Existence theorem for multivalued hyperbolic equation

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series 1: COMMENTATIONES MATHEMATICAE XXVII (1987) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXVII (1987) •

Ireneusz Kubiaczyk (Poznan)

Existence theorem for multivalued hyperbolic equation

Assume that £ is a Banach space, z0e £ , В = [xeE: \\x — z0\\ ^ r) and F : P x B -> CB(E), where P = {(x, y): x 0 ^ x ^ x 0 + a, y0 ^ y ^ y 0 + b} and CB(E) denotes the metric space of non-empty closed bounded subsets of E with the Hausdorff metric H. We shall also write D(a, B) = Inf {\\a

— b\\: b e B }.

In this paper we prove an existence theorem for multivalued partial differential equations:

(1) d2 u/dxdyeF (x, y, u(x, y))

with the initial conditions w(x, y0) = <x(x), u(x0, у) = т(у).

By a solution of equation (1) on some rectangle P' we mean an absolutely continuous mapping u(x, y) such that и satisfies (1) almost everywhere on P'.

By a. classical solution of equation (1) on some rectangle P' we mean a continuous mapping u(x, y) such that д2и/дхду is continuous and и satisfies (1) for each (x, y)eP '.

In [7] Sosulski proved an existence theorem for the solution of multivalued functional-differential equations of hyperbolic type. In this paper we prove an existence theorem for the clasical solution using an analogue of the classical Euler-Cauchy polygon method (see [3], [2]). For a single valued mapping this problem was proved by A. Alexiewicz and W. Orlicz [1], P. Negrini [5] and the others.

It will be tacitly assumed that the functions cr(x) and т(у) are defined in

<x0, x0 + a> and (Уо, Уо + Ь}, respectively, that they have continuous derivatives of the first order, and that о (x0) = т (y0) = z0.

Let C, D and M be such that ||cr(x) — a(x)|| ^ C |x — x|,

llTM -T(ÿ)ll ^ D \ y ~ y \ , IlF{x, y, z)|| ^ M for (x, y ) e P and z e B . Let a denote the Kuratowski measure of non-compactness. For properties of a see [6], [4].

We say that the multifunction F(x, y, z) is “absolutely continuous” if for any £ > 0 there exists <5 > 0 such that for any points (xf, yf, z ,)eP x B the

inequality

implies

Z (l^i+i-^l+bH-i-^l+ki+i-Zii) <<5

i — 1

Z

Theorem. Let the multifunctions F be “absolutely continuous” and bounded by M and

(2⁾ a (F(P' x V)) ^ fca(F) for every subset V of В .

Then for any /o o e F (x 0, Уо> z o) there exists on the rectangle P' a clasical solution z(x, y) of the multivalued hiperbolic equation (1) such that z ( x 0, y0)

ô2z +

= Zq and -T -r-(x0, y0) = foo f ° r апУ дх су

fo o e F ( x 0, y0, z 0), where P' = {(x, y): x0 ^ x ^ x 0 + h, y 0 ^ У < Уо+ h'}

and h, h' are such that h ^ a, W ^ b, hh'k < 1, hC + h'D + hh'M ^ r.

P ro o f. Denote by Pknl the set {(x, y): xk ^ x ^ x k + l , у* < у ^ У/+₁}, where xk = x 0 + ka/n, yt = y0 + lb/п for k, 1 = 0, 1, ..., n — 1.

Let w„ denote the mapping defined in the following way:

For (x, y )eP °°

f \ I м 0 0 ^ ч I ^{t l n} W o o /- \ , r / \ / \

u„(x, y) = w00 + ---( x - x 0) + --- — (у -У о Н /о о ^ -Х о Н у -У о ),

*i ~ * o У\~Уо

where u00 = z 0, w„1>0 = un(xu y j = a{xx), и°пл = w„(x0, y j = i(yi), /00 £ F (x o> Уо? zo)-

For (x, у ) е Р ;'

w„(x, y) = W*’° + U°’'— W00 + 'uk„ + 1’° - u k’° O.H- 1 Xk+I - X k

n

- ( x - x k) + - l и^0,1 У1+1- У- i y - y ù1

к I I

+ 1 I f i - l , j - l ( x i - X i - 1) ( y j - y j - 1) + Z f k j - A x - X k j i y j - y j - i )

i = l j = l j = 1

fc i = 1

where uk„’° = un(xk, y0) = o(xk), u°’1 = u„(x0, у,) = т(у,) and f k>l are such points that f k>ie F ( x k, yh un(xk, уг)) and

l\fk,i~fk-i,i-i\\ = ^ { f k - u i - u F (xk, y„ u„(xk, y,))),

fk.i- i e F ( x k, y , - l5 w„(xk, У!_ i))

Existence theorem for multivalued hyperbolic equation 117

and

НЛ.г- i Ул—i,ï— ill = D { f k -1,1— l > F (xk> У1 - 1 un(xk> .Vi-i)))>

f k - i , i e F (xk - u УьМ„{хк-1, У,)) and

Wfk- 1,/ fk- 1,1- ill = D{fk - 1,1-1 > F( Xk- !■> Уь Un(Xk- !■> .Vl-l))) for l, к = 1, 2, ...

This construction is possible because F ( x , y , z ) are compact sets F{x, y, z) cz F (P1 xH), where H we construct analogously as in [8],

H = z 0+ (J Ясопv a ' ( ( x 0, x 0 + h})+ (J ц с о тt'{(y 0, y 0 + h '»

o a ^ li O^n^h'

+ У Я^сопу F (P 'x H ) . As o' and x' are continuous, <t'(<(x0, x 0 + /i>) and т'(<у0, y0 + fr'» are compact. By the properties of measure of non-compactnes we have а(Я) ^ hh'kot (H). This implies that the set H is compact.

First, we prove by induction that u„ (x,y )eB for (x, y)eP'. For (x, y)eP °°

IIM„(x,

0II < lk4£i)ll(*-*o) + IH*/i)IIO;-}'o)

+ ll/ooll (x — *o)(У ~ Уо) < hC + h'D + Mhh' < r.

Now, if w„(x, y )e B for (x, y)eP%J for i ^ к — 1, j ^ / — 1, then we can show that un(x, y ) e P kn~ 1,1 и u I* ’1.

For example, for (x, y )e P *,l

■Цм„(jc, y ) - z 0|| ^ ||( x - x 0)conv(T'(<Xo, x0 + /j»||

+ IICv-To)convT, «>'o, j'o + JOMI

+ ||(x —x0)(y —y0)convF(.P' xJ5)|| ^ hC + h'D + hh'M < r.

As

u„(x, y ) e z 0+ IJ /lco n v cr'« x 0, x 0 + h})+ IJ ^ co n v i'(< y 0, Уо + h') .

+ U Ific o n v F (P' xH ) cz H . So un( x , y ) e H for each (x, y)eP'.

Also it is easy to see that

(3) ||м„(х, y)-w „(x, y)|| ^ C \x — x\ + D\y — y\ + M h\x — x\ + Mh'\y — y\.

Now we define the sequence (z„(x, y)) of functions as follows:

zn(x , У) = f k,i for (x, y ) e P k„’1, x < xk+1, у < yI+1.

Let H 0 — c o w F ( P ' хЯ ). As Я is compact the set H 0 is compact and zn{ x , y ) e H 0.

If (x, y ) e P k„’1, (x, y)eF,;j , then if i < k, j <1 we have J\z„(x, y ) - z n(x, y)\\

= НЛ.1-/Л <

< H (F (xk. u y t- u u„(xk_i, y ^ ) ) , F (xk, y b un{xk, y,))) + ...

. . . + H ( F { x h yJt un(xh yj), F (xi + 1, yj+1, un(xi + l , yJ+i))).

As |xk — Xj| < \x — x\ + 2a/n and \yt — y}\ < \y — y| + 2b/n, by (3) and absolute continuity of F we obtain that the set {z„(x, y): n = 1, 2, ...} is equioscillating, that is, for each e > 0 there exists N and <5 > 0 such that for all points (x, y) and (x, y ) e P satisfying |x — x| < <5, \y — y\ < 0 and for all n e N one has \\z„(x, y) — z n(x, ÿ)|| < г.

From a theorem of Ascoli there is a continuous function z defined on P' and a subsequence of the sequence (z„) which converges uniformly to z.

Let

* У

u(x, у) = er(x) + T (y ) - z 0+ J J z(s, t)dsdt

*oxo

Then the corresponding subsequence u„k(x, y) tends uniformly to u(x, y) and

^ ||z(x, y ) - f k,i\\ + H (F (x k, yb U„k(xk, y,)), F (x, у, м(х, у))) — 0 if nk -> со.

This shows that u(x, y) is a solution of our problem (1).

References

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[2] J. B. D ia z , On an analogue o f the Euler-Cauchy polygon method for the numerical solution o f uxy = f ( x , y, u, ux, uy), Archive Rat. Mech. Anal. (1957), 357-390.

[3] A. F. F il ip p o v , Classical solutions o f differential equations with multivalued right-hand side, Vestnik Moskov. Univ. Ser. I Mat. Meh. 22 (1967), no. 3, 16-26. English transi., SIAM J.

Control 5 (1967), 609-621.

[4] R. H. M a r tin , Nonlinear operators and differential equations in Banach spaces, New York 1976.

[5] P. N e g r in i, Sul problema di Darboux negli spazi di Banach, Bull. U. M. I. (5) 17-A (1980), 156-160.

Existence theorem for multivalued hyperbolic equation 119

[6] B. N. S a d o w s k i, Limit compact and condensing operators, Russian Math. Surveys 27 (1972), 86-144.

[7] W. S o s u ls k i, Existence theorem for generalized functional-differential equations of hyperbolic type, Comment. Math. 25 (1985), 149-152.

[8] S. S z u fla , Some remarks on ordinary differential equations in Banach spaces, Bull. Acad.

Polon. Sci., Sér. Math. Astronom. Phys. 16 (1968), 795-800.

INSTYTUT MATEMATYKI UNIWERSYTETU IM. A. MICKIEWICZA POZNAN