We investigate two boundary value problems for the second order differential equation with $p$-Laplacian $${\left(a\left(t\right){\Phi }_p\left(x^{\text{'}}\right)\right)}^{\text{'}}=b\left(t\right)F\left(x\right),t\in I=[0,\infty ),$$ where $a$, $b$ are continuous positive functions on $I$. We give necessary and sufficient conditions which guarantee the existence of a unique (or at least one) positive solution, satisfying one of the following two boundary conditions: $$\mathrm{i})x\left(0\right)=c>0,\underset{t\to \infty }{lim}x\left(t\right)=0;\mathrm{ii})x^{\text{'}}\left(0\right)=d<0,\underset{t\to \infty }{lim}x\left(t\right)=0.$$

The paper deals with the existence of multiple positive solutions for the boundary value problem $$\left\{\begin{array}{c}{\left(\varphi \left(p\left(t\right)u^{(n-1)}\right)\left(t\right)\right)}^{\text{'}}+a\left(t\right)f(t,u\left(t\right),u^{\text{'}}\left(t\right),...,u^{(n-2)}\left(t\right))=0,0<t<1,\hfill \\ u^{\left(i\right)}\left(0\right)=0,i=0,1,...,n-3,\hfill \\ u^{(n-2)}\left(0\right)=\sum _{i=1}^{m-2}{\alpha }_i{u}^{(n-2)}\left({\xi }_i\right),u^{(n-1)}\left(1\right)=0,\hfill \end{array}\right.$$ where $\varphi :ℝ\to ℝ$ is an increasing homeomorphism and a positive homomorphism with $\varphi \left(0\right)=0$. Using a fixed-point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.

We investigate the existence of positive solutions for a nonlinear second-order differential system subject to some $m$-point boundary conditions. The nonexistence of positive solutions is also studied.