Average value problems in ordinary differential equations

Abstract

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),∫a b y(x)dv = y0 where 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]≠0 and. F:F ⊂ C[a,b] → L1[a,b] is a continuous operator. © 2010 Elsevier Inc.