We study a nonlinear Volterra problem in a Banach space X: \[ (*)\quad u'(t)=f(t,u(t))+\int^{t}_{0}g(t,s,u(s))ds\quad (t\geq 0),\quad u(0)=u_ 0, \] with \(f: {\mathbb{R}}_+\times D\to X\), \(g: \Delta\times D\to X\), where \(\Delta\) is the set \(\{(t,s)\in {\mathbb{R}}^ 2\); 0\(\leq s\leq t\}\), \(D\hookrightarrow X\) is a Banach space and \(A=f_ u(0,u_ 0)\) is a generator of an analytic semigroup in \(X\) with domain \(D\). A local strict solution of (*) is obtained by a perturbation method and sharp estimates are given for the solutions of the linear evolution equation \(u'(t)=Au(t)+h(t),\) \(t\geq 0\), \(u(0)=u_ o\). We give applications to the study of local existence, uniqueness and regularity of the solutions of the Cauchy problem for a class of fully nonlinear partial integrodifferential equations of parabolic type.