On the Hyers-Ulam Stability Problem for Quadratic Multi-dimensional Mappings on the Gaussian Plane

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1 Southeast Asian Bulletin of Mathematics (00) 6: Southeast Asian Bulletin of Mathematics : Springer-Verlag 00 On the Hyers-Ulam Stability Problem f Quadratic Multi-dimensional Mappings on the Gaussian Plane John Michael Rassias Pedagogical Department, E. E., National and Capodistrian University of Athens, Section of Mathematics and Infmatics, 4, Agamemnonos Str., Aghia Paraskevi, Athens 1534, Greece jrassias@primedu.uoa.gr AMS Subject Classification (1991): 39B Abstract. In this paper we solve the Hyers-Ulam stability problem f quadratic multidimensional mappings on the Gaussian plane. Keywds: Ulam Problem, Hyers-Ulam Stability, Quadratic Weighted Means, Gaussian Plane, Quadratic Mapping, Mean Equation, Fundamental Equation 1. Introduction In 1968 S.M. Ulam [17] proposed the general stability problem: When is it true that by changing a little the hypotheses of a theem one can still assert that the thesis of the theem remains true approximately true?. Accding to P.M. Gruber [3] this kind of stability problems is of particular interest in probability they and in the case of functional equations of di erent types. In we ([5] [15]) solved the above Ulam problem f di erent mappings. In this paper we first introduce new quadratic weighted means and fundamental functional equations and then solve the Hyers-Ulam stability problem ([4], [17] [18]) f quadratic mappings Q : Z! W on the Gaussian plane C, satisfying a mean equation and the functional equation Q Xn a i z i! þ X Qða j z i a i z j Þ¼m Xn Qðz i Þ; 1ai< jan f every complex z i A Z ði ¼ 1; ;...; nþ, and fixed complex a i ði ¼ 1; ;...; nþ: 0 < m ¼ Xn ja i j 0 1 þ n n;

2 484 J.M. Rassias where Z and W are complex linear spaces (on a field of characteristic 0 ). We note that in we ([13], [14]) established the cresponding case n ¼ on the real plane R. To the best of our knowledge the afe-mentioned functional equations are established f the first time f n > and are the most general quadratic functional equations of this kind till now.. Fundamental Functional Equation of First Type Let Z and W be complex linear spaces (on a field of characteristic 0 ). Then we consider a non-linear mapping Q : Z! W satisfying the fundamental functional equation of first type Qða 1 zþþ Xn Qða j zþ¼ Xn Qða i zþ; j¼ ðþ (z ¼ conjugate of z, and a i ¼ conjugate of a i ), with 1 < m ¼ Xn ja i j 0 1 þ n n; n ¼ ; 3; 4;...; and every z A Z, as well as any fixed a ¼ða 1 ; a ;...; a n Þ A C n, a 0 0. We note that if z i, a i ði ¼ 1; Þ A R, then m ¼ X a i > 1; and equation ðþ is an identity in Z. Thus ðþ is not required. Definition.1. Let Z and W be linear spaces (on a field of characteristic 0 ). Let a ¼ða 1 ; a ;...a n Þ 0 ð0; 0;...; 0Þ with a i A C ði ¼ 1; ;...; nþ. Then a mapping Q : Z! W is called quadratic with respect to a if the functional equation Q Xn a i z i! þ X Qða j z i a i z j Þ¼m Xn Qðz i Þ; 1ai< jan ð1þ holds f every vect ðz 1 ; z ;...; z n Þ A Z n, and a fixed real a 0 0:

3 On the Hyers-Ulam Stability Problem f Quadratic Multi-dimensional Mappings < m ¼ Xn ja i j 0 1 þ n n; ([5] [15]). We note that if we set QðzÞ ¼jzj f z A C, then the mapping Q : C! R is quadratic with respect to any a A C n, a 0 0. Also if Q : C! R is quadratic with respect to any a A C n, a 0 0, then we have QðzÞ ¼Qð1Þjzj. Finally if Q : Z! W is quadratic with respect to a ¼ða 1 ; a ;...; a n Þ A C n such that 1 < m ¼ Pn ja i j 0 1 þ n n, then we have Qðm n zþ¼ðm n Þ QðzÞ; ðþ f all z A Z and all n A N. In fact, substitution of z i ¼ 0 ði ¼ 1; ;...; nþ in Eq. (1) yields Qð0Þþ n Qð0Þ ¼m½nQð0ÞŠ; mn 1 þ n Qð0Þ ¼0; Qð0Þ ¼0: ð1aþ Substituting z 1 ¼ z, z j ¼ 0 ð j ¼ ; 3;...; nþ in Eq. (1) and employing (1a) one gets that Qða 1 zþþ Xn j¼ Qða j zþþ n 1 Qð0Þ ¼m½QðzÞþðn 1ÞQð0ÞŠ; Qða 1 zþþ Xn j¼ Qða j zþ¼mqðzþ; ðaþ holds f all z A Z. Meover substitution of z i ¼ a i z ði ¼ 1; ;...; nþ in Eq. (1) and using (1a) one finds

4 486 J.M. Rassias QðmzÞþ n Qð0Þ ¼m Xn Qða i zþ; X n Qða i zþ¼m 1 QðmzÞ; ðbþ holds f all z A Z. Functional equations (a) (b) yield QðmzÞ ¼m QðzÞ ðcþ f all z A Z. Then induction on n A N with z replaced by m n 1 z yields equation (). Definition.. Let Z and W be linear spaces (on a field of characteristic 0 ). Let a ¼ða 1 ; a ;...; a n Þ A C n be 0 ð0; 0;...; 0Þ. F z A Z and set 1 < m ¼ Xn ja i j 0 1 þ n n; n ¼ ; 3; 4;...; and Q1 a ðzþ ¼Xn Qða i zþ, X n ja i j ; ð3þ 1 " #, Q a ðzþ ¼ Qða 1zÞþ Xn X n Qða j zþ j¼ ja i j : ð3þ Then the mappings Qi a ði ¼ 1; Þ : Z! W are the quadratic weighted means of second and first fm with respect to a, respectively. We note that if Q : Z! W is quadratic with respect to a A C n : a 0 0, then we have the mean functional equation with respect to a Q1 a ðzþ ¼Qa ðzþ ¼QðzÞ ½Š f every z A Z and i ¼ 1;, which is equivalent to the fundamental equation ðþ.

5 On the Hyers-Ulam Stability Problem f Quadratic Multi-dimensional Mappings 487 Theem.1. Let Z be a complex nmed linear space and let W be a complete nmed linear space (on a field of characteristic 0 ). Let a ¼ða 1 ; a ;...; a n Þ A C n : a 0 0 with 1 < m ¼ Pn ja i j 0 1 þ n n, f n ¼ ; 3; 4;...: Assume that f : Z! W is a mapping f which there exists e b 0(:¼a constant independent of z i ) such that the functional inequality f X n! a i z i þ Xn 1ai< jan f ða j z i a i z j Þ m Xn f ðz i Þ a e ð4þ holds f every vect ðz 1 ; z ;...; z n Þ A Z n. Assume in addition that f : Z! Wisa mapping such that the fundamental functional inequality k f a ðzþ f a 1 ðzþk a d, X n ja i j ; ð4þ 1 holds f every complex z A Z, d b 0(:¼a constant independent of z) and any fixed a 0 0, where " #, f a ðzþ ¼ f ða 1zÞþ Xn X n f ða j zþ j¼ ja i j ; and f1 a ðzþ ¼Xn f ða i zþ, X n are quadratic weighted means of second and first fm, with respect to a, respectively. Then the limit ja i j QðzÞ ¼ lim n!y m n f ðm n zþ ð5þ exists f every z A Z and Q : Z! W is the unique quadratic mapping with respect to a such that k f ðzþ QðzÞk a c; ð6þ and QðzÞ ¼m n Qðm n zþ; ð6aþ hold f every complex z A Z with constant c ¼ 1 m mn 1 þ n þ mðn 1Þjm n þ jþ mn 1 þ n þ n e þ m mn 1 þ n d ðm 1Þ mn 1 þ n :

6 488 J.M. Rassias F instance, f n ¼, we get m > 1 and c ¼ 1 ð3m 1Þe þ mðm 1Þd ðm 1Þ : ðm þ 1Þ Note also that f z i ¼ x i A R ði ¼ 1; ;...; nþ we don t need inequality (4) 1. In this real case the above-mentioned constant c doesn t include the term m mn 1 þ n d in the numerat. Proof of Existence in Theem.1. Substitution of z i ¼ 0 ði ¼ 1; ;...; nþ in inequality (4) yields that f ð0þþ X 1ai< jan f ð0þ m Xn f ð0þ a e; 8 1 þ n e k f ð0þk a e mn 1 þ n mn 1 þ n ; m > n >< ¼ 1 þ n ; e 1 þ n ; 1 < m < n >: ð7þ 1 < m 0 1 þ n n. Meover substituting z 1 ¼ z, z j ¼ 0 ð j ¼ ; 3;...; nþ in inequality (4) and employing (7) as well as the triangle inequality one concludes the functional inequality f ða 1 zþþ Xn j¼ f ða j zþþ n 1 f ð0þ m½fðzþþðn 1Þfð0ÞŠ a e; f a ðzþþ 1 n 1 f ð0þ fðzþ ðn 1Þfð0Þ m a e=m;

7 On the Hyers-Ulam Stability Problem f Quadratic Multi-dimensional Mappings 489 f a ðzþ f ðzþ ðn 1Þðm n þ Þ m k f a ðzþ f ðzþk a e m f ð0þ a e=m; ðn 1Þjm n þ j þ k f ð0þk; m k f a ðzþ f ðzþk a 1 mn 1 þ n þðn 1Þjm n þ j mmn 1 þ n e; ð8þ where " #, f a ðzþ ¼ f ða 1zÞþ Xn X n f ða j zþ j¼ ja i j : ð8aþ In addition replacing z i ¼ a i z ði ¼ 1; ;...; nþ in inequality (4) and using (7) as well as the triangle inequality, one gets the functional inequality f ðmzþþ n f ð0þ m Xn f ða i zþ a e; k f1 a ðzþ m f ðmzþk a e m þ 1 n m k f ð0þk; k f a 1 ðzþ m f ðmzþk a mn 1 þ n þ n m mn 1 þ n e; ð9þ where f1 a ðzþ ¼Xn f ða i zþ, X n ja i j : ð9aþ

8 490 J.M. Rassias Meover f ða 1 zþþ k f a ðzþ f 1 a ðzþk ¼ P n j¼ P n f ða j zþ Pn f ða i zþ : ja i j Employing the fundamental functional inequality (4) 1, one gets the equivalent inequality f ða 1 zþþ Xn j¼ f ða j zþ Xn f ða i zþ a d: ð4aþ 1 Functional inequalities (8) (9) and (4) 1 ( (4a) 1 ) and the triangle inequality yield the basic inequality k f ðzþ m f ðmzþk a k f ðzþ f a ðzþk þ k f a ðzþ f a 1 ðzþk þ k f a 1 ðzþ m f ðmzþk; k f ðzþ m f ðmzþk a cð1 m Þ; with 1 < m 0 1 þ n n; ð10þ where c is given in Theem.1. F instance, ifn ¼ : a 1 ¼ a ¼ 1m¼ > 1 and z i ¼ x i A R ði ¼ 1; Þ z i ¼ x i A R ði ¼ 1; Þ, then there is no d-part in c because f1 a a ðzþ ¼f ðzþ f all real z ¼ x A R. Hence c ¼ð11=6Þe. We note that in this case a better constant c ¼ ð1=þe <ð11=6þe may be found if the new substitution z 1 ¼ z ¼ x A R is applied into inequality (4) with a i ¼ 1 ði ¼ 1; Þ. In fact, k f ðxþþfð0þ 4fðxÞk a e with k f ð0þk a e=, k f ðxþ 4f ðxþk a e þkf ð0þk a 3 e, k f ðxþ f ðxþk a cð1 Þ; c ¼ 1 e: ð11þ By induction on n A N with z replaced by m n 1 z in (10) we claim that the general functional inequality k f ðzþ m n f ðm n zþk a cð1 m n Þ; ð1þ holds f every complex z A Z, all n A N, and any fixed a 0 0 such that m > 1with m 0 1 þ n n.

9 On the Hyers-Ulam Stability Problem f Quadratic Multi-dimensional Mappings 491 In fact, the basic inequality (10) with z replaced by m n 1 z yields inequality k f ðm n 1 zþ m f ðm n zþk a cð1 m Þ; km ðn 1Þ f ðm n 1 zþ m n f ðm n zþk a cðm ðn 1Þ m n Þ; ð1aþ f all z A Z. By induction hypothesis with n replaced by n 1 in (1) inequality k f ðzþ m ðn 1Þ f ðm n 1 zþk a cð1 m ðn 1Þ Þ ð1bþ holds f all z A Z. Thus functional inequalities (1a) (1b) and the triangle inequality imply k f ðzþ m n f ðm n zþk a k f ðzþ m ðn 1Þ f ðm n 1 zþk þ km ðn 1Þ f ðm n 1 zþ m n f ðm n zþk a c½ð1 m ðn 1Þ Þþðm ðn 1Þ m n ÞŠ ¼ cð1 m n Þ; completing the proof of the required inequality (1). Claim now that the sequence fg n ðzþg : g n ðzþ ¼m n f ðm n zþ converges. Note that from the general inequality (1) and the completeness of W one proves that the above sequence is a Cauchy sequence. In fact, if i > j > 0, then kg i ðzþ g j ðzþk ¼ km i f ðm i zþ m j f ðm j zþk ¼ m j km ði jþ f ðm i zþ f ðm j zþk; ð13þ f all complex z A Z and all i; j A N. Setting h ¼ m j z in (13) and employing the general inequality (1) one concludes that kg i ðzþ g j ðzþk ¼ m j km ði jþ f ðm i j hþ f ðhþk a m j cð1 m ði jþ Þ¼cðm j m i Þ < cm j ; lim kg i ðzþ g j ðzþk < c@ lim j!y i>j 0 j!y i>j m j 1 A ¼ 0;

10 49 J.M. Rassias lim kg i ðzþ g j ðzþk ¼ 0; j!y i>j ð13aþ completing the proof that the sequence fg n ðzþg converges. Hence Q ¼ QðzÞ is a well-defined mapping via the fmula (5). This means that the limit (5) exists f all complex z A Z. In addition claim that mapping Q : Z! W satisfies the functional equation (1) f all complex z i A Z ði ¼ 1; ;...; nþ. In fact, it is clear from the functional inequality (4) and the limit (5) that inequality m n f X n a i m n z i! þ X f ða j m n z i a i m n z j Þ m Xn 1ai< jan f ðm n z i Þ a m n e; ð14þ holds f all complex z i A Z ði ¼ 1; ;...; nþ, and all n A N. Therefe "!# 0 a lim X n n!y m n f m n a i z i þ m Xn lim n!y m n f ðm n z i Þ a 0; X 1ai< jan lim n!y m n f ½m n ða j z i a i z j ÞŠ Q Xn a i z i! þ X Qða j z i a i z j Þ m Xn 1ai< jan Qðz i Þ ¼ 0; the mapping Q : Z! W satisfies the functional Eq. (1) f all complex z i A Z ði ¼ 1; ;...; nþ. Thus Q is a quadratic mapping with respect to a ¼ ða 1 ; a ;...; a n Þ A C n. It is clear now from (1), n! y, and (5) that inequality (6) holds in Z, completing the existence proof of Theem.1. Proof of Uniqueness in Theem.1. Let Q 0 : Z! W be another quadratic mapping satisfying Eq. (1), such that k f ðzþ Q 0 ðzþk a c; ð6þ 0 holds f all z A Z. If there exists a quadratic mapping Q : Z! W satisfying Eq. (1), then Q 0 1 Q: ð15þ

11 On the Hyers-Ulam Stability Problem f Quadratic Multi-dimensional Mappings 493 To prove the above-mentioned uniqueness ( Eq. (15)) we employ (6a) f Q and Q 0, as well, so that Q 0 ðzþ ¼m n Q 0 ðm n zþ; ð6aþ 0 holds f all z A Z, and all n A N. Meover the triangle inequality and the functional inequalities (6) (6) 0 yield kqðm n zþ Q 0 ðm n zþk a kqðm n zþ f ðm n zþk þ k f ðm n zþ Q 0 ðm n zþk; kqðm n zþ Q 0 ðm n zþk a c; ð16þ f all complex z A Z, and all n A N. Then from (6a) (6a) 0, and (16), one proves that kqðzþ Q 0 ðzþk ¼ km n Qðm n zþ m n Q 0 ðm n zþk; kqðzþ Q 0 ðzþk a ðm n Þc; ð16aþ holds f all z A Z, and all n A N. Therefe from (16a), and n! y, one establishes 0 a lim kqðzþ Q 0 ðzþk a n!y kqðzþ Q 0 ðzþk ¼ 0; lim m n n!y c ¼ 0 QðzÞ ¼Q 0 ðzþ ð17þ f all z A Z, completing the proof of uniqueness and thus the stability of Theem Fundamental Functional Equation of Second Type Let Z and W be linear spaces (on a field of characteristic 0 ). Then we consider a non-linear mapping Q : Z! W satisfying the fundamental functional equation of second type Q a 1 þ m z Xn Q a j m z ¼ Xn j¼ Q a i m z ; ðþ

12 494 J.M. Rassias with 0 < m ¼ Xn ja i j < 1 and every z A Z, as well as any fixed a ¼ða 1 ; a ;...; a n Þ A C n, a 0 0. We note that if z i, a i ði ¼ 1; Þ A R, then 0 < m ¼ X and equation ðþ is an identity in Z. Thus ðþ is not required. Meover the functional equation holds f all complex z A Z, and all n A N: a i < 1; Qðm n zþ¼ðm n Þ QðzÞ ðþ 0 0 < m < 1: In fact substitution of z i ¼ 0 ði ¼ 1; ;...; nþ in Eq. (1) yields Qð0Þ ¼0: ð1aþ 0 Substituting z 1 ¼ z m, z j ¼ 0 ð j ¼ ; 3;...; nþ in Eq. (1) and employing (1a) 0 one finds that Q a 1 þ m z Xn j¼ Q a j m z ¼ mqðm 1 zþ; ðaþ 0 holds f all complex z A Z. In addition substituting z i ¼ a i z ði ¼ 1; ;...; nþ in Eq. (1) and employing (1a)0 m one gets that X n holds f all complex z A Z. Functional equations (a) 0 (b) 0 and ðþ yield Q a i m z ¼ m 1 QðzÞ; ðbþ 0 Qðm 1 zþ¼ðm 1 Þ QðzÞ; ðcþ 0 f all z A Z. Then induction on n A N with z replaced by m ðn 1Þ z yields equation () 0.

13 On the Hyers-Ulam Stability Problem f Quadratic Multi-dimensional Mappings 495 Definition 3.1. Let Z and W be linear spaces (on a field of characteristic 0 ). Let a ¼ða 1 ; a ;...; a n Þ A C n be 0ð0; 0;...; 0Þ and b ¼ðb 1 ; b ;...; b n Þ A C n : b i ¼ a i =m ði ¼ 1; ;...; nþ. F z A Z and set and 0 < m ¼ Xn, X n Q1 b ðzþ ¼Xn Qðb i zþ ja i j < 1 jb i j ¼ m Xn Q ða i =bþz ; ð3þ 0 1 " #, Q b ðzþ ¼ Qðb 1zÞþ Xn X n Qðb j zþ j¼ jb i j " ¼ m Q ða 1 =mþz þ Xn Q ða j =mþz # : ð3þ 0 Then the mappings Qi b ði ¼ 1; Þ : Z! W are the quadratic weighted means of second and first fm with respect to b, respectively. We note that the fundamental functional equation ðþ is equivalent to the mean functional equation with respect to b Q1 b ðzþ ¼Qb ðzþ ¼QðzÞ ½Š f every complex z A Z. Note that the functional equation (a) 0 comes from Eq. (a) if we replace z by z=m. But this z-substitution (z by z/m) does not yield Eq. () 0 directly from Eq. (). We also note that the z-substitution the a-substitution (a i by a i =m : i ¼ 1; ) does not yield Eq. (b) 0 directly from Eq. (b). Such problems in the transition from the first section to the second section arise many times in this paper. These reasons fced us to add this second section separately. Theem 3.1. Let Z be a complex nmed linear space and let W be a complete nmed linear space (on a field of characteristic 0 ). Let a ¼ða 1 ; a ;...; a n Þ A C n : a 0 0 with 0 < m ¼ Pn ja i j < 1. Assume that f : Z! W is a mapping f which there exists e b 0(:¼a constant independent of z i ) such that the functional inequality (4) holds f every vect ðz 1 ; z ;...; z n Þ A Z n. Assume in addition that f : Z! W is a mapping f which the fundamental functional inequality j¼ k f b ðzþ f 1 b Xn ðzþk a ja i j!d 0 ; ð4þ 0 1

14 496 J.M. Rassias holds f every z A Z, d 0 b 0(:¼a constant independent of z), and any fixed a 0 0, where! f1 b Xn X ðzþ ¼ ja i j n f ða i =mþz! ; and!" f b Xn ðzþ ¼ ja i j f ða 1 =mþz þ Xn f ða j =mþz # j¼ are quadratic weighted means of second and first fm with respect to b, respectively. Then the limit QðzÞ ¼ lim n!y m n f ðm n zþ ð5þ 0 exists f every z A Z and Q : Z! W is the unique quadratic mapping with respect to b ¼ðb 1 ; b ;...; b n Þ 0 0, such that holds f every z A Z with constant k f ðzþ QðzÞk a c 0 ; ð6þ 00 ( c 0 ¼ 1 m 1 þ n! þ mðn 1Þjn mjþ 1 þ n!) e þ m 1 þ n ð1 m Þ 1 þ n! d 0! : Note that f 0 < m < 1 we have 1 þ n > 0 f all n A N. Take n ¼, then 0 < m < 1 and k f ðzþ QðzÞk a 1 ð3 m Þe þ mð1 mþd 0 ð1 mþ ; ð1 þ mþ ([13]). To prove the Existence in Theem.1 it is enough to prove that the general functional inequality k f ðzþ m n f ðm n zþk a c 0 ð1 m n Þ; 0 < m < 1; ð1þ 0 holds f every z A Z and all n A N. In fact, substitution of z i ¼ 0 ði ¼ 1; ;...; nþ in inequality (4) yields, k f ð0þk a e 1 þ n! : ð7þ 0

15 On the Hyers-Ulam Stability Problem f Quadratic Multi-dimensional Mappings 497 Substituting z 1 ¼ z=m, z j ¼ 0 ð j ¼ ; 3;...; nþ in inequality (4) and (7) 0 as well as the triangle inequality one gets the functional inequality f ða 1 =mþz þ Xn j¼ f ða j =mþz þ n 1 f ð0þ m½fðm 1 zþþðn 1Þfð0ÞŠ a e; k f b ðzþ m f ðm 1 ðn 1Þjn mj zþk a m e þ k f ð0þk ; k f b ðzþ m f ðm 1 zþk a m 1þ n þðn 1Þjn mj 1 þ n e; ð8þ 0 where f b ðzþ is the quadratic weighted mean of first fm with respect to b 0 0 : 0 < m < 1. In addition replacing z i ¼ða i =mþz ði ¼ 1; ;...; nþ in inequality (4) and (7) 0 as well as the triangle inequality, one concludes the functional inequality f ðzþþ n f ð0þ m Xn f ða i =mþz a e; n 1 þ k f1 b ðzþ fðzþk a 1 þ n e; ð9þ 0 where f1 b ðzþ is the quadratic weighted mean of second fm with respect to b 0 0 : 0 < m < 1. Employing the fundamental functional inequality (4) 1 0 one gets the equivalent inequality a 1 f þ m z Xn a j f m z Xn a i f m z a d0 ; m A ð0; 1Þ; ð4aþ 0 1 j¼ f all complex z A Z.

16 498 J.M. Rassias Functional inequalities (8) 0 (9) 0 and (4) 1 0 ( (4a)0 1 ) and the triangle inequality yield the basic inequality k f ðzþ m f ðm 1 zþk a k f ðzþ f1 b ðzþk þ k f 1 b ðzþ f b ðzþk þ k f b ðzþ m f ðm 1 zþk m 1 þ n þ mðn 1Þjn mjþ 1þ n a 1þ n e þ md 0 ; k f ðzþ m f ðm 1 zþk m 1 þ n þ mðn 1Þjn mjþ 1þ n a 1þ n e þ m 1 þ n d 0 k f ðzþ m f ðm 1 zþk a c 0 ð1 m Þ; 0 < m < 1 ð10þ 0 where c is given in Theem 3.1. F instance, if we wk in R such that z i ¼ x i A R ði ¼ 1; Þ and a i ¼ 1 ði ¼ 1; Þ, then z i ¼ x i ði ¼ 1; Þ and m ¼ 1 A ð0; 1Þ. In this case there is no d0 -part in c 0 because f1 b b ðzþ ¼f ðzþ f all real z A R. Hence c0 ¼ð11=3Þc. Note that a better constant c 0 ¼ð1=Þe <ð11=3þe may be found if new substitutions z i ¼ x i ¼ x ði ¼ 1; Þ, n ¼ are applied into inequality (4) with a i ¼ 1 ði ¼ 1; Þ. Replacing now z with m ðn 1Þ z in (10) 0 one concludes that k f ðm ðn 1Þ zþ m f ðm n zþk a c 0 ð1 m Þ; km ðn 1Þ f ðm ðn 1Þ zþ m n f ðm n zþk a c 0 ðm ðn 1Þ m n Þ; ð1aþ 0 f all z A Z. By induction on n A N we claim that the general functional inequality (1) 0 holds. By induction hypothesis with n! n 1 in inequality (1) 0 holds f all z A Z. k f ðzþ m ðn 1Þ f ðm ðn 1Þ zþk a c 0 ð1 m ðn 1Þ Þ; ð1bþ 0

17 On the Hyers-Ulam Stability Problem f Quadratic Multi-dimensional Mappings 499 Thus the functional inequalities (1a) 0 (1b) 0 and the triangle inequality imply k f ðzþ m n f ðm n zþk a k f ðzþ m ðn 1Þ f ðm ðn 1Þ zþk þ km ðn 1Þ f ðm ðn 1Þ zþ m n f ðm n zþk a c 0 ½ð1 m ðn 1Þ Þþðm ðn 1Þ m n ÞŠ ¼ c 0 ð1 m n Þ; completing the proof of the required inequality (1) 0. The rest of the proof of Theem 3.1 is omitted as similar to the cresponding proof of Theem.1 ([1], [], [3], [4], [16], [17] [18]). Example. Let us take f : C! C such that f ðzþ ¼ljzj þ k, where l, k (¼complex constants): jkj a e 1 þ n, f any complex a i ði ¼ 1; ;...; nþ : 0 < m ¼ Pn ja i j < 1, and n A N : n b. Meover there exists a unique quadratic mapping Q : C! C such that QðzÞ ¼ lim n!y m n ½ljm n zj þ kš ¼ljzj ; 0 < m < 1: Therefe inequality (6) 00 holds. In fact, the condition on l; k A C such that e jkj a 1 þ n ; m A ð0; 1Þ and n A N : n b : But f ðzþ ¼ljzj þ k satisfies (4) if k þ n k k a e, if Also 1 < m 1 þ n f 0 < m < 1 and n A N : n b. In fact, (18) is equivalent to e jkj a 1 þ n :! þ mðn 1Þjn mjþ 1 þ n! 1 m ; Rðm; nþ ¼ðn 1Þm 1þ n n m 4 n f all m A R : 0 < m < 1 and any n A N : n b. ð18þ mðn 1Þjm n þ j < 0 ð19þ

18 500 J.M. Rassias To prove (19) we first establish the cases: n ¼ and n ¼ 3, separately, and then prove the general case f all m A R : 0 < m < 1 and any n A N : n b 4, as follows: Also Rðm; Þ ¼m 4 mjmj ¼ 4 < 0; as m A ð0; 1Þ: Rðm; 3Þ ¼4m m 1 mjm 1j < 0; because R m A 0; 1! ; 3 ¼ 4m m 1 þ mðm 1Þ ¼ 8m 4m 1 ¼ 4ðm 3Þðm þ 1Þ < 0; as m A 0; 1 ; and R m A! 1 ; 1 ; 3 ¼ 4m m 1 mðm 1Þ ¼ 1 < 0; as m A 1 ; 1 : Finally claim R ¼ R m A ð0; 1Þ; n A N : n b 4 ¼ ðn 1Þm 1þ n n m 4 n þ mðn 1Þðm n þ Þ < 0; where m n þ < 0 equivalently 0 < m < n 1 yields m A ð0; 1Þ f n b 4 as 1 a n 1 f n b 4. This is equivalent to R ¼½ðn 1Þþðn 1ÞŠm 1þ n n þðn 1Þðn Þ m 4 n < 0; R 1 ¼ R=ðn 1Þ ¼m ðn Þm n < 0;

19 On the Hyers-Ulam Stability Problem f Quadratic Multi-dimensional Mappings 501 R 1 ¼ðm nþðm þ 1Þ < 0: ð0þ f all n A N : n b 4, and m A ð0; 1Þ. If m 1, m are the roots of the equation of the equation with respect to m, then R 1 ¼ðm nþðm þ 1Þ ¼0; R ¼ ðn 1Þðm nþðm þ 1Þ ¼0; m 1 ¼ 1; m ¼ n=: It is clear that m A ð0; 1Þ H ðm 1 ; m Þ¼ð 1; n=þ; f all n A N : n b 4. Therefe the proof of the inequality (0) is complete. Thus the inequality (18) holds. We note that 1 þ n ¼ 1 c 0 m 1 þ n ð1 m Þ ðn 1Þðn mþ ¼ þ 1 1 m ¼ ðn n þ 1Þ ðn 1Þm > 0 ; 1 m f all n A N : n b 4 and m A ð0; 1Þ. Therefe þ mðn 1Þjn mjþ 1þ n k f ðzþ QðzÞk ¼ jkj a c 0 ; 0 < m < 1: We note that if 1 < m 0 1 þ n n, f n A N : n b, then l ¼Qð1Þ e A C and any complex constant k : jkj a mn 1 þ n :

20 50 J.M. Rassias References 1. Aczél, J.: Lectures on Functional Equations and their Applications, Academic Press, New Yk and London, Fti, G.L.: Hyers-Ulam stability of functional equations in several variables, Aequationes Math. 50, (1995). 3. Gruber, P.M.: Stability of Isometries, Trans. Amer. Math. Soc. 45, (1978). 4. Hyers, D.H.: On the stability of the linear functional equations, Proc. Nat. Acad. Sci. 7, 4 (1941). 5. Rassias, J.M.: On Approximation of Approximately Linear Mappings by Linear Mappings, J. Funct. Anal. 46, (198). 6. Rassias, J.M.: On Approximation of Approximately Linear Mappings by Linear Mappings, Bull. Sci. Math. 108, (1984). 7. Rassias, J.M.: Solution of a Problem of Ulam, J. Approx. They. 57, (1989). 8. Rassias, J.M.: Complete Solution of the Multi-dimensional Problem of Ulam, Discuss. Math. 14, (1994). 9. Rassias, J.M.: Solution of a Stability Problem of Ulam, Discuss. Math. 1, (199). 10. Rassias, J.M.: On the Stability of the Euler-Lagrange Functional Equation, Chinese J. Math. 0, (199). 11. Rassias, J.M.: On the Stability of the Non-linear Euler-Lagrange Functional Equation in Real Nmed Linear Spaces, J. Math. Phys. Sci. 8, (1994). 1. Rassias, J.M.: On the Stability of the Multi-dimensional Non-linear Euler-Lagrange Functional Equation, in Geometry, Analysis and Mechanics, pp , Wld Scientific, Singape, Rassias, J.M.: On the Stability of the General Euler-Lagrange Functional Equation, Demonstratio Math. 9, (1996). 14. Rassias, J.M.: Solution of the Ulam Stability Problem f Euler-Lagrange Quadratic Mappings, J. Math. Anal. & Appl. 0, (1998). 15. Rassias, J.M.: Solution of the Ulam Stability Problem f 3-Dimensional Euler- Lagrange Quadratic Mappings, Mathematica Balkanica, (to appear), Székelyhidi, L.: Note on Hyers theem, C. R. Math. Rep. Acad. Sci. Canada 8, (1986). 17. Ulam, S.M.: A collection of Mathematical Problems, Intersci. Publ., New Yk, 1960 (Also : Problems in modern mathematics, Wiley, New Yk, 1964). 18. Ulam, S.M.: Sets, numbers and universes, Mass. Inst. Techn. Press, Cambridge, MA