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A new iterative process for a hybrid pair of generalized asymptotically nonexpansive single-valued and generalized nonexpansive multi-valued mappings in Banach spaces
Fixed Point Theory and Applications volume 2015, Article number: 58 (2015)
Abstract
In this paper, we construct an iterative process involving a hybrid pair of a finite family of generalized asymptotically nonexpansive single-valued mappings and a finite family of generalized nonexpansive multi-valued mappings and prove weak and strong convergence theorems of the proposed iterative process in Banach spaces. Our main results extend and generalize many results in the literature.
1 Introduction
Throughout this paper we denote by \(\mathbb{N}\) the set of all positive integers. Let X be a Banach space and let D be a nonempty subset of X. Let \(CB(D)\) and \(KC(D)\) denote the families of nonempty, closed, and bounded subsets and nonempty, compact, and convex subsets of D, respectively. The Hausdorff metric on \(CB(D)\) is defined by
where \(\operatorname{dist}(x,D)=\inf\{\|x-y\|:y\in D\}\) is the distance from a point x to a subset D. Let t be a single-valued mapping of D into D and T be a multi-valued mapping of D into \(CB(D)\). The set of fixed points of t and T will be denoted by \(F(t)=\{x\in D:x=tx\}\) and \(F(T)=\{x\in D:x\in Tx\}\), respectively. A point x is called a common fixed point of t and T if \(x=tx\in Tx\).
Definition 1.1
A single-valued mapping \(t:D\to D\) is said to be generalized asymptotically nonexpansive if there exist sequences \(\{k_{n}\}\subset[1,\infty)\) and \(\{s_{n}\}\subset[0,\infty)\) with \(\lim_{n\to\infty}k_{n} =1\), \(\lim_{n\to\infty}s_{n} =0\) such that
for all \(x,y\in D\) and \(n\in\mathbb{N}\).
In the case of \(s_{n}=0\), for all \(n\in\mathbb{N}\), a single-valued mapping t is called an asymptotically nonexpansive mapping. In particular, if \(k_{n}=1\) and \(s_{n}=0\), for all \(n\in\mathbb{N}\), a single-valued mapping t reduce to a nonexpansive mapping. The fixed point property for generalized asymptotically nonexpansive single-valued mappings can be found in [1]. The following example shows that the fixed point set of a generalized asymptotically nonexpansive mapping is not necessarily closed; see also [2].
Example 1.2
([1])
Define a single-valued mapping \(t: [-\frac{2}{3},\frac{2}{3} ]\to [-\frac{2}{3},\frac{2}{3} ]\) by
Then t is generalized asymptotically nonexpansive and \(F(t)=[ { - \frac{2}{3},0} )\) which is not closed.
Definition 1.3
A multi-valued mapping \(T:D\to CB(D)\) is said to be
-
(i)
nonexpansive if \(H(Tx,Ty)\leq\|x-y\|\), for all \(x,y\in D\);
-
(ii)
quasi-nonexpansive if \(F(T)\ne\emptyset\) and \(H(Tx,Tp)\leq\|x-p\|\), for all \(x\in D\) and \(p\in F(T)\).
The study of fixed points for nonexpansive multi-valued mappings using the Hausdorff metric was initiated by Markin [3]. Different iterative processes have been used to approximate fixed points of nonexpansive and quasi-nonexpansive multi-valued mappings; in particular, Sastry and Babu [4] considered Mann and Ishikawa iterates for a multi-valued mapping T with a fixed point p and proved that these iterates converge to a fixed point q of T under certain conditions. Moreover, they illustrated that the fixed point q may be different from p. Later in 2007, Panyanak [5] generalized results of Sastry and Babu [4] to uniformly convex Banach spaces and proved a convergence theorem of Mann iterates for a mapping defined on a noncompact domain. In 2009, Shahzad and Zegeye [6] proved strong convergence theorems for the Ishikawa iteration scheme involving quasi-nonexpansive multi-valued mappings. They constructed an iterative process which removes the restriction of T, namely end-point condition, i.e., \(Tp = \{p\}\) for any \(p\in F(T)\); see also [7, 8].
In 2011, Garcia-Falset et al. [9] introduced a new condition on single-valued mappings, called condition (E), which is weaker than nonexpansiveness. Later, Abkar and Eslamian [10] used a modified condition for multi-valued mappings as follows.
Definition 1.4
A multi-valued mapping \(T:D\to CB(D)\) is said to satisfy condition (\(E_{\mu}\)) where \(\mu\geq0\) if for each \(x,y\in D\),
We say that T satisfies condition (E) whenever T satisfies (\(E_{\mu}\)) for some \(\mu\geq1\).
Remark 1.5
From the above definitions, it is clear that if T is nonexpansive, then T satisfies the condition (\(E_{1}\)).
In 2011, Sokhuma and Kaewkhao [11] introduced the following iterative process for approximating a common fixed point of a pair of a nonexpansive single-valued mapping t and a nonexpansive multi-valued mapping T:
where \(x_{1}\in D\), \(z_{n}\in Tx_{n}\), and \(0< a\leq\alpha_{n},\beta_{n}\leq b<1\). They also proved a strong convergence theorem for the iterative process (1.1) in uniformly convex Banach spaces.
In 2013, Eslamian [12] extended the results of [11, 13] in uniformly convex Banach spaces. He used the following iterative process for a pair of a finite family of asymptotically nonexpansive single-valued mappings \(\{t_{i}\}_{i=1}^{N}\) and a finite family of quasi-nonexpansive multi-valued mapping \(\{T_{i}\}_{i=1}^{N}\):
where \(x_{1}\in D\), \(z_{n}^{(i)}\in T_{i}x_{n}\), and \(\{\alpha_{n}^{(i)}\}\), \(\{\beta_{n}^{(i)}\}\) are sequences in \([0,1]\) for all \(i=1,2,\ldots,N\) such that \(\sum_{i=0}^{N}\alpha_{n}^{(i)}=\sum_{i=0}^{N}\beta_{n}^{(i)}=1\).
In this paper, motivated by the above results, we propose an iterative process for approximating a common fixed point of a pair of a finite family of generalized asymptotically nonexpansive single-valued mappings and a finite family of quasi-nonexpansive multi-valued mappings and prove weak and strong convergence theorems of the proposed iterative process in Banach spaces.
2 Preliminaries
A Banach space X is called uniformly convex if for each \(\varepsilon>0\) there is a \(\delta>0\) such that for \(x,y \in X\) with \(\Vert x \Vert \leq1\), \(\Vert y \Vert \leq1\), and \(\Vert x - y \Vert \geq\varepsilon\), \(\Vert x + y \Vert \leq2(1-\delta)\) holds. The following result was proved by Xu [14].
Proposition 2.1
Let X be a uniformly convex Banach space and let \(r > 0\). Then there exists a strictly increasing, continuous, and convex function \(g: [0,\infty) \to[0,\infty)\) with \(g(0)=0\) such that
for all \(x,y\in B_{r}=\{z\in X:\|z\|\leq r\}\) and \(\lambda\in[0,1]\).
A Banach space X is said to satisfy the Opial property (see [15]) if it is given that whenever \(\{x_{n}\}\) converges weakly to \(x\in X\),
for each \(y\in X\) with \(y \ne x\). The examples of Banach spaces which satisfy the Opial property are Hilbert spaces and all \(L^{p}[0, 2\pi]\) with \(1 < p \ne2\) fail to satisfy the Opial property.
The following results are needed for proving our results.
Definition 2.2
(see [2])
Let F be a nonempty subset of a Banach space X and let \(\{x_{n}\}\) be a sequence in X. We say that \(\{x_{n}\}\) is of monotone type (I) with respect to F if there exist sequences \(\{\delta_{n}\}\) and \(\{\varepsilon_{n}\}\) of nonnegative real numbers such that \(\sum_{n = 1}^{\infty}{\delta_{n} } < \infty\), \(\sum_{n = 1}^{\infty}{\varepsilon_{n} } < \infty\), and \(\|x_{n+1}-p\| \leq (1+\delta_{n})\|x_{n}-p\| + \varepsilon_{n}\) for all \(n\in\mathbb{N}\) and \(p\in F\).
Proposition 2.3
(see [2])
Let F be a nonempty subset of a Banach space X and let \(\{x_{n}\}\) be a sequence in X. If \(\{x_{n}\}\) is of monotone type (I) with respect to F and \(\liminf_{n\to\infty} \operatorname{dist}(x_{n},F) = 0\), then \(\lim_{n\to\infty} x_{n} = p\) for some \(p\in X\) satisfying \(\operatorname{dist}(p,F)=0\). In particular, if F is closed, then \(p\in F\).
Lemma 2.4
(see [16])
Let \(\{a_{n}\}\), \(\{b_{n}\}\), and \(\{c_{n}\}\) be sequences of nonnegative real numbers satisfy
where \(\sum_{n=1}^{\infty}b_{n}<\infty\) and \(\sum_{n=1}^{\infty}c_{n}<\infty\). Then:
-
(i)
\(\lim_{n\to\infty}a_{n}\) exists.
-
(ii)
If \(\liminf_{n\to\infty}a_{n} =0\), then \(\lim_{n\to\infty}a_{n}=0\).
Lemma 2.5
(see [17])
Let X be a uniformly convex Banach space, let \(\{\lambda_{n}\}\) be a sequence of real numbers such that \(0< a\leq\lambda_{n}\leq b<1\), for all \(n\in\mathbb{N}\), and let \(\{x_{n}\}\) and \(\{y_{n}\}\) be sequences of X satisfying, for some \(r\geq0\),
-
(i)
\(\limsup_{n\to\infty}\|x_{n}\|\leq r\),
-
(ii)
\(\limsup_{n\to\infty}\|y_{n}\|\leq r\) and
-
(iii)
\(\lim_{n\to\infty}\|\lambda_{n} x_{n} +(1-\lambda_{n})y_{n}\|= r\).
Then \(\lim_{n\to\infty}\|x_{n}-y_{n}\|=0\).
Lemma 2.6
(see [18])
Let X be a Banach space which satisfies the Opial property and \(\{x_{n}\}\) be a sequence in X. Let \(u, v\in X\) be such that \(\lim_{n\to\infty}\|x_{n} - u\|\) and \(\lim_{n\to\infty}\|x_{n} - v\|\) exist. If \(\{x_{n_{i}}\}\) and \(\{x_{n_{j}}\}\) are subsequences of \(\{x_{n}\}\) which converge weakly to u and v, respectively, then \(u = v\).
3 Main results
In this section, we prove weak and strong convergence theorems of the proposed iterative process in Banach spaces. We first note that if \(\{t_{i}\}_{i=1}^{N}\) is a finite family of generalized asymptotically nonexpansive single-valued mappings of D into itself, where D is a nonempty convex subset of a Banach space X. Then we have \(\|t_{i}^{n}x - t_{i}^{n}y\|\leq k_{n}^{(i)}\|x-y\|+s_{n}^{(i)}\), for all \(x,y\in D\) and all \(i=1,2,\ldots,N\), where \(\{k_{n}^{(i)}\}\subset[1,\infty)\) and \(\{s_{n}^{(i)}\}\subset [0,\infty)\) with \(\lim_{n\to\infty}k_{n}^{(i)}=1\) and \(\lim_{n\to\infty}s_{n}^{(i)}=0\). Put \(k_{n}=\max_{1\leq i\leq N}\{k_{n}^{(i)}\}\) and \(s_{n}=\max_{1\leq i\leq N}\{s_{n}^{(i)}\}\). It is clear that \(\lim_{n\to\infty}k_{n}=1\) and \(\lim_{n\to\infty}s_{n}=0\) and
for all \(x,y\in D\), \(i=1,2,\ldots,N\), and all \(n\in\mathbb{N}\).
In order to prove our main results, the following lemma is needed.
Lemma 3.1
Let D be a nonempty, closed, and convex subset of a Banach space X. Let \(\{t_{i}\}_{i=1}^{N}\) be a finite family of generalized asymptotically nonexpansive single-valued mappings of D into itself with sequences \(\{k_{n}\}\subset[1,\infty)\) and \(\{s_{n}\}\subset[0,\infty)\) such that \(\sum_{n=1}^{\infty}(k_{n}-1)<\infty\) and \(\sum_{n=1}^{\infty}s_{n}<\infty\). Let \(\{T_{i}\}_{i=1}^{N}\) be a finite family of quasi-nonexpansive multi-valued mappings of D into \(CB(D)\). Assume that \(\mathcal{F}=\bigcap_{i=1}^{N} F(t_{i}) \cap\bigcap_{i=1}^{N} F(T_{i})\) is nonempty closed and \(T_{i}p=\{p\}\) for all \(p\in\mathcal{F}\) and \(i=1,2,\ldots,N\). Let \(x_{1}\in D\) and the sequence \(\{x_{n}\}\) be generated by
where \(\{\alpha_{n}^{(i)}\}\) and \(\{\beta_{n}^{(i)}\}\) are sequences in \([0,1]\) for all \(i=1,2,\ldots,N\) such that \(\sum_{i=0}^{N}\alpha_{n}^{(i)}=1\) and \(\sum_{i=0}^{N}\beta_{n}^{(i)}=1\). Then \(\lim_{n\to\infty} \|x_{n}-p\|\) exists for all \(p\in \mathcal{F}\).
Proof
Let \(p\in \mathcal{F}\), for \(i=1,2,\ldots,N\), we have
By Lemma 2.4, \(\sum_{n=1}^{\infty}(k_{n}-1)<\infty\) and \(\sum_{n=1}^{\infty}s_{n}<\infty\), we conclude that \(\lim_{n\to\infty} \|x_{n}-p\|\) exists for all \(p\in\mathcal{F}\). □
Theorem 3.2
Let D be a nonempty, closed, and convex subset of a Banach space X. Let \(\{t_{i}\}_{i=1}^{N}\) be a finite family of generalized asymptotically nonexpansive single-valued mappings of D into itself with sequences \(\{k_{n}\}\subset[1,\infty)\) and \(\{s_{n}\}\subset[0,\infty)\) such that \(\sum_{n=1}^{\infty}(k_{n}-1)<\infty\) and \(\sum_{n=1}^{\infty}s_{n}<\infty\). Let \(\{T_{i}\}_{i=1}^{N}\) be a finite family of quasi-nonexpansive multi-valued mappings of D into \(CB(D)\). Assume that \(\mathcal{F}=\bigcap_{i=1}^{N} F(t_{i}) \cap\bigcap_{i=1}^{N} F(T_{i})\) is nonempty closed and \(T_{i}p=\{p\}\) for all \(p\in\mathcal{F}\) and \(i=1,2,\ldots,N\). Let \(x_{1}\in D\) and the sequence \(\{x_{n}\}\) be generated by
where \(\{\alpha_{n}^{(i)}\}\) and \(\{\beta_{n}^{(i)}\}\) are sequences in \([0,1]\) for all \(i=1,2,\ldots,N\) such that \(\sum_{i=0}^{N}\alpha_{n}^{(i)}=1\) and \(\sum_{i=0}^{N}\beta_{n}^{(i)}=1\). Then the sequence \(\{x_{n}\}\) converges strongly to a point in \(\mathcal{F}\) if and only if \(\liminf_{n\to\infty} \operatorname{dist}(x_{n},\mathcal{F})=0\).
Proof
The necessity is obvious and thus we prove only the sufficiency. Suppose that \(\liminf_{n\to\infty} \operatorname{dist}(x_{n},\mathcal{F})=0\). In the proof of Lemma 3.1, we see that the sequence \(\{x_{n}\}\) is of monotone type (I) with respect to \(\mathcal{F}\). It follows by Proposition 2.3 that \(\{x_{n}\}\) converges to a point in \(\mathcal{F}\). □
The closedness of \(\mathcal{F}=\bigcap_{i=1}^{N} F(t_{i}) \cap \bigcap_{i=1}^{N} F(T_{i})\) can be dropped if \(t_{i}\) is asymptotically nonexpansive for all \(i = 1,2,\ldots,N\). Then the following corollary is obtained directly from Theorem 3.2.
Corollary 3.3
Let D be a nonempty, closed, and convex subset of a Banach space X. Let \(\{t_{i}\}_{i=1}^{N}\) be a finite family of asymptotically nonexpansive single-valued mappings of D into itself with a sequence \(\{k_{n}\}\subset[1,\infty)\) such that \(\sum_{n=1}^{\infty}(k_{n}-1)<\infty\). Let \(\{T_{i}\}_{i=1}^{N}\) be a finite family of quasi-nonexpansive multi-valued mappings of D into \(CB(D)\). Assume that \(\mathcal{F}=\bigcap_{i=1}^{N} F(t_{i}) \cap\bigcap_{i=1}^{N} F(T_{i})\) is nonempty and \(T_{i}p=\{p\}\) for all \(p\in\mathcal{F}\) and \(i=1,2,\ldots,N\). Let \(x_{1}\in D\) and the sequence \(\{x_{n}\}\) be generated by
where \(\{\alpha_{n}^{(i)}\}\) and \(\{\beta_{n}^{(i)}\}\) are sequences in \([0,1]\) for all \(i=1,2,\ldots,N\) such that \(\sum_{i=0}^{N}\alpha_{n}^{(i)}=1\) and \(\sum_{i=0}^{N}\beta_{n}^{(i)}=1\). Then the sequence \(\{x_{n}\}\) converges strongly to a point in \(\mathcal{F}\) if and only if \(\liminf_{n\to\infty} \operatorname{dist}(x_{n},\mathcal{F})=0\).
Recall that a mapping \(t : D \to D\) is called uniformly L-Lipschitzian if there exists a constant \(L>0\) such that \(\|t^{n} x-t^{n} y\| \leq L \|x-y\|\) for all \(x,y \in D\) and \(n\in\mathbb{N}\). Next, we prove a strong convergence theorem in a uniformly convex Banach space.
Lemma 3.4
Let D be a nonempty, closed, and convex subset of a uniformly convex Banach space X. Let \(\{t_{i}\}_{i=1}^{N}\) be a finite family of uniformly L-Lipschitzian and generalized asymptotically nonexpansive single-valued mappings of D into itself with sequences \(\{k_{n}\}\subset[1,\infty)\) and \(\{s_{n}\}\subset [0,\infty)\) such that \(\sum_{n=1}^{\infty}(k_{n}-1)<\infty\) and \(\sum_{n=1}^{\infty}s_{n}<\infty\). Let \(\{T_{i}\}_{i=1}^{N}\) be a finite family of quasi-nonexpansive multi-valued mappings of D into \(CB(D)\). Assume that \(\mathcal{F}=\bigcap_{i=1}^{N} F(t_{i}) \cap \bigcap_{i=1}^{N} F(T_{i})\) is nonempty and \(T_{i}p=\{p\}\) for all \(p\in \mathcal{F}\) and \(i=1,2,\ldots,N\). Let \(x_{1}\in D\) and the sequence \(\{x_{n}\}\) be generated by
where \(\{\alpha_{n}^{(i)}\}\) and \(\{\beta_{n}^{(i)}\}\) are sequences in \([0,1]\) for all \(i=1,2,\ldots,N\) such that \(0< a\leq \alpha_{n}^{(i)},\beta_{n}^{(i)}\leq b<1\), \(\sum_{i=0}^{N}\alpha_{n}^{(i)}=1\), and \(\sum_{i=0}^{N}\beta_{n}^{(i)}=1\). Then we have the following:
-
(i)
\(\lim_{n\to\infty} \|x_{n}-z_{n}^{(i)}\|=0\) for all \(i=1,2,\ldots,N\);
-
(ii)
\(\lim_{n\to\infty} \|x_{n}-t_{i}x_{n}\|=0\) for all \(i=1,2,\ldots,N\).
Proof
(i) By Lemma 3.1, \(\lim_{n\to\infty} \|x_{n}-p\|\) exists. Put \(\lim_{n\to\infty} \|x_{n}-p\|=c\). By the definition of \(\{x_{n}\}\), we have
Then we have
By \(\lim_{n\to\infty}k_{n}= 1\) and \(\lim_{n\to\infty}s_{n}= 0\), we have
Since \(c=\lim_{n\to\infty} \|x_{n+1}-p\|=\lim_{n\to\infty} \|\alpha_{n}^{(0)} (x_{n}-p)+ \sum_{i=1}^{N}\alpha_{n}^{(i)} (t_{i}^{n}y_{n}-p)\|\), it follows by Lemma 2.5 that
Consider
This implies that
Therefore,
By (3.1), we obtain
Thus,
Since
it implies that
Hence, by Lemma 2.5, we have
(ii) Since \(t_{i}\) is generalized asymptotically nonexpansive, for all \(i=1,2,\ldots,N\), we get
By the definition of \(\{x_{n}\}\), we have \(y_{n}-x_{n}=\sum_{i=1}^{N}\beta_{n}^{(i)}(z_{n}^{(i)}-x_{n})\). This implies that
Then, by (i) and (3.2), we get
For \(i=1,2,\ldots,N\), we have
By (3.2) and (3.3), we conclude that \(\lim_{n\to\infty} \|x_{n}-t_{i}x_{n}\|=0\) for all \(i=1,2,\ldots,N\). □
Theorem 3.5
Let D be a nonempty, compact, and convex subset of a uniformly convex Banach space X. Let \(\{t_{i}\}_{i=1}^{N}\) be a finite family of uniformly L-Lipschitzian and generalized asymptotically nonexpansive single-valued mappings of D into itself with sequences \(\{k_{n}\}\subset[1,\infty)\) and \(\{s_{n}\}\subset [0,\infty)\) such that \(\sum_{n=1}^{\infty}(k_{n}-1)<\infty\) and \(\sum_{n=1}^{\infty}s_{n}<\infty\). Let \(\{T_{i}\}_{i=1}^{N}\) be a finite family of quasi-nonexpansive multi-valued mappings of D into \(CB(D)\) satisfying condition (E). Assume that \(\mathcal{F}=\bigcap_{i=1}^{N} F(t_{i}) \cap\bigcap_{i=1}^{N} F(T_{i})\) is nonempty and \(T_{i}p=\{p\}\) for all \(p\in\mathcal{F}\) and \(i=1,2,\ldots,N\). Let \(x_{1}\in D\) and the sequence \(\{x_{n}\}\) be generated by
where \(\{\alpha_{n}^{(i)}\}\) and \(\{\beta_{n}^{(i)}\}\) are sequences in \([0,1]\) for all \(i=1,2,\ldots,N\) such that \(0< a\leq \alpha_{n}^{(i)},\beta_{n}^{(i)}\leq b<1\), \(\sum_{i=0}^{N}\alpha_{n}^{(i)}=1\), and \(\sum_{i=0}^{N}\beta_{n}^{(i)}=1\). Then the sequence \(\{x_{n}\}\) converges strongly to a point in \(\mathcal{F}\).
Proof
By Lemma 3.1, we have \(\{x_{n}\}\) is bounded. Since D is compact, there exists a subsequence \(\{x_{n_{j}}\}\) of \(\{x_{n}\}\) converging strongly to \(p\in D\). By condition (E), there exists \(\mu\geq1\) such that for \(i=1,2,\ldots,N\),
Then, by Lemma 3.4(i), we have \(p\in T_{i}p\) for all \(i=1,2,\ldots,N\). So \(p\in\bigcap_{i=1}^{N}F(T_{i})\).
Since \(t_{i}\) is uniformly L-Lipschitzian, for all \(i=1,2,\ldots,N\), we have
By Lemma 3.4(ii), it implies that \(t_{i}p=p\) for all \(i=1,2,\ldots,N\). Thus, \(p\in\bigcap_{i=1}^{N}F(t_{i})\). Therefore, \(p\in\mathcal{F}\). Since \(\lim_{n\to\infty}\|x_{n}-p\|\) exists, we get \(\lim_{n\to\infty}\|x_{n}-p\|=\lim_{j\to\infty}\|x_{n_{j}}-p\|=0\). This shows that \(\{x_{n}\}\) converges strongly to a point in \(\mathcal{F}\). □
Next, we give a numerical example to support Theorem 3.5.
Example 3.6
Let \(\mathbb{R}\) be the real line with the usual norm \(|\cdot|\) and let \(D=[0,3]\). Define two single-valued mappings \(t_{1}\) and \(t_{2}\) on D as follows:
Also we define two multi-valued mappings \(T_{1}\) and \(T_{2}\) on D as follows:
Let \(\{x_{n}\}\) and \(\{y_{n}\}\) be generated by
where \(\alpha_{n}^{(0)}=\frac{3n+4}{10n}\), \(\alpha_{n}^{(1)}=\frac{2n-1}{5n}\), \(\alpha_{n}^{(2)}=\frac{3n-2}{10n}\), \(\beta_{n}^{(0)}=\frac{15n+7}{60n}\), \(\beta_{n}^{(1)}=\frac{5n-1}{20n}\), \(\beta_{n}^{(2)}=\frac{15n-2}{30n}\), for all \(n\in\mathbb{N}\). Then the sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) converge strongly to 0, where \(\{0\}=\bigcap_{i=1}^{2} F(t_{i}) \cap\bigcap_{i=1}^{2} F(T_{i})\).
Solution
It is shown in [19] that both \(t_{1}\) and \(t_{2}\) are generalized asymptotically nonexpansive single-valued mappings. Moreover, they are uniformly L-Lipschitzian mappings and \(\bigcap_{i=1}^{2} F(t_{i})=\{0\}\). It is easy to see that both \(T_{1}\) and \(T_{2}\) are quasi-nonexpansive multi-valued mappings satisfying condition (E) and \(\bigcap_{i=1}^{2} F(T_{i})=\{0\}\). Thus, \(\bigcap_{i=1}^{2} F(t_{i}) \cap\bigcap_{i=1}^{2} F(T_{i})=\{0\}\). For every \(n\in\mathbb{N}\), \(\alpha_{n}^{(0)}=\frac{3n+4}{10n}\), \(\alpha_{n}^{(1)}=\frac{2n-1}{5n}\), \(\alpha_{n}^{(2)}=\frac{3n-2}{10n}\), \(\beta_{n}^{(0)}=\frac{15n+7}{60n}\), \(\beta_{n}^{(1)}=\frac{5n-1}{20n}\), \(\beta_{n}^{(2)}=\frac{15n-2}{30n}\). Then the sequences \(\{\alpha_{n}^{(0)}\}\), \(\{\alpha_{n}^{(1)}\}\), \(\{\alpha_{n}^{(2)}\}\), \(\{\beta_{n}^{(0)}\}\), \(\{\beta_{n}^{(1)}\}\), and \(\{\beta_{n}^{(2)}\}\) satisfy all the conditions of Theorem 3.5. Put \(z_{n}^{(1)}=\frac{x_{n}}{2}\) and \(z_{n}^{(2)}=\frac{x_{n}}{3}\) for all \(n\in\mathbb{N}\). Then the algorithm (3.4) becomes
Using the algorithm (3.5) with the initial point \(x_{1}=2.5\), we have numerical results in Table 1.
Finally, we prove a weak convergence theorem in uniformly convex Banach spaces.
Theorem 3.7
Let D be a nonempty, closed, and convex subset of a uniformly convex Banach space X with the Opial property. Let \(\{t_{i}\}_{i=1}^{N}\) be a finite family of uniformly L-Lipschitzian and generalized asymptotically nonexpansive single-valued mappings of D into itself with sequences \(\{k_{n}\}\subset[1,\infty)\) and \(\{s_{n}\}\subset[0,\infty)\) such that \(\sum_{n=1}^{\infty}(k_{n}-1)<\infty\) and \(\sum_{n=1}^{\infty}s_{n}<\infty\). Let \(\{T_{i}\}_{i=1}^{N}\) be a finite family of quasi-nonexpansive multi-valued mappings of D into \(KC(D)\) satisfying the condition (E). Assume that \(\mathcal{F}=\bigcap_{i=1}^{N} F(t_{i}) \cap \bigcap_{i=1}^{N} F(T_{i})\) is nonempty and \(T_{i}p=\{p\}\) for all \(p\in \mathcal{F}\) and \(i=1,2,\ldots,N\). Let \(x_{1}\in D\) and the sequence \(\{x_{n}\}\) be generated by
where \(\{\alpha_{n}^{(i)}\}\) and \(\{\beta_{n}^{(i)}\}\) are sequences in \([0,1]\) for all \(i=1,2,\ldots,N\) such that \(0< a\leq \alpha_{n}^{(i)},\beta_{n}^{(i)}\leq b<1\), \(\sum_{i=0}^{N}\alpha_{n}^{(i)}=1\), and \(\sum_{i=0}^{N}\beta_{n}^{(i)}=1\). Then the sequence \(\{x_{n}\}\) converges weakly to a point in \(\mathcal{F}\).
Proof
By Lemma 3.1, \(\{x_{n}\}\) is bounded. Since X is uniformly convex, there exists a subsequence \(\{x_{n_{j}}\}\) of \(\{x_{n}\}\) converging weakly to \(p\in D\). By Lemma 3.4, we have \(\lim_{j\to\infty} \|x_{n_{j}}-z_{n_{j}}^{(i)}\|=0\) and \(\lim_{j\to\infty} \|x_{n_{j}}-t_{i}x_{n_{j}}\|=0\) for all \(i=1,2,\ldots,N\). We will show that \(p\in\mathcal{F}\). Since \(T_{1}p\) is compact, for all \(j\in\mathbb{N}\), we can choose \(w_{n_{j}}\in Tp\) such that \(\|x_{n_{j}}-w_{n_{j}}\|=\operatorname{dist}(x_{n_{j}},T_{1}p)\) and the sequence \(\{w_{n_{j}}\}\) has a convergent subsequence \(\{w_{n_{k}}\}\) with \(\lim_{k\to\infty}w_{n_{k}}=w\in T_{1}p\). By condition (E), we have
Then we have
This implies that
From the Opial property, we have \(p=w\in T_{1}p\). Similarly, it can be shown that \(p\in T_{i}p\) for all \(i = 2,\ldots,N\). Thus, \(p\in \bigcap_{i=1}^{N} F(T_{i})\).
Next, by mathematical induction, we can prove that, for \(i=1,2,\ldots,N\),
Indeed, it is obvious that the conclusion it true for \(m=1\). Suppose the conclusion holds for \(m\geq1\). Since \(t_{i}\) is uniformly L-Lipschitzian, we have
This shows that \(\lim_{j\to\infty}\|x_{n_{j}}-t_{i}^{m+1}x_{n_{j}}\|=0\) for all \(i=1,2,\ldots,N\). Hence, (3.6) holds.
From (3.6), we have for each \(x\in D\), \(m\in\mathbb{N}\) and \(i=1,2,\ldots,N\),
Since \(t_{i}\) is generalized asymptotically nonexpansive, we get
Then we have
By Proposition 2.1, we have
It implies that
By the Opial property and \(\{x_{n_{j}}\}\) converging weakly to p, we obtain
Then, by (3.9), we have
It implies by (3.7), (3.8), and (3.10) that
This shows that \(\lim_{m\to\infty}g(\|p-t_{i}^{m}p\|)=0\) for all \(i=1,2,\ldots,N\). Then the properties of g yield \(\lim_{m\to\infty}\|p-t_{i}^{m}p\|=0\) for all \(i=1,2,\ldots,N\). So we have
This implies that \(t_{i}p=p\) for all \(i=1,2,\ldots,N\). Thus, \(p\in \bigcap_{i=1}^{N} F(t_{i})\).
Hence, we obtain \(p\in\mathcal{F}\).
Finally, we show that \(\{x_{n}\}\) converges weakly to p. To show this, suppose not. Then there exists a subsequence \(\{x_{n_{l}}\}\) of \(\{x_{n}\}\) such that \(\{x_{n_{l}}\}\) converges weakly to \(q\in D\) and \(q\ne p\). By the same method as given above, we can prove that \(q\in \mathcal{F}\). By Lemma 3.1, \(\lim_{n\to\infty}\|x_{n}-p\|\) and \(\lim_{n\to\infty}\|x_{n}-q\|\) exist. It follows by Lemma 2.6 that \(q=p\). Thus, \(\{x_{n}\}\) converges weakly to a point in \(\mathcal{F}\). □
Remark 3.8
Theorem 3.5 extends and generalizes the results of Sokhuma and Kaewkhao [11] to a pair of a finite family of generalized asymptotically nonexpansive single-valued mappings and a finite family of quasi-nonexpansive multi-valued mappings satisfying condition (E). Theorems 3.5 and 3.7 extend and generalize the results of Eslamian [12] and Eslamian and Abkar [13] to a pair of a finite family of generalized asymptotically nonexpansive single-valued mappings and a finite family of quasi-nonexpansive multi-valued mappings satisfying condition (E).
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This paper was supported by the Thailand Research Fund under the project RTA5780007.
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Suantai, S., Phuengrattana, W. A new iterative process for a hybrid pair of generalized asymptotically nonexpansive single-valued and generalized nonexpansive multi-valued mappings in Banach spaces. Fixed Point Theory Appl 2015, 58 (2015). https://doi.org/10.1186/s13663-015-0304-7
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DOI: https://doi.org/10.1186/s13663-015-0304-7