Retract principle for neutral functional differential equations

Existence of strictly decreasing positive solutions of linear differential equations of neutral type

“…For a nice overview of topological principle, we also refer to [36]. In [18] the retract principle was modified for neutral functional differential equations. This modification should make it possible to use this in the present paper.…”

“…This modification should make it possible to use this in the present paper. Even if [18] contains an illustrative example, showing how this modification works, there is one serious problem restricting the classes of equations suitable for considering by it. Below, we explain the heart of the matter.…”

“…The rest of the paper is structured as follows. In Part 2 we give a generalization of the retract principle to neutral functional differential equations, previously developed in [18]. The main result (a criterion for the existence of a positive strictly decreasing and continuously differentiable solution of neutral differential equation (1)) is given in Part 3 where a special construction of a system of initial functions satisfying the sewing condition is also developed.…”

“…This part provides necessary background. It is mainly taken from papers [18] and [34]. Note that the underlying ideas are based, in addition to the paper of the founder T. Ważewski [38], on the so-called Razumikhin condition in the theory of stability, e.g., [30,31,32], and on Razumikhin's type of extension of Ważewski's principle by K.P.…”

“…If a set A ⊂ R × R n is given, then int A, A and ∂A denote, as usual, the interior, the closure, and the boundary of A, respectively. [18,34]) Let Λ be a topological space, let a subset Ω ⊂ R × Λ be open in R × Λ, and let x be a mapping associating with every…”