Some Remarks on the Existence of Solutions of Nonlinear Ordinary Differential Equations in Banach Spaces

摘要

The existence of solutions of the Cauchy problem x^′=f(t, x), x(0)=x_0∈E in a Banach space E are studied. We give a sufficient condition for the above equation to have a unique solution, and prove that the successive approximations of the above equation converge weakly to the unique solution