A new kind of viscosity iterative approximation algorithm is proposed for solving the fixed point problem of pseudo-contractive operators and the zero point problem of accretive operators in Banach spaces. Since continuous pseudo-contraction operators are more widely applied than nonexpansive operators and strict pseudo-contraction operators, the common solutions of fixed point problems for continuous pseudo-contractive operators and accretive operator variational inequalities are discussed in real Banach spaces with weak sequence dual mappings by using the relationship between pseudo-contraction operators and strictly nonexpansive operators. The numerical approximation algorithm of the common solution is given under the idea of viscosity iteration. The iteration sequence generated by this kind of algorithm converges strongly to a common solution of the fixed point problem for continuous pseudo-contractive operators and the variational inequality problem for accretive operators under appropriate conditions. This series of strong convergence theorems generalize and unify some conclusions of relevant literature, and complement the theory of nonlinear operators.