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Exponential estimates for solutions of half-linear differential equations



Druh Článek v odborném periodiku
Časopis / Zdroj Acta Mathematica Hungarica
Fakulta / Pracoviště MU

Pedagogická fakulta

Obor Obecná matematika
Klíčová slova half-linear differential equation; decreasing solution; increasing solution; asymptotic behavior
Popis This paper is concerned with estimates, unimprovable in a certain sense, for positive solutions to the half-linear differential equation $ (|y'|^{\alpha-1}\operatorname{\text{\rm sgn}} y')'=p(t)|y|^{\alpha-1}\operatorname{\text{\rm sgn}} y, $ where $p$ is a continuous nonnegative function on $[0,\infty)$ and $\alpha>1$. It is shown that any positive increasing solution $y$ of the equation satisfies $y(t)\ge y(0)\exp\left\{h\int_0^t p^{\frac{1}{\alpha}}(s)\,\drm s\right\}$, with $h<(\alpha-1)^{-\frac{1}{\alpha}}$, for all $t$ on the complement of a set of finite Lebesgue measure. Under an additional assumption, this estimate holds for all $t$. Further, a condition is established which guarantees that the equation has exponentially increasing solutions and exponentially decreasing solutions.
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