Volume 2008, Article ID 749392,15pages doi:10.1155/2008/749392
Research Article
Fixed Point Methods for the Generalized Stability
of Functional Equations in a Single Variable
Liviu C ˘adariu1 and Viorel Radu2
1Departamentul de Matematic˘a, Universitatea Politehnica din Timis¸oara, Piat¸a Victoriei no. 2, 300006 Timis¸oara, Romania
2Facultatea de Matematic˘a S¸i Informatic˘a, Universitatea de Vest din Timis¸oara, Bv. Vasile Pˆarvan 4, 300223 Timis¸oara, Romania
Correspondence should be addressed to Liviu C˘adariu,[email protected]
Received 4 October 2007; Accepted 14 December 2007
Recommended by Andrzej Szulkin
We discuss on the generalized Ulam-Hyers stability for functional equations in a single variable, including the nonlinear functional equations, the linear functional equations, and a generalization of functional equation for the square root spiral. The stability results have been obtained by a fixed point method. This method introduces a metrical context and shows thatthe stability is related to some fixed pointof a suitable operator.
Copyrightq2008 L. C˘adariu and V. Radu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction and preliminaries
The study of functional equations stability originated from a question of Ulam1940 concern-ing the stability of group homomorphisms is as follows.
LetG be a group endowed with a metric d.Givenε > 0,does there exist ak > 0such that for every functionf :G→Gsatisfying the inequality
dfx·y, fx·fy< ε, ∀x, y∈G, 1.1 there exists an automorphismaofG with
dfx, ax< kε, ∀x∈G? 1.2 In 1941, Hyers1gave an affirmative answer to the question of Ulam for Cauchy equa-tion in Banach spaces.
LetE1andE2be Banach spaces and let f:E1→E2be such a mapping that
for allx, y ∈E1and aδ >0, that is,fisδ-additive. Then there exists a unique additiveT :E1→E2,
which satisfies
fx−Tx≤δ, ∀x∈E1. 1.4
In fact, according to Hyers,
Tx lim
n→∞
f2nx
2n , ∀x∈E1. 1.5
For this reason, one says that the Cauchy equation isstable in the sense of Ulam-Hyers.
In2,3as well as in4–7, the stability problem with unbounded Cauchy differences is consideredsee also8,9. Their results include the following two theorems.
Theorem 1.1see1,2,4,7. Suppose thatEis a real-normed space,F is a real Banach space, and
f :E→Fis a given function, such that the following condition holds:
fxy−fx−fy
F ≤θ
xpEypE, ∀x, y∈E, 1p
for somep∈0,∞\ {1}andθ >0. Then there exists a unique additive functiona:E→Fsuch that
fx−ax
F ≤
2θ 2−2px
p
E, ∀x∈E. 2p
Also, if for eachx∈Ethe functiont→ ftxfromRtoFis continuous for each fixedx∈E, thenais linear mapping.
It is worth mentioning that the proofs used the idea conceived by Hyers. Namely, the additive functiona:E→Fis constructed, starting from the given functionf, by the following formula:
ax lim
n→∞ 1 2nf
2nx, ifp <1, 2p<1
ax lim
n→∞2 nf
x
2n
, ifp >1. 2p>1
This method is calledthe direct method or Hyers’ method.
We also mention a result concerning the stability properties with unbounded control conditions invoking products of different powers of normssee5,6,10.
Theorem 1.2. Suppose thatE is a real-normed space,F is a real Banach space, and f : E → F is a given function, such that the following condition holds
fxy−fx−fy
F ≤θx
p
E· y
q
E, ∀x, y∈E, 1p
for some fixedθ >0andp, q ∈Rsuch thatr pq / 1. Then there exists a unique additive function
L:E→Fsuch that
fx−Lx
F ≤
θ 2r−2x
r
E, ∀x∈E. 2p
If in additionf : E→F is a mapping such that the transformationt→ ftxis continuous int∈R,
Generally, whenever the constant δ in 1.3is replaced by a control function x, y → δx, ywith appropriate properties, as in3, one uses the generic termgeneralized Ulam-Hyers stabilityorstability in Ulam-Hyers-Bourgin sense.
In the general case, one uses control conditions of the form
Dfx, y≤δx, y 1.6
and thestability estimationsare of the form
fx−Sx≤εx, 1.7
whereSis a solution, that is, itverifies the functional equationDSx, y 0, and forεx, explicit
formulae are given, which depend on the controlδas well as on the equationDfx, y.
We refer the reader to the expository papers11,12or to the books13–15 see also the recent articles of Forti16,17, for supplementary details.
On the other hand, in18–25,a fixed point method was proposed, by showing that many theorems concerning the stability of Cauchy, Jensen, quadratic, cubic, quartic, and monomial functional equations are consequences of the fixed point alternative. Subsequently, the method has been successfully used, for example, in26–30. This method introduces a metrical context and shows thatthe stability is related to some fixed pointof a suitable operator.
The control conditions are responsible for three fundamental facts:
1thecontraction propertyof a Schr ¨oder-type operatorJ,
2the first two approximations,f andJf, to be at afinite distance,
3they force the fixed point ofJto be asolution of the initial equation.
Our main purpose here is to study the generalized stability for some functional equa-tions in a single variable. We prove the generalized Ulam-Hyers stability of the single variable equation
w◦f◦ηfh. 1.8
As an application of our result for1.8, the stability for the following generalized functional equation of the square root spiral
fp−1px kfx hx 1.9
is obtained.
Thereafter, we present the generalized Ulam-Hyers stability of the nonlinear equation
fx Fx, fηx. 1.10
The main result is seen to slightly extend the Ulam-Hyers stability previously given in 31, Theorem 2. As a direct consequence of this result, the generalized Ulam-Hyers stability of the linear equationfx gx·fηx hxis highlighted. Notice that in all these equations,f
Proposition 1.3cf.32or33. Suppose that a complete generalized metric spaceX, d(i.e., one for which d may assume infinite values) and a strictly contractive mapping A : X → X with the Lipschitz constant L < 1 are given. Then, for a given element x ∈ X,exactly one of the following assertions is true:
A1dAnx, An1x ∞, for all n≥0;
A2there existsksuch thatdAnx, An1x<∞, for all n≥k.
Actually, ifA2holds, then
A21the sequenceAnxis convergent to a fixed pointy∗ofA;
A22y∗is the unique fixed point ofAinY :{y∈X, dAkx, y<∞};
A23dy, y∗≤1/1−Ldy, Ay, for all y∈Y.
2. A general fixed point method
Firstly we prove, by the fixed point alternative, a stability result for the single variable equation
w◦g◦ηgh,where
i wis a Lipschitz self-mappingwith constantwof the Banach spaceY;
ii ηis a self-mapping of the nonempty setG;
iii h:G→Y is a given function;
ivthe unknown is a mappingg :G→Y, that leads to the following. Theorem 2.1. Suppose thatf:G→Y satisfies
w◦f◦ηx−fx−hx
Y ≤ψx, ∀x∈G, Cψ
with some given mappingψ:G→0,∞. If there existsL <1such that
w·ψ◦ηx≤Lψx, ∀x∈G, Hψ
then there exists a unique mappingc:G→Y which satisfies both the equation
w◦c◦ηx cx hx, ∀x∈G, Eω,η
and the estimation
fx−cx
Y ≤
ψx
1−L, ∀x∈G. Estψ
Proof. Let us consider the setE:{g :G→ Y}and introduce acomplete generalized metriconE
as usual, inf∅∞:
dg1, g2
dψ
g1, g2
infK∈R,g1x−g2xY ≤Kψx, ∀x∈G . GMψ
Now, define thenonlinearmapping
Step 1. Using the hypothesisHψit follows thatJ is strictly contractiveonE. Indeed, for any
g1, g2∈ Ewe have
dg1, g2
< K⇒g1x−g2xY ≤Kψx, ∀x∈G, Jg1x−Jg2x
Y w◦g1◦η
x−hx−w◦g2◦η
x−hxY ≤w·g1
ηx−g2
ηxY.
2.1
Therefore
Jg1x−Jg2x
Y ≤w·K·ψ
ηx≤K·L·ψx, ∀x∈G, 2.2 so thatdJg1, Jg2≤LK,which implies
dJg1, Jg2
≤Ldg1, g2
, ∀g1, g2∈ E. CCL This says thatJis astrictly contractiveself-mapping ofE,with the constantL <1.
Step 2. df, Jf<∞. In fact, using the relationCψit results thatdf, Jf≤1.
Step 3. We can apply the fixed point alternative and we obtain the existence of a mapping
c:G→Y such that the following hold.
icis a fixed point ofJ, that is,
w◦c◦ηx cx hx, ∀x∈G. Ew,η
The mappingcis the unique fixed point ofJin the set
F{g∈ E, df, g<∞}. 2.3
This says thatcis the unique mapping verifyingbothEw,ηand2.4, where
∃K <∞ such thatcx−fxY ≤Kψx, ∀x∈G. 2.4
iidJnf, c −→
n→∞0,which implies
cx lim
n→∞J
nfx, ∀x∈G, 2.5
where
Jnfx w◦Jn−1f◦ηx−hx
wwJn−2f◦η2x−h◦ηx−hx, ∀x∈G, 2.6
whence
Jnf ωωωω· · ·ωω◦f◦ηn−h◦ηn−1−h◦ηn−2− · · ·−h◦η3−h◦η2−h◦η−h.
2.7
iiiFinally,df, c≤1/1−Ldf, Jf,which implies the inequality
df, c≤ 1
1−L, 2.8
Theorem 2.1extends our recent result in34, where the generalized stability in Ulam-Hyers sense was obtained for the equation
w◦g◦ηg. 2.9
3. Applications to the generalized equation of the square root spiral
As a consequence ofTheorem 2.1, we obtain a generalized stability result for the equation
fp−1px kfx hx, ∀x∈G. 3.1
The “unknowns” are functions f : G → Y between two vector spaces while p, h are given functions, p−1 is the inverse of p, and k /0 is a fixed constant. The solution of 3.1 and a generalized stability result in Ulam-Hyers sense for the above equation are given in35, by the direct method.
A vector spaceGand a Banach spaceY will be considered.
Theorem 3.1. Let k ∈ G\ {0}and suppose thatp : G → G is bijective andh : G → Y is a given mapping. Iff :G→Y satisfies
f
p−1px k−fx−hxY ≤ψx, ∀x∈G, Sψ
with a mapping:ψ :G→0,∞for which there existsL <1such that
ψp−1px kx≤Lψx, ∀x∈G, Hψ,p
then there exists a unique mappingc:G→Y which satisfies both the equation
cp−1px kcx hx, ∀x∈G, Ep,h
and the estimation
fx−cx
Y ≤
ψx
1−L, ∀x∈G. Estψ
Moreover,
cx lim
n→∞
fp−1px nk−
n−1
i0
hp−1px ik
, ∀x∈G. 3.2
Proof. We applyTheorem 2.1, withw:Y →Y, η:G →G, ψ :G→0,∞,
wx:x, ηx:p−1px k. 3.3
Clearly,lw1 andJgx:gp−1px k−hx.
By usingSψand the hypothesisHψ,p, we immediately see thatCψandHψhold.
Since
then
Jnfx
fp−1px nk−
n−1
i0
hp−1px ik
, 3.5
whence there exists a unique mappingc:G→Y,
cx: lim
n→∞
Jnfx, ∀x∈G, 3.6
which satisfies the equationJcx cx,that is,
cp−1px kcx hx, ∀x∈G, 3.7
and the inequality
fx−cx
Y ≤
ψx
1−L, ∀x∈G. 3.8
A special caseof3.1is obtained for k1, px xn, n≥2,andhx arctan1/x. It is the so-called “nth root spiral equation”
f√nxn1fx arctan1
x. 3.9
As a consequence ofTheorem 3.1, we obtain the following generalized stability result for the above equation.
Theorem 3.2. Iff :R→Rsatisfies
f√nxn1−fx−arctan 1
x
≤ψx, ∀x∈R, 3.10
with some fixed mappingψ:R→0,∞and there existsL <1such that
ψ√nxn1≤Lψx, ∀x∈R, 3.11
then there exists a unique mappingc:R→R,
cx lim
m→∞
n √
xnm− m−1
i0
arctan√n 1
xni
, ∀x∈R, 3.12
which satisfies both3.9and the estimation
fx−cx≤ ψx
1−L, ∀x∈R. 3.13
Notice that forn 2, Jung and Sahoo36 proved in 2002 a generalized Ulam-Hyers stability result for the functional equation3.9, by using the direct method.
Corollary 3.3. Iff :R→ Rsatisfies
f√nxn1−fx−arctan1
x
≤axn, ∀x∈R, 3.14
with some fixed0< a <1, then there exists a unique mappingc:R→R,
cx lim
m→∞
n √
xnm− m−1
i0
arctan√n 1
xni
, ∀x∈R, 3.15
which satisfies both3.9and the estimation
fx−cx≤ axn
1−a, ∀x∈R. 3.16
Proof. We apply Theorem 3.2, by choosing ψx axn
0 < a < 1, n ∈ N. It is clear that the relation3.11holds, withLa <1.
Remark 3.4. A similar result of stability as inCorollary 3.3can be obtained for a control map-pingψ:R→0,∞, ψx 1/axn a >1, n∈N. The estimation relation3.16becomes
fx−cx≤ a1−xn
a−1, ∀x∈R. 3.17
4. The generalized Ulam-Hyers stability of a nonlinear equation
The Ulam-Hyers stability for the nonlinear equation
fx Fx, fηx 4.1
was discussed by Baker31. The “unknowns” are functionsf : G → Y,between two vector spaces. In this section, we will extend the Baker’s result and we will obtain the generalized stability in Ulam-Hyers sense for4.1, by using the fixed point alternative.
Let us consider a nonempty setGand a complete metric spaceY, d. Theorem 4.1. Letη:G→G, g:G→R(orC) andF :G×Y →Y. Suppose that
dFx, u, Fx, v≤gx·du, v, ∀x∈G, ∀u, v∈Y. 4.2
Iff :G→Y satisfies
dfx, Fx, fηx≤ψx, ∀x∈G, 4.3
with a mappingψ :G→0,∞for which there existsL <1such that
then there exists a unique mappingc:G→Y which satisfies both the equation
cx Fx, cηx, ∀x∈G, 4.5
and the estimation
dfx, cx≤ ψx
1−L, ∀x∈G. 4.6
Moreover,
cx lim
n→∞F
x, FFηx, . . . , Fηx,f◦ηnηx, ∀x∈G. 4.7
Proof. We use the same method as in the proof ofTheorem 2.1, namely,the fixed point alternative. Let us consider the setE: {h:G → Y}and introduce acomplete generalized metriconE
as usual, inf∅∞:
ρh1, h2
infK∈R, dh1x, h2x≤Kψx, ∀x∈G . 4.8
Now, define the mapping
J :E −→ E, Jhx:Fx, hηx. 4.9
Step 1. Using the hypothesis in4.2and4.4it follows thatJis strictly contractiveonE. Indeed, for anyh1, h2∈ Ewe have
ρh1, h2
< K⇒dh1x, h2x≤Kψx, ∀x∈G, dJh1x, Jh2xdFx, h1
ηx, Fx, h2
ηx ≤ gx·dh1
ηx, h2
ηx ≤K·gx·ψηx.
4.10
Therefore
dJh1x, Jh2x≤K·gx·ψηx≤K·L·ψx, ∀x∈G, 4.11
so thatρJh1, Jh2≤LK,which implies
ρJh1, Jh2
≤Lρh1, h2
, ∀h1, h2∈ E. 4.12
Step 3. We can apply the fixed point alternativeseeProposition 1.3, and we obtain the exis-tence of a mappingc:G→Y such that the following hold.
icis a fixed point ofJ, that is,
cx Fx, cηx, ∀x∈G. 4.13 The mappingcis the unique fixed point ofJin the set
Fh∈ E, ρf, h<∞ . 4.14
This says thatcis the unique mapping verifyingboth4.13and4.15where
∃K <∞ such thatdcx, fx≤Kψx, ∀x∈G. 4.15
iiρJnf, c −→
n→∞0,which implies
cx lim
n→∞J
nfx, ∀x∈G, 4.16
where
Jnfx Fx,Jn−1fηx
Fx, Fηx,Jn−2fηx, ∀x∈G, 4.17
hence
Jnfx Fx, FFηx, . . . Fηx,f◦ηnηx. 4.18
iiiρf, c≤1/1−Lρf, Jf,which implies the inequality
ρf, c≤ 1
1−L, 4.19
that is,4.6holds.
As a direct consequence of Theorem 4.1, the following Ulam-Hyers stability resultcf.
31, Theorem 2or37, Theorem 13for the nonlinear equation4.1is obtained.
Corollary 4.2. LetG be a nonempty set and let Y, dbe a complete metric space. Let η : G → G,
F :G×Y →Y, and0≤L <1. Suppose that
dFx, u, Fx, v≤L·du, v, ∀x∈G, ∀u, v∈Y. 4.20 Iff :G→Y satisfies
dfx, Fx, fηx≤δ, ∀x∈G, 4.21 with a fixed constantδ > 0, then there exists a unique mappingc : G → Y which satisfies both the equation
cx Fx, cηx, ∀x∈G, 4.22 and the estimation
dfx, cx≤ δ
Moreover,
cx lim
n→∞F
x, FFηx, . . . , Fηx,f◦ηnηx, ∀x∈G. 4.24
Proof. It follows byTheorem 4.1, by choosingψx δ >0. Example 4.3. If in4.1we considerF :R×1,∞→1,∞,
Fx, u ⎧ ⎨ ⎩
u1/p, ifp >1,
up, if 0< p <1, 4.25
we obtain the equation of B ¨ottcher:
fηx1/pfx, p >1, or fηxpfx, 0< p <1. 4.26
Agarwal et al. proved in37, Theorem 14that the above equations are stable in Ulam-Hyers sense, by using the result of Baker31, with
L ⎧ ⎪ ⎨ ⎪ ⎩ 1
p, ifp >1, p, if 0< p <1.
4.27
It is worth noticing that our formula4.24gives the solutions c of4.26:
cx ⎧ ⎪ ⎨ ⎪ ⎩
lim
n→∞
fηnx1/pn, ifp >1,∀x∈R,
lim
n→∞
fηnxpn, if 0< p <1, ∀x∈R. 4.28
5. The generalized Ulam-Hyers stability of a linear functional equation
In this section, we emphasize the importance ofTheorem 4.1. In fact, if
Fx, fηxgx·fηxhx, 5.1
equation4.1becomes
fx gx·fηxhx, 5.2
Theorem 5.1. Consider Ga nonempty set and Y a real (or complex) Banach space. Suppose thatη :
G→G, g:G→R(orC). Iff:G→Y satisfies
fx−gxf
ηx−hxY ≤ψx, ∀x∈G, 5.3
with some fixed mappingψ:G→0,∞and there existsL <1such that
|gx|ψ◦ηx≤Lψx, ∀x∈G, 5.4
then there exists a unique mappingc:G→Y,
cx hx lim
n→∞
fηnx·
n−1
i0
gηix
n−2
j0
hηj1x·
j
i0
gηix
, 5.5
for allx∈G,which satisfies both the equation
cx gx·cηxhx, ∀x∈G, 5.6
and the estimation
fx−cxY ≤ ψx
1−L, ∀x∈G. 5.7
Proof. We consider in Theorem 4.1 the metric d on Y, given by du, v ||u−v||Y and the function
Fx, fηx:gxfηxhx, ∀x∈G, 5.8
withg, η, has in hypothesis ofTheorem 5.1. The relation4.2holds with equality. Applying
Theorem 4.1, there exists a unique mapping c which satisfies 5.2and the estimation 5.7. Moreover,
cx lim
n→∞J
nfx, ∀x∈G, 5.9
where
Jnfx gx·Jn−1fηxhx
gx·gηx·Jn−2fη2xgx·hηxhx, ∀x∈G, 5.10
whence, for allx∈G,
Jnfx:hx fηnx·
n−1
i0
gηix
n−2
j0
hηj1x·
j
i0
gηix
. 5.11
Corollary 5.2. Consider a nonempty set G and a real (or complex) Banach space Y. Suppose that
η:G→G, g :G→R(orC), andh:G→Y are given. Iff :G→Y satisfies
fx−gxf
ηx−hxY ≤δ, ∀x∈G, 5.12
with a fixed constantδ >0and there existsL <1such that
|gx| ≤L, ∀x∈G, 5.13
then there exists a unique mappingc:G→Y,
cx hx lim
n→∞
fηnx·
n−1
i0
gηix
n−2
j0
hηj1x·
j
i0
gηix
, 5.14
for allx∈G,which satisfies both the equation
cx gx·cηxhx, ∀x∈G, 5.15
and the estimation
fx−cx
Y ≤
δ
1−L, ∀x∈G. 5.16
Remark 5.3. It is easy to see that3.1is a particular case of5.2. To prove this, it is sufficient to consider in5.2g ≡1,h:−h1,ηx:p−1px k, ∀x∈G, withpbijective onGandk /0
a fixed constant. By using the above notations,Theorem 3.1can be obtained as a consequence ofTheorem 5.1, with
cx hx lim
n→∞
fηnx·
n−1
i0
gηix
n−2
j0
hηj1x·
j
i0
gηix
−h1x lim
n→∞
fpp−1x nk−
n−1
i1
h1
pp−1x ik
lim
n→∞
fp−1px nk−
n−1
i0
h1
p−1px ik
, ∀x∈G.
5.17
Acknowledgment
The authors would like to thank the referees and the editors for their help and suggestions in improving this paper.
References
1D. H. Hyers, “On the stability of the linear functional equation,”Proceedings of the National Academy of Sciences of the United States of America, vol. 27, no. 4, pp. 222–224, 1941.
3D. G. Bourgin, “Classes of transformations and bordering transformations,”Bulletin of the American Mathematical Society, vol. 57, pp. 223–237, 1951.
4Z. Gajda, “On stability of additive mappings,”International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 431–434, 1991.
5J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,”Journal of Functional Analysis, vol. 46, no. 1, pp. 126–130, 1982.
6J. M. Rassias, “Solution of a problem of Ulam,” Journal of Approximation Theory, vol. 57, no. 3, pp. 268–273, 1989.
7T. M. Rassias, “On the stability of the linear mapping in Banach spaces,”Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.
8G. L. Forti, “An existence and stability theorem for a class of functional equations,”Stochastica, vol. 4, no. 1, pp. 23–30, 1980.
9P. G˘avrut¸a, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive map-pings,”Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431–436, 1994.
10J. M. Rassias, “On the stability of the Euler-Lagrange functional equation,” Comptes Rendus de l’ Acad´emie Bulgare des Sciences, vol. 45, no. 6, pp. 17–20, 1992.
11G. L. Forti, “Hyers-Ulam stability of functional equations in several variables,”Aequationes Mathemat-icae, vol. 50, no. 1-2, pp. 143–190, 1995.
12T. M. Rassias, “On the stability of functional equations and a problem of Ulam,” Acta Applicandae Mathematicae, vol. 62, no. 1, pp. 23–130, 2000.
13Z. Daroczy and Z. Pales, Eds.,Functional Equations—Results and Advances, vol. 3 ofAdvances in Mathe-matics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002.
14D. H. Hyers, G. Isac, and T. M. Rassias,Stability of functional equations in several variables, vol. 34 of
Progress in Nonlinear Differential Equations and Their Applications, Birkh¨auser, Basel, Switzerland, 1998. 15S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic
Press, Palm Harbor, Fla, USA, 2001.
16G. L. Forti, “Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations,”Journal of Mathematical Analysis and Applications, vol. 295, no. 1, pp. 127–133, 2004.
17G. L. Forti, “Elementary remarks on Ulam-Hyers stability of linear functional equations,” Journal of Mathematical Analysis and Applications, vol. 328, no. 1, pp. 109–118, 2007.
18L. C˘adariu and V. Radu, “Fixed points and the stability of Jensen’s functional equation,”Fixed Point Theory, vol. 4, no. 1, Article 4, p. 7, 2003.
19L. C˘adariu and V. Radu, “Fixed points and the stability of quadratic functional equations,”Analele Universit˘at¸ii de Vest din Timis¸oara, vol. 41, no. 1, pp. 25–48, 2003.
20L. C˘adariu and V. Radu, “On the stability of the Cauchy functional equation: a fixed point approach,” inIteration Theory (ECIT ’02), J. S. Ramos, D. Gronau, C. Mira, L. Reich, and A. N. Sharkovsky, Eds., vol. 346 ofGrazer Mathematische Berichte, pp. 43–52, Karl-Franzens-Univ. Graz, Graz, Austria, 2004. 21L. C˘adariu and V. Radu, “A Hyers-Ulam-Rassias stability theorem for a quartic functional equation,”
Automation Computers and Applied Mathematics, vol. 13, no. 1, pp. 31–39, 2004.
22L. C˘adariu and V. Radu, “Fixed points in generalized metric spaces and the stability of a cubic func-tional equation,” inFixed Point Theory and Applications, Y. J. Cho, J. K. Kim, and S. M. Kang, Eds., vol. 7, pp. 53–68, Nova Science Publishers, Hauppauge, NY, USA, 2007.
23L. C˘adariu and V. Radu, “The alternative of fixed point and stability results for functional equations,”
International Journal of Applied Mathematics & Statistics, vol. 7, no. Fe07, pp. 40–58, 2007.
24L. C˘adariu and V. Radu, “Remarks on the stability of monomial functional equations,” Fixed Point Theory, vol. 8, no. 2, pp. 201–218, 2007.
25V. Radu, “The fixed point alternative and the stability of functional equations,”Fixed Point Theory, vol. 4, no. 1, pp. 91–96, 2003.
26S.-M. Jung and T.-S. Kim, “A fixed point approach to the stability of the cubic functional equation,”
Bolet´ın de la Sociedad Matem´atica Mexicana. Tercera Serie, vol. 12, no. 1, pp. 51–57, 2006.
27S.-M. Jung, “A fixed point approach to the stability of isometries,”Journal of Mathematical Analysis and Applications, vol. 329, no. 2, pp. 879–890, 2007.
28S.-M. Jung, “A fixed point approach to the stability of a Volterra integral equation,”Fixed Point Theory and Applications, vol. 2007, Article ID 57064, 9, 2007.
29M. Mirzavaziri and M. S. Moslehian, “A fixed point approach to stability of a quadratic equation,”
30J. M. Rassias, “Alternative contraction principle and Ulam stability problem,”Mathematical Sciences Research Journal, vol. 9, no. 7, pp. 190–199, 2005.
31J. A. Baker, “The stability of certain functional equations,” Proceedings of the American Mathematical Society, vol. 112, no. 3, pp. 729–732, 1991.
32J. B. Diaz and B. Margolis, “A fixed point theorem of the alternative, for contractions on a generalized complete metric space,”Bulletin of the American Mathematical Society, vol. 74, pp. 305–309, 1968. 33I. A. Rus,Principii s¸i Aplicat¸ii ale Teoriei Punctului Fix, Editura Dacia, Cluj-Napoca, Romania, 1979. 34L. C˘adariu and V. Radu, “The fixed points method for the stability of some functional equations,”
Carpathian Journal of Mathematics, vol. 23, no. 1-2, pp. 63–72, 2007.
35Z. Wang, X. Chen, and B. Xu, “Generalization of functional equation for the square root spiral,”Applied Mathematics and Computation, vol. 182, no. 2, pp. 1355–1360, 2006.
36S.-M. Jung and P. K. Sahoo, “Stability of a functional equation for square root spirals,”Applied Mathe-matics Letters, vol. 15, no. 4, pp. 435–438, 2002.
37R. P. Agarwal, B. Xu, and W. Zhang, “Stability of functional equations in single variable,”Journal of Mathematical Analysis and Applications, vol. 288, no. 2, pp. 852–869, 2003.
38M. Kuczma, B. Choczewski, and R. Ger,Iterative functional equations, vol. 32 ofEncyclopedia of Mathe-matics and Its Applications, Cambridge University Press, Cambridge, UK, 1990.
