Some results on nearest points problems in Banach lattices

In this paper we prove that for two equivalent norms such that $X$ becomes an STM and LLUM space the dominated best approximation problem have the same solution. We give some conditions such that under these conditions the Fréchet differentiability of the nearest point map is equivalent to the continuity of metric projection in the dominated best approximation problem. Also we prove that these conditions are equivalent to strong solvability of the dominated best approximation problem. We prove these results in an STM, reflexive STM and UM spaces. In continuation we give some weaker conditions in the problem of best approximation. We delete the dominated condition and examine these results for a closed solid subset in Banach lattice. We prove that for two equivalent norms such that $X$ becomes an STM and LLUM space the best approximation problem have the same solution