Abstract: Abstract In this paper, we establish new results on the existence of positive periodic solutions for the following high-order neutral functional differential equation (NFDE) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable> <m:mtr> <m:mtd> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>(</m:mo> <m:mi>t</m:mi> <m:mo>)</m:mo> <m:mo>−</m:mo> <m:mi>c</m:mi> <m:mi>x</m:mi> <m:mo>(</m:mo> <m:mi>t</m:mi> <m:mo>−</m:mo> <m:mi>σ</m:mi> <m:mo>)</m:mo> <m:msup> <m:mo>)</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mn>2</m:mn> <m:mi>m</m:mi> <m:mo>)</m:mo> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:mi>f</m:mi> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>(</m:mo> <m:mi>t</m:mi> <m:mo>)</m:mo> <m:mo>)</m:mo> <m:msup> <m:mi>x</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>(</m:mo> <m:mi>t</m:mi> <m:mo>)</m:mo> <m:mo>+</m:mo> <m:mi>g</m:mi> <m:mo>(</m:mo> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>x</m:mi> <m:mo>(</m:mo> <m:mi>t</m:mi> <m:mo>−</m:mo> <m:mi>δ</m:mi> <m:mo>)</m:mo> <m:mo>)</m:mo> <m:mo>=</m:mo> <m:mi>e</m:mi> <m:mo>(</m:mo> <m:mi>t</m:mi> <m:mo>)</m:mo> <m:mo>.</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:math> $$\begin{array}{} (x(t)-cx(t-\sigma)) ^{(2m)}+f(x(t)) x'(t)+g(t,x(t-\delta))=e(t). \end{array}$$ The interesting thing is that g has a strong singularity at x = 0 and satisfies a small force condition at x = ∞, which is different from the corresponding ones known in the literature. Two examples are given to illustrate the effectiveness of our results.
Publication Year: 2018
Publication Date: 2018-03-31
Language: en
Type: article
Indexed In: ['crossref']
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Cited By Count: 2
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