Lipschitzian composition operators in some function spaces

Nonlinear

Analysis,

Theory,

Methods

Pergamon PII: SO362-546X(%)00287-2

&Applications. Vol. 30, No. 2, pp. 719-726, 1997 Proc. 2nd World Congress ofNonlinear Analysts 0 1997 Ekevier Science Lnd F’rinted in Great Britain. All rights reserved 0362-546X/97 $17.00 + 0.00

LIPSCHITZIAN COMPOSITION OPERATORS IN SOME FUNCTION SPACES JANUS2 Department

Key words

of Mathematics,

and phmses:

Technical

Composition

MATKOWSKI

University,

operator,

Wil owa

Lipschitzian

2, PL-43-309

operator,

function

Bielsko-Biala,

Banach

Poland

space,

AC(I,IR)

space, BV-space, HBlder space, C’ space. INTRODUCTION Let I = [0, l] and let J E R be a nonempty interval. By F(I, J) denote the set all the functions 4 : I +J. For a fixed two-place function f : I x J 4 IEt the mapping F : 7(I,J) -+ T(I,IEt) given

by

f’(d)(z) := f(2, d(~)>, is said to be a composition

(or Nemytski)

ZEI, 4EnI,J),

operator.

In the case when J = IR it was shown in [6] that a composition operator F mapping the function Banach space Lip(I,IR.) into itself is globally Lipschitzian with respect to the Lipschitzian norm if, and only if, 2 E I, YE& f(G Y> = dZ)Y + h(z), for some g, h E Lip(I,IR). Next this result has been extended to the Hiilder spaces H,(I,R) in [5], (21(cf. also [l], p. 194), to the space BV(I,B) o f f unctions of bounded variation in [8], to th’e space Ck (I,lR.) in [i’], to the Sobolev space W”@(l,lR), n > 1, in [12] and to some other more special function Banach spaces (cf. for instance [ll]). In section 1 of the present paper we show that the same property has the function Banach space AC (1, IR) of all absolutely continuous functions 4 : I -P IR. In fact we prove the following stronger result. Let J E IR, be an arbitrary interval, f : I x J --+ IR a fixed function, and F the composition operator generated by f. By AC(I,J) denote the set of a.ll 4 E AC(I,IR) such that 4(I) c J. If F maps AC (I, J) into AC (I,R) and there exists an L 2 0 such that

II F(4) - F(+) llx 5 L II d - d IL,

4, 11E AC(I,J),

then

f(z,

Y) = dZ)Y

+ h(z),

2 E I, y E J,

for some g, h E AC (1,lR). It turns out that the assumption J = R made in the relevant theorems from the above quoted papers is also superfluous. In section 2 we give the suitable more general results.

719

720

Second World

LIPSCHITZIAN

Congress of Nonlinear

COMPOSITION

Recall that the linear space AC (I,IR) in I, endowed with the norm

Analysts

OPERATORS

of all functions

IN THE

SPACE

4 : I + lR which

AC (I, J)

are absolutely

continuous

II 4 ILc := I 440) I + l1 I d’(t) I dt, is a Banach space. For an interval J s JR, by AC (I, J) denote the set of all 4 E AC (I, IR) such that d(1) c J. THEOREM 1. Let J c IR be an interval and F the composition function f : I x J -+ lR. Then lo. F maps the set AC (I, J) into AC (I,lR.), and 2O. there exists an L 2 0 such that

f(x,

Y) = dX)Y

generated

by a fixed

4, ti E AC(I, J),

II F(4) - F(+) II/c I L II 4 - ?I,IIAc> if and only if there exist g, h E AC (I,R)

operator

such that ZEI,~EJ.

+ h(x),

PROOF. Suppose that F satisfies conditions lo and 2O. Take y, jj E J and define 4, + : I + IR by := y, +(t) := y, t E I. Then, of course, 4 , 1c,E AC(I, J). Since

d(t)

and for all x E I,

I f(X,Y) - f(x,B) I I I f(O,Y) - f&4*) I + I f(X,Y) - m = I f(O,Y) - f(Ot8 I f I Joz; Ml,
J J

os I -gf(l,

5 I f&4 Y) - f(O, 5) I t

= I F(+)(O) - F(?1W) I +

J’ I -&VP)

Y)

Y)

- f(x7 Y>+ mid

I

- f(t, 9) dt I

?I> - f(h $1 I &

o1 I -$W, y) - f(h $1 I dt - F(lll)(t))

I dt = II F(d)

- F(ti,)

IL,

0

we infer from 2’ that I f(GY)

- f(x,$

I I: L I Y - 3 I,

ZEI,

YE J.

For every fixed y E J the constant function d(t) = y, t E 1, belongs to the set AC (I, J). Therefore, in view of lo, the function F(4) = f(., y), where f(., y)(t) := f(t, y), t E I, is absolutely continuous. It follows that the function f is continuous in the set IxJ. Take arbitrary ~1, yz, 81, 32 E J; z-cE I, n E IN, and a finite sequence (xk), lc = 1,. . . ,2n, such that 0 < x0 < Xl

< . .. <

x2,, < 1.

Second World

Let 4 be the polygonal (O,Yl),

and, similarly,

function

Nonlinear Analysts

of

the graph of which

(21,Yl),

. a-7

(Qkvy22),

let II, be the polygonal (O,Ylh

Congress

(21,Ylh

(22k+l,?/d,

function

--*,

is determined . .->

the graph

(ZZk,ji2),

72: 1

by the vertices (LYZ),

(22n,y2),

of which is determined

(~?kfl,ia,

me.7

(22n,Y2),

by the vertices

(1732).

Clearly 4 , II, E AC (I, J). Since 4, II, are constant on the intervals [0, zr], [2sn, 11, and linear intervals [zk, zkfl], k = 1,. . . ,2n- 1, by the definition of the norm ]I . [IA,-, we have

on the

II 4 - ?/,IL = I 4(O) - WX I + /ill I 4’(t) - @‘(t>I dt =

1 y1 -

7j1 1 + 2F

I 4’(t) - q’(t) I dt

J’“”

k=l

=k 2n-1

= I Yl

-

I+ c

11

I Yl

-

31 -

y2 +

y2 I

kc1

= I Y1Moreover

I + on- 1)l

I1

y1-

L&-y2

+ 82

I .

we have

II F(d) - F(ti) IIAC = I f@, WI) - f(O, NJ)) I + J,l I &O-(4 d(t)> - f(tv WI) I dt 2 I’ 1$(f(t,

c)(t)) - f(t, 4(t)))

I dt = ‘gl 1:“”

I -$f(t, b(t)> - f(4 ti(t))) l dt

2n-1 =

1 f(Zk+l,dZk+l))

c

-

- f(Zk,dhk))

f(Zk+l,d'(zk+l))

t

f(zk,+(zk))

1.

k=l

Note that for each k = 1, . . . , 2n - 1, either d(zk)=

d(Zk) = yr or b(zk) yl

+=+

d'(xk+l)

= ~2, and

= y2.

An analogous fact remains true if we replace 4 by $ and y; by yi, i = 1,2. and making Therefore, letting zk -+ z for ah k = 1,. .., 2n in the above inequality continuity of f, we get 2n-1

II F(4) - F(11) IIAC1 c

I f(&Yl)

- f(GY1)

- f(X,Y2)

-I- f(&32)

I;=1

Now applying

2’ gives (2n -

1) I f(?Yl)

- f(2731)

- f(z,Yz)+

f(Gh)

I

I*

use of the

Second World Congress of Nonlinear Analysts

122 I

for all z E I,

L(I

Yl - a1

I +e

~1, 92, yl, yZ E J. Taking y1=-,

u+v 2

-

1)

I Yl -

?-A - Y2

+

u, v E J and substituting Y2

= v,

g1 = u,

I),

32

here u-tv 2

?i2=-,

yields the inequality

en

- 1). I WC?7,

Since n E IN is arbitrary,

- ax, u) - f(x, v) I 5 L I y

I,

u, v E J, n E IN.

it follows that

2f(x, q

- f(x, u) - f(x, v) = 0,

u,v E J,

which means that for every z E I the function f(s, a) : J + IR., defined by I(z, .)(y) := f(z, y), is a continuous Jensen function. exist g(z), h(x) E IR such that

Thus (cf. M. Kuczma

f(x,

Y) = dX)Y

Y

[3], p. 316, Theorem

E J, 2) for every z E I there

y E J.

+ h(x),

By lo for every constant c E J we have f(., c) = cg + h E AC (1,lR). g and h are absolutely continuous. Since the converse implication is obvious, the proof is completed. q

It follows

Taking J = IR in this result one gets the following COROLLARY 1. Let F be the composition operator generated by a function lo. F maps the AC (I,lR) into itselfi and 2’. F is globally Lipschitzian, i.e. there exists an L > 0 such that

that the functions

f : I x J --) IR. Then

II F(4) - F(1/1)IL 5 L II 4 - II, II,tc, if and only if there exist g, h E AC (I,lR,)

f(x7 Y) =

dX)Y

such that t h(x),

2 E I, y E IR.

Let B C F(1, J) be a family of functions having the following n E IN, and a finite sequence (zk), Ic = 1,. . . ,2n, such that

property:

0 < x0 < Xl < . . . < xzn < 1, the polygonal

function @,Yl),

belongs

4 the graph of which is determined (Zl,Yl),

(Zz,Yz>,

to B. Of course B c AC (I, IR).

*. .1 (XZn-l,Yl),

by the vertices (XZWYZ),

(l,Yz),

for all yl, y2 E J;

Second World Congress of Nonlinear Analysts

723

The definition of the set B and the argument used in the proof show that the assumptions main part of Theorem 1 can be essentially weakened. In fact we have the following REMARK 1. Let J E: lR be an interval and F the composition f : I x J + IR. If F maps the set of functions B into AC(I,IR)

of the

operator generated by a function and there is an L 1 0 such that

II J’(9) - F(4) llx I L II 4 -. II, IL,

A II, E B;

then there exist g, h E AC (I, R) such that x E I, y E J.

f(z, Y) = 9(X)Y + h(s),

LIPSCHITZIAN

COMPOSITION

By BV (I, IR) denote the Banach

OPERATORS

space of all functions

IN SOME

OTHER

SPACES

4 : I + IR with the norm

II d IL” := I d(O) I + V(d), where

V(d)

denotes the total variation V(4)

of q5 over the interval := sup 2

+(Xi) -

I

I, i.e.

4(zi-1)

I>

i=l

the supremum is taken over all positive Xl < . . . < 2, = 1. For an interval

J c R,

integers

n and over all choices (2;)

by BV (I, J) d enote the set of all functions

such that 0 = zo <:

q5 E BV(I,1R)

such that

4(I) c J. Now, similarly THEOREM

f :1x

as Theorem

1, we can prove

2. Let J E IR be an interval J-+R.If

and F the composition

operator

generated

lo. F maps the set BV (I, J) into BV (I,IR), and 2O. there exists an L 2 0 such that

II J’(4) - F(+) IL I L II 9 - ?I,IL,

4, $ E WI,

then

(i) (ii)

If(z,y)--f(z,$l<

LIV-81,

x E I,

for every x E I there exist a function

f-(x,

f- (2, Y) := & and the left continuous

functions

y,g E J;

.): Y E J,

f (t, Y>,

g, h E BV (I,lR)

f-(2, Y) = dX)Y + h(x),

such that XEI,

~EJ.

J),

by a function

724

Second World Congress of Nonlinear Analysts Taking

J = IR in Theorem

2 gives the result of [8] (cf. also [l], p. 175, Theorem

REMARK 2. It is easy to verify that the counterpart for the space BV (I,lR).

of Remark

1, with the same set B, is also true

Let lR+ := [O,oo). An LY: lR+ -+ IR+ is said to be HZilder function mentary function a* : lR+ -+ lR+ defined by a’(t) are positive,

increasing

:= q

f ),

1 > 0,

6.14).

i(O)

if and Q and its comple-

:= 0,

and a(O) = 0, ~(1) = 1.

Note that (a’)’ = (Y (for some other properties For two HGlder functions (Y and /3 we write (Y < /3 if Given a HGlder function a, the HGlder functions $J E r(1, IR) for which

cf. [13]).

o(t) = 0@(t))

as t + 0.

space H,(T,IR)

consists,

by definition,

of all continuous

where w(4,s) Equipped

with

:= sup{1 4(z) - 4(z)

I: 2, z E I,

I 2 - z iI s}.

the norm

II 4 lla := I d(O) I + b(4), Ha(l,IR.)

is a Banach

For an interval

space (cf. [l], p. 182).

J C IR, by H,(I,

J) d enote the set of all functions

4 E SV(I,IR)

such that

40 c J. In a similar

way as Theorem

1 we can prove the following

THEOREM 3. Let J c IR be an interval and F the composition operator generated by a function f : I x J + IR. Suppose that (Y and p are Hijlder functions such that Q < p. Then lo. F maps the set H,(I, J) into Hp(I,R),

and 2O. there exists an L 2 0 such that

II F(4) - F(+,) Ilo I L II 4 -- ti lloo

4, 11,E &.(I, J)

if and only if there exist g, h E Hp (I, IR) such that

f(z, Y) =

dZ)Y

+ h(z),

~EI,

~EJ.

Second World

Congress

of Nonlinear

Analysts

725

Taking here J = lR and 4(t) = $(t) = t, t E IR+, we obtain the first result of that type proved in [6] for the Banach space Gp(1,IR). For J = IR and 4(t) = tp, +(t) = tq, t E R+, 0 < y 5 p 5 1 we obtain the results of [5] and [2]. For J = IR the relevant result is presented in [l]. REMARK 3. Similarly as in the case of Theorems 1 and 2 the necessity conditions in Theorem 3 can be considerably weakened. It turns out that it enough to postulate the conditions lo and 2’ of Theorem 3 for all 4, $ E B where B is a family of functions satisfying the following condition: for all z, z E I, 2 < Z; y, g E J, the polygonal function the graph of which is determined by vertices (O,Y), belongs

(2, YL

(g,id,

(LY),

to B.

Let Ic E lN, k 2 1 be fixed. By C”(I,IR) we denote the Banach continuously differentiable function 4 E 7(1, lR), equipped with the norm

space

of all

k-times

k-l

II 4 Ilk := c I d’“‘(O) I + II P 110, i=o where 11. II0 stands for the supremum norm. For an interval J E lR, by Ck (I,J) denote the set of all 4 E Ck (I,R)

such that $(I)

c J.

4. Let J c IR be an interval and F : F(I, J) -+ F(I,IR) the composition operator by a function f : I x J -+ IR. Suppose that F maps the set C”(I, J) into the space C” (I,lR), where m 2 TX, m, TZE IN. If there exists an L 2 0 such that

THEOREM generated

II F(4) - F(?1) Iln 5 L II 4 - ?I Ilm, then there exist g, h E C” (1, lR) such that

f(G The relevant

Y) = dZ)Y

+ h(z),

TEI,

YEJ.

result for J = R was proved in [7] (cf. also [l], p. 212, Theorem

REMARK 4. Also this result subset B.

can be a little

strenghened

by replacing

8.4)).

set C” (I, J) by a suitable

REMARK 5. To explain the formal structural difference in formulation of Theorem 4 and all previous Theorems 1, 2 and 3, note there exist the composition operators F mapping for instance C’ (1,lR) into itself with discontinuous generators f : I x IR --) IR (cf. [3], also [l], p. 209).

FINAL

REMARKS

The multidimensional as well as multivalued versions of Theorems l-4 are also true. Making us of the argument used in papers [9] and [lo] one can prove the counterpart of Theorems 3 for more general Halder space of vector-valued functions defined on a convex subset of a normed space.

726

Second World

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of Nonlinear

Analysts

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