We consider the existence of positive solutions of the equation $1/\lambda \left(t\right){\left(\lambda \left(t\right){\phi }_p\left(x^{\text{'}}\left(t\right)\right)\right)}^{\text{'}}+\mu f(t,x\left(t\right),x^{\text{'}}\left(t\right))=0$, where ${\phi }_p\left(s\right)={\left|s\right|}^{p-2}s$, p > 1, subject to some singular Sturm-Liouville boundary conditions. Using the Krasnosel’skiĭ fixed point theorem for operators on cones, we prove the existence of positive solutions under some structure conditions.

We propose an approach for studying positivity of Green’s operators of a nonlocal boundary value problem for the system of $n$ linear functional differential equations with the boundary conditions $n_i{x}_i-{\sum }_{j=1}^n{m}_{ij}{x}_j={\beta }_i$, $i=1,\cdots ,n$, where $n_i$ and $m_{ij}$ are linear bounded “local” and “nonlocal“ functionals, respectively, from the space of absolutely continuous functions. For instance, $n_i{x}_i=x_i\left(\omega \right)$ or $n_i{x}_i=x_i\left(0\right)-x_i\left(\omega \right)$ and $m_{ij}{x}_j={\int }_0^{\omega }k\left(s\right)x_j\left(s\right)\mathrm{d}s+{\sum }_{r=1}^{n_{ij}}{c}_{ijr}{x}_j\left(t_{ijr}\right)$ can be considered. It is demonstrated that the positivity of Green’s operator of nonlocal problem follows from the positivity of Green’s operator...

In this paper I discuss quantum systems whose Hamiltonians are non-Hermitian but whose energy levels are all real and positive. Such theories are required to be symmetric under $𝒞𝒫𝒯$, but not symmetric under $𝒫$ and $𝒯$ separately. Recently, quantum mechanical systems having such properties have been investigated in detail. In this paper I extend the results to quantum field theories. Among the systems that I discuss are $-{\phi }^4$ and $i{\phi }^3$ theories. These theories all have unexpected and remarkable properties. I discuss...