Let \(E\) be a real B-space and \(\mu\) a ball measure of noncompactness on \(E\), \(T>0\). Let \(A=\{\omega: [0,T]\times [0,\infty[\to [0,\infty[\) be continuous functions, \(\omega(t,.)\) nondecreasing, \(\omega(t,0)=0\) and such that 0 is the only continuous function on \([0,T]\) satisfying \(u(t)\leq \int^{t}_{0}\omega(s,u(s))ds\), \(u(0)=0\}\), \(C=\{\omega: ]0,T]\times [0,\infty[\to [0,\infty[\) such that for any \(\epsilon>0\) there exist \(\delta >0\) and \(t_ n\to 0\), \(t_ n>0\) and \(\rho_ n: [t_ n,T]\to [0,\infty [\) differentiable and satisfying \(\rho'_ n(t)\geq \omega(t,\rho_ n(t))\), \(\rho_ n(t)\geq \delta t_ n\), \(0<\rho_ n(t)\leq \epsilon\) for \(t\in [t_ n,T]\), \(n\in {\mathbb{N}}\}.\)
Theorem 1: Let \(\omega_ 1: [0,T]\times [0,\infty [\to [0,\infty [\) be a continuous function, \(\omega_ 1(t,.)\) nondecreasing, \(\omega_ 1(t,0)=0\) and there exist \(\omega\in C\) such that \(\omega_ 1(t,u)\leq \omega (t,u)\), \((t,u)\in]0,T]\times [0,\infty [\). Then \(\omega_ 1\in A.\)
Theorem 2: If \({\bar \omega}(t,u)=\sup \{\mu(f(t,X)): \mu(X)=u,\emptyset \neq X\subset \overline{B(x_ 0,r)}\}\), where f: [0,T]\(\times \overline{B(x_ 0,r)}\to E\) is a uniformly continuous function such that \(\mu(f(t,X))\leq \omega (t,\mu(X))\) for \(t\in [0,T]\) and for any X bounded in E, \(X\neq \emptyset\) and \(\omega\) is a given Kamke function. Then \({\bar \omega}\in A\). Theorem 3: Let f be a function bounded by \(A>0\) and as in theorem 2 and \(\omega\) is a Kamke function of C. If AT\(\leq r\) then there exists at least one solution of \(x'(t)=f(t,x(t))\) for \(t\in [0,T]\), \(x(0)=x_ 0\).