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.