The shrub is a type of Japanese holly, Ilex crenata. The browning on the branches are most likely due to a root rot. Japanese hollies are very susceptible to black root rot. Heavy soils and poor drainage make the plants susceptible.
Do you have a sprinkler system or water on a schedule? If so, this can affect the root system of the plants too. Check the drainage in the area and correct if possible. Keep mulch no thicker than two inches in depth and away from the base of the stems. Prune any dead plant material. The disease doesn't kill quickly. It affects branches here and there. It can take years to either overcome the plant or get to a point of losing their attractive form. No chemical control is recommended.