Badora (2002) proved the following stability result. Let ε and δ be nonnegative real numbers, then for every mapping f of a ring ℛ onto a Banach algebra ℬ satisfying ||f(x+y)-f(x)-f(y)||≤ε and ||f(x·y)-f(x)f(y)||≤δ for all x,y∈ℛ, there exists a unique ring homomorphism h:ℛ→ℬ such that ||f(x)-h(x)||≤ε, x∈ℛ. Moreover, b·(f(x)-h(x))=0, (f(x)-h(x))·b=0, for all x∈ℛ and all b from the algebra generated by h(ℛ). In this paper, we generalize Badora's stability result above on ring homomorphisms for Riesz algebras with extended norms.

1. Introduction

The approximation of solution of the Cauchy's equation f(x+y)=f(x)+f(y) lying near to some solution has received a lot of attention from mathematicians in the areas of modern analysis and applied mathematics. Any solution f of this equation is called an additive function. Let E and E′ be Banach spaces, and let δ be a positive number. A function f of E into E′ is called δ-additive if ∥f(x+y)-f(x)-f(y)∥<δ for all x,y∈E. In the 1940s, Ulam [1] proposed the following stability problem of this equation. Does there exist for each ε>0 a δ>0 such that, to each δ-additive function f of E into E′ there, corresponds an additive function l of E into E′ satisfying the inequality ∥f(x)-l(x)∥≤ε for each x∈E? In 1941, Hyers [2] answered this question in the affirmative way and showed that δ may be taken equal to ε. The answer of Hyers is presented in a great number of articles and books. There are several definitions and critics of the notion of this stability in the literature (see, e.g., [3, 4]). In 1949, Bourgin [5] generalized Hyers' results to the ring homomorphisms and proved the following.

Theorem 1.1.

Let ε and δ be nonnegative real numbers. Then every mapping f of a Banach algebra 𝒜 with an identity element onto a Banach algebra ℬ with an identity element satisfying
‖f(x+y)-f(x)-f(y)‖≤ε,‖f(x⋅y)-f(x)f(y)‖≤δ,
for all x,y∈𝒜, is a ring homomorphism of 𝒜 onto ℬ, that is, f(x+y)=f(x)+f(y) and f(x·y)=f(x)f(y) for all x,y∈𝒜.

Finally, Badora [6] proved the following theorem on the Bourgin's result related to stability problem (Theorem 1.1) without additional assumptions.

Theorem 1.2.

Let ℛ be a ring, let ℬ be a Banach algebra, and let ε and δ be nonnegative real numbers. Assume that f:ℛ→ℬ satisfies (1.1) for all x,y∈E. Then there exists a unique ring homomorphism h:ℛ→ℬ such that
‖f(x)-h(x)‖≤ε,x∈R.
Moreover,
b⋅(f(x)-h(x))=0,(f(x)-h(x))⋅b=0,
for all x∈ℛ and all b from the algebra generated by h(R).

The present paper is in essence a revised and extended compilation of Hyers' result and Theorem 1.2 to the Riesz algebras with extended norms. After outlining the basic information on Riesz space theory, we present the main definitions and facts concerning approximate Riesz algebra (with an extended norm)-valued ring homomorphisms.

2. Preliminaries

A real Banach space F=(F,+,·,∥·∥) endowed with a (partial) order ≤ is called a Banach lattice whenever

the order ≤ agrees with the linear operations, that is, x≤y⇒αx+z≤αy+z for all z∈F and 0≤α∈ℝ;

the order ≤ makes F a lattice, that is, for all x,y∈F, the supremum x∨y and infimum x∧y exist in F (hence, the modulus |x|:=x∨(-x) exists for each x∈F);

the norm ∥·∥ is monotonous with respect to the order ≤, that is, for all x,y∈F, |x|≤|y| implies ∥x∥≤∥y∥ (hence, ∥x∥=∥|x|∥ for all x∈F).

Recall that a (partially) ordered vector space F satisfying (1) and (2) above is called a Riesz space. C(K) the spaces of real valued continuous functions on a compact Hausdorff space K, lp-spaces, c the spaces of convergent sequences, and c0 the spaces of sequences converging to zero are natural examples of Riesz spaces under the pointwise ordering. A Riesz space F is called Archimedean if 0≤u,v∈F, and nu≤v for each n∈ℕ imply u=0. Throughout the present paper, all the Riesz spaces are assumed to be Archimedean. A subset S in a Riesz space F is said to be solid if it follows from |u|≤|v| in F and v∈S that u∈S. A solid linear subspace of a Riesz space F is called an ideal. Every subset D of a Riesz space F is included in a smallest ideal FD, called ideal generated by D. A principal ideal of a Riesz space F is any ideal generated by a singleton {u}. This ideal will be denoted by Iu. It is easy to see thatIu={v∈F:λ≥0suchthat|v|≤λ|u|}.

We assume that u is a fixed positive element in the Riesz space F. First of all, we present the following definition.

Definition 2.1.

(1) It is said that the sequence (xn) in F converges u-uniformly to the element x∈F whenever, for every ε>0, there exists n0 such that |xn0+k-x|≤εu holds for each k.

(2) It is said that the sequence (xn) in F converges relatively uniformly to x whenever xn converges u-uniformly to x for some 0≤u∈F.

When dealing with relative uniform convergence in an Archimedean Riesz space F, it is natural to associate with every positive element u∈F an extended norm ∥·∥u in F by the formula
‖x‖u=inf{λ≥0:|x|≤λu}(x∈F).

Note that ∥x∥u<∞ if and only if x∈Iu, the ideal generated by u. Also |x|≤δu if and only if ∥x∥u≤δ.

The sequence (xn) in F is called an extended u-normed Cauchy sequence, if for every ε>0 there exists k such that ∥xn+k-xm+k∥u<ε for all m,n. If every extended u-normed Cauchy sequence is convergent in F, then F is called an extended u-normed Banach lattice.

A Riesz space F is called a Riesz algebra or a lattice-ordered algebra if there exists in F an associative multiplication with the usual algebra properties such that uv≥0 for all 0≤u,v∈F.

For more detailed information about Riesz spaces, the reader can consult the book “Riesz Spaces” by Luxemburg and Zaanen [7].

3. Main Results

We begin with the following theorem concerning stability of the functional equation HofoG=f. For a function G:K→K, let us denote by G0(x)=x for x∈K, G2 the composition of G by itself and in general let Gi=Go(Gi-1) for i=1,2,….

The theorem can easily be obtained from [8] or [9]. We give the proof here for the benefit of the reader.

Theorem 3.1.

Let (Y,d) be a complete metric space, K a nonempty set and λ∈[0,∞) such that G:K→K and H:Y→Y are two given functions. Assume that f:K→Y is a function satisfying
d(HofoG(x),f(x))≤h(x),
for each x∈K and for some function h:K→[0,∞). If the function H:Y→Y satisfies the inequality
d(H(u),H(v))≤λd(u,v),u,v∈Y,
and the series
∑i=0∞λih(Gi(x))
is convergent for each x∈K, then for each integer n, one has

(1)
d(Hn+1ofoGn+1(x),HnofoGn(x))≤λnh(Gn(x)),x∈K,

(2) HnofoGn(x),(x∈K) is a Cauchy sequence. F(x)=limn→∞HnofoGn(x) exists for every x∈K, and F:K→Y is the unique function satisfying HoFoG=F and the inequality
d(F(x),f(x))≤∑i=0∞λih(Gi(x)).

Proof.

(1) Replacing x by G(x) in (3.1), we get
d(HofoG2(x),f(G(x)))≤h(G(x)).
Then by (3.2), we obtain
d(H2ofoG2(x),HofoG(x))≤λd(HofoG2(x),foG(x))≤λh(G(x)).
The proof follows by induction.

(2) Let m>n, then
d(HnofoGn(x),HmofoGm(x))≤∑i=nm-1d(Hi+1ofoGi+1(x),HiofoGi(x))≤∑i=nm-1λih(Gi(x)),
thus HnofoGn(x) is a Cauchy sequence for each x∈K and it is convergent as (Y,d) is complete. Let F(x)=limn→∞HnofoGn(x) for each x∈K.

By using (3.4), we get
d(HnofoGn(x),f(x))≤∑i=1nd(HiofoGi(x),Hi-1ofoGi-1(x))≤∑i=1nλi-1h(Gi-1(x))=∑i=0nλih(Gi(x)),
taking the limit as n goes to infinity, then we obtain (3.5). By continuity of H, we have
HoFoG(x)=H[limn→∞HnofoGn(G(x))]=limn→∞HoHnofoGn(G(x))=limn→∞Hn+1ofoGn+1(x)=F(x).
Suppose that another function F̅:K→Y satisfies HoF̅oG(x)=F̅ and (3.5). By induction it is easy to show that HnoFoGn(x)=F and HnoF̅oGn(x)=F̅. Hence for x∈K, d(F̅(x),F(x))=d(HnoF̅oGn(x),HnoFoGn(x))≤λnd(F̅oGn(x),FoGn(x))≤λnd(F̅oGn(x),foGn(x))+λnd(foGn(x),FoGn(x))≤2∑i=n∞λnh(Gi(x)).
Since for every x∈K, ∑i=n∞h(Gi(x))→0 with n→∞, this completes the proof.

Let Y be a linear space over either complex or real numbers. The operation of addition of elements x,y∈Y will be denoted, as usual, by x+y. The operation of multiplication of an element x∈Y by a scalar t will be denoted by tx. Suppose that in the linear space Y, we are given a metric d. The space (Y,d) is called a metric linear space if the operations of addition and multiplication by numbers are continuous with respect to the metric d. A metric linear space (Y,d) is called complete if every Cauchy sequence (xn) converges to an element x0∈Y, that is, limn→∞d(xn,x0)=0.

We now give the following corollary in [9] which will be useful in the sequel.

Corollary 3.2.

Let (Y,d) a complete metric linear space and K be a linear space. Suppose that there exists ξ,η∈[0,∞) such that ξη<1,
d(12x,12y)≤ξd(x,y)forx,y∈Y,χ(2x,2y)≤ηχ(x,y)forx,y∈K,
where χ:K×K→[0,∞). Let φ:K→Y satisfy
d(φ(x+y),φ(x)+φ(y))≤χ(x,y)forx,y∈K.
Then there is a unique solution F:K→Y of F(x+y)=F(x)+F(y) with
d(φ(x),F(x))≤ξχ(x,x)1-ξηforx∈K.

Proof.

From (3.12) and (3.14), we get
d(12φ(2x),φ(x))≤ξd(φ(2x),2φ(x))≤ξχ(x,x)
for x∈K. By using Theorem 3.1 with f=φ,H(z)=(1/2)z,G(x)=2x, λ=ξ, and h(x)=ξχ(x,x), the limit function F(x) exists for each x∈K and
d(f(x),F(x))≤∑i=0∞ξih(Gi(x)).

As Gi(x)=2ix and h(Gi(x))≤ξηiχ(x,x) for every x∈K, we get
d(φ(x),F(x))=d(f(x),F(x))≤ξχ(x,x)∑i=0∞(ξη)i=ξχ(x,x)1-ξη.
Next, by (3.14), for every x,y∈K we have
d(12nφ(2n(x+y)),12nφ(2nx)+12nφ(2nx))≤ηnχ(x,y)
for n∈ℕ, so letting n→∞ we obtain F(x+y)=F(x)+F(y).

Suppose F0:K→Y is also a solution of F(x+y)=F(x)+F(y) and
d(φ(x),F0(x))≤∑i=0∞ξih(Gi(x)),foreveryx∈K.
Then HoF0oG=F0, whence, by Theorem 3.1, we have F=F0 which implies the uniqueness of F.

The following theorem is an extended application of Hyers' result to the Riesz spaces.

Theorem 3.3.

Let E a linear space, F be a Riesz space equipped with an extended norm ∥·∥u such that the space (F,∥·∥u) is complete. If, for some δ>0, a map f:E→(F,∥·∥u) is δ-additive, then limit l(x)=limn→∞f(2nx)/2n exists for each x∈E. l(x) is the unique additive function satisfying the inequality ∥f(x)-l(x)∥u≤δ for all x∈E.

Now, if F is a Banach space or extended u-normed Banach lattice, then we can take d(x,y)=∥x-y∥ or d(x,y)=∥x-y∥u, χ(x,x)=ε, ξ=1/2, and η=1. We may obtain the classical Hyers' result [2] and Theorem 3.3 with such χ(x,x), ξ and η by using Corollary 3.2.

Finally, we give the following theorem which is an extended application of Badora's result (Theorem 1.2) to Riesz algebras with extended norms. For a proof, we use Theorem 3.3 and the similar techniques of Badora [6] with suitable modifications.

Theorem 3.4.

Let E be a linear algebra, and let F be a Riesz algebra with an extended norm ∥·∥u such that (F,∥·∥u) is complete. Also, let ∥·∥v be another extended norm in F weaker than ∥·∥u such that whenever

xn→x and xn·y→z in ∥·∥v, then z=x·y;

yn→y and x·yn→z in ∥·∥v, then z=x·y.

Let ε and δ be nonnegative real numbers. Assume that a map f:E→F satisfies
‖f(x+y)-f(x)-f(y)‖u≤ε,‖f(x⋅y)-f(x)f(y)‖v≤δ
for all x,y∈E. Then there exists a unique ring homomorphism h:E→F such that ||f(x)-h(x)||u≤ε, x∈E. Moreover,
b⋅(f(x)-h(x))=0,(f(x)-h(x))⋅b=0,
for all x∈E and all b from the algebra generated by h(E).Proof.

From Theorem 3.3, it follows that there exists a unique additive function h:E→F such that
‖f(x)-h(x)‖u≤ε,(x∈E).
Hence, it is enough to show that h is a multiplicative function. Using the additivity of h, it follows that
‖1nf(nx)-h(x)‖u≤1nε,(x∈E,n∈N),
which means that
h(x)=limn→∞1nf(nx),(x∈E),
with respect to ∥·∥u norm.

Let
r(x,y)=f(x⋅y)-f(x)f(y),(x,y∈E).
Then using inequality (3.22), we get
limn→∞1nr(nx,y)=0,(x,y∈E),
with respect to ∥·∥v norm.

Applying (3.26) and (3.28), we have
h(x⋅y)=limn→∞1nf(n(x⋅y))=limn→∞1nf((nx)⋅y)=limn→∞1n(f(nx)f(y)+r(nx,y))=h(x)f(y),
for all x,y∈E, since ∥·∥v is weaker than ∥·∥u. Hence, we get the following functional equation:
h(x⋅y)=h(x)f(y),(x,y∈E).
From this equation and the additivity of h, we have
h(x)f(ny)=h(x⋅(ny))=h((nx)⋅y)=h(nx)f(y)=nh(x)f(y),(x,y∈E,n∈N).
Therefore,
h(x)1nf(ny)=h(x)f(y),(x,y∈E,n∈N).
Sending n to infinity, by (3.26), we see that
h(x)h(y)=h(x)f(y),(x,y∈E).
Combining this equation with (3.30), we see that h is a multiplicative function.

Moreover, from (3.22) we get
limn→∞1nr(x,ny)=0,(x,y∈E),
with respect to ∥·∥v norm.

Thus, by (3.26) and the fact that ∥·∥v is weaker than ∥·∥u, we get that
h(x⋅y)=limn→∞1nf(n(x⋅y))=limn→∞1nf(x⋅(ny))=limn→∞(f(x)1nf(ny)+1nr(x,ny))=f(x)h(y),
for all x,y∈E. Hence, by (3.33),
f(x)h(y)=h(x⋅y)=h(x)h(y)=h(x)f(y),(x,y∈E),
so that
h(x)(f(y)-h(y))=0,(f(x)-h(x))h(y)=0,(x,y∈E)
which completes the proof.

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