We establish a Hartman type asymptotic formula for nonoscillatory solutions of the half-linear second order differential equation $${\left(r\left(t\right)\Phi \left(y^{\text{'}}\right)\right)}^{\text{'}}+c\left(t\right)\Phi \left(y\right)=0,\Phi \left(y\right):={\left|y\right|}^{p-2}y,p>1.$$

Positive solutions of the nonlinear second-order differential equation $\left(p\left(t\right)\right|x^{\text{'}}{|}^{\alpha -1}{x}^{\text{'}}{)}^{\text{'}}+q\left(t\right){\left|x\right|}^{\beta -1}x=0,\alpha >\beta >0,$ are studied under the assumption that p, q are generalized regularly varying functions. An application of the theory of regular variation gives the possibility of obtaining necessary and sufficient conditions for existence of three possible types of intermediate solutions, together with the precise information about asymptotic behavior at infinity of all solutions belonging to each type of solution classes.

In the setting of a real Hilbert space $\mathscr{H}$, we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order evolution equations            ü(t) + γ$\dot{u}$(t) + ∇ϕ(u(t)) + A(u(t)) = 0, where ∇ϕ is the gradient operator of a convex differentiable potential function ϕ: $\mathscr{H}\to ℝ$,A: $\mathscr{H}\to \mathscr{H}$ is a maximal monotone operator which is assumed to beλ-cocoercive, and γ > 0 is a damping parameter. Potential and non-potential effects are associated respectively to ∇ϕ and A. Under condition...

In the setting of a real Hilbert space $\mathscr{H}$, we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order evolution equations            ü(t) + γ$\dot{u}$(t) + ∇ϕ(u(t)) + A(u(t)) = 0, where ∇ϕ is the gradient operator of a convex differentiable potential function ϕ : $\mathscr{H}\to ℝ$, A : $\mathscr{H}\to \mathscr{H}$ is a maximal monotone operator which is assumed to be λ-cocoercive, and γ > 0 is a damping parameter. Potential and non-potential effects are associated respectively to ∇ϕ and A. Under condition...

In this paper we prove two results. The first is an extension of the result of G. D. Jones [4]: (A) Every nontrivial solution for $$\left\}\begin{array}{c}{(-1)}^n{u}^{\left(2n\right)}+f(t,u)=0,\text{in}(\alpha ,\infty ),u^{\left(i\right)}\left(\xi \right)=0,i=0,1,\cdots ,n-1,\text{and}\xi \in (\alpha ,\infty ),\hfill \end{array}\right.$$ must be unbounded, provided $f(t,z)z\ge 0$, in $E\times ℝ$ and for every bounded subset $I$, $f(t,z)$ is bounded in $E\times I$. (B) Every bounded solution for ${(-1)}^n{u}^{\left(2n\right)}+f(t,u)=0$, in $ℝ$, must be constant, provided $f(t,z)z\ge 0$ in $ℝ\times ℝ$ and for every bounded subset $I$, $f(t,z)$ is bounded in $ℝ\times I$.