## 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)

### MSC:

 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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### References:

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