In　this　paper,　we　study　a　class　of　online　non-monotone　maximization　problems　under　general　constraint　set.　In　each　round,　the　function　fed　back　by　the　environment　is　of　composite　structure:　the　sum　of　a　DR-submodular　function　and　a　concave　function.　This　setting　covers　a　wide　range　of　applications.　We　propose　a　Frank-Wolfe　type　online　algorithm　for　solving　the　considered　problem.　In　our　algorithm,　there　is　no　need　to　have　the　ability　to　obtain　exact　gradients　of　the　revealed　functions.　Adopting　the　Lyapunov　function　approach　and　variance　reduction　technique,　the　algorithm　is　shown　to　have　the　bi-criteria　competitive　ratio　(1/4,3/8)　with　sub-linear　regret　under　selecting　suitable　parameters.　©　2023　Elsevier　B.V.