Volume 14, Issue 2

Some new criteria for the oscillation of the fourth order difference equation Δ 2 (𝑎(𝑛)(Δ 2 𝑥(𝑛)) α ) + 𝑞(𝑛)𝑓(𝑥(𝑛 + 1)) = 0, where α is the ratio of two positive odd integers are established.

Sufficient conditions are established for the existence of nonoscillatory solutions to a quasilinear ordinary differential equation of higher order. For the equation with a positive potential, a criterion is established for the existence of nonoscillatory solutions with nonzero limit at infinity. In the case of even order, a criterion is obtained for all solutions at infinity to be oscillatory.

The existence of at least one oscillatory solution of a second order nonlinear differential equation with 𝑝-Laplacian is considered. The global monotonicity properties and asymptotic estimates for nonoscillatory solutions are investigated as well.

The paper is concerned with the so-called perturbed inclusion in the space of continuous functions. The right-hand side of the inclusion is represented by an algebraic sum of the values of two multi-valued maps, one of which consists of compacts and the other is not necessarily closed-valued and is a composition of a linear integral operator and multimap convex-valued with respect to switching. For such an inclusion it is proved that approximation in the space of summable functions of the values of a multimap convex-valued with respect to switching is not always a stable process. The necessary and sufficient condition for the closure of the set of approximate solutions to converge to the closure of the set of solutions for perturbed inclusion is derived.

The second order linear differential equation is considered, where 𝑞 : [0, ∞) → (0, ∞) is continuous, piecewise continuously differentiable, non-decreasing, and lim 𝑡→∞ 𝑞(𝑡) = ∞. A solution 𝑥 0 is called small if lim 𝑡→∞ 𝑥 0 (𝑡) = 0. It is known that the equation always has at least one nontrivial small solution, but, in general, it can have also solutions not small. The Armellini–Tonelli–Sansone Theorem says that if the function 𝑞 grows “regularly” in some sense, then all solutions are small. A generalization of this growth condition is given in terms only of the integral of 𝑞 and . It is proved that the result is a real generalization of the earlier theorems.

We prove the exponential stability and instability of a trivial solution of perturbed nonlinear differential systems' with various Coppel–Conti linear approximations and perturbations of higher order smallness. We also obtain upper and lower estimates for characteristic exponents of systems of linear approximation and solutions of the considered perturbed systems.

A solution of the Dirichlet problem for harmonic functions from the Smirnov class is obtained in the framework of functional spaces with a nonstandard growth condition. It is found that the domain boundary geometry influences the character of a problem solution. In the case of solvability, solutions are constructed in explicit form.

Regularity, in the sense of Karamata, (with nonoscillation as a consequence) and the precise asymptotic behaviour of solutions of two functional differential equations are studied.

We give some existence results for bounded solutions of linear and nonlinear second order difference equations. In particular, a method of lower and upper solutions for bounded solutions of some nonlinear second order difference equations is obtained, and applied to Duffing-type difference equations.

This paper investigates the singular Dirichlet problem –𝑢″ = 𝑓(𝑡, 𝑢, 𝑢′), 𝑢(0) = 0, 𝑢(𝑇) = 0, where 𝑓 satisfies the Carathéodory conditions on the set and . The function 𝑓(𝑡, 𝑥, 𝑦) can have time singularities at 𝑡 = 0 and 𝑡 = 𝑇 and space singularities at 𝑥 = 0 and 𝑦 = 0. The existence principle for the above problem is given and its application is presented here. The paper provides conditions which guarantee the existence of a solution which is positive on (0 ; T ) and which has the absolutely continuous first derivative on [0, 𝑇].

For the singularly perturbed system of three equations with one fast variable and two slow ones the problem of the emergence of duck trajectories is considered in the case with two different slow motion trajectories intersecting in a general manner.

Estimations of the number of solutions are given for various resonant cases of the boundary value problem 𝑥″ + 𝑔(𝑡, 𝑥) = 𝑓(𝑡, 𝑥, 𝑥′), 𝑥(𝑎) cos α – 𝑥′(𝑎) sin α = 0, 𝑥(𝑏) cos β – 𝑥′(𝑏) sin β = 0, where 𝑔(𝑡, 𝑥) is an asymptotically linear nonlinearity, and 𝑓 is a sublinear one. We assume that there exists at least one solution to the BVP.

The singular boundary value problem ( ϕ (𝑥″))′ = 𝑓(𝑡, 𝑥, 𝑥′, 𝑥″), 𝑥(0) = 𝑥(𝑇 1 ) = 𝑥(𝑇) = 0 is considered. Here 0 < 𝑇 1 < 𝑇, ϕ is an increasing homeomorphism from ℝ onto ℝ, positive 𝑓 satisfies the local Carathéodory conditions on [0, 𝑇] × (ℝ \ {0}) 3 and 𝑓 may be singular at the value 0 of all its phase variables. The conditions guaranteeing the solvability of the above problem are presented. The proofs are based on regularization and sequential techniques and in limit processes a combination of the Fatou theorem and the Lebesgue dominated convergence theorem is used.

Using Mawhin's continuation principle we obtain a general result on the existence of solutions to a boundary value problem for second order nonlinear vector ODEs. Applications are given to the existence of solutions which are contained in suitable bound sets with possibly non-smooth boundary.