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Review Article | Open | Published:

Nonlinear -Random Stability of an ACQ Functional Equation

Abstract

We prove the generalized Hyers-Ulam stability of the following additive-cubic-quartic functional equation: in complete latticetic random normed spaces.

1. Introduction

Random theory is a powerful hand set for modeling uncertainty and vagueness in various problems arising in the field of science and engineering. It has also very useful applications in various fields, for example, population dynamics, chaos control, computer programming, nonlinear dynamical systems, nonlinear operators, statistical convergence, and so forth. The random topology proves to be a very useful tool to deal with such situations where the use of classical theories breaks down. The usual uncertainty principle of Werner Heisenberg leads to a generalized uncertainty principle, which has been motivated by string theory and noncommutative geometry. In strong quantum gravity regime space-time points are determined in a random manner. Thus impossibility of determining the position of particles gives the space-time a random structure. Because of this random structure, position space representation of quantum mechanics breaks down, and therefore a generalized normed space of quasiposition eigenfunction is required. Hence, one needs to discuss on a new family of random norms. There are many situations where the norm of a vector is not possible to be found and the concept of random norm seems to be more suitable in such cases, that is, we can deal with such situations by modeling the inexactness through the random norm [1, 2].

The stability problem of functional equations originated from a question of Ulam [3] concerning the stability of group homomorphisms. Hyers [4] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [5] for additive mappings and by Th. M. Rassias [6] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [6] 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

(1.1)

A generalization of the Th. M. Rassias theorem was obtained by Găvruţa [7] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias approach.

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 [6, 824]).

In [25], Jun and Kim considered the following cubic functional equation:

(1.2)

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.

In [26], Lee et al. considered the following quartic functional equation:

(1.3)

It is easy to show that the function satisfies the functional equation (1.3), which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic mapping.

The study of stability of functional equations is important problem in nonlinear sciences and application in solving integral equation via VIM [2729] PDE and ODE [3034]. 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 a fundamental result in fixed point theory.

Theorem 1.1 (see [35, 36]).

Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then for each given element , either

(1.4)

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 Th. M. Rassias [37] 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 [3843]).

2. Preliminaries

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 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 by Alsina [44], Mirmostafaee and Moslehian [45] and Mirzavaziri and Moslehian [40], Miheţ and Radu [46], Miheţ et al. [47, 48], Baktash et al. [49], and Saadati et al. [50].

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 point-wise 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

(2.1)

Definition 2.1 (see [51]).

A triangular norm (-norm) on is a mapping satisfying the following conditions:

(a) (boundary condition);

(b) (commutativity);

(c) (associativity);

(d) and (monotonicity).

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

(2.2)

for all .

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

(2.3)

can also be extended to a countable operation taking for any sequence in the value

(2.4)

The limit on the right side of (2.4) exists since the sequence is nonincreasing and bounded from below.

Note that we put whenever . If is a -norm then is defined for all 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. [52]).

Definition 2.2 (see [51]).

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 , ,

(2.5)

For example,

(2.6)

for all , are continuous -representable.

Define the mapping from to by

(2.7)

Recall (see [52, 53]) that if is a given sequence in , is defined recurrently by and for .

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 2.3.

A latticetic random normed 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 .

Example 2.4.

Let and operation be defined by

(2.8)

Then is a complete lattice (see [51]). In this complete lattice, we denote its units by and . Let be a normed space. Let for all , and be a mapping defined by

(2.9)

Then is a latticetic random normed space.

If is a latticetic random normed space, then

(2.10)

is a complete system of neighborhoods of null vector for a linear topology on generated by the norm .

Definition 2.5.

Let be a latticetic random normed space.

(1)A sequence in is said to be convergent to in if, for every and , there exists a positive integer such that whenever .

(2)A sequence in is called Cauchy sequence if, for every and , there exists a positive integer such that whenever .

(3)A latticetic random normed spaces is said to be complete if and only if every Cauchy sequence in is convergent to a point in .

Theorem 2.6.

If is a latticetic random normed space and is a sequence such that , then .

Proof.

The proof is the same as classical random normed spaces, see [54].

Lemma 2.7.

Let be a latticetic random normed space and . If

(2.11)

then and .

Proof.

Let for all . Since , we have , and by (LRN1) we conclude that .

3. Generalized Hyers-Ulam Stability of the Functional Equation (1.1): An Odd Case

One can easily show that an even mapping satisfies (1.1) if and only if the even mapping is a quartic mapping, that is,

(3.1)

and that an odd mapping satisfies (1.1) if and only if the odd mapping is an additive-cubic mapping, that is,

(3.2)

It was shown in Lemma 2.2 of [55] that and are cubic and additive, respectively, and that .

For a given mapping , we define

(3.3)

for all .

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation in complete LRN-spaces: an odd case.

Theorem 3.1.

Let be a linear space, a complete LRN-space and a mapping from to is denoted by such that, for some ,

(3.4)

Let be an odd mapping satisfying

(3.5)

for all and all . Then

(3.6)

exists for each and defines a cubic mapping such that

(3.7)

for all and all .

Proof.

Letting in (3.5), we get

(3.8)

for all and all .

Replacing by in (3.5), we get

(3.9)

for all and all .

By (3.8) and (3.9),

(3.10)

for all and all . Letting and for all , we get

(3.11)

for all and all .

Consider the set

(3.12)

and introduce the generalized metric on :

(3.13)

where, as usual, . It is easy to show that is complete. (See the proof of Lemma 2.1 of [46].)

Now we consider the linear mapping such that

(3.14)

for all .

Let be given such that . Then

(3.15)

for all and all . Hence

(3.16)

for all and all . So implies that

(3.17)

This means that

(3.18)

for all .

It follows from (3.11) that

(3.19)

for all and all . So

(3.20)

By Theorem 1.1, there exists a mapping satisfying the following:

(1) is a fixed point of , that is,

(3.21)

for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set

(3.22)

This implies that is a unique mapping satisfying (3.21) such that there exists a satisfying

(3.23)

for all and all ;

(2) as . This implies the equality

(3.24)

for all ;

(3), which implies the inequality

(3.25)

This implies that inequality (3.7) holds.

From , by (3.5), we deduce that

(3.26)

and so, by (LRN3) and (3.4), we obtain

(3.27)

It follows that

(3.28)

for all , all and all . Since ,

(3.29)

for all and all . Then

(3.30)

for all and all . Thus the mapping is cubic, as desired.

Corollary 3.2.

Let and let be a real number with . Let be a normed vector space with norm and let be an LRN-space in which and . Let be an odd mapping satisfying

(3.31)

for all and all . Then

(3.32)

exists for each and defines a cubic mapping such that

(3.33)

for all and all .

Proof.

The proof follows from Theorem 3.1 by taking

(3.34)

for all . Then we can choose and we get the desired result.

Theorem 3.3.

Let be a linear space, a complete LRN-space and a mapping from to is denoted by such that, for some ,

(3.35)

Let be an odd mapping satisfying (1.1). Then

(3.36)

exists for each and defines a cubic mapping such that

(3.37)

for all and all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 3.1.

Consider the linear mapping such that

(3.38)

for all .

Let be given such that . Then

(3.39)

for all and all . Hence

(3.40)

for all and all . So implies that

(3.41)

This means that

(3.42)

for all .

It follows from (3.11) that

(3.43)

for all and all . So .

By Theorem 1.1, there exists a mapping satisfying the following:

(1) is a fixed point of , that is,

(3.44)

for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set

(3.45)

This implies that is a unique mapping satisfying (3.44) such that there exists a satisfying

(3.46)

for all and all ;

(2) as . This implies the equality

(3.47)

for all ;

(3), which implies the inequality

(3.48)

This implies that inequality (3.37) holds.

The rest of the proof is similar to the proof of Theorem 3.1.

Corollary 3.4.

Let , and let be a real number with . Let be a normed vector space with norm , and let be an LRN-space in which and . Let be an odd mapping satisfying (3.31). Then

(3.49)

exists for each and defines a cubic mapping such that

(3.50)

for all and all .

Proof.

The proof follows from Theorem 3.3 by taking

(3.51)

for all . Then we can choose , and we get the desired result.

Theorem 3.5.

Let be a linear space, an LRN-space and let be a mapping from to is denoted by such that, for some ,

(3.52)

Let be an odd mapping satisfying (3.5). Then

(3.53)

exists for each and defines an additive mapping such that

(3.54)

for all and all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 3.1.

Letting and for all in (3.10), we get

(3.55)

for all and all .

Now we consider the linear mapping such that

(3.56)

for all .

Let be given such that . Then

(3.57)

for all and all . Hence

(3.58)

for all and all . So implies that . This means that

(3.59)

for all .

It follows from (3.55) that

(3.60)

for all and all . So .

By Theorem 1.1, there exists a mapping satisfying the following:

(1) is a fixed point of , that is,

(3.61)

for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set

(3.62)

This implies that is a unique mapping satisfying (3.61) such that there exists a satisfying

(3.63)

for all and all ;

(2) as . This implies the equality

(3.64)

for all ;

(3), which implies the inequality

(3.65)

This implies that inequality (3.54) holds.

The rest of the proof is similar to the proof of Theorem 3.1.

Corollary 3.6.

Let , and let be a real number with . Let be a normed vector space with norm , and let be an LRN-space in which and . Let be an odd mapping satisfying (3.31). Then

(3.66)

exists for each and defines an additive mapping such that

(3.67)

for all and all .

Proof.

The proof follows from Theorem 3.5 by taking

(3.68)

for all . Then we can choose and we get the desired result.

Theorem 3.7.

Let be a linear space, an LRN-space and let be a mapping from to is denoted by such that, for some ,

(3.69)

Let be an odd mapping satisfying (3.5). Then

(3.70)

exists for each and defines an additive mapping such that

(3.71)

for all and all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 3.1.

Consider the linear mapping such that

(3.72)

for all .

Let be given such that . Then

(3.73)

for all and all . Hence

(3.74)

for all and all . So implies that

(3.75)

This means that

(3.76)

for all .

It follows from (3.55) that

(3.77)

for all and all . So .

By Theorem 1.1, there exists a mapping satisfying the following:

(1) is a fixed point of , that is,

(3.78)

for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set

(3.79)

This implies that is a unique mapping satisfying (3.78) such that there exists a satisfying

(3.80)

for all and all ;

(2) as . This implies the equality

(3.81)

for all ;

(3), which implies the inequality

(3.82)

This implies that inequality (3.71) holds.

The rest of the proof is similar to the proof of Theorem 3.1.

Corollary 3.8.

Let , and let be a real number with . Let be a normed vector space with norm , and let be an LRN-space in which and . Let be an odd mapping satisfying (3.31). Then

(3.83)

exists for each and defines an additive mapping such that

(3.84)

for all and all .

Proof.

The proof follows from Theorem 3.7 by taking

(3.85)

for all . Then we can choose and we get the desired result.

4. Generalized Hyers-Ulam Stability of the Functional Equation (1.1): An Even Case

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation in complete RN-spaces: an even case.

Theorem 4.1.

Let be a linear space, an LRN-space and let be a mapping from to is denoted by such that, for some ,

(4.1)

Let be an even mapping satisfying and (3.5). Then

(4.2)

exists for each and defines a quartic mapping such that

(4.3)

for all and all .

Proof.

Letting in (3.5), we get

(4.4)

for all and all .

Letting in (3.5), we get

(4.5)

for all and all .

By (4.4) and (4.5),

(4.6)

for all and all .

Consider the set

(4.7)

and introduce the generalized metric on

(4.8)

where, as usual, . It is easy to show that is complete. (See the proof of Lemma 2.1 of [46].)

Now we consider the linear mapping such that

(4.9)

for all .

Let be given such that . Then

(4.10)

for all and all . Hence

(4.11)

for all and all . So implies that

(4.12)

This means that

(4.13)

for all .

It follows from (4.6) that

(4.14)

for all and all . So .

By Theorem 1.1, there exists a mapping satisfying the following:

(1) is a fixed point of , that is,

(4.15)

for all . Since is even, is an even mapping. The mapping is a unique fixed point of in the set

(4.16)

This implies that is a unique mapping satisfying (4.15) such that there exists a satisfying

(4.17)

for all and all ;

(2) as . This implies the equality

(4.18)

for all ;

(3), which implies the inequality

(4.19)

This implies that inequality (4.3) holds.

The rest of the proof is similar to the proof of Theorem 3.1.

Corollary 4.2.

Let , and let be a real number with . Let be a normed vector space with norm , and let be an LRN-space in which and . Let be an even mapping satisfying and (3.31). Then

(4.20)

exists for each and defines a quartic mapping such that

(4.21)

for all and all .

Proof.

The proof follows from Theorem 4.1 by taking

(4.22)

for all . Then we can choose , and we get the desired result.

Similarly, we can obtain the following. We will omit the proof.

Theorem 4.3.

Let be a linear space, an LRN-space and let be a mapping from to is denoted by such that, for some ,

(4.23)

Let be an even mapping satisfying and (3.5). Then

(4.24)

exists for each and defines a quartic mapping such that

(4.25)

for all and all .

Corollary 4.4.

Let , and let be a real number with . Let be a normed vector space with norm , and let be an LRN-space in which and . Let be an even mapping satisfying and (3.31). Then

(4.26)

exists for each and defines a quartic mapping such that

(4.27)

for all and all .

Proof.

The proof follows from Theorem 4.3 by taking

(4.28)

for all . Then we can choose , and we get the desired result.

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Correspondence to Reza Saadati.

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Keywords

  • Functional Equation
  • Unique Mapping
  • Additive Mapping
  • Unique Fixed Point
  • Generalize Uncertainty Principle