The Vallée-Poussin theorem for a class of functional-differential equations (Q1390473)

The two-point boundary value problem \[ x^{(n)}(t)+\int_a^bx(s) d_sR(t,s)=f(t),\quad t\in [a,b], \] with \(x^{(i)}(a)=0\), \(i=0,\dots,n-k-1\); \(x^{(j)}(b)=0\), \(j=0,\dots,k-1\); \(k\in\{1,2,\dots,n-1\}\), is considered, where \(R: [a,b]\times [a,b] \to \mathbb{R}^1\) is a merasurable function, the complete variation \(\bigvee _{s=a}^bR(t,s)\) and \(f(t)\) being summable on \([a,b]\). Results concerning the existence of a unique solution are proven. Moreover, its estimates are given and properties of sign of Green functions are discussed.