Asymptotic properties of solutions to linear nonhomogeneous differential equations with variable delays and variable coefficients

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

doi:10.3934/era.2026279

Electronic Research Archive

2026, Volume 34, Issue 9: 6382-6403. doi: 10.3934/era.2026279

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Asymptotic properties of solutions to linear nonhomogeneous differential equations with variable delays and variable coefficients

1.
Department of Mathematics, The Faculty of Education, Kocaeli University, Kocaeli 41001, Turkey
2.
Department of Computer Engineering, The Faculty of Engineering and Natural Science, Kocaeli Health and Technology University, Kocaeli 41275, Turkey

Received: 07 March 2026 Revised: 15 June 2026 Accepted: 02 July 2026 Published: 20 July 2026

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.
- delay differential equation,
- nonhomogeneous equation,
- asymptotic behavior,
- characteristic equation
Citation: Ali Fuat Yeniçerioğlu, Ahmet Küçük, Cüneyt Yazıcı, Vildan Yazıcı. Asymptotic properties of solutions to linear nonhomogeneous differential equations with variable delays and variable coefficients[J]. Electronic Research Archive, 2026, 34(9): 6382-6403. doi: 10.3934/era.2026279

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Abstract

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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