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$\,\!\!\!$\SCField{BD:}  745

$\,\!\!\!$\SCField{SG:}  34

$\,\!\!\!$\SCField{RF:}  039

$\,\!\!\!$\SCField{AU:}  Aubin, Jean-Pierre; Da Prato, Giuseppe

$\,\!\!\!$\SCField{TI:}  Contingent solutions to the center manifold equation.

$\,\!\!\!$\SCField{SO:}  Ann. Inst. Henri Poincar{\'e}, Anal. Non Lin{\'e}aire 9, No. 1, 
13-28 (1992).

$\,\!\!\!$\SCField{LA:}  English

$\,\!\!\!$\SCField{PY:}  1992

$\,\!\!\!$\SCField{CC:}  34C30 35R70 35A35

$\,\!\!\!$\SCField{UT:}  ordinary differential equations; center manifold; partial differential 
inclusions; contingent derivative; global bounded and Lipschitzian 
contingent solution; viscosity method; ordinary differential equations; 
center manifold; partial differential inclusions; contingent derivative; 
global bounded and Lipschitzian contingent solution; viscosity methode.

$\,\!\!\!$\SCField{AB:}  For a system of ordinary differential equations 
$$x'(t)=f(x(t),y(t)),{\kern8pt}{\kern8pt}y'(t)=-\lambda y(t)+g(x(t),y(t)), 
$$


 where  $X,Y $  are finite dimensional vector spaces,  $f:X\times Y\rightarrow 
X $  and  $g:X\times Y\rightarrow Y $  are Lipschitzian maps and  $\lambda 
>0 $ , the authors characterize a center manifold  $r:X\rightarrow 
Y $  as a solution of the quasi-linear first order system of ``contingent'' 
partial differential inclusions (1)  $\lambda r(x)\in g(x,r(x))-Dr(x)(f(x,r(x))) 
$  (where  $D $  denotes the contingent derivative). They show that 
there exists a global bounded and Lipschitzian contingent solution 
to (1) if  $\parallel g(x,y)\parallel \leq c(1+\parallel y\parallel 
) $  is satisfied and  $\lambda  $  is large enough. Furthermore they 
prove that this solution can be approximated in the spirit of the ``viscosity 
method'' by a solution  $r_{\varepsilon } $  to the second-order system 
 $\lambda r(x)=\varepsilon \Delta r(x)-r'(x)f(x,r(x))+g(x,r(x)) $  
when  $\varepsilon \rightarrow 0 $.

$\,\!\!\!$\SCField{RV:}  W. M{\"u}ller (Berlin) %% $j. $ 
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