The behavior of solutions of linear integro-differential equations with unbounded delay

More on the Behavior of Solutions to Linear Integrodifferential Equations with Unbounded Delay

“…As it has been shown by Kordonis, Philos and Purnaras [20], the fact that x satisfies (E) for t b 0 is equivalent to the fact that z satisfies By taking into account (3.4) and (3.5) as well as the definition of Uðl 0 ; l 1 ; fÞ and V ðl 0 ; l 1 ; fÞ, we conclude that the double inequality in the conclusion of the theorem can equivalently be written as follows inf sa0 wðsÞ a wðtÞ a sup sa0 wðsÞ for all t b 0:…”

“…Motivated by the results in [29,30], we continue here the study in [18,20] and we establish some further results on the behavior of solutions of linear neutral integrodi¤erential equations with unbounded delay, and especially of linear (non-neutral) integrodi¤erential equations with unbounded delay. Our results will be derived by the use of two distinct real roots of the corresponding characteristic equations.…”

“…Very recently, Kordonis, Philos and Purnaras [20] obtained some results on the behavior of solutions of linear neutral integrodi¤erential equations with unbounded delay. More precisely, in [20], a useful inequality for solutions is established, via an appropriate real root of the corresponding characteristic equation, and then this inequality is used to obtain an estimate for solutions that leads to a stability criterion.…”

“…More precisely, in [20], a useful inequality for solutions is established, via an appropriate real root of the corresponding characteristic equation, and then this inequality is used to obtain an estimate for solutions that leads to a stability criterion. The results in [20] extend and improve previous ones given by Kordonis and Philos [18] for the special case of (nonneutral) integrodi¤erential equations with unbounded delay.…”

“…More precisely, in [20], a useful inequality for solutions is established, via an appropriate real root of the corresponding characteristic equation, and then this inequality is used to obtain an estimate for solutions that leads to a stability criterion. The results in [20] extend and improve previous ones given by Kordonis and Philos [18] for the special case of (nonneutral) integrodi¤erential equations with unbounded delay. The work in the papers [18] and [20] is motivated by some asymptotic and stability results for delay di¤erential equations or neutral delay di¤erential equations, which have bounded delay, given by Driver [6], Driver, Sasser and Slater [8], Frasson and Verduyn Lunel [9], Graef and Qian [12], Kordonis, Niyianni and Philos [17], Philos [26], and Philos and Purnaras [27,28] (see, also, the papers by Arino and Pituk [1], Driver [7], and Gyö ri [14] for some closely related results).…”