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Journal of Applied Mathematics

Volume 2012 (2012), Article ID 902931, 45 pages

http://dx.doi.org/10.1155/2012/902931

## Nonlinear Random Stability via Fixed-Point Method

^{1}Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea^{2}Department of Mathematics and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea^{3}Department of Mathematics, Iran University of Science and Technology, Behshahr, Iran

Received 31 October 2011; Accepted 22 December 2011

Academic Editor: Yeong-Cheng Liou

Copyright © 2012 Yeol Je Cho et al. 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 the following additive-quadratic-cubic-quartic functional equation in various complete random normed spaces.

#### 1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias [4] has provided a lot of influence in the development of what we call * generalized Hyers-Ulam stability* or as *Hyers-Ulam-Rassias stability* of functional equations. A generalization of the Rassias theorem was obtained by Găvruţa [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias approach.

The functional equation

is called a * quadratic functional equation*. In particular, every solution of the quadratic functional equation is said to be a * quadratic mapping*. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Cholewa [6] for mappings , where is a normed space and is a Banach space. Czerwik [7] proved the generalized Hyers-Ulam stability of the quadratic functional equation. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [8–12]).

In [13], Jun and Kim consider the following cubic functional equation:

It is easy to show that the function satisfies the functional equation (1.2), which is called a * cubic functional equation*, and every solution of the cubic functional equation is said to be a * cubic mapping*.

Considered the following quartic functional equation
It is easy to show that the function satisfies the functional equation, which is called a * quartic functional equation*, and every solution of the quartic functional equation is said to be a * quartic mapping*. One can easily show that an odd mapping satisfies the additive-quadratic-cubic-quadratic functional equation
if and only if it is an additive-cubic mapping, that is,

It was shown in Lemma 2.2 of [14] that and are cubic and additive, respectively, and that .

One can easily show that an even mapping satisfies (1.4) if and only if it is a quadratic-quartic mapping, that is,

Also and are quartic and quadratic, respectively, and .

For a given mapping , we define

for all .

Let be a set. A function is called a * generalized metric * on if satisfies(1) if and only if ,(2) for all ,(3) for all .

We recall the fixed-point alternative of Diaz and Margolis.

Theorem 1.1 (see [15, 16]). *Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant , then for each given element , either
**
for all nonnegative integers or there exists a positive integer such that*(1)* for all ,*(2)*the sequence converges to a fixed point of ,*(3)* is the unique fixed point of in the set ,*(4)* for all .*

In 1996, Isac and Rassias [17] were the first to provide applications of stability theory of functional equations for the proof of new fixed-point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [18–21]).

#### 2. Preliminaries

In the sequel, we adopt the usual terminology, notations, and conventions of the theory of random normed spaces, as in [22–26]. Throughout this paper, is the space of all probability distribution functions, that is, the space of all mappings , such taht is left continuous, nondecreasing on , and . is a subset of consisting of all functions for which , where denotes the left limit of the function at the point , that is, . The space is partially ordered by the usual pointwise ordering of functions, that is, if and only if for all in . The maximal element for in this order is the distribution function given by

A *triangular norm* (shortly *-norm*) is a binary operation on the unit interval , that is, a function , such that for all the following four axioms satisfied:(T1) (commutativity),(T2) (associativity),(T3) (boundary condition),(T4) whenever (monotonicity).

Basic examples are the Łukasiewicz -norm for all and the -norms , where , ,

If is a -norm, then is defined for every and by 1, if and if . A is said to be *of Hadžić type* (we denote by ) if the family is equicontinuous at (cf. [27]).

Other important triangular norms are the following (see [28]):(1)The * Sugeno-Weber family * is defined by , and
if .(2)The * Domby family * is defined by if , if , and
if .(3)The * Aczel-Alsina family * is defined by if , if and
if .

A -norm can be extended (by associativity) in a unique way to an -array operation taking for the value defined by

can also be extended to a countable operation taking for any sequence in the value The limit on the right side of (6.4) exists since the sequence is nonincreasing and bounded from below.

Proposition 2.1 (see [28]). *We have the following.*(1)*For , the following implication holds:
*(2)*If is of Hadžić type, then
for every sequence in such that .*(3)*If , then
*(4)*If , then
*

*Definition 2.2 (see [26]). *A *Random normed space* (briefly, RN-space) is a triple , where is a vector space, is a continuous norm, and is a mapping from into such that, the following conditions hold: (RN1) for all if and only if ,(RN2) for all , and ,(RN3) for all and .

*Definition 2.3. *Let be an RN-space.(1)A sequence in is said to be *convergent* to in if, for every and , there exists positive integer such that whenever .(2)A sequence in is called a *Cauchy sequence* if, for every and , there exists positive integer such that whenever .(3)An RN-space is said to be *complete* if and only if every Cauchy sequence in is convergent to a point in . A complete RN-space is said to be random Banach space.

Theorem 2.4 (see [25]). *If is an RN-space and is a sequence such that , then almost everywhere.*

The theory of random normed spaces (RN-spaces) is important as a generalization of deterministic result of linear normed spaces and also in the study of random operator equations. The RN-spaces may also provide us with the appropriate tools to study the geometry of nuclear physics and have important application in quantum particle physics. The generalized Hyers-Ulam stability of different functional equations in random normed spaces, RN-spaces, and fuzzy normed spaces has been recently studied [20, 24, 29–39].

#### 3. Non-Archimedean Random Normed Space

By a *non-Archimedean field,* we mean a field equipped with a function (valuation) from into such that if and only if , , and for all . Clearly, and for all . By the *trivial valuation*, we mean the mapping taking everything but 0 into 1 and . Let be a vector space over a field with a non-Archimedean nontrivial valuation . A function is called a *non-Archimedean norm* if it satisfies the following conditions:(NAN1) if and only if ,(NAN2) for any and , ,(NAN3) the strong triangle inequality (ultrametric), namely,

then is called a *non-Archimedean normed space*. Due to the fact that

a sequence is a Cauchy sequence if and only if converges to zero in 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 [40] discovered the -adic numbers of as a number theoretical analogues of power series in complex analysis. Fix a prime number . For any nonzero rational number , there exists a unique integer such that , where and are integers not divisible by . Then defines a non-Archimedean norm on . The completion of with respect to the metric is denoted by , which is called the *-adic number field*.

Throughout the paper, we assume that is a vector space and is a complete non-Archimedean normed space.

*Definition 3.1. **A non-Archimedean random normed space* (briefly, non-Archimedean RN-space) is a triple , where is a linear space over a non-Archimedean field , is a continuous -norm, and is a mapping from into such that the following conditions hold:(NA-RN1) for all if and only if ,(NA-RN2) for all , , and ,(NA-RN3) for all and .

It is easy to see that if (NA-RN3) holds, then so is(RN3).

As a classical example, if is a non-Archimedean normed linear space, then the triple , where

is a non-Archimedean RN-space.

*Example 3.2. *Let be a non-Archimedean normed linear space. Define
then is a non-Archimedean RN-space.

*Definition 3.3. *Let be a non-Archimedean RN-space. Let be a sequence in , then is said to be *convergent* if there exists such that
for all . In that case, is called the *limit* of the sequence .

A sequence in is called a *Cauchy sequence *if for each and each there exists such that for all and all , we have .

If each Cauchy sequence is convergent, then the random norm is said to be *complete* and the non-Archimedean RN-space is called a non-Archimedean *random Banach space*.

*Remark 3.4 (see [41]). *Let be a non-Archimedean RN-space, then
So, the sequence is a Cauchy sequence if for each and there exists such that for all ,

#### 4. Generalized Ulam-Hyers Stability for a Quartic Functional Equation in Non-Archimedean RN-Spaces of Functional Equation (1.4): An Odd Case

Let be a non-Archimedean field, let be a vector space over , and let be a non-Archimedean random Banach space over .

Next, we define a random approximately AQCQ mapping. Let be a distribution function on such that is nondecreasing and

*Definition 4.1. *A mapping is said to be -*approximately AQCQ* if

In this section, we assume that in (i.e., characteristic of is not 2). Our main result, in this section, is the following.

We prove the generalized Hyers-Ulam stability of the functional equation in non-Archimedean random spaces, an odd case.

Theorem 4.2. *Let be a non-Archimedean field, let be a vector space over and let be a non-Archimedean random Banach space over . Let be an odd mapping and -approximately AQCQ mapping. If for some , , and some integer , with ,
**
then there exists a unique cubic mapping such that
**
for all and , where
*

*Proof. *Letting in (4.2), we get
for all and . Replacing by in (4.2), we get
for all and . By (4.7) and (4.8), we have
for all and . Letting and for all in (4.9), we get
for all and . Now, we show by induction on that for all , and ,
Putting in (4.11), we obtain (4.10). Assume that (4.11) holds for some . Replacing by in (4.10), we get
Since ,
for all and . Thus, (4.11) holds for all . In particular,
Replacing by in (4.14) and using inequality (4.3), we obtain
Then
Hence
Since
then
is a Cauchy sequence in the non-Archimedean random Banach space . Hence we can define a mapping such that

Next for each , and ,
Therefore,
By letting , we obtain
So,
This proves (4.5). From , by (4.2), we deduce that
and so, by (NA-RN3) and (4.2), we obtain
It follows that
for all , , and . Since
for all and , by Theorem 2.4, we deduce that
for all and . Thus, the mapping satisfies (1.4).

Now, we have
for all . Since the mapping is cubic (see Lemma 2.2 of [14]), from the equality , we deduce that the mapping is cubic.

Corollary 4.3. *Let be a non-Archimedean field, let be a vector space over , and let be a non-Archimedean random Banach space over under a t-norm . Let be an odd and -approximately AQCQ mapping. If, for some , , and some integer , , with ,
**
then there exists a unique cubic mapping such that
**
for all and .*

*Proof. *Since
and is of Hadžić type, from Proposition 2.1, it follows that
Now, we can apply Theorem 4.2 to obtain the result.

*Example 4.4. *Let be non-Archimedean random normed space in which
And let be a complete non-Archimedean random normed space (see Example 3.2). Define
It is easy to see that (4.3) holds for . Also, since
we have
Let be an odd and -approximately AQCQ mapping. Thus, all the conditions of Theorem 4.2 hold, and so there exists a unique cubic mapping such that

Theorem 4.5. *Let be a non-Archimedean field, let be a vector space over , and let be a non-Archimedean random Banach space over . Let be an odd mapping and -approximately AQCQ mapping. If for some , , and some integer , with ,
**
then there exists a unique additive mapping such that
**
for all and , where
*

*Proof. *Letting and for all in (4.9), we get
for all and .

The rest of the proof is similar to the proof of Theorem 4.2.

Corollary 4.6. *Let be a non-Archimedean field, let be a vector space over , and let be a non-Archimedean random Banach space over under a t-norm . Let be an odd and -approximately AQCQ mapping. If, for some , and some integer , with ,
**
then there exists a unique additive mapping such that
**
for all and .*

*Proof. *Since
and is of Hadžić type, from Proposition 2.1, it follows that
Now, we can apply Theorem 4.5 to obtain the result.

*Example 4.7. *Let non-Archimedean random normed space in which
and let be a complete non-Archimedean random normed space (see Example 3.2). Define
It is easy to see that (4.3) holds for . Also, since
we have
Let be an odd and -approximately AQCQ mapping. Thus, all the conditions of Theorem 4.2 hold, and so there exists a unique additive mapping such that

#### 5. Generalized Hyers-Ulam Stability of the Functional Equation (1.4) in Non-Archimedean Random Normed Spaces: An Even Case

Now, we prove the generalized Hyers-Ulam stability of the functional equation in non-Archimedean Banach spaces, an even case.

Theorem 5.1. *Let be a non-Archimedean field, let be a vector space over , and let be a non-Archimedean random Banach space over . Let be an even mapping, , and -approximately AQCQ mapping. If for some , , and some integer , with ,
**
then there exists a unique quartic mapping such that
**
for all and , where
*

*Proof. *Letting in (4.2), we get
for all and . Replacing by in (4.2), we get
for all and . By (5.4) and (5.5), we have
for all and . Letting and for all in (5.6), we get
for all and .

The rest of the proof is similar to the proof of Theorem 4.2.

Corollary 5.2. *Let be a non-Archimedean field, let be a vector space over , and let be a non-Archimedean random Banach space over under a t-norm . Let be an even, , and -approximately AQCQ mapping. If, for some , , and some integer , , with ,
**
then there exists a unique quartic mapping such that
**
for all and .*

*Proof. *Since
and is of Hadžić type, from Proposition 2.1, it follows that
Now, we can apply Theorem 5.1 to obtain the result.

*Example 5.3. *Let be non-Archimedean random normed space in which
And let be a complete non-Archimedean random normed space (see Example 3.2). Define
It is easy to see that (4.3) holds for . Also, since
we have
Let be an even, , and -approximately AQCQ mapping. Thus all the conditions of Theorem 5.1 hold, and so there exists a unique quartic mapping such that

Theorem 5.4. *Let be a non-Archimedean field, let be a vector space over and let be a non-Archimedean random Banach space over . Let be an even mapping, and -approximately AQCQ mapping. If for some , , and some integer , with ,
**
then there exists a unique quadratic mapping such that
**
for all and , where
*

*Proof. *Letting and for all in (5.6), we get
for all and .

The rest of the proof is similar to the proof of Theorem 5.1.

Corollary 5.5. *Let be a non-Archimedean field, let be a vector space over , and let be a non-Archimedean random Banach space over under a t-norm . Let be an even, , and -approximately AQCQ mapping. If, for some , , and some integer , , with ,
**
then there exists a unique quadratic mapping such that
**
for all and .*

*Proof. *Since
and is of Hadžić type, from Proposition 2.1, it follows that
Now, we can apply Theorem 5.4 to obtain the result.

*Example 5.6. *Let be a non-Archimedean random normed space in which
And let be a complete non-Archimedean random normed space (see Example 3.2). Define
It is easy to see that (4.3) holds for . Also, since
we have
Let be an even, , and -approximately AQCQ mapping. Thus, all the conditions of Theorem 5.4 hold, and so there exists a unique quadratic mapping such that

#### 6. Latticetic Random Normed Space

Let be a complete lattice, that is, a partially ordered set in which every nonempty subset admits supremum and infimum, and , . The space of latticetic random distribution functions, denoted by , is defined as the set of all mappings such that is left continuous and nondecreasing on , , .

is defined as , where denotes the left limit of the function at the point . The space is partially ordered by the usual pointwise ordering of functions, that is, if and only if for all in . The maximal element for in this order is the distribution function given by

In Section 2, we defined -norms on , and now we extend -norms on a complete lattice.

*Definition 6.1 (see [42]). *A *triangular norm* (-norm) on is a mapping satisfying the following conditions:(a) (boundary condition);(b) (commutativity);(c) (associativity);(d) (monotonicity).

Let be a sequence in converges to (equipped order topology). The -norm is said to be a *continuous **-norm* if
for all .

A -norm can be extended (by associativity) in a unique way to an -array operation taking for the value defined by

can also be extended to a countable operation taking for any sequence in the value The limit on the right side of (6.4) exists since the sequence is nonincreasing and bounded from below.

Note that we put whenever . If is a -norm, then is defined for every and by 1 if and if . A -norm is said to be *of Hadžić type*, (we denote by ) if the family is equicontinuous at (cf. [27]).

*Definition 6.2 (see [42]). * A continuous -norm on is said to be * continuous **–representable* if there exist a continuous -norm and a continuous -conorm on such that, for all , ,

For example, for all , are continuous -representable. Define the mapping from to by

Recall (see [27, 28]) that if is a given sequence in , is defined recurrently by and for all .

A negation on is any decreasing mapping satisfying and . If , for all , then is called an *involutive negation*. In the following, is endowed with a (fixed) negation .

*Definition 6.3. *A *latticetic random normed space* (in short LRN-space) is a triple , where is a vector space and is a mapping from into such that the following conditions hold: (LRN1) for all if and only if , (LRN2) for all in , and , (LRN3) for all and .

We note that from (LPN2) it follows that for all and .

*Example 6.4. *Let and operation be defined by
then is a complete lattice (see [42]). In this complete lattice, we denote its units by and . Let be a normed space. Let for all , and be a mapping defined by
then is a latticetic random normed spaces.

If is a latticetic random normed space, then
is a complete system of neighborhoods of null vector for a linear topology on generated by the norm .

*Definition 6.5. *Let be a latticetic random normed spaces.(1)A sequence in