On approximation of approximately quadratic mappings by quadratic mappings

Prace Naukowe Uniwersytetu Śląskiego nr 1999, Katowice

O N A P P R O X I M A T I O N O F A P P R O X I M A T E L Y Q U A D R A T I C M A P P I N G S B Y Q U A D R A T I C M A P P I N G S

J O H N M I C H A E L R A S S I A S

A b s t r a c t . In this paper we establish an approximation of approximately quadratic mappings by quadratic mappings, which solves the pertinent Ulam stability problem.

I n t r o d u c t i o n

In 1940 S. M . Ulam [34] proposed before the Mathematics Club of the University of Wisconsin a number of interesting open problems, one of which is the following problem: Give conditions in order for a linear mapping near an approximately linear mapping to exist. In 1968 S. M . Ulam [34] proposed the general problem: When is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true. In 1978 P. M . Gruber [7] proposed the Ulam type problem: Suppose a mathematical object satisfies a certain property appro­

ximately. Is it then possible to approximate this object by objects, satisfying the property exactly? According to P. M . Gruber [7] this kind of stability problems is of particular interest in probability theory and in the case of functional equations of different types. In 1982-2000 we ([15]-[28]) solved the above-mentioned Ulam problem, or equivalently the Ulam type problem for linear mappings as well as for quadratic, cubic and quartic mappings and established analogous stability problems. In this paper we introduce the

Received: 25.05.2000. Revised: 31.10.2000.

A M S (1991) subject classification: Primary 39B.

Key words and phrases: Ulam problem, Ulam type problem, general Ulam problem, qua­

dratic mapping, approximately quadratic mapping, approximation, Ulam stability problem, normed linear space, complete normed linear space.

5 *

following quadratic functional equation

(*) Q(a^xxi + a²x²) + Q{a²xi - aix²) = (a] + a\)[Q{x-ⁱ) + Q(x²)]

with quadratic mappings Q : X -4 Y satisfying condition Q(0) = 0 if m = a\ + <Ą > 0 such that X and V are real linear spaces, and then establish an approximation of approximately quadratic mappings / : X —)• Y, with /(O) = 0 (if m = 1), by quadratic mappings Q : X Y, such that the corresponding approximately quadratic functional inequality

(**)

\\f(a¹x^l+a²x²) + f(a²x^l-a^xx²) - (a\ +4)[f(^xi) + />2)]|| < c | | * i | |^{r i} \\x²\\^r* holds with a constant c > 0 (independent of xi,x² C X), and any fixed pair (ai,a²) of reals a< ^ 0 (i = 1,2) and (rj, r²) of reals r ; ^ 0 (i = 1,2):

h = { ( r , m ) € R² : r < 2 , m > 1 or r > 2,0 < m < 1}, J² = {(r, m) € R² : r < 2,0 < m < 1 or r > 2, m > 1}, or

I³ = { ( r , m ) € R² : r < 2, TO = 1 = 2a² : a^x = a² = a = 2~5}

hold, where m — a\ Ą- a\ > 0 and r = r\ + r² ^ 0. However, we have established the following case: r* = 0 (i — 1,2) such that r = 0 [23].

Note that m^{r _ 2} < 1 if (r, m) € / i , m^{2 _ r} < 1 if (r, m) € I², and 2^{r _ 2} < 1 if ( r , r » = l ) G /³.

It is useful for the following, to observe that, from (*) with x\ = x² = 0, and 0 < m ^ 1 we get

2 ( m - 1)Q(0) = 0, or

( i ) g ( o ) = o.

D E F I N I T I O N 1.1. Let X and Y be real linear spaces. Then a mapping Q : X —ł Y is called quadratic, if (*) holds for every vector ( x i , x²) 6 X² .

For every x € R set Q(x) = x². Then the mapping Q : R -4 R is quadratic. Finally let F : X² -> Y be a bilinear mapping. Set Q(x) = F(x, x) for every x G X. Then Q : X -> V is quadratic.

Denote (2)

Q(z) = <

f ^{q (} ' f f l g H , i f ( r , m = a²+ a²) € /¹ ( a² + a².) [Q ( ^ * ) + Q *)] . if {r,m = a² + a²) € 7²

for all x £ X.

Now, claim that for n € N = { 0 , 1 , 2 , . . . }

{

For n = 0, it is trivial. From (1), (2) and (*), with ajj = a^{x (i = 1,2), we obtain

Q(mx) = m[Q(aix) + Q(a<ix)], or

(4) Q(a;) = m-²Q(mx),

if 7i holds. Besides from (1), (2) and (*), with x^x =x,x² = 0, we get Q(a^tx) +Q(a²x) = mQ(x),

or

(5) Q(*) = Q(x), if h holds. Therefore from (4) and (5) we have

(6) Q{x) = m-²Q{mx),

which is (3) for n = 1, if h holds. Similarly, from (1), (2) and (*), with Xi = ^x (i = 1,2), we obtain

(7) Q(x) = Q(x)

if h holds. Besides from (1), (2) and (*), with x^x = %,x² = 0, we get Q (^x) +Q (^-x) = mQim-'x),

\ m J ^V in / or

(8) Q{x) = m²Q{m-^xx)

if h holds. Therefore from (7) and (8) we have

(9) Q(x) = m²Q(m-¹x),

which is (3) for n = 1, if I² holds. Also, with x^x = x² = x in (*) and cii = a² = a = 2^{- 1}/², we obtain

Q(2^²x) = 2Q(x), or

(10) Q{x) = 2-^lQ{2¹'²x),

which is (3) for n = 1, if 1% holds.

Assume (3) is true and from (6), with mⁿx on place of x, we get:

(11) Q(mⁿ⁺¹x) = m²Q(mⁿx) = m²(mⁿ)²Q(x) = (mⁿ⁺¹)²Q(x).

Similarly, with m~ⁿx on place of z , we get:

Q (m-^{n+1)x) = m-²Q{m-ⁿx) = m~²(m~ⁿ)²Q(z)

(12) ²

= ( m - <^{n + 1}> ) Q(x).

Also, with (2a)ⁿx (= 2ⁿl²x) on place of x, we get:

Q h ^ x ) = Q ( ( 2 a )^{n + 1}x ) = 2^lQ ((2a)ⁿx)

= 2¹(2ⁿ)Q(x) = 2ⁿ⁺¹Q(x) = (2 3 ) Q(x).

These formulas (11), (12) and (13) by induction, prove formula (3), ([l]-[6], [8]-[14], [29]-[33]).

Quadratic functional stability

T H E O R E M 2.1. Let X andY be normed linear spaces. Assume that Y is complete. Assume in addition that f : X —tY satisfies functional inequality (**), such that /(0) = 0 (ifm>0).

Define

{

2-ⁿf(2ⁿ'²x), if(r,m=l)€h

for all x 6 X and ^{B 6 N .}

Then the limit

(14) Q{x) = lim fⁿ{x)

n—+00

exists for all x £ X and Q : X —t Y is the unique quadratic mapping, such that Q(0) = 0 (if m > 1) and

{

1C/{m^r - m²) , if ( r , m ) € /² c/(2-2^r"), i f ( r , m = l ) 6 /³ holds for all x £ X, c > 0 (constant independent of x £ X ) and 7 =

| a i |^{r i}| o²|^{r 2} > 0 .

Existence

P R O O F . It is useful for the following, to observe that, from (**) with x i = xi = 0 and 0 < m ^ 1, we get

2 | m - 1|||/(0)|| < 0 ,

or

(16) 1(0) = 0.

Now claim that for n £ (17)

n/ ( * ) - / n ( x ) n < n*ir

(1 - m*'-*)) , if (r,m) € A : m^r~² < 1

^ £ _ ^ ( , _ ^mn ( 2 - r ) ) , ^if (^r, ^m) £ 7² : m²" ' < 1

2 Zf ^ r (1 - 2"<'-²>/²) , if (r, m = 1) £ J³ : 2^r~² < 1.

For n = 0, it is trivial.

Denote (18)

i ' f(aix)+f[a²x)

if ( r , m = + a|) 6 /1 if ( r , m = a² + a?,) € 7²

for all x £ X.

From (16), (18) and (**), with x ; = a;x (i = 1,2), we obtain

| | / ( m x ) - m[f(aix) + / ( a²x ) ] | | < ⁷c | | x | |^r,

or

(19) \\m-²f(mx)-f(x)\\< I i | | x | r ,

if 7i holds. Besides from (16), (18) and (**), with x\ = x,x² = 0, we get

| | / ( a i * ) + / ( a²a r ) - m / ( a O | | < 0, or

(20) / ( * ) = / ( * ) , if Ą holds. Therefore from (19) and (20) we have

(21) | | / ( x ) - m -²/ ( m x ) | | < ^\\x\\^r = ^ ^ ( 1 - n O | | * | r , ^yc m* — m¹

which is (17) for n= 1, if I^x holds.

Similarly, from (16), (18) and (**), with x^t = %x (i = 1,2), we obtain

(22)

!!/(*)-/(*)!!<-Sinn

if I² holds. Besides from (16), (18) and (**), with x^x = — , x² = 0, we get

< o ,

or

(23) f{x) = m²f(m-¹x),

if I² holds. Therefore from (22) and (23) we have

(24) \\f(x) - m*f(m-*x)\\ < £\\z\\' = ^ ^ ( 1 - m⁸- ) | | * | r^f which is (17) for n = 1, if 7² holds.

Also, with x\ = x² = x in (**) and Oi = a² = a = 2^{- 1}/², we obtain

| | / ( 2 a x ) - 2 / ( x ) | | < c | | x | r , or

(25)

| | / ( x ) - 2 - V ( 2^ł/²a : ) | j = | | / ( x ) - 2 -¹/ ( ( 2 a )¹x ) | |

^ il | x | | r =2 ^ ^^{[ 1}-2 ( r _ 2 ) / 2 ] l | a : | | r'

which is (17) for n = 1, if I3 holds.

Assume (17) is true if (r, in) G I\. From (21), with mⁿx on place of x, and the triangle inequality, we have

(26)

| | / ( x ) - f^n+l(x)\\ = ||/(x) - m-^+Vf(mⁿ⁺ⁱx)\\

< ||/(a!) - m-²ⁿf{mⁿx)\\ + \\m-²ⁿf{mⁿx) - m -²<^{n + 1}) / ( m^{n + 1}x ) | |

1 0 - [(1 - m^{n ( r}-²) ) + m "^{2 n}( l - m^r-²) m "^r] | | x | |^r

<

m" — in' 7c m* — rn'

( l - ^m( » + D ( » - 2 ) ) | |^a. | | r^j

if I\ holds.

Similarly assume (17) is true if (r,in) (E I². From (24), with m~ⁿx on place of x, and the triangle inequality, we have

(27)

| | / ( x ) - /^{n +}, ( x ) | | = | | / ( x ) - m²( » +¹) / ( m - ( " +¹) x ) | |

< \\f(x) - m²ⁿf{m-ⁿx)\\ + \\m²ⁿf(m-ⁿx) - m²<^{n + 1}> f{m^ⁿ⁺^x)\\

^ r7c 2 7c

m^r — m²

(1 - mⁿ(²-^r> + m^{2 n}( l - m²-^r) m -^{n r}] ||a;||^r

(¹_^m( n+ D ( 2 - r )⁾| |^a. | | r⁾

if I² holds.

Also, assume (17) is true if ( r , m = 1) € I3. From

(25),

(2a)"x

(= ²

^/

^x)

(28)

| | / ( x ) - /^{n + 1}( x ) | | = | | / ( x ) -

2-<" >/

^{(2^x)}

= ||/(x)-2-("+

)/((2ar+

x)||

< | | / ( x ) -

2-"/((2a)"x)||

||2-"/((2a)"x)

2-("

>/((2a)

x)||

< { [ l - 2 " ( r - 2 ) / 2 + 2"

2 - 2^r/² l _²(^r-²) /²] ( 2 a )^{n r}} ||x||^r

if I3 holds.

Therefore inequalities (26), (27) and (28) prove inequality (17) for any n e N.

Claim now that the sequence {/ⁿ(x)} converges.

To do this it suffices to prove that it is a Cauchy sequence. Inequality

(17) is involved if (r, m) G h. In fact, if i> j > 0, and h\ = raJx, we have:

Wfii*) ~ /,•(*)II = Wm-^fim'x) - m - « / K ' x ) | | .

= m - ^ ' U m - ^ ' - ^ / K ' - ^ , ) - / ( M l =

= m-V\\fⁱ-^j(h¹)-f(h,)\\<

(29) = m -aJ | | / n | |r , T C (1 - ro<*'->Hr-2>)

= ^m ( r - 2 ) i _ 7 £ _⁽¹ _ ^m( i - j ) ( r - 2 h | |^x| . r 7?ł2 - 7 Hr V ' " "

< 2 7° m ^ - ^ H a r i r - 0,

mi — mr j->oo

if h holds: mr~2 < 1.

Similarly, if h² = m~^x in 7², we have:

\\fi(x) - /•,(*)„ = | | m27 ( m - x ) - m « / ( m - ^ ) | |

= m2^ | | m2(i- ^ / ( m -( ,-J')/i 2) - / ( M l

(30) < ^ro(2-r)j TC ⁽¹ _ ^m(i-m-r)^)Mr

< ^{r 7 C 2}ml²-^rV\\x\\^r0.

if I² holds: m²"^r < 1.

Also, if h³ = 2^²x in 7³, we have:

(31)

- m \ \ = l|2-7(2^{< / a}«) - 2 - V ( 2 -? /^ ) | |

= 2- i | |²- ( ^ )^{/ ( 2}( « - i ) / 2^{/ l 3 )}_/ ( / l 3 )| |

= 2 ^ | | /i_j( /i 3) - / ( M U < 2 -j| | ^ 3 | r 2 - 3 ^ ( 1 - 2 ( ' - ^ "2> /2)

= 2 " i / a a r W ( 1 " 2( <-i ) (-a , / a) l l * i r < 2 r W2"i / 2 | | 3 f | | ri r c o0' if 7³ holds: 2^r~² < 1.

Then inequalities (29), (30) and (31) define a mapping Q : X -4 Y, given by (14).

Claim that from (**) and (14) we can get (*), or equivalently that the afore-mentioned well-defined mapping Q : X —> Y is quadratic.

In fact, it is clear from the functional inequality (**) and the limit (14) for (r, m) C I\ that the following functional inequality

m~2n\\f(a\mnx\ + a2mna ;2) + / ( a27 ? in£ i — a\mnX2) -(a21+a2)[f(mnxl) + f(mnx2)]\\

< 7 n -^{2 n}c | | 7 nⁿx¹| |^{r i}| | 7 7 *ⁿx²| |^{r 2},

holds for all vectors (x^u x²) € X², and all n € N with fⁿ(x) = m ²ⁿf(mⁿx) I\ holds. Therefore

lim fⁿ{a\X\ + a²x²) + lim / „ ( a²x i - a j x²)

n-*oo n-*oo

~ (^al + ^al) I^{l i m} fn{x\) + Hm / n ( x²)

< ( lim ^m» ( r - 2 ) \ c H x ^ r M I ^ I P = 0,

\n-+oo / because mr 2 < 1 or

(32) \\Q(a^lXl + a²x²) + Q(a²x^x - o n , ) - {a\ + a\){Q{x^x) + Q(x²)]\\ = 0,

or mapping Q satisfies the quadratic equation (*).

Similarly, from (**) and (14) for (r, m) € I2 we get that m²ⁿ | | / ( a i m^{- n}x i -+- a²m~ⁿx²) + f(a²m~ⁿx^l — a\m~ⁿx²)

"(a? +4) [ / ( « » -^Ba : i ) + / ( m -ⁿa :²) ] | | < m²ⁿc\\^m-ⁿx,\p||m-"x²|p, holds for all vectors ( x i , x2) C X2, and all n € N w i t h /n( x ) = m2nf(m~nx) : 72 holds. Thus

lim fn(a\x\ + a²x²) + lim / „ ( a²x i - a^tx²) n-»oo n—voo

- ( a² + a²) [ lim /ⁿ( x^a) + l i m fⁿ{x²)

< ( lim mⁿ^-^rA c | | x¹| |^{r i}| | a ;²| r² = 0, because m2_r < 1, or (32) holds or mapping Q satisfies (*).

Also, from (**) and (14) for (r, in = 1) G h we obtain that

2 ~n / ( a i 2n/2X ! + a22 " /2x2) + f{a22n/2xt - a , 2n/2x2)

- ( « i + « 2 ) [ / ( 2^{B / 2}* i ) + / ( 2^{n / 3}*²) ] | < 2 -ⁿc | | 2 " /²x i | |^r' | | 2ⁿ/²x²| | ^ ,

holds for all vectors (n, x²) 6 X², and all n € N with fⁿ(x) - 2^{_ n}/ ( 2ⁿ/²x ) A holds. Hence

lim /ⁿ( a i X ! +a²x²) + l i m fⁿ(a²xi - aix²) n-+oo n-+oo

-(a\+a\) lim /n( x i ) + l i m / „ ( x2)

< f lim 2n<r-2^2) c | | x1| |r> | | x2| |r 2 = 0,

because 2r~2 < 1, or (32) holds or mapping Q satisfies (*).

Therefore (32) holds if /_,• (j = 1,2,3) hold or mapping Q satisfies (*), completing the proof that Q is a quadratic mapping in X.

It is now clear from (17) with n —>• oo, as well as formula (14) that inequality (15) holds in X . This completes the existence proof of the above theorem 2.1.

Uniqueness

Let Q' : X —> Y be a quadratic mapping satisfying (15), as well as Q.

Then Q' = Q.

P R O O F . Remember both Q and Q' satisfy (3) for (r, m) G I\, too. Then for every x G X and n G N ,

(33)

\\Q(x) - Q'(x)\\ = \\m-²ⁿQ(mⁿx) - m-²ⁿQ'(mⁿx)\\

< m-²ⁿ {\\Q(mⁿx) - f(rnⁿx)\\ + \\Q'(mⁿx) - f{mⁿx)\\}

< m~2n }1 C \\mnx\\r = ^wn ( r - 2 ) ²7 C n u r Q as n -> oo,

if h holds: mr~2 < 1.

Similarly for (r, m) G h, we establish (34)

||Q(x) - Q'(x)\\ = \\m2nQ(m-nx) - m2nQ'{m,-nx)\\

< m2n {||Q(Tn-"x) - / ( m -nx ) | | + \\Q'(m-nx) - / ( m -nx ) | | )

< m^{2n 2 l C 0}| | m -^mg i r = ^m" ( 2 - r ) ²T C .. , .^{r Q} • _ ^

if J2 holds: m2"r < 1.

Also for (r, m = 1) G ^3, we get (35)

\\Q(x) - Q'(x)\\ = | | 2 - " Q ( 2 " /². T ) - 2 - " Q ' ( 2 " /²x ) | |

< 2~n {\\Q(2n/2x) - /(2"/2x)|| + ||Q'(2"/2x) - /(2"/2x)||}

if 7³ holds: 2^r~² < 1. Thus from (33), (34) and (35) we find Q{x) = Q'(x) for all x G X. This completes the proof of uniqueness and the stability of equation (*).

Query. What is the situation in the above theorem 2.1 either in case r = 2 or for m = 1 when a\ ^ a²?

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P E D A G O G I C A L D E P A R T M E N T E . E . ,

N A T I O N A L A N D C A P O D I S T R I A N U N I V E R S I T Y O F A T H E N S S E C T I O N O F M A T H E M A T I C S A N D I N F O R M A T I C S ,

4 , A G A M E M N O N O S S T R . A G H I A P A R A S K E V I , A T H E N S 1 5 3 4 2 G R E E C E

e - m a i l : j r a s s i a s Q p r i m e d u . u o a . g r