definition of a valuation mapping is replaced with a strong triangle inequality (ultrametric): , then the valuation is said to be non-Archimedean. In any non-Archimedean field we have and for all .
Throughout this paper, we assume that is a valued field, and are vector spaces over , are fixed with ( if ) and n is a positive integer. Moreover, stands for the set of all positive integers and (respectively, ) denotes the set of all reals (respectively, rationals).
A mapping is called a general multi-Euler-Lagrange quadratic mapping if it satisfies the general Euler-Lagrange quadratic equations in each of their n arguments:

for all and all . Letting in (1.1), we get . Putting in (1.1), we have
Replacing by and by in (1.1), respectively, we obtain

From (1.2) and (1.3), one gets
for all and all . If in (1.1), then we have

Letting in (1.5), we obtain
for all and all .
The study of stability problems for functional equations is related to a question of Ulam [30] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [13]. The result of Hyers was generalized by Aoki [2] for approximate additive mappings and by Rassias [27] for approximate linear mappings by allowing the Cauchy difference operator to be controlled by . In 1994, a further generalization was obtained by Găvruţa [9], who replaced by a general control function . We refer the reader to see, for instance, [1, 4–7, 14–16, 18, 20, 22, 23, 25, 26, 28, 31–37] for more information on different aspects of stability of functional equations. On the other hand, for some outcomes on the stability of multi-quadratic and Euler-Lagrange-type quadratic mappings we refer the reader to [7, 11, 24].
The main purpose of this paper is to prove the generalized Hyers-Ulam stability of multi-Euler-Lagrange quadratic functional equation (1.1) in complete non-Archimedean fuzzy normed spaces over a field with valuation using the direct and the fixed point methods.
2 Preliminaries
We recall the notion of non-Archimedean fuzzy normed spaces over a field with valuation and some preliminary results (see for instance [3, 22, 23, 31, 32]). For more details the reader is referred to [3, 22].
Definition 2.1 Let be a linear space over a field with a non-Archimedean valuation . A function is said to be a non-Archimedean norm if it satisfies the following conditions:
-
(i)
if and only if ;
-
(ii)
, , ;
-
(iii)
the strong triangle inequality
Then is called a non-Archimedean normed space. By a complete non-Archimedean normed space, we mean one in which every Cauchy sequence is convergent.
In 1897, Hensel discovered the p-adic numbers as a number-theoretical analogue of power series in complex analysis. Let p be a prime number. For any nonzero rational number a, there exists a unique integer r such that , where m and n are integers not divisible by p. Then defines a non-Archimedean norm on . The completion of with respect to the metric is denoted by which is called the p-adic number field. Note that if , then for each integer n but .
During the last three decades, p-adic numbers have gained the interest of physicists for their research, in particular, into problems deriving from quantum physics, p-adic strings, and superstrings (see for instance [21]).
A triangular norm (shorter t-norm, [29]) is a binary operation which satisfies the following conditions: (a) T is commutative and associative; (b) for all ; (c) whenever and for all . Basic examples of continuous t-norms are the Łukasiewicz t-norm , , the product t-norm , and the strongest triangular norm , . A t-norm is called continuous if it is continuous with respect to the product topology on the set .
A t-norm T can be extended (by associativity) in a unique way to an m-array operation taking for , the value defined recurrently by and for . T can also be extended to a countable operation, taking for any sequence in , the value is defined as . The limit exists since the sequence is non-increasing and bounded from below. is defined as .
Definition 2.2 A t-norm T is said to be of Hadžić-type (H-type, we denote by ) if a family of functions for all is equicontinuous at , that is, for all there exists such that
The t-norm is a t-norm of Hadžić-type. Other important triangular norms we refer the reader to [12].
Proposition 2.3 (see [12])
-
(1)
If or , then
-
(2)
If T is of Hadžić-type, then
for every sequence in such that .
Definition 2.4 (see [22])
Let be a linear space over a valued field and T be a continuous t-norm. A function is said to be a non-Archimedean fuzzy Menger norm on if for all and all :
(N1) for all ;
(N2) if and only if , ;
(N3) if ;
(N4) , ;
(N5) .
If N is a non-Archimedean fuzzy Menger norm on , then the triple is called a non-Archimedean fuzzy normed space. It should be noticed that from the condition (N4) it follows that
for every and , that is, is non-decreasing for every x. This implies . If (N4) holds, then so does
(N6) .
We repeatedly use the fact , , , which is deduced from (N3). We also note that Definition 2.4 is more general than the definition of a non-Archimedean Menger norm in [23, 31], where only fields with a non-Archimedean valuation have been considered.
Definition 2.5 Let be a non-Archimedean fuzzy normed space. Let be a sequence in . Then is said to be convergent if there exists such that for all . In that case, x is called the limit of the sequence and we denote it by . The sequence in is said to be a Cauchy sequence if for all and . If every Cauchy sequence in is convergent, then the space is called a complete non-Archimedean fuzzy normed space.
Example 2.6 Let be a real (or non-Archimedean) normed space. For each , consider
Then is a non-Archimedean fuzzy normed space.
Example 2.7 (see [22])
Let be a real normed space. Then the triple , where
is a non-Archimedean fuzzy normed space. Moreover, if is complete, then is complete and therefore it is a complete non-Archimedean fuzzy normed space over an Archimedean valued field.
Let Ω be a set. A function is called a generalized metric on Ω if d satisfies
-
(1)
if and only if ; (2) , ; (3) , .
For explicitly later use, we recall the following result by Diaz and Margolis [8].
Theorem 2.8 Let be a complete generalized metric space and be a strictly contractive mapping with Lipschitz constant , that is
If there exists a nonnegative integer such that for an , then
-
(1)
the sequence converges to a fixed point of J;
-
(2)
is the unique fixed point of J in the set ,
-
(3)
if , then
3 Stability of the functional equation (1.1): a direct method
Throughout this section, using a direct method, we prove the stability of Eq. (1.1) in complete non-Archimedean fuzzy normed spaces.
Theorem 3.1 Let be a valued field, be a vector space over and be a complete non-Archimedean fuzzy normed space over . Assume also that, for every , is a mapping such that

and

for all and . If is a mapping satisfying
and

for all , and , then for every there exists a unique general multi-Euler-Lagrange quadratic mapping such that

for all and .
Proof Fix , , and . Putting in (3.4), we get

Replacing by and by in (3.4), respectively, we have

From (3.6) and (3.7), one gets

Therefore one can get

and thus from (3.2) it follows that is a Cauchy sequence in a complete non-Archimedean fuzzy normed space. Hence, we can define a mapping such that
Next, for each with , we have

Therefore,

Letting in this inequality, we obtain (3.5). Now, fix also , from (3.1) and (3.4) it follows that

Next, fix , , and assume, without loss of generality, that (the same arguments apply to the case where ). From (3.1) and (3.4), it follows that

Hence the mapping is a general multi-Euler-Lagrange quadratic mapping. Let us finally assume that is another multi-Euler-Lagrange quadratic mapping satisfying (3.5). Then, by (1.4), (3.5) and (3.2), it follows that

and therefore . □
For , we get the following result.
Theorem 3.2 Let be a valued field, be a vector space over and be a complete non-Archimedean fuzzy normed space over . Assume also that, for every , is a mapping such that

and

for all and . If is a mapping satisfying (3.3) and

for all , and , then for every there exists a unique general multi-Euler-Lagrange quadratic mapping satisfying the functional equation (1.5) and such that

for all and .
Proof Fix , , and . Putting in (3.11), we get

Hence,

Therefore one can get

and thus by (3.10) it follows that is a Cauchy sequence in a complete non-Archimedean fuzzy normed space. Hence, we can define a mapping such that
Using (3.13) and induction, one can show that for any we have

Therefore,

Letting in this inequality, we obtain (3.12). The rest of the proof of this theorem is omitted as being similar to the corresponding that of Theorem 3.1. □
Let be a complete non-Archimedean fuzzy normed space over a non-Archimedean field . In any such space, a sequence is Cauchy if and only if converges to zero. Analysis similar to that in the proof of Theorem 3.2 gives the following.
Theorem 3.3 Let be a non-Archimedean field, be a vector space over and be a complete non-Archimedean fuzzy normed space over . Assume also that, for every , is a mapping such that (3.9) holds and
for all and . If is a mapping satisfying (3.3) and (3.11), then for every there exists a unique general multi-Euler-Lagrange quadratic mapping satisfying (1.5) and (3.12).
Remark 3.4 Let and be a commutative group, Theorems 3.1-3.3 also hold. For , consider the non-Archimedean fuzzy normed space defined as in Example 2.6, Theorem 3.3 yields Theorem 2 in [7]. If , then in Theorems 3.2-3.3 and which is a singular case of Theorem 3.1.
Analysis similar to that in the proof of Theorem 3.1 gives the following.
Theorem 3.5 Let be a valued field, be a vector space over and be a complete non-Archimedean fuzzy normed space over . Assume also that, for every , is a mapping such that

and

for all and . If is a mapping satisfying (3.3) and (3.4), then for every there exists a unique general multi-Euler-Lagrange quadratic mapping such that

for all and .
Corollary 3.6 Let be a non-Archimedean field with , be a normed space over and let be a complete non-Archimedean fuzzy normed space over under a t-norm . Assume also that and is a function such that and for all .
If is a mapping satisfying (3.3) and

for all , and , then for every there exists a unique general multi-Euler-Lagrange quadratic mapping such that

for all and .
Proof Fix , and . Let be defined by . Then we can apply Theorem 3.5 to obtain the result. □
Remark 3.7 Let and . Then the mapping given by , satisfies and for all .
4 Stability of the functional equation (1.1): a fixed point method
Throughout this section, we prove the stability of Eq. (1.1) in complete non-Archimedean fuzzy normed spaces using the fixed point method.
Theorem 4.1 Let be a valued field, be a vector space over and be a complete non-Archimedean fuzzy normed space over . Assume also that, for every , is a mapping such that (3.1) holds and

for an . If is a mapping satisfying (3.3) and (3.4), then for every there exists a unique general multi-Euler-Lagrange quadratic mapping such that

for all and .
Proof Fix an . Consider the set and introduce the generalized metric on Ω:
A standard verification (see for instance [19]) shows that is a complete generalized metric space. We now define an operator by
Let and with . Then
which together with (4.1) gives
and consequently, , which means that the operator is strictly contractive. Moreover, from (3.8) it follows that
and thus . Therefore, by Theorem 2.8, has a unique fixed point in the set such that
and
Furthermore, from the fact that , Theorem 2.8, and , we get
and (4.2) follows. Similar to the proof of Theorem 3.1, one can prove that the mapping is also general multi-Euler-Lagrange quadratic.
Let us finally assume that is a general multi-Euler-Lagrange quadratic mapping satisfying condition (4.2). Then fulfills (4.3), and therefore, it is a fixed point of the operator . Moreover, by (4.2), we have , and consequently . Theorem 2.8 shows that . □
Similar to Theorem 4.1, one can prove the following result.
Theorem 4.2 Let be a valued field, be a vector space over and be a complete non-Archimedean fuzzy normed space over . Assume also that, for every , is a mapping such that (3.14) holds and

for an . If is a mapping satisfying (3.3) and (3.4), then for every there exists a unique general multi-Euler-Lagrange quadratic mapping such that
for all and .
Remark 4.3 Similar to the proof of Corollary 3.6, one can deduce from Theorem 4.2 an analog of Corollary 3.6.
As applications of Theorems 4.1 and 4.2 , we get the following corollaries.
Corollary 4.4 Let be a real normed space, be a real Banach space and be the complete non-Archimedean fuzzy normed space defined as in the second example in the preliminaries. Let such that , or . If is a mapping satisfying (3.3) and (3.4), where
then for every there exists a unique general multi-Euler-Lagrange quadratic mapping such that

for all and .
Proof Fix , , and assume that , (the same arguments apply to the case where , ). Then we can choose and apply Theorem 4.2 to obtain the result. For , , or , , the corollary follows from Theorem 4.1. □
Corollary 4.5 Let be a real normed space and be a real Banach space (or be a non-Archimedean normed space and be a complete non-Archimedean normed space over a non-Archimedean field , respectively). Let and . If is a mapping satisfying (3.3) and

then for every there exists a unique general multi-Euler-Lagrange quadratic mapping such that
for all .
Proof Consider the non-Archimedean fuzzy normed space defined as in the first example in the preliminaries, be defined by
and apply Theorems 4.1 and 4.2.
The following example shows that the Hyers-Ulam stability problem for the case of was excluded in Corollary 4.5. □
Example 4.6 Let be defined by
Consider the function be defined by
for all , where . Then f satisfies the functional inequality

for all , but there do not exist a general multi-Euler-Lagrange quadratic function and a constant such that for all .
It is clear that f is bounded by on . If or , then
Now suppose that . Then there exists an integer such that
Hence
for all . From the definition of f and the inequality (4.6), we obtain that f satisfies (4.5). Now, we claim that the functional equation (1.1) is not stable for in Corollary 4.5. Suppose, on the contrary, that there exist a general multi-Euler-Lagrange quadratic function and a constant such that for all . Then there exists a constant such that for all rational numbers x. So, we obtain that
for all rational numbers x. Let with . If x is a rational number in , then for all , and for this x we get
which contradicts (4.7).
Corollary 4.7 Let be a non-Archimedean field with , be a normed space over and be a complete non-Archimedean normed space over . Let such that . If is a mapping satisfying (3.3) and

then for every there exists a unique general multi-Euler-Lagrange quadratic mapping such that
for all .
Proof Fix , and . Let be defined by . Consider the non-Archimedean fuzzy normed space defined as in Example 2.6, and apply Theorem 4.2. □
Corollary 4.8 Let be a real normed space and be a real Banach space. Let such that , or . If is a mapping satisfying (3.3) and

then, for every , there exists a unique general multi-Euler-Lagrange quadratic mapping such that

for all .
Proof Fix , and . Let be defined by
Consider the non-Archimedean fuzzy normed space defined as in Example 2.6, and apply Theorems 4.1 and 4.2. □
Remark 4.9 Theorems 4.1 and 4.2 can be regarded as a generalization of the classical stability result in the framework of normed spaces (see [14]). For and , Corollary 4.8 yields the main theorem in [17]. The generalized Hyers-Ulam stability problem for the case of was excluded in Corollary 4.8 (see[10]).
Note that by (4.4) one can get

Letting in this inequality, we obtain (4.9). Thus Corollary 4.8 is a singular case of Corollary 4.4. This study indeed presents a relationship between three various disciplines: the theory of non-Archimedean fuzzy normed spaces, the theory of stability of functional equations and the fixed point theory.
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The first author was supported by the National Natural Science Foundation of China (NNSFC) (Grant No. 11171022).
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All authors carried out the proof. All authors conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.
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Xu, T.Z., Rassias, J.M. Stability of general multi-Euler-Lagrange quadratic functional equations in non-Archimedean fuzzy normed spaces. Adv Differ Equ 2012, 119 (2012). https://doi.org/10.1186/1687-1847-2012-119
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DOI: https://doi.org/10.1186/1687-1847-2012-119
Keywords
- stability of general multi-Euler-Lagrange quadratic functional equation
- direct method
- fixed point method
- non-Archimedean fuzzy normed space