New existence results are presented for the two point singular “resonant” boundary value problem $\frac{1}{p}{\left(p{y}^{\text{'}}\right)}^{\text{'}}+ry+{\lambda }_{m}qy=f(t,y,p{y}^{\text{'}})$ a.eȯn $[0,1]$ with $y$ satisfying Sturm Liouville or Periodic boundary conditions. Here ${\lambda }_m$ is the ${(m+1)}^{st}$ eigenvalue of $\frac{1}{pq}[{\left(p{u}^{\text{'}}\right)}^{\text{'}}+rpu]+\lambda u=0$ a.eȯn $[0,1]$ with $u$ satisfying Sturm Liouville or Periodic boundary data.

Sufficient conditions are given for the solvability of an impulsive Dirichlet boundary value problem to forced nonlinear differential equations involving the combination of viscous and dry frictions. Apart from the solvability, also the explicit estimates of solutions and their derivatives are obtained. As an application, an illustrative example is given, and the corresponding numerical solution is obtained by applying Matlab software.

We establish Vallée Poussin type disconjugacy and disfocality criteria for the half-linear second order differential equation $u^{\text{'}\text{'}}={p\left(t\right)\left|u\right|}^{\alpha }{\left|u^{\text{'}}\right|}^{1-\alpha }sgnu+g\left(t\right)u^{\text{'}}$, where α ∈ (0,1] and the functions $p,g\in L_{loc}(a,b)$ are allowed to have singularities at the end points t = a, t = b of the interval under consideration.