Recursive procedure in the stability of Fréchet polynomials
© Dăianu; licensee Springer. 2014
Received: 4 March 2013
Accepted: 17 December 2013
Published: 15 January 2014
By means of a new stability result, established for symmetric and multi-additive mappings, and using the concepts of stability couple and of stability chain, we prove, by a recursive procedure, the generalized stability of two of Fréchet’s polynomial equations. We also give a new functional characterization of generalized polynomials and a new approach to solving the generalized stability of the monomial equation.
MSC:39B82, 39B52, 20M15, 65Q30.
which is equivalent to equation (1.2) [9, 10], but apparently more restrictive than equation (1.1). The only results of generalized stability in the sense of Bourgin and Găvruţa for equation (1.2) have recently been obtained by Jun and Kim  and by Lee [12, 13] for .
The difficulties that arise in the proof of a criterion of the generalized stability for such equations with differences for an arbitrary n are caused by the necessity of inventing a recursive procedure of determining the control functions and the monomial components of the approximated polynomial.
Throughout this paper we assume: ℕ is the set of nonnegative integers, is an integer, M is an abelian monoid under addition, B is a Banach space with the norm , and is the vector space of all functions from M to B.
For all we have .
Let . The function is a j-monomial if for all (we agree that ). If , the j-monomial m verifies the relation , , .
We say that the mapping is an n-polynomial if for any there exists a j-monomial such that .
where is the 0-monomial defined by .
The concept of Hyers-Ulam stability of a functional equation has surfaced as a consequence of the first answer given by Hyers (for Cauchy’s equation on Banach spaces ) to a question posed by Ulam in 1940 about the stability of group morphisms. The concept was extended by Aoki , Bourgin , Rassias , Găvruţă  and others. Here we consider that a functional equation is stable if it admits a nontrivial stability couple.
If, in addition, a is the unique mapping with these properties, we say that is a strong stability couple.
If is a stability couple, φ is called a control function. If for any constant and positive function φ there is a stability couple for equation , we say that this equation is stable in the Hyers-Ulam sense. If S is a normed vector space and there is a nontrivial stability couple (i.e. ) such that the control function φ is defined with the help of the norm from S, we say that the equation is stable in the Aoki-Rassias sense (see  and  for the origin of the eponymies).
We recall only two classic stability results, reformulated in terms of stability couples.
Theorem 2.4 
and . Then the pair is a strong stability couple for Cauchy’s functional equation , .
Theorem 2.5 
The functional equation (1.2) is stable in the Hyers-Ulam sense: if M is an -divisible abelian group and , then there exists a positive constant such that is a stability couple for equation (1.2).
The functional characterization of the real polynomial functions of degree less than or equal to n with the continuous solutions of equation (1.1), or of equation (1.2) - seen as generalizations of Cauchy’s equation - was realized by M. Fréchet in  and . Fundamental studies of Fréchet’s equations (1.2) and (1.4) on more general structures can be found in [9, 10, 14, 15, 19]. Some classical works on the stability of Cauchy’s equation are [1, 2, 16–18, 20, 21]. The stability of some particular polynomials has been studied by a great number of authors . Aoki-Rassias type theorems for equation (1.4) are given in  and . For some results on stability of multi-additive mappings we refer the reader to [8, 25, 26], on stability of monomials to [8, 27–33] and on stability of other different kinds of polynomials to [7, 8, 26, 34–44].
3 Stability of symmetric and n-additive mappings
In the proving of the generalized stability - part of the existence - the following lemma is very useful.
Lemma 3.1 Let be a sequence in B, let be a sequence in and such that . If for all , then is a convergent sequence and , where .
Since , it follows that is a Cauchy sequence. Let . Then, for and in the previous inequality, we obtain . □
The following result is crucial in determining of strong stability couples for the functional equations (2.1), (1.1), (1.2), and (1.5).
The operators , , and the set defined in the following lines will play a key role in building concrete stability couples for equations (2.1), (1.1), (1.2), and (1.5).
is a nontrivial set.
Now, we are able to prove that is a set of strong stability couples for the functional equation (2.1), where the operator acts on the vector space of symmetric functions from to B. The following theorem extends Găvruţă’s result from Theorem 2.4.
whence , , i.e. a is a symmetric and n-additive mapping that satisfies (3.2) and is a n-monomial that satisfies (3.3).
it follows that .
as ; therefore . □
Proof It is sufficient to remark that , , and . □
4 Stability of the equation
The recurrence , , is an essential tool in this section. First we complete Theorem 2.2.
Theorem 4.1 The function is an n-polynomial if and only if .
Proof If p is an n-polynomial then for any there exists a j-monomial such that . Then, for all we have , hence , . Therefore , . From Theorem 2.2 it follows that , , or, equivalently, .
is a j-monomial for . But , and defines a 0-monomial. Consequently and, therefore, p is an n-polynomial. □
The central idea in justifying the fact that a pair is a stability couple for equation (1.1) is the existence of a stability chain between the mappings φ and Φ.
Definition 4.2 We say that is a stability chain between the functions and if is a stability couple for the equation , , on , and on .
Remark Theorem 3.3 provides stability chains: if for all , then is a stability chain between the and .
Stability chains provide stability couples for equation (1.1).
Theorem 4.3 If there exists a stability chain between and , then is a stability couple for equation (1.1).
where is a symmetric and j-additive mapping, . According to Theorem 2.1, is an n-polynomial, and, from Theorem 4.2, it follows that . Consequently, is a stability couple for equation (1.1). □
The following theorem provides a technique of building strong stability couples for equation (1.1) and is the main result of this section.
Proof Since , , from Theorem 3.3 and Theorem 4.3 it follows that is a stability chain between and Φ, the pair is a stability couple for equation (1.1), and there exists an n-polynomial p satisfying (4.2).
From for and it follows that (4.2) is satisfied for the n-polynomial , where , and , .
hence , and (4.4) is proved for .
Therefore and the alternative is completely proved.
By reverse induction, we obtain for . So and the theorem is proved. □
Remark The condition imposed in the previous theorem is needed to ensure the uniqueness of the 0-monomial . In fact, as , we can consider, without affecting the generality, in inequality (4.1).
The following consequence provides a class of strong stability couples for equation (1.1), and a technique for building stability chains.
the pairs are strong stability couples for equation (1.1). Moreover, if and is a function satisfying (4.1), then procedure (4.3) defines the unique n-polynomial p that verifies (4.2).
Since it follows immediately that and from Theorem 4.4, we obtain the conclusion. □
The following consequence is a stability result for equation (1.1) in the sense of Hyers-Ulam.
where , for and .
Proof Let , . Then , and , . By recurrence we obtain , and, from the previous corollary, we obtain the conclusion. □
The functional equation (1.1) is stable in the Aoki-Rassias sense, as can be seen from the following corollary.
The j-monomial , is given in (4.3), and , .
and, by recurrence, we finally obtain , . □
5 Stability of the equation
Further, we use the following conventions:
M is a uniquely -divisible and commutative group. If and we denote by the unique solution of the equation .
Let , and let be a bijection. For any , the linear system(5.1)
with the unknowns has a unique solution denoted by , .
If is a function, , and is a bijection, then is the mapping defined by
The following lemma establishes a fundamental connection between the behavior of the operators and .
for all , where , is the solution of system (5.1); consequently, for and in the previous inequality, we obtain , . □
Then is a (strong) stability couple for equation (1.2). If, in addition, the pair verifies the conditions of Theorem 4.4 and is a mapping that satisfies (5.2), then procedure (4.3) gives the unique n-polynomial p that verifies (4.2).
The consequences of Theorem 4.4 and the previous theorem provide specific classes of strong stability couples for Fréchet’s second functional equation as follows.
and the monomial components of p can be calculated with procedure (4.3).
Proof It is sufficient to note that , , and that we can apply Corollary 4.5 for . □
Applying Corollary 4.6, we obtain an improvement of Theorem 2.5.
where , for and . The monomial components of p can be calculated with procedure (4.3).
Proof Let and defined by if and . Then . Defining , we have , , and , . From Corollary 4.6 it follows that is a strong stability couple for equation (1.1); hence, from Theorem 5.2, it follows that is a strong stability couple for equation (1.2) and that procedure (4.3) can be applied in this case. □
The flexibility of working with stability couples is illustrated by the following Aoki-Rassias type result.
Applying Theorem 5.2 for , and Corollary 4.7 for , it follows that procedure (4.3) gives the unique n-polynomial p that satisfies (5.3). □
The n-monomial m is given by .
Letting in (5.7) and taking into account that , we obtain , .
as . Therefore, m is the only n-monomial that satisfies (5.5). □
6 Future work
Suppose that for all , is an arbitrary function, , for all , and . Then is a strong stability couple for equation (1.4) (see Theorem 4.4).
New stability couples for equations (2.1), (1.1), (1.2), and (1.5) can be determined using the ideas of the above theory, but replacing the operator with the operator defined by , where is a function for which , , and M is a commutative 2-divisible monoid (see also ).
The main results of this research paper are:
the first proofs of generalized stability for two of the best known functional equations: the Fréchet polynomial equations;
a proof of the equivalence of these two equations;
a very general iterative technique for solving the stability of polynomial equations that can be applied to other similar problems;
extensions and improvements of some known results of Hyers-Ulam type;
a new technique for proving the generalized stability of the monomial equation.
I would like to thank to the referees for careful reading of this paper and their useful comments.
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