Consider the homogeneous equation $$ u'(t)=\ell (u)(t)\qquad \mbox {for a.e. } t\in [a,b] $$ where $\ell \colon C([a,b];\Bbb R)\to L([a,b];\Bbb R)$ is a linear bounded operator. The efficient conditions guaranteeing that the solution set to the equation considered is one-dimensional, generated by a positive monotone function, are established. The results obtained are applied to get new efficient conditions sufficient for the solvability of a class of boundary value problems for first order linear functional differential equations.
We consider an elliptic pseudodifferential equation in a multi-dimensional cone, and using the wave factorization concept for an elliptic symbol we describe a general solution of such equation in Sobolev-Slobodetskii spaces. This general solution depends on some arbitrary functions, their quantity being determined by an index of the wave factorization. For identifying these arbitrary functions one needs some additional conditions, for example, boundary conditions. Simple boundary value problems, related to Dirichlet and Neumann boundary conditions, are considered. A certain integral representation for this case is given.
In this paper, sufficient conditions have been obtained for oscillation of solutions of a class of $n$th order linear neutral delay-differential equations. Some of these results have been used to study oscillatory behaviour of solutions of a class of boundary value problems for neutral hyperbolic partial differential equations.