where J = [O, m). Let X be a Banach space. The operator A(t), t E J, is assumed to be linear with domain and range in X, F is defined on J x X with values in X, and U is a bounded linear operator on a Banach function space on X. The vector b E X is fixed. The author studied in [I] problems of the form ((E), (B)) with X = R”. The method employed there involved the application ofthe Leray-Schauder theorem to a single integral equation arising from (E) and (B). Of course, this can only be achieved in the non-resonance case, i.e., the case in which the problem F s 0, I = 0 has only the zero solution. Our purpose here is to obtain existence results for problems (E) and (B) by the method of [l]. The reader is referred also to the paper of Ward [2], where problems of the form (E) and (B) are handled on finite intervals. Ward used the Schauder-Tychonov theorem on a suitable Banach function space. In the application of the Leray-Schauder theorem, we introduce a parameter ~(0 < p < 1) as follows: