Abstract
In this article we modify an iteration process to prove strong convergence and Δ— convergence theorems for a finite family of nonexpansive multivalued mappings in hyperbolic spaces. The results presented here extend some existing results in the literature.
1 Introduction
Many important problems of mathematics, including boundary value problems for nonlinear ordinary or partial differential equation, can be translated in terms of a fixed point equation T x = x for a given mapping T on a Banach space. The class of nonexpansive mappings contains contractions as a subclass and its study has remained a popular area of research ever since its introduction. The iterative construction of fixed points of these mappings is a fascinating field of research. The fixed point problem for one or a family of nonexpansive mappings has been studied in Banach spaces, metric spaces and hyperbolic spaces [1-12].
Most of the fundamental early results discovered for nonexpansive mappings were done in the context of Banach spaces. It is then natural to try to develop a similar theory in the nonlinear spaces. The closest class of sets considered was the class of hyperbolic spaces that enjoys convexity properties very similar to the linear one. This class of metric spaces includes all normed vector spaces, Hadamard manifolds, as well as the Hilbert ball and the cartesian product of Hilbert balls.
Multivalued mappings arise in optimal control theory, especially inclusions and related subjects like game theory and economics. In physics, multivalued mappings play an increasingly important role. They form the mathematical basis for Dirac’s magnetic monopoles, for the theory of defects in crystals and the resulting plasticity of materials, for vortices in superfluids and superconductors, and for phase transitions in these systems, for instance melting and quark confinement. They are the origin of gauge field structures in many branches of physics.
In 2009 Yildirim and Ozdemir [13] used the following iteration to approximate fixed point of nonself asymptotically nonexpansive mappings.
Let x1 ∈ C and {xn}be the sequence generated by
where y0n = xn. For k = 3 the iterative process (1) is reduced to SP iteration which is defined by Phuengrattana and Suantai [14] in 2011 and iteration process of Thianwan [15, 16] for k = 2. Also, the iterative process (1) is the generalized form of the modified Mann (one-step) iterative process which is given by Schu [17].
In 2010 Kettapun et all [18] studied the iteration process (1) for self mapping in Banach spaces. Recently, Gunduz and Akbulut [19] studied this iteration for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces by using the following modified version of it.
where αin ∈ [0, 1], for all i = 1,2,...,k and any x1∈ C.
Now, we use the iteration (2) for a finite family of nonexpansive multivalued mappings in hyperbolic spaces and get some convergence results.
Let E be a hyperbolic space and D be a nonempty convex subset of E. Let {Ti : i = 1,2,...,k} be a family of multivalued mappings such that Ti : D → P(D)and
where
2 Preliminaries
Now we need to give some notions about the concept of hyperbolic spaces and multivalued mappings.
A hyperbolic space is a triple (X, d, W)such that (X, d)is a metric space and W : X × X × [0,1]→ X is a mapping satisfying the following conditions.
(W1) d(z, W (x, y, α)≤ (1 - α) d (z, x) + αd(z, y),
(W2) d(W (x, y, α), W(x, y, β)) = |α - β| d (x, y),
(W3) W(x, y) = W(y, x,(1 - α)),
(W4) d(W (x, z, α), W(y, w, α)) ≤ (1 - α) d (x, y) + αd (z, w) for all x, y, z, w ∈ X and α, β ∈[0,1].
Let D ⊂ X if W(x, y, α) ∈ D for all x; y ∈ K and α ∈ [0,1], then D is called convex. If (X, d, W)satisfies only (W1), it is reduced to the convex metric space introduced by Takahashi [20] which incorporates all normed linear spaces,
A hyperbolic space (X, d, W)is said to be uniformly convex [26] if there exists a δ ∈ (0,1] such that
for all u, x, y ∈X, r > 0 and ∊ ∈ (0,2].
A map η : (0, ∞ × (0,2] → (0,1] which satisfies such aδ = η(r, ∊) for given r > 0 and ∊ ∈ (0,2], is called modulus of uniform convexity. We call η monotone if it decreases with r for a fixed ∊.
Let (X, d)be a metric space and K be a nonempty subset of X, K is said to be proximinal if there exists an element y ∈ K such that
for each x ∈ X. The collection of all nonempty compact subsets of K, the collection of all nonempty closed bounded subsets and nonempty proximinal bounded subsets of K are denoted by C(K),CB(K)and P(K)respectively. The Hausdorff metric H on CB(X)is defined by
for all A, B ∈ CB(X): Let T : K → CB(X)be a multivalued mapping. An element x ∈K is said to be a fixed point of T if x ∈ Tx: A multivalued mapping T : K → CB(X)is said to be nonexpansive if
Now,we need to give some definitions and notations to mention the concept of convergences in hyperbolic spaces.
Let {xn} be a bounded sequence in a hyperbolic space (X, d, W). Let r be a continuous functional r(·,{xn}): W X →[0, ∞) given by
The asymptotic radius r({xn}) of {xn}is given by
The asymptotic center AK.({xn}) of a bounded sequence {xn}with respect to K ⊂X is the set
If the asymptotic center is taken with respect to X, then it is simply denoted by A({xn}).
([26, Proposition 3.3]). Let (X, d, W) be a complete uniformly convex hyperbolic space. Every bounded sequence {xn} in X has a unique asymptotic center with respect to any nonempty closed convex subset C of X.
Recall that if x is the unique asymptotic center of {un}for every subsequence {un}of {xn}then the sequence {xn} in X is said to be Δ—converge to x ∈ X. In this case, we write Δ — limn→∞xn = x and call x the Δ-limit of {xn}.
This concept in general metric spaces was coined by Lim [3] and Kirk and Panyanak [27].
([28, Lemma 2.5]). Let (X,d,W) be a uniformly convex hyperbolic space with monotone modulus of uniform convexity η. Let x ∈ E and {αn} be a sequence in [b,c] for some b,c ∈ (0,1). If {xn} and {yn} are sequences in X such that lim supn→∞d (xn, x) ≤r, lim supn→∞d(yn, x) ≤r and limn→∞d(W(xn, yn, αn),x) = r for some r ≥ 0, then limn→∞d(xn, yn) = 0.
([28, Lemma 2.6]). Let D be a nonempty closed subset of a uniformly convex hyperbolic space X and {xn} be a bounded sequence in D such that A({xn}) = {y} and r({xn}) = ρ. If {ym} is another sequence in D such that limn→∞r(ym,{xn}) = ρ, then limm→∞ym = y.
([29, Lemma 1]). Let Τ : D → Ρ (D) be a multivalued mapping andPT(x) = {y ∈Tx : d(x, y) = d(x, T x)}. Then the fallowings are equivalent.
(1) x ∈ F (Τ), that is, x ∈ Τ x,
(2) PT(χ) = {x}, that is, x = y for each y ∈ PT(x),
(3) x ∈ F (ΡT), that is, x ∈ PT(x)- Moreover, F (Τ) = F(PT).
([30, p. 480]). Let Α, Β ∈ CB(X) and a ∈ A. If η > 0, then there exists b ∈ Β such that d(a, b) ≤ Η(Α,Β) + η.
3 Main results
Now we give two useful lemmas to prove our main results.
Let D be a nonempty closed convex subset of a hyperbolic space X and {Ti : i = 1,2,... ,k} be a family of nonexpansive multivalued mappings such that
(1)
(2)
(3)
(4)
(5)
(6)
Let p ∈ F.
(1). For i = 1, 2, 3,... ,k, we have
(2). In a similar way with part (1), we get
(3). For i = 1, 2, 3,..., k, we have
(4). We prove this item in three parts. Firstly
Secondly, we assume that d(yjn, p) ≤ d(xn, p) holds for some 1 ≤ j ≤ k — 2. Then
Lastly,
So, by induction, we get
for all i = 1, 2,... ,k - 1.
(5). By part (4), we have
(6). By part (5), we get
for i = 1,2,... ,k. Thus, we obtain limn→∞d(xn, p)exists for each p ∈ F.
Let D be a nonempty closed subset of a uniformly convex hyperbolic space X and Ti : D →Ρ (D) be a family of multivalued mappings such that
for i = 1,2,... ,k.
By Lemma 3.1, limn→∞d(xn, p)exists for each p ∈ F. Therefore, for a c ≥ 0, we have
By taking lim sup on both sides of (10), we have
for all i = 1,2,... ,k - 1.
So, by (6) and (13), we get
forall i = 1,2,..., k.
Since limn→∞d(xn+1, p) = c, we have
We claim that,
for all j = 2,3,..., k . By Lemma 3.1 we get
for all i = 1, 2 ,..., k – 1. Therefore, from (16), we obtain
for i = 2,3,..., k it. By (3), (13) and (17), we obtain
Using (13), (14) and Lemma 2.2, we get
for all i = 1,2,... ,k. From (3), we get
By (18), we have
for i = 1,2,... ,k. Since
by taking i = 1 in (18), we get
By known triangle inequality,
for all i = 1, 2,..., k - 1. It follows by (19) and (20) that
For i = 1,2, 3,..., k
From (18) and (21), we conclude
Let D be a nonempty closed convex subset of a complete uniformly convex hyperbolic space E with monotone modulus of uniform convexity η. Let Ti,
Proof. Let p ∈ F. Then
To prove this, we take another sequence {zm}in PT1(v) Then,
this gives
Let D be a nonempty closed convex subset of a hyperbolic space E and Let Ti,
If {xn}converges to p ∈ F, then limn→∞d(xn, p) = 0. since 0 ≤ d(xn,F) ≤d(xn, p), we have limn→∞d(xn, F) = 0. Conversely, suppose that limn→∞ inf d(xn, F) = 0. By Lemma 3.1, we have
which implies
This gives that limn→∞d(xn, F)exists. Therefore, by the hypothesis of our theorem, lim infn→∞d(xn, F) = 0. Thus we have limn→∞d(xn, F) = 0. Let us show that {xn}is a Cauchy sequence in D. Let m, n, ∈ ℕ and assume m > n. Then it follows that d(xm, p) ≤d(xn, p)for all p ∈ F. Thus we get,
Taking inf on the set F, we have d(xm, xn)≤ d(xn, F). We show that {xn}is a Cauchy sequence in D. By taking as m, n → ∞ in the inequality d(xm, xn)≤ d(xn, F). So, it converges to a q ∈ D. Now it is left to show that
q ∈ F (Τ1). Indeed by
This implies
which shows
Now we give the definition of condition (B) of Senter and Dotson for a finite family of multivalued mappings to complete the proof of the following theorem.
([31]). The multivalued nonexpansive mappingsΤ1, Τ2,...,Tk : D → CB(D) are said to satisfy condition (B). If there exists a nondecreasing function ƒ : [0, ∞) → [0, ∞) with f(0) = 0, f(r) > 0 for all r ∈ (0, ∞) such that
A map Τ : D → Ρ(D) is called semi-compact if any bounded sequence {xn} satisfying
Let D be a nonempty closed convex subset of a complete uniformly convex hyperbolic space E with monotone modulus of uniform convexity η. Let Ti,
Proof. By Lemma 3.1, limn→∞d(xn, p)exists for all p ∈ F. We call it c for some c ≥ 0.Then if c = 0,proof is completed. Assume c > 0. Now d (xn+1, p) ≤d (xn, p)gives that
which means that
and so limn→∞ (d(xn, F)) = 0. By the properties of ƒ, we get limn→∞d(xn, F) = 0. Finally by applying Theorem 3.4, we obtain the result.
Let D be a nonempty closed convex subset of a complete uniformly convex hyperbolic space E with monotone modulus of uniform convexity η and let Ti,
Acknowledgement
The authors would like to thank the reviewers for their valuable comments and suggestions to improve the quality of the paper.
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