In this paper, we first prove a weak convergence theorem by Mann’s iteration for a commutative family of positively homogeneous nonexpansive mappings in a Banach space. Next, using the shrinking projection method defined by Takahashi, Takeuchi and Kubota, we prove a strong convergence theorem for such a family of the mappings. These results are new even if the mappings are linear and contractive.

In this paper, the convergence of Zakharov-Kuznetsov (ZK) equation by homotopy analysis method (HAM) is investigated. A theorem is proved to guarantee the convergence of HAM and to nd the series solution of this equation via a reliable algorithm.

In this paper, by means of the concept of attractive points of a nonlinear mapping, we prove strong convergence theorem of the Ishikawa iteration for an (α, β)−generalized hybrid mapping in a uniformly convex Banach space, and obtain weak convergence theorem of the Ishikawa iteration for such a mapping in a Hilbert space.