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 (F-IVIP) by using the Fan-Knaster-Kuratowski-Mazurkiewicz Math. Ann. 142 (1961), 305 -- 310] and Lin's Bull. Austral. Math. Soc. 34 (1986), 107 -- 117] under some suitable assumptions without the monotonicity.