ON THE GENERAL QUADRATIC FUNCTIONAL EQUATION

Bol. So. Mat. Mexiana (3) Vol. 11, 2005 ON THE GENERAL QUADRATIC FUNCTIONAL EQUATION JOHN MICHAEL RASSIAS Abstrat. In 1940 and in 1964 S. M. Ulam proposed the general problem: When is it true that by hanging a little the hypotheses of a theorem one an still assert that the thesis of the theorem remains true or approximately true?. In 1941 D. H. Hyers solved this stability problem for linear mappings. In 1951 D. G. Bourgin was the seond author to treat the same problem for additive mappings. Aording to P. M. Gruber (1978) this kind of stability problems are of partiular interest in probability theory and in the ase of funtional equations of different types. In 1981 F. Skof was the first author to solve the Ulam problem for quadrati mappings. In 1982 2002 we solved the above Ulam problem for linear and nonlinear mappings and established analogous stability problems even on restrited domains. Further, we applied some of our reent results to the asymptoti behavior of funtional equations of different types. The purpose of this paper is the stability result for generalized quadrati mappings. 1. Introdution In 1940 and in 1964 S. M. Ulam [27] proposed the general problem: When is it true that by hanging a little the hypotheses of a theorem one an still assert that the thesis of the theorem remains true or approximately true?. In 1941 D. H. Hyers [13] solved this stability problem for linear mappings. In 1951 D. G. Bourgin [3] was the seond author to treat the same problem for additive mappings. Aording to P. M. Gruber [12] (1978), this kind of stability problems are of partiular interest in probability theory and in the ase of funtional equations of different types. In 1978 Th. M. Rassias [22] employed Hyers ideas to new additive mappings. In 1981 and 1983 F. Skof [23], [24] was the first author to solve the Ulam problem for quadrati mappings. In 1982-2002 we ([16], [17], [18], [19], [20], [21]) solved the above Ulam problem for linear and nonlinear mappings and established analogous stability problems on restrited domains (see also [14]). Further, we applied some of our reent results to the asymptoti behavior of funtional equations of different types. In 1999 P. Gavruta [11] answered a question of ours [16] onerning the stability of Cauhy equation. In 1996 and 1998 we [19], [20] solved the Ulam stability problem for quadrati mappings Q : X Y satisfying the funtional equation Q(a 1 x 1 + a 2 x 2 )+Q(a 2 x 1 a 1 x 2 )=(a 2 1 + a 2 2)[Q(x 1 )+Q(x 2 )] for every x 1, x 2 X, and fixed reals a 1, a 2 0, where X and Y are real normed linear spaes. The purpose of this paper is the stability result for generalized 2000 Mathematis Subjet Classifiation: 39B. Keywords and phrases: Ulam problem, stability, general quadrati mapping. 259

260 JOHN MICHAEL RASSIAS quadrati mappings Q : X Y satisfying Q(0) = 0 and the following quadrati funtional equation ( ) (*) Q a i x i + Q(a j x i a i x j )=m Q(x i ) for every x i X(i =1, 2,...,p), and fixed a i 0(i =1, 2,...,p),a i R (i = 1, 2,...,p), where R := set of reals and p is arbitrary but fixed and equals to 2, 3, 4,...,suh that 0 <m= ai 2. If X and Y are normed linear spaes and Y is omplete, then we establish an approximation of approximately quadrati mappings f : X Y by quadrati mappings Q : X Y, suh that f(0) = 0 and the orresponding approximately quadrati funtional inequality ( ) (**) f a i x i + ( )[ )] f(a j x i a i x j ) ai 2 f(x i x i r i holds with onstants 0 (independent of x i X : i =1, 2,...,p), and any fixed reals a i and r i > 0(i =1, 2,...,p). Denote I 1 = {(r, m) R 2 :0<r<2, m>1 or r>2, 0 <m<1}, I 2 = {(r, m) R 2 :0<r<2, 0 <m<1 or r>2, m>1}, I 3 = {(r, m) R 2 :0<r<2, m=1=pb 2,a i = b = p 1/2 : i =1, 2,...,p}, I 4 = {(r, m) R 2 : r>2, m=1=pb 2,a i = b = p 1/2 : i =2,...,p}, where r = r i > 0, where p is arbitrary but fixed and equals to 2, 3, 4,... Note that m r 2 < 1if(r, m) I 1,m 2 r < 1if(r, m) I 2,p r 2 < 1if(r, m) I 3, and p 2 r < 1if(r, m) I 4. Also denote γ = a i r i > 0. Also denote m 2n f(m n x), if (r, m) I 1 m 2n f(m n x), if (r, m) I f n (x) = 2 p n f(p n/2 x), if (r, m) I 3 p n f(p n/2 x), if (r, m) I 4 for all x X and n N : p =2, 3, 4,... Definition (1.1). Let X and Y be real normed linear spaes. Let a =(a 1,a 2,...,a p ) (0, 0,...,0) with a i R (i =1, 2,...,p). Then a mapping Q : X Y ( p ) 1/2 is alled quadrati with respet to a : a = ai 2, if the generalized quadrati funtional equation (*) holds for every x i X(i =1, 2,...,p). Denote Q(a i x)/ ai 2, if (r, m) I 1, (1.2) Q(x) = [ p ( ai 2) Q ( a i x/ p ) ] a2 i, if (r, m) I 2, for all x X.

ON THE GENERAL QUADRATIC FUNCTIONAL EQUATION 261 2. Quadrati funtional stability Theorem (2.1). Let X and Y be normed linear spaes. Assume that Y is omplete. Assume in addition that the mapping f : X Y satisfies f(0) = 0 and the approximately quadrati funtional inequality (**) for every x i X (i = 1, 2,...,p). Ifr 2and p 2, then the limit (2.2) Q(x) = lim f n(x) exists for all x X and Q : X Y is the unique quadrati mapping suh that /(m 2 m r ), if (r, m) I 1 (2.3) f(x) Q(x) x r /(m r m 2 ), if (r, m) I 2 /(p p r/2 ), if (r, m) I 3 /(p r/2 p), if (r, m) I 4 holds for all x X. Proof. From the hypotheses of this theorem, the following ondition (2.4) f(0) = 0 is useful to hold. We laim for eah n N that (1 m n(r 2) ), if (r, m) I m 2 m r 1, (2.5) f(x) f n (x) x r m r m 2 (1 m n(2 r) ), if (r, m) I 2, p p r/2 (1 p n(r 2)/2 ), if (r, m) I 3, p r/2 p (1 pn(2 r)/2 ), if (r, m) I 4 for all x X. By replaing Q, Q of (1.2) with f, f, respetively, one denotes: f(a i x)/ ai 2, if (r, m) I 1 (2.6) f(x) = [ p ( )] ( ai 2) f a i x/, if (r, m) I 2 ai 2 holds for all x X. From (2.4), (2.6) and (**), with x i = a i x (i =1, 2,...,p), we obtain ( ) p f(mx)+ f(0) m f(a i x) 2 x r, or [ f(mx) m f(a i x)] x r, or (2.7) m 2 f(mx) f(x) m 2 x r, if I 1 holds. Besides from (2.4), (2.6) and (**), with x 1 = x, x j = 0 (j =2, 3,...,p), we get 1x)+ f(a j x) m[f(x)+(p 1)f(0)] j=2

262 JOHN MICHAEL RASSIAS or f(a i x) mf(x) 0, or (2.8) f(x) =f(x), if I 1 holds. Therefore from (2) and (2.8) we have (2.9) f(x) m 2 f(mx) m 2 x r = m 2 m r (1 mr 2 ) x r, whih is (2.5) for n =1, if I 1 holds. Similarly, from (2.4), (2.6) and (**), with x i = a i m x(i =1, 2,...,p), we obtain ( ) p ( f(x)+ ai ) f(0) m f 2 m x m r x r, or (2.10) f(x) f(x) m r x r, if I 2 holds. Further from (2.4), (2.6) and (**), with x 1 = x m,x j = 0 (j =2, 3,...,p), we get f ( a1 m x ) + f j=2 f ( aj m x ) ( ai m x ) m[f(m 1 x)+(p 1)f(0)] 0, mf(m 1 x) 0, or (2.11) f(x) =m 2 f(m 1 x), if I 2 holds. Therefore from (2.10) and (2.11) we have (2.12) f(x) m 2 f(m 1 x) m r x r = m r m 2 (1 m2 r ) x r, whih is (2.5) for n =1,ifI 2 holds. Also, with x i = x (i =1, 2,...,p) in (**) and a i = b = p 1/2 (i =1, 2,...,p), we obtain f(p 1/2 x) pf(x) x r, or (2.13) f(x) p 1 f(p 1/2 x) [ p x r = 1 p (r 2)/2] x r, p p r/2 whih is (2.5) for n =1,ifI 3 holds. In addition, with x i = p 1/2 x (i =1, 2,...,p) in (**) and a i = b = p 1/2 (i =1, 2,...,p), we obtain f(x) pf(p 1/2 x) p r/2 x r, or (2.13a) f(x) pf(p 1/2 x) p r/2 x r = p r/2 p [1 p(2 r)/2 ] x r, whih is (2.5) for n =1,ifI 4 holds. or

ON THE GENERAL QUADRATIC FUNCTIONAL EQUATION 263 Assume (2.5) is true if (r, m) I 1. From (2.9), with m n x in plae of x, and from the triangle inequality, we have f(x) f n+1 (x) = f(x) m 2(n+1) f(m n+1 x) f(x) m 2n f(m n x) + m 2n f(m n x) m 2(n+1) f(m n+1 x) (2.14) m 2 m r [(1 mn(r 2) )+m 2n (1 m r 2 )m nr ] x r = m 2 m r (1 m(n+1)(r 2) ) x r, if I 1 holds. Similarly assume (2.5) is true if (r, m) I 2. From (2.12), with m n x in plae of x, and the triangle inequality, we have f(x) f n+1 (x) = f(x) m 2(n+1) f(m (n+1) x) f(x) m 2n f(m n x) + m 2n f(m n x) m 2(n+1) f(m (n+1) x) (2.15) m r m 2 [(1 mn(2 r) )+m 2n (1 m 2 r )m nr ] x r = m r m 2 (1 m(n+1)(2 r) ) x r, if I 2 holds. Also, assume (2.5) is true if (r, m) I 3. From (2.13), with (pb) n x (= p n/2 x) in plae of x, and the triangle inequality, we have ( ) f(x) f n+1 (x) = f(x) p (n+1) f p n+1 2 x = f(x) p (n+1) f((pb) n+1 x) f(x) p n f((pb) n x) + p n f((pb) n x) p (n+1) f((pb) n+1 x) (2.16) p p {[1 r/2 pn(r 2)/2 ]+p n [1 p (r 2)/2 ](pb) nr } x r = p p [1 r/2 p(n+1)(r 2)/2 ] x r, if I 3 holds. In addition, assume (2.5) is true if (r, m) I 4. From (2.13a), with (pb) n x (= p n/2 x) in plae of x, and the triangle inequality, we have ( ) f(x) f n+1 (x) = f(x) p n+1 f p n+1 2 x = f(x) p n+1 f((pb) (n+1) x) f(x) p n f((pb) n x) + p n f((pb) n x) p n+1 f((pb) (n+1) x) (2.16a) p r/2 p {[1 pn(2 r)/2 ]+p n [1 p (2 r)/2 ](pb) nr } x r = p r/2 p [1 p(n+1)(2 r)/2 ] x r, if I 4 holds. Therefore inequalities (2.14), (2.15) and (2.16) and (2.16a) prove inequality (2.5) for any n N.

264 JOHN MICHAEL RASSIAS We laim now that the sequene {f n (x)} onverges. To do this it suffies to prove that it is a Cauhy sequene. Inequality (2.5) is involved if (r, m) I 1. In fat, if i>j>0and h 1 = m j x, we have: f i (x) f j (x) = m 2i f(m i x) m 2j f(m j x) =m 2j m 2(i j) f(m i j h 1 ) f(h 1 ) m 2j m 2 m r (1 m(i j)(r 2) ) h 1 r (2.17) < m 2 m r m 2j h 1 r 0, j if I 1 holds: m r 2 < 1. Similarly, if h 2 = m j x in I 2, we have: f i (x) f j (x) = m 2i f(m i x) m 2j f(m j x) =m 2j m 2(i j) f(m (i j) h 2 ) f(h 2 ) m 2j m r m 2 (1 m(i j)(2 r) ) h 2 r (2.18) < m r m 2 m2j h 2 r 0, j if I 2 holds: m 2 r < 1. Also, if h 3 = p j/2 x in I 3, we have: f i (x) f j (x) = p i f(p i/2 x) p j f(p j/2 x) = p j p (i j) f(p (i j)/2 h 3 ) f(h 3 ) p j p p (1 r/2 p(i j)(r 2)/2 ) h 3 r (2.19) < p p r/2 p j h 3 r 0, j if I 3 holds: p r 2 < 1. In addition, if h 4 = p j/2 x in I 4, we have: f i (x) f j (x) = p i f(p i/2 x) p j f(p j/2 x) = p j p i j f(p (i j)/2 h 4 ) f(h 4 ) (2.19a) p j p r/2 p (1 p(i j)(2 r)/2 ) h 4 r < p r/2 p pj h 4 r 0, j if I 4 holds: p 2 r < 1. Then inequalities (2.17), (2.18) and (2.19) and (2.19a) define a mapping Q : X Y in p variables x i X (i =1, 2,...,p), given by (2.2). Claim that from (**) and (2.2) we an get (*), or equivalently that the aforementioned well defined mapping Q : X Y is quadrati with respet to a ( 0). In fat, it is lear from the funtional inequality (**) and the limit (2.2) for (r, m) I 1 that the following funtional inequality ( m 2n f p ) ( p )[ p a i m n x i + f(a j m n x i a i m n x j ) f(m n x i )] m 2n m n x i r i, holds for all vetors (x 1,x 2,...,x p ) X p, and all n N with p =2, 3, 4,...and f n (x) =m 2n f(m n x):i 1 holds. Therefore ( ) lim f ( )[ n a i x i + lim f n (a j x i a i x j ) a 2 i lim f n(x i )] a 2 i

ON THE GENERAL QUADRATIC FUNCTIONAL EQUATION 265 ( lim mn(r 2) ) x i r i =0, beause m r 2 < 1or ( ) (2.20) Q a i x i + ( )[ Q(a j x i a i x j ) ai 2 Q(x i )] =0, i.e., the mapping Q satisfies the quadrati funtional equation (*). Similarly, from (**) and (2.2) for (r, m) I 2 we get that ( ) m 2n f a i m n x i + ( )[ f(a j m n x i a j m n x j )= f(m n x i )] m 2n m n x i r i, holds for all vetors (x 1,x 2,...,x p ) X p, and all n N with f n (x) =m 2n f(m n x): I 2 holds. Thus ( ) lim f ( )[ n a i x i + lim f n (a j x i a i x j ) a 2 i lim f n(x i )] ( lim mn(2 r) ) x i r i =0, beause m 2 r < 1, i.e., (2.20) holds and the mapping Q satisfies (*). Also, from (**) and (2.2) for (r, m) I 3 we obtain that ( ) p n f a i p n/2 x i + ( )[ f(a i p n/2 x j a j p n/2 x i ) f(p n/2 x i )] p n p n/2 x i r i, holds for all vetors (x 1,x 2,...,x p ) X p, and all n N with f n (x) =p n f(p n/2 x): I 3 holds. Hene ( ) lim f ( )[ n a i x i + lim f n (a j x i a i x j ) a 2 i lim f n(x i )] ( lim pn(r 2)/2 ) x i r i =0, beause p r 2 < 1, i.e., (2.20) holds and the mapping Q satisfies (*). In addition, from (**) and (2.2) for (r, m) I 4 we obtain that ( ) p n f a i p n/2 x i + f(a j p n/2 x i a i p n/2 x j ) ( a 2 i a 2 i a 2 i )[ f(p n/2 x i )] p n p n/2 x i r i,

266 JOHN MICHAEL RASSIAS holds for all vetors (x 1,x 2,...,x p ) X p, and all n N with f n (x) =p n f(p n/2 x): I 4 holds. Hene lim f n ( ) a i x i + lim ( f n (a j x i a i x j ) a 2 i )[ lim f n(x i )] ( lim pn(2 r)/2 ) x i r i =0, beause p 2 r < 1, i.e., (2.20) holds and the mapping Q satisfies (*). Therefore (2.20) holds if I j (j =1, 2, 3, 4) hold or the mapping Q satisfies the quadrati funtional equation (*), ompleting the proof that Q is a quadrati mapping with respet to a in X. It is now lear from (2.5) with n, as well as from the formula (2.2) that the funtional inequality (2.3) holds in X. This ompletes the existene proof of the afore-mentioned Theorem (2.1). It remains to prove the uniqueness: Let Q : X Y be a quadrati mapping with respet to a satisfying (2.3), as well as Q. Then Q = Q. In fat, the ondition m 2n Q(m n x), if (r, m) I 1, m 2n Q(m n x), if (r, m) I (2.21) Q(x) = 2, p n Q(p n/2 x), if (r, m) I 3, p n Q(p n/2 x), if (r, m) I 4 holds for all x X and n N where p is arbitrary but fixed and equals 2, 3, 4,..., as a onsequene of (2.5) with = 0. Remember Q satisfies (2.21) as well for (r, m) I 1. Then for every x X and n N, Q(x) Q (x) = m 2n Q(m n x) m 2n Q (m n x) } (2.22) m { Q(m 2n n x) f(m n x) + Q (m n x) f(m n x) m 2n 2 m 2 m r mn x r = m n(r 2) 2 m 2 m r x r 0 as n, if I 1 holds: m r 2 < 1. Similarly for (r, m) I 2, we establish Q(x) Q (x) = m 2n Q(m n x) m 2n Q (m n x) } (2.23) m { Q(m 2n n x) f(m n x) + Q (m n x) f(m n x) if I 2 holds: m 2 r < 1. m 2n 2 m r m 2 m n x r = m n(2 r) 2 m r m 2 x r 0, as n,

ON THE GENERAL QUADRATIC FUNCTIONAL EQUATION 267 Also for (r, m) I 3,weget Q(x) Q (x) = p n Q(p n/2 x) p n Q (p n/2 x) } (2.24) p { Q(p n n/2 x) f(p n/2 x) + Q (p n/2 x) f(p n/2 x) p n 2 p p r/2 pn/2 x r = p n(r 2)/2 2 p p r/2 x r 0, as n, if I 3 holds: p r 2 < 1. In addition, for (r, m) I 4,weget Q(x) Q (x) = p n Q(p n/2 x) p n Q (p n/2 x) } (2.25) p { Q(p n n/2 x) f(p n/2 x) + Q (p n/2 x) f(p n/2 x) p n 2 p r/2 p p n/2 x r = p n(2 r)/2 2 p r/2 p x r 0, as n, if I 4 holds: p 2 r < 1. Thus from (2.22), (2.23), (2.24) and (2.25) we find Q(x) =Q (x) for all x X. This ompletes the proof of the uniqueness and stability of the quadrati funtional equation (*). Aknowledgement We are grateful to the anonymous referees for their valuable omments. Reeived November 24, 2004 Final version reeived June 14, 2005 Pedagogial Department, E. E. National and Capodistrian University of Athens Setion of Mathematis and Informatis 4, Agamemnonos Str. Aghia Paraskevi, Athens 15342 Greee jrassias@primedu.uoa.gr; jrassias@tellas.gr Referenes [1] J. Azél, Letures on funtional equations and their appliations, Aademi Press, New York and London, 1966. [2] C. Borelli and G. L. Forti, On a general Hyers Ulam stability result, Internat. J. Math. Si. 18 (1995), 229 236. [3] D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. So. 57 (1951), 223 237. [4] P. W. Cholewa, Remarks on the stability of funtional equations, Aequationes Math. 27 (1984), 76 86. [5] St. Czerwik, On the stability of the quadrati mapping in normed spaes, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59 64.

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