This paper is concerned with the following Cauchy problem \[ \begin{cases} x^{\prime}=f(t, x), \\ x(0)=0, \end{cases} \] where \(f: I \times B \rightarrow E\) is a bounded continuous function. First, some sufficient conditions for the convergence of the sequence of successive approximations to the unique solution of first order Cauchy problem in a Banach space are discussed.
This paper provides an analogue of Constantin’s criterion for the considered problem when the function \(f\) taking values in the Banach space \(E\). The crucial ingredient in the proof of this aritcle’s main result is the Kuratowski measure of noncompactness. Authors also use the ball measure of noncompactness.
Main results:
Let \(u: I \rightarrow[0, \infty)\) be an absolutely continuous function with \(u(0)=0\) and \(u^{\prime}(t)>0\) a.e. on \(I\). Assume that \(f: I \times B \rightarrow E\) is a bounded continuous function and:
(H1) \(\|f(t, x)-f(t, y)\| \leq \frac{u^{\prime}(t)}{u(t)} \omega(\|x-y\|), \quad \text{for a.e.} t \in(0, a], x, y \in B\), where \(\omega \in \mathcal{F}_{2 b}\),
(H2) \(\lim _{t \rightarrow 0^{+}} \frac{\|f(t, x)\|}{u^{\prime}(t)}=0\) uniformly in \(x\) with \(\|x\| \leq b\). Then the considered problem has a unique solution \(x^{*}\) defined on J. Moreover, the successive approximations \(x_{n}\), given by \[ x_{0}=0, \quad x_{n}=F\left(x_{n-1}\right) \quad \text{ for } n \in \mathbb{N}, \] converge uniformly on \(J\) to \(x^{*}\).
One example illustrates the obtained theoretical results.