Ulam type stability problems for alternative homomorphisms

We introduce an alternative homomorphism with respect to binary operations and investigate the Ulam type stability problem for such a mapping. The obtained results apply to Ulam type stability problems for several important functional equations.


Introduction
In , SM Ulam proposed the following stability problem: Given an approximately additive mapping, can one find the strictly additive mapping near it? A year later, DH Hyers gave an affirmative answer to this problem for additive mappings between Banach spaces. Subsequently many mathematicians came to deal with this problem (cf. [-]).
We introduce an alternative homomorphism from a set X with two binary operations • and * to another set E with two binary operations and defined by f (x • y) f (x * y) = f (x) f (y) (∀x, y ∈ X), and we investigate the Ulam type stability problem for such a mapping when E is a complete metric space. In particular, if s t = s for all s, t ∈ E, then our results imply the stability results obtained in []. Also the method used in the paper have already applied for some other equations (cf. [-]).

One consequence of Banach's fixed point theorem
A fixed point theorem has played an important role in the stability problem (cf. []). The authors used an easy consequence of Banach's fixed point theorem in []. It will serve again in this paper. Here we review it.
Let X be a set and (E, d) a complete metric space. Fix two mappings f : X → E and ϕ : X → R + , where R + denotes the set of all nonnegative real numbers. Denote by f ,ϕ the set of all mappings u : X → E such that there exists a finite constant K u satisfying d u(x), f (x) ≤ K u ϕ(x) (∀x ∈ X). ©2014 Takahasi et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. http://www.journalofinequalitiesandapplications.com/content/2014/1/228 For any u, v ∈ f ,ϕ , we define Then ( f ,ϕ , ρ f ,ϕ ) is a complete metric space which contains f . Now, fix three mappings σ : X → X, τ : E → E and ε : X × X → R + . For any mapping u : X → E, we define the mapping T σ ,τ u : X → E by Also, we consider three quantities: If α σ ,ε < ∞, β σ ,ϕ < ∞ and γ τ < ∞, then we have respectively. We will use these inequalities throughout this paper.
We now state our fixed point theorem.
for all x ∈ X.

A stability of alternative homomorphisms
Let (X, •, * ) be a set X with two binary operations • and * . Let (E, d, , ) be a complete metric space (E, d) with two binary operations and . Given f : X → E, we consider the following commutative diagram: () http://www.journalofinequalitiesandapplications.com/content/2014/1/228 This means that In particular, if s t = s for all s, t ∈ E, then () and () become In other words, f is a homomorphism from X to E. Thus, if a mapping f : X → E satisfies (), then we say that f is an alternative homomorphism.
In this section, we establish two general settings, on which we can give an affirmative answer to the Ulam type stability problem for the commutative diagram (). These settings have a property such as duality, that is, each of them works as a complement of the other.
Let us describe the first setting. For ε : X × X → R + and δ : X → R + , we consider the following three conditions: (i) The square operator x → x • x is an automorphism of X with respect to • and * . We denote by σ the inverse mapping of this automorphism. (ii) The binary operations and on E are continuous. The square operator τ : s → s s is an endomorphism of E with respect to and . (iii) α ≡ α σ ,ε < ∞, β ≡ β σ ,δ < ∞, γ ≡ γ τ < ∞ and γ max{α, β} < . Under the above conditions, we show the Ulam type stability for the commutative diagram (), as follows.
Theorem  Let (X, •, * ) and (E, d, , ) be as above. Suppose that four mappings σ : X → X, τ : E → E, ε : X × X → R + and δ : X → R + satisfy (i), (ii), and (iii). If a mapping f : then there exists a mapping f ∞ : Moreover, if a mapping g : X → E satisfies (), (), and then g = f ∞ . http://www.journalofinequalitiesandapplications.com/content/2014/1/228 Proof For simplicity, we write T = T σ ,τ . We note that α, β, and γ are finite by (iii). Suppose that f : X → E satisfies () and (). Put ϕ(x) = αε(x, x)+δ(x) for all x ∈ X. To apply Lemma A to f and ϕ, we first observe that Tf ∈ f ,ϕ . Fix x ∈ X. Replacing x and y in () by σ x, we get Since Using this and (), we have Hence Tf ∈ f ,ϕ and ρ f ,ϕ (Tf , f ) ≤ . We next estimate the quantity β σ ,ϕ . For x ∈ X, we have Hence β σ ,ϕ ≤ max{α, β} and β σ ,ϕ γ τ ≤ γ max{α, β} <  by (iii). Thus we can apply Lemma A. As a consequence, T has a unique fixed point f ∞ ∈ f ,ϕ . Moreover, and for all x ∈ X. Since ρ f ,ϕ (Tf , f ) ≤  and β σ ,ϕ γ τ ≤ γ max{α, β} < , () implies (). http://www.journalofinequalitiesandapplications.com/content/2014/1/228 Here we show (). If x, y ∈ X and n ∈ N, then we have We will see that the right hand side of () tends to  as n → ∞. The first and third terms on the right hand side tend to  as n → ∞, because of () and the continuity of and in (ii). Moreover, the second term, say A n (x, y), is estimated as follows: By (i), (ii), and (), we have where τ n and σ n denote the n-fold compositions of endomorphisms τ and σ , respectively. Since γ α <  by (iii), it follows that A n (x, y) →  as n → ∞. Thus the right hand side of () tends to , and we obtain (). Next, we show (). For x ∈ X, we replace x and y in () by σ x to get we obtain (). Finally, we show the last statement. Since g satisfies () and (), we have for all x ∈ X. This says that g is a fixed point of T. Also, by (), we have g ∈ f ,ϕ . Thus the uniqueness of a fixed point of T in f ,ϕ implies that g = f ∞ . http://www.journalofinequalitiesandapplications.com/content/2014/1/228 The next corollary is obtained in [].
Corollary  ([, Corollary .]) Let X be a set with a binary operation • such that the square operation x → x • x is an automorphism of X with respect to • and E a complete metric space with a continuous binary operation such that the square operation τ : s → s s is an endomorphism of E with respect to . Let ε : X × X → R + and suppose that α ≡ α σ ,ε < ∞, γ ≡ γ τ < ∞ and γ α < , where σ denotes the inverse mapping of the square then there exists a unique mapping f ∞ : X → E such that for all x, y ∈ X.
Proof Consider the case that * = • and s t = s for s, t ∈ E, in Theorem . In this case, τ is clearly an endomorphism of E with respect to . Therefore the corollary follows immediately from Theorem  with δ = .
Now we turn to another setting. Let (X, •, * ) and (E, d, , ) be as in the first part of this section. For ε : X × X → R + and δ : X → R + , we consider the following three conditions: (iv) The square operatorσ : x → x • x is an endomorphism of X with respect to • and * .
(v) The binary operations and on E are continuous. The square operator s → s s is an automorphism of E with respect to and . We denote byτ the inverse mapping of this automorphism. (vi)α ≡ ασ ,ε < ∞,β ≡ βσ ,δ < ∞,γ ≡ γτ < ∞, andγ max{α,β} < . Under the above conditions, we show the Ulam type stability for the commutative diagram (), as follows.
Here we show (). If x, y ∈ X and n ∈ N, then we have Letting n → ∞, the first and third terms on the right hand side tend to , because of () and the continuity of and in (v). Moreover, the second term, sayÃ n (x, y), is estimated http://www.journalofinequalitiesandapplications.com/content/2014/1/228 as follows: By (iv), (v), and (), *  y) ,τ n f σ n x τ n f σ n y = d τ n f σ n x •σ n y τ n f σ n x * σ n y ,τ n f σ n x f σ n y = d τ n f σ n x •σ n y f σ n x * σ n y ,τ n f σ n x f σ n y ≤γ n d f σ n x •σ n y f σ n x * σ n y , f σ n x f σ n y ≤γ n ε σ n x,σ n y ≤γ nαn ε(x, y), whereτ n andσ n denote the n-fold compositions of endomorphismsτ andσ , respectively. Sinceγα <  by (vi), it follows thatÃ n (x, y) →  as n → ∞. Thus we obtain (). Next, we show (). Replacing y in () by x, we have Also sincẽ it follows that Combining with (), we obtain (). Finally, we show the last statement. Since g satisfies () and (), we have for all x ∈ X. This says that g is a fixed point ofT. Also, by (), we have g ∈ f ,φ . Hence the uniqueness of a fixed point ofT in f ,φ implies that g = f ∞ .
The next corollary is obtained in [].

Corollary  ([, Corollary .]) Let X be a set with a binary operation
• such that the square operationσ : x → x • x is an endomorphism of X with respect to • and E a complete metric space with a continuous binary operation such that the square operation s → s s is an automorphism of E with respect to . Let ε : X × X → R + and suppose that α ≡ ασ ,ε < ∞,γ ≡ γτ < ∞ andγα < , whereτ denotes the inverse mapping of the square operation s → s s. If a mapping f : then there exists a unique mapping f ∞ : X → E such that for all x, y ∈ X. http://www.journalofinequalitiesandapplications.com/content/2014/1/228 Proof Consider the case that * = • and s t = s for s, t ∈ E, in Theorem . Thenτ is clearly an endomorphism of E with respect to . Therefore the corollary follows immediately from Theorem  with δ = .

Application I
The Ulam type stability problem for Euler-Lagrange type additive mappings has been investigated in []. Here we take up the following Euler-Lagrange type mapping f : X → E satisfying where X is a complex normed space, E a complex Banach space and a, b ∈ C with a + b = .
The following is an Ulam type stability result for this mapping.
then there exists a unique mapping f ∞ : X → E satisfying () and Proof Put u = -x, v = -y for each x, y ∈ X. Under these transformations, () changes into the following estimate: In this case, we can easily see that the square operator u → u • u is an endomorphism of X with respect to • and * . Also since a + b = , this endomorphism is bijective and so automorphic. We denote by σ the inverse mapping of this automorphism. Moreover, we define s t = -  (a +b)(s +t), s t =   (s + t) for each s, t ∈ E. Then we can also see that the binary operations and on E are continuous and the square operator τ : s → s s is an automorphism of E with respect to and . Note that () changes into the following: Also, since s s = s for all s ∈ E, it follows that for all x ∈ X and then () holds with δ = . Moreover, β σ ,δ =  must hold with δ = . It is also obvious that γ τ = |a + b| from the definition of τ . We also note that α σ ,ε  ≤ K from the http://www.journalofinequalitiesandapplications.com/content/2014/1/228 second condition of (vii) and hence γ τ α σ ,ε  ≤ |a + b|K <  from the first condition of (vii). Therefore, by Theorem , there exists a unique mapping f ∞ : X → E such that namely, () holds and and so () holds.
The following is also an Ulam type stability result for the mapping satisfying ().
. If a mapping f : X → E satisfies (), then there exists a unique mapping f ∞ : X → E satisfying () and Proof As observed in the proof of Corollary , () changes into (). Now we define In this case, we can easily see that the square operatorσ : u → u • u is an endomorphism of X with respect to • and * . Moreover, we define s t = -  (a + b)(s + t), s t =   (s + t) for each s, t ∈ E. Then we can also see that the binary operations and on E are continuous and the square operator s → s s is an endomorphism of E with respect to and . Also since a + b = , this endomorphism is bijective and so automorphic. We denote byτ the inverse mapping of this automorphism. Note that () changes into (). Since x • x = x * x (∀x ∈ X) and s s = s (∀s ∈ E), it follows for all x ∈ X and then () holds with δ = .
Moreover, βσ ,δ =  must hold with δ = . It is also obvious that γτ = |a + b| - from the definition ofτ . We also note that ασ ,ε  ≤ K from the second condition of (viii) and hence γτ ασ ,ε  ≤ |a + b| - K <  from the first condition of (viii).
Therefore, by Theorem , there exists a unique mapping f ∞ : X → E such that namely, () holds and and so () holds.
In this section, we deal with the Ulam type stability problem for Equation (). Put x • y = x + y and x * y = xy for each x, y ∈ X. Moreover, put s t = s + t and s t = max{s, t} for each s, t ∈ R. Then () changes into (). Also we can easily see that the square operatioñ σ : x → x • x is endomorphic with respect to • and * and that the square operator s → s s is automorphic with respect to and . Denote byτ the inverse mapping of this automorphism. In this case, it is obvious thatτ (s) =   s for each s ∈ R and hence γτ = /. Now let ε be a nonnegative constant and suppose that f : X → R satisfies Put δ = ε. By () and (), we obtain and hence () holds. Moreover, note that ασ ,ε = βσ ,δ =  since ε and δ are constant. Then Lemma B and Theorem  easily imply the following.
Corollary  Let X be an Abelian group and ε a nonnegative constant. If f : X → R satisfies (), then there exists an additive mapping π : X → R such that f (x)π(x) ≤ ε (∀x ∈ X).