Oscillation criteria for third-order hybrid trinomial advanced differential equations with mixed sign terms

Abstract This paper investigates the oscillation properties of solutions to a class of hybrid type third-order trinomial advanced differential equations of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:msup> <m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:msub> <m:mrow> <m:mi>η</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:msub> <m:mrow> <m:mi>η</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mrow> <m:mo>′</m:mo> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mo>′</m:mo> </m:mrow> </m:msup> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mo>′</m:mo> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>ϕ</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>μ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>ϕ</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>δ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>.</m:mo> </m:math> $${\left({\eta }_{2}(t){\left({\eta }_{1}(t){\mu }^{\prime }(t)\right)}^{\prime }\right)}^{\prime }+{\phi }_{1}(t)\mu (t)-{\phi }_{2}(t){\mu }^{\alpha }(\delta (t))=0.$$ To obtain the results, we first transform the considered trinomial equation into an equivalent binomial form using positive solutions of the related auxiliary second-and third-order equations. Then, by applying comparison theorems and the integral averaging technique, new sufficient conditions for the oscillation of all solutions of the equation under consideration are derived. The theoretical findings are illustrated with examples that highlight the novelty and the effectiveness of the proposed criteria.