Using the fixed point method, we prove the generalized Hyers-Ulam stability of C∗-algebra homomorphisms and of generalized derivations on C∗-algebras for the following functional equation of Apollonius
type _i=1n_f(z_xi)=_(1/n)___1_i<j_n_f(xi+xj)+nf(z_(1/n2)_i=1n_xi).

1. Introduction and Preliminaries

A classical question in the theory of functional equations is the following:
“when is it true that a function, which approximately satisfies a functional
equation ℰ, must be close to an exact solution of ℰ?” If the problem accepts a solution, we say that the
equation ℰ is stable. Such a problem was formulated by Ulam [1] in 1940 and solved in the next year for the
Cauchy functional equation by Hyers [2]. It
gave rise to the stability theory for functional equations. The result of Hyers was
extended by Aoki [3] in 1950 by considering
the unbounded Cauchy differences. In 1978, Rassias [4] proved that the additive mapping T, obtained by Hyers or Aoki, is linear if, in addition, for each x∈E, the mapping f(tx) is continuous in t∈ℝ. Găvruţ a [5]
generalized the Rassias' result. Following the techniques of the proof of the
corollary of Hyers [2], we observed that
Hyers introduced (in 1941) the following Hyers continuity condition about the
continuity of the mapping for each fixed point and then he proved homogeneity of
degree one and, therefore, the famous linearity. This condition has been assumed
further till now, through the complete Hyers direct method, in order to prove
linearity for generalized Hyers-Ulam stability problem forms (see [6]). Beginning around 1980, the stability
problems of several functional equations and approximate homomorphisms have been
extensively investigated by a number of authors and there are many interesting
results concerning this problem (see [7–21]).

Rassias [22], following the spirit of the
innovative approach of Hyers [2], Aoki [3], and Rassias [4] for the unbounded Cauchy difference, proved a similar stability
theorem in which he replaced the factor ∥x∥p+∥y∥p by ∥x∥p⋅∥y∥q for p,q∈ℝ with p+q≠1 (see also [23, 24] for a number of other new results).

In 2003, Cădariu and Radu applied the fixed-point method to the investigation
of the Jensen functional equation [25] (see
also [8, 26–30]). They could
present a short and a simple proof (different of the “direct
method,” initiated by Hyers in 1941) for the generalized
Hyers-Ulam stability of Jensen functional equation [25], for Cauchy functional equation [8], and for quadratic functional equation [26].

The following functional equation:Q(x+y)+Q(x−y)=2Q(x)+2Q(y)is called a quadratic functional equation, and every
solution of (1.1) is said to be a
quadratic mapping. Skof [31] proved the Hyers-Ulam stability of the quadratic functional equation
(1.1) for mappings f:E1→E2, where E1 is a normed space and E2 is a Banach space. In [32],
Czerwik proved the Hyers-Ulam stability of the quadratic functional equation (1.1). Borelli and Forti [33] generalized the stability result of the
quadratic functional equation (1.1). Jun and Lee [34] proved the
Hyers-Ulam stability of the Pexiderized quadratic equationf(x+y)+g(x−y)=2h(x)+2k(y)for mappings f,g,h, and k. The stability problem of the quadratic equation has been
extensively investigated by some mathematicians [35].

In an inner product space, the equality∥z−x∥2+∥z−y∥2=12∥x−y∥2+2∥z−x+y2∥2holds, then it is called the Apollonius' identity.
The following functional equation, which was motivated by this
equation,Q(z−x)+Q(z−y)=12Q(x−y)+2Q(z−x+y2),holds, then it is called quadratic (see [36]). For this reason, the functional equation
(1.4) is called a
quadratic functional equation of Apollonius type, and each
solution of the functional equation (1.4) is said to be a quadratic mapping of Apollonius
type. The quadratic functional equation and several other functional
equations are useful to characterize inner product spaces [37].

In [36], Park and Rassias introduced and
investigated a functional equation, which is called a generalized Apollonius
type quadratic functional equation. In [38], Najati introduced and investigated a functional equation, which is
called a quadratic functional equation of n-Apollonius type. Recently in [39], Park and Rassias introduced and investigated the following
functional equation:f(z−x)+f(z−y)=−12f(x+y)+2f(z−x+y4)which is called an Apollonius type additive functional
equation, and whose solution is called an Apollonius type
additive mapping. In [40], Park
introduced and investigated a functional equation, which is called a
generalized Apollonius-Jensen type additive functional equation
and whose solution is said to be a generalized Apollonius-Jensen type
additive mapping.

In this paper, employing the above equality (1.5), for a fixed positive integer n≥2, we introduce a new functional equation, which is called an
additive functional equation of n-Apollonius type and whose solution is said to be an
additive mapping of n-Apollonius type;∑i=1nf(z−xi)=−1n∑1≤i<j≤nf(xi+xj)+nf(z−1n2∑i=1nxi).

We will adopt the idea of Cădariu and Radu [8, 25, 28] to prove the generalized Hyers-Ulam stability results of C*-algebra homomorphisms as well as to prove the generalized
Ulam-Hyers stability of generalized derivations on C*-algebra for additive functional equation of n-Apollonius type.

We recall two fundamental results in fixed-point theory.

Theorem 1.1 (see [<xref ref-type="bibr" rid="B6">25</xref>]).

Let (X,d) be a complete metric space and let J:X→X be strictly contractive, that is,d(Jx,Jy)≤Lf(x,y),∀x,y∈Xfor some Lipschitz constant L<1. Then, the following hold:

the mapping J has a unique fixed point x*=Jx*;

the fixed point x* is globally attractive, that is, limn→∞Jnx=x* for any starting point x∈X;

one has the following estimation inequalities: d(Jnx,x*)≤Lnd(x,x*),d(Jnx,x*)≤11−Ld(Jnx,Jn+1x),d(x,x*)≤11−Ld(x,Jx) for all nonnegative integers n and all x∈X.

Let X be a set. A function d:X×X→[0,∞] is called a generalized metric on X if d satisfies the following:

d(x,y)=0 if and only if x=y;

d(x,y)=d(y,x) for all x,y∈X;

d(x,z)≤d(x,y)+d(y,z) for all x,y,z∈X.

Theorem 1.2 (see [<xref ref-type="bibr" rid="B10">41</xref>]).

Let (X,d) be a complete generalized metric space and let J:X→X be a strictly contractive mapping with Lipschitz constant L<1. Then for each given element x∈X, eitherd(Jnx,Jn+1x)=∞for all nonnegative integers n or there exists a positive integer n0 such that the following hold:

d(Jnx,Jn+1x)<∞ for all n≥n0;

he sequence {Jnx} converges to a fixed point y* of J;

y* is the unique fixed point of J in the set Y={y∈X∣d(Jn0x,y)<∞};

d(y,y*)≤(1/(1−L))d(y,Jy)for ally∈Y.

Throughout this paper, assume that A is a C*-algebra with norm ∥⋅∥A and that B is a C*-algebra with norm ∥⋅∥B.

2. Stability of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M80"><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mo>*</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>-Algebra HomomorphismsLemma 2.1.

Let X and Y be real-vector spaces. A mapping f:X→Y satisfies (1.6) for all x1,…,xn,z if and only if the mapping f is additive.

Proof.

Letting x1=⋯=xn=z=0 in (1.6), we
get that f(0)=0. Let j and k be fixed integers with 1≤j<k≤n. Setting xi=0 for all 1≤i≤n,i≠j,k in (1.6), we
havef(z−xj)+f(z−xk)+(n−2)f(z)=−1nf(xj+xk)−n−2n(f(xj)+f(xk))+nf(z−1n2(xj+xk))for all xj,xk,z∈X. Replacing xj by −xj and xk by xj in (2.1),
respectively, we getf(z+xj)+f(z−xj)=−n−2n(f(−xj)+f(xj))+2f(z)for all xj,z∈X. Putting z=0 in (2.2), we
conclude that f(−xj)=−f(xj) for all xj∈X. This means that f is an odd function. Letting xk=z=0 in (2.1) and
using the oddness of f, we obtain thatf(1n2xj)=1n2f(xj),f(n2xj)=n2f(xj)for all xj∈X. Letting z=0 in (2.1),
using the oddness of f and (2.3), we
havef(xj+xk)=f(xj)+f(xk)for all xj,xk∈X. Therefore, f:X→Y is an additive mapping.

The converse is obviously true.

For a given mapping f:A→B and for a fixed positive integer n≥2, we defineCμf(z,x1,…,xn):=∑i=1nμf(z−xi)+1n∑1≤i<j≤nf(μxi+μxj)−nf(μz−1n2∑i=1nμxi)for all μ∈𝕋1:={ν∈ℂ:|ν|=1} and all z,x1,…,xn∈A.

We prove the generalized Hyers-Ulam stability of C*-algebra homomorphisms for the functional equation Cμf(z,x1,…,xn)=0.

Theorem 2.2.

Let f:A→B be a mapping satisfying f(0)=0 for which there exists a function φ:An+1→[0,∞) such that ∑j=0∞(n2n2−1)2jφ((n2−1n2)jz,(n2−1n2)jx1,…,(n2−1n2)jxn)<∞,∥Cμf(z,x1,…,xn)∥B≤φ(z,x1,…,xn),∥f(xy)−f(x)f(y)∥B≤φ(x,y,0,…,0︸n−1times),∥f(x*)−f(x)*∥B≤φ(x,…,x︸n+1times) for all μ∈𝕋1 and all x,y,z,x1,…,xn∈A. If for some 1≤j≤n there exists a Lipschitz constant L<1 such that φ(x,0,…,0,x︸jth,0,…,0)≤n2−1n2Lφ(n2n2−1x,0,…,0,n2n2−1x︸jth,0,…,0) for all x∈A, then there exists a unique C*-algebra homomorphism H:A→B such that ∥f(x)−H(x)∥B≤n(n2−1)×(1−L)φ(x,0,…,0,x︸jth,0,…,0)for all x∈A.

Proof.

Consider the setX:={g:A⟶B,g(0)=0}and introduce the generalized metric on X:d(g,h)=inf{C∈ℝ+:∥g(x)−h(x)∥B≤Cφ(x,0,…,0,x︸jth,0,…,0)∀x∈A}.It is easy to show that (X,d) is complete.

For convenience, setφj(x,y):=φ(x,0,…,0,y︸jth,0,…,0)for all x,y∈A and all 1≤j≤n.

Now we consider the linear mapping J:X→X such thatJg(x):=nαg(αnx)for all x∈A, where α=(n2−1)/n.

For any g,h∈X, we haved(g,h)<C⟹∥g(x)−h(x)∥B≤Cφj(x,x)∀x∈A⟹∥nαg(αnx)−nαh(αnx)∥B≤nαCφj(αnx,αnx)⟹∥nαg(αnx)−nαh(αnx)∥B≤LCφj(x,x)⟹d(Jg,Jh)≤LC.

Therefore, we see thatd(Jg,Jh)≤Ld(g,h),∀g,h∈A.This means J is a strictly contractive self-mapping of X, with the Lipschitz constant L.

Letting μ=1,z=xj=x, and for each 1≤k≤n with k≠j,xk=0 in (2.7), we
get∥αf(x)−nf(αnx)∥B≤φj(x,x)for all x∈A. So∥f(x)−nαf(αnx)∥B≤1αφj(x,x)for all x∈A. Hence d(f,Jf)≤1/α.

By Theorem 1.2, there exists a
mapping H:A→B such that the following hold:

(1) H is a fixed point of J, that is,H(αnx)=αnH(x)for all x∈A; the mapping H is a unique fixed point of J in the setY={g∈X:d(f,g)<∞};and this implies that H is a unique mapping satisfying (2.20) such that there exists C∈(0,∞) satisfying∥H(x)−f(x)∥B≤Cφj(x,x)for all x∈A.

(2) d(Jmf,H)→0 as m→∞; and this implies the equalitylimm→∞(nα)mf((αn)mx)=H(x)for all x∈A;

(3) d(f,H)≤(1/(1−L))d(f,Jf), which implies the inequalityd(f,H)≤1α−αL;and this implies that the inequality (2.11) holds.

It follows from (2.6), (2.7), and (2.23) that∥∑i=1nH(z−xi)+1n∑1≤i<j≤nH(xi+xj)−nH(z−1n2∑i=1nxi)∥B=limm→∞(nα)m∥∑i=1nf((αn)m(z−xi))+1n∑1≤i<j≤nf((αn)m(xi+xj))−nf((αn)mz−(αn)m×1n2∑i=1nxi)∥B≤limm→∞(nα)mφ((αn)mz,(αn)mx1,…,(αn)mxn)≤limm→∞(nα)2mφ((αn)mz,(αn)mx1,…,(αn)mxn)=0for all x1,…,xn,z∈A. So∑i=1nH(z−xi)=−1n∑1≤i<j≤nH(xi+xj)+nH(z−1n2∑i=1nxi)for all x1,…,xn,z∈A. By Lemma 2.1,
the mapping H:A→B is Cauchy additive, that is, H(x+y)=H(x)+H(y) for all x,y∈A.

By a similar method to the proof of [14],
one can show that the mapping H:A→B is ℂ-linear.

It follows from (2.8)
that∥H(xy)−H(x)H(y)∥B=limm→∞(nα)2m∥f((αn)2mxy)−f((αn)mx)f((αn)my)∥B≤limm→∞(nα)2mφ((αn)mx,(αn)my,0,…,0︸n−1times)=0for all x,y∈A. SoH(xy)=H(x)H(y)for all x,y∈A.

It follows from (2.9)
that∥H(x*)−H(x)*∥B=limm→∞(nα)m∥f((αn)mx*)−f((αn)mx)*∥B≤limm→∞(nα)mφ((αn)mx,…,(αn)mx︸n+1times)≤limm→∞(nα)2mφ((αn)mx,…,(αn)mx︸n+1times)=0for all x∈A. SoH(x*)=H(x)*for all x∈A.

Thus H:A→B is a C*-algebra homomorphism satisfying (2.11) as desired.

Corollary 2.3.

Let r>2 and θ be nonnegative real numbers, and let f:A→B be a mapping such that ∥Cμf(z,x1,…,xn)∥B≤θ(∥z∥Ar+∑i=1n∥xi∥Ar),∥f(xy)−f(x)f(y)∥B≤θ(∥x∥Ar+∥y∥Ar),∥f(x*)−f(x)*∥B≤(n+1)θ∥x∥Ar for all μ∈𝕋1 and all x,y,z∈A. Then there exists a unique C*-algebra homomorphism H:A→B such that∥f(x)−H(x)∥B≤2n(n2−1)−rθ(n2−1)1−r−n2(1−r)∥x∥Arfor all x∈A.

Proof.

The proof follows from Theorem 2.2
by takingφ(z,x1,…,xn):=θ(∥z∥Ar+∑i=1n∥xi∥Ar)for all x,y,z∈A. It follows from (2.31) that f(0)=0. We can choose L=(n2/(n2−1))1−r to get the desired result.

Theorem 2.4.

Let f:A→B be a mapping satisfying f(0)=0 for which there exists a function φ:An+1→[0,∞) satisfying (2.7), (2.8), and
(2.9) such
that∑j=0∞(n2−1n2)jφ((n2n2−1)jz,(n2n2−1)jx1,…,(n2n2−1)jxn)<∞for all z,x1,…,xn∈A. If for some 1≤j≤n there exists a Lipschitz constant L<1 such thatφ(x,0,…,0,x︸jth,0,…,0)≤n2n2−1Lφ(n2−1n2x,0,…,0,n2−1n2x︸jth,0,…,0)for all x∈A, then there exists a unique C*-algebra homomorphism H:A→B such that∥f(x)−H(x)∥B≤nL(n2−1)×(1−L)φ(x,0,…,0,x︸jth,0,…,0)for all x∈A.

Proof.

Similar to proof of Theorem (2.2), we consider the linear mapping J:X→X such thatJg(x):=αng(nαx)for all x∈A, where α=(n2−1)/n. We can conclude that J is a strictly contractive self mapping of X with the Lipschitz constant L.

It follows from (2.18)
that∥f(x)−αnf(nαx)∥B≤1nφj(nαx,nαx)≤Lαφj(x,x)for all x∈A. Hence, d(f,Jf)≤(L/α).

By Theorem 1.2, there exists a
mapping H:A→B such that the following hold:

(1) H is a fixed point of J, that is,H(nαx)=nαH(x)for all x∈A; the mapping H is a unique fixed point of J in the setY={g∈X:d(f,g)<∞};and this implies that H is a unique mapping satisfying (2.41) such that there exists C∈(0,∞) satisfying∥H(x)−f(x)∥B≤Cφj(x,x)for all x∈A;

(2) d(Jmf,H)→0 as m→∞; and this implies the equalitylimm→∞(αn)mf((αn)mx)=H(x)for all x∈A;

(3) d(f,H)≤(1/(1−L))d(f,Jf), which implies the inequalityd(f,H)≤Lα−αL,which implies that the inequality (2.38) holds.

The rest of the proof is similar to the proof of Theorem 2.2.

Corollary 2.5.

Let r<1 and θ be nonnegative real numbers, and let f:A→B be a mapping satisfying (2.31), (2.32), and (2.33).
Then there exists a unique C*-algebra homomorphism H:A→B such that∥f(x)−H(x)∥B≤2n(n2−1)r−2Lθ(n2−1)r−1−n2(r−1)∥x∥Arfor all x∈A and L=(n2/(n2−1))r−1.

Proof.

The proof follows from Theorem 2.4
by takingφ(z,x1,…,xn):=θ(∥z∥Ar+∑i=1n∥xi∥Ar)for all z,x1,…,xn∈A. It follows from (2.31) that f(0)=0. We can choose L=(n2/(n2−1))r−1 to get the desired result.

3. Stability of Generalized Derivations on <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M269"><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mo>*</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>-Algebras

For a given mapping f:A→A and for a fixed positive integer n≥2, we defineCμf(z,x1,…,xn):=∑i=1nμf(z−xi)+1n∑1≤i<j≤nf(μxi+μxj)−nf(μz−1n2∑i=1nμxi)for all μ∈𝕋1 and all z,x1,…,xn∈A.

Definition 3.1

(see [42]). A generalized derivation δ:A→A is involutive ℂ-linear and fulfills

δ(xyz)=δ(xy)z−xδ(y)z+xδ(yz)for all x,y,z∈A.

We prove the generalized Hyers-Ulam stability of derivations on C*-algebras for the functional equation Cμf(z,x1,…,xn)=0.

Theorem 3.2.

Let f:A→A be a mapping satisfying f(0)=0 for which there exists a function φ:An+1→[0,∞) such that ∑j=0∞(n2n2−1)3jφ((n2−1n2)jz,(n2−1n2)jx1,…,(n2−1n2)jxn)<∞,∥Cμf(x1,…,xn,z)∥A≤φ(z,x1,…,xn),∥f(xyz)−f(xy)z+xf(y)z−xf(yz)∥A≤φ(x,y,z,0,…,0︸n−2times),∥f(x*)−f(x)*∥A≤φ(x,…,x︸n+1times) for all μ∈𝕋1 and all x,y,z,x1,…,xn∈A. If for some 1≤j≤n there exists a Lipschitz constant L<1 such thatφ(x,0,…,0,x︸jth,0,…,0)≤n2−1n2Lφ(n2n2−1x,0,…,0,n2n2−1x︸jth,0,…,0)for all x∈A, then there exists a unique generalized derivation δ:A→A such that∥f(x)−δ(x)∥A≤n(n2−1)×(1−L)φ(x,0,…,0,x︸jth,0,…,0)for all x∈A.

Proof.

By the same reasoning as in the proof of Theorem 2.2, there exists a unique involutive ℂ-linear mapping δ:A→A satisfying (3.7). The mapping δ:A→A is given byδ(x)=(nα)mf((nα)mx)for all x∈A.

It follows from (3.4)
that∥δ(xyz)−δ(xy)z+xδ(y)z−xδ(yz)∥A=limm→∞(nα)3m∥f((αn)3mxyz)−f((αn)2mxy)⋅(αn)mz+(αn)mxf((αn)my)⋅(αn)mz−(αn)mxf((αn)2myz)∥A≤limm→∞(nα)3mφ((αn)mx,(αn)my,(αn)mz,0,…,0︸n−2times)=0for all x,y,z∈A. So δ(xyz)=δ(xy)z−xδ(y)z+xδ(yz) for all x,y,z∈A. Thus δ:A→A is a generalized derivation satisfying (3.7).

Theorem 3.3.

Let f:A→A be a mapping satisfying f(0)=0 for which there exists a function φ:An+1→[0,∞) satisfying (2.36),(3.3), (3.4) and (3.5) for all x,y,z,x1,…,xn∈A. If for some 1≤j≤n there exists a Lipschitz constant L<1 such thatφ(x,0,…,0,x︸jth,0,…,0)≤n2n2−1Lφ(n2−1n2x,0,…,0,n2−1n2x︸jth,0,…,0)for all x∈A, then there exists a unique generalized derivation δ:A→A such that∥f(x)−δ(x)∥B≤nL(n2−1)×(1−L)φ(x,0,…,0,x︸jth,0,…,0)for all x∈A.

Proof.

The proof is similar to the proofs of Theorems 2.4 and 3.2.

Acknowledgments

This paper is based on final report of the research project of the Ph.D. thesis in
University of Tabriz and the third author was supported by Grant no.
F01-2006-000-10111-0 from the Korea Science and Engineering Foundation. The authors
would like to thank the referees for a number of valuable suggestions regarding a
previous version of this paper.

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