Average value problems in ordinary differential equations

Using Schauderʹs fixed point theorem, with the help of an integral representation in ‘Sharp conditions for weighted 1-dimensional Poincaré inequalities’, Indiana Univ. Math. J., 49 (2000) 143–175, by Chua and Wheeden, we obtain existence and uniqueness theorems and ‘continuous dependence of average condition’ for average value problem:y′=F(x,y), under the usual conditions for initial value problem. We also extend our existence and uniqueness theorems in the case where v is just a signed measure with v[a,b]≠0 and