Let K be a nonempty subset of a p -uniformly convex Banach space E , G a left reversible semitopological semigroup, and 𝒮 = { T t : t ∈ G } a generalized Lipschitzian semigroup of K into itself, that is, for s ∈ G , ‖ T s x − T s y ‖ ≤ a s ‖ x − y ‖ + b s ( ‖ x − T s x ‖ + ‖ y − T s y ‖ ) + c s ( ‖ x − T s y ‖ + ‖ y − T s x ‖ ) , for x , y ∈ K where 0$"> a s , b s , c s > 0 such that there exists a t 1 ∈ G such that b s + c s < 1 for all s ≽ t 1 . It is proved that if there exists a closed subset C of K such that ⋂ s co ¯ { T t x : t ≽ s } ⊂ C for all x ∈ K , then 𝒮 with [ ( α + β ) p ( α p ⋅ 2 p − 1 − 1 ) / ( c p − 2 p − 1 β p ) ⋅ N p ] 1 / p < 1 has a common fixed point, where α = lim sup s ( a s + b s + c s ) / ( 1 - b s - c s ) and β = lim sup s ( 2 b s + 2 c s ) / ( 1 - b s - c s ) .