We investigate two boundary value problems for the second order differential equation with $p$-Laplacian $${\left(a\left(t\right){\Phi }_p\left(x^{\text{'}}\right)\right)}^{\text{'}}=b\left(t\right)F\left(x\right),t\in I=[0,\infty ),$$ where $a$, $b$ are continuous positive functions on $I$. We give necessary and sufficient conditions which guarantee the existence of a unique (or at least one) positive solution, satisfying one of the following two boundary conditions: $$\mathrm{i})x\left(0\right)=c>0,\underset{t\to \infty }{lim}x\left(t\right)=0;\mathrm{ii})x^{\text{'}}\left(0\right)=d<0,\underset{t\to \infty }{lim}x\left(t\right)=0.$$

In the paper it is shown that each solution $u(r,\alpha )$ ot the initial value problem (2), (3) has a finite limit for $r\to \infty$, and an asymptotic formula for the nontrivial solution $u(r,\alpha )$ tending to 0 is given. Further, the existence of such a solutions is established by examining the number of zeros of two different solutions $u(r,\overline{\alpha })$, $u(r,\widehat{\alpha })$.

Sufficient conditions are formulated for existence of non-oscillatory solutions to the equation $$y^{\left(n\right)}+\sum _{j=0}^{n-1}{a}_j\left(x\right)y^{\left(j\right)}+p\left(x\right){\left|y\right|}^k\mathrm{sgn}y=0$$ with $n\ge 1$, real (not necessarily natural) $k>1$, and continuous functions $p\left(x\right)$ and $a_j\left(x\right)$ defined in a neighborhood of $+\infty$. For this equation with positive potential $p\left(x\right)$ a criterion is formulated for existence of non-oscillatory solutions with non-zero limit at infinity. In the case of even order, a criterion is obtained for all solutions of this equation at infinity to be oscillatory. Sufficient conditions are obtained...

Mostramos la existencia de dos curvas de datos iniciales (x0, v0) para las cuales las soluciones x(t) correspondientes del problema de Cauchy asociado a la ecuación xtt + |xt|α-1 xt + x = 0, supuesto α ∈ (0,1), se anulan idénticamente después de un tiempo finito. Mediante métodos asintóticos y argumentos de comparación mostramos que para muchos otros datos iniciales las soluciones decaen a 0, en un tiempo infinito, como t-α / (1-α).