In this paper we consider a linear operator on an unbounded interval associated with a matrix linear Hamiltonian system. We characterize its Friedrichs extension in terms of the recessive system of solutions at infinity. This generalizes a similar result obtained by Marletta and Zettl for linear operators defined by even order Sturm-Liouville differential equations.

Suppose that the function $q\left(t\right)$ in the differential equation (1) $y^{\text{'}\text{'}}+q\left(t\right)y=0$ is decreasing on $(b,\infty )$ where $b\ge 0$. We give conditions on $q$ which ensure that (1) has a pair of solutions $y_1\left(t\right),y_2\left(t\right)$ such that the $n$-th derivative ($n\ge 1$) of the function $p\left(t\right)=y_1^2\left(t\right)+y_2^2\left(t\right)$ has the sign ${(-1)}^{n+1}$ for sufficiently large $t$ and that the higher differences of a sequence related to the zeros of solutions of (1) are ultimately regular in sign.

We revisit the concept of Stepanov--Orlicz almost periodic functions introduced by Hillmann in terms of Bochner transform. Some structural properties of these functions are investigated. A particular attention is paid to the Nemytskii operator between spaces of Stepanov--Orlicz almost periodic functions. Finally, we establish an existence and uniqueness result of Bohr almost periodic mild solution to a class of semilinear evolution equations with Stepanov--Orlicz almost periodic forcing term.

We formulate nonuniform nonresonance criteria for certain third order differential systems of the form $X^{{}^{\text{'}\text{'}\text{'}}}+A{X}^{{}^{\text{'}\text{'}}}+G(t,X^{{}^{\text{'}}})+CX=P\left(t\right)$, which further improves upon our recent results in [12], given under sharp nonresonance considerations. The work also provides extensions and generalisations to the results of Ezeilo and Omari [5], and Minhós [9] from the scalar to the vector situations.

In this paper, we shall give sufficient conditions for the ultimate boundedness of solutions for some system of third order non-linear ordinary differential equations of the form $$\stackrel{\text{t}o8pt...}{Xwidth0ptheight5.46pt}+F\left(\ddot{X}\right)+G\left(\dot{X}\right)+H\left(X\right)=P(t,X,\dot{X},\ddot{X})$$ where $X,F\left(\ddot{X}\right)$, $G\left(\dot{X}\right)$, $H\left(X\right)$, $P(t,X,\dot{X},\ddot{X})$ are real $n$-vectors with $F,G$, $H:{ℝ}^n\to {ℝ}^n$ and $P:ℝ\times {ℝ}^n\times {ℝ}^n\times {ℝ}^n\to {ℝ}^n$ continuous in their respective arguments. We do not necessarily require that $F\left(\ddot{X}\right),G\left(\dot{X}\right)$ and $H\left(X\right)$ are differentiable. Using the basic tools of a complete Lyapunov Function, earlier results are generalized.

In the paper a comparison theory of Sturm-Picone type is developed for the pair of nonlinear second-order ordinary differential equations first of which is the quasilinear differential equation with an oscillatory forcing term and the second is the so-called half-linear differential equation. Use is made of a new nonlinear version of the Picone’s formula.