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Common fixed points of Gnonexpansive mappings on Banach spaces with a graph
Fixed Point Theory and Applications volume 2016, Article number: 87 (2016)
Abstract
In this paper, we prove the weak and strong convergence of a sequence \(\{x_{n}\}\) generated by the Ishikawa iteration to some common fixed points of two Gnonexpansive mappings defined on a Banach space endowed with a graph.
Introduction
In 1922, Banach proved a remarkable and powerful result called the Banach contraction principle. Because of its fruitful applications, the principle has been generalized in many directions. The recent version of the theorem was given in Banach spaces endowed with a graph. In 2008, Jachymski [1] gave a generalization of the Banach contraction principle to mappings on a metric space endowed with a graph. In 2012, Aleomraninejad et al. [2] presented some iterative scheme results for Gcontractive and Gnonexpansive mappings on graphs. In 2015, Alfuraidan and Khamsi [3] defined the concept of Gmonotone nonexpansive multivalued mappings defined on a metric space with a graph. In the same year, Alfuraidan [4] gave a new definition of the Gcontraction and obtained sufficient conditions for the existence of fixed points for multivalued mappings on a metric space with a graph, and also in [5], he proved the existence of a fixed point of monotone nonexpansive mapping defined in a Banach space endowed with a graph. Recently, Tiammee et al. [6] proved Browder’s convergence theorem for Gnonexpansive mapping in a Banach space with a directed graph. They also proved the strong convergence of the Halpern iteration for a Gnonexpansive mapping.
Inspired by all aforementioned references, the author proves strong and weak convergence theorems for Gnonexpansive mappings using the Ishikawa iteration generated from arbitrary \(x_{0} \) in a closed convex subset C of a uniformly convex Banach space X endowed with a graph.
Preliminaries
In this section, we recall some standard graph notations and terminology and also some needed results.
Let \((X, d)\) be a metric space, and \(\triangle= \{(x, x)  x \in X\}\). Consider a directed graph G for which the set \(V(G)\) of its vertices coincides with X and the set \(E(G)\) of its edges contains all loops. Assume that G has no parallel edges. Then \(G = (V(G), E(G))\), and by assigning to each edge the distance between its vertices, G may be treated as a weighted graph.
Definition 2.1
The conversion of a graph G is the graph obtained from G by reversing the direction of edges denoted by \(G^{1}\), and
Definition 2.2
Let x and y be vertices of a graph G. A path in G from x to y of length N (\(N \in\mathbb{N} \cup\{0\}\)) is a sequence \(\{ x_{i}\}_{i=0}^{N}\) of \(N+1\) vertices for which
Definition 2.3
A graph G is said to be connected if there is a path between any two vertices of the graph G.
Definition 2.4
A directed graph \(G=(V(G), E(G))\) is said to be transitive if, for any \(x,y,z \in V(G)\) such that \((x,y)\) and \((y,z)\) are in \(E(G)\), we have \((x,z) \in E(G)\).
The definition of a Gnonexpansive mapping is given as follows.
Definition 2.5
Let C be a nonempty convex subset of a Banach space X, and \(G =(V(G), E(G))\) a directed graph such that \(V(G) =C\). Then a mapping \(T: C \to C\) is Gnonexpansive (see [3], Definition 2.3(iii)) if it satisfies the following conditions.

(i)
T is edgepreserving.

(ii)
\(\Tx Ty\ \leq\xy\\) whenever \((x,y) \in E(G)\) for any \(x,y \in C\).
Definition 2.6
([7])
Let C be a nonempty closed convex subset of a real uniformly convex Banach space X. The mappings \(T_{i}\) (\(i = 1,2\)) on C are said to satisfy Condition B if there exists a nondecreasing function \(f : [0, \infty) \rightarrow [0, \infty)\) with \(f(0) = 0\) and \(f(r) > 0\) for all \(r > 0\) such that, for all \(x \in C\),
where \(F = F(T_{1}) \cap F(T_{2}) \) and \(F(T_{i})\) (\(i=1,2\)) are the sets of fixed points of \(T_{i}\).
Definition 2.7
([7])
Let C be a subset of a metric space \((X, d)\). A mapping T is semicompact if for a sequence \(\{x_{n}\}\) in C with \(\lim_{n \rightarrow\infty} d(x_{n}, Tx_{n}) = 0\), there exists a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{i}} \rightarrow p \in C\).
Definition 2.8
A Banach space X is said to satisfy Opial’s property if the following inequality holds for any distinct elements x and y in X and for each sequence \(\{x_{n}\}\) weakly convergent to x:
Definition 2.9
Let X be a Banach space. A mapping T with domain D and range R in X is demiclosed at 0 if, for any sequence \(\{x_{n}\}\) in D such that \(\{x_{n}\}\) converges weakly to \(x \in D\) and \(\{Tx_{n}\} \) converges strongly to 0, we have \(Tx = 0\).
Lemma 2.10
([8])
Let X be a uniformly convex Banach space, and \(\{\alpha_{n}\}\) a sequence in \([\delta, 1  \delta]\) for some \(\delta\in(0,1)\). Suppose that sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) in X are such that \(\limsup_{n \to\infty} \x_{n}\ \leq c\), \(\limsup_{n \to\infty} \ y_{n}\ \leq c\) and \(\limsup_{n \to\infty} \\alpha x_{n} + (1\alpha _{n})y_{n}\ = c\) for some \(c \geq0\). Then \(\lim_{n \to\infty} \ x_{n} y_{n}\ = 0\).
Lemma 2.11
([9])
Let X be a Banach space, and \(R>1\) be a fixed number. Then X is uniformly convex if and only if there exists a continuous, strictly increasing, and convex function \(g : [0,\infty) \to[0,\infty)\) with \(g(0) =0\) such that
for all \(x,y \in B_{R}(0) = \{x \in X \x\ \leq R\}\) and \(\lambda\in[0,1]\).
Lemma 2.12
([10])
Let X be a Banach space that satisfies Opial’s property, and let \(\{ x_{n}\}\) be a sequence in X. Let x, y in X be such that \(\lim_{n \to\infty} \x_{n}  x\\) and \(\lim_{n \to\infty} \x_{n}  y\\) exist. If \(\{x_{n_{j}}\}\) and \(\{x_{n_{k}}\}\) are subsequences of \(\{ x_{n}\}\) that converge weakly to x and y, respectively, then \(x=y\).
Main results
Throughout the section, we let C be a nonempty closed convex subset of a Banach space X endowed with a directed graph G such that \(V(G) = C\) and \(E(G)\) is convex. We also suppose that the graph G is transitive. The mappings \(T_{i}\) (\(i =1,2\)) are Gnonexpansive from C to C with \(F = F(T_{1}) \cap F(T_{2})\) nonempty. Let \(\{x_{n}\}\) be a sequence generated from arbitrary \(x_{0} \in C\),
where \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are real sequences in \([0,1]\).
We first begin by proving the following useful results.
Proposition 3.1
Let \(z_{0} \in F\) be such that \((x_{0}, z_{0})\), \((y_{0}, z_{0})\), \((z_{0}, x_{0})\), and \((z_{0}, y_{0})\) are in \(E(G)\). Then \((x_{n}, z_{0})\), \((y_{n}, z_{0})\), \((z_{0}, x_{n})\), \((z_{0}, y_{n})\), and \((x_{n}, y_{n})\) are in \(E(G)\).
Proof
We divide the proof into three parts. In the first part, with the assumption \((x_{0}, z_{0}), (y_{0}, z_{0}) \in E(G)\), we will show by induction that \((x_{n}, z_{0}), (y_{n}, z_{0}) \in E(G)\). Then, with the assumption \((z_{0}, x_{0}), (z_{0}, y_{0})\in E(G)\), we will again prove by induction that \((z_{0}, x_{n}), (z_{0}, y_{n}) \in E(G)\). In the third part, we combine these two results using transitivity of G to get the statement in the proposition. Let \((x_{0},z_{0})\) and \((y_{0},z_{0}) \in E(G)\). Then \((T_{1}y_{0}, z_{0}), (T_{2}x_{0}, z_{0}) \in E(G)\) since \(T_{i}\) (\(i = 1,2\)) are edgepreserving. By the convexity of \(E(G)\) and \((T_{1}y_{0}, z_{0}), (x_{0},z_{0}) \in E(G)\), we have \((x_{1},z_{0}) \in E(G)\). Then, by edgepreserving of \(T_{2}\), \((T_{2}x_{1}, z_{0}) \in E(G)\). Again, by the convexity of \(E(G)\) and \((T_{2}x_{1}, z_{0}), (x_{1},z_{0}) \in E(G)\), we get \((y_{1},z_{0}) \in E(G)\) and then \((T_{1}y_{1},z_{0}) \in E(G)\). Next, we assume that \((x_{k}, z_{0}), (y_{k},z_{0}) \in E(G)\). Then \((T_{2}x_{k}, z_{0}), (T_{1}y_{k}, z_{0}) \in E(G)\) since \(T_{i}\) (\(i=1,2\)) are edgepreserving. Since \(E(G)\) is convex, \((x_{k+1},z_{0}) \in E(G)\). Indeed,
Since \(T_{2}\) is edgepreserving, \((T_{2}x_{k+1}, z_{0}) \in E(G)\). Using the convexity of \(E(G)\), we get \((y_{k+1}, z_{0}) \in E(G)\). To be explicit,
Hence, by induction, \((x_{n},z_{0}), (y_{n},z_{0}) \in E(G)\) for all \(n \geq1\). Using a similar argument, we can show that \((z_{0},x_{n}), (z_{0},y_{n}) \in E(G)\) under the assumption that \((z_{0},x_{0}), (z_{0},y_{0}) \in E(G)\). Therefore, \((x_{n},y_{n}) \in E(G)\) by the transitivity of G. □
Lemma 3.2
Let \(z_{0} \in F\). Suppose that \((x_{0}, z_{0}), (y_{0}, z_{0}), (z_{0}, x_{0}), (z_{0}, y_{0})\in E(G)\) for arbitrary \(x_{0}\) in C. Then \(\lim_{n \to\infty} \x_{n} z_{0}\\) exists.
Proof
Notice that
Thus, \(\lim_{n \to\infty} \x_{n}z_{0}\\) exists. In particular, the sequence \(\{x_{n}\}\) is bounded. □
Lemma 3.3
If X is uniformly convex, \(\{\alpha_{n}\}, \{\beta_{n}\} \subset [\delta, 1\delta]\) for some \(\delta\in(0,\frac{1}{2})\), and \((x_{0}, z_{0}), (y_{0}, z_{0}), (z_{0}, x_{0}), (z_{0}, y_{0})\in E(G)\) for arbitrary \(x_{0}\) in C and \(z_{0} \in F\), then
Proof
Let \(z_{0} \in F\). Then, by the boundedness of \(\{x_{n}\}\) and \(\{T_{2}x_{n}\}\) there exists \(r > 0\) such that \(x_{n}  z_{0}, y_{n} z_{0} \in B_{r}(0)\) for all \(n \geq1\). Put \(c = \lim_{n \to\infty} \x_{n}  z_{0}\\). If \(c = 0\), then by the Gnonexpansiveness of \(T_{i}\) (\(i=1,2\)) we have
Therefore, the result follows. Suppose that \(c > 0\). Hence, by Lemma 2.11 together with the Gnonexpansiveness of \(T_{2}\), we have
Thus,
Notice also that
Thus,
This implies that \(\lim_{n \to\infty} g (\T_{1} y_{n} x_{n}\) =0\), and since g is strictly increasing and continuous at 0,
Since \(T_{1}\) is Gnonexpansive, we have
Taking lim inf yields
Hence, we have
Since
and
by Lemma 2.10 we have
By the Gnonexpansiveness of \(T_{1}\) together with \(\x_{n} y_{n}\ \leq\T_{2}x_{n} x_{n}\\) we have
Using (1) and (2), \(\lim_{n \to\infty} \T_{1}x_{n}  x_{n}\ =0\). Hence, the lemma is proved. □
Lemma 3.4
Suppose that X satisfies the Opial’s property and that \((x_{0},z_{0})\), \((y_{0},z_{0})\) are in \(E(G)\) for \(z_{0} \in F\) and arbitrary \(x_{0} \in C\). Then \(I T_{i}\) (\(i=1,2\)) are demiclosed.
Proof
Suppose that \(\{x_{n}\}\) is a sequence in C that converges weakly to q. From Lemma 3.3 we have \(\lim_{n \to \infty}\x_{n} T_{i}x_{n}\ =0\). Suppose for contradiction that \(q \neq T_{i}q\). Then, by Opial’s property we have
a contradiction. Hence, \(T_{i}q = q\), so the conclusion holds. □
Theorem 3.5
Suppose X is uniformly convex, \(\{\alpha_{n}\} , \{\beta_{n}\} \subset[\delta, 1\delta]\) for some \(\delta\in(0,\frac{1}{2})\), \(T_{i}\) (\(i =1,2\)) satisfy Condition B, F is dominated by \(x_{0}\), F dominates \(x_{0}\), and \((x_{0}, z), (y_{0}, z), (z, x_{0}), (z, y_{0})\in E(G)\) for each \(z \in F\) and arbitrary \(x_{0}\in C\). Then \(\{ x_{n}\}\) converges strongly to a common fixed point of \(T_{i}\).
Proof
Let \(z\in F\). Recall the following facts from Lemma 3.2:

(i)
\(\{x_{n}\}\) is bounded;

(ii)
\(\lim_{n \to\infty} \x_{n}  z\\) exists;

(iii)
\(\x_{n+1} z\ \leq\x_{n} z\\) for all \(n \geq1\).
They imply that
Thus \(\lim_{n \to\infty} d(x_{n}, F)\) exists. Since each \(T_{i}\) (\(i = 1,2\)) satisfies Condition B and \(\lim_{n \to\infty}\x_{n}  T_{i}x_{n}\ =0\), we have
and then
Hence, there are a subsequence \(\{x_{n_{j}}\}\) of \(\{x_{n}\}\) and a sequence \(\{z_{j}\} \subset F\) satisfying
Put \(n_{j+1} = n_{j} + k\) for some \(k \geq1\). Then
Hence,
so that \(\{z_{j}\}\) is a Cauchy sequence. We assume that \(z_{j} \to q \in C\) as \(n \to\infty\). Since F is closed, \(q \in F\). Hence, we have \(x_{n_{j}} \to q\) as \(j \to\infty\), and since \(\lim_{n \to\infty } \x_{n} q\\) exists, the conclusion follows. □
Theorem 3.6
Suppose that X is uniformly convex, \(\{\alpha_{n}\}, \{\beta_{n}\} \subset[\delta, 1\delta]\) for some \(\delta\in(0,\frac{1}{2})\), one of \(T_{i}\) (\(i =1,2\)) is semicompact, F is dominated by \(x_{0}\), F dominates \(x_{0}\), and \((x_{0}, z_{0}), (y_{0}, z_{0}), (z_{0}, x_{0}), (z_{0}, y_{0})\in E(G)\) for \(z_{0} \in F\) and arbitrary \(x_{0} \in C\). Then \(\{x_{n}\}\) converges strongly to a common fixed point of \(T_{i}\).
Proof
Suppose that \(T_{2}\) is semicompact; by Lemma 3.2 and Lemma 3.3 we have a bounded sequence \(\{x_{n}\}\), and \(\lim_{n \to\infty} \x_{n} T_{i}x_{n}\ = 0\). Hence, by the semicompactness of \(T_{2}\) there exist \(q \in C\) and a subsequence \(\{x_{n_{j}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{j}} \to q\) as \(j \to\infty\) and \(\lim_{n \to\infty} \x_{n_{j}} T_{i}x_{n_{j}}\ = 0\). Notice that
Hence, \(q \in F\). Since \(\lim_{n \to\infty} d(x_{n}, F)=0\), it follows, by repeating the same argument as in the proof of Theorem 3.5, that \(\{x_{n}\}\) converges strongly to a common fixed point of \(T_{i}\) (\(i = 1,2\)), and the proof is complete. □
Theorem 3.7
Suppose that X is uniformly convex, \(\{\alpha_{n}\}, \{\beta_{n}\} \subset[\delta, 1\delta]\) for some \(\delta\in(0,\frac{1}{2})\). If X satisfies Opial’s property, \(IT_{i}\) is demiclosed at zero for each i, F is dominated by \(x_{0}\), F dominates \(x_{0}\), and \((x_{0}, z_{0}), (y_{0}, z_{0}), (z_{0}, x_{0}), (z_{0}, y_{0})\in E(G)\) for \(z_{0} \in F\) and arbitrary \(x_{0} \in C\), then \(\{x_{n}\}\) converges weakly to a common fixed point of \(T_{i}\).
Proof
Note that by Lemma 3.2, for each \(q \in F\),
Let \(\{x_{n_{k}}\}\) and \(\{x_{n_{j}}\}\) be subsequences of the sequence \(\{x_{n}\}\) with two weak limits \(q_{1}\) and \(q_{2}\), respectively. Notice that, by Lemma 3.3,
Hence, \(T_{i}q_{1} = q_{1}\) and \(T_{i}q_{2} = q_{2}\). By Lemma 3.4 we have \(q_{1}, q_{2} \in F\). In particular, \(q_{1} =q_{2}\) by Lemma 2.12. Therefore, \(\{x_{n}\}\) converges weakly to a common fixed point in F. □
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Acknowledgements
The author is grateful to Professor Suthep Suantai for valuable suggestion and comments. The author would also like to thank the anonymous reviewers for their helpful comments.
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Tripak, O. Common fixed points of Gnonexpansive mappings on Banach spaces with a graph. Fixed Point Theory Appl 2016, 87 (2016). https://doi.org/10.1186/s1366301605784
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MSC
 47H09
 47E10
 47H10
Keywords
 common fixed point
 Gnonexpansive mappings
 Ishikawa iteration
 Banach space
 directed graph