Let　U　and　V　be　Banach　spaces,　L　and　N　be　non-linear　operators　from　U　into　V.　L　is　said　to　be　Lipschitz　if　L-1(L)　:=　sup{parallel　toLx　-　Lyparallel　to　.　parallel　tox　-　yparallel　to(-1)　:　x　not　equal　y}　is　finite.　In　this　paper,　we　give　some　basic　properties　of　Lipschitz　operators　and　then　discuss　the　unique　solvability,　exact　solvability,　approximate　solvability　of　the　operator　equations　Lx　=　y　and　Lx　+　Nx　=　y.　Under　some　conditions　we　prove　the　equivalence　of　these　solvabilities.　We　also　give　an　estimation　for　the　relative-errors　of　the　solutions　of　these　two　systems　and　an　application　of　our　method　to　a　non-linear　control　system.