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Keywords:
singular ordinary differential equation of the second order; time singularities; unbounded domain; asymptotic properties; Kneser solutions; damped solutions; non-oscillatory solutions
singular ordinary differential equation of the second order; time singularities; unbounded domain; asymptotic properties; Kneser solutions; damped solutions; non-oscillatory solutions
Summary:
We investigate an asymptotic behaviour of damped non-oscillatory solutions of the initial value problem with a time singularity $\left( p(t)u^{\prime }(t) \right)^{\prime } + p(t)f ( u(t) )=0$, $u(0)=u_0$, $u^{\prime }(0)=0$ on the unbounded domain $[0,\infty )$. Function $f$ is locally Lipschitz continuous on $\mathbb {R}$ and has at least three zeros $L_0 <0$, $0$ and $L>0$. The initial value $u_0\in (L_0, L)\setminus \lbrace 0\rbrace $. Function $p$ is continuous on $[0,\infty ),$ has a positive continuous derivative on $(0,\infty )$ and $p(0)=0$. Asymptotic formulas for damped non-oscillatory solutions and their first derivatives are derived under some additional assumptions. Further, we provide conditions for functions $p$ and $f$, which guarantee the existence of Kneser solutions.
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