In this paper, the F-implicit complementarity problem (F-ICP) and F-implicit variational inequality problem (F-IVIP) are introduced and studied. The equivalence between (F-ICP) and (F-IVIP) is presented under certain assumptions. Furthermore, we derive some new existence theorems of solutions for (F-ICP) and (FIVIP) by using the Fan-Knaster-Kuratowski-Mazurkiewicz theorem [K. Fan: Math. Ann. 142 (1961), 305-310] and Lin's result [T. C. Lin: Bull. Austral. Math. Soc. 34 (1986), 107- 117] under some suitable assumptions without the monotonicity.