Non-convex integral functionals on Musielak-Orlicz spaces

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series E COMMENTATIONES MATHEMATICAE XXX (1990) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXX (1990)

Ry sz a r d Pl u c ie n n ik

(Poznan),

and

(Harbin) Non-convex integral functionals on Musielak-Orlicz spaces

Abstract. Non-convex integral functionals on Musielak-Orlicz spaces are discussed. A re­

presentation for the generalized gradient of an integral functional is given under some conditions.

A sufficient condition for the lower semicontinuity of a functional is given.

Introduction and preliminaries. It is well known that integral functionals on a Banach space are important non-linear functionals, which can be applied in optimization theory. In 1972 A. D. Ioffe and V. L. Levin (see [1]) discussed the subdifferential of a convex integral functional on L00. In 1979 A. Kozek extended these results and studied convex integral functionals on Orlicz spaces of vector-valued functions. In this paper we discuss integral functionals, without the convexity assumption, on Musielak-Orlicz spaces of vector-valued functions.

Let (T, I , p) be a measure space with a non-negative, complete, non-atomic and o--finite measure; (X , || • Ц*) denotes a separable real Banach space and (Y, || • ||y) its dual space. We denote by M(X) = M(T, I , X ) the linear space obtained from the set of all strongly /i-measurable functions x: T-> X by identifying the functions which are equal д-almost everywhere, and similarly for M(Y). Moreover, let <y, x> stand for the value of the functional ye Y at the point x eX . Obviously, for xeM (X ) and yeM(Y) the function <y(-), x(-)> is jU-measurable.

A function Ф: Tx [0, oo)->[0, oo) is called a Musielak-Orlicz function iff (i) for a.e. t e T the function Ф(г, •): [0, oo)->[0, oo) is convex, left-continuous on (0, oo), Ф(Г, 0) = 0, lim„_,.00Ф(^ и) = со and there exists a0 > 0 such that Ф(г, u0) < oo;

(ii) Ф ( , и ): T— > [0, oo) is /i-measurable for every we[0, oo).

The functional / ф: M (X)->[0, oo) defined by 1ф(х, X) = J Ф(г, \\x(t)\\x)dfi(t)

T

ls a convex pseudomodular.

The Musielak-Orlicz space is defined by

ЬФ(Т, X) = {

M(X): / ф(гх, X) < oo for some r > 0).

~ Commentationes Math. 30.1

11

The set

d o m /ф =

{

M(X ): 1

(

,

is called the Musielak-0riiez class.

For every Musielak-Orlicz function Ф, we define the complementary function W:

by

Ф(£, v) = sup {uv — <P(t, и);

[0, oo)}

for every teT,

;

[0, oo). The function Ф is a. Musielak-Orlicz function, too.

Moreover,

U V

Ф(£, u) = j (p(t, s)ds and • T(t, v) = j ij/{t, s)ds,

о

where q>(t, •) and iJ/(t, •) are mutually right inverse functions. Both q>(t, •) and IJ/(t,-) are increasing for a.e. teT.

The Luxemburg and Orlicz norms are introduced as follows:

||.х||(ф) = inf{r > 0: 1ф(х/г, X) ^ 1}, . \\х\[ф = sup \\ <y(f), x(t))dp(t)\.

/ Ф ( у . Г К 1 T

For any

L0(T,

and y

L^T , Y) the function <y(

T-»(

is integrable and

|J <y(f), x(t))dp(t)I ^

|j <y(f), x{t))dp(t)I ^

T T

(see [3]).

Co n d i t i o n

A. A Musielak-Orlicz function Ф(£, и

is said to satisfy Condition A if there exist К ^ 1 and a positive ^-measurable function ô(t) on T such that

j Ф(£, 3(t))dp(t) < oo and Ф(£, 2и) ^ КФ(1, и)

T

for a.e. t e T and u ^ 0(t) (see [4]).

Let F be a subset of a Banach space U. We say that a function / : satisfies the Lipschitz condition if there exists a constant К > 0 such that l / M - / ( / ) l ^ K lly -y 'llu for еуегУ У.У'еК

We say that / satisfies the Lipschitz condition near x e U if there exists s > 0 such that / satisfies the Lipschitz condition on x + eBi , where B. is the unit ball of U.

We say that / satisfies the local Lipschitz condition if it satisfies the Lipschitz condition near every xe U .

De f in it io n 1.

Let

U -+R satisfy the Lipschitz condition near xe U . The expression

, 0/ ч г f ( y + tv)-f(y) TT

/

v) = limsup —--- ;--- , for v e u ,

V - * X

t l O

t

Non-convex integral functionals on Musielak-Orlicz spaces 115

is called the generalized directional derivative at x along v. The expression df{x) = /° (x , v) ^ <£, r>, for all veU},

where U* is the dual of U, is called the generalized gradient of / at x (cf. [5]).

De f in it io n

2. A function / : U -> R is said to be regular at x if the Gateaux derivative f'{x, v) of / at x along v exists and f ( x , v) = /° (x , v) for every v e U (cf. [5]).

An integral functional on. ЬФ(Т, X) is defined by f(x) = J f(x(t))dp(t)

T

for every х е Ь ф(Т,Х), where /(• ) : X ^ R (teT) is a class of functions on X and t-+ft(x) is measurable for every x e X .

Results. For simplicity, we introduce

Co n d i t io n

(M). The function

is measurable for every x e X .

Co n d i t io n

(A). There exists a function K (-)eL r (T, R) such that

\ft(xl) - f t(x2)\ ^ \K(t)\ H x j-x J * for every t e T and for all x l5 x 2eX.

Co n d i t io n

(B). For every teT, the function / ( • ) satisfies the Lipschitz condition on X and there exists c > 0 such that

£ed/f(x) => U\\Y ^ c{\+(p(t, ||x||*)}

for all t

T and

Le m m a 1.

Let tp(t,-) be the right derivative of

,

for any fixed t e T and иЕЬф(Т, R).

(i) ||м||ф ^ 1 implies (p(-,|u(-)|)edom 1^ and ||<p(-, \u( • ^ 1.

(Ü) INL ^ 1 implies | г Ф (с\u(t)\)dp(t) ^ ||м||ф.

(iii) If Ф satisfies Condition A, then for any m > 0, there exists M > 0 such that ||и||ф ^ ш implies (/>(•, |n(-)|)e L^(T, R) and \\q)(-, \u{ • )|)||((p) ^ M.

Proof, (i) By the definition of Luxemburg norm, it is easy to show that 1Мф ^ 1 implies (p(-,\u(-)\)edomI4, and

/ (р[Ф(-,|п(-)|), R] < 1.

If not, we would have

1 < j T [t, (p(t,\u(t)\)]dp(t) ^

T

Without loss of generality, we may suppose that the integral is finite.

116 R. Pl uci e nni k, S. Tian and Y. Wang

Then

I V <p(t,\u(t) 1)

’ Iv [(p(-,\u{‘)\), Я] d/n(t)

< 1

/*.!>(•> MOI), R] 1 î 'f t , q>(t,\u(t)\)]dii(t) = 1.

Therefore, we get a contradiction as follows:

1 =

<

1

-MOI), к ] 1

Ivb>(' ,M -)I),R ] 1

^ [ N ’ MOI). R]

J (p(t,\u(t)\)] dn(t)

T

j[ÿ(Mu(t)|)+ï'[t><p(t,|«(t)l)]]^W

T

$\u(t)\(p(tt \u(t)\)dn(t) ^ ||м||ф,

which implies (i).

(ii) By (i) and using the estimate from the proof of (i), we have j Ф(г, \u(t)\)dfi{t) < 1

М

( - ,М - ) М ]

j <P(t,\u(t)\)dfi{t)

^ J..| ^ . I [#>N01)+!?(*> </?(t,|M(t)l))] N ( 0 < \\и\\ф, which finishes the proof of (ii).

(iii) In view of (i), we can suppose that m > 1. By the monotonicity of (p(t,•) for a.e. teT, we have

2|u(r)|

\u(t)\(p(t,\u(t)\) ^ j (p{t,s)ds

2|u(f)|

^ J (p(t, s)ds = Ф(т 2|м(01) о

for a.e. teT. Hence

¥>[>, v(t,|u(t)|)] = |w(t)l l«(OI)—Ф(г,l«(t)l)

«S Ф(£,

|ы(г)|)+Ф(г,|и(

)|) « 24>(t, 2|»(t)|)

for a.e. teT. Since Ф satisfies Condition A, there exist К ^ 1 and <5(г) defined on T such that

j Ф(С S(t))dfi(t) < oo and 0(t, 2и) ^ КФ(

, u) for a.e. teT,

T

provided и ^ S(t). Hence, choosing an integer n0 > 1 such that 2 < 2m < 2"°,

Non-convex integral functionals on Musielak-Orlicz spaces ^{1 1 7}

we have Ф(1, 2mu) ^ Ф(1, 2nou) ^ К"°Ф(1, u), whenever и ^ 5(1) for a.e. teT.

Let

Tx = {teT: \u(t)\ < mô(t)} and T2 = T \ ^ . Therefore

/^[<i!)(-,|M(-)|), Я] = J *^[r, <p(l,|w(l)|)]^u(l) ^ 2 |Ф(1, 2 \u(t)\)dp(t)

T T

= 2 j Ф[1, 2\u{t)\\dp{t) + 2 j Ф[1, 2m |u(l)|/m]d/i(f)

T i T 2

^ 2 j Ф(1, 2mô(t))dp.{t) + 2Kno J 0[t,\u{t)\jm\dp{t)

T i r 2

^ 2K"°{j Ф(1, 5(0)^(l) + l} = M < oo,

T

which completes the proof of (iii).

Le m m a

2. Let х( -) е Ь ф(Т,Х) and let f{-) satisfy the local Lipschitz condition and Condition (M). Define

f t(v) = f?(x(t), u)

for each fixed v e X and for any teT. Then the function t-*ft(v) is measurable.

P roof. Since /,(•) satisfies the local Lipschitz condition, we may express /,°(x(f), v) as the upper limit of

(1) ft(y(t) + lv)-ft{y(t))

A

with 2 j0 taking only rational values and y(t)-+x(t) taking values in a count­

able dense subset (xj,® t of X. But (1) defines a measurable function of t by Condition (M). Thus ff(x(t), v), being a countable upper limit of measurable functions of t, is measurable in t.

Le m m a 3. (i)

For х ( - ) е Ь ф(Т, X), we have

М ф = sup j |j/(f)l \\x(t)\\xdp(t) = III X III ф.

I v ( ÿ , R ) ^ 1 T

(ii) For х( -) е Ь ф(Т,Х) and ÿ ( - ) e LV{T, R), we have J 1Я01 \\x{t)\\xdp{t) < III

III(«

) IWL>

T

where |||ÿ |||m = inf{r > 0: Iv [ÿ/r, R] ^ 1}.

Proof, (i). By the definition of || • ||ф, it is easy to see that

\\х\\ф= sup J <y(l), x{t)ydp(t) ^ sup S\\y{t)\\Y\\x{t)\\xdn(t)

I < r ( y , Y ) ^ l T 1 чг(у,у) ^ 1 T

= sup Ш 0 1 У * ( 0 М / Ф К sup j |ÿ(t)| \\x{t)\\x dp(t).

I v l II y(-) II Y, R) ^ 1 г I r i y . l t ) ^ 1 T

118 R. Pl uci e nni k, S. Ti an and Y. Wang

For every уеЬ^(Т, R) such that 1т{у, R) ^ 1 and for every x(-)eM(X) choose

(-)

M(Y) such that <z(f), x(t)> = ||x(t)||^ and \\z{t)\\Y = 1 for a.e. teT.

Define y(t) = z(t)\y(t)\ for teT. Then I^(y, У) = /^(ÿ, R) ^ 1 and j (y(t), x(t)}dp(t) = j \ÿ{t)\(z(t), x(t)}dg(t) = j |ÿ(t)| \\x(t)\\xdfi(t),

T T T

which implies that

||х||ф = sup j\y(t)\\\x{t)\\x dp(t).

I v ( y , R ) ^ 1 T

(ii) In virtue of part (i) of Lemma 3 and by the Holder inequality we have Л

(0И1*(0М

*(0 < lllÿlllc'P)HMH® = Illy 111(«р)||х||ф,

т

and the proof is complete.

Now, we will prove the main theorem.

Th e o r e m 1.

I f f ( - ) satisfies Condition (M) and Condition (A), then (i) / satisfies the Lipschitz condition on ЬФ(Т, X);

(ii) For every х е Ь ф(Т, X )

(2) df(x) c= J ôf,(x{t))dp(t);

T

(iii) I f for each teT, f ( - ) is regular at x(t), then f is regular at x, and

(3) df{x) = $dft(x{t))dg(t),

T

where

d/,(x(t)) = {yeL ^T , У): ff(x(t)) ^ <y(f), v(t)}, for all v e L 0(T, X)}.

P roof, (i) Let x l5 х 2е Ь ф{Т, X). Then, by Lemma 3 and Condition (A), we have

\f(x1) - f ( x 2)\ ^ J |/f(x1(t))-/f(x2(0)|^(r)

T

< f \K(t)\\\x1( t ) - x 2{t)\\x dfi{t) ^ IIIXHIm ||xj —х2||ф = Ц \х 1- х 2\\ф.

T

(ii) For every х е Ь ф(Т, X), Я > 0 and v e L 0(T, X), by Condition (A) and Lemma 3, we have

f(y{t) + b ( t ) ) - f t(y(t))

Я *sl*(t)INOIIx,

where у е Ь ф(Т, X), y->x in ЬФ(Т, X) and

\ \K(t)\ \\v(t)\\x dp(t) ^ HI K ||| m 1М1

< °o.

N on-convex integral functionals on Musielak-Orlicz spaces 119

It follows from the Fatou lemma and (i) that

(5) -0 0 < /■>(*, ,) - iimsup + w

y-> X

^

ХЮ

< Iimsup

/ . H + ^ M ) - / , W )Jai(0

y(t)->x(t) J A

A | 0 ■

= 1 /° ( x (f)> v(t))dfi(t),

T

where in the fourth term y(t)->x(f) almost everywhere. According to the definition of the generalized gradient, in view of (5), we have

(6) J /,°(х(г), И 0 Ж 0 3= / V ») 3 v)

T

for any Çedf(x) and for all

L0(T,X).

Let f t{v) = f t°(x{t),v) for v e X . By Lemma 2, the function t-*ft{v) is jU-measurable for every v e X . Define

f(v) = J

t "

for all

L0(T, X). Then (5) can be written as follows:

for all

L0(T,X). In virtue of Proposition 2.11 from [5], f(v) is a finite convex function and £ed/(0), where 5/(0) is the subdifferential of / at 0. By Theorem 3.1 and Corollary 3.11 in [2] there exist ÇeL^T, У) and a singular functional ,9 such that

<£, v} = j <£(f), v(t)}dfi(t) + &(v)

T

for all

L0(T, X), where (e d /t(0) = ôft(x(t)) and #(i>) ^ 0 for all vEdomf.

Since f(v ) is finite for all

L0{T, X), we have 5 = 0. Hence

<£> v) = f <((0> v(t)>dn(t)

T

for all

L0(T, X), i.e.

f e $ dft(x(t))d f(t)

T

and so

df(x) c J dft(x{t))dfi{t).

T

(iii) For any

L0(T, X), by the assumptions, we have

f t'{x{t), v(t)) = f t°(x(t), v(t)).

120 R. P l u c i e n n i k , S. T i a n a n d Y. W a n g

According to (5) and the Fatou lemma, we have

f°{x, v) ^ J f t°(x{t), v{t))dp{t) = J//(x(t), v{t))dp{t)

= jiim fl*M+Mo)-/.(*w^(t)

Y Я k О A

Я “ ^ 0 j* A

= liminf/(X + ;f ~ / ( x ) ^ / 4 x , 1,),

A->0 Л

i.e. f ( x , v) = f°(x,.v) for all v e L 0(T, X ).

Let £e j r 5/t(v:(t))d)u(t). Then there exist £(£)e df{x{t)) for a.e. te T such that

<£, t?> = j <C(0, v(t))dfi(t) < J / t°(x(f), ^ (0 )^ (0 = f°(x, v),

г г

so £ed/(x). From this and by part (ii) of our theorem, we have d/(x) = J 5/((x(t))d/r(t),

г

which finishes the proof of Theorem 1.

Th e o r e m

2. I f Ф(г, и) satisfies Condition Л and /Д-) satisfies Condition (M) and Condition (B), then

(i) / satisfies the Lipschitz condition on any hounded subset of L0(T, X );

(ii) For ebery x e L 0(T,X)

df(x) c= j df(x(t))dp(t);

T

(iii) I f for each teT, f ( - ) is regular at x(t), then f is regular at x and df(x) = J df(x(t))dp(t).

T

P roof, (i). Let x e L 0{T, X) and ||х||ф ^ m. For any u e L 0(T, X) such that

||w ||ф ^ m, by the mean-value theorem (cf. Th. 2.3.7 in [5]), we have (7) f(u{t))-f(x{t)) = <c(t), u(t)-x(t)),

where Ç(t)edft(x*[t)), x*(f) = 0(t)u(t) + (l — 0(t))x(t) and 0(t)e[0, 1] for all teT. By the monotonicity of '(p(t, • ) and Condition (B), there is a c > 0 such that

IIÉWIIr ^ c [ l+ ( ? ( t , ||х*(011лг)] = c[l+ç>(f,||O (t)M (O + (l-0 (O )^ (O |U )]

c[\ + (p(ti max(||u(t)||Xi ИОН*))]

< c[l +(p(t, \\u(t)\\x) + (p(t, ||x(t)||*)].

Non-convex integral functionals on Musielak-Orlicz spaces 121

Define

(8) y(t) = c[\ +(p(t, \\u(t)\\x) + (p(t, Цх(ОИл-)].

By Lemmas 1 and 2 and Holder’s inequality, it follows from (7) and (8) that

\f(u)-f(x)\ ^ $\ft(u(t))-ft(x(tj)\dfi{t)

T

= ^ j U(t)\\y\\u(t)-x{t)\\xdfi(t)

T T

< j 1Я01 \\u(t)-x{t)\\x dfi{t) ^ L\\u-x\\0,

T

where L = |||y ||| (<F) depends only on m.

(ii) For

L 0(T, X ), у е Ь ф(Т, X), y->x and v e L 0{T, X), by the mean- value theorem, we have

(9) \ft(y(t) + AV(t))-ft(y(t))\ Ш , Н Ф

x = \< m , яо>1 for any X > 0, where Ç(t)edft(x*(t)) and

x*{t) = 0(t)(y(O + ^(O) + (l ~0(t))y(t) = X0(t)v{t) + y{t).

By the same proof as for (8) there exists a ÿ e L (f,(T, R) such that ||£(£)||y < y(t) and hence, by (9), we have

\ft(y(t) + Xv(t))-ft(y(tj)\ ^ |ÿ(OI \\v(t)\\x . Since

ЛЯ01 \\v(t)\\xdn(t) ^ Illy III (У)И1ф <

,

T

using the Fatou lemma, we obtain a formula analogous to (5). Therefore, repeating the argument in the proof of Th. 1 (ii), we have

df{x) c { df(x(t))d[i(t).

T

(iii) The proof is almost the same as the proof of Theorem 1 (iii).

Now, we will discuss the lower semicontinuity of the functional / To this end we will introduce some general definition concerning this fact.

De f in it io n 3.

A functional

U

R on a Banach space U is called lower semicontinuous if and only if its epigraph

Epi / = {(x, a)e U x R: / ( x ) ^ a | is closed in U xR.

Le m m a 4.

I f xn-+x in ЬФ(Т, X) and Ф satisfies Condition A, then

xn{t) — ->-0 almost everywhere in T.

1 2 2 R. P l u c i e n n i k , S. T i a n a n d Y. W a n g

P ro o f. Passing to a subsequence, we may assume that ||хи — х||ф < 2 "for all natural n. For each natural k, let

An,k = {teTk: ||xB(t)-x(0llx > V4»

where (Tk) is an increasing sequence of measurable subsets of T such that

00

[j Tk = T and ft{Tk) <oo for к = 1, 2, ...

k = 1

Set 00 00

^oo,к 0 LJ fe­

rn =ln=m Since \\хлп,к\\Ф < \\к(х„-х)\\ф < k/2n, we have

l lz w ll* ^ X \\Хлп,к\\ф+\\Х и Лп1к\\ф<к/2т~1 + \\х AnJ #

n — m n > j n > j

for each j ^ m ^ 1. Since Ф satisfies Condition Д, it follows that I I X u ^ fcU ° as j -кх),

n > j

so \\Хлх,к\\ф = 0 and hence р{Аж^ = 0. Putting к = 1 ,2 ,..., we obtain 00 00 00

r f U П U

iix „(0 -x (0 iix ^ i/fc }] = o,

fc = 1 m = 1 n = m

which shows that ||x„(f) — х(г)||х ->0 almost everywhere in T.

Th e o r e m

3. I f /Д-) satisfies Condition (M) and/Д-) is lower semicontinuous on X for almost every teT, then

(i) / is lower semicontinuous on L0(T, X)\

(ii) For every х е Ь ф(Т, X) the set function A^>f(%Ax) is countably additive on the o-algebra Z, provided /ДО) = 0 and f is proper, i.e. f { x ) > — oo for every х е Ь ф(Т,Х).

P roof, (i) Let (x„, a„) be a sequence in Epi/ convergent to (x, a) in L0( T , X ) x R , i.e. ||x„ — х||ф->0 and |a„ — a|-»0 as n->oo and /(x „ )^ a „ for every natural n. By Definition 3 it is sufficient to show that / (x) ^ a. By Lemma 4 we have ||x„(t) — x(t)||x ->0 a.e. in T. In view of the lower semicon­

tinuity of / ( • ) for a.e. teT, we have

f t(x{t)) ^ lim inf f(x(t)) a.e. in T, so by the Fatou lemma

/(* ) = I ^ J liminf/(x„(0)dju(0 ^ lim inf f f t(xn(t))dp(t) ^ a.

T T л ~ * oc f\ oo j

Non-convex integral functionals on Musielak-Orlicz spaces 123

(ii) Let х е Ь ф(Т, X) and A = |J* =1 An with pairwise disjoint Ane l . Then we have

ЯХл х) = Z / (Х

{

)+/(Х

) (n> 1),

i = l

where Bn = By the lower semicontinuity of /, we have lim inf/(xBnx) ^ /(0) = 0.

Moreover, it is easy to prove that ||/в„х||ф|0 as n->cc. It follows that

П П

f (xAx) > limsup £ f(XAiX) + HminSf(xBnx) > limsup £ f ( x Aix )•

Л 00 j = l П-+СС HGO

On the other hand, since П

II z О и * - Х л * ) | | ф = \\Хвпх \\Ф1 0 , i = 1

we have

f(XAx) ^ liminf/ ( £ x^.x) = Hminf £ /Ou,*)-

n-* со j = l л oo г = 1

Thus

/ ( b * ) = Z /O u * )- i= 1

References

[1] A. D. I o f fe and V. L. L e v in , Subdifferentials of convex functions, Trans. Moscow Math. Soc.

26 (1972), 1-72.

[2] A. K o z e k , Convex integral functionals on Orlicz spaces, Comment. Math. 21 (1979), 109 135.

[3] R. P lu c ie n n ik , Conditions for compactness o f integral operator on Musielak-Orlicz space of vector-valued functions, Publ. Math. 35 (1988), 1-9.

[4] W u C o n g x in , W a n g T in g fu , C h en S h u t a o and W a n g Y u w en , Geometry of Orlicz Spaces, Harbin, 1986 (in Chinese).

[5] H. F. C la rk , Optim and Nonsmooth Analysis, Wiley-Interscience, New York 1983.

INSTYTUT MATEMATYKI, POLITECHNIKA POZNANSK.A

INSTITUTE OF MATHEMATICS, POZNAN TECHNICAL UNIVERSITY PIOTROWO ЗА, 60-965 POZNAN, POLAND

DEPARTMENT O F MATHEMATICS

HARBIN UNIVERSITY OF SCIENCE AND TECHNOLOGY 22 XUEFU ROAD, HARBIN, CHINA