The paper deals with the boundary value problem $$u^{\text{'}\text{'}}+ku=g\left(u\right)+e\left(t\right),u\left(0\right)=u\left(2\pi \right),u^{\text{'}}\left(0\right)=u^{\text{'}}\left(2\pi \right),$$ where $k\in ℝ$, $gI\mapsto ℝ$ is continuous, $e\in 𝕃J$ and ${lim}_{x\to 0+}{\int }_x^{1}g\left(s\right)\text{d}s=\infty .$ In particular, the existence and multiplicity results are obtained by using the method of lower and upper functions which are constructed as solutions of related auxiliary linear problems.

The linear differential equation $\left(q\right):y^{\text{'}\text{'}}=q\left(t\right)y$ with the uniformly almost-periodic function $q$ is considered. Necessary and sufficient conditions which guarantee that all bounded (on $ℝ$) solutions of $\left(q\right)$ are uniformly almost-periodic functions are presented. The conditions are stated by a phase of $\left(q\right)$. Next, a class of equations of the type $\left(q\right)$ whose all non-trivial solutions are bounded and not uniformly almost-periodic is given. Finally, uniformly almost-periodic solutions of the non-homogeneous differential...

The paper deals with the impulsive boundary value problem $$\frac{d}{dt}\left[\phi \left(y^{\text{'}}\left(t\right)\right)\right]=f(t,y\left(t\right),y^{\text{'}}\left(t\right)),y\left(0\right)=y\left(T\right),y^{\text{'}}\left(0\right)=y^{\text{'}}\left(T\right),y(t_i+)=J_i\left(y\left(t_i\right)\right),y^{\text{'}}(t_i+)=M_i\left(y^{\text{'}}\left(t_i\right)\right),i=1,...m.$$ The method of lower and upper solutions is directly applied to obtain the results for this problems whose right-hand sides either fulfil conditions of the sign type or satisfy one-sided growth conditions.

The problem on the existence of a positive in the interval $]a,b[$ solution of the boundary value problem $$u^{\text{'}\text{'}}=f(t,u)+g(t,u)u^{\text{'}};u(a+)=0,u(b-)=0$$ is considered, where the functions $f$ and $g]a,b[\times ]0,+\infty [\to ℝ$ satisfy the local Carathéodory conditions. The possibility for the functions $f$ and $g$ to have singularities in the first argument (for $t=a$ and $t=b$) and in the phase variable (for $u=0$) is not excluded. Sufficient and, in some cases, necessary and sufficient conditions for the solvability of that problem are established.