Since　the　introduction　of　R-duals　by　Casazza,　Kutyniok　and　Lammers　with　the　motivation　to　obtain　a　general　version　of　duality　principle　in　Gabor　analysis,　various　R-duals　and　some　relaxations　of　the　R-dual　setup　have　been　introduced　and　studied　by　some　mathematicians.　They　provide　a　powerful　tool　in　the　analysis　of　duality　relations　in　general　frame　theory,　and　are　far　beyond　the　duality　principle　in　Gabor　analysis.　In　this　paper,　we　introduce　the　concept　of　generalized　weak　R-dual　based　on　a　pair　of　frames　which　is　a　relaxation　of　the　R-dual　setup.　Using　generalized　weak　R-duals,　we　characterize　the　frame　properties　of　a　sequence　and　the　equivalence　between　two　frames,　prove　that　the　generalized　weak　R-duals　of　frames　(Riesz　bases)　are　frame　sequences　(frames),　and　present　a　coefficient　expression　corresponding　to　the　canonical　duals　of　generalized　weak　R-duals.　Some　examples　are　provided　to　illustrate　the　generality　of　the　theory.