Stability and convergence of a new composite implicit iterative sequence in Banach spaces

The purpose of this paper is to study stability and strong convergence of asymptotically pseudocontractive mappings by using a new composite implicit iteration process in an arbitrary real Banach space. The results in this paper improve and extend the corresponding results in the literature.

Throughout this paper we assume that E is an arbitrary real Banach space and \(E^{*}\) denotes the dual space of E. The normalized duality map \(J: E\to2^{E^{*}}\) is defined by

where \(\langle\cdot,\cdot\rangle\) denotes the generalized duality pairing between elements of E and \(E^{*}\). If \(E^{*}\) is strictly convex, then J is single-valued.

We first recall some definitions and conclusions.

Definition 1.1

Let \(T: D(T)\subset E\to E\) be a mapping.

1. (1)

   T is said to be asymptotically nonexpansive (see [1]) if there exists a sequence \(\{k_{n}\}\subset [1,\infty)\), \(\lim_{n\to\infty}k_{n}=1\) such that

   $$ \bigl\Vert T^{n}x-T^{n}y\bigr\Vert \leq k_{n} \|x-y\|, \quad \forall x, y\in D(T), n\geq1; $$
2. (2)

   T is said to be asymptotically pseudocontractive (see [2]) with sequence \(\{k_{n}\}\subset[0, \infty)\), if and only if \(\lim_{n\to\infty }k_{n}=1\), for all \(n\geq1\), \(x, y\in D(T)\) and there exists \(j(x-y)\in J(x-y)\) such that

   $$ \bigl\langle T^{n}x-T^{n}y, j(x-y)\bigr\rangle \leq k_{n}\|x-y\|^{2}; $$
3. (3)

   T is said to be strictly asymptotically pseudocontractive with sequence \(\{k_{n}\}\subset[0, \infty)\), if and only if \(\lim_{n\to \infty}k_{n}=k\in(0, 1)\), for all \(n\geq1\), \(x, y\in D(T)\) and there exists \(j(x-y)\in J(x-y)\) such that

   $$ \bigl\langle T^{n}x-T^{n}y, j(x-y)\bigr\rangle \leq k_{n}\|x-y\|^{2}; $$
4. (4)

   T is said to be uniformly L-Lipschitzian (see [3]) if there exists a constant \(L> 0\) such that

   $$ \bigl\Vert T^{n}x-T^{n}y\bigr\Vert \leq L\|x-y\| \quad \mbox{for all } x, y\in D(T), n\ge1. $$

It is easy to see that every asymptotically nonexpansive mapping is uniformly L-Lipschitzian and asymptotically pseudocontractive. In [4], Rhoades constructed an example to show that the class of asymptotically pseudocontractive mappings properly contains the class of asymptotically nonexpansive mappings.

The class of asymptotically pseudocontractive mappings has been studied by several authors (see [2, 4–7] and the references cited therein) by using the modified Mann iteration process (see [8]) and the modified Ishikawa iteration process (see [9]). Schu [5] proved the following theorem.

Theorem SC

Let H be a Hilbert space, \(K\subset H\) nonempty bounded closed convex, \(L>0\), \(T: K\to K\) completely continuous, uniformly L-Lipschitzian and asymptotically pseudocontractive with sequence \(\{k_{n}\}\subset[1, \infty)\), \(q_{n}=2k_{n}-1\), \(\forall n\ge1\), \(\sum_{n=1}^{\infty}(q_{n}^{2}-1)<\infty\). \(\{\alpha_{n}\}\subset[0, 1]\), \(\epsilon <\alpha_{n}\le b\), \(\forall n\ge1\), and some \(b\in (0, L^{-2}[(1+L^{2})^{1/2}-1])\), \(x_{1}\in K\), for all \(n\ge1\), define

Then \(\{x_{n}\}\) converges to some fixed point of T.

The recursion formula (1.1) is a modification of the well-known Mann iteration process (see, e.g., [8]).

Recently, Chang [6] extended Theorem SC to real uniformly smooth Banach spaces and proved the following theorem.

Theorem CH

Let E be a real uniformly smooth Banach space, K a nonempty bounded closed convex subset of E, \(T: K\to K\) be an asymptotically pseudocontractive mapping with sequence \(\{k_{n}\}\subset[1, \infty)\), \(\lim_{n\to\infty}k_{n}=1\), and \(F(T)=\{x\in K: Tx=x\}\neq\emptyset\). Let \(\{\alpha_{n}\}\subset[0, 1]\) satisfy the following conditions: (i) \(\lim_{n\to\infty}\alpha _{n}=0\), (ii) \(\sum_{n=1}^{\infty}\alpha_{n}=\infty\). For an arbitrary \(x_{0}\in K\), let \(\{x_{n}\}\) be iteratively defined by (1.1). If there exists a strictly increasing function \(\phi: [0, \infty)\to[0, \infty)\), \(\phi(0)=0\) such that \(\langle T^{n}x_{n}-x^{*}, j(x_{n}-x^{*}) \rangle \le k_{n}\lVert x_{n}-x^{*}\rVert^{2}-\phi(\lVert x_{n}-x^{*}\rVert)\), \(\forall n\ge 1\), then \(x_{n}\to x^{*}\in F(T)\).

Let K be nonempty closed convex subset of E and \(\{T_{i}\}_{i=1}^{r}\) be a finite family of nonexpansive mappings from K into itself (i.e., \(\lVert T_{i}x-T_{i}y\rVert\le \lVert x-y\rVert\) for \(x, y\in K\) and \(i=1, 2, \ldots, r\)). In 2001, Xu and Ori [10] introduced the following implicit iteration process. For an arbitrary \(x_{0}\in K\) and \(\{\alpha_{n}\}_{n=1}^{\infty}\subset[0, 1)\), the sequence \(\{x_{n}\}_{n=1}^{\infty}\) is generated as follows:

The scheme can be expressed in compact form by

Using this iteration, they proved that the sequence \(\{x_{n}\}\) converges weakly to a common fixed point of the finite family of nonexpansive mappings \(\{T_{i}\}_{i=1}^{r}\) in Hilbert spaces under certain conditions. Since then, the construction of fixed points for nonexpansive mappings and strictly pseudocontractive mappings and some other mappings via the implicit iterative algorithm has been extensively investigated by many authors (see, e.g., [10–16] and the references cited therein). An implicit process is generally desirable when no explicit scheme is available. Such a process is generally used as a ‘tool’ to establish the convergence of an explicit scheme.

In 2006, Chang et al. [13] introduced another implicit iteration process with error. In the sense of [13], the implicit iteration process with errors for a finite family of asymptotically nonexpansive mappings \(\{T_{i}\}_{i=1}^{r}\) is generated from an arbitrary \(x_{0}\in K\) by

where \(n=(k(n)-1)r+i(n)\) with \(i(n)\in\{1,2,\ldots, r\}\) and \(k(n)\in \mathbb{N}\) (the positive integer set) and \(k(n)\to\infty\), as \(n\to \infty\). \(\{\alpha_{n}\}_{n=1}^{\infty}\) is a suitable sequence in \([0,1]\) and \(\{u_{n}\}\subset K\) is such that \(\sum_{n=1}^{\infty}\|u_{n}\|<\infty\). They extended the results of [10] from Hilbert spaces to more general uniformly convex Banach spaces and from nonexpansive mappings to asymptotically nonexpansive mappings.

It is clear that even if K is a nonempty convex subset of E and \(\{u_{n}\}\subset K\) is such that \(\sum_{n=1}^{\infty}\|u_{n}\|<\infty\), then the implicit iterative sequence with errors in the sense of [13] need not be well defined, i.e., \(\{x_{n}\}_{n=1}^{\infty}\) may fail to be in K. More precisely, the conditions imposed on the error terms are not compatible with the randomness of the occurrence of errors.

In [16], Thakur proposed another modified composite implicit iteration process for a finite family of asymptotically nonexpansive mappings as follows:

where \(n=(k-1)N+i\), \(i=i(n)\in\{1,2,\ldots, N\}\), \(k=k(n)\ge1\) is some positive integer and \(k(n)\to\infty\) as \(n\to\infty\).

Inspired and motivated by these facts, we introduce a new modified composite implicit iteration process for an asymptotically pseudocontractive mappings as follows:

Unlike iteration methods with errors of [13], our iteration process (1.4) is always well defined, that is, \(\{x_{n}\}\) is always in K if K is convex subset of E.

If \(\{\beta_{n}\}=\{0\}\) for all \(n\ge1\), (1.4) becomes the explicit form as follows:

Equation (1.5) is the modified Mann iterative process (see, e.g., [8]).

Stability results established in metric space, normed linear space, and Banach space settings are available in the literature. There are several authors whose contributions are of colossal value in the study of stability of the fixed point iterative procedures: Imoru and Olatinwo [17], Olatinwo and Postolache [18], Akewe and Okeke [19].

Harder and Hicks [20] mentioned that the study of the stability of iterative schemes is useful for both theoretical and numerical investigations. Consequently, several authors have studied the stability of iterative schemes for various types of nonlinear mappings (see, e.g., [20–25] and the references cited therein).

The purpose of this paper is to study the stability and convergence of the composite implicit iterative sequence for an asymptotically pseudocontractive mapping in arbitrary real Banach spaces.

Let K be a nonempty convex subset of E and \(T: K\to K\) be a mapping, \(x_{0}\in K\) and \(\{x_{n}\}\subset K\) defined by

where f is a continuous mapping. Suppose that \(F(T)\neq\emptyset\) and \(x_{n}\to p\in F(T)\). Let \(\{s_{n}\}\) be any bounded sequence in K and \(\{\varepsilon _{n}\}\) a sequence in \([0, \infty)\) defined by

If \(\varepsilon _{n}\to0\) implies that \(s_{n}\to p\), then the iterative sequence \(\{x_{n}\}\) defined by (1.6) is said to be T-stable. If \(\sum_{n=1}^{\infty}\varepsilon _{n} <\infty\) implies that \(s_{n}\to p\), then the sequence \(\{x_{n}\}\) defined by (1.6) is said to be almost T-stable. An example in [25] represents an iterative sequence which is almost T-stable but not T-stable.

In the sequel, \(F(T)=\{x\in K: Tx=x\}\) denotes the set of fixed points of T. We give the stability definition for the sequence \(\{x_{n}\}\) defined by (1.4).

Definition 2.1

Let \(\{x_{n}\}\) be the sequence defined by (1.4) such that \(x_{n}\to p\in F(T)\). Let \(\{s_{n}\}\) be any bounded sequence in K. Define a sequence \(\{\varepsilon _{n}\}\) by

If \(\varepsilon _{n}\to0\) implies that \(s_{n}\to p\), then the sequence \(\{x_{n}\}\) is said to be T-stable. If \(\varepsilon _{n}/\alpha_{n}\to0\) implies that \(s_{n}\to p\), then the sequence \(\{x_{n}\}\) is said to be weakly T-stable. If \(\sum_{n=1}^{\infty}\varepsilon _{n}<\infty\) implies that \(s_{n}\to p\), then the sequence \(\{x_{n}\}\) is said to be almost T-stable.

The following lemmas will be needed in proving our main results.

Lemma 2.2

(Lemma 2 of [26])

Let \(\{a_{n}\}\), \(\{b_{n}\}\), \(\{ c_{n}\}\) be nonnegative real sequences satisfying the following conditions:

where \(\{t_{n}\}\) is a sequence in \([0, 1]\) such that \(\sum_{n=1}^{\infty}t_{n}=\infty\), \(\lim_{n\to\infty}b_{n}=0\) and \(\sum_{n=1}^{\infty}c_{n}<\infty\). Then \(a_{n}\to0\) (\(n\to\infty\)).

We denote Φ := {\(\phi \mid \phi: [0, \infty)\to[0, \infty)\) be a nondecreasing function such that \(\phi(t)=0\) if and only if \(t=0\)}.

Lemma 2.3

(Lemma 2.1 of [27])

Let \(\{\theta_{n}\}\) be a sequence of nonnegative real numbers, \(\{\lambda_{n}\}\) be a real sequence satisfying

and let \(\phi\in\Phi\). If there exists a positive integer \(n_{0}\) such that

for some \(q>1\), all \(n\ge n_{0}\) and \(\lim_{n\to\infty}\sigma_{n}=0\), then \(\lim_{n\to\infty}\theta_{n}=0\).

Lemma 2.4

(Lemma 1.1 of [28])

Let E be a real Banach space, \(T: E\to E\) a mapping, and λ any positive real number. Then for any \(x, y\in E\) and \(k>0\),

whenever

Lemma 2.5

Let E be a normed linear space then for all \(x, y\in E\) and for all \(j(x+y)\in J(x+y)\), the following inequality holds:

First we give two auxiliary lemmas.

Lemma 2.6

Let K be a nonempty convex subset of an arbitrary real Banach space E and \(T: K\to K \) an asymptotically pseudocontractive mapping with \(\{k_{n}\}\subset[0, \infty)\) and let \(F(T)\neq\emptyset\). If \(\{x_{n}\}\) is the iterative sequence defined by (1.4) and \(p\in F(T)\), then

where

Proof

Since T is an asymptotically pseudocontractive mapping with \(\{k_{n}\}\subset[0, \infty)\), there exists \(j(x_{n+1}-p)\in J(x_{n+1}-p)\) for \(x_{n+1}\in K\) and \(p\in F(T)\) such that

Then we have

Thus it follows from Lemma 2.4 that

It follows from (1.4) that

Since \(p\in F(T)\), we have

From (2.4) and (2.5), we have

Therefore from (2.3) we have

i.e.,

Since \((1+\alpha_{n})^{-1}\le1\) and \((1+\alpha_{n})^{-1}\le1-\alpha_{n}+\alpha_{n}^{2}\), from (2.7), we have

where \(b_{n}=\lVert T^{n}x_{n+1}-T^{n}y_{n}\rVert+\alpha_{n}(k_{n}+1)\lVert x_{n}-T^{n}y_{n}\rVert\). This completes the proof. □

Theorem 3.1

Let K be a nonempty convex subset of an arbitrary real Banach space E and \(T: K\to K\) be a uniformly Lipschitzian (with a Lipschitzian constant \(L> 0\)) and strictly asymptotically pseudocontractive mapping with sequence \(\{k_{n}\}\subset[0, \infty)\), \(k_{n}\to k\in(0, 1)\), and let \(F(T)\neq\emptyset\). Assume that \(\{x_{n}\}\), \(\{y_{n}\}\) are the sequences defined by (1.4), \(\{\alpha_{n}\}, \{\beta_{n}\} \subset[0, 1]\) satisfy the following conditions:

If \(\{x_{n}\}\) is bounded in K, then

1. (1)

   \(\{x_{n}\}\) converges strongly to the unique common fixed point p of T;

2. (2)

   \(\{x_{n}\}\) is both almost T-stable and weakly T-stable.

Proof

Assume that \(p_{1}, p_{2}\in F(T)\). Since T is a strictly asymptotically pseudocontractive mapping, there exists \(j(p_{1}-p_{2})\in J(p_{1}-p_{2})\) such that

Letting \(n\to\infty\) we have \(\lVert p_{1}-p_{2}\rVert^{2}\le k\lVert p_{1}-p_{2}\rVert^{2}\), \(k\in (0, 1)\). This implies that \(p_{1}=p_{2}\).

By the strictly asymptotically pseudocontractive property of T, similar to (2.8), we have

where \(b_{n}=\lVert T^{n}x_{n+1}-T^{n}y_{n}\rVert+\alpha_{n}(k_{n}+1)\lVert x_{n}-T^{n}y_{n}\rVert\). Since \(\alpha_{n}\to0\), \(k_{n}\to k\in(0, 1)\), and

Since \(\lim_{n\to\infty}(1-k_{n}-\alpha_{n})=1-k\ge\delta\in(0, 1-k)\), there exists a natural number \(n_{1}\) such that \((1-k_{n}-\alpha_{n})\ge\delta\) for all \(n\ge n_{1}\). Thus we have

Substituting (3.2) into (3.1) we have

Next we will prove that \(b_{n}\to0\) (\(n\to\infty\)).

Since \(T: K\to K\) is uniformly Lipschitzian with a Lipschitzian constant \(L> 0\), for all \(x, y\in K\), we have

This implies that \(\{T^{n}x_{n+1}\}\) is bounded in K since \(\{x_{n}\}\) is bounded. Similarly, \(\{T^{n}x_{n}\}\) is bounded sequence in K. It follows from (1.4) that

This implies that \(\{y_{n}\}\) is bounded. Thus by the uniform Lipschitzianness of T, we have

This implies that \(\{T^{n}y_{n}\}\) is also bounded. From (1.4) and (3.4), we have

Therefore we have

Observing (3.6), \(\{x_{n}\}\), \(\{T^{n}y_{n}\}\) bounded in K and \(\alpha_{n}\to0\) (\(n\to\infty\)), we know that \(b_{n}\to 0\) (\(n\to\infty\)). By Lemma 2.2 and (3.3), we have \(x_{n}\to p\) (\(n\to\infty\)). The conclusion (1) is proved.

Next we prove the conclusion (2). For any bounded sequence \(\{s_{n}\}\subset K\) defined by (2.1) and \(p\in F(T)\), we have

where

It follows from (3.8) that

By using a similar method to that given in proving (2.5)-(2.8), (3.2), we can prove that

where \(b_{n}= \lVert T^{n}p_{n}-T^{n}z_{n} \rVert+\alpha_{n}(k_{n}+1) \lVert s_{n}-T^{n}z_{n} \rVert\). Since T is uniformly Lipschitzian with Lipschitzian constant \(L> 0\), we have \(\lVert T^{n}s_{n}-p \rVert\le L \lVert s_{n}-p \rVert\). This implies that the sequence \(\{T^{n}s_{n}\}\) is bounded since \(\{s_{n}\}\) is bounded in K. By the same method as in proving (3.4)-(3.6), we can prove that the sequence \(\{T^{n}z_{n}\}\) is bounded and \(\lVert T^{n}p_{n}-T^{n}z_{n} \rVert\to0\) (\(n\to\infty\)). Therefore we have \(b_{n}\to0\) (\(n\to\infty\)). Substituting (3.9) into (3.7) we have

If \(\sum_{n=1}^{\infty}\varepsilon _{n}<\infty\), taking \(a_{n}= \lVert s_{n}-p \rVert\), \(t_{n}=\alpha_{n}\delta\), \(c_{n}=\varepsilon _{n}\) in Lemma 2.2, from (3.10), we have \(s_{n}\to p\) (\(n\to\infty\)), i.e., \(\{x_{n}\}\) is almost T-stable.

If \(\varepsilon _{n}/\alpha_{n}\to0\), taking \(a_{n}= \lVert s_{n}-p \rVert \), \(t_{n}=\alpha_{n}\delta\), \(c_{n}=0\) in Lemma 2.2, from (3.10), we have \(s_{n}\to p\) (\(n\to\infty\)), i.e., \(\{x_{n}\}\) is weakly T-stable. This completes the proof. □

Example 3.2

Let \(E=R=(-\infty, \infty)\) with the usual norm. Take \(K=[0, 1]\) and define \(T: K\to K\) by

for all \(n\ge0\). Then \(F(T)=\{0\}\) and T is not continuous at \(x=1\). We can verify that

Thus \(T^{2}\) is continuous in K and \(T^{2}K\subset[0, 2^{-n}]\) for all \(n\ge1\). Then for any \(x\in K\), there exists \(j(x-0)\in J(x-0)\) satisfying

for all \(n\ge1\). That is, T is a strictly asymptotically pseudocontractive mapping.

Example 3.3

Let \(E=[0, 1]\). Define \(T: E\to E\) by \(Tx=\frac{x}{5}\), where E has the usual norm. Then \(F(T)=\{0\}\) and T is a strictly asymptotically pseudocontractive mapping with \(k_{n}=\frac{1}{5}\). Consider the following conditions:

Let \(\{x_{n}\}\) be the sequence defined by (1.4). So

We have the results in Table 1.

Therefore, the conditions of Theorem 3.1 are fulfilled.

Theorem 3.4

Let K be a nonempty convex subset of an arbitrary real Banach space E, \(T: K\to K\) be a uniformly Lipschitzian (with a Lipschitzian constant \(L> 0\)) and asymptotically pseudocontractive mapping with sequence \(\{k_{n}\}\subset[1, \infty)\), \(\lim_{n\to\infty}k_{n}=1\) and let \(F(T)\neq\emptyset\) and \(p\in F(T)\). Let \(\{x_{n}\}\), \(\{y_{n}\}\) be the sequences defined by (1.4). Assume that \(\{\alpha_{n}\}, \{\beta_{n}\}\subset[0, 1]\) satisfy the following conditions:

If \(\{x_{n}\}\) is bounded in K and there exists a strictly increasing function \(\phi: [0, \infty)\to[0, \infty)\) with \(\phi(0)=0\) such that

where \(j(x_{n+1}-p)\in J(x_{n+1}-p)\). Then \(\{x_{n}\}\) converges strongly to the fixed point of T.

Proof

Since \(\{x_{n}\}\) is bounded in K, then \(M=\sup_{n\ge1}\{ \lVert x_{n}-p \rVert\}<\infty\) for \(p\in F(T)\). It follows from Lemma 2.5 and (1.4) that there exists \(j(x_{n+1}-p)\in J(x_{n+1}-p)\) such that

where \(d_{n}=\langle T^{n}x_{n+1}-p, j(x_{n+1}-p)\rangle -k_{n} \lVert x_{n+1}-p \rVert^{2}+\phi( \lVert x_{n+1}-p \rVert)\), \(e_{n}= \lVert T^{n}y_{n}-T^{n}x_{n+1} \rVert\). From (3.6), we have \(e_{n}\to 0\) as \(n\to\infty\). It follows from (3.11) that

Notice that \(\lim_{n\to\infty}(1-2\alpha_{n} k_{n})=1>0\), without loss of generality, we assume that \((1-2\alpha_{n} k_{n})>0\) for all \(n\ge1\). Therefore, we have

where \(\lambda_{n}'=[2(k_{n}-1)+\alpha_{n}]M^{2}+2e_{n}M\). Taking \(q=2\), \(\theta_{n}= \lVert x_{n}-p \rVert\), \(\lambda_{n}=2\alpha_{n}\), \(\sigma _{n}={\lambda_{n}'+2d_{n} \over 2(1-2\alpha_{n} k_{n})}\) in Lemma 2.3, from (3.12), we have \(x_{n}\to p\) as \(n\to\infty\). This completes the proof. □

If \(\{\beta_{n}\}=\{0\}\) for all \(n\ge1\) in (1.4), it follows from Theorem 3.4 that we have the following result.

Theorem 3.5

Let K be a nonempty convex subset of an arbitrary real Banach space E, \(T: K\to K \) be a uniformly Lipschitzian (with a Lipschitzian constant \(L> 0\)) and asymptotically pseudocontractive mapping with sequence \(\{k_{n}\}\subset[1, \infty)\), \(\lim_{n\to\infty}k_{n}=1\) and let \(F(T)\neq\emptyset\) and \(p\in F(T)\), \(\{x_{n}\}\) be the sequence defined by (1.5). Let \(\{\alpha_{n}\}\subset[0, 1]\) satisfy the following conditions: (i) \(\lim_{n\to\infty}\alpha_{n}=0\), (ii) \(\sum_{n=1}^{\infty}\alpha _{n}=\infty\). If \(\{x_{n}\}\) is bounded in K and there exists a strictly increasing function \(\phi: [0, \infty)\to[0, \infty)\) with \(\phi(0)=0\) such that

where \(j(x_{n+1}-p)\in J(x_{n+1}-p)\), \(\forall n\ge1\), then \(x_{n}\to p\) as \(n\to\infty\).

Remark 3.6

Theorem 3.5 extends Theorem CH from real uniformly smooth Banach space to arbitrary real Banach space. The requirement that K be bounded closed imposed in Theorem CH is stronger than the requirement that \(\{x_{n}\}\) be bounded imposed in Theorem 3.5. The condition

in Theorem CH is replaced by

in Theorem 3.5.

Remark 3.7

We remark that if the error terms are added in (1.4) and are assumed to be bounded, then the results of this paper still hold. On carefully reading Thakur’s work [16], we discovered that there are gaps in the proof of Lemma 2.1 in [16]. In (2.2) and (2.3) of Lemma 2.1, one cannot deduce \(\sum_{n=1}^{\infty}\sigma_{n}<\infty\) from \(\sum_{n=1}^{\infty}\mu_{n}<\infty\), \(\sum_{n=1}^{\infty}d_{n}<\infty\). Thus, his main results would not hold.

The authors thank the referees for the comments that helped improve the presentation of this article. This work was supported by the Humanity and Social Science Planning Foundation of Ministry of Education of China (Grant No. 14YJAZH095), the National Natural Science Foundation of China (Grant No. 61374081), the High-level Talents Project in Guangdong Province (Grant No. 2014011), and the Natural Science Foundation of Guangdong Province (Grant Nos. S2013010013034, 2015A030313485).

Affiliations

1. School of Applied Mathematics, Guangdong University of Technology, Guangzhou, 510520, China

   • Liping Yang
   •  & Weiming Kong

Corresponding author

Correspondence to Liping Yang.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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