On a Singular Cauchy Problem for Functional Differential Equations with Non-Increasing Non-Linearities

Abstract

Abstract. We obtain general conditions su‰cient for the solvability of a singular Cauchy problem for functional di¤erential equations with non-increasing nonlinearities.

On the solvability of linear singular functional differential equations

“…The results archived in this work is close to [20,21,22,23,44,45,46,40,41,42], however a special choice of "model" equations (1.1) and (5.2) allows to consider more general singularities.…”

“…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].…”

Slowly growing solutions of singular linear functional differential systems

“…Problems similar to (1.1)–(1.3), including the singular Cauchy problem for various classes of functional differential equations, are studied, in particular, in 2,5,6,11,19,21–24,26. A problem on Carathéodory solutions of the functional differential system (2.4) possessing properties of type (1.2) is considered in 1.…”

A singular problem for functional differential equations with decreasing non‐linearities

“…The present note is a continuation of Pylypenko and Rontó 30 and contains an improved version of a theorem stated there. The problem formulation has been inspired, in particular, by a relation to boundary value problems with conditions at infinity and integral boundary conditions on unbounded intervals 31–36 …”