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# Erratum to: Nonlinear quasi-contractions in non-normal cone metric spaces

*Fixed Point Theory and Applications*
**volume 2014**, Article number: 196 (2014)

## Abstract

In the note we correct some errors that appeared in the article (Jiang and Li in Fixed Point Theory Appl. 2014:165, 2014).

**MSC:**06A07, 47H10.

## Correction

Upon critical examination of the main results and their proofs in [1], we note some critical errors under the conditions of the main theorem and its proof in our article [1].

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 [[1], Theorem 1].

**Theorem 1** *Let* (X,d) *be a complete cone metric space over a solid cone* *P* *of a Banach space* (E,\parallel \cdot \parallel ) *and* T:X\to X *a quasi*-*contraction* (*i*.*e*., *there exists a mapping* A:P\to P *such that*

*where* u\in \{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)\}). *Assume that* A:P\to P *is a nondecreasing*, *continuous and subadditive* (*i*.*e*., A(u+v)\u2aafAu+Av *for each* u,v\in P) *mapping with* A\theta =\theta *such that*

*If* *B* *is continuous at* *θ*, *where* Bu={\sum}_{i=0}^{\mathrm{\infty}}{A}^{i}u *for each* u\in P, *then* *T* *has a unique fixed point* {x}^{\ast}\in X, *and for each* {x}_{0}\in X, *the Picard iterative sequence* \{{x}_{n}\} *converges to* {x}^{\ast}, *where* {x}_{n}={T}^{n}{x}_{0} *for each* *n*.

**Remark 1** In the case that the normed vector space (E,\parallel \cdot \parallel ) is complete, if (2) holds then the mapping *B* is well defined. In fact, fix u\in P, let {s}_{n}={\sum}_{i=0}^{n}{A}^{i}u and {S}_{n}={\sum}_{i=0}^{n}\parallel {A}^{i}u\parallel. By (2), we get {lim}_{n\to \mathrm{\infty}}{S}_{n}={\sum}_{i=0}^{\mathrm{\infty}}\parallel {A}^{i}u\parallel and hence \{{S}_{n}\} is a Cauchy sequence of reals. Note that \parallel {s}_{m}-{s}_{n}\parallel =\parallel {\sum}_{i=n+1}^{m}{A}^{i}u\parallel \le {\sum}_{i=n+1}^{m}\parallel {A}^{i}u\parallel ={S}_{m}-{S}_{n} for each m>n, then \{{s}_{n}\} is a Cauchy sequence in *E*. Moreover, by the completeness of *E*, \{{s}_{n}\} is convergent. This implies that Bu={lim}_{n\to \mathrm{\infty}}{\sum}_{i=0}^{n}{A}^{i}u for each *u*, *i.e.*, *B* is well defined. However, in [[1], Theorem 1] the normed vector space *E* is not assumed to be complete, then \{{s}_{n}\} may not be convergent, and consequently, *B* may be not meaningful.

**Remark 2** (i) In [1] the authors claim that (see (4) in [1])

which plays an important role in the proof of [[1], Theorem 1]. However, if *A* is a nonlinear mapping, the above claim may not hold. For example, let E=P=[0,a] and A(t)={t}^{2} for each t\in P, where 0<a<1. It is clear that A:P\to P is nonlinear. Note that {A}^{0}(t)=t and {A}^{i}(t)={t}^{{2}^{i}} (i=1,2,3,\dots), then {\sum}_{i=0}^{\mathrm{\infty}}{A}^{i}(t)=t+{\sum}_{i=1}^{\mathrm{\infty}}{t}^{{2}^{i}}\le t+{\sum}_{i=1}^{\mathrm{\infty}}{t}^{2i}=t+\frac{{t}^{2}}{1-{t}^{2}} for each t\in [0,a], and hence {\sum}_{i=0}^{\mathrm{\infty}}{A}^{i}(t) is convergent for each t\in [0,a]. This implies that the function B(t) is well defined, where B(t)={\sum}_{i=0}^{\mathrm{\infty}}{A}^{i}(t) for each t\in [0,a]. For each t\in [0,a], we have AB(t)=\sqrt{t+{\sum}_{i=1}^{\mathrm{\infty}}{t}^{{2}^{i}}} and BA(t)={\sum}_{i=1}^{\mathrm{\infty}}{t}^{{2}^{i}}. Suppose that there exists {t}_{0}\in (0,\frac{\sqrt{2}}{2}] such that AB({t}_{0})=BA({t}_{0}), and set b=BA({t}_{0}). Then we have 0<b\le \frac{{t}_{0}^{2}}{1-{t}_{0}^{2}} and {t}_{0}+b={b}^{2}. Solve the equation {t}_{0}+b={b}^{2}, then b=\frac{1+\sqrt{1+4{t}_{0}}}{2} by b>0. Thus we get 1<\frac{1+\sqrt{1+4{t}_{0}}}{2}=b\le \frac{{t}_{0}^{2}}{1-{t}_{0}^{2}}\le 1, a contradiction. Hence BA(t)\ne AB(t) for each t\in (0,\frac{\sqrt{2}}{2}]. This shows that (3) does not hold.

(ii) Note that *A* is not confined to a linear mapping in [[1], Theorem 1], then from (i) we know that (3) may not hold, and consequently, the proof of [[1], Theorem 1] is not finished yet. In order to complete its proof, we add the continuity of *A* to Theorem 1.

(iii) Suppose that *E* is a Banach space and *A* is a continuous and subadditive mapping such that (2) is satisfied, then by Remark 1 we get

In what follows, we shall complete the proof of Theorem 1 by using (4) instead of (3). Since there are too many changes required for the proof of [[1], Theorem 1], we present the full proof of Theorem 1 as follows.

*Proof of Theorem 1* It follows from (2) and Remark 1 that the mapping *B* is well defined. Clearly, *B* is a nondecreasing and subadditive mappings with B(P)\subset P and B\theta =\theta since *A* is nondecreasing and subadditive, A(P)\subset (P) and A\theta =\theta. We claim that for each n\ge 1,

In the following we shall show this claim by induction.

If n=1, then i=j=1, and so the claim is trivial.

Assume that (5) holds for *n*. To prove (5) holds for n+1, it suffices to show

By (1),

where

Consider the case that {i}_{0}=1.

If u=d({x}_{0},{x}_{n}), then by the triangle inequality, the nondecreasing property, subadditivity of *A*, the definition of *B*, (4), (5), and (7)

*i.e.*, (6) holds.

If u=d({x}_{0},{x}_{1}), then by the definition of *B* and (7)

*i.e.*, (6) holds.

If u=d({x}_{0},{x}_{n+1}), then by the triangle inequality, the nondecreasing property and subadditivity of *A*, and (7)

Acting on the above inequality with *B*, by the nondecreasing property and subadditivity of *B*

which together with the definition of *B* implies that

*i.e.*, (6) holds.

If u=d({x}_{n},{x}_{{i}_{0}}), then by the definition and the nondecreasing property of *A*, (4), (5), and (7)

*i.e.*, (6) holds.

If u=d({x}_{n},{x}_{n+1}), we set {i}_{1}=n-1, and then by (7)

Consider the case that 2\le {i}_{0}\le n.

If u=d({x}_{{i}_{0}-1},{x}_{n}), or u=d({x}_{{i}_{0}-1},{x}_{{i}_{0}}), or d({x}_{n},{x}_{{i}_{0}}), then by the definition and the nondecreasing property of *A*, (4), (5), and (7)

*i.e.*, (6) holds.

If u=d({x}_{n},{x}_{n+1}), or u=d({x}_{{i}_{0}-1},{x}_{n+1}), we set {i}_{1}=n, or {i}_{1}={i}_{0}-1\ge 1, respectively, and then (8) follows.

From the above discussions of both cases, we find the result that either (6) holds, and so the proof of our claim is complete, or there exists {i}_{1}\in \{1,2,\dots ,n\} such that (8) holds. For the latter situation, continuing in a similar way, it will be found as a result that either

which together with the definition and the nondecreasing property of *A*, (4), and (8), forces that

*i.e.*, (6) holds, and so the proof of our claim is complete; or there exists {i}_{2}\in \{1,2,\dots ,n\} such that

If the above procedure ends by the *k* th step with k\le n-1, that is, there exist k+1 integers {i}_{0},{i}_{1},\dots ,{i}_{k}\in \{1,2,\dots ,n\} such that

then by the nondecreasing property of *A* and (4)

*i.e.* (6) holds, and so the proof of our claim is complete.

If the above procedure continues more than *n* steps, then there exist n+1 integers {i}_{0},{i}_{1},{i}_{n}\in \{1,2,\dots ,n\} such that

It is clear that {i}_{0},{i}_{1},{i}_{n}\in \{1,2,\dots ,n\} implies there exist two integers k,l\in \{0,1,2,\dots ,n\} with k<l such that {i}_{k}={i}_{l}, then by the nondecreasing property of *A* and (9)

Acting on (10) with *B*, by the nondecreasing property of *B* we get

which together with the definition of *B* implies that

*i.e.*, (6) holds. The proof of our claim is complete.

Note that *B* and *A* are nondecreasing and continuous at *θ*, B\theta =\theta and A\theta =\theta, then it follows from Lemma 3 of [1] that for each \{{u}_{n}\}\in P,

provided that {u}_{n}\stackrel{w}{\to}\theta. By (2), we get

Then in analogy to the proof of [[1], Theorem 1], by (5), (11), (12) we can show that

and there exists some {x}^{\ast}\in X such that

By (1),

where u\in \{d({x}_{n},{x}^{\ast}),d({x}_{n},{x}_{n+1}),d({x}^{\ast},T{x}^{\ast}),d({x}_{n},T{x}^{\ast}),d({x}^{\ast},{x}_{n+1})\}.

If u=d({x}_{n},{x}^{\ast}), or u=d({x}_{n},{x}_{n+1}), or u=d({x}^{\ast},{x}_{n+1}), then it follows from (11), (13), and (14) that Au\stackrel{w}{\to}\theta and hence d(T{x}^{\ast},{x}^{\ast})=\theta by (15).

If u=d({x}^{\ast},T{x}^{\ast}), then by (15)

Acting on the above inequality with *B*, by the nondecreasing and subadditivity of *B* we get

which together with the definition of *B* implies that

and hence d({x}^{\ast},T{x}^{\ast})=\theta since Bd({x}_{n+1},{x}^{\ast})\stackrel{w}{\to}\theta by (11) and (14).

If u=d({x}_{n},T{x}^{\ast}), then, by the triangle inequality, the nondecreasing property, and subadditivity of *A* and (15), we get

Acting on the above inequality with *B*, then by the nondecreasing property and subadditivity of *B*

which together with the definition of *B* implies that

and hence d({x}^{\ast},T{x}^{\ast})=\theta since Bd({x}_{n+1},{x}^{\ast})\stackrel{w}{\to}\theta and BAd({x}_{n},{x}^{\ast})\stackrel{w}{\to}\theta by (11) and (14). This shows that {x}^{\ast} is a fixed point of *T*.

If *x* is another fixed point of *T*, then by (1)

where u\in \{d(x,{x}^{\ast}),d(x,Tx),d({x}^{\ast},T{x}^{\ast}),d(x,T{x}^{\ast}),d({x}^{\ast},Tx)\}. If u=d(x,Tx), or u=d({x}^{\ast},T{x}^{\ast}), then u=\theta, and hence d(x,{x}^{\ast})=\theta since A\theta =\theta. If u=d(x,{x}^{\ast}), or u=d(x,T{x}^{\ast}) or u=d({x}^{\ast},Tx), then we must have u=d(x,{x}^{\ast}), and hence d(x,{x}^{\ast})\u2aafAd(x,{x}^{\ast}). Acting on it with *B*, by the nondecreasing property of *B* we get Bd(x,{x}^{\ast})\u2aafBAd(x,{x}^{\ast}). Moreover, by the definition of *B*, we have d(x,{x}^{\ast})=Bd(x,{x}^{\ast})-BAd(x,{x}^{\ast})\u2aaf\theta and hence d(x,{x}^{\ast})=\theta. This shows {x}^{\ast} is the unique fixed point of *T*. The proof is complete. □

## References

Jiang S, Li Z: Nonlinear quasi-contraction in non-normal cone metric spaces.

*Fixed Point Theory Appl.*2014., 2014: Article ID 165

## Acknowledgements

The work was supported by Natural Science Foundation of China (11161022, 11461029, 71462015), Natural Science Foundation of Jiangxi Province (20122BAB201015, 20142BCB23013, 20143ACB21012), Educational Department of Jiangxi Province (KJLD14034), and Program for Excellent Youth Talents of JXUFE (201201).

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The online version of the original article can be found at 10.1186/1687-1812-2014-165

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Li, Z., Jiang, S. Erratum to: Nonlinear quasi-contractions in non-normal cone metric spaces.
*Fixed Point Theory Appl* **2014, **196 (2014). https://doi.org/10.1186/1687-1812-2014-196

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DOI: https://doi.org/10.1186/1687-1812-2014-196

### Keywords

- nonlinear quasi-contraction
- cone metric space