This work aims to generalization of (????, ????)-Berinde-????-contraction to(????, ????)-Berinde-(????,????)-rational contraction and to (????, ????)-Berinde-(????,????)-weak contraction in I-metric space and metric space and establish new fixed point results in I-metric space and its corresponding versions in metric space. Also, generalization of T-cyclic (????, ????, ????, ????)-contraction to ((????, ????, ????), (????, ????, ????), ????, ????)-T-cyclic contraction and T-cyclic (????, ????, ????, ????)-rational contraction to ((????, ????, ????), (????, ????, ????), ????, ????)-T-cyclic rational contraction took placeand establish new common fixed point theorems in I-metric spaceand its corresponding versions in metric space.