An existence theorem for the generalized hyperbolic equation z'xyeF(x,y

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXX (1990)

Ma r i a n Da w i d o w s k i

and

(Poznan)

An existence theorem for the generalized hyperbolic equation

z'xy e F ( x , y , z) in Banach space

Abstract. It is shown that the Darboux problem for the generalized hyperbolic equation z”ye F ( x , y , z ) has a solution, where F has compact values in a separable Banach space, is measurable in (x, y), integrably bounded and Lipschitzian in z, or continuous. Our result is proved via the contraction principle for multifunctions of Covitz and Nadler, using the method of Himmelberg-Van Vleck ([8]) and via Sadovskii fixed point theorem for multifunctions, and it is a generalization of some result of Sosulski ([12]).

Introduction. Throughout the whole paper <£, | • |> is a separable Banach space, and Com p£, CConvF, C1B£ the family of all nonempty compact, compact convex and closed bounded subsets of E. The Hausdorff metric H on Comp E is given by

H(A, B) = max (sup {h(a, B): aeA}, sup {h(b, A): be В}}

where A,

Comp E and

h{a, B) = inf{|a — b\: beB}.

For Л e Comp Is

STS = H(A, {0}) = sup {Sa|: aeA}.

In the space Cl E of all closed subsets of E we can introduce analogously the Hausdorff pseudometric, which takes in addition the value oo.

Let 7] = [0, a] and T2 = [0, fr] be closed bounded intervals, and let T= 7\ x T2 be the Lebesgue measurable space. A multifunction F: T->Comp£ is said to be strongly measurable if there exists a sequence {^nlnejv of simple functions such that \imn^ ж H(Fn{t), F(t)) = 0 for almost all teT.

A multifunction F: T->Comp£ is measurable (weakly measurable) iff F~1(B) = {teT: F(t) n В ф 0} is Lebesgue measurable for each closed (open) subset В of E.

The three measurability concepts are equivalent (see [7], Th. 3,1; or [3],

Th. III.2; and [11], Th. IV.26).

By L(T, E) we shall denote the space of all Lebesgue-Bochner integrable functions / : T-*E endowed with the norm

II/II

= Я If ( x , y)\dxdy.

T

C(T, E) is the space of all continuous functions / : T ^ E .

Existence theorems. Given F : Tx E -> Comp E we shall study the existence of solutions for the Darboux problem

z'xy(x, y)eF(x, y, z{x, y)),

z(x, 0) = u(x), z(0, y) = u(y) for (x, y)e T,

where и: [0 ,a]-+E, v: [0, b]-*E are absolutely continuous functions and u(0) = v(0) = z0.

A function T->E is called a solution of (*) if it is absolutely continuous in T, f (x, 0) =

£(0, y) = v(y) and

^ - ^ Ç ( x , y)eF(x,

Ç{x, у)) x almost everywhere in T.

Th e o r e m

1. Let F

T x E -> C o m p £ satisfy the following conditions

(i) F(-, z): T->-Comp£ is measurable for each z e E , (ii) there exists a function к

L(T, R +) such that

H(F(x, y, z j , F (x , y, z2)) ^ k(x, y)’\z1- z 2\

(i.e. F{x,y,-): T ^ E is Lipschitzian with constant k{x, y)), (iii) there exists a function m eL(T, R +) such that

IF(x, y, z)I < m(x, y) for each z e E .

I f J J r /c(x, y)dxdy < 1, then the problem (*) has a solution on T.

P ro o f. Let SC = { /

L(T, E): |f ( x , y)| ^ m(x, y) for (x, y)eT}. The set SC is closed in L(T, E). We define I: 3C->C(T, E) by

* ^У

I(f){x, y) = M(x) + u (y)-z0 + J J /( t, s)dtds.

о 0

We associate with F a new multifunction Ф defined on SC by H f ) = {geST: g(x, y)eF(x, y, I{f){x, y))}.

We next show that Ф has closed nonempty values. In fact, by a Scorza Dragoni type theorem (see [6]) for any e > 0 there exists a compact subset TE of T such that p { T \T J < e and F\TeXE is continuous. Hence F(-, / ( / ) ( ‘))1

is con­

tinuous for each feSC and in virtue of Lusin’s theorem F (•,/(/)(•)) is

measurable on T (see [11], IV,4). By the Kuratowski-Nardzewski theorem

(see [9], Th. 1) the multifunction F(-, /(/)(•)) has a measurable selector g and Ig(x, y)| ^ IF(x, y, y))| ^ m(x, y) for (x, y)eT. Thus Ф(/) is nonempty for each f еЖ.

Now suppose that

is a sequence in Ф(/) which converges to g0eL(T, E). Then д0еЖ and there is a subsequence g„k of gn which converges almost everywhere on T to g0. Thus for almost all (x, y)e T, g„k{x, y)->c/0 (x, y).

But g„k{x, y)eF(x, y, I(f)(x, y)) and- F(x, y, I(f)(x, y)) is compact. Hence g0(x, y)eF(x, y, /(/)(x , y)) a.e. and Ф(/) is closed.

Next we show that Ф: Ж -» Cl Ж is a contractive multifunction. To see this, let иеФ (/). Then v is measurable and y(x, y)eF(x, y, /(/)(x , y)). By Filip­

pov’s lemma (see [13], Lemma 2.1.4) there exists a measurable selector w of F(-, I(g)(-)) such that

|y(x, y) w(x, у)I = h(v(x, y), F(x, y, I(g)(x, y)))

< H(F(x, y, I(f)(x, y)), F(x, y, I{g)(x, y)))

^ k(x, y)|/(/)(x, y)-I(g)(x, y)|

* у

= Цх, y ) |f |( / ( L s)-g(t, s))dtds\

о 0

x

у

^ fc(x, y ) f j |/ ( t , s)-g(t, s)| d t d s ^ k ( x , y)||/-gr||L.

о 0

Hence

||i?-w||L ^ jjk ( x , y)dxdy• \lf-g\\L = K \\f-g \\L.

о 0

From this and the analogous inequality obtained by interchanging the roles of / and g we get

0{Ф(Л, Ф{д)) < K \\f—g\\L,

where D is the Hausdorff metric on Cl Ж corresponding to || • ||L and К < 1.

Finally, we apply the contraction principle of Covitz and Nadler (see [3], Corollary 3) to Ф to obtain a function/ such that/е Ф ( /) . Let q> = /( /) . Then (p is the desired solution of (*) on the rectangle T.

R em ark 1. If the functions к and m in hypotheses (ii) and (iii) are replaced by constants К and M, respectively, then Ф maps the closed ball В of L(T, E) into C1B. Further, if we renorm L(T, E) by letting

ll/IL = ess sup {I f(x , y)|: (x, y)eT}, then it is not difficult to show that

£>Ж(Ф(Л, Ф(д))< K \ \ f - g ||со,

where Dm is the Hausdorff metric corresponding to ||-|L -

T

2. Assume (i)— (üi)- I f k(x, y) = k(x)l(y), where /ceL(Ti, R +) and le L (T 2, R +), then (*) has a solution on T.

P ro o f. The proof is a slight modification of that of Theorem 1. Let

* у

K(x) = J k(t)dt and L(y) = j l{s)ds.

о о

We replace the usual norm || • ||L in L(T, E) by the following one (proposed by A. Bielecki):

II f\\B = $$е~ЦК{х)+Ш)\/(х, y)\ dxdy,

T

where X is any constant greater than one. The norms || • \\L and || • ||B are equivalent.

Analogously to the proof of Theorem 1 we see that for any иеФ (/) there exists we<I>(g) such that

X У

\v{x, y )-w (x , у)I ^ k (x )l(y)\\\f(t, s) — g(t, s)\dtds

о 0

for (x, y)e T.

Hence by the Lebesgue-Fubini theorem ([11], IV, §8, Th. 91)

\\v — w||B ^ J j (e~МК(Х)+Ш)k(x)l{y) J J I f{t, s) — g(t, s)| dtds)dxdy

00 00

= jd x Jdy J d t§e~*{Kix) + L{y))k(x)l(y)\f(t, s) — g(t, s)| ds

0 0 0 0

= § dx § dt j dy §(.. .)ds = § d tj dx j d s j(.. .)dy

o o o o о t о s

= J dt§ ds^ds J (.. •) dy

0 0 t s

= §dt$^-(e~XL(s) — e~XL(b))-\-(e~*K(t) — e~ma))\f(t, s) — g(t, s)|ds

о о Л- X

^ d t$е~МК(г)+Ш)\ № ’ s)lds = T

I I /-

II в-

From this and the analogous inequality obtained by interchanging the roles of / and g we get

where DB is the Hausdorff metric generated by || • ||B.

R em ark 2. Assume (i)— (iii). If k{x, у) = К = const, then (*) has a solution on T.

Theorem 3 below is a modification of Sadovskii’s result [10].

Assume that the function Ф: 2 has the following properties:

(а) Ф (Х и{х}) = Ф(Х),

(Д) if I c f then Ф{Х) ^ Ф(У), (у) Ф(

(Х)) = Ф(Х),

(<5) if Ф{Х) = 0, then X is relatively compact subset of E, for every point x in E and any subsets X , Y of E.

T

3. Let SC be a nonempty closed, bounded, convex subset of a Banach space and suppose T: SC -> conv SC is continuous and

Ф(Т[Х]) < Ф{Х) for each subset X a SC for which Ф(Х) > 0.

Then T has a fixed point in SC.

P ro o f. By Michael’s Selection Theorem [2], there exists a continuous selection. This selection satisfies the assumptions of Sadovskii’s fixed point theorem, so has a fixed point. This fixed point is a fixed point of T.

Let a be Kuratowski’s measure of noncompactness and P = [0, a] x [0, b].

T

4. Let F: P x £ -> C C o n v £ satisfy the following conditions:

(a) F(-, z): P-»C C onvP is measurable for each z e E , (b) F(x, y, ■): E-+CConvE is continuous for each (x , y ) e P , (c) there exists a function meL(P, R +) such that

IF(x, y, z)I ^ m(x, y) for each z e E , (x, y)eP, (d) there exist functions h: P-+R + , p: R+ ->R+ such that

(4) k e L (P ,R ),

(iij p is nondecreasing and

Kp{t) < t for t > 0 , where K = J J k{t, s)dtds,

p

(iiij) for every e > 0 and every subset Z a E, there exists a closed subset В с P such that p{P\B) < e and for every closed subset Q с В

a(F(QxZ)) ^ sup {k{t, s): (t, s)eQ} ■ p(ct{Z)).

Then the problem (*) has a solution on P.

P ro o f. We define mappings I : L{P, E)-+C{P, E) and S : C(P, E)

~+2L(P'E) by setting for each f e L ( P , E)

x у

/(/)(x , y) = n(x) + v (y)-z0 + J J /(f, S)dtds,

о 0

and for each /е С ( Р , E)

S{f) = [g: g is a measurable selection of P(-,/(•))}.

Next we define a mapping T : C(P, £) -> 2C(P,E) by T (/) = /(£(/)). If /is a fixed point of T, then / is a solution of (*).

We will show that T is continuous. First we show that S is continuous.

Let fe C ( P , E). By a Scorza Dragoni type theorem and Lusin’s theorem F (-,/(-)) is measurable on P. By the Kuratowski-Nardzewski theorem the multifunction F (-,/(-)) has a measurable selector, so S ( f ) is nonempty.

As F is closed, convex, bounded, the sets S ( f ) are closed, convex and bounded.

Let HL denote the Hausdorff metric in C1BL(P, E).

Let /„, f 0eC (P, E) (ncN), ||/n- / o||c->0 as

-> oo. By Ascoli’s theorem, there exists a compact set A a E such that f n(x, y)eA for n e N u {0} and (x, y)eP.

By the Scorza Dragoni theorem, for any r\ > 0 there exists a closed subset c P such that fi(P\Pv) < tj and F\PriXE is continuous. For any e > 0, let rj > 0 be such that

j j m(x, y)dxdy <

p\p v

As F\Pr>xA is uniformly continuous, there exists ô = <5(e) > 0 such that for (x', y'), (x", y")eP, z', z " e A we have

H(F(x', ÿ , z'), F(x", y", z")) < 8/(3M(P)) if |x '-x " | <

|y '-y " | < S, \z'-z"\ < b .

Let N be such that \\f„—f 0\\c < <5(e) for n > N. By Filippov’s lemma [13]

for gne S ( fn) there exists g0e S ( f0) such that

I

)I =

(x > >0))

< H(F(x, y , f n(x, y)), F(x, y , f 0{x, y)))

< ffi/(3p{P)) for (x , У)еРг, and n > N, 42m (x, y) for (x, y )e P \P

Hence

g

II0».-0

II

<

И

у )dxdy ^ 8 for n

N.

Analogously for g0e S ( f 0) there exists gneS {fn) such that \\g0-g„\\L < 8 for n > N. So HL(S(fn), S (f0)) < e for n > N m , this shows that S is continuous.

Let

M0 = sup{|u(x)|: x

<0, a>} + sup{|i>(y)|: ye<0, b)}+ \z0\ + $ $m(x, y)dxdy,

p

p xy = {(x, y): 0 < x о с ', 0 < у < У}, Q = (Px-y APx y ) u (Px y \P x.y .), œ(ô) = sup | |

(

') —

+ \v(ÿ) — i>(y")|

+ j j m(t, s)dtds: |x' —

< <3, \y'— y"| < <5}.

Q

Let Ж denote the set of all functions / : P-+E with modulus of continuity (о and equibounded by M 0. Ж is a closed convex subset of C(P, E).

By the identity of Aumann and Bochner integrals (see [13], p. 13) and Arzelà’s theorem I(S{f )) is a compact subset of C(P, E) for any / eC{P, E).

Since I is continuous, T — I o S : C{P, E)-*conv Ж is continuous.

Let a be Kuratowski’s measure of noncompactness in the Banach space E.

For X а Ж we define

Ф(Х)= sup cc({f{x, y): f e X } ) .

(x,y)eP

It is easy to see that Ф satisfies conditions (a)-(<5) from Sadovskii’s theorem.

We prove that Ф satisfies Ф(Т{Х)) < Ф(Х) for X с Ж and Ф{Х) > 0. For X с Ж we have

ct{X[Pxy]) = sup [ct(X(t, s)): (t, s)ePxy}.

For e > 0, let Ô = <5(e) be such that

s)dtds < £ for А c Pxy and pt(A) < <5.

A

In virtue of Lusin’s theorem there exists a closed subset c Pxy such that

< & and the function к is continuous on Bl . Now by assumption (d)(iii1) there exists a closed subset B2 c Pxy such that fi(Pxy\ B 2) <

and

a (F(Q xX{Pxyj)) < sup {k(t, s): (t, s)e Q} ■ p(<x(X{Pxy))) for each closed subset Q c= B2.

Let В = n B2, A = Pxy\B. Then fi(A) < 3. As к is uniformly con­

tinuous on B, for e' > 0 there exist rj > 0 such that

\k(t', s') — k(t", s")| < £r if (f, s")eB and < ц, |s '-s " | < rj.

We divide the intervals <0, x ) and <0, y) : 0 < x l < ... < x„ = x,

so that max{|xf

and max

1 ^ j < m) < ц.

Let Qij = « х г_ !, xf> x <jj_ x n В and let (L, s}) be a point in QtJ such that

k(th Sj) = sup {k(t, s): (t , s)eQij}.

For h e T ( X ) there exist / e X and g e S { f) such that h(x, y) = n(x) + u(y) — z0+ j j g(t, s)dtds

P x y

= u(x) + v(y) — z0 + $jg{t, s)dtds + §§g(t, sjdtds.

A В

As

f f

Z

^{) conv}

^x

в i j

If

s)dtds\ ^ J f m(t, s) dt ds < c,

Л A

\{$$g(t, s)dtds: g e S (f), f e X } \ ^ e,

A

T(X)(x, y) c= ф ) + ф ) - г 0 + {((д(г, s)dtds: g e S { f ) , f e X }

A

+ {$Sg(t, s) dt ds: g e S (f), f e X } ,

в

by the properties of the measure of noncompactness and assumption (d)(iii1), we obtain

ct(T{X){x, y)) ^ a({j|6f(t, s)dtds: g e S ( f ) , f e X } )

+ <*({$$g(t, s)dtds: g e S {f), f eX})

в

« 2г + Х 1 л Ш - « И е Ух AT(F„))) i j

< 2e + £ Z ^ (ô o W i> Sj)'p(ot(X{Pxy))) i j

^

g(P)+ J j k(t, s)dtds).

P x y

As e' > 0 is arbitrary, we have

« № (

, у)) « 2г + К-р(а(А-(Р„))).

By Ambrosetti’s lemma and as e > 0 is arbitrary, we obtain Ф(Т(Х)) = sup {а (T(X){x,

(x, y)cP}

К р(Ф(Х )) < Ф{Х)

for Ф(Х) > 0.

By Theorem 3, T has a fixed point and this ends the proof.

References

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[2] J. P. A u b in and A. C e llin a , Differential Inclusions, Springer, Berlin 1984.

[3] C. C a s t a in g and M. V a la d ie r , Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer, Berlin 1977.

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[8] —, F. S. V an V le c k , Lipschitzian generalized differential equations, Rend. Sem Mat. Univ.

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[11] L. S c h w a r tz , Analyse Mathématique, vol. 1, Hermann, Paris 1967 (in French) or PWN, Warszawa 1979 (in Polish).

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4 — Commentationes Math. 30.1