Modular approximation in

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXX (1991) ROCZNIKI POLSKTEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXX (1991)

An d r z e j Ka s p e r s k i

(Gliwice)

Modular approximation in X \ by a filtered family of sublinear operators and convex operators

Abstract. We introduce the notion of boundedness of a filtered family (TJ of operators in a space of multifunctions. This notion is used to get convergence theorems for families of sublinear operators and families of convex operators.

1. Introduction. Let If(Q, I , g) be the Musielak-Orlicz function space generated by a modular g(x) = JQ(p(t, \x(t)\)dg. Let X be the space of all multifunctions F from Q to 2R such that F(t) is nonempty and compact for every teQ. Two multifunctions F , G e X such that F(t) = G(t) for g-a.e. teQ will be treated as the same element of X. Let F e X . Define

f{F)(t) = min x, f(F)(t) = maxx for teQ ,

xeF (t) xeF (t)

Xjp = {F eX : F(t) is convex for every te Q and

f(F), /(F ) eI?(Q, Z,

)}.

2. A general theorem. Let V be an abstract nonempty set and let

be a filter of subsets of V.

De f in it io n 1.

A function

tends to zero with respect to

written g(v)^>0, if for every e > 0 there is a set VeY' such that \g(v)\ < s for all v e V

De f in it io n

2. An operator A: will be called q-sublinear if for

both f = f and f = f we have

\f(A(aF + bG))(t)\ « |a/(A (F))(t)| + |a/H(F))W | + \bf(A(G))(t)\ + \bf(A(G))(t)\

for all F, G e X I and a, b e R and every teQ.

De f in it io n

3. A family T = (Tv)vev of operators Tv: will be called

'V'-bounded if there exist positive constants k x, ..., k8 and a function g: F-»R +

such that g(v)^> 0 and for all F, G eX^, there is a set V f ^ e V for which

332 A. K a s p e r s k i

e{a(f(Tv(F )) -f (T v(Gj))) klB(ak2(f ( F } - f ( G ) ) ) + ks e(ak4(f(F) - 7 (G)))+g(v), e ( a ( 7 ( T M - 7 ( T v(G)))) k 5e(ak6(_f(F)-_f(G)))

+ k1ÿ(aki (J (F)- J (G))) + g(v) for all v e VFtG and every a > 0.

R em ark 1. If

is convex, then kt , k3, k5, k7 may always be taken to be 1.

De f in i t i o n

4. Let F

X\, for every

V and let F

X J. We write Fv - ^ f F if for every г > 0 and every a > 0 there exists Ve 'V' such that

Q(a(f(Fv) - f ( F ) ) ) < e and g{a(f(Fv)-f(F))) < e for every v e V

De f in i t i o n

5. Let S cz X*. Let

= {F

X^,: Fv^ F for some F

S ,

V\.

The set S ^

will be called the (<p, i^)-closure of S in X£.

Th e o r e m

1. Suppose the family T = (Tv)veV of q-sublinear operators Tv:

Хф-+Хф is 1V-bounded. Let S0 cz X* and let TV{F) ~ ^ 0 for every F

S0. Let S be the set of all finite linear combinations of elements of S0. Then Tv(F)^j^0 for every F

S ^

-

P ro o f. First, note that the assertion holds for all F

S, since supposing F = c1F 1 + ... +cnFn with F te S 0 we have, writing c = Y

=

lc,-l> f°r a > 0

e ( a f ( T v(F))) « t e (2ac/(T„(F1)))+ £ е(2ас/(Г„(Г,)))^0.

i — 1 i = 1

Now, let a, e > 0 be arbitrary and let F

S ^

be given. Then there exist G

S and

such that

g{2ak2( f ( F )- f( G )) ) < e/(4kj), e(2ak4(f(F)-J(G))) < e/(4fc3), e(2a/(T„(G))) < e/4, g(v) < e/4

for every

V1, where we may assume kt , k3 ^ 1. Let VF<G be chosen for (Tv)vev and F, G according to the definition of У -boundedness. Then we have

Q(af(Tv(F)j) < e(2a( _f{Tv(F)j)-_f(T„(Gj)) + e(2af(Tv(G)))

=£

к 1 е(2ак2(/(Р)-7Щ ) + кзе(2ак47(Р)-/(С)))

+ g(v) + e {2af(Tl,(G))).

Approximation in X* by a family of sublinear operators 333

Taking V = Vl n VF<G, we obtain Q(af(Tv{F))) < s for all veV. We prove analogously that there exists Ve'F' such that g(af(Tv(F))) < e for every ve V.

Hence Tv(F)~ÿ*0 because V0 = V n V e 'F '.

Definition

6. An operator A: Xl~> X l will be called m-sublinear if for all F, GeX* and a, b eR

max(| f(A (a F + bG))(t)|, \ J (-4(aF + bG))(t)|)

m ax(|a/(A(F))(r)| + \ b f (/4(G))(t)|, |a/(.4(F))<t)| + |6/(^(G))(t)|).

Theorem 2.

Let T = (Tv)veV be a У'-bounded family of m-sublinear operators Tv: X ^ X ^ . Let S0 cz X* and let Tv{F)~p*0 for every F e S 0. Let S be the set of all finite linear combinations of elements of S0. Then Tv(F)~y^0 for every F e S 9rir.

The proof is quite analogous to that of Theorem 1 and we omit it.

Definition

7. An operator C: will be called convex if for both / = / and / = / we have

/(C (aF + (l-a)G ))(t) « af(C(Fj)(t) + (l-a)f(C(G))(t) for all F, G e X j and

[0, 1] and every teQ.

Theorem 3.

Let T

{Tv)veV be a 'V'-bounded family of convex operators Tv: Xl~+ Xl such that for every F e X * , f (Tv(F))(t) ^ / (F)(t) and f ( T v(F))(t) ^ / (F)(t) for every te Q and every veV. Let S0 a X], and let Tv(F)~j^F for every F e S 0. Let now S be the set of all finite convex combinations of elements of S0. Then Tv(F)~y>F for every F e S (Ptir.

P ro o f. First, the assertion holds for all FeS, since supposing F = cxF t + ... + cnFn with F {-e S0, ct ^ 0, i = 1, ..., n, cx + ... + c„ = 1 we have for both f = f and / = /

<?(a(/(7;(F))-/(F))) « C(a( f с ,(/(В Д )) ))- £ c,(/(F,)))

£ = 1 £ = 1

= e(a( t ъ Ш Ш ) ) - Л Р , ) ) ) )

£= 1

« Î

for every a > 0.

The remainder of the proof is quite analogous to that of Theorem 1 from [2] (see also the proof of Theorem 2 from [1]) and we omit it.

^ — Comment. Math. 30.2

334 A. K a s p e r s k i

R e m ark 2. Let the assumptions of Theorem 3 hold and assume additionally that F(t) = — F(t) for all F e S 0 and teQ. If, moreover, Tv(aF) = aTv{F) for all a e R , F e X ^ and veV, then we can take for S the set of all finite linear combinations of elements of S0.

References

[1] A. K a s p e r s k i, Modular approximation by a filtered family o f sublinear operators, Comment.

Math. 27 (1987), 109-114.

[2] —, Modular approximation in X \ by a filtered family of “linear operators”, this fascicle, 335-341.

[3] J. M u s ie la k , Orlicz Spaces and Modular Spaces, Springer, Berlin 1983.

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