The notion of bounded pseudo-amenability was introduced by Y. Choi and et al.
[CGZ]. In this paper, similarly, we define bounded pseudo-contractibility
and then investigate bounded pseudoamenability and contractibility of
various classes of Banach algebras including ones related to locally compact
groups and discrete semigroups. We also introduce a multiplier bounded
version of approximate biprojectivity for Banach algebras and determine its
relation to bounded pseudo-amenability and contractibility.