We present a new approach to solving boundary value problems on noncompact intervals for second order differential equations in case of nonlocal conditions. Then we apply it to some problems in which an initial condition, an asymptotic condition and a global condition is present. The abstract method is based on the solvability of two auxiliary boundary value problems on compact and on noncompact intervals, and uses some continuity arguments and analysis in the phase space. As shown in the applications, Kneser-type properties of solutions on compact intervals and a priori bounds of solutions on noncompact intervals are key ingredients for the solvability of the problems considered, as well as the properties of principal solutions of an associated half-linear equation. The application of this method leads to some new existence results, which complement and extend some previous ones in the literature.
We propose results about sign-constancy of Green's functions to impulsive nonlocal boundary value problems in a form of theorems about differential inequalities. One of the ideas of our approach is to construct Green's functions of boundary value problems for simple auxiliary differential equations with impulses. Careful analysis of these Green's functions allows us to get conclusions about the sign-constancy of Green's functions to given functional differential boundary value problems, using the technique of theorems about differential and integral inequalities and estimates of spectral radii of the corresponding compact operators in the space of essential bounded functions.
A couple (σ, τ ) of lower and upper slopes for the resonant second order boundary value problem x ′′ = f(t, x, x′ ), x(0) = 0, x ′ (1) = ∫ 1 0 x ′ (s) dg(s), with g increasing on [0, 1] such that ∫ 1 0 dg = 1, is a couple of functions σ, τ ∈ C 1 ([0, 1]) such that σ(t) ≤ τ (t) for all t ∈ [0, 1], σ ′ (t) ≥ f(t, x, σ(t)), σ(1) ≤ ∫ 1 0 σ(s) dg(s), τ ′ (t) ≤ f(t, x, τ (t)), τ (1) ≥ ∫ 1 0 τ (s) dg(s), in the stripe ∫ t 0 σ(s) ds ≤ x ≤ ∫ t 0 τ (s) ds and t ∈ [0, 1]. It is proved that the existence of such a couple (σ, τ ) implies the existence and localization of a solution to the boundary value problem. Multiplicity results are also obtained.
We propose an approach for studying positivity of Green’s operators of a nonlocal boundary value problem for the system of n linear functional differential equations with the boundary conditions nixi − ∑n j=1 mijxj = βi , i = 1, . . . , n, where ni and mij are linear bounded ''local” and ''nonlocal” functionals, respectively, from the space of absolutely continuous functions. For instance, nixi = xi(ω) or nixi = xi(0) − xi(ω) and mijxj = ∫ ω 0 k(s)xj(s) ds + ∑nij r=1 cijrxj (tijr) can be considered. It is demonstrated that the positivity of Green’s operator of nonlocal problem follows from the positivity of Green’s operator for auxiliary ''local'' problem which consists of a “close” equation and the local conditions nixi = αi , i = 1, . . . , n.