The implicit midpoint rule for nonexpansive mappings in 2-uniformly convex hyperbolic spaces

Abstract

The purpose of this paper is to introduce the implicit midpoint rule (IMR) of nonexpansive mappings in 2- uniformly convex hyperbolic spaces and study its convergence. Strong and △-convergence theorems based on this algorithm are proved in this new setting. The results obtained hold concurrently in uniformly convex Banach spaces, CAT(0) spaces and Hilbert spaces as special cases.

Citation

Fukhar-ud-din, H. and Khan, A.R. (2020), "The implicit midpoint rule for nonexpansive mappings in 2-uniformly convex hyperbolic spaces", Arab Journal of Mathematical Sciences, Vol. 26 No. 1/2, pp. 95-105. https://doi.org/10.1016/j.ajmsc.2019.02.002

1. Introduction

The iterative methods for approximating fixed points of nonexpansive mappings have received a great attention due to the fact that in many practical problems, the controlling operators are nonexpansive (cf. [16]). The iterative methods of Mann [17] and Halpern [9] are very popular (see also [20]). An implicit iterative method was proposed [25] and studied in [7,12]. The IMR is a powerful numerical method for solving ordinary differential equations and differential algebraic equations. For related works in this context, we refer the reader to [2,5,20,22].

For the ordinary differential equation

(1.1)y′(t)=g(t), y0=y(0),

IMR generates a sequence {yn} via the relation

1h(yn+1−yn)=g(yn+1+yn2)

where h>0 is a step size. It is well known that if g:ℝk→ℝk is Lipschitzian continuous and sufficiently smooth, then the sequence {yn} converges to the exact solution of (1.1) as h→0 uniformly over t∈[0,a] for any fixed a>0.

Based on the above fact, Alghamdi et al. [1] presented the following IMR for nonexpansive mappings in the setting of a Hilbert space H:

(1.2)yn+1=(1−tn)yn+tnT(yn+1+yn2)

where tn∈(0,1) and T:H→H is a nonexpansive mapping and established weak convergence of (1.2) to the fixed point of T under some control conditions on {tn}.

The extension of a linear version of a known result (usually in Banach spaces or Hilbert spaces) to metric spaces is very important. As an IMR for nonexpansive mappings involves general convex combinations, so we need some convex structure in a metric space to define an IMR on a nonlinear domain.

Let C be a nonempty subset of a metric space (M,d) and T:C→C a mapping. Set F(T)={x∈M:Tx=x}. The mapping T is: (i) nonexpansive if d(Tx,Ty)≤d(x,y) for all x,y∈C (ii) quasi-nonexpansive if d(Tx,y)≤d(x,y) for all x∈C and y∈F(T) (iii) semi-compact if for any bounded sequence {xn} in C satisfying d(xn,Txn)→0, there exists a subsequence {xni} of {xn} such that xni→x∈C (iv) completely continuous if every bounded sequence {xn} in C implies that {Txn} has a convergent subsequence. A sequence {xn} is Fejér monotone with respect to a subset C of M if d(xn+1,x)≤d(xn,x) for all x∈C.

For a bounded sequence {xn} in a metric space M, set

r(x,{xn})=limsupn→∞d(x,xn)

for all x∈M.

The asymptotic radius of {xn} with respect to C⊆M is defined as

r({xn})=infx∈Cr(x,{xn}).

A point y∈C is called the asymptotic centre of {xn} with respect to C⊆M if

r(y,{xn})≤r(x,{xn}) for all x∈C.

The set of all asymptotic centres of {xn} is denoted by A({xn}).

A sequence {xn} in M, is △-convergent to x∈M (△−limnxn=x) if x is the unique asymptotic centre of {un} for every subsequence {un} of {xn}. It has been observed that △-convergence in metric spaces constitutes an analogue of weak convergence in Hilbert spaces and both coincide in Hilbert spaces.

Let (M,d) be a metric space. Suppose that there exists a family F of metric segments such that any two points x,y in M are endpoints of a unique metric segment [x,y]∈ F ([x,y] is an isometric image of the real line interval [0,d(x,y)]). We denote by z the unique point αx⊕(1−α)y of [x,y] which satisfies

d(x,z)=(1−α)d(x,y) and d(z,y)=αd(x,y) for α∈I=[0,1].

Such metric spaces are usually called convex metric spaces [18]. A convex metric space M is hyperbolic if

(1.3)d(αx⊕(1−α)y,αz⊕(1−α)w)≤αd(x,z)+(1−α)d(y,w)

for all x,y,z,w∈M and α∈I.

For z=w, the hyperbolic inequality reduces to convex structure of Takahashi [23]

d(αx⊕(1−α)y,z)≤αd(x,z)+(1−α)d(y,z).

A nonempty subset C of a hyperbolic space M is convex if αx⊕(1−α)y∈C for all x,y∈C and α∈I. A few examples of nonlinear hyperbolic spaces are Hadamard manifolds [4], the Hilbert open unit ball equipped with the hyperbolic metric [8] and the CAT(0) spaces [14,15] while normed spaces and their subsets are linear hyperbolic spaces. Throughout this paper, we denote 12x⊕12y by x⊕y2.

A hyperbolic space M is uniformly convex if

δ(r,ε)=inf{1−1rd(a,x⊕y2):d(a,x)≤r,d(a,y)≤r,d(x,y)≥rε}>0,

for any a∈M, r>0 and ε>0.

Xu [24], extensively used the concept of p-uniform convexity; its nonlinear version in hyperbolic spaces for p=2 has been introduced by Khamsi and Khan [13] as under:

For a fixed a∈M,r>0,ε>0, define

ψ(r,ε)=inf{12d(a,x)2+12d(a,y)2−d(a,x⊕y2)2}

where the infimum is taken over all x,y∈M such that d(a,x)≤r,d(a,y)≤r and d(x,y)≥rε.

We say that M is 2-uniformly convex if

cM=inf{ψ(r,ε)r2ε2:r>0,ε>0} >0.

It has been shown in [13] that any CAT(0) space is 2-uniformly convex hyperbolic space with cM=14.

From now onwards we assume that M is a uniformly convex hyperbolic space with the property that for every s≥0,ε>0, there exists η(s,ε)>0 depending on s and ε such that δ(r,ε)>η(s,ε)>0 for any r>s.

Using the concept of metric segment [x,y], we translate (1.2) for nonexpansive mappings in a hyperbolic space as follows:

(1.4)x0=x∈C,xn+1=αnT(xn⊕xn+12)⊕(1−αn)xn,

where {αn} is the sequence in (0,1) satisfying (C1): liminfn→∞αn>0 and (C2): αn+12≤λαn2 for some λ>0.

The following known results are needed in the sequel.

Lemma 1.1 ([3]). Let C be a nonempty closed subset of a complete metric space (M,d) and {xn} be a Fejér monotone with respect to C. Then {xn} strongly converges to x∈C if and only if limn→∞d(xn,C)=0.

Lemma 1.2 ([6]). Let C be a nonempty closed and convex subset of a complete uniformly convex hyperbolic space M. Then every bounded sequence {yn} inM has a unique asymptotic centre with respect to C that lies in C.

Lemma 1.3 ([10]). Suppose that M is a 2-uniformly convex hyperbolic space. Then for any θ∈(0,1), we have that

d(u,θx⊕(1−θ)y)2≤θd(u,x)2+(1−θ)d(u,y)2−4cMmin{θ2,(1−θ)2}d(x,y)2,

for all u,x,y∈M and cM is the number as given above.

Our purpose in this paper is to approximate fixed point of nonexpansive mappings using iterative method (1.4) in a 2-uniformly convex hyperbolic spaces. This work provides a unified approach to convergence results in Hilbert spaces, uniformly convex Banach spaces and CAT(0) spaces.

2. Convergence in 2-uniformly convex hyperbolic spaces

Lemma 2.1. Let C be a nonempty convex subset of a complete hyperbolic space M and T:C→C a nonexpansive mapping. Then the sequence {xn} in (1.4) is well defined.

Proof. Define S:C→C by

Sx=α0T(x0⊕x2)⊕(1−α0)x0.

With the help of (1.3), we have

d(Sx,Sy)=d(α0T(x0⊕x2)⊕(1−α0)x0,α0T(x0⊕y2)⊕(1−α0)x0)     ≤α0d(T(x0⊕x2),T(x0⊕y2))     ≤α0d(x0⊕x2,x0⊕y2)     ≤α02d(x,y).

This gives that S is a contraction with contraction constant α02∈(0,1). Therefore by Banach contraction principle, there is a unique element x1∈C such that x1=Sx1=α0T(x0⊕x12)⊕(1−α0)x0. Hence x1 is achieved. Similarly, we can find x2 and so on. So in general,

xn+1=αnT(xn⊕xn+12)⊕(1−αn)xn.□

Lemma 2.2. Let C be a nonempty convex subset of a complete 2-uniformly convex hyperbolic space M and T:C→C a nonexpansive mapping such that F(T)≠φ. Then for the sequence {xn} in (1.4), we have the following: (i) limn→∞d(xn,p) exists for all p∈F(T)

• (ii) ∑n=1∞αnd(xn,xn+1) <∞

• (iii) ∑n=1∞αn2(1−αn)2d(xn,T(xn⊕xn+12))2<∞.

Proof. Let p∈F(T). Applying Lemma 1.3 to (1.4), we have that

d(xn+1,p)2=d(αnT(xn⊕xn+12)⊕(1−αn)xn,p)2     ≤αnd(T(xn⊕xn+12),p)2+(1−αn)d(xn,p)2     −4cMmin{αn2,(1−αn)2}d(xn,T(xn⊕xn+12))2     ≤αnd(xn⊕xn+12,p)2+(1−αn)d(xn,p)2     −4cMαn2(1−αn)2d(xn,T(xn⊕xn+12))2     ≤αnd(12d(xn,p)2+12d(xn+1,p)2−CM4d(xn,xn+1)2)     +(1−αn)d(xn,p)2−4cMαn2(1−αn)2d(xn,T(xn⊕xn+12))2.

That is,

(1−αn2)d(xn+1,p)≤(1−αn2)d(xn,p)−αnCM4d(xn,xn+1)2−4cMαn2(1−αn)2d(xn,T(xn⊕xn+12))2

which further implies that

d(xn+1,p)≤d(xn,p)−αnCM2(2−αn)d(xn,xn+1).−8cMαn2(1−αn)22(2−αn)d(xn,T(xn⊕xn+12))2.

The above inequality provides the following three inequalities:

(2.1)d(xn+1,p)≤d(xn,p),

(2.2)αnCM2(2−αn)d(xn,xn+1)≤d(xn,p)−d(xn+1,p)

and

(2.3)8cMαn2(1−αn)22(2−αn)d(xn,T(xn⊕xn+12))2≤d(xn,p)−d(xn+1,p).

From (2.1), it follows that limn→∞d(xn,p) exists, that is, (i) holds.

Since αn∈(0,1), therefore αn≤αn2(2−αn). Hence (2.2) becomes

(2.4)αnd(xn,xn+1)≤1CM[d(xn,p)−d(xn+1,p)].

Let m≥1 be any positive integer. Then from (2.4), we have that

∑n=1mαnd(xn,xn+1)≤1CM[d(x1,p)−d(xm+1,p)]≤d(x1,p)CM.

Let m→∞. Then

∑n=1∞αnd(xn,xn+1)≤d(x1,p)CM<∞.

That is,

∑n=1∞αnd(xn,xn+1)<∞,

proving (ii). Similarly, from (2.3), we have

∑n=1∞αn2(1−αn)2d(xn,T(xn⊕xn+12))2<∞.□

Lemma 2.3. Let C be a nonempty convex subset of a complete 2-uniformly convex hyperbolic space M and T:C→C a nonexpansive mapping such that F(T)≠φ. Then for the sequence {xn} in (1.4), we have that limn→∞d(xn,xn+1)=0.

Proof. Consider

d(xn+1,xn+2)=d(αn+1T(xn+1⊕xn+22)⊕(1−αn+1)xn+1,xn+1)      ≤αn+1d(xn+1,T(xn+1⊕xn+22))      ≤αn+1d(xn+1,T(xn⊕xn+12))      +αn+1d(T(xn⊕xn+12),T(xn+1⊕xn+22))      ≤αn+1d(xn+1,T(xn⊕xn+12))      +αn+1d(xn⊕xn+12,xn+1⊕xn+22)      ≤αn+1(1−αn)d(xn,T(xn⊕xn+12))      +αn+1d(xn⊕xn+12,xn+1⊕xn+22)      ≤αn+1(1−αn)d(xn,T(xn⊕xn+12))      +αn+12(d(xn,xn+1)+d(xn+1,xn+2)).

Therefore

(1−αn+12)d(xn+2,xn+1)≤αn+1(1−αn)d(xn,T(xn⊕xn+12))+αn+12d(xn,xn+1)

which further implies that

d(xn+1,xn+2)≤2αn+1(1−αn)2−αn+1d(xn,T(xn⊕xn+12))      +αn+12−αn+1d(xn,xn+1)      ≤2αn+1(1−αn)d(xn,T(xn⊕xn+12))      +αn+1d(xn,xn+1).

For some A>0,B>0 and using the assumption αn+12≤λαn2, we further derive that

d(xn+1,xn+2)2≤4Aαn+12(1−αn)2d(xn,T(xn⊕xn+12))2      +Bαn+12d(xn,xn+1)2     ≤4Aλαn+12(1−αn)2d(xn,T(xn⊕xn+12))2      +Bαn+12d(xn,xn+1)2     ≤4Aλαn2(1−αn)2d(xn,T(xn⊕xn+12))2      +Bλαn2d(xn,xn+1)2     ≤4Aλαn2(1−αn)2d(xn,T(xn⊕xn+12))2      +Bλαnd(xn,xn+1)2.

Hence by Lemma 2.2(ii)–(iii), we have that

∑n=1∞d(xn+1,xn+2)2<∞.

This in turn implies that

(2.5)limn→∞d(xn,xn+1)=0.□

Lemma 2.4. Let C be a nonempty closed and convex subset of a complete 2-uniformly convex hyperbolic space M and T:C→C a nonexpansive mapping such that F(T)≠φ. Then for the sequence {xn} in (1.4), we have that limn→∞d(xn,Txn)=0.

Proof. The condition liminfn→∞αn>0 implies that 0<1αn≤1α for sufficiently large n. The inequality

d(xn,T(xn⊕xn+12))≤d(xn,xn+1)+d(xn+1,T(xn⊕xn+12))≤d(xn,xn+1)+(1−αn)d(xn,T(xn⊕xn+12))

implies that

d(xn,T(xn⊕xn+12))≤1αnd(xn,xn+1)≤1αd(xn,xn+1).

By taking limsupn→∞ on both sides in the above inequality and then appealing to Lemma 2.3, we get that

(2.6)limn→∞d(xn,T(xn⊕xn+12))=0.

Finally, the inequality

d(xn,Txn)≤d(xn,T(xn⊕xn+12))+d(T(xn⊕xn+12),Txn)    ≤d(xn,T(xn⊕xn+12))+d(xn⊕xn+12,xn)    ≤d(xn,T(xn⊕xn+12))+12d(xn+1,xn)

together with (2.5) and (2.6) provides that

(2.7)limn→∞d(xn,Txn)=0.□

The following concept is needed to establish strong convergence of (1.4).

Let f be a nondecreasing function on [0,∞) with f(0)=0 and f(t)>0 for all t∈(0,∞). Then the mapping T:C→C with F(T)≠φ, satisfies condition (A) [21] if

d(x,Tx)≥f(d(x,F(T))) for x∈C,

where d(x,F(T))=inf{d(x,y):y∈F(T)}.

Using condition(A) and Lemma 2.4, we obtain the following strong convergence result.

Theorem 2.5. Let C be a nonempty closed and convex subset of a complete 2-uniformly convex hyperbolic space M and T:C→C a nonexpansive mapping such that F(T)≠φ. If the mapping T:C→C satisfies condition(A), then the sequence {xn} in (1.4), strongly converges to a fixed point of T.

Proof. By Lemma 2.4, limn→∞d(xn,Txn)=0. Now condition(A) implies that limn→∞d(xn,F(T))=0. Finally, by Lemma 1.1, {xn} strongly converges to a fixed point of T.□

Here are our other strong convergence results.

Theorem 2.6. Let C be a nonempty closed and convex subset of a complete 2-uniformly convex hyperbolic space M and T:C→C a nonexpansive mapping such that F(T)≠φ. If T is semi-compact, then the sequence {xn} in (1.4) strongly converges to a fixed point of T.

Proof. By Lemma 2.4, we have that limn→∞d(xn,Txn)=0. Since limn→∞d(xn,p) exists for each p∈F(T), {xn} is bounded. As limn→∞d(xn,Txn)=0 and T is semi-compact, so there is a subsequence {xni} of {xn} such that xni→q∈C and hence Txni→Tq. Therefore, limi→∞d(xni,Txni)=0 implies that d(Tq,q)=0. That is, q∈F(T). Since limn→∞d(xn,p) exists and xni→q, xn→q.□

Theorem 2.7. Let C be a nonempty closed and convex subset of a complete 2-uniformly convex hyperbolic space M and T:C→C a nonexpansive mapping such that F(T)≠φ. If T is completely continuous, then the sequence {xn} in (1.4), strongly converges to a fixed point of T.

Proof. Since {xn} is bounded and T is completely continuous, {Txn} has a convergent subsequence say {Txni}. Therefore by (2.7), {xni} converges. Let limi→∞xni=υ. By continuity of T and (2.7), we have that Tυ=υ. By Lemma 2.2, limn→∞d(xn,υ) exists and so {xn} strongly converges to υ.□

We now present our △-convergence result.

Theorem 2.8. Let C be a nonempty closed and convex subset of a complete 2-uniformly convex hyperbolic space M and T:C→C a nonexpansive mapping such that F(T)≠φ. Then the sequence {xn} in (1.4), △-converges to a fixed point of T.

Proof. It follows from Lemma 2.1 that {xn} is bounded in C. By Lemma 1.2, {xn} has a unique asymptotic centre, that is, AC({xn})={y}. Let {wn} be any subsequence of {xn} such that AC({wn})={w}. We claim that w∈F(T). By Lemma 2.4, we have that

limn→∞d(wn,Twn)=0.

The nonexpansive mapping T satisfies the following inequality:

d(wn,Tw)≤d(wn,Twn)+d(wn,w)

which further implies that

limsupn→∞d(wn,Tw)≤limsupn→∞d(wn,Twn)+limsupn→∞d(wn,w)=limsupn→∞d(wn,w).

By the uniqueness of asymptotic centre, we have Tw=w. Therefore F(T)≠φ. If y≠w, then by the uniqueness of asymptotic centre and the fact that limn→∞d(xn,x) exists for each x∈F(T), we have that

limsupn→∞d(wn,w)<limsupn→∞d(wn,y)      ≤limsupn→∞d(xn,y)      <limsupn→∞d(xn,w)      =limsupn→∞d(wn,w).

This is a contradiction and therefore y=w. This proves that {xn}, △-converges to x∈F(T).□

Remark 2.9. (1) All the results of this paper instantly hold in Hilbert spaces, uniformly convex Banach spaces satisfying Opial property and CAT(0) spaces; (2) The results of Alghamdi et al. [1] are corollaries of our corresponding results; (3) The interested reader is referred to [11] for another notion of p-uniformly convex metric spaces; (4) The two control conditions: (C1)and (C2) in our algorithm (1.4) are satisfied by the sequence αn=1−1n+1.

3. Application

We know that L2[0,1] is a Hilbert space and hence it is a 2-uniformly convex hyperbolic space. Suppose that h:[0,1]→[0,1] and F:[0,1]×[0,1]×ℝ→ℝ are continuous functions and F satisfies the Lipschitz continuity condition, i.e.,

|F(t,λ,x)−F(t,s,y)|≤|x−y|for t,s∈[0,1] and x,y∈ℝ.

Consider a Fredholm integral equation of the form

(3.1)x(t)=h(t)+∫01F(t,s,x(s))ds for t∈[0,1].

It has been shown in [19] that the solution of Eq. (3.1) exists in L2[0,1]. To find an approximate solution of this equation, we define S:L2[0,1]→L2[0,1] by

Sx(t)=h(t)+∫01F(t,s,x(s))ds for t∈[0,1].

For x,y∈L2[0,1], we calculate

||Sx−Sy||2=∫01|Sx(t)−Sy(t)|2dt    =∫01|∫01(F(t,s,x(s))−F(t,s,y(s)))ds|2dt    ≤∫01|∫01|x(s)−y(s)|ds|2dt    ≤∫01|x(s)−y(s)|2ds=||x−y||2.

So S is nonexpansive. For any function x0∈L2[0,1], we define a sequence of functions {xn} in L2[0,1] by

xn+1=αnS(xn+xn+12)+(1−αn)xn

where αn∈(0,1) such that liminfn→∞αn>0 and αn+12≤λαn2 for some λ>0. Now by Theorem 2.8, {xn} weakly converges to the fixed point of S which is a solution of Eq. (3.1).