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In this paper we prove existence theorems for integro - differential equations xΔ(t) = f(t, x(t), ∫ 0 t k(t, s, x(s))Δs), x(0) = x0 t ∊ Ia = [0, a] ∩ T, a ∊ R+, where T denotes a time scale (nonempty closed subset of real numbers R), Ia is a time scale interval. Functions f, k are Carathéodory functions with values in a Banach space E and the integral is taken in the sense of Henstock-Kurzweil delta integral, which generalizes the Henstock-Kurzweil integral. Additionally, functions f and k satisfy some boundary conditions and conditions expressed in terms of measures of noncompactness. Moreover, we prove an Ambrosetti type lemma on a time scale. [ABSTRACT FROM AUTHOR]