  The Hartree-Fock-Bogolyubov (HFB) plus proton-neutron quasiparticle random
phase approximation (pnQRPA) approach based on Skyrme interaction is applied to
study the nuclear $\beta^+$/EC decay for nuclei near the proton magic numbers
$Z=$20, 28, and 50. With properly selected Skyrme interactions that include the
tensor terms, and the $T=0$ pairing interaction, the experimental $\beta^+$/EC
decay half-lives of these nuclei can be systematically reproduced quite well.
It is shown that the tensor and $T=0$ pairing interactions play different roles
in different nuclei. The specific effect is relevant to the configurations that
contribute to the decay. The attractive and repulsive properties of tensor
interaction for the GT state is also studied. The present results indicate that
it works repulsively on the GT states dominated by the configurations from $\pi
j_\gtrless$ to $\nu j_\gtrless$ orbits, while working attractively for those
mainly composed of the configurations from $\pi j_\gtrless$ to $\nu j_\lessgtr$
orbits.
