On a boundary value problem for a differential equation with variant retarded argument (Q1294424)

The authors consider the following boundary value problem \[ x'' (t)+a(t) x\bigl(t- \tau(t)\bigr) =f(t),\quad x(t)= \varphi(t),\;\lambda_0 \leq t \leq 0, \quad x(T)= x_T,\tag{1} \] with \(0\leq t\leq T\) and \(a(t)\), \(f(t)\), \(\tau (t)\geq 0\), \(0\leq t\leq T\) and \(\varphi (t)\), \(\lambda_0\leq t\leq 0\), are known continuous functions. The equivalent integral equation they had obtained is a Fredholm-Volterra integral equation. The Fredholm operator included is an operator with a degenerated kernel. For proving the existence and the uniqueness of the problem they apply the ordinary successive method and modified successive method. One of the two main results in the paper is the following theorem: Suppose that \(a=a(t)\) is a continuous function in \([0,T]\) and \(l=T^2 \|a \|/8<1\). Then problem (1) has a unique solution and the approximations converge to the solution to (1) and the speed of convergence is determined by \(\|x_n-x \|\leq l^n \|x_0- x\|\).