Average value problems in ordinary differential equations

Using Schauder's 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),y′=F(x,y),∫aby(x)dv=y0where v is any probability measure on [a,b] 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]≠0v[a,b]≠0 andF:F⊂C[a,b]→L1[a,b] is a continuous operator.F:F⊂C[a,b]→L1[a,b] is a continuous operator.