A technique for proving existence of solutions (Q678683)

Problems of the form \[ x'(t)= f(t,x(t)),\qquad t\in[0,T], \quad x(0)\neq x_0,\tag{IVP} \] where \(T\in \mathbb{R}^+\), \(x_0\in \mathbb{R}^N\), and \(f:\mathbb{R}\times \mathbb{R}^N\to \mathbb{R}^N\) are considered and the existence of Carathéodory solutions of (IVP) is studied. In Section 2, the author gives requirements on \(f\) under which a sequence (1) considered converges (passing to a subsequence if necessary). Finally, in Section 3 the author presents assumptions under which the sequence (1) can be used to construct a Carathéodory solution of (IVP).