Exponential solutions of equation ẏ(t)=β(t)[y(t−δ)−y(t−τ)]

Abstract

Exponential solutions of equatioṅAsymptotic behaviour of solutions of first-order differential equation with two deviating arguments of the forṁis discussed for t → ∞. A criterion for representing solutions in exponential form is proved. As consequences, inequalities for such solutions are given. Connections with known results are discussed and a sufficient condition for existence of unbounded solutions, generalizing previous ones, is derived. An illustrative example is considered, too.

Convergence of the solutions of the equation y˙(t)=β(t)[y(<

“…But recently, the problem of asymptotic convergence or divergence of solutions of linear delayed equations has received much attention. Let us mention at least investigations [1][2][3][4][5][6][7][8][9][10][11]. Comparing the known investigations with the results presented we can see that our results can be applied when we deal with the so called critical case and give more exact sufficient conditions for this case.…”

Discrete and Differential Equations in Applied Mathematics

“…In applications it is important to know if all solutions of given equation are for t → ϩ∞ convergent to a finite limit or if there are divergent solutions. Such problems are solved in [16,18] for equations of the form with arbitrary positive and continuous function β and with positive constants δ, τ, τ Ͼ δ. New results on existence of solutions of singular initial problems of systems of differential equations were derived.…”

Convergence of solutions of an impulsive differential system with a piecewise constant argument

“…Bereketoglu and Pituk [9] gave su¢ cient conditions for the asymptotic constancy of solutions of nonhomogeneous linear delay di¤erential equations with unbounded delay and they also computed the limits of solutions in terms of the initial conditions and a special matrix solution of the corresponding adjoint equation. In [12] Diblik and Ruzickova studied the asymptotic behavior of the solutions of the …rst order di¤erential equation containing two delays. Bereketoglu and Karakoc [4] obtained su¢ cient conditions for the asymptotic constancy of solutions for an impulsive di¤erential equations.…”