We have studied the phase diagrams of the one-dimensional spin-1 Blume-Capel model with anisotropy constant D, in which equivalent-neighbor ferromagnetic interactions of strength -J are superimposed on nearest-neighbor antiferromagnetic interactions of strength K. A rich critical behavior is found due to the competing interactions. At zero temperature two ordered phases exist in the D/J-K/J plane, namely the ferromagnetic (F) and the antiferromagnetic one (AF). For lower values of D/J(D/J<0.25) these two ordered phases are separated by the point Kc=0.25J. For 0.25<D/J≤0.50, the paramagnetic phase P emerges in a region separated between the lines determined by D/J=0.5-K/J and D/J=K/J. For D/J>0.5, only phases AF and F exist and are separated by a line given by D/J=K/J. At finite temperatures, we found that the ferromagnetic region of the phase diagram in the kBT/J-D/J plane is enriched by another ferromagnetic phase F′ above a first-order line for 0.195<K/J<0.250. This first-order line, which separates phases F and F′, begins at a coexistence point, where phases F,F′, and P coexist, and ends at an ordered critical point. Similarly, we found that the phase F′ is present in the phase diagram in the kBT/J-K/J plane for 0.228<D/J<0.286.