A hybrid iterative algorithm with Meir-Keeler contraction is presented for solving the fixed point problem of the pseudocontractive mappings and the variational inequalities. Strong convergence analysis is given as ${\text{l}\text{i}\text{m}}_{n\to \mathrm{\infty }}d(ST{x}_n,TS{x}_n)$.

We introduce the notions of $(\alpha -\varphi -\psi )$-weaker Meir-Keeler contractive mappings and $(\alpha -\phi )$-stronger Meir-Keeler contractive mappings. We discuss the existence of periodic points in the setting of generalized quasi-metric spaces. Our results improve, extend, and generalize several results in the literature.