A new product integral representation for differential equations in separable Banach spaces (Q1378389)

Let \(X\) be a separable Banach space and let \(f: [0,\infty)\times X\to X\) be a function satisfying the conditions: (i) for any \(T>0\) the function \(f(t,x)\) is uniformly continuous on \([0,T]\) for every \(x\in X\), (ii) \(x\to f(t,x)\) is locally Lipschitzian uniformly in \(t\) on bounded intervals, (iii) \(f\) is bounded on \([0,T]\times X\) for any \(T>0\). Consider the initial value problem (1) \(x'= f(t,x)\), \(x(0)= x_0\). Assume that \(t_{\max}>0\) is such a point that the mild solution \(x(t)\) of (1) exists on \([0,t_{\max})\). Under the above assumptions it is proved that for any \(T\), \(0\leq T<t_{\max}\) the following product formula holds: \[ x(T)= \lim _{n\to\infty} \prod^n_{i= 0} \Biggl[I+{T\over n} f\Biggl(i{T\over n}\Biggr)\Biggr] x_0. \]