Skip to main content

Ulam type stability problems for alternative homomorphisms

Abstract

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.

MSC: Primary 39B82; secondary 47H10.

1 Introduction

In 1940, 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. [15]).

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(xy)f(xy)=f(x)f(y)(x,yX),

and we investigate the Ulam type stability problem for such a mapping when E is a complete metric space. In particular, if st=s for all s,tE, then our results imply the stability results obtained in [6]. Also the method used in the paper have already applied for some other equations (cf. [715]).

One consequence of Banach’s fixed point theorem

A fixed point theorem has played an important role in the stability problem (cf. [16]). The authors used an easy consequence of Banach’s fixed point theorem in [6]. 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:XE and φ:X R + , where R + denotes the set of all nonnegative real numbers. Denote by Δ f , φ the set of all mappings u:XE such that there exists a finite constant K u satisfying

d ( u ( x ) , f ( x ) ) K u φ(x)(xX).

For any u,v Δ f , φ , we define

ρ f , φ (u,v)=inf { K 0 : d ( u ( x ) , v ( x ) ) K φ ( x ) ( x X ) } .

Then ( Δ f , φ , ρ f , φ ) is a complete metric space which contains f.

Now, fix three mappings σ:XX, τ:EE and ε:X×X R + . For any mapping u:XE, we define the mapping T σ , τ u:XE by

( T σ , τ u)(x)=τ ( u ( σ x ) ) (xX).

Also, we consider three quantities:

α σ , ε = inf { K 0 : ε ( σ x , σ y ) K ε ( x , y ) ( x , y X ) } , β σ , φ = inf { K 0 : φ ( σ x ) K φ ( x ) ( x X ) } , γ τ = inf { K 0 : d ( τ s , τ t ) K d ( s , t ) ( s , t E ) } .

If α σ , ε <, β σ , φ < and γ τ <, then we have

ε ( σ x , σ y ) α σ , ε ε ( x , y ) ( x , y X ) , φ ( σ x ) β σ , φ φ ( x ) ( x X ) , d ( τ s , τ t ) γ τ d ( s , t ) ( s , t E ) ,

respectively. We will use these inequalities throughout this paper.

We now state our fixed point theorem.

Lemma A ([[6], Proposition 2.1])

Let X be a set and (E,d) a complete metric space. Suppose that four mappings f:XE, φ:X R + , σ:XX and τ:EE satisfy

T σ , τ f Δ f , φ , β σ , φ <, γ τ <and β σ , φ γ τ <1.

Then T σ , τ ( Δ f , φ ) Δ f , φ and T σ , τ has a unique fixed point f in Δ f , φ . Moreover,

lim n d ( ( T σ , τ n f ) ( x ) , f ( x ) ) =0andd ( f ( x ) , f ( x ) ) ρ f , φ ( T σ , τ f , f ) 1 β σ , φ γ τ φ(x)

for all xX.

2 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:XE, we consider the following commutative diagram:

(1)

This means that

f(xy)f(xy)=f(x)f(y)(x,yX).
(2)

In particular, if st=s for all s,tE, then (1) and (2) become

and

f(xy)=f(x)f(y)(x,yX).

In other words, f is a homomorphism from X to E. Thus, if a mapping f:XE satisfies (2), 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 (1). 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:

  1. (i)

    The square operator xxx is an automorphism of X with respect to and . We denote by σ the inverse mapping of this automorphism.

  2. (ii)

    The binary operations and on E are continuous. The square operator τ:sss is an endomorphism of E with respect to and .

  3. (iii)

    α α σ , ε <, β β σ , δ <, γ γ τ < and γmax{α,β}<1.

Under the above conditions, we show the Ulam type stability for the commutative diagram (1), as follows.

Theorem 1 Let (X,,) and (E,d,,) be as above. Suppose that four mappings σ:XX, τ:EE, ε:X×X R + and δ:X R + satisfy (i), (ii), and (iii). If a mapping f:XE satisfies

d ( f ( x y ) f ( x y ) , f ( x ) f ( y ) ) ε(x,y)(x,yX),
(3)
d ( f ( x ) f ( σ x σ x ) , f ( x ) ) δ(x)(xX),
(4)

then there exists a mapping f :XE such that

f (xy) f (xy)= f (x) f (y)(x,yX),
(5)
f (x) f (σxσx)= f (x)(xX),
(6)
d ( f ( x ) , f ( x ) ) α ε ( x , x ) + δ ( x ) 1 γ max { α , β } (xX).
(7)

Moreover, if a mapping g:XE satisfies (5), (6), and

K g 0:d ( f ( x ) , g ( x ) ) K g { α ε ( x , x ) + δ ( x ) } (xX),
(8)

then g= f .

Proof For simplicity, we write T= T σ , τ . We note that α, β, and γ are finite by (iii). Suppose that f:XE satisfies (3) and (4). Put φ(x)=αε(x,x)+δ(x) for all xX. To apply Lemma A to f and φ, we first observe that Tf Δ f , φ . Fix xX. Replacing x and y in (3) by σx, we get

d ( f ( σ x σ x ) f ( σ x σ x ) , f ( σ x ) f ( σ x ) ) ε(σx,σx).

Since

σ x σ x = σ 1 ( σ x ) = x , f ( σ x ) f ( σ x ) = τ ( f ( σ x ) ) = ( T f ) ( x ) ,

and

ε(σx,σx)αε(x,x),

it follows that

d ( f ( x ) f ( σ x σ x ) , ( T f ) ( x ) ) αε(x,x).

Using this and (4), we have

d ( ( T f ) ( x ) , f ( x ) ) d ( ( T f ) ( x ) , f ( x ) f ( σ x σ x ) ) + d ( f ( x ) f ( σ x σ x ) , f ( x ) ) α ε ( x , x ) + δ ( x ) = φ ( x ) .

Hence Tf Δ f , φ and ρ f , φ (Tf,f)1.

We next estimate the quantity β σ , φ . For xX, we have

φ ( σ x ) = α ε ( σ x , σ x ) + δ ( σ x ) α 2 ε ( x , x ) + β δ ( x ) max { α , β } ( α ε ( x , x ) + δ ( x ) ) = max { α , β } φ ( x ) .

Hence β σ , φ max{α,β} and β σ , φ γ τ γmax{α,β}<1 by (iii).

Thus we can apply Lemma A. As a consequence, T has a unique fixed point f Δ f , φ . Moreover,

lim n d ( ( T n f ) ( x ) , f ( x ) ) =0
(9)

and

d ( f ( x ) , f ( x ) ) ρ f , φ ( T f , f ) 1 β σ , φ γ τ φ(x)
(10)

for all xX. Since ρ f , φ (Tf,f)1 and β σ , φ γ τ γmax{α,β}<1, (10) implies (7).

Here we show (5). If x,yX and nN, then we have

d ( f ( x y ) f ( x y ) , f ( x ) f ( y ) ) d ( f ( x y ) f ( x y ) , ( T n f ) ( x y ) ( T n f ) ( x y ) ) + d ( ( T n f ) ( x y ) ( T n f ) ( x y ) , ( T n f ) ( x ) ( T n f ) ( y ) ) + d ( ( T n f ) ( x ) ( T n f ) ( y ) , f ( x ) f ( y ) ) .
(11)

We will see that the right hand side of (11) tends to 0 as n. The first and third terms on the right hand side tend to 0 as n, because of (9) and the continuity of and in (ii). Moreover, the second term, say A n (x,y), is estimated as follows: By (i), (ii), and (3), we have

A n ( x , y ) = d ( τ n ( f ( σ n ( x y ) ) ) τ n ( f ( σ n ( x 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 n α n ε ( x , y ) ,

where τ n and σ n denote the n-fold compositions of endomorphisms τ and σ, respectively. Since γα<1 by (iii), it follows that A n (x,y)0 as n. Thus the right hand side of (11) tends to 0, and we obtain (5).

Next, we show (6). For xX, we replace x and y in (5) by σx to get

f (σxσx) f (σxσx)= f (σx) f (σx).

Since σxσx=x and

f (σx) f (σx)=τ ( f ( σ x ) ) =(T f )(x)= f (x),

we obtain (6).

Finally, we show the last statement. Since g satisfies (5) and (6), we have

( T g ) ( x ) = τ ( g ( σ x ) ) = g ( σ x ) g ( σ x ) = g ( σ x σ x ) g ( σ x σ x ) = g ( x ) g ( σ x σ x ) = g ( x )

for all xX. This says that g is a fixed point of T. Also, by (8), we have g Δ f , φ . Thus the uniqueness of a fixed point of T in Δ f , φ implies that g= f . □

The next corollary is obtained in [6].

Corollary 1 ([[6], Corollary 3.2])

Let X be a set with a binary operation such that the square operation xxx 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:sss is an endomorphism of E with respect to . Let ε:X×X R + and suppose that α α σ , ε <, γ γ τ < and γα<1, where σ denotes the inverse mapping of the square operation xxx. If a mapping f:XE satisfies

d ( f ( x y ) , f ( x ) f ( y ) ) ε(x,y)(x,yX),

then there exists a unique mapping f :XE such that

f (xy)= f (x) f (y)andd ( f ( x ) , f ( x ) ) α 1 α γ ε(x,x)

for all x,yX.

Proof Consider the case that  =  and st=s for s,tE, in Theorem 1. In this case, τ is clearly an endomorphism of E with respect to . Therefore the corollary follows immediately from Theorem 1 with δ=0. □

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:

  1. (iv)

    The square operator σ ˜ :xxx is an endomorphism of X with respect to and .

  2. (v)

    The binary operations and on E are continuous. The square operator sss is an automorphism of E with respect to and . We denote by τ ˜ the inverse mapping of this automorphism.

  3. (vi)

    α ˜ α σ ˜ , ε <, β ˜ β σ ˜ , δ <, γ ˜ γ τ ˜ <, and γ ˜ max{ α ˜ , β ˜ }<1.

Under the above conditions, we show the Ulam type stability for the commutative diagram (1), as follows.

Theorem 2 Let (X,,) and (E,d,,) be as above. Suppose that four mappings σ ˜ :XX, τ ˜ :EE, ε:X×X R + and δ:X R + satisfy (iv), (v), and (vi). If a mapping f:XE satisfies (3) and

d ( f ( x x ) f ( x x ) , f ( x x ) ) δ(x)(xX),
(12)

then there exists a mapping f :XE satisfying (5)

f (xx) f (xx)= f (xx)(xX),
(13)
d ( f ( x ) , f ( x ) ) γ ˜ { ε ( x , x ) + δ ( x ) } 1 γ ˜ max { α ˜ , β ˜ } (xX).
(14)

Moreover, if a mapping g:XE satisfies (13), (14), and

K g 0:d ( f ( x ) , g ( x ) ) K g γ ˜ { ε ( x , x ) + δ ( x ) } (xX),
(15)

then g= f .

Proof For simplicity, we write T ˜ = T σ ˜ , τ ˜ , that is, ( T ˜ f)(x)= τ ˜ (f( σ ˜ x)) for xX. We note that α ˜ , β ˜ and γ ˜ are finite by (vi). Suppose that f:XE satisfies (3) and (12). Put φ ˜ (x)= γ ˜ {ε(x,x)+δ(x)} for all xX. To apply Lemma A to f and φ ˜ , we first observe that T ˜ f Δ f , φ ˜ . Fix xX. Since τ ˜ (f(x)f(x))=f(x), it follows from (3) and (12) that

d ( ( T ˜ f ) ( x ) , f ( x ) ) = d ( τ ˜ ( f ( σ ˜ x ) ) , f ( x ) ) = d ( τ ˜ ( f ( x x ) ) , τ ˜ ( f ( x ) f ( x ) ) ) γ ˜ d ( f ( x x ) , f ( x ) f ( x ) ) γ ˜ { d ( f ( x x ) , f ( x x ) f ( x x ) ) + d ( f ( x x ) f ( x x ) , f ( x ) f ( x ) ) } γ ˜ { δ ( x ) + ε ( x , x ) } = φ ˜ ( x ) .

Hence T ˜ f Δ f , φ ˜ and ρ f , φ ˜ ( T ˜ f,f)1.

We next estimate the quantity β σ ˜ , φ ˜ . For xX, we have

φ ˜ ( σ ˜ x ) = γ ˜ { ε ( σ ˜ x , σ ˜ x ) + δ ( σ ˜ x ) } γ ˜ { α ˜ ε ( x , x ) + β ˜ δ ( x ) } γ ˜ max { α ˜ , β ˜ } { ε ( x , x ) + δ ( x ) } = max { α ˜ , β ˜ } φ ˜ ( x ) .

Hence β σ ˜ , φ ˜ max{ α ˜ , β ˜ } and β σ ˜ , φ ˜ γ τ ˜ γ ˜ max{ α ˜ , β ˜ }<1 by (vi).

Thus we can apply Lemma A. As a consequence, T ˜ has a unique fixed point f Δ f , φ ˜ . Moreover,

lim n d ( ( T ˜ n f ) ( x ) , f ( x ) ) =0
(16)

and

d ( f ( x ) , f ( x ) ) ρ f , φ ˜ ( T ˜ f , f ) 1 β σ ˜ , φ ˜ γ τ ˜ φ ˜ (x)
(17)

for all xX. Since ρ f , φ ˜ ( T ˜ f,f)1 and β σ ˜ , φ ˜ γ τ ˜ γ ˜ max{ α ˜ , β ˜ }<1, (17) implies (14).

Here we show (5). If x,yX and nN, then we have

d ( f ( x y ) f ( x y ) , f ( x ) f ( y ) ) d ( f ( x y ) f ( x y ) , ( T ˜ n f ) ( x y ) ( T ˜ n f ) ( x y ) ) + d ( ( T ˜ n f ) ( x y ) ( T ˜ n f ) ( x y ) , ( T ˜ n f ) ( x ) ( T ˜ n f ) ( y ) ) + d ( ( T ˜ n f ) ( x ) ( T ˜ n f ) ( y ) , f ( x ) f ( y ) ) .

Letting n, the first and third terms on the right hand side tend to 0, because of (16) and the continuity of and in (v). Moreover, the second term, say A ˜ n (x,y), is estimated as follows: By (iv), (v), and (3),

A ˜ n ( x , y ) = d ( τ ˜ n ( f ( σ ˜ n ( x y ) ) ) τ ˜ n ( f ( σ ˜ n ( x 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 ˜ n α ˜ n ε ( x , y ) ,

where τ ˜ n and σ ˜ n denote the n-fold compositions of endomorphisms τ ˜ and σ ˜ , respectively. Since γ ˜ α ˜ <1 by (vi), it follows that A ˜ n (x,y)0 as n. Thus we obtain (5).

Next, we show (13). Replacing y in (5) by x, we have

f (xx) f (xx)= f (x) f (x).
(18)

Also since

τ ˜ ( f ( x x ) ) = τ ˜ ( f ( σ ˜ x ) ) =( T ˜ f )(x)= f (x)= τ ˜ ( f ( x ) f ( x ) ) ,

it follows that

f (xx)= f (x) f (x).

Combining with (18), we obtain (13).

Finally, we show the last statement. Since g satisfies (14) and (13), we have

g( σ ˜ x)=g(xx)=g(xx)g(xx)=g(x)g(x)= τ ˜ 1 ( g ( x ) ) ,

that is, ( T ˜ g)(x)=g(x) for all xX. This says that g is a fixed point of T ˜ . Also, by (15), we have g Δ f , φ ˜ . Hence the uniqueness of a fixed point of T ˜ in Δ f , φ ˜ implies that g= f . □

The next corollary is obtained in [6].

Corollary 2 ([[6], Corollary 3.5])

Let X be a set with a binary operation such that the square operation σ ˜ :xxx is an endomorphism of X with respect to and E a complete metric space with a continuous binary operation such that the square operation sss is an automorphism of E with respect to . Let ε:X×X R + and suppose that α ˜ α σ ˜ , ε <, γ ˜ γ τ ˜ < and γ ˜ α ˜ <1, where τ ˜ denotes the inverse mapping of the square operation sss. If a mapping f:XE satisfies

d ( f ( x y ) , f ( x ) f ( y ) ) ε(x,y)(x,yX),

then there exists a unique mapping f :XE such that

f (xy)= f (x) f (y)andd ( f ( x ) , f ( x ) ) γ ˜ 1 α ˜ γ ˜ ε(x,x)

for all x,yX.

Proof Consider the case that  =  and st=s for s,tE, in Theorem 2. Then τ ˜ is clearly an endomorphism of E with respect to . Therefore the corollary follows immediately from Theorem 2 with δ=0. □

3 Application I

The Ulam type stability problem for Euler-Lagrange type additive mappings has been investigated in [17]. Here we take up the following Euler-Lagrange type mapping f:XE satisfying

f(ax+by)+f(bx+ay)+(a+b) ( f ( x ) + f ( y ) ) =0(x,yX),
(19)

where X is a complex normed space, E a complex Banach space and a,bC with a+b0. The following is an Ulam type stability result for this mapping.

Corollary 3 (cf. [[17], Theorem 2.1])

Let ε:X×X R + and suppose that

(vii) K0:|a+b|K<1 and ε(x,y)Kε((a+b)x,(a+b)y) (x,yX).

If a mapping f:XE satisfies

f ( a x + b y ) + f ( b x + a y ) + ( a + b ) ( f ( x ) + f ( y ) ) ε(x,y)(x,yX),
(20)

then there exists a unique mapping f :XE satisfying (19) and

f ( x ) f ( x ) K 2 ( 1 | a + b | K ) ε(x,x)(xX).
(21)

Proof Put u=x, v=y for each x,yX. Under these transformations, (20) changes into the following estimate:

1 2 { f ( a u b v ) + f ( b u a v ) } + a + b 2 { f ( u ) + f ( v ) } ε 1 (u,v)(u,vX),
(22)

where ε 1 (u,v)= 1 2 ε(u,v) (u,vX).

Now we define uv=aubv, uv=buav for each u,vX. In this case, we can easily see that the square operator uuu is an endomorphism of X with respect to and . Also since a+b0, this endomorphism is bijective and so automorphic. We denote by σ the inverse mapping of this automorphism. Moreover, we define st= 1 2 (a+b)(s+t), st= 1 2 (s+t) for each s,tE. Then we can also see that the binary operations and on E are continuous and the square operator τ:sss is an automorphism of E with respect to and . Note that (22) changes into the following:

f ( u v ) f ( u v ) f ( u ) f ( v ) ε 1 (u,v)(u,vX).
(23)

Since xx=xx for all xX, it follows that σxσx=σxσx= σ 1 σx=x for all xX. Also, since ss=s for all sE, it follows that f(x)f(σxσx)=f(x)f(x)=f(x) for all xX and then (4) holds with δ=0. Moreover, β σ , δ =0 must hold with δ=0. It is also obvious that γ τ =|a+b| from the definition of τ. We also note that α σ , ε 1 K from the second condition of (vii) and hence γ τ α σ , ε 1 |a+b|K<1 from the first condition of (vii). Therefore, by Theorem 1, there exists a unique mapping f :XE such that

f (uv) f (uv)= f (u) f (v)(u,vX),

namely, (19) holds and

f ( u ) f ( u ) α σ , ε 1 ε 1 ( u , u ) 1 γ τ max { α σ , ε 1 , β σ , δ } K 2 ( 1 | a + b | K ) ε(u,u)(uX),

and so (21) holds. □

The following is also an Ulam type stability result for the mapping satisfying (19).

Corollary 4 (cf. [[17], Theorem 2.2])

Let ε:X×X R + and suppose that

  1. (viii)

    K0:K<|a+b| and ε((a+b)x,(a+b)y)Kε(x,y) (x,yX).

If a mapping f:XE satisfies (20), then there exists a unique mapping f :XE satisfying (19) and

f ( x ) f ( x ) 1 2 ( | a + b | K ) ε(x,x)(xX).
(24)

Proof As observed in the proof of Corollary 3, (20) changes into (22). Now we define uv=aubv, uv=buav for each u,vX. In this case, we can easily see that the square operator σ ˜ :uuu is an endomorphism of X with respect to and . Moreover, we define st= 1 2 (a+b)(s+t), st= 1 2 (s+t) for each s,tE. Then we can also see that the binary operations and on E are continuous and the square operator sss is an endomorphism of E with respect to and . Also since a+b0, this endomorphism is bijective and so automorphic. We denote by τ ˜ the inverse mapping of this automorphism. Note that (22) changes into (23). Since xx=xx (xX) and ss=s (sE), it follows that f(xx)f(xx)=f(xx) for all xX and then (12) holds with δ=0.

Moreover, β σ ˜ , δ =0 must hold with δ=0. It is also obvious that γ τ ˜ = | a + b | 1 from the definition of τ ˜ . We also note that α σ ˜ , ε 1 K from the second condition of (viii) and hence γ τ ˜ α σ ˜ , ε 1 | a + b | 1 K<1 from the first condition of (viii).

Therefore, by Theorem 2, there exists a unique mapping f :XE such that

f (uv) f (uv)= f (u) f (v)(u,vX),

namely, (19) holds and

f ( u ) f ( u ) γ τ ˜ ε 1 ( u , u ) 1 γ τ ˜ max { α σ ˜ , ε 1 , β σ ˜ , δ } | a + b | 1 2 ( 1 | a + b | 1 K ) ε ( u , u ) = 1 2 ( | a + b | K ) ε ( u , u ) ( u X ) ,

and so (24) holds. □

Corollary 5 (cf. [[17], Corollary 2.3])

Suppose that |a+b|1, δ,p,q0 and p+q1. If a mapping f:XE satisfies

f ( a x + b y ) + f ( b x + a y ) + ( a + b ) { f ( x ) + f ( y ) } δ x p y q

for all x,yX, then there exists a unique mapping f :XE satisfying (19) and

f ( x ) f ( x ) δ 2 ( | | a + b | p + q | a + b | | x p + q (xX).

Proof Put ε(x,y)=δ x p y q for each x,yX.

  1. (a)

    The case where either

    { | a + b | > 1 , p + q > 1 ,

or

{ | a + b | < 1 , p + q < 1 .

Put K= | a + b | ( p + q ) . Then K satisfies (vii). Note also that

K 2 ( 1 | a + b | K ) ε(x,x)= δ 2 ( | a + b | p + q | a + b | ) x p + q

for all xX. Then the desired result follows from Corollary 3.

  1. (b)

    The case where either

    { | a + b | > 1 , p + q < 1 ,

or

{ | a + b | < 1 , p + q > 1 .

Put K= | a + b | p + q . Then K satisfies (viii). Note also that

1 2 ( | a + b | K ) ε(x,x)= δ 2 ( | a + b | | a + b | p + q ) x p + q

for all xX. Then the desired result follows from Corollary 4. □

4 Application II

Let (X,+) be an Abelian group. In [18], the following result has been shown by A. Simon and P. Volkmann.

Lemma B ([[18], Théorème 1)]

A mapping f:XR satisfies

max { f ( x + y ) , f ( x y ) } =f(x)+f(y)(x,yX),
(25)

if and only if f(x)=|π(x)| (xX) for some additive function π:XR.

In this section, we deal with the Ulam type stability problem for Equation (25). Put xy=x+y and xy=xy for each x,yX. Moreover, put st=s+t and st=max{s,t} for each s,tR. Then (25) changes into (2). Also we can easily see that the square operation σ ˜ :xxx is endomorphic with respect to and and that the square operator sss is automorphic with respect to and . Denote by τ ˜ the inverse mapping of this automorphism. In this case, it is obvious that τ ˜ (s)= 1 2 s for each sR and hence γ τ ˜ =1/2.

Now let ε be a nonnegative constant and suppose that f:XR satisfies

| max { f ( x + y ) , f ( x y ) } { f ( x ) + f ( y ) } | ε(x,yX).
(26)

Putting x=y=0 in (26), we obtain

| f ( 0 ) | ε.
(27)

Also, putting x=y in (26), we obtain

ε+f(0)ε+max { f ( x + x ) , f ( 0 ) } 2f(x)(xX).
(28)

Combining (27) and (28), we obtain

εf(x)(xX).
(29)

Put δ=2ε. By (27) and (28), we obtain

0max { f ( x + x ) , f ( 0 ) } f(x+x)ε+ε=δ(xX),

and hence (12) holds. Moreover, note that α σ ˜ , ε = β σ ˜ , δ =1 since ε and δ are constant. Then Lemma B and Theorem 2 easily imply the following.

Corollary 6 Let X be an Abelian group and ε a nonnegative constant. If f:XR satisfies (26), then there exists an additive mapping π:XR such that

| f ( x ) | π ( x ) | | 3ε(xX).

For the related results, see [19, 20].

References

  1. 1.

    Brillouët-Belluot N, Brzdȩk J, Ciepliński K: On some recent developments in Ulam’s type stability. Abstr. Appl. Anal. 2012. Article ID 716936, 2012: Article ID 716936

    Google Scholar 

  2. 2.

    Brzdȩk J: Hyperstability of the Cauchy equation on restricted domains. Acta Math. Hung. 2013, 141: 58-67. 10.1007/s10474-013-0302-3

    MathSciNet  Article  Google Scholar 

  3. 3.

    Brzdȩk J, Ciepliński K: Hyperstability and superstability. Abstr. Appl. Anal. 2013. Article ID 401756, 2013: Article ID 401756

    Google Scholar 

  4. 4.

    Gajda Z: On stability of additive mappings. Int. J. Math. Math. Sci. 1991, 14: 431-434. 10.1155/S016117129100056X

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Jung S-M Springer Optimization and Its Applications 48. In Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, New York; 2011.

    Google Scholar 

  6. 6.

    Takahasi S-E, Miura T, Takagi H: On a Hyers-Ulam-Aoki-Rassias type stability and a fixed point theorem. J. Nonlinear Convex Anal. 2010, 11: 423-439.

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Bahyrycz A, Piszczek M: Hyperstability of the Jensen functional equation. Acta Math. Hung. 2014, 142: 353-365. 10.1007/s10474-013-0347-3

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Brzdȩk J: Stability of the equation of the p -Wright affine functions. Aequ. Math. 2013, 85: 497-503. 10.1007/s00010-012-0152-z

    MathSciNet  Article  Google Scholar 

  9. 9.

    Brzdȩk J: A hyperstability result for the Cauchy equation. Bull. Aust. Math. Soc. 2014, 89: 33-40. 10.1017/S0004972713000683

    MathSciNet  Article  Google Scholar 

  10. 10.

    Gilányi A, Kaiser Z, Páles Z: Estimates to the stability of functional equations. Aequ. Math. 2007, 73: 125-143. 10.1007/s00010-006-2854-6

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    Kim GH: On the stability of functional equations with square-symmetric operation. Math. Inequal. Appl. 2001, 4: 257-266.

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Kim GH: Addendum to ‘On the stability of functional equations on square-symmetric groupoid’. Nonlinear Anal. 2005, 62: 365-381. 10.1016/j.na.2004.12.003

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Páles Z: Hyers-Ulam stability of the Cauchy functional equation on square-symmetric groupoids. Publ. Math. (Debr.) 2001, 58: 651-666.

    MATH  MathSciNet  Google Scholar 

  14. 14.

    Páles Z, Volkmann P, Luce RD: Hyers-Ulam stability of functional equations with square symmetric operations. Proc. Natl. Acad. Sci. USA 1998, 95: 12772-12775. 10.1073/pnas.95.22.12772

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Piszczek M: Remark on hyperstability of the general linear equation. Aequ. Math. 2013. 10.1007/s00010-013-0214-x

    Google Scholar 

  16. 16.

    Ciepliński K: Applications of fixed point theorems to the Hyers-Ulam stability of functional equations-a survey. Ann. Funct. Anal. 2012, 3: 151-164. 10.15352/afa/1399900032

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Kim H-M, Jun K-W, Rassias JM: Extended stability problem for alternative Cauchy-Jensen mappings. J. Inequal. Pure Appl. Math. 2007. Article ID 120, 8: Article ID 120

    Google Scholar 

  18. 18.

    Simon A, Volkmann P: Caractérisation du module d’une fonction á l’aide d’une équation fonctionnelle. Aequ. Math. 1994, 47: 60-68. 10.1007/BF01838140

    Article  Google Scholar 

  19. 19.

    Gilányi A, Nagatou K, Volkmann P: Stability of a functional equation coming from the characterization of the absolute value of additive functions. Ann. Funct. Anal. 2010, 1: 1-6.

    Article  MATH  MathSciNet  Google Scholar 

  20. 20.

    Jarczyk W, Volkmann P: On functional equations in connection with the absolute value of additive functions. Series Mathematicae Catoviciensis et Debreceniensis 2010, 32: 1-11.

    Google Scholar 

Download references

Acknowledgements

The authors are deeply grateful to the referees for careful reading of the paper and for the helpful suggestions and comments. All authors are partially supported by Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Sin-Ei Takahasi.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to this paper. They read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Takahasi, SE., Tsukada, M., Miura, T. et al. Ulam type stability problems for alternative homomorphisms. J Inequal Appl 2014, 228 (2014). https://doi.org/10.1186/1029-242X-2014-228

Download citation

Keywords

  • Ulam type stability
  • homomorphism
  • binary operation
  • fixed point theorem