On the application of some measures of noncompactness to existence theorems for an \(m\)th order differential equation (Q2777554)

Let \(I= [0,a]\) and let \(B= \{x\in E:\|x\|\leq b\}\) be the ball in a Banach space \(E\). Here, an existence result is given for the Cauchy problem NEWLINE\[NEWLINEx^{(m)}= f(t, x),\quad x(0)= 0,\quad x^{(i)}(0)= x_i,\quad i= 1,2,\dots, m-1,NEWLINE\]NEWLINE where \(f: I\times B\to E\) is a bounded uniformly continuous function and \(x_i\in E\) for \(i= 1,2,\dots, m-1\). Moreover, it is assumed that \(f\) satisfies the condition of the form \(\mu(f(t,X))\leq u^{(m)}(t)\mu(X)/u(t)\), where \(\mu\) is a measure of noncompactness defined axiomatically, and \(u= u(t)\) is a function satisfying some extra condition.