Integro-differential systems with a degenerate matrix multiplying the derivative (Q1874951)

This paper is concerned with the existence of solutions to the system of integro-differential equations with a degenerate matrix multiplying the derivative \[ A(t)x'(t)+B(t)x(t)+\int_{0}^{t}K(t,\tau,x(\tau)) d\tau=f(t), \quad t\in [0,1], \] \[ x(0)=a, \] where \(A(t)\) and \(B(t)\) are given \(n\times n\) matrices, \(K:\mathbb{R}^{n+2}\to \mathbb{R}^{n}\) and \(f(t)\) is a given \(n\)-dimensional vector function. The degeneracy is understood as the relation \(\operatorname {det} A(t)\equiv 0.\) A numerical solution method based on Euler's implicit method and a quadrature formula using left rectangles are suggested.