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FUZZY STABILITY OF AN ADDITIVE-QUADRATIC FUNCTIONAL EQUATION WITH THE FIXED POINT ALTERNATIVE
FUZZY STABILITY OF AN ADDITIVE-QUADRATIC FUNCTIONAL EQUATION WITH THE FIXED POINT ALTERNATIVE
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2015. Aug, 22(3): 285-298
Copyright © 2015, Korean Society of Mathematical Education
  • Received : July 06, 2015
  • Accepted : July 21, 2015
  • Published : August 31, 2015
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About the Authors
JEONG PIL SEO
OHSANG HIGH SCHOOL, GUMI 730-842, KYEONGSANGBUK DO, KOREAEmail address:sjp4829@hanmail.net
SUNGJIN LEE
DEPARTMENT OF MATHEMATICS, DAEJIN UNIVERSITY, KYEONGGI 487-711, KOREAEmail address:hyper@daejin.ac.kr
REZA SAADATI
DEPARTMENT OF MATHEMATICS, IRAN UNIVERSITY OF SCIENCE AND TECHNOLOGY, TEHRAN, IRANEmail address:rsaadati@eml.cc

Abstract
In [41] , Th.M. Rassias proved that the norm defined over a real vector space V is induced by an inner product if and only if for a fixed positive integer l holds for all x 1 , ⋯ , x 2l V . For the above equality, we can define the following functional equation Using the fixed point method, we prove the Hyers-Ulam stability of the functional equation (0.1) in fuzzy Banach spaces.
Keywords
1. INTRODUCTION AND PRELIMINARIES
The stability problem of functional equations originated from a question of Ulam [51] concerning the stability of group homomorphisms. Hyers [13] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Th.M. Rassias [40] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by Găvruta [12] by replacing the unbounded Cauchy difference by a general control function in the spirit of the Th.M. Rassias’ approach.
The functional equation
  • f(x+y) +f(x−y) = 2f(x) + 2f(y)
is called a quadratic functional equation . In particular, every solution of the quadratic functional equation is said to be a quadratic mapping . A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [50] for mappings f : X Y , where X is a normed space and Y is a Banach space. Cholewa [8] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [9] proved the Hyers-Ulam stability of the quadratic functional equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [14 , 16 , 17 , 31 , 32 , 33 , 34 , 35 , 37 , 38 , 39 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49] ).
In [33] , Park, Lee and Shin proved that an even mapping f : V W satisfies the functional equation (0.1) if and only if the even mapping f : V W is quadratic. Moreover, they proved the Hyers-Ulam stability of the quadratic functional equation (0.1) in real Banach spaces.
Katsaras [18] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [11 , 20 , 52] . In particular, Bag and Samanta [2] , following Cheng and Mordeson [7] , gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [19] . They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [3] .
We use the definition of fuzzy normed spaces given in [2 , 24 , 25] to investigate a fuzzy version of the Hyers-Ulam stability for the functional equation (0.1) in the fuzzy normed vector space setting.
Definition 1.1 ( [2 , 24 , 25 , 26] ). Let X be a real vector space. A function N : X × ℝ → [0, 1] is called a fuzzy norm on X if for all x , y X and all s , t ∈ ℝ,
  • (N1)N(x,t) = 0 fort≤ 0;
  • (N2)x= 0 if and only ifN(x,t) = 1 for allt> 0;
  • (N3)ifc≠= 0;
  • (N4)N(x+y,s+t) ≥ min{N(x,s),N(y,t)};
  • (N5)N(x, ·) is a non-decreasing function of ℝ and limt→∞N(x,t) = 1;
  • (N6) forx≠= 0,N(x, ·) is continuous on ℝ.
The pair ( X , N ) is called a fuzzy normed vector space .
The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [24 , 27] .
Definition 1.2 ( [2 , 24 , 25 , 26] ). Let ( X , N ) be a fuzzy normed vector space. A sequence { x n } in X is said to be convergent or converge if there exists an x X such that lim n→∞ N ( x n x , t ) = 1 for all t > 0. In this case, x is called the limit of the sequence { x n } and we denote it by N -lim n→∞ x n = x .
Definition 1.3 ( [2 , 24 , 25] ). Let ( X , N ) be a fuzzy normed vector space. A sequence { x n } in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n 0 ∈ ℕ such that for all n n 0 and all p > 0, we have N ( x n+p x n , t ) > 1 − ε .
It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space .
We say that a mapping f : X Y between fuzzy normed vector spaces X and Y is continuous at a point x 0 X if for each sequence { x n } converging to x 0 in X , then the sequence { f ( x n )} converges to f ( x 0 ). If f : X Y is continuous at each x X , then f : X Y is said to be continuous on X (see [3] ).
Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies
  • (1)d(x,y) = 0 if and only ifx=y;
  • (2)d(x,y) =d(y,x) for allx,y∈X;
  • (3)d(x,z) ≤d(x,y) +d(y,z) for allx,y,z∈X.
  • We recall a fundamental result in fixed point theory.
Theorem 1.4 ( [4 , 10] ). 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 , either
  • d(Jnx,Jn+1x) = ∞
for all nonnegative integers n or there exists a positive integer n 0 such that
  • (1), ∀n≥n0;
  • (2)the sequence{Jnx}converges to a fixed point y* ofJ;
  • (3)y*is the unique fixed point of J in the set Y= {y∈X|d(Jn0x,y) < ∞};
  • (4)for all y∈Y.
In 1996, G. Isac and Th.M. Rassias [15] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [5 , 6 , 27 , 29 , 30 , 36] ).
Starting with the paper [24] , the stability of some functional equations in the framework of fuzzy normed spaces or random normed spaces has been investigated (see e.g., [21 , 22 , 23 , 24 , 25 , 26 , 27 , 28] ).
This paper is organized as follows: In Section 2, we prove the Hyers-Ulam stability of the functional equation (0.1) in fuzzy Banach spaces for an odd case. In Section 3, we prove the Hyers-Ulam stability of the functional equation (0.1) in fuzzy Banach spaces for an even case.
Throughout this paper, assume that X is a vector space and that ( Y , N ) is a fuzzy Banach space. Let l be a fixed positive integer.
2. HYERS-ULAM STABILITY OF THE FUNCTIONAL EQUATION (0.1): AN ODD CASE
For a given mapping f : X Y , we define
PPT Slide
Lager Image
for all x 1 , ⋯ , x 2l X .
Using the fixed point method, we prove the Hyers-Ulam stability of the functional equation C f ( x 1 , ⋯ , x 2l ) = 0 in fuzzy Banach spaces: an odd case.
Theorem 2.1 . Let φ : X 2l → [0,∞) and
PPT Slide
Lager Image
be functions such that there exists an L < 1 with
PPT Slide
Lager Image
for all x 1 , ⋯ , x 2l X . Let f : X Y be an odd mapping satisfying
PPT Slide
Lager Image
for all x 1 , ⋯ , x 2l X and all t > 0. Then
PPT Slide
Lager Image
exists for each x X and defines an additive mapping A : X Y such that
PPT Slide
Lager Image
for all x X and all t > 0.
Proof . Letting x 1 = ⋯ = x l = x and x l+1 = ⋯ = x 2l = 0 in (2.1), we get
PPT Slide
Lager Image
for all x X .
Consider the set
PPT Slide
Lager Image
and introduce the generalized metric on S :
PPT Slide
Lager Image
where, as usual, inf ϕ = +∞. It is easy to show that ( S , d ) is complete. (See the proof of Lemma 2.1 of [22] .)
Now we consider the linear mapping J : S S such that
PPT Slide
Lager Image
for all x X .
Let g , h S be given such that d ( g , h ) = ε . Then
PPT Slide
Lager Image
for all x X and all t > 0. Hence
PPT Slide
Lager Image
for all x X and all t > 0. So d ( g , h ) = ε implies that d ( Jg, Jh ) ≤ L ε . This means that
PPT Slide
Lager Image
for all g, h S .
It follows from (2.3) that
PPT Slide
Lager Image
By Theorem 1.4, there exists a mapping A : X Y satisfying the following:
(1) A is a fixed point of J , i.e.,
PPT Slide
Lager Image
for all x X . Since f : X Y is odd, A : X Y is an odd mapping. The mapping A is a unique fixed point of J in the set
PPT Slide
Lager Image
This implies that A is a unique mapping satisfying (2.4) such that there exists a µ ∈ (0, ∞) satisfying
PPT Slide
Lager Image
for all x X ;
(2) d ( J n f, A ) → 0 as n → ∞. This implies the equality
PPT Slide
Lager Image
for all x X ;
(3)
PPT Slide
Lager Image
, which implies the inequality
PPT Slide
Lager Image
This implies that the inequality (2.2) holds.
By (2.1),
PPT Slide
Lager Image
for all x 1 , ⋯ , x 2l X , all t > 0 and all n ∈ ℕ. So
PPT Slide
Lager Image
for all x 1 , ⋯ , x 2l X , all t > 0 and all n ∈ ℕ. Since
PPT Slide
Lager Image
for all x 1 , ⋯ , x 2l X and all t > 0,
PPT Slide
Lager Image
for all x 1 , ⋯ , x 2l X and all t > 0. Thus CA ( x 1 , ⋯ , x 2l ) = 0. Since A is odd, it follows from Lemma 2.1 of [35] that the mapping A : X Y is additive, as desired.              ☐
Corollary 2.2. Let θ ≥ 0 and let p be a real number with p > 1. Let X be a normed vector space with norm ∥ · ∥. Let f : X Y be an odd mapping satisfying
PPT Slide
Lager Image
for all x 1 , ⋯ , x 2l X and all t > 0. Then
PPT Slide
Lager Image
exists for each x X and defines an additive mapping A : X Y such that
PPT Slide
Lager Image
for all x X and all t > 0.
Proof . The proof follows from Theorem 2.1 by taking
PPT Slide
Lager Image
for all x 1 , ⋯ , x 2l X . Then we can choose L = 2 1−p and we get the desired result.              ☐
Theorem 2.3. Let φ : X 2l → [0,∞) and
PPT Slide
Lager Image
be functions such that there exists an L < 1 with
PPT Slide
Lager Image
for all x 1 , ⋯ , x 2l X . Let f : X Y be an odd mapping satisfying (2.1). Then
PPT Slide
Lager Image
exists for each x X and defines an additive mapping A : X Y such that
PPT Slide
Lager Image
for all x X and all t > 0.
Proof . Let ( S , d ) be the generalized metric space defined in the proof of Theorem 2.1.
Consider the linear mapping J : S S such that
PPT Slide
Lager Image
for all x X .
It follows from (2.3) that
PPT Slide
Lager Image
for all x X and all t > 0. Thus
PPT Slide
Lager Image
for all x X and all t > 0. So
PPT Slide
Lager Image
.
The rest of the proof is similar to the proof of Theorem 2.1.               ☐
Corollary 2.4. Let θ ≥ 0 and let p be a real number with 0 < p < 1. Let X be a normed vector space with norm ∥ · ∥. Let f : X Y be an odd mapping satisfying (2.5). Then
PPT Slide
Lager Image
exists for each x X and defines an additive mapping A : X Y such that
PPT Slide
Lager Image
for all x X and all t >0.
Proof . The proof follows from Theorem 2.3 by taking
PPT Slide
Lager Image
for all x 1 , ⋯ , x 2l X . Then we can choose L = 2 p−1 and we get the desired result.               ☐
3. HYERS-ULAM STABILITY OF THE FUNCTIONAL EQUATION (0.1): AN EVEN CASE
In this section, using the fixed point method, we prove the Hyers-Ulam stability of the functional equation Cf ( x 1 , ⋯ , x 2l ) = 0 in fuzzy Banach spaces: an even case.
Theorem 3.1. Let φ : X 2l → [0,∞) and
PPT Slide
Lager Image
be functions such that there exists an L < 1 with
PPT Slide
Lager Image
for all x 1 , ⋯ , x 2l X . Let f : X Y be an even mapping satisfying f (0) = 0 and (2.1). Then
PPT Slide
Lager Image
exists for each x X and defines a quadratic mapping Q : X Y such that
PPT Slide
Lager Image
for all x X and all t > 0.
Proof . Letting x 1 = ⋯ = x l = x and x l+1 = ⋯ = x 2l = 0 in (2.1), we get
PPT Slide
Lager Image
for all x X .
Let ( S , d ) be the generalized metric space defined in the proof of Theorem 2.1. Now we consider the linear mapping J : S S such that
PPT Slide
Lager Image
for all x X .
Let g, h S be given such that d(g, h) = ε . Then
PPT Slide
Lager Image
for all x X and all t > 0. Hence
PPT Slide
Lager Image
for all x X and all t > 0. So d ( g, h ) = ε implies that d ( Jg, Jh ) ≤ L ε . This means that
PPT Slide
Lager Image
for all g, h S .
It follows from (3.2) that
PPT Slide
Lager Image
By Theorem 1.4, there exists a mapping Q : X Y satisfying the following:
(1) Q is a fixed point of J , i.e.,
PPT Slide
Lager Image
for all x X . Since f : X Y is even, Q : X Y is an even mapping. The mapping Q is a unique fixed point of J in the set
PPT Slide
Lager Image
This implies that Q is a unique mapping satisfying (3.3) such that there exists a µ ∈ (0, ∞) satisfying
PPT Slide
Lager Image
for all x X ;
(2) d ( J n f , Q ) → 0 as n → ∞. This implies the equality
PPT Slide
Lager Image
for all x X ;
(3)
PPT Slide
Lager Image
, which implies the inequality
PPT Slide
Lager Image
This implies that the inequality (3.1) holds.
The rest of the proof is similar to the proof of Theorem 2.1.               ☐
Corollary 3.2. Let θ ≥ 0 and let p be a real number with p > 2. Let X be a normed vector space with norm ∥ · ∥. Let f : X Y be an even mapping satisfying f (0) = 0 and (2.5). Then
PPT Slide
Lager Image
exists for each x X and defines a quadratic mapping Q : X Y such that
PPT Slide
Lager Image
for all x X and all t > 0.
Proof . The proof follows from Theorem 3.1 by taking
PPT Slide
Lager Image
for all x 1 , ⋯ , x 2l X . Then we can choose L = 2 2−p and we get the desired result.               ☐
Similarly, we can obtain the following. We will omit the proof.
Theorem 3.3. Let φ : X 2l → [0,∞) and
PPT Slide
Lager Image
be functions such that there exists an L < 1 with
PPT Slide
Lager Image
for all x 1 , ⋯ , x 2l X . Let f : X Y be an even mapping satisfying f (0) = 0 and (2.1). Then
PPT Slide
Lager Image
exists for each x X and defines a quadratic mapping Q : X Y such that
PPT Slide
Lager Image
for all x X and all t > 0.
Corollary 3.4. Let θ ≥ 0 and let p be a real number with 0 < p < 2. Let X be a normed vector space with norm ∥ · ∥. Let f : X Y be an even mapping satisfying f (0) = 0 and (2.5). Then
PPT Slide
Lager Image
exists for each x X and defines a quadratic mapping Q : X Y such that
PPT Slide
Lager Image
for all x X and all t > 0.
Proof . The proof follows from Theorem 3.3 by taking
PPT Slide
Lager Image
for all x 1 , ⋯ , x 2l X . Then we can choose L = 2 p−2 and we get the desired result.               ☐
References
Aoki T. 1950 On the stability of the linear transformation in Banach spaces J. Math. Soc. Japan 2 64 - 66    DOI : 10.2969/jmsj/00210064
Bag T. , Samanta S.K. 2003 Finite dimensional fuzzy normed linear spaces J. Fuzzy Math. 11 687 - 705
Bag T. , Samanta S.K. 2005 Fuzzy bounded linear operators Fuzzy Sets and Systems 151 513 - 547    DOI : 10.1016/j.fss.2004.05.004
Cădariu L. , Radu V. 2003 Fixed points and the stability of Jensen’s functional equation J. Inequal. Pure Appl. Math. Art. ID 4 4 (1)
Cădariu L. , Radu V. 2004 On the stability of the Cauchy functional equation: a fixed point approach Grazer Math. Ber. 346 43 - 52
Cădariu L. , Radu V. 2008 Fixed point methods for the generalized stability of functional equations in a single variable Fixed Point Theory and Applications Art. ID 749392 2008
Cheng S.C. , Mordeson J.M. 1994 Fuzzy linear operators and fuzzy normed linear spaces Bull. Calcutta Math. Soc. 86 429 - 436
Cholewa P.W. 1984 Remarks on the stability of functional equations Aequationes Math 27 76 - 86    DOI : 10.1007/BF02192660
Czerwik S. 1992 On the stability of the quadratic mapping in normed spaces Abh. Math. Sem. Univ. Hamburg 62 59 - 64    DOI : 10.1007/BF02941618
Diaz J. , Margolis B. 1968 A fixed point theorem of the alternative for contractions on a generalized complete metric space Bull. Amer. Math. Soc. 74 305 - 309    DOI : 10.1090/S0002-9904-1968-11933-0
Felbin C. 1992 Finite dimensional fuzzy normed linear spaces Fuzzy Sets and Systems 48 239 - 248    DOI : 10.1016/0165-0114(92)90338-5
Găvruta P. 1994 A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings J. Math. Anal. Appl. 184 431 - 436    DOI : 10.1006/jmaa.1994.1211
Hyers D.H. 1941 On the stability of the linear functional equation Proc. Nat. Acad. Sci. U.S.A 27 222 - 224    DOI : 10.1073/pnas.27.4.222
Hyers D.H. , Isac G. , Rassias M. 1998 Stability of Functional Equations in Several Variables Birkhäuser Basel
Isac G. , Rassias M. 1996 Stability of ψ-additive mappings: Appications to nonlinear analysis Internat. J. Math. Math. Sci. 19 219 - 228    DOI : 10.1155/S0161171296000324
Jung S. 2001 Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis Hadronic Press lnc. Palm Harbor, Florida
Jung S. 2009 Approximation of analytic functions by Hermite functions Bull. Sci. Math. 133 756 - 764    DOI : 10.1016/j.bulsci.2007.11.001
Katsaras A.K. 1984 Fuzzy topological vector spaces II Fuzzy Sets and Systems 12 143 - 154    DOI : 10.1016/0165-0114(84)90034-4
Kramosil I. , Michalek J. 1975 Fuzzy metric and statistical metric spaces Kybernetica 11 326 - 334
Krishna S.V. , Sarma K.K.M. 1994 Separation of fuzzy normed linear spaces Fuzzy Setsand Systems 63 207 - 217    DOI : 10.1016/0165-0114(94)90351-4
Miheţ D. 2009 The fixed point method for fuzzy stability of the Jensen functional equation Fuzzy Sets and Systems 160 1663 - 1667    DOI : 10.1016/j.fss.2008.06.014
Miheţ D. , Radu V. 2008 On the stability of the additive Cauchy functional equation in random normed spaces J. Math. Anal. Appl. 343 567 - 572    DOI : 10.1016/j.jmaa.2008.01.100
Mirmostafaee A.K. 2009 A fixed point approach to almost quartic mappings in quasi fuzzy normed spaces Fuzzy Sets and Systems 160 1653 - 1662    DOI : 10.1016/j.fss.2009.01.011
Mirmostafaee A.K. , Mirzavaziri M. , Moslehian M.S. 2008 Fuzzy stability of the Jensen functional equation Fuzzy Sets and Systems 159 730 - 738    DOI : 10.1016/j.fss.2007.07.011
Mirmostafaee A.K. , Moslehian M.S. 2008 Fuzzy versions of Hyers-Ulam-Rassias theorem Fuzzy Sets and Systems 159 720 - 729    DOI : 10.1016/j.fss.2007.09.016
Mirmostafaee A.K. , Moslehian M.S. 2008 Fuzzy approximately cubic mappings Inform. Sci. 178 3791 - 3798    DOI : 10.1016/j.ins.2008.05.032
Mirzavaziri M. , Moslehian M.S. 2006 A fixed point approach to stability of a quadratic equation Bull. Braz. Math. Soc. 37 361 - 376    DOI : 10.1007/s00574-006-0016-z
Mirmostafaee A.K. , Moslehian M.S. 2009 Fuzzy stability of additive mappings in nonArchimedean fuzzy normed spaces Fuzzy Sets and Systems 160 1643 - 1652    DOI : 10.1016/j.fss.2008.10.011
Park C. 2007 Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras Fixed Point Theory and Applications Art. ID 50175 2007
Park C. 2008 Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach Fixed Point Theory and Applications Art. ID 493751 2008
Park C. , Cho Y. , Han M. 2007 Functional inequalities associated with Jordan-von Neumann type additive functional equations J. Inequal. Appl. Art. ID 41820 2007
Park C. , Cui J. 2007 Generalized stability of C*-ternary quadratic mappings Abstract Appl. Anal. Art. ID 23282 2007
Park C. , Lee J. , Shin D. Quadratic mappings associated with inner product spaces (preprint)
Park C. , Najati A. 2007 Homomorphisms and derivations in C*-algebras Abstract Appl. Anal. Art. ID 80630 2007
Park C. , Park W. , Najati A. 2009 Functional equations related to inner product spaces Abstract Appl. Anal. Art. ID 907121 2009
Radu V. 2003 The fixed point alternative and the stability of functional equations Fixed Point Theory 4 91 - 96
Rassias J.M. 1984 On approximation of approximately linear mappings by linear mappings Bull. Sci. Math. 108 445 - 446
Rassias J.M. 2007 Refined Hyers-Ulam approximation of approximately Jensen type mappings Bull. Sci. Math. 131 89 - 98    DOI : 10.1016/j.bulsci.2006.03.011
Rassias J.M. , Rassias M.J. 2005 Asymptotic behavior of alternative Jensen and Jensen type functional equations Bull. Sci. Math. 129 545 - 558    DOI : 10.1016/j.bulsci.2005.02.001
Rassias M. 1978 On the stability of the linear mapping in Banach spaces Proc. Amer. Math. Soc. 72 297 - 300    DOI : 10.1090/S0002-9939-1978-0507327-1
Rassias M. 1984 New characterizations of inner product spaces Bull. Sci. Math. 108 95 - 99
Rassias M. 1990 Problem 16; 2. Report of the 27th International Symp. on Functional Equations Aequationes Math. 39 292 - 293; 309
Rassias M. 1998 On the stability of the quadratic functional equation and its applications Studia Univ. Babes-Bolyai XLIII 89 - 124
Rassias M. 2000 The problem of S.M. Ulam for approximately multiplicative mappings J. Math. Anal. Appl. 246 352 - 378    DOI : 10.1006/jmaa.2000.6788
Rassias M. 2000 On the stability of functional equations in Banach spaces J. Math. Anal. Appl. 251 264 - 284    DOI : 10.1006/jmaa.2000.7046
Rassias M. 2000 On the stability of functional equations and a problem of Ulam Acta Appl. Math. 62 23 - 130    DOI : 10.1023/A:1006499223572
Rassias M. , Šemrl P. 1992 On the behaviour of mappings which do not satisfy Hyers- Ulam stability Proc. Amer. Math. Soc. 114 989 - 993    DOI : 10.1090/S0002-9939-1992-1059634-1
Rassias M. , Šemrl P. 1993 On the Hyers-Ulam stability of linear mappings J. Math. Anal. Appl. 173 325 - 338    DOI : 10.1006/jmaa.1993.1070
Rassias M. , Shibata: K. 1998 Variational problem of some quadratic functionals in complex analysis J. Math. Anal. Appl. 228 234 - 253    DOI : 10.1006/jmaa.1998.6129
Skof F. 1983 locali e approssimazione di operatori Rend. Sem. Mat. Fis. Milano 53 113 - 129    DOI : 10.1007/BF02924890
Ulam S.M. 1960 A Collection of the Mathematical Problems Interscience Publ New York
Xiao J.Z. , Zhu X.H. 2003 Fuzzy normed spaces of operators and its completeness Fuzzy Sets and Systems 133 389 - 399    DOI : 10.1016/S0165-0114(02)00274-9