NONLINEAR BOUNDARY VALUE PROBLEMS 13 l/(ft*)lftp(0. Those conditions , sometimes called the Carathiodory conditions, are in particular satisfied if / i s continuou s and takes bounded sets into bounded sets, the latter condition being a con- sequence of th e former when r = 0 , in which case of cours e C is naturally identifie d wit h Rn. I t is easy to verify tha t the (Nemicky) operator F defined o n OJ r , R n ) b y the relation maps C(/r, R n ) int o Ll(I, R n ). Let g\ C x C —• C be a completely continuous mapping and consider the associated map- ping defined o n C(Ir, R n ) b y G: x H » g(x0, x x ). We can then consider the nonlinear boundary value problem x'(t)=f(f9xt), tei, (Lii) MxQ +Nx x =g(x Qt x x ). Defining X, Z an d L a s in §3 , and the mapping N o n X b y Nx = (Fx , G*) , we see at once that (1.11) is equivalent to the abstract equation Lx = Nx an d we want to sho w that under the condition s of § §3 an d 4 for L an d N, N is L-completely continuou s on X. Le t E be a bounded subset of X an d let us first show that Q T N{E) an d Kp Q N(E) ar e relatively compact. I t is not too difficul t fo r the first one if we notice tha t (/ - T) has a finite di- mensional range because Im( M + NT X ) ha s a finite codimension . Now , using (1.10) we ob- tain (KPS,QTNXW = CW0 ) + (r f ((W + NT x )sl(Gx - NS x Fx))){fi) i f 0 t 1 , = ( M + NT x )sl(Gx - NS^xXt) i f -r r 0 . If p 0 is such that F C B[0, p] C .Y, it follows fro m the definition o f S t an d Carath^odory conditions that (S F(E))(0) C X an d S}F(E) C C are relatively compac t (Arzela-Ascoli theorem) , so that, using the complet e continuit y o f G the same is true in X for KP Q N(E). T o show that KpStQTN i s continuous, let (x n ) b e a sequence i n X whic h converges to x\ by th e compactness properties above, we can assume, going if necessary t o a subsequence, that ((S Fxn)(0)) an d (Sj/V) converg e to ^ an d z on the othe r hand, for each s E /, (x£ ) converges in C to x5 s o that, by the Carathdodory conditions , (/(s, *") ) converge s to /(s, x s ) i n /?" an d is dominated by a L1 function using the Lebesgu e convergenc e theo - rem, i t follows that , for each t GI yt s E [-/• , 0] , (StFxn)(Q) - (S t FxX0), (S.Fx^is) (S^X*) , so that,}* = (S Fx)(0), x = SjF x an d the limits are thus independent fro m th e subsequence by a classical argument, this implies that (S Fxn)(0) converge s to (S Fx)(Q) in X an d (S { Fxn) converges t o S x Fx i n C , implying i n tur n that , i n X t (T(M + NT l )slNS1Fxn)(0) con - verges t o (T(M + NT^NS^xXO). A s (r(( M + NTJ^Gx^XO) converge s t o (r((M + NTx)glGxy)(0)9 b y the complete continuit y o f G and the continuity o f the othe r
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