Analytical and numerical investigation of mixed-type functional differential equations

This paper is concerned with the approximate solution of a linear non-autonomous functional differential equation, with both advanced and delayed arguments. We search for a solution x ( t ) , defined for t ∈ [ − 1 , k ] , ( k ∈ N ), that satisfies this equation almost everywhere on [ 0 , k − 1 ] and assumes specified values on the intervals [ − 1 , 0 ] and ( k − 1 , k ] . We provide a discussion of existence and uniqueness theory for the problems under consideration and describe numerical algorithms for their solution, giving an analysis of their convergence.