In　recent　years,　submodularity　has　been　found　in　a　wide　range　of　connections　and　applications　with　different　scientific　fields.　However,　many　applications　in　practice　do　not　fully　meet　the　characteristics　of　diminishing　returns.　In　this　paper,　we　consider　the　problem　of　maximizing　unconstrained　non-negative　weakly-monotone　non-submodular　set　function.　The　generic　submodularity　ratio　gamma　is　a　bridge　connecting　the　non-negative　monotone　functions　and　the　submodular　functions,　and　no　longer　applicable　to　the　non-monotone　functions.　We　study　a　class　of　non-monotone　functions,　define　as　the　weakly-monotone　function,　redefine　the　submodular　ratio　related　to　it　and　name　it　weaklymonotone　submodularity　ratio　(gamma)　over　cap,　propose　a　deterministic　double　greedy　algorithm,　which　implements　the　(gamma)　over　cap/(gamma)　over　cap　+2　approximation　of　the　maximizing　unconstrained　non-negative　weakly-monotone　function　problem.　When　(gamma)　over　cap.　=　1,　the　algorithm　achieves　an　approximate　guarantee　of　1/3,　achieving　the　same　ratio　as　the　deterministic　algorithm　for　the　unconstrained　submodular　maximization　problem.