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Halpern type iteration with multiple anchor points in a Hadamard space
Journal of Inequalities and Applications volume 2015, Article number: 182 (2015)
Abstract
In this paper, we consider an approximation sequence of a common fixed point generated by Halpern type iteration with a finite family of nonexpansive mappings in a Hadamard space. We propose another style of Halpern type iteration with multiple anchor points and prove that it converges strongly to a common fixed point.
1 Introduction
The problem of finding a fixed point of nonexpansive mappings is one of the most important problems in nonlinear analysis and it has been investigated by many researchers with various approaches. In 1992, Wittmann [1] obtained that a Halpern type iteration with nonexpansive mapping is strongly convergent to a fixed point in a Hilbert space. Later, Shimizu and Takahashi [2] showed that a Halpern type iteration with two nonexpansive mappings converges strongly to a common fixed point in a Hilbert space. Moreover, Kimura et al. [3] proved an approximation of common fixed points of a finite family of nonexpansive mappings in a uniformly convex Banach space whose norm is Gâteaux differentiable.
On the other hand, in 2010, Saejung [4] introduced a Halpern type iteration with a nonexpansive mapping approximating a fixed point in a Hadamard space, and also proved the following theorem.
Theorem 1.1
Let X be a Hadamard space. Let \(T_{1}, T_{2}, \ldots, T_{N} : X \to X\) be nonexpansive mappings with \(\bigcap_{i = 1}^{N} F(T_{i}) \neq\emptyset\), and let \(u, x_{1} \in C\) be arbitrarily chosen. Define an iterative sequence \(\{x_{n}\}\) by
for all \(n \in\mathbb{N}\), where \(\{\alpha_{n}\}\) is a sequence in \(] 0, 1 [ \) such that \(\lim_{n \to\infty} \alpha_{n} = 0\), \(\sum^{\infty}_{n = 1} \alpha_{n} = \infty\), and \(\sum^{\infty }_{n = 1} |\alpha_{n + 1} - \alpha_{n}| < \infty\). Suppose, in addition, that
Then \(\{x_{n}\}\) converges to \(z \in\bigcap_{i = 1}^{N} F(T_{i})\) which is nearest to u.
In this paper, we introduce an approximation theorem of common fixed points of nonexpansive mappings in a Hadamard space. We produce the iterative sequence \(\{x_{n}\}\) as follows. Let \(u_{1}, u_{2}, \ldots, u_{r}\), \(x_{1}\) be arbitrary points in a Hadamard space, and let \(\{x_{n}\}\) be iteratively generated by
for all \(n \in\mathbb{N}\), where \(\{\alpha_{n}\}\) is a sequence under the same conditions of Theorem 1.1, and, for all \(k = 1, 2, \ldots, r - 1\), \(\{\beta^{k}_{n}\}\) are sequences in \([a, b] \subset \, ] 0, 1 [ \). This iterative sequence is another type of convex combination for Theorem 1.1. Furthermore, the anchor point of known Halpern type iteration is single, however our iterative sequence has multiple anchor points. Then we show that \(\{x_{n}\}\) converges strongly to a common fixed point. In a Hilbert space, \(\{x_{n}\} \) is convergent to the nearest point to \(\sum_{i = 1}^{r} \gamma^{i} u_{i}\) in the set of common fixed points of \(\{T_{i}\}\), where \(\gamma^{i} \in\, ] 0, 1 [ \) for all \(i = 1, 2, \ldots, r\), and \(\sum_{i = 1}^{r} \gamma^{i} = 1\); see [3]. However, it is not always true in a Hadamard space.
2 Preliminaries
Let \((X, d)\) be a metric space. For \(x, y \in X\), a mapping \(c: [0, l] \to X\) is called a geodesic with endpoints x, y if c satisfies \(c(0) = x\), \(c(l) = y\) and \(d(c(u), c(v)) = |u - v|\) for \(u, v \in[0, l]\). If a geodesic with endpoints x, y exists for any \(x, y \in X\), then we call X a geodesic metric space. Moreover, if a geodesic exists uniquely for each \(x, y \in X\), then we call X a uniquely geodesic space. A Hadamard space, which is defined below, is a uniquely geodesic space.
Let X be a uniquely geodesic space. For \(x, y \in X\), the image of a geodesic c with endpoints x, y is called a geodesic segment joining x and y, and is denoted by \([x, y]\). A geodesic triangle \(\bigtriangleup(x_{1}, x_{2}, x_{3})\) with vertices \(x_{1}\), \(x_{2}\), \(x_{3}\) in X is the union of geodesic segments joining each pair of vertices. A comparison triangle \(\overline{\bigtriangleup}(\bar{x}_{1},\bar {x}_{2},\bar{x}_{3})\) in \(\mathbb{R}^{2}\) for \(\bigtriangleup(x_{1}, x_{2}, x_{3})\) is a triangle such that \(d(x_{i}, x_{j}) = \|\bar{x}_{i} - \bar{x}_{j}\| \) for all \(i, j = 1, 2, 3\). If, for any \(p, q \in\bigtriangleup(x_{1}, x_{2}, x_{3})\) and their comparison points \(\bar{p}, \bar{q} \in \overline{\bigtriangleup}(\bar{x}_{1},\bar{x}_{2},\bar{x}_{3})\), the inequality
is satisfied for all triangle in X, then X is called a \(\operatorname{CAT}(0)\) space, and this inequality is called the \(\operatorname{CAT}(0)\) inequality. A Hadamard space is defined as a complete \(\operatorname{CAT}(0)\) space.
Let X be a Hadamard space. For \(t \in[0, 1]\) and \(x, y \in X\), there exists unique \(z \in[x, y]\) such that \(d(x, z) = (1 - t)d(x, y)\) and \(d(z, y) = td(x, y)\). We denote z by \(tx \oplus(1 - t)y\). From the \(\operatorname{CAT}(0)\) inequality, we obtain the following lemma. This lemma plays an important role in this paper.
Lemma 2.1
Let X be a Hadamard space. Then, for any \(x, y, z \in X\) and \(t \in\, ] 0, 1 [ \), it follows that
By this lemma, it is easy to see the following result.
Lemma 2.2
Let \(\{x_{n}\}\), \(\{y_{n}\}\) be bounded sequences of a Hadamard space X. For \(\{\alpha_{n}\} \subset \, ] 0, 1 [ \), define a sequence \(\{z_{n}\}\) by \(z_{n} = \alpha_{n}x_{n} \oplus(1 - \alpha_{n})y_{n}\). Then \(\{z_{n}\}\) is bounded.
For more details on Hadamard spaces, see [5].
Let T be a mapping from X into itself. T is called a nonexpansive mapping if the inequality \(d(Tx, Ty) \leq d(x, y)\) is satisfied for any \(x,y \in X\). A point \(z \in X\) is called a fixed point of T if \(Tz = z\) holds. We denote the set of all fixed points of T by \(F(T)\). A subset \(C \subset X\) is said to be convex if, for any \(x, y \in C\), \([x, y]\) is included in C. We know that \(F(T)\) is a closed convex subset of X if T is nonexpansive.
Let \(\{x_{n}\}\) be a bounded sequence in a metric space X. For any \(x \in X\), we put
Then, if there exists \(x \in X\) such that \(r(x, \{x_{n}\}) = r(\{x_{n}\})\), we call x an asymptotic center of \(\{x_{n}\}\). Moreover if, for any subsequence of \(\{x_{n}\}\), each asymptotic center is a unique point x, we say that \(\{x_{n}\}\) is Δ-convergent to x. We know that any bounded sequence \(\{x_{n}\}\) in a Hadamard space has a Δ-converging subsequence; see [6, 7].
3 Halpern type iteration with multiple anchor points
In this section, we introduce some lemmas and show the main theorem.
Lemma 3.1
(Aoyama-Kimura-Takahashi-Toyoda [8], Xu [9])
Let \(\{s_{n}\}\) be a sequence of nonnegative real numbers, \(\{\alpha_{n}\} \) be a sequence in \([0, 1]\) with \(\sum_{n = 1}^{\infty}\alpha_{n} = \infty\), \(\{u_{n}\}\) be a sequence of nonnegative real numbers with \(\sum_{n = 1}^{\infty} u_{n} < \infty\), and \(\{t_{n}\}\) be a sequence of real numbers with \(\limsup_{n \to\infty} t_{n} \leq0\). Suppose that
Then \(\lim_{n \to\infty}s_{n} = 0\).
Lemma 3.2
Let \(\{a_{n}\}\) be a sequence of real numbers with \(\sum_{n = 1}^{\infty}\lvert a_{n + 1} - a_{n} \rvert < \infty\). Then \(\{a_{n}\}\) is convergent.
Lemma 3.3
(Seajung [4])
Let X be a Hadamard space and \(T, S : X \to X\) be nonexpansive mappings with \(F(T) \cap F(S) \neq\emptyset\). For any \(\beta\in\, ] 0, 1 [ \), define a mapping U by \(Ux = \beta Tx \oplus(1 - \beta)Sx\) for all \(x \in X\). Then U is a nonexpansive mapping such that \(F(U) = F(T) \cap F(S)\).
Lemma 3.4
(He-Fang-López-Li [10])
Let X be a Hadamard space and \(\{x_{n}\}\) be a bounded sequence of X. If \(\{x_{n}\}\) is Δ-convergent to \(x \in X\), then
for all \(u \in X\).
Lemma 3.5
(Kirk-Panyanak [7])
Let X be a Hadamard space and \(T : X \to X\) be a nonexpansive mapping. Suppose \(\{x_{n}\} \subset X\) is Δ-convergent to \(x \in X\). If \(d(x_{n}, Tx_{n}) \to0\), then x is an element of \(F(T)\).
Lemma 3.6
(Mayer [11])
Let X be a Hadamard space and \(g : X \to\mathbb{R}\cup\{+\infty\}\). If g is convex and lower semicontinuous, then g is bounded from below on bounded subsets of X. Furthermore, g attains its infimum on nonempty bounded convex closed subsets of X. The resulting minimizer is unique if g is strictly convex.
Using Lemma 3.6, we get the following result.
Corollary 3.7
Let X be a Hadamard space. For any \(u_{1}, u_{2}, \ldots, u_{n} \in X\) and \(\beta^{1}, \beta^{2}, \ldots, \beta^{n} \in\, ] 0, 1 [ \) with \(\sum_{i = 1}^{n} \beta^{i} = 1\), define a function \(g : X \to\mathbb{R}\) by
for all \(x \in X\). Then g attains its infimum on a nonempty closed convex subset C of X, and its minimizer is unique.
Proof
Let p be an element of X. Since \(g(x) \to\infty\) as \(d(x, p) \to \infty\), there exists a nonempty bounded closed convex set D such that the minimizers of g on C and D are identical.
For \(x, y \in X\) with \(x \neq y\) and \(t \in\, ] 0, 1 [ \), we have
Thus g is a strictly convex, and by Lemma 3.6 we get the desired result. □
Now we can obtain the main theorem for a finite family of nonexpansive mappings with multiple anchor points.
Theorem 3.8
Let X be a Hadamard space and \(T_{1}, T_{2}, \ldots, T_{r} : X \to X\) be nonexpansive mappings with \(F = \bigcap_{i = 1}^{r} F(T_{i}) \neq \emptyset\). Let \(u_{1}, u_{2}, \ldots, u_{r}\), \(x_{1}\) be arbitrary points in X and let \(\{x_{n}\}\) be iteratively generated by
for all \(n \in\mathbb{N}\), where \(\{\alpha_{n}\}\) is a sequence in \(] 0, 1 [ \) such that
and, for all \(k = 1, 2, \ldots, r - 1\), \(\{\beta^{k}_{n}\}\) are sequences in \([a, b] \subset \, ] 0, 1 [ \) such that
Then \(\{x_{n}\}\) converges to \(x_{0} \in F\) which is the unique minimizer of \(g(x) = \sum_{i = 1}^{r} \gamma_{i}d(u_{i}, x)^{2}\) on F, where \(\gamma_{k} = \beta^{k - 1}\prod_{j = k}^{r - 1} (1 - \beta^{j})\) for \(k = 1, 2, \ldots, r - 1\) and \(\gamma_{r} = \beta^{r - 1}\) for \(\beta^{0} = 1\) and \(\beta^{i} = \lim_{n \to\infty}\beta^{i}_{n}\) for \(i = 1, 2, \ldots, r - 1\).
For the sake of simplicity, we will prove only the case for triple mappings, that is, the following theorem. The proof of Theorem 3.8 is omitted as it can be deduced by similar arguments.
Theorem 3.9
Let X be a Hadamard space and \(R, S, T : X \to X\) be nonexpansive mappings with \(F = F(R) \cap F(S) \cap F(T) \neq\emptyset\). Let u, v, w, \(x_{1}\) be arbitrary points in X and let \(\{x_{n}\}\) be iteratively generated by
for all \(n \in\mathbb{N}\), where \(\{\alpha_{n}\}\) is a sequence in \(] 0, 1 [ \) such that
and \(\{\beta_{n}\}\), \(\{\gamma_{n}\}\) are sequences in \([a, b] \subset \, ] 0, 1 [ \) such that
Then \(\{x_{n}\}\) converges to \(x_{0} \in F\) which is a minimizer of \(g(x) = \beta d(u, x)^{2} + (1 - \beta)(\gamma d(v, x)^{2} + (1 - \gamma)d(w, x)^{2})\) on F, where \(\beta= \lim_{n \to\infty}\beta_{n}\) and \(\gamma= \lim_{n \to\infty}\gamma_{n}\).
Proof
Let \(y_{n} = \gamma_{n}s_{n} \oplus(1 - \gamma_{n})t_{n}\) for all \(n \in \mathbb{N}\). We first show \(\{x_{n}\}\) is bounded. Let \(p \in F\). Then
Putting \(M = \max\{d(u, p)^{2}, d(v, p)^{2}, d(w, p)^{2}\}\), we have
By induction, we get
and hence we have \(\{x_{n}\}\) is bounded. Since R, S, and T are all nonexpansive, we get \(\{Rx_{n}\}\), \(\{Sx_{n}\}\), \(\{Tx_{n}\}\) are bounded. Moreover, by Lemma 2.2, we also have that \(\{r_{n}\}\), \(\{s_{n}\}\), \(\{ t_{n}\}\), \(\{y_{n}\}\) are bounded sequences.
Next, we show that \(d(x_{n + 1}, x_{n}) \to0\). Using the \(\operatorname{CAT}(0)\) inequality, we obtain
From this result, we also get
Therefore, we get
Using conditions (ii), (iii), (iv), (v), and Lemma 3.1, we have
From conditions (iv), (v) and Lemma 3.2, there exist \(\beta, \gamma\in\, ] 0, 1 [ \) such that \(\beta_{n} \to\beta\) and \(\gamma _{n} \to\gamma\). We put \(Ux = \beta Rx \oplus(1 - \beta)Qx\) for all \(x \in X\), where \(Qx = \gamma Sx \oplus(1 - \gamma)Tx\). From Lemma 3.3, we have that the mapping Q is nonexpansive with \(F(Q) = F(S) \cap F(T)\). Similarly, we have that U is nonexpansive with \(F(U) = F(R) \cap F(Q) = F\).
We show that \(d(Ux_{n}, x_{n}) \to0\). Let \(q_{n} = \alpha_{n}u \oplus(1 - \alpha_{n})Qx_{n}\). Then, using the \(\operatorname{CAT}(0)\) inequality, we have
Since \(\{\beta_{n}\}\) converges to β, by condition (i), we get
Put \(t'_{n} = \alpha_{n}v \oplus(1 - \alpha_{n})Tx_{n}\). Then, using this result, we have
By the \(\operatorname{CAT}(0)\) inequality, we get
Since \(\gamma_{n} \to\gamma\), we have
Therefore, by condition (i), we have
Consequently, we get
Suppose p is an element of F. Then we get
Thus, we have
and hence we get
Furthermore, we obtain that
By the same procedure, it follows that
and hence we get
Therefore, we have that
Define a function g on X by \(g(x) = \beta d(u, x)^{2} + (1 - \beta )h(x)\) for all \(x \in X\), where \(h(x) = \gamma d(v, x)^{2} + (1 - \gamma )d(w, x)^{2}\). From Corollary 3.7, there exists \(x_{0} \in F\) which is the unique minimizer of g on F. Then we have
Put
Since \(\beta_{n} \to\beta\) and \(\gamma_{n} \to\gamma\), we get
Moreover, since \(d(Rx_{n}, x_{n})\), \(d(Sx_{n}, x_{n})\) and \(d(Tx_{n}, x_{n})\) converges to 0, we also get
Therefore, we obtain that
and hence
Since \(\{x_{n}\}\) is bounded, there exists a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) such that
and \(\{x_{n_{i}}\}\) is Δ-convergent to some \(x \in X\). From Lemma 3.4, we have that
Since \(d(Ux_{n}, x_{n}) \to0\), x is an element of F by Lemma 3.5. Moreover, since \(x_{0}\) is a minimizer of g on F, we have that
Hence, by Lemma 3.1, \(\{x_{n}\}\) converges to \(x_{0}\) in F. □
For any points in a Hadamard space, we know that there exists a unique point in any closed convex subset which is the nearest of each subset to the point. Thus, we obtain the following corollary.
Corollary 3.10
Let X be a Hadamard space and \(T_{1}, T_{2}, \ldots, T_{r} : X \to X\) be nonexpansive mappings with \(F = \bigcap_{i = 1}^{r} F(T_{i}) \neq \emptyset\). Suppose u, \(x_{1}\) are arbitrary points in X and \(\{x_{n}\}\) is iteratively generated by
for all natural number n, where \(\{\alpha_{n}\}\) is a sequence in \(] 0, 1 [ \) such that
and, for all \(k = 1, 2, \ldots, r - 1\), \(\{\beta^{k}_{n}\}\) are sequences in \([a, b] \subset \, ] 0, 1 [ \) such that
Then \(\{x_{n}\}\) converges to \(x_{0}\) in F which is the nearest point of F to u.
Proof
Let \(x_{0} \in F\) be the nearest point to u. Then we have that
From Theorem 3.8, \(\{x_{n}\}\) converges to the minimizer of \(g(x) = d(u, x)^{2}\) on F. Therefore, \(\{x_{n}\}\) is convergent to \(x_{0}\). □
Remark
It will be interesting to consider similar results for an amenable semigroup of nonexpansive mappings using asymptotic invariant nets as in [12] for Hadamard spaces.
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Kimura, Y., Wada, H. Halpern type iteration with multiple anchor points in a Hadamard space. J Inequal Appl 2015, 182 (2015). https://doi.org/10.1186/s13660-015-0693-5
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DOI: https://doi.org/10.1186/s13660-015-0693-5
MSC
- 47H09
Keywords
- fixed point
- approximation theorem
- nonexpansive mapping
- geodesic space
- \(\operatorname{CAT}(0)\)