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Abstract
We construct an iteration scheme involving a hybrid pair of mappings, one a single-valued asymptotically nonexpansive mapping t and the other a multivalued nonexpansive mapping T, in a complete CAT(0) space. In the process, we remove a restricted condition (called the end-point condition) from results of Akkasriworn and Sokhuma [1] and and use this to prove some convergence theorems. The results concur with analogues for Banach spaces from Uddin et al. [16].
Fixed point theory in CAT(0) spaces was first studied by Kirk [6, 8] who showed that every nonexpansive mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then, the existence problem of fixed point and the Δ–convergence problem of iterative sequences to a fixed point for nonexpansive mappings and asymptotically nonexpansive mappings in a CAT(0) space have been extensively developed with many papers published.
Let (X, d) be a geodesic metric space. 2^{K} is denoted as the family of nonempty subsets of K, FB(K) is the collection of all nonempty closed bounded subsets of K and KC(K) is the collection of all nonempty compact convex subsets of K. A subset K of X is called proximinal if for each x ∈ X there exists an element k ∈ K such that
A mapping t : K → K is called asymptotically nonexpansive if there is a sequence {k_{n}} of positive numbers with the property $\underset{n\to \infty}{\mathrm{lim}}{k}_{n}=1$ such that
A point x is called a fixed point for a multivalued mapping T if x ∈ Tx.
Then, I − T is strongly demiclosed if for every sequence {x_{n}} in K which converges to x ∈ K and such that lim_{n}_{→∞}d(x_{n}, Tx_{n}) = 0, then x ∈ T(x).
For every continuous mapping T : K → 2^{K}, I − T is strongly demiclosed but the converse is not true. Notice also that if T satisfies condition (E), then I − T is strongly demiclosed.
The notation Fix(T) stands for the set of fixed points of a mapping T and Fix(t) ∩ Fix(T) stands for the set of common fixed points of t and T. A precise point x is called a common fixed point of t and T if x = tx ∈ Tx.
In 2009, Laokul and Panyanak [9] defined the iterative and proved the Δ–convergence for nonexpansive mapping in CAT(0) spaces as follows:
Let C be a nonempty closed convex subset of a complete CAT(0) space and t : C → C be a nonexpansive mapping with Fix(t) := {x ∈ C : tx = x} ≠ ∅︀. Suppose {x_{n}} is generated iteratively by x_{1} ∈ C,
for all n ∈ ℕ, where {α_{n}} and {β_{n}} are real sequences in [0, 1] such that one of the following two conditions is satisfied:
(i) α_{n} ∈ [a, b] and β_{n} ∈ [0, b] for some a, b with 0 < a ≤ b < 1,
(ii) α_{n} ∈ [a, 1] and β_{n} ∈ [a, b] for some a, b with 0 < a ≤ b < 1,
Then the sequence {x_{n}} is Δ–convergent to a fixed point of t.
In 2010, Sokhuma and Kaewkhao [15] proved the convergence theorem for a common fixed point in Banach spaces as follows.
Let E be a nonempty compact convex subset of a uniformly convex Banach space X, and t : E → E and T : E → KC(E) be a single-valued nonexpansive mapping and a multivalued nonexpansive mapping, respectively. Assume in addition that Fix(t) ∩ Fix(T) ≠ ∅︀ and Tw = {w} for all w ∈ Fix(t) ∩ Fix(T). Suppose {x_{n}} is generated iterative by x_{1} ∈ E,
for all n ∈ ℕ where z_{n} ∈ Tx_{n} and {α_{n}}, {β_{n}} are sequences of positive numbers satisfying 0 < a ≤ α_{n}, β_{n} ≤ b < 1. Then the sequence {x_{n}} converges strongly to a common fixed point of t and T.
In 2013, Sokhuma [14] proved the convergence theorem for a common fixed point in CAT(0) spaces as follows.
Let K be a nonempty compact convex subset of a complete CAT(0) space X and t : K → K and T : K → FC(K) a single-valued nonexpansive mapping and a multivalued nonexpansive mapping respectively and Fix(t) ∩ Fix(T) ≠ ∅︀ satisfying Tw = {w} for all w ∈ Fix(t) ∩ Fix(T). Let {x_{n}} is generated iterative by x_{1} ∈ K,
for all n ∈ ℕ where z_{n} ∈ Tx_{n} and {α_{n}}, {β_{n}} are sequences of positive numbers satisfying 0 < a ≤ α_{n}, β_{n} ≤ b < 1. Then the sequence {x_{n}} converges strongly to a common fixed point of t and T.
In 2013, Laowang and Panyanak proved the convergence theorem for a common fixed point in CAT(0) spaces as follows.
Let C be a nonempty bounded closed convex subset of a complete CAT(0) space X. Let f : C → C be a pointwise asymtotically nonexpansive mapping, and g : C → C a quasi-nonexpansive mapping, and let T : C → KC(C) be a multivalued mapping satisfying conditions (E) and C_{λ} for some λ ∈ (0, 1). If f, g and T are pairwise commuting, then there exists a point z ∈ C such that z = f(z) = g(z) ∈ T(z).
In 2015, Akkasriworn and Sokhuma [1] proved the convergence theorem for a common fixed point in a complete CAT(0) space as follows.
Theorem 1.2
Let E be a nonempty bounded closed convex subset of a complete CAT(0) space X, t : E → E and T : E → FB(E) an asymptotically nonexpansive mapping and a multivalued nonexpansive mapping, respectively. Assume that t and T are commuting and Fix(t) ∩ Fix(T) ≠ ∅︀ satisfying Tw = {w} for all w ∈ Fix(t) ∩ Fix(T) and $\sum _{n=1}^{\infty}({k}_{n}-1)}<\infty $. Let {x_{n}} be the sequence of the modified Ishikawa iterates defined by
for all n ∈ ℕ where z_{n} ∈ Tt^{n}x_{n} and {α_{n}}, {β_{n}} ∈ [0, 1]. Then {x_{n}} is Δ–convergent to a common fixed point of t and T.
In 2016, Uddin and Imdad [17] introduced the following iteration scheme:
Let K be a nonempty closed, bounded and convex subset of Banach space X, let f : K → K be a single-valued nonexpansive mapping and let T : K → FB(K) be a multivalued nonexpansive mapping with Fix(f) ∩ Fix(T) ≠ ∅︀ such that P_{T} is a nonexpansive mapping. The sequence {x_{n}} of the modified Ishikawa iteration is defined by
where x_{0} ∈ K, z_{n} ∈ P_{T} (x_{n}) and 0 < a ≤ α_{n}, β_{n} ≤ b < 1. Then {x_{n}} converges strongly to a common fixed point of f and T.
The Ishikawa iteration method was studied with respect to a pair of single-valued asymptotically nonexpansive mapping and a multivalued nonexpansive mapping. It also established the convergence theorem of a sequence from such process in a nonempty bounded closed convex subset of a complete CAT(0) space. A restricted condition (called end-point condition) in Akkasriworn and Sokhuma’s results was removed [1].
Here, an iteration method modifying the above was introduced and called the Ishikawa iteration method
Definition 1.3
Let K be a nonempty bounded closed convex subset of a complete CAT(0) space X, t : K → K be a single-valued asymptotically nonexpansive mapping and T : K → PB(K) be a multivalued nonexpansive mapping where P_{T} (x) = {y ∈ Tx : d(x, y) = dist(x, Tx)}. For fixed x_{1} ∈ K the sequence {x_{n}} of the Ishikawa iteration is defined by
for all n ∈ ℕ where z_{n} ∈ P_{T} (t^{n}x_{n}) and {α_{n}}, {β_{n}} ∈ (0, 1).
2. Preliminaries
Relevant basic definitions followed previous research results and iterative methods were used frequently.
Let (X, d) be a metric space. A geodesic path joining x ∈ X to y ∈ X is a map c from a closed interval [0, s] ⊂ ℝ to X such that c(0) = x, c(s) = y, and d(c(t), c(u)) = |t − u| for all t, u ∈ [0, s]. In particular, c is an isometry and d(x, y) = s. The image α of c is called a geodesic (or metric) segment joining x and y. When it is unique this geodesic segment is denoted by [x, y]. The space (X, d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X. A subset Y ⊆ X is said to be convex if Y includes every geodesic segment joining any two of its points.
A geodesic triangle Δ(x_{1}, x_{2}, x_{3}) in a geodesic metric space (X, d) consists of three points x_{1}, x_{2}, x_{3} in X (the vertices of Δ) and a geodesic segment between each pair of vertices (the edges of Δ). A comparison triangle for the geodesic triangle Δ(x_{1}, x_{2}, x_{3}) in (X, d) is a triangle Δ̄(x_{1}, x_{2}, x_{3}) := Δ(χ̄_{1}, χ̄_{2}, χ̄_{3}) in the Euclidean plane such that for i, j ∈ {1, 2, 3}.
A geodesic space is said to be a CAT(0) space if all geodesic triangles of appropriate size satisfy the following comparison axiom.
CAT(0): Let Δ be a geodesic triangle in X and let Δ̄ be a comparison triangle for Δ. Then Δ is said to satisfy the CAT(0) inequality if for all x, y ∈ Δ and all comparison points χ̄, ȳ ∈ Δ̄, .
If x, y_{1}, y_{2} are points in a CAT(0) space and
This is the (CN) inequality of Bruhat and Tits [3]. A geodesic space is a CAT(0) space if and only if it satisfies the (CN) inequality [2].
The following results and methods deal with the concept of asymptotic centres. Let K be a nonempty closed convex subset of a CAT(0) space X and {x_{n}} be a bounded sequence in X. For x ∈ X, define the asymptotic radius of {x_{n}} at x as the number
The number r and the set A are called the asymptotic radius and asymptotic centre of {x_{n}} relative to K respectively.
If X is a complete CAT(0) space and K is a closed convex subset of X, then A(K, {x_{n}}) consists of exactly one point. A sequence {x_{n}} in CAT(0) space X is said to be Δ–convergent to x ∈ X if x is the unique asymptotic centre of every subsequence of {x_{n}}. A bounded sequence {x_{n}} is said to be regular with respect to K if for every subsequence {${x}_{n}^{\prime}$}, we get
$$r(K,\{{x}_{n}\})=r(K,\{{x}_{n}^{\prime}\}).$$
The definition of Δ–convergence is presented below.
A sequence {x_{n}} in a CAT(0) space X is said to be Δ–convergent to x ∈ X if x is the unique asymptotic centre of {u_{n}} for every subsequence {u_{n}} of {x_{n}}. In this case,$\mathrm{\Delta}-\underset{n\to \infty}{\mathrm{lim}}{x}_{n}=x$and x is the Δ – limit of {x_{n}}.
Some elementary facts about CAT(0) spaces which will be used in the proofs of the main results are stated. The following lemma can be found in [4, 5, 8].
(i) For x, y ∈ X and u ∈ [0, 1], there exists a unique point z ∈ [x, y] such that$$d(x,z)=ud(x,y)\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}and\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}d(y,z)=(1-u)d(x,y).$$
he notation (1 − u)x ⊕ ty is used for the unique point z satisfying (2.2).
(ii) For x, y, z ∈ X and u ∈ [0, 1], $$d((1-u)x\oplus uy,z)\le (1-u)d(x,z)+ud(y,z).$$
The existence of fixed points for asymptotically nonexpansive mappings in CAT(0) spaces was proved by Kirk [7] as the following result.
Theorem 2.5
Let K be a nonempty bounded closed and convex subset of a complete CAT(0) space X and let t : K → K be asymptotically nonexpansive. Then t has a fixed point.
Let X be a complete CAT(0) space and K be a nonempty bounded closed and convex subset of X and t : K → K be an asymptotically nonexpansive mapping. Then I − t is demiclosed at 0.
Let K be a closed and convex subset of a complete CAT(0) space X and let t : K → X be an asymptotically nonexpansive mapping. Let {x_{n}} be a bounded sequence in K such that $\underset{n\to \infty}{\mathrm{lim}}\hspace{0.17em}d(t{x}_{n},{x}_{n})=0$and $\mathrm{\Delta}-\underset{n\to \infty}{\mathrm{lim}}{x}_{n}=w$. Then tw = w.
Let X be a complete CAT(0) space and let x ∈ X. Suppose {α_{n}} is a sequence in [a, b] for some a, b ∈ (0, 1) and {x_{n}}, {y_{n}} are sequences in X such that $\underset{n\to \infty}{\mathrm{lim}\hspace{0.17em}\mathrm{sup}}\hspace{0.17em}d({x}_{n},x)\le r,\underset{n\to \infty}{\mathrm{lim}\hspace{0.17em}\mathrm{sup}}\hspace{0.17em}d({y}_{n},x)\le r$, and $\underset{n\to \infty}{\mathrm{lim}}\hspace{0.17em}d((1-{\alpha}_{n}){x}_{n}\oplus {\alpha}_{n}{y}_{n},x)=r$ for some r ≥ 0. Then $\underset{n\to \infty}{\mathrm{lim}}\hspace{0.17em}d({x}_{n},{y}_{n})=0$.
The following fact is well-known.
Lemma 2.9
Let X be a CAT(0) space, K be a nonempty compact convex subset of X and {x_{n}} be a sequence in K. Then,
Let {a_{n}} and {b_{n}} be two sequences of nonnegative numbers such that
$${a}_{n+1}\le (1+{b}_{n}){a}_{n},$$
for all n ≥ 1. If $\sum _{n=1}^{\infty}{b}_{n}$ converges, then $\underset{n\to \infty}{\mathrm{lim}}{a}_{n}$ exists. In particular, if there is a subsequence of {a_{n}} which converges to 0 then $\underset{n\to \infty}{\mathrm{lim}}{a}_{n}=0$.
3. Main Results
The following lemmas play very important roles in this section.
Lemma 3.1
Let K be a nonempty bounded closed convex subset of a complete CAT(0) space X, T : K → PB(K) be a multivalued mapping, and P_{T} (x) = {y ∈ Tx : d(x, y) = dist(x, Tx)}. Then the following are equivalent
(1) x ∈ Fix(T), that is x ∈ Tx;
(2) P_{T} (x) = {x}, that is x = y for each y ∈ P_{T} (x);
(3) x ∈ Fix(P_{T}), that is x ∈ P_{T} (x).
Further, Fix(T) = Fix(P_{T}).
Proof
(1) implies (2). Since x ∈ Tx, then d(x, Tx) = 0. Therefore, for any y ∈ P_{T} (x), d(x, y) = dist(x, Tx) = 0 and so x = y. That is, P_{T} (x) = {x}.
(2) implies (3). Since P_{T} (x) = {x}, then x ∈ Fix(P_{T}) and hence x ∈ P_{T} (x).
(3) implies (1). Since x ∈ Fix(P_{T}), then x ∈ P_{T} (x). Therefore, d(x, x) = dist(x, Tx) = 0 and so x ∈ Tx by the closedness of Tx.
This implies that Fix(T) = Fix(P_{T}).
Lemma 3.2
Let K be a nonempty bounded closed convex subset of a complete CAT(0) space X, t : K → K and T : K → PB(K) a single-valued asymptotically nonexpansive mapping and a multivalued nonexpansive mapping respectively with Fix(t) ∩ Fix(T) ≠ ∅︀ such that P_{T} is nonexpansive and $\sum _{n=1}^{\infty}({k}_{n}-1)}<\infty $. Let {x_{n}} be the sequence of Ishikawa iterates defined by (1.2). Then $\underset{n\to \infty}{\mathrm{lim}}\hspace{0.17em}d({x}_{n},w)$ exists for all w ∈ Fix(t) ∩ Fix(T).
Proof
Let x_{1} ∈ K and w ∈ Fix(t)∩Fix(T), in the view of Lemma 3.1, w ∈ P_{T}(w) = {w}. Now consider,
By condition $\sum _{n=1}^{\infty}({k}_{n}-1)}<\infty $ and Lemma 2.10, which implies that $\underset{n\to \infty}{\mathrm{lim}}\hspace{0.17em}d({x}_{n},w)$ exists.
Lemma 3.3
Let K be a nonempty bounded closed convex subset of a complete CAT(0) space X, t : K → K and T : K → PB(K) a single-valued asymptotically nonexpansive mapping and a multivalued nonexpansive mapping respectively with Fix(t) ∩ Fix(T) ≠ ∅︀ such that P_{T} is nonexpansive and $\sum _{n=1}^{\infty}({k}_{n}-1)}<\infty $. Let {x_{n}} be the sequence of Ishikawa iterates defined by (1.2). Then $\underset{n\to \infty}{\mathrm{lim}}\hspace{0.17em}d({t}^{n}{y}_{n},{x}_{n})=0$.
Proof
Let x_{1} ∈ K and w ∈ Fix(t) ∩ Fix(T), in view of Lemma 3.1, w ∈ P_{T} (w) = {w}. From Lemma 3.2, $\underset{n\to \infty}{\mathrm{lim}}\hspace{0.17em}d({x}_{n},w)=c$ is set. Now consider,
Since $c=\underset{n\to \infty}{\mathrm{lim}}\hspace{0.17em}d({x}_{n+1},w)=\underset{n\to \infty}{\mathrm{lim}}\hspace{0.17em}d((1-{\alpha}_{n}){x}_{n}\oplus {\alpha}_{n}{t}^{n}{y}_{n},w)$, it implies by Lemma 2.8 that $\underset{n\to \infty}{\mathrm{lim}}\hspace{0.17em}d({t}^{n}{y}_{n},{x}_{n})=0$.
Lemma 3.4
Let K be a nonempty bounded closed convex subset of a complete CAT(0) space X, t : K → K and T : K → PB(K) a single-valued asymptotically nonexpansive mapping and a multivalued nonexpansive mapping respectively with Fix(t) ∩ Fix(T) ≠ ∅︀ such that P_{T} is nonexpansive and $\sum _{n=1}^{\infty}({k}_{n}-1)}<\infty $. Let {x_{n}} be the sequence of Ishikawa iterates defined by (1.2). Then $\underset{n\to \infty}{\mathrm{lim}}\hspace{0.17em}d({x}_{n},{z}_{n})=0$.
Proof
Let x_{1} ∈ K and w ∈ Fix(t) ∩ Fix(T), in view of Lemma 3.1, w ∈ P_{T} (w) = {w}. Consider,
Let K be a nonempty bounded closed convex subset of a complete CAT(0) space X, t : K → K and T : K → PB(K) a single-valued asymptotically nonexpansive mapping and a multivalued nonexpansive mapping respectively with Fix(t) ∩ Fix(T) ≠ ∅︀ such that P_{T} is nonexpansive and $\sum _{n=1}^{\infty}({k}_{n}-1)}<\infty $. Let {x_{n}} be the sequence of Ishikawa iterates defined by (1.2). Then $\underset{n\to \infty}{\mathrm{lim}}\hspace{0.17em}d({t}^{n}{x}_{n},{x}_{n})=0$.
Hence, by Lemmas 3.3 and 3.4,$\underset{n\to \infty}{\mathrm{lim}}\hspace{0.17em}d({t}^{n}{x}_{n},{x}_{n})=0$.
Lemma 3.6
Let K be a nonempty bounded closed convex subset of a complete CAT(0) space X, t : K → K and T : K → PB(K) a single-valued asymptotically nonexpansive mapping and a multivalued nonexpansive mapping respectively with Fix(t) ∩ Fix(T) ≠ ∅︀ such that P_{T} is nonexpansive and $\sum _{n=1}^{\infty}({k}_{n}-1)}<\infty $. Let {x_{n}} be the sequence of Ishikawa iterates defined by (1.2). Then $\underset{n\to \infty}{\mathrm{lim}}\hspace{0.17em}d(t{x}_{n},{x}_{n})=0$.
Let K be a nonempty bounded closed convex subset of a complete CAT(0) space X, t : K → K and T : K → PB(K) a single-valued asymptotically nonexpansive mapping and a multivalued nonexpansive mapping respectively with Fix(t) ∩ Fix(T) ≠ ∅︀ such that P_{T} is nonexpansive and $\sum _{n=1}^{\infty}({k}_{n}-1)}<\infty $. Let {x_{n}} be the sequence of Ishikawa iterates defined by (1.2). Then {x_{n}} is Δ–convergent to y implies y ∈ Fix(t) ∩ Fix(T).
Proof
Since {x_{n}} is Δ–convergent to y. From Lemma 3.6,
It follows that y ∈ Fix(P_{T}) then y ∈ Fix(T). Therefore y ∈ Fix(t) ∩ Fix(T) as desired.
Theorem 3.8
Let K be a nonempty bounded closed convex subset of a complete CAT(0) space X, t : K → K and T : K → PB(K) a single-valued asymptotically nonexpansive mapping and a multivalued nonexpansive mapping respectively with Fix(t) ∩ Fix(T) ≠ ∅︀ such that P_{T} is nonexpansive and $\sum _{n=1}^{\infty}({k}_{n}-1)}<\infty $. Let {x_{n}} be the sequence of Ishikawa iterates defined by (1.2). Then {x_{n}} is Δ–convergent to a common fixed point of t and T.
Proof
Since Lemma 3.6 guarantees that {x_{n}} is bounded and $\underset{n\to \infty}{\mathrm{lim}}\hspace{0.17em}d(t{x}_{n},{x}_{n})=0$. So, let ω_{w}(x_{n}) := ∪A({u_{n}}) where the union is taken over all subsequences {u_{n}} of {x_{n}}. If ω_{w}(x_{n}) ⊂ Fix(t) ∩ Fix(T), then there exists a subsequence {u_{n}} of {x_{n}} such that A({u_{n}}) = {u}. By Lemmas 1.2 and 1.3 there exists a subsequence {v_{n}} of {u_{n}} such that $\mathrm{\Delta}-\underset{n\to \infty}{\mathrm{lim}}{v}_{n}=v\in K$. Since
$\underset{n\to \infty}{\mathrm{lim}}\hspace{0.17em}d(t{v}_{n},{v}_{n})=0$, it follows that v ∈ Fix(t). Since,
It follows that v ∈ Fix(P_{T}) and v ∈ Fix(T) by Lemma 3.1. Therefore v ∈ Fix(t) ∩ Fix(T) as desired. Suppose that u ≠ v, since t is a single-valued asymptotically nonexpansive mapping and v ∈ Fix(t) ∩ Fix(T),$\underset{n\to \infty}{\mathrm{lim}}\hspace{0.17em}d({x}_{n},v)$ exists by Lemma 3.2. Then by the uniqueness of asymptotic centres,
a contradiction, and hence u = v ∈ Fix(t) ∩ Fix(T).
To show that {x_{n}} is Δ–convergent to a common fixed point of t and T, it suffices to show that ω_{w}(x_{n}) consists of exactly one point. Let {u_{n}} be a subsequence of {x_{n}}. By Lemmas 1.2 and 1.3 there exists a subsequence {v_{n}} of {u_{n}} such that $\mathrm{\Delta}-\underset{n\to \infty}{\mathrm{lim}}{v}_{n}=v\in K$. Let A({u_{n}}) = {u} and A({x_{n}}) = {x}. It has seen that u = v and v ∈ Fix(t) ∩ Fix(T).
It can complete the proof by showing that x = v. Suppose not, since $\underset{n\to \infty}{\mathrm{lim}}\hspace{0.17em}d({x}_{n},v)$ exists, then by the uniqueness of asymptotic centres,
a contradiction, and hence the conclusion follows.
Acknowledgements
I would like to thank the Institute for Research and Development, Phranakhon Rajabhat University, for financial support.
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