On Two-point Boundary Value Problems for Second-order Difference Equation

In this paper, we aim to investigate the difference equation \begin{align*} \Delta^{2}y(t-1)+|y(t)|=0, \ \ \ \ \ t\in[1,T]_{\mathbb{Z}} \end{align*} with different boundary conditions $y(0)=0$ or $\Delta y(0)=0$ and $y(T+1)=B$ or $\Delta y(T)=B$,\ where\ $T\geq 1$ is an integer and $B\in\mathbb{R}$. We will show that how the values of $T$ and $B$ influence the existence and uniqueness of the solutions to the about problem. In details, for the different problems, the $TB$-plane explicitly divided into different parts according to the number of the solutions to the above problems. These parts of $TB$-plane for the value of $T$ and $B$ guarantee the uniqueness, the existence and the nonexistence of solutions respectively.     

Synthetic Applications of the beta-Lithiation of beta-Aryl Secondary Amides: Diastereoselective and Enantioselective Substitutions

The sequence of beta-lithiation and electrophilic substitution of beta-aryl secondary amides is reported. The lithiations occur regioselectively at the beta-position, and the resulting lithiated intermediates can be reacted with a wide range of electrophiles to give substituted products. Reactions of beta-lithiated amides bearing an alpha-substituent provide substituted products with high diastereoselectivity. Electrophilic substitutions of beta-lithiated N-methylamides in the presence of the chiral diamine (-)-sparteine provide highly enantioenriched products. The methodology is used to synthesize enantioenriched beta-aryl beta-substituted amides, acids, and lactones.