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

Abstract

ABSTRACT. General conditions for the unique solvability of a non-linear nonlocal boundary-value problem for systems of non-linear functional differential equations are obtained. 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.

Measure Functional Differential Equations in the Space of Functions of Bounded Variation

“…The statements formulated above express fairly general properties of (1) and extend, in particular, the corresponding results of [25,27,29,31].…”

“…The general character of the object represented by (1) suggests a natural idea to examine its solvability by comparing it to simpler linear equations with suitable properties. Here, we show that such statements can indeed be obtained rather easily by analogy to [24][25][26]. The key assumption is that certain linear operators associated with the nonlinear operator appearing in (1) possess the following property.…”

On the Unique Solvability of a Nonlinear Nonlocal Boundary-Value Problem for Systems of Second-Order Functional Differential Equations

“…Proof of Theorem 3.2. This statement is proved similarly to [5] (Theorem 2). It is obvious, that for arbitrary functions u and v from W 2 with property (3.1), condition (3.5) is equivalent to the relation 3.12), one can check that the operators l i : W 2 → L 1 , i = 1, 2, defined by the formulae…”

On a Class of Functional Differential Equations with Symmetries

“…We need auxiliary propositions based on the results of [8,12,13]. Consider the two-point boundary value problem…”

“…wheref is defined according to (17) andĪ 0 is as in (11) and (12). A solution of Equation (55) is defined as an absolutely continuous function satisfying (55) almost everywhere onĪ 0 .…”