On Mann-type iteration method for a family of hemicontractive mappings in Hilbert spaces

for all $x,y\in H$.

for all $x\in H$ and $x^{\ast }\in F(T)$. It is easy to see that the class of pseudocontractive mappings with fixed points is a subclass of the class of hemicontractive mappings.

The following example, due to Rhoades [3], shows that the inclusion is proper. For any $x\in [0,1]$, define a mapping $T:[0,1]\to [0,1]$ by $Tx={(1-x^{\frac{2}{3}})}^{\frac{3}{2}}$. It is shown in [4] that T is not Lipschitz and so T cannot be nonexpansive. A straightforward computation (see [5]) shows that T is pseudocontractive. For the importance of fixed points of pseudocontractive mappings, the reader may refer to [1].

In the last ten years or so, numerous papers have been published on the iterative approximation of fixed points of Lipschitz strongly pseudocontractive (and, correspondingly, Lipschitz strongly accretive) mappings using the Mann iteration process (see, for example, [6]). The results which were known only in Hilbert spaces and only for Lipschitz mappings have been extended to more general Banach spaces (see [3–5, 7–33]) and the references cited therein).

In 1974, Ishikawa [34] introduced an iteration process which, in some sense, is more general than Mann iteration and which converges, under this setting, to a fixed point of T. He proved the following theorem.

In [35], Qihou extended Theorem 1.1 to a slightly more general class of Lipschitz hemicontractive mappings and, in [25], Reich proved, under the setting of Theorem 1.1, the convergence of the recursion formula (1.3) to a fixed point of T, when T is a continuous hemicontractive mapping, under an additional hypothesis that the number of fixed points of T is finite. The iteration process (1.3) is generally referred to as the Ishikawa iteration process in light of Ishikawa [34]. Another iteration process which has been studied extensively in connection with fixed points of pseudocontractive mappings is the following.

Let K be a nonempty convex subset of E and $T:K\to K$ be a mapping.

The iteration process (1.4) is generally referred to as the Mann iteration process in light of [36].

In 1995, Liu [37] introduced the iteration process with errors as follows.

for each $n\ge 1$, where $\{{\alpha }_n\}$, $\{{\beta }_n\}$ are the sequences in $[0,1]$ satisfying appropriate conditions and $\sum \parallel u_n\parallel <\mathrm{\infty }$, $\sum \parallel v_n\parallel <\mathrm{\infty }$, is called the Ishikawa iteration process with errors.

for each $n\ge 1$, where $\{{\alpha }_n\}$ is a sequence in $[0,1]$ satisfying appropriate conditions and $\sum \parallel u_n\parallel <\mathrm{\infty }$, is called the Mann iteration process with errors.

While it is known that the consideration of error terms in the iterative processes (1.5), (1.6) is an important part of the theory, it is also clear that the iterative processes with errors introduced by Liu in (I-a) and (I-b) are unsatisfactory. The occurrence of errors is random so the conditions imposed on the error terms in (I-a) and (I-b), which imply, in particular, that they tend to zero as n tends to infinity, are unreasonable. In 1997, Xu [32] introduced the following more satisfactory definitions.

for each $n\ge 1$, where $\{u_n\}$, $\{v_n\}$ are the bounded sequences in K and $\{a_n\}$, $\{b_n\}$, $\{c_n\}$, $\{a_n^{\mathrm{\prime }}\}$, $\{b_n^{\mathrm{\prime }}\}$ and $\{c_n^{\mathrm{\prime }}\}$ are the sequences in $[0,1]$ such that $a_n+b_n+c_n=a_n^{\mathrm{\prime }}+b_n^{\mathrm{\prime }}+c_n^{\mathrm{\prime }}=1$ for each $n\ge 1$, is called the Ishikawa iteration sequence with errors in the sense of Xu.

for each $n\ge 1$ is called the Mann iteration sequence with errors in the sense of Xu.

We remark that if K is bounded (as is generally the case), then the error terms $u_n$, $v_n$ are arbitrary in K.

In [11], Chidume and Chika Moore proved the following theorem.

for each $n\ge 1$, where $\{u_n\}$ and $\{v_n\}$ are the arbitrary sequences in K. Then $\{x_n\}$ converges strongly to a fixed point of T.

They also gave the following remark in [11].

Remark 1.1 (1) In connection with the iterative approximation of fixed points of pseudocontractive mappings, the following question is still open.

for each $n\ge 1$ does not converge to zero. Since the T in this example is not continuous, the above question remains open.

In [14], Chidume and Mutangadura provide an example of a Lipschitz pseudocontractive mapping with a unique fixed point for which the Mann iteration sequence failed to converge and they stated that ‘This resolves a long standing open problem’. However, in [38, 39], Rafiq provided affirmative answers to the above questions (see also [40]) and proved the following result.

for each $n\ge 1$. Then $\{x_n\}$ converges strongly to a fixed point of T.

The purpose of this paper is to introduce the following Mann-type implicit iteration process associated with a family of continuous hemicontractive mappings to have a strong convergence in the setting of Hilbert spaces.

for each $n\ge 1$, where ${\alpha }_n,{\beta }_n^i\in [0,1]$, $i=1,2,\dots ,k$, are such that ${\alpha }_n+{\sum }_{i=1}^k{\beta }_n^i=1$ and some appropriate conditions hold.

In the sequel, we will use following results.

Lemma 2.1 [29]

Lemma 2.2 [31]

Now, we prove our main results.

where $\alpha ,{\beta }^i\in [0,1]$, $i=1,2,\dots ,k$, and $\alpha +{\sum }_{i=1}^k{\beta }^i=1$.

Substituting (2.4) and (2.5) in (2.3), we get

This completes the proof. □

Remark 2.1 Lemma 2.2 is now the special case of our result.

Theorem 2.1 Let K be a compact convex subset of a real Hilbert space H and $T_i:K\to K$, $i=1,2,\dots ,k$, be a family of continuous hemicontractive mappings. Let ${\alpha }_n,{\beta }_n^i\in [0,1]$ be such that ${\alpha }_n+{\sum }_{i=1}^k{\beta }_n^i=1$ and satisfying $\{{\alpha }_n\},{\beta }_n^i\subset [\delta ,1-\delta ]$ for some $\delta \in (0,1)$, $i=1,2,\dots ,k$.

Then, for arbitrary $x_0\in K$, the sequence $\{x_n\}$ defined by (1.9) converges strongly to a common fixed point in ${\bigcap }_{i=1}^{k}F(T_i)\ne \mathrm{\varnothing }$.

Substituting (2.6) in (2.7), we get

(2.8)

and, in view of (2.11) and Lemma 2.1, we conclude that $\parallel x_n-y^{\ast }\parallel \to 0$ as $n\to \mathrm{\infty }$, that is, $x_n\to y^{\ast }$ as $n\to \mathrm{\infty }$. This completes the proof. □

Theorem 2.2 Let H, K, $T_i$, $i=1,2,\dots ,k$, be as in Theorem  2.1 and ${\alpha }_n,{\beta }_n^i\in [0,1]$ be such that ${\alpha }_n+{\sum }_{i=1}^k{\beta }_n^i=1$ and satisfying $\{{\alpha }_n\},{\beta }_n^i\subset [\delta ,1-\delta ]$ for some $\delta \in (0,1)$, $i=1,2,\dots ,k$.

for each $n\ge 0$ converges strongly to a common fixed point in ${\bigcap }_{i=1}^{k}F(T_i)\ne \mathrm{\varnothing }$.

The set $K\cup T(K)$ is compact and so the sequence $\{\parallel x_n-T_i{x}_n\parallel \}$ is bounded. The rest of the argument follows exactly as in the proof of Theorem 2.1. This completes the proof. □

Theorem 2.3 Let K be a compact convex subset of a real Hilbert space H and $T_i:K\to K$, $i=1,2,\dots ,k$, be a family of Lipschitz hemicontractive mappings. Let ${\alpha }_n,{\beta }_n^i\in [0,1]$ be such that ${\alpha }_n+{\sum }_{i=1}^k{\beta }_n^i=1$ and satisfying $\{{\alpha }_n\},{\beta }_n^i\subset [\delta ,1-\delta ]$ for some $\delta \in (0,1)$, $i=1,2,\dots ,k$.

Then, for arbitrary $x_0\in K$, the sequence $\{x_n\}$ defined by (1.9) converges strongly to a common fixed point in ${\bigcap }_{i=1}^{k}F(T_i)\ne \mathrm{\varnothing }$.

Theorem 2.4 Let H, K, $T_i$, $i=1,2,\dots ,k$, be as in Theorem  2.3 and ${\alpha }_n,{\beta }_n^i\in [0,1]$ be such that ${\alpha }_n+{\sum }_{i=1}^k{\beta }_n^i=1$ and satisfying $\{{\alpha }_n\},{\beta }_n^i\subset [\delta ,1-\delta ]$ for some $\delta \in (0,1)$, $i=1,2,\dots ,k$.

for each $n\ge 1$ converges strongly to a common fixed point in ${\bigcap }_{i=1}^{k}F(T_i)\ne \mathrm{\varnothing }$.

Example For $k=2$, we can choose the following control parameters: ${\alpha }_n=\frac{1}{4}-\frac{1}{{(n+2)}^2}$, ${\beta }_n^1=\frac{1}{4}$ and ${\beta }_n^2=\frac{1}{2}+\frac{1}{{(n+2)}^2}$.