Yeniçerioğlu, Ali Fuat; Küçük, Ahmet; Yazıcı, Cüneyt; Yazıcı, Vildan

In this paper, we examine the asymptotic properties of solutions to first-order linear nonhomogeneous differential equations with variable delays and variable coefficients. Initially, a unique global solution to the relevant characteristic equation is obtained. Subsequently, the asymptotic properties of the solutions to the delay equation are demonstrated through the utilization of the solution of the characteristic equation. Finally, upon examination of specific cases pertaining to the variable coefficients of the delay differential equation, three discrete results are obtained. The findings are expressed as the solution of the characteristic equation. In addition, seven examples are given to demonstrate the hypothesis of the main theorem.

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Fuat Yeniçerioğlu, Ali; Yazıcı, Vildan; Yazıcı, Cüneyt

This article concerns first-order linear nonhomogeneous neutral delay differential equations with periodic coefficients and constant delays, where the coefficients share a common period and the delays are multiples of this period. First, we obtain periodic solutions of linear nonhomogeneous neutral delay differential equations using the variation of parameters method. These periodic solutions are expressed analytically. Two examples demonstrating the applicability of our results are also included. Second, we investigate the asymptotic behavior and estimation of solutions to linear nonhomogeneous neutral delay differential equations. The results are obtained using an appropriate real root of the relevant characteristic equation. Three examples are given to illustrate our results. Finally, we present the special case of first-order linear nonhomogeneous neutral delay differential equations with constant coefficients and constant delays, and provide an interesting example.

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