Remark on a. difference equation

ROCZNIKI POLSKIEGO TO W AR ZYSTW A MATEMATYCZNEGO Seria I : PRACE M ATEMATYCZNE I X (1965)

M. Kuczma (Kraków)

Remark on a. difference equation

Introduction. In the present note we are concerned with the differ­

ence equation

(1 ) g{x + l ) — g{x) = tp{x),

where g[x) is the unknown function. W. Krull ([3]) and, independently, the author of the present paper ([4]) have proved the following theorem:

Theorem 1 . I f the function <p(x) is concave in an interval (a, oo),

a > — oo, and fulfils the condition

(2) lim[ę>(w + 1 ) — q>(n)] = 0,

then for every y0e( — oo, oo) there exists exactly one convex function g(x) satisfying equation (1 ) in (a, oo) and fulfilling the initial condition

(3) д ( щ ) = У о , xo€{ a , oo).

This function is given by the formula: (4) g{x) = у0 + { о с х 0)<р{х0)

-OO

n = 0

In fact, the above theorem has been proved in [3], [4] for a = 0 and x0 = 1 , but the same argument applies also in the general case (cf. also [6]).

Eecently we have proved a theorem ([8], theorem 1 1 .1 ; cf. also [6], [7]) of which the following is a particular case:

Theorem 2. I f the function <p(x) is defined almost everywhere {a. e.) in (a , oo), monotonie (1) and fulfils the condition

(5) lim (p{x) = 0,

then there exists a one-parameter family (with the additive constant y 0) of monotonie functions g(x) which are defined a. e. in (a, oo) and satisfy equation (1) a. e. in (a, oo). These functions are given by the formula

OO

(6) g{x) = y 0- £ { ( p { x + n ) - ( p { x 0 + n)}, n = 0

where x 0 is a point from (a, oo). The family (6) is unique in the sense that if a monotonie function g(x) is defined and satisfies equation (1 ) a. e. in (a , oo), then g(x) is equal a. e. in (a, oo) to one of the func­ tions (6).

In the present note we are going to prove that Theorems 1 and 2 are equivalent, by which we mean that each can be directly derived from the other. The implication Theorem 2 - » Theorem 1 seems to be of interest, for if one needs for some purpose both theorems, then it would be reasonable to prove Theorem 2 (and rather a more general theorem, as in [8]) and then deduce Theorem 1 from it, instead of presenting the longer independent proof. The converse implication seems to be less interesting, for the direct argument gives a stronger result than Theorem 2 above.

Throughout the paper we shall make frequent use of the properties of convex and concave functions. These properties are proved, for example, in [1] (p. 41-65) and [2].

§ 1. We begin by proving some lemmas. OO

Lemma 1. Let £ f n { % ) be a convergent series of convex (concave)

func-00

tions in an interval (a ,b ), and suppose that the series £ /»(# ) converges w=0

absolutely a.e. in (a,b). Then

(7) ( £ f n ( x ) ) ' = ] ? f n (x) a.e. in ( a ,b ).

П=0 n=0

oo

P r o o f. We take a, /5, a < a < {} < b, such that the series £ \ f n(a)\ n= 0

oo

and £\fn(P)\ converge. The functions n=0

Fn(co) = fn(oc) + x(\fn(a)\+\fn(P)\) are increasing in <a, jS>, since

F'n(x) = f n(<c)+\fn(a)\ + \fn(P)\ > 0 a.e. in <o, /5)

Remark on a difference equation 3

functions F'n{x), being convex (concave) jnst like / п(ж), are absolutely continuous ([2], p. 15). On account of the theorem of Fubini ([9], p. 403)

oo oo ( ^ F n(x)}' = ^Fn(s o) a.e. in <a,|S>, n —0 » = 0 i.e. OO 00 ( 2 fn(oo) + x ]?(\fn{a)\+ \fn(P)\))' 71 = 0 77 = 0 00 oo

= J£7 »(®) + ^(|/«(a)|H-|/h(jS)|) a.e. in <a,^>.

w = 0 w = 0

Hence

OO 00

a.e. in < a ,0>.

n = o « . = 0

Letting a -> a, /?->&, we obtain (7).

Lemma 2. I / 9?(a?) is a monotonie function, defined a.e. in an inter­ val (a, 00) and fulfilling condition (5), then the series (2)

00 » i + i

(8) ]?[<Р(Я + П)— J q>(x0 + t)dt], X , X0e(a, oo),

71=0 n

converges absolutely a.e. in (a, 00).

P r o o f. For an arbitrary set A a (a , 00) and an integer h we denote by A -F(fc) the set of numbers x + l c where x ranges over the set A.

Let В be the set of xe(a, 00) at which the function cp (x) is not defined and put

+00

(») E = ( « , с о ) - ( J [■» + (*)]• '

k — — 00

The set (a, 00) — E has measure zero. We shall show that series (8) con­ verges absolutely in the set E.

We take an arbitrary x e E . Then cp(x-\-n) is defined for every n. For the argument’s sake we assume that cp{x) is decreasing and x Then

71+1

cp (Xq ~j- n - j - 1 — 0) ^ J cp (a?Q - ( - 1) dt cp (a?o-f- n -j- 0) (^)

and

О < (p(%-\-n) — 9?(a?o + ^ + 0) < (р(х-\-п)~ J (р(оо0+^Ш

П

^ (р{х -\-7l) —(р (Xо Н- ^ “f" 1 ~ ' 0). Thus it is enough to show that the series

oo

( 1 0 ) £ » ( ж + h) — <p(®0 + П + 1 — 0 ) ]

n =о

converges. Let N be an integer such that x-\-N > ж0 + 1. Consequently

(1 1 ) <p(x-\-n) — (p(x0-\-nJr l — 0) ^<p.(x-\-n) — <p(x-\-N-±-n). Now oo (12) JT l<p{x + n) — <p(x + N-\-n)] n = 0 V — 1 oo =

2 2

converge in view of (5). This shows, according to (12) and (11), that se­ ries (10) converges, which was to be proved.

§ 2. Now we assume that Theorem 1 holds true and we shall prove Theorem 2.

Let cp(x) be a monotonie function, defined a.e. in an interval (a, oo)

and fulfilling condition (6). We may assume that (p (x) is decreasing, for otherwise we consider the equation

- 0( ® + i ) — [ —д Ш = — 9>(®)

instead of equation (1). Consequently, according to (5), the function <p (x) must be non-negative.

We put

X

Ф(х) = J <p(t)dt , x 1e(a1 oo). *1

The function Ф(х) is defined and concave in (a, oo). By (5) we get n+ 1

0 < Ф{тьАг 1) — Ф(ть) = J cp (t) dt < 0) -> 0 as n - > o o .

П

Remark on a difference equation

On account of Theorem 1 there exists a convex function G(x), satisfying in (a, oo) the equation

(13) G{x + 1) — G{x) = Ф{х) and we have G(x) = Y - f (x — х 0)Ф(х0) — OO — {Ф{х + n) — Ф(x 0 -j- n) — (x — x 0) [Ф(x0 + n + 1 ) — Ф(x0 + %)]}, n=0 Y being a constant.

Let В be the set of xe(a, oo) at which the function G(x) is not differ­ entiable. В is a set of measure zero. Define the set E by formula (9). Thus (a, o o) — E has measure zero and for every x e E the functions G{x), G(xĄ-1) and (in view of (13)) also Ф(х) are differentiable. Hence it follows that the functions (4)

(14) g(x) = G'(x) — G'{x0) + y0, :)/0e ( - o o , + o o ),

are defined and increasing in E, i.e. almost everywhere in (a, oo). Accord­ ing to (13) functions (14) satisfy equation (1) in E.

On the other hand, let a function gx(x) be defined and monotonie a.e. in (a, oo) and satisfy equation (1) a.e. in (a, oo). Since 99(0?) > 0 , gx(x) must be increasing (cf. (1 )). Let g0(x) be the function obtained from (14) by putting y0 — gx{x0). Thus

(15) gx(x0) = g0(x0).

The function

(16) Gx(x) = f gx(t)dt

xi

is defined and convex in (a, 00). Integrating the relation

we get whence, writing gx( x + l ) - g x(x) = <p(x) xx + l Gx( x + 1 ) — Gx{x)— f gx(t)dt = Ф(х), £Cj ! 1 G = f g S ) M

and

(17) _ G2(x) — 6?! (a?) — cx,

we obtain

G2{x + 1) — G2(x) = Ф(х).

The function G2{x) is convex in (a, oo), just like Gx(x), and so, by Theo­ rem 1 , it may differ from G{x) only by an additive constant:

(18) G2(x) = G (x) + C.

By (16) and (17) we have

G2(x) = gx(x) — c a.e. in {a, oo), and by (14) and (15)

G'(x) = go(x ) + & (xo) — 9i(x0) a.e. in (a, oo), whence in view of (18)

9i (®) = 9o 0») + (a?0) - gx (®0) + о a.e. in (a, oo).

Setting in the above relation x — x0 and taking into account (15), we obtain

G'{Xo) - g x{x0) + c = 0 ,

i.e.

9i(x) = & (®) a.e. in (a, oo).

It remains to prove formula (6). By Lemmas 1 and 2 we have

oo n+i

G'(x) = Ф(х0) — ^ {?>(#+ n) — f <p{oc0 + t)dt} a.e. in {a, oo).

n = 0 n

Hence

OO

g(x) = G'(x) — G’ (x0) + Уо = Vo— £ {(p{x+n)-cp{x0+ n ) } a.e. in (a , oo). n = 0

This completes the proof.

§ 3. Now we assume Theorem 2 to be true and we shall prove Theo­ rem 1 .

Let cp(x) be a concave function defined in (a, oo) and fulfilhng con­ dition (2). It is then differentiable a.e. in (a, oo) and the function

Ф(х) — <p'(x)

is decreasing. Moreover, we have for x e ( n , n - f l ) for which cp' (x) exists (cf. [1 ])

Remark on a difference equation 7

whence by (2) ИшФ(ж) = 0. By Theorem 2 equation (13) has an

in-X —~^0O

creasing solution

oo

(19) G(x) = Y — ^ {Ф(х-\-п) — Ф(х1-\-п)}

n = 0 '

(xx is a point from (a , oo)) . Here we take Xq+I 00 (20) Т = (р(ж0)+

f У

+

(The point x x may be chosen fairly arbitrarily, in particular it may be taken outside (x 0, a?0 + l ) . Since the function Ф(х) is decreasing, the terms Ф(х-\-п) — Ф{хх-\-п) have a constant sign in (a?0, ж0 + 1 ) and consequently we may integrate term by term; cf. [9], p. 294.)

The function

X

9(00) = y0+ f (r(t)dt x0 is convex in (a, 00). Integrating (13) we get

*0+1

g { x + l ) — g(x) — f G(t)dt = <p(x) — <p{x0), x 0

i.e. according to (20)

g ( x + l ) — g(x) = <p(x).

On the other hand, let gx(x) be a convex solution of (1), fulfilling condition (3). Then Gx(x) = g[(x) is an increasing solution of equation (13) and by Theorem 2

Gx{x) — G(x)Jr G a.e. in (a, 00),

where C is a constant. Since gx(x), being convex, is absolutely contin­ uous, we have

X

(21) <94 (ж) = y 0 + f G{t)dt+C(x — x0) = g(x) + C(x — x 0).

x 0

Moreover, the functions g(x) and gx{x) both satisfy equation (1), whence (22) g ( ^ + l ) - g ( x ) = g1( x + l ) - g 1(x).

Inserting (21) into (22) we obtain (7 = 0, i.e. gx(x) = g(x).

References

[1] К . В о игЪ ak i, Les structures fondamentales de Vanalyse. Livre IV . Fonc-

tions d'une variable reelle, Chap. 1, 2, 3, Paris 1958.

[2] M. А. К р а с н о с е л е к и й и Я. Ъ. Р у т и ц к и й , Выпуклые функции и пространства Орлича, Москва 1958.

[3] W . K r u ll, BemerJcungen zur Differenzengleichung g (x + 1) — g (x) = <p{x) , Math. Nachr. 1 (1948), pp. 365-376.

[4] M. K u c z m a , O równaniu funkcyjnym g (x-\-1) — g(x) =cp( x) , Zeszyty N a ­ ukowe Uniwersytetu Jagiellońskiego, Mat-Fiz-Chem. 4 (1958), pp. 2 7 -3 8 .

[5] — On convex solutions of the functional equation g [а (ж)]— g(x) =<p(x), Publ. Math. Debrecen 6 (1959), pp. 4 0 -4 7 .

[6] — Bemarques sur quelques theoremes de J. Anastassiadis, Bull. Sci. Math. (2) 84 (1960), pp. 98-102 .

[7] — Sur une equation fonctionnelle, Mathematica, Cluj, 3 (26) (1961), pp. 7 9 -8 7 .