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In this paper, we obtain a new fixed point theorem for weak contraction mappings on complete generalized metric spaces. Our result is different from that of (ψ,φ)-weakly contractive mappings which had been obtained by Lakzian–Samet’s [Appl. Math. Lett. 25 (2012) 902–906]. Moreover, an example is given to illustrate the usability of the obtained result.

We prove the equivalence of the convergence of the Mann and Ishikawa iterations with errors for uniformly continuous generalized Φ-pseudocontractive mappings in normed linear spaces. Our results extend and improve the corresponding results of Xu, 1998, Kim et al., 2009, Ofoedu, 2006, Chidume and Zegeye, 2004, Chidume, 2001, Chang et al. 2002, Liu, 1995, Hirano and huang, 2003, C. E. Chidume and C. O. Chidume, 2005, and huang, 2007.

We prove the equivalence of the convergence of the Mann and multistep iterations with errors for uniformly continuous generalized weak Φ-pseduocontractive mappings in Banach spaces. We also obtain the convergence results of Mann and multistep iterations with errors. Our results extend and improve the corresponding results.