Convergence results for solutions of a first-order differential equation

We consider the first order differential problem: ( P n ) { u ′ ( t ) = f n ( t , u ( t ) ) , for almost every t ∈ [ 0 , 1 ] , u ( 0 ) = 0. Under certain conditions on the functions f n , the problem ( P n ) admits a unique solution u n ∈ W 1 ; 1 ( [ 0 ; 1 ] ; E ) . In this paper, we propose to study the limit behavior of sequences ( u n ) n ∈ N and ( u ′ n ) n ∈ N , when the sequence ( f n ) n ∈ N is subject to different growing conditions.