In this paper we develop the monotone method in the presence of upper and lower solutions for the $2$nd order Lidstone boundary value problem $$u^{\left(2n\right)}\left(t\right)=f(t,u\left(t\right),u^{\text{'}\text{'}}\left(t\right),\cdots ,u^{\left(2\right(n-1\left)\right)}\left(t\right)),0<t<1,u^{\left(2i\right)}\left(0\right)=u^{\left(2i\right)}\left(1\right)=0,0\le i\le n-1,$$ where $f[0,1]\times {ℝ}^n\to ℝ$ is continuous. We obtain sufficient conditions on $f$ to guarantee the existence of solutions between a lower solution and an upper solution for the higher order boundary value problem.

In this paper, we develop monotone iterative technique to obtain the extremal solutions of a second order periodic boundary value problem (PBVP) with impulsive effects. We present a maximum principle for ``impulsive functions'' and then we use it to develop the monotone iterative method. Finally, we consider the monotone iterates as orbits of a (discrete) dynamical system.

We describe the nonlinear limit-point/limit-circle problem for the $n$-th order differential equation $$y^{\left(n\right)}+r\left(t\right)f(y,y^{\text{'}},\cdots ,y^{(n-1)})=0.$$ The results are then applied to higher order linear and nonlinear equations. A discussion of fourth order equations is included, and some directions for further research are indicated.

We prove an Ambrosetti–Prodi type result for the periodic solutions of the equation $\left(\right|u^{\text{'}}{|}^{p-2}{u}^{\text{'}}{\left)\right)}^{\text{'}}+f\left(u\right)u^{\text{'}}+g(x,u)=t$, when $f$ is arbitrary and $g(x,u)\to +\infty$ or $g(x,u)\to -\infty$ when $\left|u\right|\to \infty$. The proof uses upper and lower solutions and the Leray–Schauder degree.

We study the question of the unique solvability of the periodic type problem for the second order linear integro-differential equation with distributed argument deviation $$u^{\text{'}\text{'}}\left(t\right)=p_0\left(t\right)u\left(t\right)+{\int }_0^{\omega }p(t,s)u\left(\tau (t,s)\right)\mathrm{d}s+q\left(t\right),$$ and on the basis of the obtained results by the a priori boundedness principle we prove the new results on the solvability of periodic type problem for the second order nonlinear functional differential equations, which are close to the linear integro-differential equations. The proved results are optimal in some sense.

In the paper the singular Cauchy-Nicoletti problem for the system ot two ordinary differential equations is considered. New sufficient conditions for solvability of this problem are proved. In the proofs the topological method is applied. Some comparisons with known results are also given in the paper.

In this note we derive a type of a three critical point theorem which we further apply to investigate the multiplicity of solutions to discrete anisotropic problems with two parameters.