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In this work, we consider boundary-value problems of the form $$f(t, x, x', x'') = 0, 0 0$$ , where the scalar function f(t, x, p, q) may be singular at x = 0. As far as we know, the solvability of the singular boundary-value problems of this form has not been treated yet. Here we try to fill in this gap. Examples illustrating our main result are included.

In 1952, for the wave equation,Protter formulated some boundary value problems (BVPs), which are multidimensional analogues of Darboux problems on the plane. He studied these problems in a3D domainΩ0,bounded by two characteristic conesΣ1andΣ2,0and a plane regionΣ0. What is the situation around these BVPs now after 50 years? It is well known that, for the infinite number of smooth functions in the right-hand side of the equation, these problems do not have classical solutions. Popivanov and Schneider (1995) discovered the reason of this fact for the cases of Dirichlet's or Neumann's conditions onΣ0. In the present paper, we consider the case of third BVP onΣ0and obtain the existence of many singular solutions for the wave equation. Especially, for Protter's problems inℝ3, it is shown here that for anyn∈ℕthere exists aCn(Ω¯0)- right-hand side function, for which the corresponding unique generalized solution belongs toCn(Ω¯0\O),but has a strong power-type singularity of ordernat the pointO. This singularity is isolated only at the vertexOof the characteristic coneΣ2,0and does not propagate along the cone.

For the differential equation where 𝑝𝑖 ∈ 𝐿 loc (𝑅+; 𝑅+), τ 𝑖 ∈ 𝐶(𝑅+; 𝑅+), τ 𝑖(𝑡) ≤ 𝑡 for 𝑡 ∈ 𝑅+, , optimal integral conditions for the oscillation of all solutions are established.

A neutral differential equation of the form (x(t) + μ(t)x(ρ(t)))(n) + f(t, x(τ 1(t)), . . . , x(τm (t))) = 0 is considered, where μ, ρ, τi : R + → R (i = 1, . . . , m) are continuous functions, 0 ≤ μ(t) ≤ 1, ρ(t) ≤ t for t ∈ R +, and the function f : R + × Rm → R satisfies the local Carathéodory conditions. Sufficient conditions are given for the considered equation to have the so-called "weak" properties A and B.

Consider the nth-order neutral differential equation [formula] where n ≧ 1, J , K are initial segments of natural numbers, and pi, τi, qk, σk ∈ R for i ∈ J and k ∈ K . A necessary and sufficient condition for the oscillation of all solutions of (E) is that its characteristic equation [formula] has no real roots. It is to be emphasized that the coefficients and the arguments are real numbers and also that the method of proof has the advantage that it results in easily verifiable sufficient conditions (in terms of the coefficients and the arguments only) for the oscillation of all solutions of Eq. (E).

Consider the nth-order neutral differential equation where n ≥ 1, δ = ±1, I, K are initial segments of natural numbers, pi, τi, σk ∈ R and qk ≥ 0 for i ∈ I and k ∈ K. Then a necessary and sufficient condition for the oscillation of all solutions of (E) is that its characteristic equation has no real roots. The method of proof has the advantage that it results in easily verifiable sufficient conditions (in terms of the coefficients and the arguments only) for the oscillation of all solutionso of Equation (E).

This paper is dealing with the oscillatory properties of first order advanced impulsive differential equations and corresponding to them inequalities with variable coefficients and advanced argument. The established sufficient conditions ensure the oscillation of every solution of this type of equations.

Consider the second- order neutral delay differential equation $$\frac{{d^2 }}{{dt^2 }}[y(t) + P(t - \tau )] + Q(t)y(t - \sigma ) = 0,t \geqq t_0 ,$$ (1) where P, Q e C([t0g, ∞), R) and the delays τ and σ are nonnegative real numbers. We examined the asymptotic behavior of the nonoscillatory solutions of eq. (1) and obtained sufficient conditions for the oscillation of (i) all solutions of eq. (1); (ii) all bounded solutions of eq. (1); (iii) all unbounded solutions of eq. (1).

Consider the neutral equation with several positive and negative coefficients dd[tx(t)+∑Ipix(t−τi)−∑Jrjx(t−ϱj)]+∑Kqkx(t−σk)=0, (1) where I, J, and K are initial segments of natural numbers, pi, τi, rj, ϱj, qk are positive numbers, and σk ⩾ 0 for i ϵ I, j ϵ J, and k ϵ K. Then a necessary and sufficient condition for the oscillation of all solutions of (1) is that its characteristic equation λ+λ∑Ipie−λτi−λ∑Irje−λϱj+∑Kqke−λσk=0 has no real roots.

For the equation (⋆) below we give a condition which guarantees the existence of a bounded nonoscillatory solution x with\(\mathop {\lim }\limits_{x \to \infty } {\text{ }}x(t) = c,{\text{ }}c \in R - \{ 0\} .\)