## A note on the application of topological transversality to nonlinear differential equations in Hilbert spaces.(English)Zbl 0684.34063

The existence of a solution of the initial value problem $(1)\quad y'=f(t,y),\quad t\in [0,T],\quad y(0)=0,$ where y takes values in a real Hilbert space H and $$f: [0,T]\to H$$ is continuous, is studied. A brief and interesting proof of the existence of a solution of (1) is given using the definition of essential map and the topological transversality theorem. Under the assumption $$\| f(t,y)\|\leq \psi(\| y\|)$$ with $$\psi: [0,+\infty)\to (0,+\infty)$$ and other regularity properties of f the existence of at least one solution in $$C^ 1([0,T],H)$$ is proved. Moreover under the assumptions that ensure the existence of the solution of (1), the solution y exists in $$C^ 1([0,T],H)$$ where T is given in terms of $$\psi$$, i.e. $$T<\int^{+\infty}_{0}du/\psi (u).$$
Reviewer: S.Totaro

### MSC:

 34G20 Nonlinear differential equations in abstract spaces

### Keywords:

Hilbert space; essential map; topological transversality
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