In　this　paper,　we　investigate　the　maximization　of　the　differences　between　a　nonnegative　monotone　diminishing　return　submodular　(DR-submodular)　function　and　a　nonnegative　linear　function　on　the　integer　lattice.　As　it　is　almost　unapproximable　for　maximizing　a　submodular　function　without　the　condition　of　nonnegative,　we　provide　weak　(bifactor)　approximation　algorithms　for　this　problem　in　two　online　settings,　respectively.　For　the　unconstrained　online　model,　we　combine　the　ideas　of　single-threshold　greedy,　binary　search　and　function　scaling　to　give　an　efficient　algorithm　with　a　1/2　weak　approximation　ratio.　For　the　online　streaming　model　subject　to　a　cardinality　constraint,　we　provide　a　one-pass　(3-5)/2　weak　approximation　ratio　streaming　algorithm.　Its　memory　complexity　is　(klogk/Ε),　and　the　update　time　for　per　element　is　(log2k/Ε).　©　Operations　Research　Society　of　China,　Periodicals　Agency　of　Shanghai　University,　Science　Press,　and　Springer-Verlag　GmbH　Germany,　part　of　Springer　Nature　2022.