Correction: Nonlinear quasi-contractions in non-normal cone metric spaces

In the note we correct some errors that appeared in the article (Jiang and Li in Fixed Correction Upon critical examination of the main results and their proofs in [], we note some critical errors under the conditions of the main theorem and its proof in our article []. In this note, we would like to supplement some essential conditions, which will ensure that the mapping B is well defined, to achieve our claim. The following theorem is a slight modification of [, Theorem ]. Theorem  Let (X, d) be a complete cone metric space over a solid cone P of a Banach space (E, · ·) and T : X → X a quasi-contraction (i.e., there exists a mapping A : P → P such that d(Tx, Ty) ≤ Au, ∀x, y ∈ X, () where u ∈ {d(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx)}). Assume that A : P → P is a nonde-creasing, continuous and subadditive (i.e., A(u + v) Au + Av for each u, v ∈ P) mapping with Aθ = θ such that ∞ i= A i u < ∞, ∀u ∈ P. () If B is continuous at θ , where Bu = ∞ i= A i u for each u ∈ P, then T has a unique fixed point x * ∈ X, and for each x  ∈ X, the Picard iterative sequence {x n } converges to x * , where x n = T n x  for each n. Remark  In the case that the normed vector space (E, · ·) is complete, if () holds then the mapping B is well defined. In fact, fix u ∈ P, let s n = n i= A i u and S n = n i= A i u. By (), we get lim n→∞ S n = ∞ i= A i u and hence {S n } is a Cauchy sequence of reals. Note that s m – s n = m i=n+ A i u ≤ m i=n+ A i u = S m – S n for each m > n, then {s n } is a Cauchy ©2014 Li and Jiang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided …