Differential Delayed Equations - Asymptotic Behavior of Solutions and Positive Solutions

Brief Summary of Basic Notions For a b R , ∈ , a b < , let ([ ] ) n C a b R , , be the Banach space of the continuous functions from the interval [ ] a b , into n R equipped with the supremum norm | ⋅ | . In the case 0 a r = − < , 0 b = , we shall denote this space as r C , that is, ([ 0] ) n r C C r R := − , , and put 0 sup ( ) r r σ φ φ σ − ≤ ≤ || || = | | for r C φ∈ . If R σ ∈ , 0 A ≥ and ([ ] ) n y C r A R σ σ ∈ − , + , then for each [ ] t A σ σ ∈ , + , we define the function t r y C ∈ by ( ) ( ) t y y t θ θ := + , [ 0] r θ ∈ − , . In the contribution presented we will investigate a class of linear differential equations with delay. In order to derive new results we need an auxiliary nonlinear statement. At first we consider a nonlinear retarded functional differential equation ( ) ( ) t y t f t y = , & (1) where n f R : Ωa , r R C Ω⊆ × is a continuous quasi-bounded map which satisfies a local Lipschitz condition with respect to the second argument. We assume that the derivatives in the system (1) are at least right-sided. A function y is said to be a solution of system (1) on [ ) r A σ σ − , + with 0 A > if ([ ) ) n y C r A R σ σ ∈ − , + , , ( ) t t y , ∈Ω for [ ) t A σ σ ∈ , + , and y satisfies (1) on [ ) A σ σ , + . In view of the above conditions, each element ( ) σ φ , ∈Ω determines a unique noncontinuable solution ( ) y σ φ , of the system (1) through ( ) σ φ , ∈Ω on its maximal existence interval. This solution depends continuously on the initial data [13]. Mentioned auxiliary result concerning the system (1) is given in part 3.