Oscillation and nonoscillation criteria for the higher order self-adjoint differential equation $${(-1)}^n{\left(t^{\alpha }{y}^{\left(n\right)}\right)}^{\left(n\right)}+q\left(t\right)y=0(*)$$ are established. In these criteria, equation $(*)$ is viewed as a perturbation of the conditionally oscillatory equation $${(-1)}^n{\left(t^{\alpha }{y}^{\left(n\right)}\right)}^{\left(n\right)}-\frac{{\mu }_{n,\alpha }}{t^{2n-\alpha }}y=0,$$ where ${\mu }_{n,\alpha }$ is the critical constant in conditional oscillation. Some open problems in the theory of conditionally oscillatory, even order, self-adjoint equations are also discussed.

The paper deals with the periodic boundary value problem (1) $L_{4}x\left(t\right)+a\left(t\right)x\left(t\right)\in F(t,x\left(t\right))$, $t\in J=[a,b]$, (2) $L_{i}x\left(a\right)=L_{i}x\left(b\right)$, $i=0,1,2,3$, where $L_{0}x\left(t\right)=a_{0}x\left(t\right)$, $L_{i}x\left(t\right)=a_i\left(t\right)L_{i-1}x\left(t\right)$, $i=1,2,3,4$, $a_0\left(t\right)=a_4\left(t\right)=1$, $a_i\left(t\right)$, $i=1,2,3$ and $a\left(t\right)$ are continuous on $J$, $a\left(t\right)\ge 0$, $a_i\left(t\right)>0$, $i=1,2$, $a_1\left(t\right)=a_3\left(t\right)·F(t,x):J\times R\to${nonempty convex compact subsets of $R$}, $R=(-\infty ,\infty )$. The existence of such periodic solution is proven via Ky Fan’s fixed point theorem.

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.

We study the singular periodic boundary value problem of the form $${\left(\phi \left(u^{\text{'}}\right)\right)}^{\text{'}}+h\left(u\right)u^{\text{'}}=g\left(u\right)+e\left(t\right),u\left(0\right)=u\left(T\right),u^{\text{'}}\left(0\right)=u^{\text{'}}\left(T\right),$$ where $\phi ℝ\to ℝ$ is an increasing and odd homeomorphism such that $\phi \left(ℝ\right)=ℝ,$$h\in C[0,\infty ...$

By using the least action principle and minimax methods in critical point theory, some existence theorems for periodic solutions of second order Hamiltonian systems are obtained.