A functional equation related to inner product spaces in non-Archimedean L -random normed spaces

In this paper, we prove the stability of a functional equation related to inner product spaces in non-Archimedean -random normed spaces.

MSC: 46S10, 39B52, 47S10, 26E30, 12J25.

One of the most interesting questions in the theory of functional analysis concerning the Ulam stability problem of functional equations is as follows: When is it true that a mapping satisfying a functional equation approximately must be close to an exact solution of the given functional equation?

The first stability problem concerning group homomorphisms was raised by Ulam [1] in 1940 and affirmatively solved by Hyers [2]. The result of Hyers was generalized by Aoki [3] for approximate additive mappings and by ThM Rassias [4] for approximate linear mappings by allowing the difference Cauchy equation to be controlled by . In 1994, a generalization of the ThM Rassias‘ theorem was obtained by Gǎvruta [5], who replaced by a general control function .

Quadratic functional equations were used to characterize inner product spaces [6]. A square norm on an inner product space satisfies the parallelogram equality . The functional equation

is related to a symmetric bi-additive mapping [7,8]. It is natural that this equation is called a quadratic functional equation, and every solution of the quadratic equation (1.1) is said to be a quadratic mapping.

It was shown by ThM Rassias [9] that the norm defined over a real vector space X is induced by an inner product if and only if for a fixed integer

for all .

Let be a field. A non-Archimedean absolute value on is a function such that for any we have

(i) and equality holds if and only if ,

(ii) ,

(iii) .

The condition (iii) is called the strict triangle inequality. By (ii), we have . Thus, by induction, it follows from (iii) that for each integer n. We always assume in addition that is non-trivial, i.e., that there is an such that .

Let X be a linear space over a scalar field with a non-Archimedean non-trivial valuation . A function is a non-Archimedean norm (valuation) if it satisfies the following conditions:

(NA1) if and only if ;

(NA2) for all and ;

(NA3) the strong triangle inequality (ultra-metric); namely,

Then is called a non-Archimedean space.

Thanks to the inequality

a sequence is Cauchy in X if and only if converges to zero in a non-Archimedean space. By a complete non-Archimedean space, we mean a non-Archimedean space in which every Cauchy sequence is convergent.

In 1897, Hensel [10] introduced a normed space, which does not have the Archimedean property.

During the last three decades, the theory of non-Archimedean spaces has gained the interest of physicists for their research in particular in problems coming from quantum physics, p-adic strings, and superstrings [11]. Although many results in the classical normed space theory have a non-Archimedean counterpart, but their proofs are essentially different and require an entirely new kind of intuition [12-16].

The main objective of this paper is to prove the Hyers-Ulam stability of the following functional equation related to inner product spaces:

(, ) in non-Archimedean normed spaces. Interesting new results concerning functional equations related to inner product spaces have recently been obtained by Najati and ThM Rassias [18] as well as for the fuzzy stability of a functional equation related to inner product spaces by Park [19] and Gordji and Khodaei [20]. During the last decades, several stability problems for various functional equations have been investigated by many mathematicians; [21-56].

The theory of random normed spaces (RN-spaces) is important as a generalization of the 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 applications in quantum particle physics. The Hyers-Ulam stability of different functional equations in RN-spaces and fuzzy normed spaces has been recently studied by Alsina [57], Mirmostafaee, Mirzavaziri, and Moslehian [58,59], Miheţ and Radu [60], Miheţ, Saadati, and Vaezpour [61,62], Baktash et al.[63], Najati [64], and Saadati et al.[65].

Let be a complete lattice, that is, a partially ordered set in which every non-empty 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 F is left continuous, non-decreasing on and , .

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

Definition 2.1[66]

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

(1) (: boundary condition);

(2) (: commutativity);

(3) (: associativity);

(4) (: monotonicity).

Let be a sequence in L converging to (equipped the order topology). The t-norm is called a continuoust-norm if

for any .

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

The t-norm can also be extended to a countable operation taking, for any sequence in L, the value

The limit on the right side of (2.1) exists since the sequence is non-increasing and bounded from below.

Note that we put whenever . If T is a t-norm then, for all and , is defined by 1 if and if . A t-norm T is said to be of Hadžić-type (we denote by ) if the family is equi-continuous at (see [67]).

Definition 2.2[66]

A continuous t-norm on is said to be continuoust-representable if there exist a continuous t-norm ∗ and a continuous t-co-norm ⋄ on such that, for all , ,

For example,

and

for all , are continuous t-representable.

Define the mapping from to L by

Recall (see [67,68]) that, if is a given sequence in L, then 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 2.3 A latticetic random normed space is a triple , where X is a vector space and μ is a mapping from X into satisfying the following conditions:

(LRN1) for all if and only if ;

(LRN2) for all x in X, and ;

(LRN3) for all and .

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

Example 2.4 Let and an operation be defined by

Then is a complete lattice (see [66]). In this complete lattice, we denote its units by and . Let be a normed space. Let for all , and μ be a mapping defined by

Then is a latticetic random normed space.

If is a latticetic random normed space, then we have

is a complete system of neighborhoods of null vector for a linear topology on X generated by the norm F, where

Definition 2.5 Let be a latticetic random normed space.

(1) A sequence in X is said to be convergent to a point if, for any and , there exists a positive integer N such that for all .

(2) A sequence in X is called a Cauchy sequence if, for any and , there exists a positive integer N such that for all .

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

Theorem 2.6Ifis a latticetic random normed space andis a sequence such that, then.

Proof The proof is the same as in classical random normed spaces (see [17]). □

Lemma 2.7Letbe a latticetic random normed space and. If

thenand.

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

In the rest of this paper, unless otherwise explicitly stated, we will assume that G is an additive group and that X is a complete non-Archimedean latticetic random space. For convenience, we use the following abbreviation for a given mapping :

for all , where is a fixed integer.

Lemma 3.1[18]

Letandbe real vector spaces. If an odd mappingsatisfies the functional equation (1.2), thenfis additive.

Let be a non-Archimedean field, a vector space over and a non-Archimedean complete LRN-space over . In the following theorem, we prove the Hyers-Ulam stability of the functional equation (1.2) in non-Archimedean latticetic random spaces for an odd mapping case.

Theorem 3.2Letbe a non-Archimedean field anda non-Archimedean complete LRN-space over. Letbe a distribution function such that

for all, and

exists for all, where

for all. Suppose that an odd mappingsatisfies the inequality

for alland. Then there exists an additive mappingsuch that

for alland, and if

thenAis a unique additive mapping satisfying (3.5).

Proof Letting , () in (3.4) and using the oddness of f, we obtain that

for all and . Interchanging with in (3.7) and using the oddness of f, we get

for all and . It follows from (3.7) and (3.8) that

for all and . Setting , , () in (3.4) and using the oddness of f, we get

for all and . It follows from (3.9) and (3.10) that

for all and . Putting , () in (3.4), we obtain

for all and . It follows from (3.11) and (3.12) that

for all and . Replacing and by and in (3.13), respectively, we obtain

for all and . Hence,

for all and . Replacing x by in (3.14), we have

for all and . It follows from (3.1) and (3.15) that the sequence is Cauchy. Since X is complete, we conclude that is convergent. So one can define the mapping by for all . It follows from (3.14) and (3.15) that

for all and all and . By taking m to approach infinity in (3.16) and using (3.2), one gets (3.5). By (3.1) and (3.4), we obtain

for all and . Thus the mapping A satisfies (1.2). By Lemma 3.1, A is additive.

If is another additive mapping satisfying (3.5), then

for all , thus, . □

Corollary 3.3Letbe a function satisfying

(i) for all,

(ii) .

Letand letbe an LRN-space in which. Suppose that an odd mappingsatisfies the inequality

for alland. Then there exists a unique additive mappingsuch that

for alland.

Proof Defining by , we have

for all and . So, we have

and

for all and . It follows from (3.3) that

Applying Theorem 3.2, we conclude that

for all and . □

Lemma 3.4[18]

Letandbe real vector spaces. If an even mappingsatisfies the functional equation (1.2), thenfis quadratic.

In the following theorem, we prove the Hyers-Ulam stability of the functional equation (1.2) in non-Archimedean LRN-spaces for an even mapping case.

Theorem 3.5Letbe a function such that

for all, and

exists for allandwhere

and

for alland. Suppose that an even mappingwithsatisfies the inequality (3.4) for alland. Then there exists a quadratic mappingsuch that

for all, and if

thenQis a unique quadratic mapping satisfying (3.21).

Proof Replacing by , and by () in (3.4) and using the evenness of f, we obtain

for all and . Interchanging with in (3.23) and using the evenness of f, we obtain

for all and . It follows from (3.23) and (3.24) that

for all and . Setting , , () in (3.4) and using the evenness of f, we obtain

for all and . So, it follows from (3.25) and (3.26) that

for all and . Setting , in (3.27), we obtain

for all and . Putting , () in (3.4), one obtains

for all and . It follows from (3.28) and (3.29) that

for all and . Letting and replacing by in (3.26), we get

for all and . It follows from (3.28) and (3.31) that

for all and . It follows from (3.30) and (3.32) that

for all and . Setting , () in (3.4), we obtain

for all and . It follows from (3.33) and (3.34) that

for all and . Thus,

for all and . Replacing x by in (3.36), we have

for all and . It follows from (3.17) and (3.37) that the sequence is Cauchy. Since X is complete, we conclude that is convergent. So, one can define the mapping by for all . By using induction, it follows from (3.36) and (3.37) that

for all and all and . By taking m to approach infinity in (3.38) and using (3.18), one gets (3.21).

The rest of proof is similar to the proof of Theorem 3.2. □

Corollary 3.6Letbe a function satisfying

(i) for all,

(ii) for.

Letand letbe a LRN-space in which. Suppose that an even mappingwithsatisfies the inequality

for alland. Then there exists a unique quadratic mappingsuch that

for alland.

Proof Defining by , we have

for all and . We have

and

for all and . It follows from (3.20) that

Hence, by using (3.19), we obtain

for all and . Applying Theorem 3.5, we conclude the required result. □

Lemma 3.7[18]

Letandbe real vector spaces. A mappingsatisfies (1.2) if and only if there exist a symmetric bi-additive mappingand an additive mappingsuch thatfor all.

Now, we are ready to prove the main theorem concerning the Hyers-Ulam stability problem for the functional equation (1.2) in non-Archimedean spaces.

Theorem 3.8Letbe a function satisfying (3.1) for all, andandexist for alland, whereandare defined as in Theorems 3.2 and 3.5. Suppose that a mappingwithsatisfies the inequality (3.4) for all. Then there exist an additive mappingand a quadratic mappingsuch that

for alland. If

thenAis a unique additive mapping andQis a unique quadratic mapping satisfying (3.39).

Proof Let for all . Then

for all and . By Theorem 3.5, there exists a quadratic mapping such that

for all and . Also, let for all . By Theorem 3.2, there exists an additive mapping such that

for all and . Hence (3.39) follows from (3.40) and (3.41).

The rest of proof is trivial. □

Corollary 3.9Letbe a function satisfying

(i) for all,

(ii) for.

Let, be an LRN-space in whichand letsatisfy

for all, and. Then there exist a unique additive mappingand a unique quadratic mappingsuch that

for alland.

Proof The result follows from Corollaries 3.6 and 3.3. □

The authors declare that they have no competing interests.

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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