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Let $D$ be a strong digraph on $n\geq 4$ vertices. In [2, J. Graph Theory 22(2) (1996) 181-187)], J. Bang-Jensen, G. Gutin and H. Li proved the followingtheorems: If (*) $d(x)+d(y)\geq 2n-1$ and $min \{d(x), d(y)\}\geq n-1$ forevery pair of non-adjacent vertices $x, y$ with a common in-neighbour or (**)$min \{d^+(x)+ d^-(y),d^-(x)+ d^+(y)\}\geq n$ for every pair of non-adjacentvertices $x, y$ with a common in-neighbour or a common out-neighbour, then $D$is hamiltonian.  In this paper we show that: (i) if $D$ satisfies the condition (*) and theminimum semi-degree of $D$ at least two or (ii) if $D$ is not directed cycleand satisfies the condition (**), then either $D$ contains a cycle of length$n-1$ or $n$ is even and $D$ is isomorphic to complete bipartite digraph or tocomplete bipartite digraph minus one arc.

On longest non-Hamiltonian Cycles in Digraphs with the Conditions of Bang-Jensen, Gutin and Li