We present several results dealing with the asymptotic behaviour of a real twodimensional system x ′ (t) = A(t)x(t) + ∑ Pm k=1 Bk(t)x(θk(t)) + h(t, x(t), x(θ1(t)), . . . , x(θm(t))) with bounded nonconstant delays t − θk(t) ≥ 0 satisfying limt→∞ θk(t) = ∞, under the assumption of instability. Here A, Bk and h are supposed to be matrix functions and a vector function, respectively. The conditions for the instable properties of solutions together with the conditions for the existence of bounded solutions are given. The methods are based on the transformation of the real system considered to one equation with complex-valued coefficients. Asymptotic properties are studied by means of a suitable Lyapunov-Krasovskii functional and the Wa˙zewski topological principle. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one constant or nonconstant delay were studied.
In this paper we investigate the asymptotic properties of all solutions of the delay differential equation y'(x)=a(x)y(\tau(x))+b(x)y(x),\qquad x\in I=[x_0,\infty). We set up conditions under which every solution of this equation can be represented in terms of a solution of the differential equation z'(x)=b(x)z(x),\qquad x\in I and a solution of the functional equation |a(x)|\varphi(\tau(x))=|b(x)|\varphi(x),\qquad x\in I.
In the paper we consider the difference equation of neutral type (E) ∆3 [x(n) − p(n)x(σ(n))] + q(n)f(x(τ (n))) = 0, n ∈ N(n0 ), where p, q : N(n0 ) → R+; σ, τ : N → Z, σ is strictly increasing and lim n→∞ σ(n) = ∞; τ is nondecreasing and lim n→∞ τ (n) = ∞, f : R → R, xf(x) > 0. We examine the following two cases: 0 < p(n) ≤ λ ∗ < 1, σ(n) = n − k, τ (n) = n − l, and 1 < λ∗ ≤ p(n), σ(n) = n + k, τ (n) = n + l, where k, l are positive integers. We obtain sufficient conditions under which all nonoscillatory solutions of the above equation tend to zero as n → ∞ with a weaker assumption on q than the usual assumption ∑∞ i=n0 q(i) = ∞ that is used in literature.
We study the limit behavior of certain classes of dependent random sequences (processes) which do not possess the Markov property. Assuming these processes depend on a control parameter we show that the optimization of the control can be reduced to a problem of nonlinear optimization. Under certain hypotheses we establish the stability of such optimization problems.
Asymptotic properties of the half-linear difference equation (∗) ∆(an|∆xn| α sgn ∆xn) = bn|xn+1| α sgn xn+1 are investigated by means of some summation criteria. Recessive solutions and the Riccati difference equation associated to (∗) are considered too. Our approach is based on a classification of solutions of (∗) and on some summation inequalities for double series, which can be used also in other different contexts.
The asymptotic behaviour of the solutions is studied for a real unstable twodimensional system x ' (t) = A(t)x(t) + B(t)x(t − r) + h(t, x(t), x(t − r)), where r > 0 is a constant delay. It is supposed that A, B and h are matrix functions and a vector function, respectively. Our results complement those of Kalas [Nonlinear Anal. 62(2) (2005), 207–224], where the conditions for the existence of bounded solutions or solutions tending to the origin as t → ∞ are given. The method of investigation is based on the transformation of the real system considered to one equation with complex-valued coefficients. Asymptotic properties of this equation are studied by means of a suitable Lyapunov-Krasovskii functional and by virtue of the Wazewski topological principle. Stability and asymptotic behaviour of the solutions for the stable case of the equation considered were studied in Kalas and Baráková [J. Math. Anal. Appl. 269(1) (2002), 278–300].
The paper discusses the asymptotic properties of solutions of the scalar functional differential equation \[ y^{\prime }(x)=ay(\tau (x))+by(x),\qquad x\in [x_0,\infty ) \] of the advanced type. We show that, given a specific asymptotic behaviour, there is a (unique) solution $y(x)$ which behaves in this way.
We study the asymptotic behavior of the solutions of a differential equation with unbounded delay. The results presented are based on the first Lyapunov method, which is often used to construct solutions of ordinary differential equations in the form of power series. This technique cannot be applied to delayed equations and hence we express the solution as an asymptotic expansion. The existence of a solution is proved by the retract method.