A generalization of compatible maps of type ( A ) called “biased maps of type ( A ) ” is introduced and used to prove fixed point theorems for certain contractions of four maps. Extensions of known results are thereby obtained, i.e., the results of Pathak, Prasad, Jungck et al. are improved. Some problems on convergence of self-maps and fixed points are also discussed. Further, we use our main results to show the existence of solutions of nonlinear integral equations.