In fact, for each
\(n \ge1\), let us define a mapping
\(T_{n}: X \to X\) by
$$T_{n} (x)= \alpha_{n} f\bigl(J_{\lambda}(x_{n}) \bigr) \oplus(1- \alpha_{n})T^{n}\bigl(\beta_{n} J_{\lambda}(x_{n}) \oplus(1- \beta_{n})J_{\lambda}(x) \bigr). $$
Since
T is asymptotically nonexpansive and
\(J_{\lambda}\) is nonexpansive, we have
$$\begin{aligned} d(T_{n} x , T_{n} y) &= d\bigl( \alpha_{n} f\bigl(J_{\lambda}(x_{n})\bigr) \oplus(1- \alpha _{n})T^{n}\bigl(\beta_{n} J_{\lambda}(x_{n}) \oplus(1- \beta_{n})J_{\lambda}(x) \bigr), \\ &\qquad\alpha_{n} f\bigl(J_{\lambda}(x_{n})\bigr) \oplus(1- \alpha_{n})T^{n}\bigl(\beta_{n} J_{\lambda}(x_{n}) \oplus(1- \beta_{n})J_{\lambda}(y) \bigr)\bigr) \\ & \le(1-\alpha_{n}) d\bigl(T^{n}\bigl(\beta_{n} J_{\lambda}(x_{n}) \oplus(1- \beta _{n})J_{\lambda}(x) \bigr), T^{n}\bigl(\beta_{n} J_{\lambda}(x_{n}) \oplus(1- \beta _{n})J_{\lambda}(y)\bigr)\bigr) \\ & \le(1-\alpha_{n}) k_{n} d\bigl(\beta_{n} J_{\lambda}(x_{n}) \oplus(1- \beta _{n})J_{\lambda}(x), \beta_{n} J_{\lambda}(x_{n}) \oplus(1- \beta _{n})J_{\lambda}(y)\bigr) \\ & \le(1-\alpha_{n}) k_{n} (1- \beta_{n}) d \bigl(J_{\lambda}(x), J_{\lambda}(y)\bigr) \\ & \le(1-\alpha_{n}) k_{n} (1- \beta_{n}) d(x, y) \\ &\le(1-\alpha_{n}) k_{n} d(x, y). \end{aligned} $$
By condition (iii), for any given
\(0 < \epsilon< 1-\gamma\) there exists
\(n_{0} \ge1\) such that for any
\(n \ge n_{0}\) we have
\(k_{n} - 1 < \alpha_{n} \epsilon< \alpha_{n} (1- \gamma) \le\alpha_{n} (k_{n} - \gamma)\), i.e.,
\((1 -\alpha_{n})k_{n} < 1 - \alpha_{n} \gamma< 1\). Therefore for any
\(n \ge n_{0}\),
\(T_{n} : X \to X\) is a contractive mapping. By the Banach contraction principle, there exists a unique fixed point
\(x_{n+1} \in X\) of
\(T_{n}\) for each
\(n \ge n_{0}\). Without loss of generality, in the sequel, we can assume that the following is true for all
\(n \ge1\):
$$ \left \{ \textstyle\begin{array}{l} k_{n} - 1 < \alpha_{n} \epsilon,\\ \frac{k_{n} - 1}{\alpha_{n}} < 1- \gamma,\\ (1 -\alpha_{n})k_{n} < 1 - \alpha_{n} \gamma< 1, \end{array}\displaystyle \right . \quad\forall n \ge1. $$
(3.3)
Therefore
\(\{x_{n}\}\) is well defined.
In fact, for each
\(p \in\Gamma: = \operatorname{Fix}(t) \cap A^{-1}(0)\) we have
$$\begin{aligned} d(x_{n+1}, p) & = d\bigl(\alpha_{n} f(y_{n}) \oplus(1- \alpha_{n})T^{n}\bigl( \beta_{n} y_{n} \oplus(1- \beta_{n})y_{n+1} \bigr), p\bigr) \\ & \le\alpha_{n} d\bigl(f(y_{n}), p\bigr) +(1- \alpha_{n}) d\bigl(T^{n}\bigl(\beta_{n} y_{n} \oplus(1- \beta_{n})y_{n+1}\bigr), p\bigr) \\ & \le\alpha_{n} \bigl\{ d\bigl(f\bigl(J_{\lambda}(x_{n}) \bigr), f(p)\bigr) + d\bigl(f(p), p\bigr)\bigr\} \\ & \quad{} + (1- \alpha_{n})k_{n} d\bigl(\beta_{n} y_{n} \oplus(1- \beta_{n})y_{n+1}, p\bigr) \\ & \le\alpha_{n} \gamma d\bigl(J_{\lambda}(x_{n}), p\bigr) + \alpha_{n} d\bigl(f(p), p\bigr) \\ & \quad{} + (1- \alpha_{n})k_{n} d\bigl(\beta_{n} y_{n} \oplus(1- \beta_{n})y_{n+1}, p\bigr) \\ & \le\alpha_{n} \gamma d(x_{n}, p) + \alpha_{n} d \bigl(f(p), p\bigr) \\ & \quad{} + (1- \alpha_{n})k_{n} \bigl\{ \beta_{n} d(x_{n}, p) + (1- \beta_{n}) d(x_{n+1}, p)\bigr\} . \end{aligned} $$
After simplifying, and by using (
3.3) we have
$$\begin{aligned} d(x_{n+1}, p) & \le\frac{\alpha_{n} \gamma+ k_{n} \beta_{n} - \alpha_{n} k_{n} \beta_{n}}{1-(1-\alpha_{n} - \beta_{n} +\alpha_{n}\beta_{n})k_{n}} d(x_{n}, p) \\ & \quad{} + \frac{\alpha_{n}}{1-(1-\alpha_{n} - \beta_{n} +\alpha_{n}\beta _{n})k_{n}} d\bigl(f(p), p\bigr) \\ & = \biggl(1 + \frac{(k_{n} -1) - \alpha_{n} k_{n} + \alpha_{n} \gamma}{1-(1-\alpha_{n} - \beta_{n} +\alpha_{n}\beta_{n})k_{n}}\biggr) d(x_{n}, p) \\ & \quad{} + \frac{\alpha_{n}}{1-(1-\alpha_{n} - \beta_{n} +\alpha_{n}\beta _{n})k_{n}} d\bigl(f(p), p\bigr) \\ & \le\biggl(1 + \frac{(\alpha_{n} \epsilon- \alpha_{n} k_{n} + \alpha_{n} \gamma )}{1-(1-\alpha_{n} - \beta_{n} +\alpha_{n}\beta_{n})k_{n}}\biggr) d(x_{n}, p) \\ & \quad{} + \frac{\alpha_{n}}{1-(1-\alpha_{n} - \beta_{n} +\alpha_{n}\beta _{n})k_{n}} d\bigl(f(p), p\bigr) \\ & = \biggl(1 - \frac{k_{n} - \epsilon- \gamma}{1-(1-\alpha_{n} - \beta_{n} +\alpha _{n}\beta_{n})k_{n}}\biggr)\alpha_{n} d(x_{n}, p) \\ & \quad{} + \frac{\alpha_{n}}{1-(1-\alpha_{n} - \beta_{n} +\alpha_{n}\beta _{n})k_{n}} d\bigl(f(p), p\bigr) \\ & \le\biggl\{ 1- \frac{(1-\epsilon- \gamma)\alpha_{n}}{\alpha_{n} + \beta_{n} - \alpha_{n} \beta_{n}}\biggr\} d(x_{n}, p) \\ & \quad{} + \frac{(1 - \gamma- \epsilon)\alpha_{n}}{(1 - \gamma- \epsilon)(\alpha_{n} + \beta_{n} - \alpha_{n} \beta_{n}) } d\bigl(f(p), p\bigr) \\ & \le \max \biggl\{ d(x_{n}, p), \frac{d(f(p), p)}{1 - \gamma- \epsilon}\biggr\} . \end{aligned} $$
By induction we can prove that
$$d(x_{n}, p) \le \max\biggl\{ d(x_{1}, p), \frac{d(f(p), p)}{1 - \gamma- \epsilon} \biggr\} . $$
This implies that the sequence
\(\{x_{n}\}\) is bounded, so
\(\{y_{n}\}\),
\(\{ f(y_{n})\}\) and
\(\{T^{n}(\beta_{n} y_{n} \oplus (1- \beta_{n})y_{n+1})\}\) are also bounded.
(III)
Next we define a sequence
\(\{w_{n}\}\)
by
$$ w_{n} = \alpha_{n} f(J_{\lambda}w_{n}) \oplus(1- \alpha_{n})T^{n}(J_{\lambda}w_{n}), \quad\forall n \ge1. $$
(3.4)
By a similar method to the proof of Sect.
1, we can also prove that the sequence
\(\{w_{n}\}\) is well defined and bounded.
In fact, it follows from (
3.1) and (
3.4) that
$$\begin{aligned} d(x_{n+1}, w_{n}) &= d\bigl( \alpha_{n} f\bigl(J_{\lambda}(x_{n})\bigr) \oplus(1- \alpha _{n})T^{n}\bigl(\beta_{n} J_{\lambda}(x_{n}) \oplus(1- \beta_{n})J_{\lambda }(x_{n+1}) \bigr), \\ & \quad{}\alpha_{n} f(J_{\lambda}w_{n}) \oplus(1- \alpha_{n})T^{n}(J_{\lambda}w_{n})\bigr) \\ & \le\alpha_{n} d\bigl(f\bigl(J_{\lambda}(x_{n}) \bigr),f(J_{\lambda}w_{n})\bigr)\\ &\quad{} + (1- \alpha_{n}) d \bigl(T^{n}\bigl(\beta_{n} J_{\lambda}(x_{n}) \oplus(1- \beta_{n})J_{\lambda }(x_{n+1})\bigr), T^{n}(J_{\lambda}w_{n})\bigr) \\ & \le\alpha_{n} \gamma d(x_{n}, w_{n}) \\ &\quad{} + (1- \alpha_{n}) k_{n} \bigl\{ \beta_{n} d \bigl(J_{\lambda}(x_{n}), J_{\lambda}w_{n}\bigr)+ (1- \beta_{n}) d\bigl(J_{\lambda }(x_{n+1}), J_{\lambda}w_{n}\bigr)\bigr\} \\ & \le\alpha_{n} \gamma d(x_{n}, w_{n}) + (1- \alpha_{n}) k_{n} \bigl\{ \beta_{n} d(x_{n}, w_{n}) + (1- \beta_{n}) d(x_{n+1}, w_{n})\bigr\} . \end{aligned} $$
After simplifying, and using (
3.3), we have
$$ \begin{aligned}[b] d(x_{n+1}, w_{n}) & \le \frac{\alpha_{n} \gamma+\beta_{n} k_{n} - \alpha_{n} \beta_{n} k_{n}}{1 - (1 - \alpha_{n} - \beta_{n} + \alpha_{n} \beta_{n})k_{n}} d(x_{n}, w_{n}) \\ &= \biggl\{ 1 - \frac{-(k_{n} -1 -\alpha_{n} k_{n} + \alpha_{n} \gamma)}{1 - (1 - \alpha_{n} - \beta_{n} + \alpha_{n} \beta_{n})k_{n}}\biggr\} d(x_{n}, w_{n}) \\ &\le\biggl\{ 1 - \frac{-(\alpha_{n} \epsilon-\alpha_{n} k_{n} + \alpha_{n} \gamma )}{1 - (1 - \alpha_{n} - \beta_{n} + \alpha_{n} \beta_{n})k_{n}}\biggr\} d(x_{n}, w_{n}) \\ & = \biggl\{ 1 - \frac{(k_{n} - \gamma-\epsilon)\alpha_{n}}{1 - (1 - \alpha_{n} - \beta_{n} + \alpha_{n} \beta_{n})k_{n}}\biggr\} d(x_{n}, w_{n}) \\ & \le\biggl(1 - \frac{(1 - \gamma-\epsilon)\alpha_{n}}{\alpha_{n} + \beta_{n} - \alpha_{n} \beta_{n}}\biggr)d(x_{n}, w_{n}) \\ & \le\bigl(1 - (1 - \gamma-\epsilon)\alpha_{n}\bigr) \bigl[d(x_{n}, w_{n-1}) + d( w_{n-1}, w_{n}) \bigr]. \end{aligned} $$
(3.6)
In order to use Lemma
2.9, it should be proved that
$$ \limsup_{n \to\infty} \frac{d(w_{n-1}, w_{n})}{(1 - \gamma-\epsilon )\alpha_{n}} = 0. $$
(3.7)
Indeed, it follows from Lemma
2.1 that
$$\begin{aligned} d(w_{n}, w_{n -1}) & = d\bigl( \alpha_{n} f(J_{\lambda}w_{n}) \oplus(1- \alpha _{n})T^{n}(J_{\lambda}w_{n}), \\ & \quad{} \alpha_{n-1} f(J_{\lambda}w_{n-1}) \oplus(1- \alpha _{n-1})T^{{n-1}}(J_{\lambda}w_{n-1})\bigr) \\ & \le d\bigl(\alpha_{n} f(J_{\lambda}w_{n}) \oplus(1- \alpha_{n})T^{n}(J_{\lambda}w_{n}), \alpha_{n} f(J_{\lambda}w_{n}) \oplus(1- \alpha_{n})T^{n}(J_{\lambda}w_{n-1})\bigr) \\ & \quad{} + d\bigl( \alpha_{n} f(J_{\lambda}w_{n}) \oplus(1- \alpha _{n})T^{n}(J_{\lambda}w_{n-1}), \alpha_{n} f(J_{\lambda}w_{n-1}) \oplus(1- \alpha_{n})T^{n}(J_{\lambda}w_{n-1}) \bigr) \\ & \quad{} + d\bigl(\alpha_{n} f(J_{\lambda}w_{n-1}) \oplus(1- \alpha _{n})T^{n}(J_{\lambda}w_{n-1}), \\ & \quad{} \alpha_{n-1} f(J_{\lambda}w_{n-1}) \oplus(1- \alpha _{n-1})T^{{n-1}}(J_{\lambda}w_{n-1})\bigr) \\ &\le (1- \alpha_{n-1})d\bigl(T^{n}(J_{\lambda}w_{n}), T^{{n}}(J_{\lambda}w_{n-1})\bigr) \\ & \quad{} + \alpha_{n} d\bigl( f(J_{\lambda}w_{n}), f(J_{\lambda}w_{n-1})\bigr) + \vert \alpha_{n} - \alpha_{n-1} \vert d\bigl(f(J_{\lambda}w_{n-1}), T^{n}(J_{\lambda}w_{n-1})\bigr) \\ &\le(1- \alpha_{n-1})k_{n} d( w_{n}, w_{n-1}) + \alpha_{n} \gamma d( w_{n}, w_{n-1}) + \vert \alpha_{n} - \alpha_{n-1} \vert M^{*}, \end{aligned}$$
where
\(M^{*} = \sup_{n \ge1} d(f(J_{\lambda}w_{n-1}), T^{n}(J_{\lambda}w_{n-1}))\). After simplifying and using (
3.3) we have
$$\begin{aligned} d(w_{n}, w_{n -1}) & \le \frac{1}{-(k_{n} -1 - \alpha_{n} k_{n} + \alpha_{n} \gamma)} \vert \alpha_{n} - \alpha_{n-1} \vert M^{*} \\ & \le\frac{1}{-(\epsilon- k_{n} + \gamma)\alpha_{n}} \vert \alpha_{n} - \alpha _{n-1} \vert M^{*} \\ & \le\frac{1}{(1 - \epsilon- \gamma)\alpha_{n}} \vert \alpha_{n} - \alpha _{n-1} \vert M^{*}. \end{aligned} $$
By the condition (iv) we have
$$\limsup_{n \to\infty} \frac{d(w_{n}, w_{n -1})}{(1 - \epsilon- \gamma )\alpha_{n}} \le\limsup _{n \to\infty} \frac{ \vert \alpha_{n} - \alpha _{n-1} \vert }{(1 - \epsilon- \gamma)^{2}\alpha_{n}^{2}} M^{*} = 0. $$
This implies that (
3.7) is true. By Lemma
2.9 and (
3.6), we get
$$ \lim_{n \to\infty} d(x_{n+1}, w_{n}) = 0. $$
(3.8)
In fact, it follows from (
3.1) and (
3.4) that
$$ \begin{aligned}[b] d\bigl(w_{n}, T^{n} J_{\lambda}(w_{n})\bigr)& = d\bigl(\alpha_{n} f(J_{\lambda}w_{n}) \oplus(1- \alpha_{n})T^{n}(J_{\lambda}w_{n}),T^{n} \bigl(J_{\lambda}(w_{n})\bigr)\bigr) \\ & \le\alpha_{n} d\bigl( f(J_{\lambda}w_{n}), T^{n} \bigl(J_{\lambda}(w_{n})\bigr)\bigr) \to0. \end{aligned} $$
(3.10)
Also for each
\(p \in\Gamma\), it follows from (
2.4) that
$$ \begin{aligned}[b] d^{2}(w_{n}, p) & = d^{2} \bigl(\alpha_{n} f(J_{\lambda}w_{n}) \oplus(1- \alpha _{n})T^{n}(J_{\lambda}w_{n}), p\bigr) \\ & \le\alpha_{n} d^{2}\bigl(f(J_{\lambda}w_{n}), p\bigr) + (1-\alpha_{n}) d^{2} \bigl(T^{n}(J_{\lambda}w_{n}), p\bigr) \\ & \le\alpha_{n} d^{2}\bigl(f(J_{\lambda}w_{n}), p\bigr) + (1-\alpha_{n})k_{n}^{2} d^{2}\bigl(J_{\lambda}(w_{n}), p\bigr). \end{aligned} $$
(3.11)
After simplifying, we have
$$ - d^{2}\bigl(J_{\lambda}(w_{n}), p\bigr)\le\frac{1}{(1-\alpha_{n})k_{n}^{2}} \bigl\{ \alpha_{n} d^{2}\bigl(f(J_{\lambda}w_{n}), p\bigr) - d^{2}( w_{n}, p)\bigr\} . $$
(3.12)
Again since
\(J_{\lambda}\) is firmly nonexpansive, we have
$$\begin{aligned} d^{2}\bigl(J_{\lambda}(w_{n}), p\bigr) &\le\bigl\langle \overrightarrow{J_{\lambda}(w_{n})p} , \overrightarrow{w_{n} p} \bigr\rangle \\ & = \frac{1}{2} \bigl\{ d^{2}\bigl(J_{\lambda}(w_{n}), p\bigr) + d^{2}(p, w_{n}) - d^{2} \bigl(J_{\lambda}(w_{n}), w_{n}\bigr)\bigr\} . \end{aligned} $$
This together with (
3.12) shows that
$$ \begin{aligned}[b] d^{2}\bigl(J_{\lambda}(w_{n}), w_{n}\bigr) & \le d^{2}(p, w_{n}) - d^{2} \bigl(J_{\lambda}(w_{n}), p\bigr) \\ & \le d^{2}(p, w_{n}) + \frac{1}{(1-\alpha_{n})k_{n}^{2}} \bigl\{ \alpha_{n} d^{2}\bigl(f(J_{\lambda}w_{n}), p \bigr) - d^{2}(w_{n}, p)\bigr\} \to0. \end{aligned} $$
(3.13)
From (
3.10) and (
3.13) one gets
$$ \lim_{n \to\infty} d\bigl(J_{\lambda}(w_{n}), T^{n} J_{\lambda}(w_{n})\bigr) = 0. $$
(3.14)
By the assumption that
T is uniformly asymptotic regularity, from (
3.14) we obtain
$$ \begin{aligned}[b] d\bigl(J_{\lambda}(w_{n}), TJ_{\lambda}(w_{n})\bigr)& \le d\bigl(J_{\lambda}(w_{n}), T^{n} J_{\lambda}(w_{n})\bigr) + d\bigl(T^{n} J_{\lambda}(w_{n}), T^{n+1} J_{\lambda}(w_{n}) \bigr) \\ & \quad{} + d\bigl(T^{n+1} J_{\lambda}(w_{n}), TJ_{\lambda}(w_{n})\bigr) \\ & \le(1 + k_{1})d\bigl(J_{\lambda}(w_{n}), T^{n} J_{\lambda}(w_{n})\bigr) + d\bigl(T^{n} J_{\lambda}(w_{n}), T^{n+1} J_{\lambda}(w_{n})\bigr) \\&\to0 \quad(\text{as } n \to\infty). \end{aligned} $$
(3.15)
In fact, it follows from Lemma
2.1(4) that
$$\begin{aligned} d^{2}\bigl(w_{n}, x^{*}\bigr) & = d^{2}\bigl(\alpha_{n} f(J_{\lambda}w_{n}) \oplus(1- \alpha _{n})T^{n}(J_{\lambda}w_{n}), x^{*}\bigr) \\ &\le\alpha_{n}^{2} d^{2}\bigl(f(J_{\lambda}w_{n}), x^{*}\bigr) + (1-\alpha_{n})^{2} d^{2}\bigl(T^{n}(J_{\lambda}w_{n}), x^{*}\bigr) \\ & \quad{} + 2\alpha_{n} (1-\alpha_{n}) \bigl\langle \overrightarrow{f(J_{\lambda}w_{n}) x^{*}}, \overrightarrow{T^{n}(J_{\lambda}w_{n}) x^{*}} \bigr\rangle \\ &\le\alpha_{n}^{2} d^{2}\bigl(f(J_{\lambda}w_{n}), x^{*}\bigr) + (1-\alpha_{n})^{2} k_{n}^{2} d^{2}\bigl(J_{\lambda}w_{n}, x^{*}\bigr) \\ & \quad{} + 2\alpha_{n} (1-\alpha_{n}) \bigl\{ \bigl\langle \overrightarrow {f(J_{\lambda}w_{n}) x^{*}}, \overrightarrow{T^{n} \bigl(J_{\lambda}(w_{n})\bigr)J_{\lambda}(w_{n})} \bigr\rangle \\ & \quad{} + \bigl\langle \overrightarrow{ f(J_{\lambda}w_{n}) f \bigl(x^{*}\bigr)}, \overrightarrow{J_{\lambda}(w_{n})x^{*}} \bigr\rangle + \bigl\langle \overrightarrow {f\bigl(x^{*}\bigr) x^{*}}, \overrightarrow{J_{\lambda}(w_{n})x^{*}} \bigr\rangle \bigr\} \\ &\le\alpha_{n}^{2} d^{2}\bigl(f(J_{\lambda}w_{n}), x^{*}\bigr) + (1-\alpha_{n})^{2} k_{n}^{2} d^{2}\bigl(w_{n}, x^{*}\bigr) \\ & \quad{} + 2\alpha_{n} (1-\alpha_{n})\bigl\{ d \bigl(f(J_{\lambda}w_{n}), x^{*}\bigr) d\bigl(T^{n} \bigl(J_{\lambda}(w_{n})\bigr), J_{\lambda}(w_{n}) \bigr) \\ & \quad{} + \gamma d^{2}\bigl(w_{n}, x^{*}\bigr) + \bigl\langle \overrightarrow{f\bigl(x^{*}\bigr) x^{*}}, \overrightarrow{J_{\lambda}(w_{n})x^{*}} \bigr\rangle \bigr\} . \end{aligned}$$
After simplifying, we have
$$ \begin{aligned}[b] d^{2}\bigl(w_{n}, x^{*}\bigr) & \le\frac{\alpha_{n}}{1 - 2\alpha_{n}(1-\alpha_{n})\gamma- (1- \alpha_{n})^{2} k_{n}^{2}} \bigl\{ \alpha_{n} d^{2} \bigl(f(J_{\lambda}w_{n}), x^{*}\bigr) \\ & \quad{} + 2(1-\alpha_{n}) \bigl[ d\bigl(f(J_{\lambda}w_{n}), x^{*}\bigr) d\bigl(T^{n}\bigl(J_{\lambda}(w_{n})\bigr), J_{\lambda}(w_{n})\bigr) \\ & \quad{} + \bigl\langle \overrightarrow{f\bigl(x^{*}\bigr) x^{*}}, \overrightarrow {J_{\lambda}(w_{n})x^{*}} \bigr\rangle \bigr]\bigr\} . \end{aligned} $$
(3.16)
We use
$$ \begin{aligned}[b] & \frac{\alpha_{n}}{1 - 2\alpha_{n}(1-\alpha_{n})\gamma- (1- \alpha_{n})^{2} k_{n}^{2}} \\ &\quad= \frac{1}{(2-\alpha_{n})k_{n}^{2} - 2(1-\alpha_{n})\gamma+ \frac{(1 - k_{n})(1 + k_{n})}{\alpha_{n}}} \to\frac{1}{2(1-\gamma)} \quad(\text{as } n \to\infty). \end{aligned} $$
(3.17)
Again since
\(\{J_{\lambda}(w_{n_{i}})\}\) Δ-converges to
\(x^{*} \in \Gamma\), by Lemma
2.2(2), we have
$$ \lim_{n_{i} \to\infty}\bigl\langle \overrightarrow{f\bigl(x^{*}\bigr) x^{*}}, \overrightarrow{J_{\lambda}(w_{n_{i}})x^{*}} \bigr\rangle = \limsup _{n_{i} \to \infty}\bigl\langle \overrightarrow{f\bigl(x^{*}\bigr) x^{*}}, \overrightarrow{J_{\lambda}(w_{n_{i}})x^{*}} \bigr\rangle \le0. $$
(3.18)
It follows from (
3.14), (
3.16), (
3.17) and (
3.18) that
$$ \lim_{n_{i} \to\infty} d\bigl(w_{n_{i}}, x^{*}\bigr) =0. $$
(3.19)
Next we prove that
\(x^{*}\) is a solution of variational inequality (
3.9). In fact, for any
\(q \in\Gamma\), it follows from Lemma
2.1(4) that (for the sake of convenience we denote
\(\{w_{n_{i}}\}\) by
\(\{w_{i}\}\))
$$\begin{aligned} d^{2}(w_{i},q) & = d^{2}\bigl(\alpha_{n} f(J_{\lambda}w_{i}) \oplus(1- \alpha _{i})T^{i}(J_{\lambda}w_{i}), q\bigr) \\ & \le\alpha_{i} d^{2}\bigl( f(J_{\lambda}w_{i}), q\bigr) + (1- \alpha_{i}) d^{2}\bigl( T^{i}(J_{\lambda}w_{i}), q\bigr) \\ & \quad{} -\alpha_{i} (1- \alpha_{i}) d^{2} \bigl(f(J_{\lambda}w_{i}),T^{i}(J_{\lambda}w_{i})\bigr) \\ & \le\alpha_{i} d^{2}\bigl( f(J_{\lambda}w_{i}), q\bigr) + (1- \alpha_{i})k_{i}^{2} d^{2}( w_{i}, q) \\ & \quad{} -\alpha_{i} (1- \alpha_{i}) d^{2} \bigl(f(J_{\lambda}w_{i}),T^{i}(J_{\lambda}w_{i})\bigr). \end{aligned} $$
After simplifying we have
$$ d^{2}(w_{i},q) \le\frac{1}{\frac{1-k_{i}^{2}}{\alpha_{i}} + k_{i}^{2}}\bigl\{ d^{2} \bigl( f(J_{\lambda}w_{i}), q\bigr) - (1- \alpha_{i}) d^{2}\bigl(f(J_{\lambda}w_{i}),T^{i}(J_{\lambda}w_{i})\bigr)\bigr\} . $$
(3.20)
On the other hand, it follows from (
3.19) and (
3.13) that
\(w_{i} \to x^{*}\) and
\(J_{\lambda}(w_{i}) \to x^{*}\) (as
\(i \to\infty\)). Hence
\(f(J_{\lambda}(w_{i})) \to f(x^{*})\). Again by (
3.14) and condition (iii),
\(T^{i}(J_{\lambda}w_{i}) \to x^{*}\) and
\(\frac{1}{\frac{1-k_{i}^{2}}{\alpha_{i}} + k_{i}^{2}} \to1\) (as
\(i \to\infty\)). Letting
\(i \to\infty\) in (
3.20) we have
$$d^{2}\bigl(x^{*},q\bigr) \le d^{2}\bigl(f\bigl(x^{*}\bigr), q \bigr) - d^{2}\bigl(f\bigl(x^{*}\bigr), x^{*}\bigr), $$
i.e.,
$$0 \le d^{2}\bigl(f\bigl(x^{*}\bigr), q\bigr) - d^{2}\bigl(f \bigl(x^{*}\bigr), x^{*}\bigr) - d^{2}\bigl(x^{*},q\bigr). $$
Hence we have
$$\bigl\langle \overrightarrow{x^{*} f\bigl(x^{*}\bigr)}, \overrightarrow{q x^{*}}\bigr\rangle = \frac{1}{2}\bigl\{ d^{2}\bigl(f\bigl(x^{*}\bigr), q\bigr) - d^{2}\bigl(f\bigl(x^{*}\bigr), x^{*}\bigr) - d^{2}\bigl(x^{*},q \bigr)\bigr\} \ge0,\quad \forall q \in\Gamma, $$
i.e.,
\(x^{*}\) is a solution of variational inequality (
3.9). If there exists another subsequence
\(\{w_{n_{k}}\}\) of
\(\{w_{n}\}\) which Δ-converges to
\(y^{*}\). By the same argument, we know that
\(y^{*} \in\Gamma\) which solves the variational inequality (
3.9). Therefore we have
$$\begin{gathered} \bigl\langle \overrightarrow{x^{*} f\bigl(x^{*}\bigr)}, \overrightarrow{y^{*} x^{*}}\bigr\rangle \ge0, \\ \bigl\langle \overrightarrow{y^{*} f\bigl(y^{*}\bigr)}, \overrightarrow{x^{*} y^{*}} \bigr\rangle \ge0. \end{gathered} $$
Adding the above two inequalities, we obtain
$$\begin{aligned} 0 & \le \bigl\langle \overrightarrow{x^{*} f\bigl(x^{*} \bigr)}, \overrightarrow{y^{*} x^{*}} \bigr\rangle - \bigl\langle \overrightarrow{y^{*} f \bigl(y^{*}\bigr)}, \overrightarrow{ y^{*}x^{*}}\bigr\rangle \\ & = \bigl\langle \overrightarrow{x^{*} f\bigl(y^{*}\bigr)}, \overrightarrow{y^{*} x^{*}} \bigr\rangle + \bigl\langle \overrightarrow{f\bigl(y^{*}\bigr)f\bigl(x^{*}\bigr)}, \overrightarrow{y^{*} x^{*}}\bigr\rangle \\ & \quad{} - \bigl\langle \overrightarrow{y^{*} x^{*}}, \overrightarrow{y^{*} x^{*}} \bigr\rangle - \bigl\langle \overrightarrow{x^{*} f\bigl(y^{*}\bigr)}, \overrightarrow{y^{*} x^{*}} \bigr\rangle \\ & \le\gamma d^{2}\bigl(y^{*}, x^{*}\bigr) - d^{2}\bigl(y^{*}, x^{*} \bigr) < 0. \end{aligned} $$
This is a contradiction, and so
\(x^{*} = y^{*}\). Hence
\(\{w_{n}\}\) converges strongly to
\(x^{*}\). By (
3.8) one shows that
\(\{x_{n}\}\) converges strongly to
\(x^{*}\).