ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXV П985) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATEMATYCZNE XXV (1985)
Wl a d y s l a w So s u l s k i (Zielona Gôra)
Existence theorem for generalized
functional-differential equations of hyperbolic type
1. Introduction. Let (A", d) be a metric space and denote by Cl (A) the space of all non-empty closed and convex subsets of A. Define, for fixed А c= X and e > 0,
J e(A) = { x s X : d(x, a) ^ e for some a e A } and let
h(A, B) = inf (e > 0: A c : J e(B) and В c J E(A)}, if the infimum exists,
oo, otherwise,
for A, B e Cl (A). It can be verified that h is a generalized metric for Cl (A).
Hence it follows, in particular, that if A is bounded, then h is a metric on Cl (A). In this case, h is called the Hausdorff metric. In a general case h will be called the generalized Hausdorff metric. It is not difficult to observe that Hausdorff metric can be defined by
h{A, B) = max (sup Dist(x, A), sup Dist(x, B))
x e B x e A
for A, B eC onv(A ), where Conv(A) denotes family of all non-empty com
pact and convex subsets of A and Dist(x, A) is a distance between a point x e X and a set zleC onv(A ), i.e.,
D ist(x, A) = inf {d(x, a): a s A}.
Let a > 0 and b > 0 be given, P = [0, a] x [0, b], and let us denote by С (P) and L2(P) Banach and Hilbert spaces respectively of all continuous or square Lebesgue integrable functions, of P into R n with usual norms || j| and || ||p.
Furthermore, we use the space CX(P) of all functions и : P -> R n such that u( ,y): [0, a~\-+Rn is measurable for fixed y e [0 , b], m(x, •): [0, b]
-♦ R” is continuous for fixed x e [0, a] and such that JuL = f max |u(x, y)\dx < oo.
О Уе[0.Ь]
150 M. Sos ul s ki
We will consider CX(P) together with the norm defined above. Similarly we can define the space Cy{P) with the norm |- |y. It was proved by Deimling ([1]) that (СХ(Р), И*) and (Cy(P), |- |y) are Banach spaces.
Let F: P x C(P) x CX{P) x Cy(P) -* Conv(P") be a multivalued mapping satisfying the following Carathéodory type conditions:
(i) F( , \ z, u, v) is measurable for fixed (z, u, v)eC(P) x C x(P) x C y(P), (ii) F{x, y, is continuous for fixed (x, y)eP,
(iii) there exists a square Lebesgue integrable function m: P -» R such that h ( F ( x , y , z , u, v), (0 } )^ m (x , y) for (z, u, v)eC(P) x C x(P) x Cy{P) and almost all (x, y) e P.
Furthermore, it will be assumed that F(x, y, z, -, •) is Lipschitz con
tinuous, uniformly with respect to zeC (P ), i.e.,
(iv) there exists a square Lebesgue integrable function к : P -*■ R such that
h(F(x, y, z, u, v), F(x, y, z, U, v)) ^ k(x, y)\u — û\x + \v — v\y
for z e C (P ); u , ü e C x (P); v, v e C y(P) and almost all (x , y ) e P .
We will consider now a generalized functional-differential equation of the form
(1) zxy(x, y ) e F ( x , y, z, zx, zy) for almost all (x, y ) e P
with the initial Darboux conditions, (2) z(x, 0) = <t(x), z(0, y) = r(y), where cr: [0, u ]-> P ",
t: [0, b~] -+ R n are absolutely continuous functions such that cr'eLfO, a),
t'gL(0, b).
We will say that F : P x C(P) x CX{P) x Cy(P) -* ConvfP") has the Volterra’s property if P (x, y, z, u, v) = P (x, y, z, m, v) for every (x, y ) e P and (z, и, V), (z, ü, v ) e C ( P ) x CX(P) x Cy(P) such that z|[0iJe] x[0,y] = z\[0,X] xto,y],
wl[0,x]x[0)y] = «1[0,*]x[0,yj, ^Ito.jc]x[0,y] = «l[0.x]x[0,y], where for given w: P -> P "
and (x, y) eP , w|[0 х] [0 у] denotes the restriction of w to the rectangle [0, x] x x [0, у] с P.
In the-sequel we will need the following Lemma ([3]).
Le m m a 1. Suppose A is a non-empty, closed, bounded and convex subset of a Hilbert space (У, || -||) and let Г be an operator with domain A and range in a Banach space (X , | -|). Suppose further that G: А х Г ( А ) ->С1(Л) is such that
1° G(-, y) is a contraction, uniformly with respect to у е Г ( А ) , 2° G(x, •) is continuous on Г (A) in the relative topology.
I f Г is completely continuous, then there exists x e A such that
x eG(x, P (x)).
Functional-differential equations o f hyperbolic type 151
2. Existence theorem. Now we can prove the main result of this paper.
T heorem 1. Suppose F : P x C ( P ) x C x (P) x Cy (P) -> Conv (R n) satisfies the Carathéodory conditions (i)-(iii) and the Lipschitz condition (iv) with Lipschitz function к such that 2y/\P\-\\k\\p < 1. Let o: [0, a] -> R n, i: [0, b]
-» R n be absolutely continuous functions such that a' and t' are Lebesgue integrable on [0, a] and [0, b], respectively. Then (1)— (2) has at least one solution.
P ro o f. Let A be a subset of L2(P) containing all function w e L 2(P) such that |w(x, y)| ^ m(x, y) for almost all (x, y)eP. It is not difficult to see that A is a closed, convex and bounded subset of L2{P). Define on A an operator Г of the form
дс у
r{w){x, y) = a (x) + z (>>) - о (0) + j j w(s, t)dsdt.
0 0
It is easy to see that Г is a completely continuous mapping of A into C(P).
Let У X
J 1 M (x , y) = (T'(x)+ J w(x, t)dt, J 2(w)(x, y) = J’ W(s, y)ds,
о 0
and consider now equation
(3) w(x, y ) e F ( x , y, r(w), J 1{w), J 2(w)) for almost all (x, y ) e P . It is clear that for each w eA satisfying (3), the function z = r ( w ) satisfies (1И2).
Let G(w, v) denote, for fixed (w, v)eA x C(P), a subset of L2{P) contain
ing all measurable selectors of F (-, -, v, J i(w), J 2(w)). If is easy verifed ([2], [4] ) that G(w,v)ï£ 0 for each (w, v ) e A x C ( P ) . Furthermore, it is not difficult to see that G(w, v) is a closed and convex subset of A.
Then G: A x C { P ) ->C1(A). Now, we will show that G(-, v) is a contrac
tion, uniform with respect to veC( P) . Indeed, let v eC( P) be fixed and let wb w2e A . For each yeG (w 1, r), we have y(x, y )e F (x , y, v, J y i w ^ , J 2(w1)) for almost all (x, y) eP.
Since
h(F(x, y, v, Ji(w j), J 2(w1)), F(x, y, v, J 2(w2)))
^ к { X , y) (I J1 (w j - J ! (w2)|x -(-1J 2 (Wj) - J 2 (w2)|y)
a y у
= к (x, y)(J max |<7,(x)+ J wt {x, t ) d t - o ’( x ) - j w2(x, t)dt\ +
0 ye[0,fe] о 0
b x x
+ J max |t'0 0 + J Wi (s, y ) d s - F { y ) - j w2(s, y)ds\)
О хе[0,а] о 0
a b
^ 2k{x, y) J J Iw ^x, y) — w2(x, y)\dxdy ^ 2k(x, y) * ' Hw! ~w2IIj*
152 M. S os u l s k i
for almost every (x, y)eP, then there exists AeG( w2, v) such that |y(x, y) —
— A(x, y)| ^ 2k(x, y)'y/\P\ *||>^1 — w2||p for almost all (x, y)eP. Thus ||y — - 2 ||p « 2 4/iP|-||/c||p-||vv1- w 2||p. Hence it follows H(G(wly v), G(w2, v))
« г у й - р н p'Hwj — w2||p, where H denotes the Hausdorff metric in С1(Л) generated by the norm ||-||p. In a similar way, we can verify that G(w, •) is continuous for fixed a s A. Therefore, in virtue of Lemma 1, there exists we A such that w e G (w, Г (w)) which means that w(x, y ) e F ( x , y, F(w), J x (w), J 2(w)) for almost all (x, y) eP . Hence it follows that z = T(w) satisfies (l)-(2) and the proof is complete.
R e m a rk . If F : P x C{P) x CX(P) x Cy(P) ->Conv(R") has the Volterra’s property, then in Theorem 1, the condition 2||/с||рч/|Р | < 1 can be omitted.
References
[1] K. D e im lin g , A Carathéodory theory for systems o f integral equations, Ann. di Math.
(1969).
[2] C. J. H im m e lb e r g , F. S. V an V leck , Lipschitzian generalized differential equations, Rend. Sem. Math. Univ. Padove 48 (1973).
[3] M. K is ie le w ic z , Generalized functional-differential equations o f neutral type, Ann. Polon.
Math. 42 (to appear).
[4 ] K. K u r a t o w s k i, C. R y ll- N a r d z e w s k i, A general theorem on selectors, Bull. Acad.
Polon. Sci. 13 (1965).
