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The purpose of this paper is to study the periodic boundary value problem -u''(t) = f(t,u(t),u'(t)), u(0) = u(2π), u'(0) = u'(2π) when f satisfies the Carathéodory conditions. We show that a generalized upper and lower solution method is still valid, and develop a monotone iterative technique for finding minimal and maximal solutions.

We study certain subgroups of the Hopf group-coalgebra automorphism group of Radford’s $\pi$-biproduct. Firstly, we discuss the endomorphism monoid ${\mathrm{End}}_{\pi \text{-Hopf}}(A\times H,p)$ and the automorphism group ${\mathrm{Aut}}_{\pi \text{-Hopf}}(A\times H,p)$ of Radford’s $\pi$-biproduct $A\times H={\{A\times H_{\alpha }\}}_{\alpha \in \pi }$, and prove that the automorphism has a factorization closely related to the factors $A$ and $H={\left\{H_{\alpha }\right\}}_{\alpha \in \pi }$. What’s more interesting is that a pair of maps $(F_L,F_R)$ can be used to describe a family of mappings $F={\left\{F_{\alpha }\right\}}_{\alpha \in \pi }$. Secondly, we consider the relationship between the automorphism group ${\mathrm{Aut}}_{\pi \text{-Hopf}}(A\times H,p)$ and the automorphism group ${\mathrm{Aut}}_{\pi \text{-}𝒴𝒟\text{-Hopf}}\left(A\right)$ of $A$, and...