proceedings of the
american mathematical society
Volume 112, Number 3, July 1991
THE STABILITY OF CERTAIN FUNCTIONAL EQUATIONS
JOHN A. BAKER
(Communicated by Kenneth R. Meyer)
Abstract.
The aim of this paper is to prove the stability (in the sense of Ulam)
of the functional equation:
f(t) = a(t) + ß(t)f(<j>(t)),
where a and ß are given complex valued functions defined on a nonempty
set S such that sup{|/?(f)| : t e S} < 1 and <j>is a given mapping of 5 into
itself.
INTRODUCTION
Questions concerning the stability of functional equations seem to have originated with S. M. Ulam in the 1940s (see [2, 3] and the survey paper of Hyers
[4]). One of the first assertions to be proved in this direction is the following
result, essentially due to Hyers, that answered a question of Ulam.
Theorem. Suppose S is an additive semigroup, E is a Banach space, f : S —>E,
â > 0, and
(1)
||/(*+jO-/(*)-/(y)ll<¿
for all x, y eS.
Then there is a unique function a : S -» E such that
(2)
a(x + y) = a(x) + a(y) for all x, y e S,
and
(3)
\\f(x)-a(x)\\<S
forxeS.
This assertion is usually summarized by saying that the Cauchy functional
equation (2) is stable (in the sense of Ulam).
The proof proceeds as follows. First note that
(4)
||/(2x)/2-/(Jc)||<f
for all x 6 5.
Received by the editors January 2, 1990.
1980 Mathematics Subject Classification (1985 Revision). Primary 39B10, 26D20; Secondary
39B70, 47H10.
Key words and phrases. Functional equations, stability, fixed points.
©1991 American Mathematical Society
0002-9939/91 $1.00+ $.25 per page
729
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730
J. A. BAKER
Then one proves, by induction, that {f(2nx)/2n}°^=x is a Cauchy sequence in
E. Denoting its limit by a(x), one then proves that (2) and (3) hold. The
same idea can be used to prove the stability of the equation:
(5)
f(2x)/2 = f(x).
In fact the same technique can be used to prove the
Proposition. Suppose S is a nonempty set, 4>'■S -» S, E is a Banach space, À
is a scalar, |A|< 1, ô > 0, and f : S —>E such that
(6)
Uf(4>(x))-f(x)\\<ô
forallxeS.
Then there exists a unique h : S —»E such that
(7)
kh((j)(x)) = h(x) forallxeS,
and
(8)
\\f(x)-h(x)\\<ô/(\-\X\)
forallxeS.
Proof. Let c¡> denote the kth. iterate of cf>for k = 0, 1, 2, ... . Thus 4>(x) =
x ,<f>(x) = (f)(x), <f>(x) = (¡>(<¡)(x)),
etc. for all x e S.
By induction, it follows from (6) that
Uk+lf((f>k+l(x))-t.kf((t>k(x))\\< \À\kô for all x e S and k = 0,1, 2.
Hence, if m and n are nonnegative integers and m < n , then
u"/(«v» - *"/»"M)«<pi""'+■••+WV
(9)
SJä"*
1 — \A\
fora,,«*.
Thus {Xnf((t>n(x))}^=x is a Cauchy sequence in E for each x e S. Let h(x) =
lim^^ A"f(4>n(x)) for x e S. Then (7) clearly holds.
If we choose m = 0 in (9) we find that
||A"/(Ax)) - f(x)\\ <S/(\-
|A|) for all x € S,
and all n = 1, 2, ... , and this implies (8). D
Most known stability theorems for functional equations concern functional
equations in "several variables" in the sense of Aczél [1]. Equation (2) is the
best-loved representative of this class. Equation (7), whose stability we have
just proved, is often referred to as a functional equation in a "single variable."
Such equations are the subject of the book by Kuczma [5]. The aim of this
paper is to generalize the above proposition by proving a stability result for the
general functional equation:
f(t) = F(t,fX<j>(t)).
We refer the reader to [5] for numerous results and references concerning this
equation and particular cases thereof.
Main results
Our analysis is based on the following variant of Banach's fixed point theorem
that is likely included among the thousands of published generalizations thereof.
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STABILITYOF CERTAIN FUNCTIONAL EQUATIONS
731
Theorem 1. Suppose (Y, p) is a complete metric space and T : Y —>Y is a
contraction (for some À e [0, 1), p(T(x), T(y)) < Ap(x, y) for all x, y e Y).
Also suppose that u e Y, ô > 0, and
p(u, T(u)) < ô.
Then there exists a unique p e Y such that p = T(p).
Moreover, p(u, p) <
ô/(\-X).
Proof. Define a sequence {y„}^l0 in Y by decreeing that y0 = u and yn+x =
T(yn) for all n = 0, 1, 2, ... . According to the well-known proof of Banach's
fixed point theorem, {y„}^l, converges, say to p , p is the unique fixed point
of T, and
An
p(y„,P) < y^jP&i
>yo) forn = 0, 1, 2, ... .
Thus
P(u,p)<p(u,
so that p(u,p)
T(u)) + p(T(u),p)<S
< ô/(l -A).
+ p(T(u), T(p))<ô
+ Àp(u,p)
D
Theorem 2. Suppose S is a nonempty set; (X, d) is a complete metric space;
4>:S -> S, F :SxX ^ X, 0 < A< 1 ; and
d(F(t, u), F(t, v)) <Ad(u, v) for all t e S and all u, v eS.
Also suppose that g : S —>X, ô > 0, and
(10)
d(g(t),F(t,g(ct>(t))))<ô
for all teS.
Then there is a unique function f : S —>X such that
(11)
f(t) = F(t,f(4>(t))) for allteS,
and
(12)
d(f(t),g(t))<S/(l-A)
for all teS.
Proof. Let Y = {a : S -> X\sxxp{d(a(t), g(t))\t e S} < +oc}. For a,b e Y
define p(a, b) = sup{d(a(i), b(t))\t e S}. Then g e Y, p is a metric on
Y, and convergence with respect to p means uniform convergence on 5 with
respect to d . Moreover, the completeness of X with respect to d implies the
completeness of Y with respect to p .
For a e Y define T(a) : S -* X by
(T(a))(t) = F(t,a(4>(t)))
Then T maps Y into Y. If a,beY
for t e S.
then for all teS,
d((T(a))(t), (T(b))(t)) = d(F(t, a(4>(t))),F(t, b(<p(t))))
<Ad(a(4>(t)),b(<t>(t)))
< Ap(a o (p, b o cf>)
<Ap(a,b).
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732
J. A. BAKER
Thus,
p(T(a), T(b))<Ap(a,b)
for all a, b e Y.
But (10) means that p(g, T(g)) < ô . Hence, according to Theorem 1, there is
a unique / in Y such that / = T(f) and p(g ,f)<ô/(\-A).
and (12) hold. D
That is, (11)
We will now use Theorem 2 to deduce stability assertions concerning the
functional equation:
f(t) = a(t) + ß(t)f(<p(t)).
Theorem 3. Suppose S is a nonempty set; E is a real (or complex) Banach
space; <j>
: S -> S, a: S -►E, ß :S ^ R (or C), 0 < A < 1; and \ß(t)\ < A
for all t e S. Also suppose that g : S -* E, â > 0, and
(13)
\\g(t)-{a(t) + ß(t)g(4>(t))}\\<o for all teS.
Then there exists a unique function f : S —>E such that
(14)
f(t) = a(t) + ß(t)f(<p(t))
and
(15)
\\f(t)-g(t)\\<S/(l-A)
for allteS.
Proof. The result follows from Theorem 2 by letting
F(t,x)
= a(t) + ß(t)x
for (t, x) eSxE.
D
Similarly we can prove
Theorem 4. Suppose S is a nonempty set; A is a Banach algebra; <f>
: S -* S,
a:S
-> A, ß : S - A; 0 < A < 1, \\ß(t)\\ < À for all t e S, g : S -» A,
ô > 0; and (13) holds. Then there is a unique function f : S —>A satisfying
(14) and(\5).
Acknowledgment
A referee has noted that our proposition is a particular case of Theorem 1
and Corollary 3 of G. L. Forti, An existence and stability theorem for a class of
functional equations, Stochastica 4 (1980), 23-30.
References
1. J. Aczél, Lectures on functional equations and their applications, Academic Press, New York,
1966.
2. Michael Albert and J. A. Baker, Functions with bounded nth differences, Ann. Polon. Math.
42 (1983), 93-103.
3. J. A. Baker, The stability of the cosine equation, Proc. Amer. Math. Soc. 80 (1980), 411-416.
4. D. H. Hyers, The stability of homomorphisms and related topics, Global Analysis-Analysis
on Manifolds (T. M. Rassias, ed.), Teubner-Texte zur Mathematik, Band 57, Teubner
Verlagsgesellschaft, Leipzig, 1983, pp. 140-153.
5. M. Kuczma, Functional equations in a single variable, Monographs Math., vol. 46, PWN,
Warszawa, 1968.
Department
of Pure Mathematics,
N2L3G1 Canada
University
of Waterloo,
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Waterloo,
Ontario,