Contingent solutions to the center manifold equation. (English) Zbl 0745.34039

For a system of ordinary differential equations \[ x'(t)=f(x(t),y(t)),\qquad y'(t)=-\lambda y(t)+g(x(t),y(t)), \] where \(X,Y\) are finite dimensional vector spaces, \(f:X\times Y\to X\) and \(g:X\times Y\to Y\) are Lipschitzian maps and \(\lambda>0\), the authors characterize a center manifold \(r:X\to Y\) 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 \(\| g(x,y)\|\leq c(1+\| y\|)\) 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 \(\varepsilon\to 0\).
Reviewer: W.Müller (Berlin)


34C30 Manifolds of solutions of ODE (MSC2000)
35R70 PDEs with multivalued right-hand sides
35A35 Theoretical approximation in context of PDEs
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