Let $cal{H}$ and $cal{K}$ be Hilbert spaces and let T, $ ilde{T}$ = T + ${delta}T$ be bounded operators from $cal{H}$ into $cal{K}$. In this article, two facts related to the perturbation bounds are studied. The first one is to find the upper bound of $parallel ilde{T}^+;-;T^+parallel$ which extends the results obtained by the second author and enriches the perturbation theory for the Moore-Penrose inverse. The other one is to develop explicit representations of projectors $parallel ilde{T} ilde{T}^+;-;TT^+parallel$ and $parallel ilde{T}^+ ilde{T};-;T^+Tparallel$. In addition, some spectral cases related to these results are analyzed.