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Here we put Xi(t) ~f x(t , Yi(t)) (i = > t O (with Xl(t O+ 0) = X2(t O+ 0) = x(t O , ν。) = xO). Next we solve the equation x = x(t , y) with respect to y for x ε (Xl(t) , X2(t)) , t > t O (t - t O small). Then we get three solutions y = 9i(t , X) (i = 1, 2, 3) with gl(t , X) < def g2(t , X) < g3(t , X). Define Ui(t , X)~' V(t , gi(t , X)) (i 二 1 , 2 , 3). This means that 1, 2). Then Xl(t) < X2(t) for t the "solution" constructed by the characteristic method takes three values for x (Xl(t) , 句 (t)) ε (t > t O with t - t Osmall).

Proof. i) As (θxjθν) (t , ν。) d I θz\ 一(一川 > 0 for t < t O and (θxjθν )(tO , yO) = 0 , we have Oa , 二二土 (t勺。 , V O ) τ (tO , dt 飞 θν) l(t , y)=(tO , yO)θx uy yO).

Next we choose an open V of L so that the Jacobian (Dx/Dy)(t , y) does not vanish on V < t O} , and ~f that the equation x = x(t , y) can be uniquely solved with resp田t to y for any (t , x) in U 乞f{(t, z):Z =z(tJ) , (tJ)ε V} (theorem of inverse functions). We write the solution by y = y(t , x). Here we define u(t , x) ~f 叫 t , y(t , x)). 14) in the domain U. 2. 1臼5 时叫) line町 ordinary θ'pdθ的 i, j with respe 时ct = 1, 2,... , 叫 , (j we get a system of = 1, 2,... 10). As this system ofequations is linear , we 用…用 mu 用…用 bh 阶句部 Udy -n =n t 二:: for any o.

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