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(L) L [ y ] s y + ^ = /, (F) F[y] = c where y e <^([a, b], E\ fe<£{[a, b], £), A e V([a, b\ L(£)), F e L[<#([a9 b\ E), E] and ceE. When the system has one and only one solution, for any fe^([a9b],E) and ceE, we show that it has a Green function, that is, a function G:[a,b] x [a,b]-+L(E,E") such that ye<#([a,blE) is the solution of L[y] = ƒ and F[y] = 0 if and only if y(t) = ƒ* G(t, s)f(s) ds. We exhibit the relations between G, A and F. (F) is called a lateral condition ; initial conditions and boundary conditions are particular instances of lateral conditions. The construction of G uses a Riemann-Stieltjes integral representation for F, given in §1.

The green function of a linear differential equation with lateral condition