Asian Research Journal of Mathematics

The objectives of this paper is to investigate the Cauchy problem of the type ( ℒx)(t) x(t) – p(ṫ )x(t) = f(t). x(a) = α, tϵ[a, b]  and to  establishes  the necessary and sufficient conditions for it solvability if  the  matrix  are sum able. Thus the above equations can be re written as . Where X is a fundamental matrix such that X(a) is the identity matrix,  also  can be represented as the general solution of the equation of the type ℒx=f.  So the studying equations can be written as a boundary value problem of linear functional differential equations of the form ℒx=f, IX= α. The Green Operators was used to established the conditions that guarantee uniquely solvable bounded value problem of the type defined above. The paper also considered the case where the boundary value problems continuous dependence of parameters, to established conditions that guarantee uniquely solvability of the equations of the form ℒ_ox = f, ℒ_ox = α and ℒ_kx = f,ℒ_kx = α  With the establishment of these two arguments, the objectives of this paper was established.