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Recursive procedure in the stability of Fréchet polynomials
Advances in Difference Equations volume 2014, Article number: 16 (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.
(see  for the original equation) and
(see , for the original equation), where is the unknown function, M is an abelian monoid, B is a real (or complex) Banach space, n is a positive integer and, for any mapping , is the symmetric function defined by
The Hyers-Ulam stability of equation (1.2) has been studied intensively. Starting with the fundamental work of Whitney [5, 6] and Hyers , in 1983 Albert and Baker  proved the Hyers-Ulam stability of equation (1.2) using the Hyers-Ulam stability of the equation
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.
In this paper we eliminate these impediments with the help of two new instruments: the stability couples - which rigorously define the concept of the stability of a functional equation - and the stability chains - necessary for studying the stability of equation (1.1). First we prove - through the so-called direct method (of Hyers) - a new generalized stability theorem for multi-additive and symmetric functions in the spirit of Bourgin and Găvruţa. We use this result to justify a recursive procedure of solving the stability of equation (1.1) and we prove the equivalence of equations (1.1), (1.2), and (1.4). We then give a general technique of solving the stability of equation (1.2) using the stability of equation (1.1). As a consequence, we obtain stability results of Hyers-Ulam type which extend and improve the above mentioned results, and Aoki-Rassias type stability results for equations (1.1) and (1.2). Finally, we give a new technique for proving the generalized stability of the monomial equation
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 , the linear operators are defined by
For all we have .
The n th iteration of , denoted , verifies the identity
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 .
If is a symmetric function, we define by , for , and for by
The symmetric mapping is n-additive if and only if
The operator defined in equation (1.3) can be described by the difference operators as follows:
Since , and for and we have , and it follows that
Theorem 2.1 If is a symmetric and n-additive mapping then the function defined by is an n-monomial,
Theorem 2.2 The function is an n-polynomial if and only if it verifies one of the equivalent equations (1.2) and (1.4). In these conditions, for any there is a symmetric and i-additive mapping 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.
Definition 2.3 Let , be a functional equation, where is a mapping, S is a nonempty set and is the unknown function. Let also and be two mappings. The pair is called a stability couple for the functional equation , if for all for which
there exists a function such that and
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 
Let G be an abelian group, a function such that
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 .
Proof According to the hypothesis it follows that
Let j be an arbitrary positive integer. Then, by using the triangle inequality and the above inequality, we obtain
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).
Lemma 3.2 Let K be a commutative semigroup, and let be a function such that , . Then
Proof For we have
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).
Let be a function. Then is the mapping defined by if , and, for , by
If , , then we define the mapping by
From Lemma 3.2 it follows that , . Therefore
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.
Theorem 3.3 Let and be a symmetric function satisfying the inequality
Then defines the unique symmetric and n-additive function such that
Moreover, is the unique n-monomial from for which
Proof Let . Putting in (3.1) we get , hence
But g is a symmetric function; therefore
Replacing y by we get
Now, from Lemma 3.1 (for , , and ) it follows that is a convergent sequence in the Banach space B, and its limit, , satisfies (3.2). But g is symmetric, therefore a is a symmetric function, too. From (3.1), the definition of the set , and Lemma 3.2 it follows that
whence , , i.e. a is a symmetric and n-additive mapping that satisfies (3.2) and is a n-monomial that satisfies (3.3).
In order to justify the uniqueness of a, we consider the symmetric and n-additive mapping for which , ; since for any , , we have
it follows that .
Finally, if m is an n-monomial that satisfies (3.3), that is, , , since , , it follows that
as ; therefore . □
Corollary 3.4 Let M be a linear normed space, , and let be defined by if , and if . Suppose that is a symmetric function such that
Then defines the unique symmetric and n-additive mapping for which
and is the unique n-monomial that satisfies the inequality
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, .
Conversely, suppose that . Then
hence the symmetric function is n-additive. From Theorem 2.1 it follows that is an n-monomial and . Therefore
analogously (for ), is an -monomial and
By recurrence we finally obtain , where
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).
Proof Let be a stability chain between the functions and Φ. Let also be a mapping such that , . Since , we have
Because is a stability couple for equation (2.1) and is a symmetric mapping, it follows that there is a symmetric and n-additive mapping , such that , . According to Theorem 2.1 we have , hence
By reverse induction we finally obtain
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.
Theorem 4.4 Suppose that for all , for all , and . If is a function satisfying
then there exists a unique n-polynomial such that
Moreover, , where is a j-monomial for any , , and the monomials , , can be obtained by recurrence: let ; then, for , we have the alternative
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).
We successively apply Theorem 3.3 and Theorem 2.1 (as in the proof of Theorem 4.3) for justifying procedure (4.3). From (4.1) we have
Since , from Theorem 3.3 it follows that
defines an n-monomial satisfying the inequality
where . Analogously, (if ) and, from Theorem 3.3, it follows that
defines an -monomial satisfying the inequality
where . By recurrence, we finally find that
defines an additive function such that
where . Since , it follows that and . Therefore the above inequality becomes
From for and it follows that (4.2) is satisfied for the n-polynomial , where , and , .
We prove now by reverse induction that
for and . Let . Since is a j-monomial and we have
We prove relations (4.4) for . First, we remark that inequality (4.2) becomes
Replacing here x by , multiplying both members of this inequality by and taking into account relations (4.5), we obtain
hence , and (4.4) is proved for .
Since , (4.2) becomes
By recurrence, we finally obtain , hence
Therefore and the alternative is completely proved.
We have yet to show that the only n-polynomial satisfying (4.2) is . Suppose that is a j-polynomial, , and verifies (4.2), i.e. , . It follows immediately that
Putting here , we obtain and thus
Again, replacing here x by and multiplying both members of this inequality by , we obtain
as , namely , and inequality becomes
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.
Corollary 4.5 Let be a mapping such that , . Then for all functions for which
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).
Proof We successively apply Lemma 3.2. Let . Then . Because , , it follows that . By reverse induction, we obtain
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.
Corollary 4.6 Let . If is a function such that
then procedure (4.3) gives the unique n-polynomial p satisfying the condition
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.
Corollary 4.7 Let M be a normed space, and . If and
then there exists a unique n-polynomial such that
The j-monomial , is given in (4.3), and , .
Proof Let and . According to Corollary 4.5, it is sufficient to show that , . It is straightforward to verify that
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
and is the function defined by
The following lemma establishes a fundamental connection between the behavior of the operators and .
Lemma 5.1 Let , be a bijection, let be a mapping and let be a function satisfying
Proof The central idea of this proof is to work with the operator in the direct product . Let . Then
From this, using the triangle inequality, hypothesis (5.2), and the definition of , we obtain , , or, equivalently,
Since for all , it follows immediately that, for all , we have
for all , where , is the solution of system (5.1); consequently, for and in the previous inequality, we obtain , . □
Theorem 5.2 Let , be a bijection, let be a mapping, and let be a (strong) stability couple for equation (1.1) such that
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.
Corollary 5.3 Let , be a bijection, let be a function such that
and let be a mapping satisfying the conditions
for all . If satisfies (5.2), then there exists a unique n-polynomial p for which
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.
Corollary 5.4 If and verifies the inequality
then there exists a unique n-polynomial p that satisfies the inequality
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.
Corollary 5.5 Let M be a rational and normed vector space, , and . If is a function satisfying
then procedure (4.3) defines the unique n-polynomial for which
Proof Let , , and . Then for all . Let . The system (5.1) has the solution
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). □
Theorem 5.6 Let , be a bijection, let be a function so that
and . If is a mapping such that
then there exists a unique n-monomial that verifies the inequality
The n-monomial m is given by .
Proof Let us first note that from , it follows that
and from Lemma 3.2 it follows that . For all we have ; hence, from Lemma 5.1 we have , , or, equivalently,
From Theorem 3.3, it follows that there exists a unique n-additive mapping so that , , or, equivalently,
Let . Then
and, from (5.4) and (5.6), it follows that the n-monomial m satisfies (5.5). But ; therefore, from (5.5) we have for all
Letting in (5.7) and taking into account that , we obtain , .
Finally, if is an n-monomial that verifies (5.5), then . Since , we have
as . Therefore, m is the only n-monomial that satisfies (5.5). □
6 Future work
As future work we propose two unsolved problems.
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.
Bourgin D: Classes of transformations and bordering transformations. Bull. Am. Math. Soc. 1951, 57: 223–237. 10.1090/S0002-9904-1951-09511-7
Găvruţă P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 1994, 184: 431–438. 10.1006/jmaa.1994.1211
Fréchet M: Une definition fonctionnelle des polynômes. Nouv. Ann. Math. 1909, 9: 145–182.
Fréchet M: Les polynômes abstraits. J. Math. Pures Appl. 1929, 8: 71–92.
Whitney H: On functions with bounded n th differences. J. Math. Pures Appl. 1957, 36(9):87–95.
Whitney H: On bounded functions with bounded n th differences. Proc. Am. Math. Soc. 1959, 10: 480–481.
Hyers D: Transformations with bounded m -th differences. Pac. J. Math. 1961, 11: 591–602. 10.2140/pjm.1961.11.591
Albert M, Baker J: Functions with bounded n th differences. Ann. Pol. Math. 1983, 43: 93–103.
Mazur S, Orlicz W: Grundlegende Eigenschaften der polynomischen Operationen, I. Stud. Math. 1934, 5: 50–88.
Mazur S, Orlicz W: Grundlegende Eigenschaften der polynomischen Operationen, II. Stud. Math. 1934, 5: 179–189.
Jun KW, Kim HM: On the Hyers-Ulam-Rassias stability of a general cubic functional equation. Math. Inequal. Appl. 2003, 8(2):289–302.
Lee YH: On the generalized Hyers-Ulam stability of the generalized polynomial function of degree 3. Tamsui Oxford Univ. J. Math. Sci. 2008, 24(4):429–444.
Lee YH: On the Hyers-Ulam-Rassias stability of the generalized polynomial function of degree 2. J. Chungcheong Math. Soc. 2009, 22(2):201–210.
Hyers D: Polynomial operators. In Topics in Mathematical Analysis. Edited by: Rassias T. World Scientific, Teaneck; 1989:410–444.
Djoković D:A representation theorem for and its applications. Ann. Pol. Math. 1969, 22: 189–198.
Hyers D: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224. 10.1073/pnas.27.4.222
Aoki T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 1950, 2: 84–88.
Rassias T: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1
Lijn G: Les polynômes abstraits. Bull. Sci. Math. 1940, 84: 55–80. 102–112, 183–198
Gajda Z: On stability of additive mappings. Int. J. Math. Sci. 1991, 14: 431–434. 10.1155/S016117129100056X
Rassias T, Šemrl P: On the behavior of mappings which do not satisfy Hyers-Ulam stability. Proc. Am. Math. Soc. 1992, 114: 989–993. 10.1090/S0002-9939-1992-1059634-1
Jung SM Springer Optimization and Its Applications 48. In Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, New York; 2011.
Borelli C, Invernizzi C: Sulla stabilitá dell’equazione funzionale dei polinomi. Rend. Semin. Mat. (Torino) 1999, 57(3):197–208.
Kaiser, Z: Stability of functional equations in abstract structures. PhD thesis, University of Debrecen (2005)
Ciepliński K: Generalized stability of multi-additive mappings. Appl. Math. Lett. 2010, 23: 1291–1294. 10.1016/j.aml.2010.06.015
Gajda Z: Local stability of the functional equation characterizing polynomial functions. Ann. Pol. Math. 1990, 52(2):119–137.
Cădariu L, Radu V: Stability properties for monomial functional equations. An. Univ. Vest. Timiş., Ser. Mat.-Inform. 2005, 43(1):23–38.
Cădariu L, Radu V: Remarks on the stability of monomial functional equations. Fixed Point Theory 2007, 8(2):201–218.
Eungrasamee T, Udomkavanich P, Nakmahachalasint P: Generalized stability of classical polynomial functional equation of order n . Adv. Differ. Equ. 2012. 10.1186/1687-1847-2012-135
Gilányi A: Hyers-Ulam stability of monomial functional equations on a general domain. Proc. Natl. Acad. Sci. USA 1999, 19: 10588–10590.
Gilányi A: On the stability of monomial functional equations. Publ. Math. (Debr.) 2000, 58(1–2):201–212.
Lee YH: On the stability of the monomial functional equation. Bull. Korean Math. Soc. 2008, 45(2):397–403. 10.4134/BKMS.2008.45.2.397
Mirmostafaee A: Stability of the monomial functional equation in quasi normed spaces. Bull. Korean Math. Soc. 2010, 47(4):777–785. 10.4134/BKMS.2010.47.4.777
Baker J: A general functional equation and its stability. Proc. Am. Math. Soc. 2005, 133: 1857–1864.
Cholewa P: Almost approximately polynomial functions. In Nonlinear Analysis. World Scientific, Singapore; 1987:127–136.
Gajda Z: A solution to a problem of J. Schwaiger. Aequ. Math. 1987, 32(1):38–44. 10.1007/BF02311297
Ger R: On almost polynomial functions. Colloq. Math. 1971, 24: 95–101.
Kominek Z: A few remarks on almost C -polynomial functions. Math. Slovaca 2005, 55(5):555–561.
Popa D, Raşa I: The Fréchet functional equation with application to the stability of certain operators. J. Approx. Theory 2012, 184(1):138–144.
Schwaiger J: Functional equations for homogeneous polynomials arising from multilinear mapings and their stability. Ann. Math. Sil. 1994, 8: 157–171.
Székelyhidi L: Note on a stability theorem. Can. Math. Bull. 1982, 25(4):500–501. 10.4153/CMB-1982-074-0
Székelyhidi L: Fréchet’s equation and Hyers theorem on noncommutative semigroups. Ann. Pol. Math. 1988, 48: 183–189.
Tabor J: k -proper families and almost approximately polynomial functions. Glas. Mat. 2001, 38(58):177–191.
Wolna D: The stability of monomial functions on a restricted domain. Aequ. Math. 2006, 72(1–2):100–109. 10.1007/s00010-006-2832-z
I would like to thank to the referees for careful reading of this paper and their useful comments.
The author declares that he has no competing interests.
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Cite this article
Dăianu, D.M. Recursive procedure in the stability of Fréchet polynomials. Adv Differ Equ 2014, 16 (2014). https://doi.org/10.1186/1687-1847-2014-16
- multi-additive mapping
- Fréchet polynomial