The initial value problem for hyperbolic equations $d^{2}u\left(t\right)/d{t}^2+Au\left(t\right)=f\left(t\right)\left(0\le t\le 1\right),u\left(0\right)=\varphi ,u^{\prime }\left(0\right)=\psi$, in a Hilbert space $H$ is considered. The first and second order accuracy difference schemes generated by the integer power of $A$ approximately solving this initial value problem are presented. The stability estimates for the solution of these difference schemes are obtained.

We consider an extension of the best approximation operator from an Orlicz space $L^{\varphi }$ to the space $L^{{\varphi }^{\prime }}$, where ${\varphi }^{\prime }$ denotes the derivative of $\varphi$, and we prove a weak-type inequality in this space. Further, we obtain some strong inequalities for suitable $L^{\psi }$ spaces.

We show that infinite dimensional geometric moduli introduced by Milman are strongly related to nearly uniform convexity and nearly uniform smoothness. An application of those moduli to fixed point theory is given.