In this paper, by the strongly positive linear bounded operator technique, a new generalized Mann-type hybrid composite extragradient CQ iterative algorithm is first constructed. Then by using the algorithm, we find a common element of the set of solutions of the variational inequality problem for a monotone, Lipschitz continuous mapping, the set of zeros of two families of finite maximal monotone mappings and the set of fixed points of an asymptotically κ-strict pseudocontractive mappings in the intermediate sense in a real Hilbert space. Finally, we prove the strong convergence of the iterative sequences, which extends and improves the corresponding previous works.