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We establish Hartman-Wintner type criteria for the half-linear second order differential equation $${\left(r\left(t\right)\Phi \left(x^{\text{'}}\right)\right)}^{\text{'}}+c\left(t\right)\Phi \left(x\right)=0,\Phi \left(x\right)={\left|x\right|}^{p-2}x,p>1,$$ where this equation is viewed as a perturbation of another equation of the same form.

We establish Hille-Wintner type comparison criteria for the half-linear second order differential equation $${\left(r\left(t\right)\Phi \left(x^{\text{'}}\right)\right)}^{\text{'}}+c\left(t\right)\Phi \left(x\right)=0,\Phi \left(x\right)={\left|x\right|}^{p-2}x,p>1,$$ where this equation is viewed as a perturbation of another equation of the same form.