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


34G20 Nonlinear differential equations in abstract spaces
Full Text: DOI