Convergence theorems for a family of multivalued nonexpansive mappings in hyperbolic spaces

Osman Alagoz; Birol Gunduz; Sezgin Akbulut

doi:10.1515/math-2016-0095

Article Open Access

Convergence theorems for a family of multivalued nonexpansive mappings in hyperbolic spaces

Osman Alagoz , Birol Gunduz and Sezgin Akbulut

Published/Copyright: December 24, 2016

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$Open Mathematics$

From the journal Open Mathematics Volume 14 Issue 1

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.

Key words: Hyperbolic spaces; Δ— convergence; Nonexpansive multivalued mapping

MSC 2010: 47H10; 49M05; 54H25

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 {x_n}be the sequence generated by

(1)xn+1=(1−αkn)y(k−1)n+αknTkny(k−1)n,y(k−1)n=(1−α(k−1)n)y(k−2)n+α(k−1)nTk−1ny(k−2)n,⋮y2n=(1−α2n)y1n+α2nT2ny1n,y1n=(1−α1n)y0n+α1nT1ny0n

where y_0n = x_n. 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.

(2)xn+1=WTkny(k−1)n,y(k−1)n,αkny(k−1)n=WTk−1ny(k−1)n,y(k−1)n,α(k−1)n⋮y2n=WT2ny1n,y1n,α2ny1n=WT1ny0n,y0n,α1n

where α_in ∈ [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 {T_i : i = 1,2,...,k} be a family of multivalued mappings such that T_i : D → P(D)and PTi(x)={y∈Tix:d(x,y)=d(x,Tix)} is a nonexpansive mappping. Suppose that α_in ∈ [0,1], for all n = 1, 2,... and i = 1, 2,...,k for x₀ ∈ D and let {x_n}be the sequence generated by the following algorithm;

(4)xn+1=Wu(k−1)n,y(k−1)n,αkny(k−1)n=Wu(k−2)n,y(k−2)n,α(k−1)n⋮y2n=Wu1n,y1n,α2ny1n=Wu0n,y0n,α1n

where uin∈PTi+1(yin)fori=0,1,2,…,k−1andy0n=xn.

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, R-trees and the Hilbert Ball with the Hyperbolic metric [21]. If (X, d, W)satisfies (W1)-(W3) then it is called the hyperbolic space in the sense of Goebel and Kirk [22]. After Itoh [23] gave the condition (III) that is equivalent (W4) condition, Reich and Shafrir [24] and Kirk [25] defined their notions of hyperbolic space by using Itoh’s condition.

A hyperbolic space (X, d, W)is said to be uniformly convex [26] if there exists a δ ∈ (0,1] such that

d(x,u)≤rd(y,u)≤rd(x,y)≥ϵr⇒d(W(x,y,12),u)≤(1−δ)r

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

d(x,y)=d(x,K):=infz∈Kd(x,z)

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

H(A,B):=maxsupx∈A⁡d(x,B),supy∈B⁡d(y,A)

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

H(Tx,Ty)≤d(x,y),∀x,y∈K.

Now,we need to give some definitions and notations to mention the concept of convergences in hyperbolic spaces.

Let {x_n} be a bounded sequence in a hyperbolic space (X, d, W). Let r be a continuous functional r(·,{x_n}): W X →[0, ∞) given by

r(x,{xn})=limsupn→∞⁡d(x,xn).

The asymptotic radius r({x_n}) of {x_n}is given by

r({xn})=inf{r(x,{xn}):x∈X}.

The asymptotic center A_K.({x_n}) of a bounded sequence {x_n}with respect to K ⊂X is the set

AK({xn})={x∈X:r(x,{xn})≤r(y,{xn}),∀y∈K}.

If the asymptotic center is taken with respect to X, then it is simply denoted by A({x_n}).

Lemma 2.1

([26, Proposition 3.3]). Let (X, d, W) be a complete uniformly convex hyperbolic space. Every bounded sequence {x_n} 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 {u_n}for every subsequence {u_n}of {x_n}then the sequence {x_n} in X is said to be Δ—converge to x ∈ X. In this case, we write Δ — lim_n_→∞x_n = x and call x the Δ-limit of {x_n}.

This concept in general metric spaces was coined by Lim [3] and Kirk and Panyanak [27].

Lemma 2.2

([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 {α_n} be a sequence in [b,c] for some b,c ∈ (0,1). If {x_n} and {y_n} are sequences in X such that lim sup_n_→∞d (x_n, x) ≤r, lim sup_n_→∞d(y_n, x) ≤r and lim_n_→∞d(W(x_n, y_n, α_n),x) = r for some r ≥ 0, then lim_n_→∞d(x_n, y_n) = 0.

Lemma 2.3

([28, Lemma 2.6]). Let D be a nonempty closed subset of a uniformly convex hyperbolic space X and {x_n} be a bounded sequence in D such that A({x_n}) = {y} and r({x_n}) = ρ. If {y_m} is another sequence in D such that lim_n_→∞r(y_m,{x_n}) = ρ, then lim_m_→∞y_m = y.

Lemma 2.4

([29, Lemma 1]). Let Τ : D → Ρ (D) be a multivalued mapping andP_T(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) P_T(χ) = {x}, that is, x = y for each y ∈ P_T(x),

(3) x ∈ F (Ρ_T), that is, x ∈ P_T(x)- Moreover, F (Τ) = F(P_T).

Lemma 2.5

([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.

Lemma 3.1

Let D be a nonempty closed convex subset of a hyperbolic space X and {T_i : i = 1,2,... ,k} be a family of nonexpansive multivalued mappings such that PTi is nonexpansive mapping and p ∈F(T_i) for i = 1, 2,..., k. Suppose that F=⋂i=1kFTi≠∅, and the iterative sequence {x_n} is defined by (3 Then for p ∈ F, we get

(1) d(xn,PTixn)≤2d(xn,p) for all i = 1,2,... ,k,

(2) d(y(i−1)n,u(i−1)n)≤2d(y(i−1)n,p) for all i = 1,2,... ,k,

(3) d(u(i−1)n,p)≤d(y(i−1)n,p) for all i = 1,2,... ,k,

(4) d(yin,p)≤d(xn,p) for all i = 1,2,... ,k - 1,

(5) d(xn+1,p)≤d(xn,p),,

(6) limn→∞d(xn,p)exists,

Proof

Let p ∈ F.

(1). For i = 1, 2, 3,... ,k, we have

(4)d(xn,PTixn)≤d(xn,p)+d(p,PTi(xn))≤d(xn,p)+H(PTi(p),PTi(xn))≤d(xn,p)+d(p,xn)=2d(xn,p).

(2). In a similar way with part (1), we get

(5)d(y(i−1)n,u(i−1)n)≤d(y(i−1)n,p)+d(p,u(i−1)n)≤d(y(i−1)n,p)+d(p,PTi(y(i−1)n))≤d(y(i−1)n,p)+H(PTi(p),PTi(y(i−1)n))≤d(y(i−1)n,p)+d(p,y(i−1)n)=2d(y(i−1)n,p).

(3). For i = 1, 2, 3,..., k, we have

(6)d(u(i−1)n,p)≤d(u(i−1)n,PTi(p))≤H(PTi(y(i−1)n),PTi(p))≤d(y(i−1)n,p).

(4). We prove this item in three parts. Firstly

(7)d(y1n,p)=d(W(u0n,y0nα1n),p)≤(1−α1n)d(u0n,p)+α1nd(y0n,p)≤(1−α1n)d(u0n,PT1(p))+α1nd(y0n,p)≤(1−α1n)H(PT1(y0n),PT1(p))+α1nd(y0n,p)≤(1−α1n)d(y0n,P)+α1nd(y0n,p)=d(xn,p)

Secondly, we assume that d(y_jn, p) ≤ d(x_n, p) holds for some 1 ≤ j ≤ k — 2. Then

(8)d(y(j+1)n,p)=d(W(ujn,yjn,α(j+1)n),p)≤(1−α(j+1)n)d(ujn,p)+α(j+1)nd(yjn,p)≤(1−α(j+1)n)H(PT(j+1)n(yjn),PT(j+1)n(p))+α(j+1)nd(yjn,p)≤(1−α(j+1)nd(yjn,p)+α(j+1)nd(yjn,p)≤d(yjn,p)≤d(xn,p).

Lastly,

(9)d(y(k−1)n,p)=d(W(u(k−2)n,yy(k−2)n,α(k−1)n),p)≤(1−α(k−1)n)d(u(k−2)n,p)+α(k−1)nd(y(k−2)n,p)≤(1−α(k−1)n)d(u(k−2)n,PTk−1(p))+α(k−1)nd(y(k−2)n,p)≤(1−α(k−1)n)H(PTk−1(y(k−2)n),PTk−1(p))+α(k−1)nd(y(k−2)n,p)≤(1−α(k−1)n)d(y(k−2)n,p)+α(k−1)nd(y(k−2)n,p)=d(y(k−2)n,p).

So, by induction, we get

(10)d(yin,p)≤d(xn,p)

for all i = 1, 2,... ,k - 1.

(5). By part (4), we have

(11)d(xn+1,p)=d(W(u(k−1)n,y(k−1)n,αkn),p)≤(1−αkn)d(u(k−1)n,p)+αknd(y(k−1)n,p)≤(1−αkn)d(u(k−1)n,pTk(p)+αknd(y(k−1)n),p)≤(1−αkn)H(pTk(y(k−1)n),pTk(p))+αknd(y(k−1)n,p)=(1−αkn)d(y(k−1)n,p)+αknd(y(k−1)n,p)=d(y(k−1)n,p)≤d(xn,p).

(6). By part (5), we get

d(xn+1,p)≤d(xn,p)

for i = 1,2,... ,k. Thus, we obtain lim_n_→∞d(x_n, p)exists for each p ∈ F.

Lemma 3.2

Let D be a nonempty closed subset of a uniformly convex hyperbolic space X and T_i : D →Ρ (D) be a family of multivalued mappings such that PTi is nonexpansive mapping for i = 1,2,...,k with a set of F ≠ ∅. Then for the iterative process {x_n} defined in (3), we have

limn→∞⁡d(xn,PTixn)=0

for i = 1,2,... ,k.

Proof

By Lemma 3.1, lim_n_→∞d(x_n, p)exists for each p ∈ F. Therefore, for a c ≥ 0, we have

(12)limn→∞⁡d(xn,p)=c.

By taking lim sup on both sides of (10), we have

(13)limsupn→∞⁡dyin,p≤c

for all i = 1,2,... ,k - 1.

So, by (6) and (13), we get

(14)limsupn→∞⁡dui−1n,p≤c

forall i = 1,2,..., k.

Since lim_n_→∞d(x_n+₁, p) = c, we have limn→∞d(Wuk−1n,yk−1n,αkn,p)=c.. From Lemma 2.2, (13) and (14), we have

limn→∞⁡dyk−1n,uk−1n=0.

We claim that,

(15)limn→∞⁡dyj−1n,uj−1n=0

for all j = 2,3,..., k . By Lemma 3.1 we get

(16)dxn+1,p≤dyin,p

for all i = 1, 2 ,..., k – 1. Therefore, from (16), we obtain

(17)c≤liminfn→∞⁡dyi−1n,p

for i = 2,3,..., k it. By (3), (13) and (17), we obtain

limn→∞⁡d(W(uj−2n,yj−2n,αj−1n),p)=limn→∞⁡d(y(j−1)n,p)=c.

Using (13), (14) and Lemma 2.2, we get limn→∞duj−2n,yj−2n=0. So, by induction

(18)limn→∞⁡d(y(i−1)n,u(i−1)n)=0

for all i = 1,2,... ,k. From (3), we get

d(yin,y(i−1)n)=d(W(u(i−1)n,yi−1n,αin),yi−1n)≤(1−αin)d(u(i−1)n,yi−1n).

By (18), we have

(19)limn→∞⁡d(yin,yi−1n)=0

for i = 1,2,... ,k. Since

d(xn,y1n)=d(xn,W(u0n,y0n,α1n)≤(1−α1n)d(xn,u0n)+α1nd(xn,y0n)=(1−α1n)d(xn,u0n).

by taking i = 1 in (18), we get

(20)limn→∞⁡d(xn,y1n)=0.

By known triangle inequality,

d(xn,yin)≤d(xn,y1n)+d(y1n,y2n)+⋯+d(y(i−1)n,yin).

for all i = 1, 2,..., k - 1. It follows by (19) and (20) that

(21)limn→∞d(xn,yin)=0.

For i = 1,2, 3,..., k

d(xn,PTixn)≤d(xn,y(i−1)n)+d(y(i−1)n,u(i−1)n)+d(ui−1n,PTixn)≤d(xn,y(i−1)n)+d(y(i−1)n,u(i−1)n)+H(PTiy(i−1)n,PTixn)≤2d(xn,y(i−1)n)+d(y(i−1)n,u(i−1)n).

From (18) and (21), we conclude

limn→∞⁡d(xn,PTixn)=0.

Theorem 3.3

Let D be a nonempty closed convex subset of a complete uniformly convex hyperbolic space E with monotone modulus of uniform convexity η. Let T_i, PTi and F be as in Lemma 3.2, Then the iterative process {x_n}, Δ— converges to p in F.

Proof

Proof. Let p ∈ F. Then p∈FTi=FPTi, for i = 1,2,... ,k. By the Lemma 3.1, {x_n}is bounded and so lim_n_→∞d(x_n, p)exists. Thus {x_n}has a unique asymptotic center. In other words, we have A({x_n}) = {x_n}. Let {v_n}be a subsequence of {x_n}such that A({v_n}) = {v}. From Lemma 3.2, we get limn→∞vn,PT1vn=0. We claim that υ is a fixed point of PT1.

To prove this, we take another sequence {z_m}in P_T1(v) Then,

r(zm,{vn})=limsupn→∞⁡d(zm,vn)≤limn→∞⁡{d(zm,PT1(vn)+d(PT1(vn),vn)}≤limn→∞⁡{H(PT1(v),PT1(vn))+d(PT1(vn),vn))}≤limsupn→∞⁡d(v,vn)=r(v,{vn})

this gives |r(zm,{vn})−r(v,{vn})|→0 for m →. By Lemma 2.3, we get lim_m→∞_→∞z_m = v. Note that T₁(υ) ∈ P(D)being proximinal is closed, hence PT1v is either closed or bounded. Consequently limm→∞zm=v∈PT1v. Similarly v∈PT2v,v∈PT3v…v∈PTkv. So ν ∈ F.

Theorem 3.4

Let D be a nonempty closed convex subset of a hyperbolic space E and Let T_i, PTi and F be as in Lemma 3.2 and let {x_n} be the iterative process defined in 3, then {x_n} converges to p in F if and only if lim_n_→∞d(x_n, F) = 0.

Proof

If {x_n}converges to p ∈ F, then lim_n_→∞d(x_n, p) = 0. since 0 ≤ d(x_n,F) ≤d(x_n, p), we have lim_n_→∞d(x_n, F) = 0. Conversely, suppose that lim_n_→∞ inf d(x_n, F) = 0. By Lemma 3.1, we have

d(xn+1,p)≤d(xn,p)

which implies

d(xn+1,F)≤d(xn,F).

This gives that lim_n_→∞d(x_n, F)exists. Therefore, by the hypothesis of our theorem, lim inf_n_→∞d(x_n, F) = 0. Thus we have lim_n_→∞d(x_n, F) = 0. Let us show that {x_n}is a Cauchy sequence in D. Let m, n, ∈ ℕ and assume m > n. Then it follows that d(x_m, p) ≤d(x_n, p)for all p ∈ F. Thus we get,

d(xm,xn)≤d(xm,p)+d(p,xn)≤2d(xn,p).

Taking inf on the set F, we have d(x_m, x_n)≤ d(x_n, F). We show that {x_n}is a Cauchy sequence in D. By taking as m, n → ∞ in the inequality d(x_m, x_n)≤ d(x_n, F). So, it converges to a q ∈ D. Now it is left to show that

q ∈ F (Τ₁). Indeed by d(xn,F(PT1))=infy∈F(PT1)d(xn,y).So for each ∊ > 0, there exists pn(ϵ)∈F(PT1) such that,

d(xn,pn(ϵ))<d(xn,F(PT1))+ϵ2.

This implies limn→∞d(xn,pn(ϵ))≤ϵ2.Sinced(pn,(ϵ)q)≤d(xn,p0(ϵ))+d(xn,q) it follows that limn→∞d(pn(ϵ),q)≤ϵ2. Finally

d(PT1(q),q)≤d(q,pn(ϵ))+d(pn(ϵ),PT1(q))≤d(q,pn(ϵ))+H(PT1(pn(ϵ)),PT1(q))≤2d(pn(ϵ),q)

which shows d(PT1(q),q)<ϵ.So,d(PT1(q),q)=0. In a similar way, we get for any i = 1, 2..., k we obtain d(PTi(q),q)=0. Since F is closed, q ∈ F.

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.

Definition 3.5

([31]). The multivalued nonexpansive mappingsΤ₁, Τ₂,...,T_k : 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

d(xn,Tixn)≥f(d(xn,F)),F≠∅.

Definition 3.6

A map Τ : D → Ρ(D) is called semi-compact if any bounded sequence {x_n} satisfying d(xn,Txn)→0as n → ∞ has a convergent subsequence.

Theorem 3.7

Let D be a nonempty closed convex subset of a complete uniformly convex hyperbolic space E with monotone modulus of uniform convexity η. Let T_i, PTi and F be as in Lemma 3.2. Suppose that each PTi satisfies condition (B). Then the iterative process {x_n} defined in (3) converges strongly to p ∈F.

Proof

Proof. By Lemma 3.1, lim_n_→∞d(x_n, 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 (x_n+₁, p) ≤d (x_n, p)gives that

infp∈F(Ti)d(xn+1,p)≤infp∈F(Ti)d(xn,p)

which means that d(xn+1,F)≤d(xn,F).So,limn→∞d(xn,F) exists. By using the condition (B)and Lemma 3.2 we obtain,

limn→∞f(d(xn,F))≤limn→∞d(xn,PTi(xn)→0

and so lim_n_→∞ (d(x_n, F)) = 0. By the properties of ƒ, we get lim_n_→∞d(x_n, F) = 0. Finally by applying Theorem 3.4, we obtain the result.

Theorem 3.8

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 T_i, PTi F be as in Lemma 3.2. Suppose that PTi is semi-compact then the iterative process {x_n} defined in (3) converges strongly to p ∈F.

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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Received: 2016-8-3

Accepted: 2016-10-27

Published Online: 2016-12-24

Published in Print: 2016-1-1

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