We study the turnpike property for the nonconvex optimal control problems described by the differential inclusion $\dot{x}\in a\left(x\right)$. We study the infinite horizon problem of maximizing the functional ${\int }_0^{T}u\left(x\left(t\right)\right)dt$ as $T$ grows to infinity. The turnpike theorem is proved for the case when a turnpike set consists of several optimal stationary points.

We consider the minimization problem $f\left(x\right)\to min$, $x\in K$, where $K$ is a closed subset of an ordered Banach space $X$ and $f$ belongs to a space of increasing lower semicontinuous functions on $K$. In our previous work, we showed that the complement of the set of all functions $f$, for which the corresponding minimization problem has a solution, is of the first category. In the present paper we show that this complement is also a $\sigma$-porous set.

We study the nonlinear two-parameter problem $-u^{″}\left(x\right)+\lambda u{\left(x\right)}^q=\mu u{\left(x\right)}^p$, $u\left(x\right)>0$, $x\in \left(0,1\right)$, $u\left(0\right)=u\left(1\right)=0$. Here, $1<q<p$ are constants and $\lambda ,\mu >0$ are parameters. We establish precise asymptotic formulas with exact second term for variational eigencurve $\mu \left(\lambda \right)$ as $\lambda \to \infty$. We emphasize that the critical case concerning the decaying rate of the second term is $p=\left(3q-1\right)/2$ and this kind of criticality is new for two-parameter problems.