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Stability and bounded of solutions to non-autonomous delay differential equations of third order. (English) Zbl 1215.34079

Summary: We obtain some sufficient conditions to guarantee the uniform asymptotic stability of zero solution and bounded of all solutions to non-autonomous delay differential equation of third order

$\stackrel{⃛}{x}\left(t\right)+a\left(t\right)\varphi \left(\stackrel{˙}{x}\left(t\right)\right)\stackrel{¨}{x}\left(t\right)+b\left(t\right)\psi \left(\stackrel{˙}{x}\left(t\right)\right)+c\left(t\right)h\left(x\left(t-r\right)\right)=p\left(t,x\left(t\right),x\left(t-r\right),\stackrel{˙}{x}\left(t\right),\stackrel{˙}{x}\left(t-r\right),\stackrel{¨}{x}\left(t\right)\right),$

when $p\left(t,x\left(t\right),x\left(t-r\right),\stackrel{˙}{x}\left(t\right),\stackrel{˙}{x}\left(t-r\right),\stackrel{¨}{x}\left(t\right)\right)=0$ and $p\left(t,x\left(t\right),x\left(t-r\right),\stackrel{˙}{x}\left(t\right),\stackrel{˙}{x}\left(t-r\right),\stackrel{¨}{x}\left(t\right)\right)\ne 0$, respectively. By using the Lyapunov functional approach, we prove two new results on the subject.

##### MSC:
 34K12 Growth, boundedness, comparison of solutions of functional-differential equations 34K20 Stability theory of functional-differential equations
##### References:
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