On Singular Solutions of Linear Functional Differential Equations with Negative Coefficients

Abstract

The problem on solutions with specified growth for linear functional differential equations with negative coefficients is treated by using two-sided monotone iterations. New theorems on the existence and localisation of such solutions are established.

Slowly growing solutions of singular linear functional differential systems

“…The proof of Lemma 7.6 for n = 1 can be found in 18. The considerations allowing one to pass to higher dimensions are quite straightforward, and we omit them.…”

“…Proof. It follows from 18, Lemma 5.8] that K is a regular cone in X . In particular, K is normal (see, e.g., 13, Theorem 1.6]).…”

“…Proof. It is proved in 18, Lemma 5.10] that, under assumption (4.1), T ( X )⊆ X . On the other hand, it is obvious from (7.27) that (Tu)(b) = 0 for any u from X .…”

On the solvability of linear singular functional differential equations

“…A locally absolutely continuous on (0, 1] function 𝑥 : (0, 1] → R is called a solution of (1.1) if it satisfies this equation almost everywhere on (0, 1]. Since (1.1) is not singular in each interval [𝜀, 1] (𝜀 ∈ (0, 1)), then any its solution has a representation The Cauchy problem for singular equations and problems with weighted initial equations are considered, in particular, in [20,21,22,23,44,45,46,2,40,41,42]. In [20,21,22,23,44,45,46] for nonlinear singular functional differential equations, the conditions for the solvability of the Cauchy problem and problems with weighted initial conditions were obtained (including the many-dimensional case).…”

“…We can interpret the works [40,41,42] as a research of singular equations in the space of solutions of the "model" singular equation ẋ(𝑡) = 𝑝(𝑡)𝑓 (𝑡), 𝑡 ∈ [0, 1], (1.3) for all 𝑓 ∈ L, where 𝑝 : (0, 1] → (0, +∞) is a non-increasing continuous function such that lim 𝑡→0+ 𝑝(𝑡) = +∞. Any function 𝑥 : (0, 1] → R such that 𝑥 ∈ AC[𝜀, 1] for each 𝜀(0, 1) is called a solution of (1.3) if it satisfies the equation everywhere on [0, 1].…”