General conditions for the unique solvability of initial-value problem for nonlinear functional differential equations

Abstract

We establish general conditions for the unique solvability of the Cauchy problem for systems of nonlinear functional differential equations. Statement of the ProblemWe consider the initial-value problemwhere -R , k = 1, 2, … , n, are continuous (generally speaking, nonlinear) operators, and c k nThe aim of the present paper is to establish general conditions for the unique solvability of problem (1), (2) under the assumption that the nonlinearities in the system of equations (1) can be estimated by using certain linear operators that generate uniquely solvable initial-value problems with positive Green operators. Generally speaking, the problem of finding these operators is not simple, but, as shown below, once they are found, the results of the linear theory can be used for the investigation of the solvability of the nonlinear problem (1), (2). Main DefinitionsWe understand a solution of the initial-value problem (1), (2) in the sense of the following standard definition (see, e.g., [1]): Definition 1. An absolutely continuous vector function u = u k k n ( ) =1 : a b , [ ] → R n that satisfies equality (1) almost everywhere on a b , [ ] and possesses property (2) at the point a is called a solution of problem (1), (2).For what follows, we need the notion of positivity of a linear operator defined on the space of vector functions with absolutely continuous components.

Unique solvability of a non-linear non-local boundary-value problem for systems of non-linear functional differential equations

“…The main goal of this paper is to establish new general condition sufficient for the unique solvability of the non-local non-linear boundary-value problem (2) for the non-linear functional differential equations (1). For non-linear functional differential systems determined by operators that may be defined on the space of the absolutely continuous functions only, we prove several new theorems close to some results of [2,5,7,14,15,13,12].…”

“…The main theorems established here generalize some recent results from [5,14,3,6,15] and are proved by using an abstract theorem from [9].…”

“…The main goal of this paper is to establish new general condition sufficient for the unique solvability of the non-local non-linear boundary-value problem (2) for the non-linear functional differential equations (1). For non-linear functional differential systems determined by operators that may be defined on the space of the absolutely continuous functions only, we prove several new theorems close to some results of [2,5,7,14,15,13,12].The main theorems established here generalize some recent results from [5,14,3,6,15] and are proved by using an abstract theorem from [9].2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 34K10. K e y w o r d s: non-linear boundary-value problem, functional differential equation, non-local condition, unique solvability, differential inequality.…”

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.…”

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

“…The presence of a certain weight function is thus expected when dealing with the problems of this type. As such a weight function, our definition of a solution of equation (1) uses the same function h with properties (3) and (4) that appears in condition (3). For example, function (5) turns out to be a solution of equation (6) if hðtÞ :¼ t g , t A ð0; 1, with g > 4.…”