Conditions for the existence and uniqueness of bounded solutions of nonlinear differential equations

Abstract

We establish conditions required for the existence and uniqueness of bounded solutions of the Main Notation, Object of Investigation, and ResultsBy C 0 we denote a Banach space of functions x = x(t) continuous and bounded in R and taking values from R with normBy C 1 we denote a Banach space of functions x C ∈ 0 such that the derivative of each function is an element of the space C 0 with normFurther, by C we denote the set of all continuous functions y : R → R, M is the set of all strictly monotone functions g : R → R such that the set of values R g ( ) of each of these functions coincides with R, and F is the set of all continuous functions g : R → R such that, for each of these functions, R g ( ) = R and lim ( )An operator L : C 1 → C 0 is defined as follows:where x

Method of local linear approximation in the theory of bounded solutions of nonlinear differential equations

“…This inequality is similar to (8) and, by virtue of (20) and (23), coincides with (21). Therefore, by The- …”

“…The following assertion is important for our subsequent investigation: Theorem 8 [21]. The differential operator L f : By using this theorem, we can readily prove the following assertion:…”

“…In [21][22][23], we study the conditions of existence of bounded solutions of this equation and their properties. For Eq.…”

Необходимые и достаточные условия существования и $\varepsilon$-единственности ограниченных решений нелинейного уравнения $x'=f(x)-h(t)$

“…В [1]- [3] и [5] получены необходимые и достаточные условия существования, а также условия существования и единственности ограниченных и существенно ограниченных реше-ний этого уравнения. Необходимые и достаточные условия липшицевой обратимости оператора / − в пространствах 0 и (R, R), 1 ∞, приведены в [4], [6].…”

Invertibility of the nonlinear operator $\left( {{\mathcal{L}}x} \right)\left( t \right) = H\left( {x\left( t \right),\;\frac{{dx\left( t \right)}} {{dt}}} \right)$ in the space of functions bounded on the axis

“…In the spaces C 0 and C 1 , we consider the balls S i r = {x : x C i ≤ r}, i = 0, 1, of radius r. The following statement on a bounded sequence of elements of the space C 1 is important: Lemma 1 [12]. For every sequence of functions x n ∈ S 0 r ∩ S 1 R , n ∈ N, where r and R are arbitrary positive numbers, there exist a strictly increasing sequence of natural numbers n k , k ∈ N, and a function x ∈ S 0 r such that…”