Weak nearly uniform soothness and worth property of $\psi$-direct sums of Banach spaces

We shall characterize the weak nearly uniform smoothness of the $\psi$-direct sum $X{\oplus }_{\psi }Y$ of Banach spaces $X$ and $Y$. The Schur and WORTH properties will be also characterized. As a consequence we shall see in the ${\ell }_{\mathrm{\infty }}$-sums of Banach spaces there are many examples of Banach spaces with the fixed point property which are not uniformly non-square.