New and improved oscillation criteria of third-order half-linear delay differential equations via canonical transform

Abstract In this paper, the third-order delay differential equation of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:mo maxsize="1.623em" minsize="1.623em">(</m:mo> </m:mrow> <m:msub> <m:mi>a</m:mi> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> <m:mrow> <m:mo maxsize="1.2em" minsize="1.2em">(</m:mo> </m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>a</m:mi> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> <m:msup> <m:mi>x</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> <m:mrow> <m:msup> <m:mo stretchy="false">)</m:mo> <m:mo>′</m:mo> </m:msup> </m:mrow> <m:msup> <m:mrow> <m:mo maxsize="1.2em" minsize="1.2em">)</m:mo> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msup> <m:msup> <m:mrow> <m:mo maxsize="1.623em" minsize="1.623em">)</m:mo> </m:mrow> <m:mo>′</m:mo> </m:msup> <m:mo>+</m:mo> <m:mi>b</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> <m:msup> <m:mi>x</m:mi> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msup> <m:mo stretchy="false">(</m:mo> <m:mi>σ</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo stretchy="false">)</m:mo> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:math> $$\Big(a_{2}(t)\big((a_{1}(t)x'(t))'\big)^{\alpha}\Big)'+b(t)x^{\alpha}(\sigma(t))=0$$ under the conditions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:munderover> <m:mo>∫</m:mo> <m:mrow> <m:msub> <m:mi>t</m:mi> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mrow> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:munderover> </m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:msubsup> <m:mi>a</m:mi> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mrow> <m:mo>/</m:mo> </m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msubsup> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mfrac> <m:mtext>d</m:mtext> <m:mi>t</m:mi> <m:mo>=</m:mo> <m:mi mathvariant="normal">∞</m:mi> <m:mspace width="1em"/> </m:math> $\int\limits_{t_{0}}^{\infty}\frac{1}{a_{2}^{1/\alpha}(t)}\text{d} t =\infty \quad$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:munderover> <m:mo>∫</m:mo> <m:mrow> <m:msub> <m:mi>t</m:mi> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mrow> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:munderover> </m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:msub> <m:mi>a</m:mi> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mfrac> <m:mtext>d</m:mtext> <m:mi>t</m:mi> <m:mo>&lt;</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:math> $\int\limits_{t_{0}}^{\infty} \frac{1}{a_{1}(t)}\text{d} t &lt;\infty$ is studied. Our newly proposed approach allow us to reduce the number of conditions required for all solutions of the considered equation to be oscillatory. The results established here are new, improve and complement some oscillation results known for the studied equation. Examples are provided to show the strength and novelty of the results obtained.