Electron. J. Differential Equations, Vol. 2018 (2018), No. 177, pp. 1-9.

Sufficient conditions for the existence of non-oscillatory solutions to first-order differential equations with multiple advanced arguments

Juio G. Dix

Abstract:
This article concerns the existence of non-oscillatory solutions to the equation
$$ x'(t)=\sum_{k=1}^m a_k(t) x(h_k(t)), $$
where $a_k\geq 0$ and $h_k(t)\geq t$. We generalize existing results and then give an answer to the open question stated in [4]. Moreover we present a new condition based on the integral of $(\sum a_k)^2$.

Submitted October 5, 2018. Published October 31, 2018.
Math Subject Classifications: 34K06, 34K11.
Key Words: Advanced linear differential equation; positive solution; oscillatory solution.

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Juio G. Dix
Department of Mathematics
Texas State University
601 University Drive
San Marcos, TX 78666, USA
email: jd01@txstate.edu

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