Strong and weak convergence of Ishikawa iterations for best proximity pairs

Moosa Gabeleh 1 , S. I. Ezhil Manna 2 , A. Anthony Eldred 2 ,  and Olivier Olela Otafudu 3
  • 1 Department of Mathematics, Ayatollah Boroujerdi University, Boroujerd, Iran
  • 2 PG and Research Department of Mathematics, St.Joseph’s college, Trichy, India
  • 3 School of Mathematics, University of the Witwatersrand, 2050, Johannesburg, South Africa
Moosa Gabeleh, S. I. Ezhil Manna, A. Anthony Eldred and Olivier Olela Otafudu

Abstract

Let A and B be nonempty subsets of a normed linear space X. A mapping T : ABAB is said to be a noncyclic relatively nonexpansive mapping if T(A) ⊆ A, T(B) ⊆ B and ∥TxTy∥ ≤ ∥xy∥ for all (x, y) ∈ A × B. A best proximity pair for such a mapping T is a point (p, q) ∈ A × B such that p = Tp, q = Tq and d(p, q) = dist(A, B). In this work, we introduce a geometric notion of proximal Opiaľs condition on a nonempty, closed and convex pair of subsets of strictly convex Banach spaces. By using this geometric notion, we study the strong and weak convergence of the Ishikawa iterative scheme for noncyclic relatively nonexpansive mappings in uniformly convex Banach spaces. We also establish a best proximity pair theorem for noncyclic contraction type mappings in the setting of strictly convex Banach spaces.

1 Introduction

Let X be a normed linear space. A self-mapping T : XX is said to be nonexpansive provided that ∥TxTy∥ ≤ ∥xy∥. It is well known that if A is a nonempty, compact and convex subset of a Banach space X, then every nonexpansive mapping of A into itself has a fixed point.

In 1965, Kirk proved that if A is a nonempty, weakly compact and convex subset of a Banach space X with a geometric property, called normal structure, then every nonexpansive self-mapping T : AA has a fixed point (Kirk’s fixed point theorem [1]).

Now, suppose that (A, B) is a nonempty pair of subsets of a normed linear space X. A mapping T : ABAB is said to be noncyclic relatively nonexpansive if T is noncyclic, that is, T(A) ⊆ A, T(B) ⊆ B, and ∥TxTy∥ ≤ ∥xy∥ for all (x, y) ∈ A × B. Under this weaker assumption over T w.r.t. nonexpansiveness, the existence of the so-called best proximity pair, that is, a point (p, q) ∈ A × B such that

p=Tp,q=Tqandd(p,q)=dist(A,B),

was first studied in [2] as below.

Theorem 1.1

(See Theorem 2.2 of [2]) Let (A, B) be a nonempty, bounded, closed and convex pair of subsets of a uniformly convex Banach space X, and suppose T : ABAB is a noncyclic relatively nonexpansive mapping. Then T has a best proximity pair.

An interesting observation about Theorem 1.1 is that the mapping T may not be continuous whereas the existence of two fixed points of T which estimates the distance between two sets A and B is guaranteed (see also [3, 4] for more information).

In addition to the existence result of best proximity pairs for noncyclic relatively nonexpansive mappings, the convergence of Krasnoselskii’s iteration process for such mappings was discussed as follows.

Theorem 1.2

(Theorem 2.3 of [2]) Let (A, B) be a nonempty, bounded, closed and convex pair in a uniformly convex Banach space X and T : ABAB be a noncyclic relatively nonexpansive mapping. Let x0A0 and define xn+1=xn+Txn2. ThenxnTxn∥ → 0. Moreover, if T(A) is compact, then {xn} converges to a fixed point of T.

The current paper is organized as follows: in Section 2, we recall some notions and notations which will be used in our coming discussion. We also introduce a geometric concept of proximal Opiaľs condition on a nonempty, closed and convex pair of subsets of strictly convex Banach spaces. In Section 3, we improve Theorem 1.2 and prove strong and weak convergence theorems for noncyclic relatively nonexpansive mappings in uniformly convex Banach spaces. Finally, in Section 4, we establish a best proximity pair theorem for noncyclic contraction type mappings in the setting of strictly convex Banach spaces. We also present some appropriate examples to illustrate our main conclusions.

2 Preliminaries

To describe our results, we need some definitions and notations.

Definition 2.1

A Banach space X is said to be

  1. uniformly convex if there exists a strictly increasing function δ : [0, 2] → [0, 1] such that the following implication holds for all x, y, pX, R > 0 and r ∈ (0, 2R]:
    xpR,ypR,xyrx+y2p(1δ(rR))R;
  2. strictly convex if the following implication holds x, y, pX and R > 0:
    xpR,ypR,xyx+y2p<R.

It is well known that Hilbert spaces and lp spaces (1 < p < ∞) are uniformly convex Banach spaces. Also, the Banach space l1 with the norm

|x|=x1+x2,xl1,

where, ∥.∥1 and ∥.∥2 are the norms on l1 and l2, respectively is strictly convex but not uniformly convex (see [5] for more details).

The following lemma gives a suitable property for characterization of uniformly convex Banach spaces.

Lemma 2.2

[6] A Banach space X is uniformly convex if and only if for each fixed number r > 0, there exits a continuous strictly increasing function φ:[0, ∞) → [0, ∞), φ(t) = 0 ⇔ t = 0, such that

λx+(1λ)y2λx2+(1λ)y2λ(1λ)φ(xy),

for all λ ∈ [0, 1] and all x, yX such thatx∥ ≤ r andy∥ ≤ r.

We also refer to the following auxiliary lemma.

Lemma 2.3

Consider a strictly increasing function ϕ : [0, ∞) → [0, ∞) with ϕ(0) = 0. If a sequence {rn} in [0, ∞) satisfies limn→∞ ϕ(rn) = 0, then limn→∞ rn = 0.

Another appropriate property of uniformly convex Banach spaces will be explained in the next lemma.

Lemma 2.4

[7] Let (A, B) be a nonempty and closed pair in a uniformly convex Banach space X such that A is convex. Let {xn} and {zn} be sequences in A and {yn} be a sequence in B such that limn→∞xnyn∥ = dist(A, B) and limn→∞znyn∥ = dist(A, B), then we have limn→∞xnzn∥ = 0.

We shall say that a pair (A, B) of subsets of a Banach space X satisfies a property if both A and B satisfy that property. For example, (A, B) is convex if and only if both A and B are convex; (A, B) ⊆ (C, D) ⇔ AC, and BD. We shall also adopt the notation

δx(A)=sup{d(x,y):yA}for allxX,δ(A,B)=sup{δx(B):xA},diam(A)=δ(A,A).

The closed and convex hull of a set A will be denoted by conv(A). Also, 𝓑(p, r) will denote the closed ball in the space X centered at pX with radius r > 0.

Given (A, B) a pair of nonempty subsets of a Banach space, if ∥xy∥ = dist(A, B), xA, yB then y (x) is called a proximal point of the point x (y). Also, the proximal pair of (A, B) will be denoted by (A0, B0) which is given by

A0={xA:xy=dist(A,B) for some yB},B0={yB:xy=dist(A,B) for some xA}.

Proximal pairs may be empty but, in particular, if (A, B) is a nonempty, bounded, closed and convex pair in a reflexive Banach space X, then (A0, B0) is also nonempty, closed and convex.

For a noncyclic mapping T : ABAB the set of all best proximity pairs of T will be denoted by ProxA×B(T).

Suppose A is a nonempty and convex subset of X. A self-mapping T : AA is said to be affine if

T(λx+(1λ)y)=λTx+(1λ)Ty,

for any x, yA and 0 < λ < 1. Also, a mapping T : ABAB is called affine provided that both TA and TB are affine, where (A, B) is a convex pair in X.

Definition 2.5

A pair (A, B) in a Banach space is said to be proximinal if A = A0 and B = B0.

For a nonempty subset A of X a metric projection operator 𝓟A : X → 2A is defined as

PA(x):={yA:xy=dist({x},A)},

where 2A denotes the set of all subsets of A. It is well known that if A is a nonempty, closed and convex subset of a reflexive and strictly convex Banach space X, then the metric projection 𝓟A is single valued from X to A.

Next result will be used in the sequel.

Proposition 2.6

[2, 8] Let (A, B) be a nonempty, bounded, closed and convex pair in a reflexive and strictly convex Banach space X. Define 𝓟:A0B0A0B0 as

P(x)=PA0(x)ifxB0,PB0(x)ifxA0.

Then the following statements hold:

  1. x − 𝓟x∥ = dist(A, B) for any xA0B0 and 𝓟(A0) ⊆ B0, 𝓟(B0) ⊆ A0.
  2. 𝓟 is an isometry, that is, ∥𝓟x − 𝓟y∥ = ∥xyfor all (x, y) ∈ A0 × B0.
  3. 𝓟 is affine.

Definition 2.7

[9] Let (A, B) be a pair of nonempty subsets of a metric space (X, d) with A0 ≠ ∅. The pair (A, B) is said to have the P-property if and only if

d(x1,y1)=dist(A,B)d(x2,y2)=dist(A,B)d(x1,x2)=d(y1,y2),

whenever x1, x2A0 and y1, y2B0.

Next theorem plays an important role in our next conclusions.

Lemma 2.8

[10, 11] Every, nonempty, bounded, closed and convex pair in a uniformly convex Banach space X has the P-property.

We finish this section by introducing a notion of proximal Opiaľs condition. We recall that a Banach space X is said to satisfy Opiaľs condition ([12]) if for each sequence {xn} in X with xnw x we have

lim supnxnx<lim supnxnyyXwithyx.

It is well known that finite dimensional Banach spaces, Hilbert spaces and lp spaces (1 < p < ∞) satisfy Opiaľs condition.

Definition 2.9

Let (A, B) be a nonempty, closed, convex and proximinal pair in a strictly convex Banach space X. Then (A, B) is said to satisfy proximal Opiaľs condition if for every sequence {xn} in A(respectively in B) with xnw uA(respectively xnw uB) we have

lim supnxnPu<lim supnxny,yPuB(respectivelyyPuA).

We mention that the condition of proximinality of the closed and convex pair (A, B) in the above definition is essential even if the considered Banach space X is Hilbert. Let us illustrate this fact with the following example.

Example 2.1

Consider the Hilbert space X = l2 with the canonical basis {en} and let

A=con¯{e2n:nN},B=con¯{2e1,e3}.

Then dist(A, B) = 1. Also, the closed and convex pair (A, B) is not proximinal. Notice that e2nw0. Let u = 0 and y=14e1+78e3. Then 𝓟u = e3 and we have

lim supnxnPu=lim supne2ne3=2>1+116+4964=lim supne2n14e1+78e3=lim supnxny,

that is, (A, B) does not have proximal Opiaľs condition.

Definition 2.10

[13] A nonempty, closed and convex pair (A, B) in a strictly convex Banach space X is said to have Pythagorean property if for each (x, y) in A0 × B0 we have

xy2=xPx2+Pxy2,xy2=xPy2+Pyy2.

Proposition 2.11

Let (A, B) be a nonempty, closed, convex and proximinal pair in a uniformly convex Banach space X which has the Opial property. If (A, B) posses Pythagorean property, then it satisfies proximal Opiaľs condition.

Proof

Let {xn} be a sequence in A such that xnwu and y ≠ 𝓟u in B. By Pythagorean property

xnPu2=xnu2+uPu2,xny2=xnPy2+yPy2.

Notice that ∥u − 𝓟u∥ = ∥y − 𝓟y∥ = dist(A, B). Since X has the Opial property, lim supn→∞xnu∥ < lim supn→∞xn − 𝓟y∥ which implies that

lim supnxny2=lim supnxnPy2+dist(A,B)2>lim supnxnu2+dist(A,B)2=lim supnxnPu2,

and the result follows.□

Next corollaries are straightforward consequences of Proposition 2.11.

Corollary 2.12

Let (A, B) be a nonempty, bounded, closed and convex pair in a Hilbert space ℍ. Then the pair (A0, B0) has the proximal Opiaľs condition.

Corollary 2.13

Let (A, B) be a nonempty, bounded, closed and convex pair in lp (1 < p < ∞) spaces such that (A, B) has the Pythagorean property. Then the pair (A0, B0) has the proximal Opiaľs condition.

3 Convergence results

We begin our results of this section by improving Theorem 1.2 as follows.

Theorem 3.1

Let (A, B) be a nonempty, bounded, closed and convex pair in a uniformly convex Banach space and T : ABAB be a noncyclic relatively nonexpansive mapping such that T(A) is compact. Let x0A0 and y0B0 be a unique proximal point of x0. Define

xn+1=xn+Txn2,yn+1=yn+Tyn2.

ThenxnTxn∥ → 0, ∥ynTyn∥ → 0 and the sequence {(xn,yn)} converges to a best proximity pair of T.

Proof

It follows from the proof of Theorem 1.2 that ∥xnTxn∥ → 0 and that the sequence {xn} converges to a fixed point of T in A, say pA. Besides,

x1y1=x0+Tx02y0+Ty0212x0y0+12Tx0Ty0dist(A,B).

Equivalently, and by induction we obtain ∥xnyn∥ = dist(A, B) for all n ∈ ℕ. Also for each n, ∥TxnTyn∥ = dist(A, B) and ∥Txnyn∥ → dist(A, B). Using Lemma 2.4, ∥ynTyn∥ → 0. Since B is bounded and X is reflexive, there exists a subsequence {ynk} of the sequence {yn} so that ynkw qB. Thus xnkynkw pq. It now follows from the lower semi-continuity of norm that

pqlim infkxnkynk=dist(A,B).

By Lemma 2.8 we conclude that ∥xnp∥ = ∥ynq∥ for all n ∈ ℕ. Thereby, ynq. We finish the proof by showing that q is a fixed point of T in B. Indeed,

dist(A,B)=pqpTq=TpTqpq=dist(A,B).

Again by the fact that (A, B) has the P-property, we conclude that q = Tq.□

In what follows we prove the strong and weak convergence results of Ishikawa iteration scheme for noncyclic mappings. For more information regarding Ishikawa iteration scheme we refer to [14, 15].

Let (A, B) be a nonempty, closed and convex pair in a strictly convex Banach space X. For x1A0, put x1 := 𝓟(x1) ∈ B0. Define the sequence pair {(xn,xn)} as follows:

xn+1=(1αn)xn+αnTyn,xn+1=(1αn)xn+αnT(yn);yn=(1βn)xn+βnT(xn),yn=(1βn)xn+βnT(xn)n=1,2,3.,

where {αn} and {βn} are sequences in [0, 1] satisfying one of the following conditions:

(C) 0 < ϵαn(1 − αn) and βn → 0 as n → ∞,

(D) 0 < ϵαn ≤ 1 and 0 < ϵβn(1 − βn).

Lemma 3.2

Let (A, B) be a nonempty, closed and convex pair in a uniformly convex Banach space X such that either A or B is bounded and let T : ABAB be a noncyclic relatively nonexpansive mapping. Suppose {xn} and {xn} are given by (1). Then limn→∞xnqexists for all q ∈ Fix(T) ∩ B0 and limn→∞ xnp exists for all p ∈ Fix(T) ∩ A0, where Fix(T) denotes the set of all fixed points of the mapping T.

Proof

Notice that (A0, B0) is nonempty, closed and convex. For any q ∈ Fix(T) ∩ B0 we have

xn+1q=αnTyn+(1αn)xnqαnTynq+(1αn)xnqαnynq+(1αn)xnqαnβnTxn+(1βn)xnq+(1αn)xnqαnβnxnq+αn(1βn)xnq+(1αn)xnq=xnq.

This implies {∥ xnq ∥} is non-increasing and hence limn→∞xnq∥ exists. Similarly, we can show that limn→∞ xnp exists for all p ∈ Fix(T) ∩ A0 and hence the lemma.□

Here, we establish the first main result of this section.

Theorem 3.3

Let (A, B) be a nonempty, bounded, closed and convex pair in a uniformly convex Banach space X and let T : ABAB be a noncyclic relatively nonexpansive mapping. For x1A0 let x1B0 be a unique proximal point of x1. Assume {xn} and {xn} are given by (1) where {αn}, {βn} satisfy either (C) or (D). Then

limnxnTxn=0,limnxnTxn=0.

Further the sequence pair {(xn,xn)} converges to a best proximity pair of T if T(A) lies in a compact subset.

Proof

By Theorem 1.1, Fix(T) ∩ B0 is nonempty. Let p ∈ Fix(T) ∩ B0. Then from Lemma 2.2 there exists continuous strictly increasing function φ:[0, ∞) → [0, ∞) such that

xn+1p2=αnTyn+(1αn)xnp2=αn(Tynp)+(1αn)(xnp)2αnTynp2+(1αn)xnp2αn(1αn)φ(Tynxn)αnynp2+(1αn)xnp2αn(1αn)φ(Tynxn)=αnβnTxn+(1βn)xnp2+(1αn)xnp2αn(1αn)φ(Tynxn)αnβnTxnp2+αn(1βn)xnp2αnβn(1βn)φ(Txnxn)+(1αn)xnp2αn(1αn)φ(Tynxn)αnβnxnp2+αn(1βn)xnp2αnβn(1βn)φ(Txnxn)+(1αn)xnp2αn(1αn)φ(Tynxn)=xnp2αnβn(1βn)φ(Txnxn)αn(1αn)φ(Tynxn).

From the above, we can deduce the following inequalities:

αn(1αn)φ(Tynxn)xnp2xn+1p2,

αnβn(1βn)φ(Txnxn)xnp2xn+1p2.

We consider two following cases.

  1. CASE ISuppose αn and βn satisfy (C). From (2) we conclude that
    n=1mαn(1αn)φ(Tynxn)x1p2xm+1p2.
    Letting m → ∞, we obtain n=1 αn (1 − αn)φ(∥ Tynxn ∥) < ∞. In view of the fact that αn(1 − αn) ≥ ϵ, it results that limn φ(∥ Tynxn ∥) = 0. Therefore, ∥ Tynxn ∥ → 0. Since 𝓟 is affine and isometry and 𝓟T = T𝓟 on A0B0,
    TxnPxnTxnPTyn+PTynPxn=TxnT(Pyn)+TynxnxnPyn+TynxnxnP{(1βn)xn+βnTxn}+Tynxn=xnPxn+βn(PxnPTxn)+TynxnxnPxn+βnPxnPTxn+Tynxn=xnPxn+βnxnTxn+Tynxn.
    Thus,
    TxnPxnxnPxn+βnxnTxn+Tynxn.
    If n → ∞ in above relation, we obtain
    limnTxnPxnlimnxnPxn=dist(A,B).
    Lemma 2.4 implies that limn→∞Txnxn ∥ → 0.
  2. CASE IISuppose the sequences {αn} and {βn} satisfy (D). By (3) we deduce that
    n=1mαnβn(1βn)φ(Txnxn)x1p2xm+1p2.
    As m → ∞, we get n=1 αn βn (1 − βn)φ(∥ Txnxn∥) < ∞. On account of the facts that αnβn (1 − βn) ≥ ϵ2, one can be sure that lim supn φ(∥ Txnxn ∥) = 0 which means lim supnTxnxn∥ = 0.Therefore, ∥ xnTxn ∥ → 0 in both the cases.Now, suppose that T(A) lies in a compact subset. Then {Txn} has a convergent subsequence {Txnk}, converging to some point uA0. Since limn→∞Txnk-xnk ∥ = 0, xnku. Besides, from T(𝓟u) = 𝓟(Tu) we have
    |TxnkP(Tu)=TxnkT(Pu)xnkPudist(A,B),
    which deduces that TxnkTu. Thus Tu = u and so T(𝓟u) = 𝓟u. Therefore by Lemma 3.2, limnxn −𝓟u∥ exists and
    limnxnPu=limkxnkPu=uPu=dist(A,B),
    which implies xnu ∈ Fix(T) ∩ A. Besides, by the assumption x1x1 = dist(A, B). We now have
    x2x2=(1α1)x1+α1Ty1((1α1)x1+α1Ty1)(1α1)x1x1+α1Ty1Ty1(1α1)x1x1+α1y1y1=(1α1)x1x1+α1(1β1)x1+β1Tx1((1β1)x1+β1Tx1)(1α1)x1x1+α1x1x1=x1x1=dist(A,B).
    Continuing this process and by induction we obtain xnxn = dist(A, B) for each n ∈ ℕ. By a similar argument of aforesaid discussion we conclude that
    xnTxn0andxnvFix(T)B.
    On the other hand,
    uv=limnxnxn=dist(A,B),
    which deduces that (u, v) ∈ ProxA×B(T).□

Theorem 3.4

Let (A, B) be a nonempty, bounded, closed and convex pair in a uniformly convex Banach space X with (A0, B0) satisfying proximal Opiaľs condition and T : ABAB be a noncyclic relatively nonexpansive mapping. Then IT is demi-closed at zero, i.e., for each sequence pair {(xn,xn)} in (A0, B0) with xnxn = dist(A, B) for each n ∈ ℕ if {(xn,xn)} converges weakly to (x, y) and {(Txnxn,Txnxn)} converges to (0, 0) then (x, y) ∈ ProxA×B(T).

Proof

Let {xn} weakly converge to x in A0 and xnTxn → 0. Since 𝓟Tx = T(𝓟x) for all xA0,

xnPTxxnTxn+TxnPTx=xnTxn+TxnT(Px)xnTxn+xnPx.

Thereby,

lim supnxnPTxlim supnxnPx.

Due to the fact that (A0, B0) has the proximal Opiaľs condition, we must have Tx = x. Equivalently, we can prove that Ty = y. Moreover,

xylim infnxnxn=dist(A,B),

that is, (x, y) ∈ ProxA×B(T).□

Next theorem guarantees the weak convergence of the iterative sequence defined in (1) to finding best proximity pairs of noncyclic relatively nonexpansive mappings in uniformly convex Banach spaces.

Theorem 3.5

Let (A, B) be a nonempty, bounded, closed and convex pair in a uniformly convex Banach space X with (A0, B0) satisfying proximal Opiaľs condition and let T : ABAB be a noncyclic relatively nonexpansive mapping. Let x1A0 and x1B0 be a unique proximal point of x1. Suppose {xn} and {xn} are given by (1) where {αn}, {βn} satisfy either condition (C) or (D). Then {(xn,xn)} converges weakly to a best proximity pair of T.

Proof

We assert that {xn} weakly converges to a member of Fix(T) ∩ A0. Let {xnk} and {xmk} be subsequences of {xn} converging weakly to u and v respectively, such that uv. Since ∥ xnkTxnk ∥ and ∥ xmkTxmk ∥ converge to 0, by Theorem 3.4 Tu = u and Tv = v. From Lemma 3.2, limn→∞xn − 𝓟u∥ and limn→∞xn − 𝓟v∥ exist. It now follows from the proximal Opiaľs condition that

lim supnxnPu=lim supnxnkPu<lim supnxnkPv=lim supnxmkPv<lim supnxmkPu=lim supnxnPu,

which is a contradiction. Thus u = v and so {xn} converges weakly to u in A0. Similarly we can show that {xn} converges weakly to a member u′ in Fix(T) ∩ B0. By Theorem 3.3, xnxn = dist(A, B) for each n ∈ ℕ and that limn→∞xnTxn ∥ = 0 and limn→∞ xnTxn = 0. Therefore, from Theorem 3.4, (u, u′) ∈ ProxA×B(T) and this completes the proof of theorem.□

4 Noncyclic contraction type mappings

We begin our main results of this section with the following well known fixed point theorem which is the main idea for coming discussions.

Theorem 4.1

Let (X, d) be a complete metric space and T : XX be a continuous self-mapping such that

d(Tx,T2x)αd(x,Tx),xX,

where α ∈ [0, 1), then T has a fixed point.

In fact, the condition on T ensures that {Tn (x)} is a Cauchy sequence for each xX, and continuity does the rest.

Here, we state the following existence theorems of best proximity pairs for two different classes of noncyclic mappings.

Theorem 4.2

[16] Let (A, B) be a nonempty weakly compact convex pair in a strictly convex Banach space X. Assume that T : ABAB is a noncyclic contraction mapping, that is, T is noncyclic on AB and

TxTyrxy+(1r)dist(A,B),

for some r ∈ [0, 1) and for all (x, y) ∈ A × B. Then ProxA×B(T) is nonempty.

Let (A, B) be a nonempty pair in a normed linear space X. For a noncyclic mapping T : ABAB and (x, y) ∈ A × B, we set

O(x,):={x,Tx,T2x,},O(y,):={y,Ty,T2y,}.

We note that (𝓞(x, ∞),𝓞(y, ∞)) ⊆ (A, B).

Definition 4.3

[17] Let (A, B) be a nonempty pair of subsets of a normed linear space X. A mapping T : ABAB is said to be a generalized noncyclic relatively nonexpansive mapping provided that T is noncyclic on AB and

TxTy=xy,forall(x,y)A×Bwithxy=dist(A,B),

and

TxTyαxy+(1α)min{δx(O(Ty,)),δy(O(Tx,))},

for some α ∈ [0, 1] and for all (x, y) ∈ A × B withxy∥ > dist(A, B).

Theorem 4.4

(see Theorem 3.2 of [17]) Let (A, B) be a nonempty, bounded, closed and convex pair in a uniformly convex Banach space X. Suppose that T : ABAB is a generalized noncyclic relatively nonexpansive mapping. Then T has a best proximity pair.

In this section, we establish a best proximity pair result under some sufficient conditions. For this purpose, we introduce a new class of noncyclic contraction mappings as below.

Definition 4.5

Let (A, B) be nonempty pair of subsets of a metric space (X, d). A mapping T : ABAB is said to be a noncyclic contraction type mapping if there exists α ∈ [0, 1) such that

d(Tx,T2y)αd(x,Ty)+(1α)dist(A,B),

for all (x, y) ∈ A × B.

It is clear that every noncyclic contraction mapping is noncyclic contraction type mapping in the sense of Definition 4.5. The next example shows that the reverse implication does not hold.

Example 4.1

Let X be the real Banach space l2 renormed according to

x=max{x2,2x},

where, ∥x denotes the l-norm and ∥x2 the l2 norm. Let {en} be the canonical basis of l2. Note that this norm is equivalent to ∥⋅∥2 and so, (X, ∥ ⋅ ∥) is a reflexive Banach space. Consider

A={x:=e1+e2}andB={y=(yn):y3=1,y2}.

Set u := e1 + e3, v := e2 + e3 and w := e3 + e4. Then u, v, w are tree distinct elements in B and we have ∥xu∥ = ∥xv∥ = 2. Also, for each y = (y1, y2, 1, y4, …) ∈ B we have ∥y22 which implies that ∑i≠3yi2 ≤ 1 and since ∥y ≤ 1, we conclude that ∣yi∣ ≤ 1 for each i ∈ ℕ. Thus, for all yB we have ∥xy∥ ≥ 2 which deduces that dist(A, B) = 2. Let T : ABAB be a mapping defined as follows

Tx=x,and for eachyB,Ty=vifyu,wify=u.

Then T is noncyclic and for each α ∈ [0, 1) and yB we have T2y = v and so,

TxT2y=2=α2+(1α)2αxTy+(1α)dist(A,B),

that is, T is noncyclic contraction type mapping. We also note that if y = u, then

TxTy=(e1+e2)(e3+e4)=2>2=xy.

Therefore, T is not a generalized noncyclic relatively nonexpansive mapping.

Next lemma will be used in our main result of this section.

Lemma 4.6

[18] Let (K1, K2) be a nonempty pair of subsets of a Banach space X. Then

δ(K1,K2)=δ(conv¯(K1),conv¯(K2)).

The following theorem guarantees the existence of best proximity pairs for noncyclic contraction type mappings which are affine.

Theorem 4.7

(compare with Theorems 4.2, 4.4) Let (A, B) be a nonempty, weakly compact and convex pair of subsets of a strictly convex Banach space X, and let T : ABAB be a noncyclic contraction type mapping. Assume that TA, TB are affine. Then T has a best proximity pair.

Proof

Assume that 𝔈 denotes the collection of all nonempty, closed and convex pair (E, F) ⊆ (A, B) such that T is noncyclic on EF. Then (A, B) ∈ 𝔈 ≠ ∅. By using Zorn’s Lemma we get an element say (K1, K2) which is minimal with respect to being nonempty, closed, convex and Tinvariant. Note that (conv(T(K1)),conv(T(K2))) ⊆ (K1, K2) is a nonempty, bounded, closed and convex pair in X. Further,

T(conv¯(T(K1)))T(K1)conv¯(T(K1)),

and also,

T(conv¯(T(K2)))conv¯(T(K2)),

that is, T is noncyclic on conv(T(K1)) ∪ conv(T(K2)). It now follows from the minimality of (K1, K2) that

conv¯(T(K1))=K1,conv¯(T(K2))=K2.

We now assert that

conv¯(T2(K1))=K1andconv¯(T2(K2))=K2.

Since T(K1) ⊆ K1, we have T2(K1) ⊆ T(K1) and so, conv(T2(K1)) ⊆ conv(T(K1)) = K1. Then T[conv(T2(K1))] ⊆ T[conv(T(K1))]. We show that T[conv(T(K1))] ⊆ conv(T2(K1)). Suppose that uT[conv(T(K1))]. Then there exists an element u′ ∈ conv(T(K1)) such that u = Tu′. By the fact that u′ ∈ conv(T(K1)), we have u′ = i=1n αiTxi, where 0 ≤ αi ≤ 1 and i=1n αi = 1 and xiK1 for i = 1, 2, …, n. Since TA is affine, we obtain

u=Tu=T(i=1nαiTxi)=i=1nαiT(Txi)=i=1nαiT2xi,

which implies that uconv(T2(K1)). Therefore, T[conv(T2(K1))] ⊆ conv(T2(K1)). By the similar manner, we have T[conv(T2(K2))] ⊆ conv(T2(K2)), that is, T is noncyclic on conv(T2(K1)) ∪ conv(T2(K2)). Minimality of (K1, K2) implies that conv(T2(K1)) = K1 and conv(T2(K2)) = K2. Let xK1. Then for each yK2 we have

TxT2yαxTy+(1α)dist(A,B)αδx(K2)+(1α)dist(A,B).

Thus T2y ∈ 𝓑(Tx; αδx(K2) + (1 − α)dist(A, B)) for each yK2 and so, T2(K2) ⊆ 𝓑(Tx; αδx(K2) + (1 − α)dist(A, B)). Therefore,

K2=conv¯(T2(K2))B(Tx;αδx(K2)+(1α)dist(A,B)).

Hence, ∥yTx∥ ≤ αδx(K2) + (1 − α)dist(A, B) for each yK2 and

δTx(K2)αδx(K2)+(1α)dist(A,B)αδ(K1,K2)+(1α)dist(A,B),xK1.

This implies that

δ(T(K1),K2)=supxK1δTx(K2)αδ(K1,K2)+(1α)dist(A,B).

It now follows from Lemma 4.6 that

δ(K1,K2)=δ(conv¯(T(K1)),conv¯(K2))=δ(T(K1),K2)αδ(K1,K2)+(1α)dist(A,B).

Therefore,

δ(K1,K2)=dist(A,B)(=dist(K1,K2)).

Strictly convexity of the Banach space X yields that both K1 and K2 must be singleton and so ProxA×B(T) ≠ ∅.□

Next fixed point result is concluded from Theorem 4.7.

Corollary 4.8

Let (A, B) be a nonempty, weakly compact and convex pair of subsets of a strictly convex Banach space X, and let T : ABAB be a noncyclic mapping and suppose there exists α ∈ [0, 1) such that

TxT2yαxTy,

for each (x, y) ∈ A × B. If TA, TB are affine, then AB is nonempty and T has a fixed point in AB.

Example 4.2

Consider the real Banach space l2 with the canonical basis {en}. Let

A={x1e1:0x11}andB={x1e1+x2e2:0x1,x21}.

Define T : ABAB with T(x1e1) = 0 and T(x1e1 + x2e2) = x2e1. Then T is noncyclic on AB and it is easy to see that TA and TB are affine. Let x := x1e1 and y := y1e1 + y2e2. Thus for each α ∈ [0, 1) we have

TxT2y=0αxTy.

That is, T is a noncyclic contraction type mapping. Now, by Corollary 4.8, T has a fixed point. Note that existence of fixed point for T cannot be obtained from Theorem 4.1. Indeed, if z := ze1 + ze2, then

TzT2z=ze10=zzTz.

References

  • [1]

    William A. Kirk, A fixed point theorem for mapping which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006.

    • Crossref
    • Export Citation
  • [2]

    A. Anthony Eldred, William A. Kirk, and P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Math. 171 (2005), no. 3, 283–293.

    • Crossref
    • Export Citation
  • [3]

    Dan Butnariu, Simeon Reich, and Alexander J. Zaslavski, Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Applied Anal. 7 (2001), no. 2, 151–174.

  • [4]

    William A. Kirk, Simeon Reich, and P. Veeramani, Proximinal retracts and best proximity pair theorems, Numer. Funct. Anal. Optim. 24 (2003), no. 7-8, 851–862.

    • Crossref
    • Export Citation
  • [5]

    Vaclav E. Zizler, On some rotundity and smoothness properties of Banach spaces, Dissertationes Mat. (Rozprawy Mat.) 87 (1971), 33 pages.

  • [6]

    Hong-Kun Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991), no. 12, 1127–1138.

    • Crossref
    • Export Citation
  • [7]

    A. Anthony Eldred and P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2006), no. 2, 1001–1006.

    • Crossref
    • Export Citation
  • [8]

    Moosa Gabeleh, Common best proximity pairs in strictly convex Banach spaces, Georgian Math. J. 24 (2017), no. 3, 363–372.

    • Crossref
    • Export Citation
  • [9]

    V. Sankar Raj, A best proximity point theorem for weakly contractive non-self-mappings, Nonlinear Anal. 74 (2011), no. 14, 4804–4808.

    • Crossref
    • Export Citation
  • [10]

    Ali Abkar and Moosa Gabeleh, Global optimal solutions of noncyclic mappings in metric spaces, J. Optim. Theory Appl. 153 (2012), no. 2, 298–305.

    • Crossref
    • Export Citation
  • [11]

    V. Sankar Raj and A. Anthony Eldred, Characterizaton of strictly convex spaces and applications, J. Optim. Theory Appl. 160 (2014), no. 2, 703–710.

    • Crossref
    • Export Citation
  • [12]

    Zdzislaw Opial, Weak convergence of the sequences of approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), no. 4, 591–597.

    • Crossref
    • Export Citation
  • [13]

    Rafael Espínola, G. Sankara Raju Kosuru, and P. Veeramani, Pythagorean property and best proximity point theorems, J. Optim. Theory Appl. 164 (2015), 534–550.

    • Crossref
    • Export Citation
  • [14]

    Abdul Rahim Khan and Hafiz Fukhar-ud-din, Weak convergence of Ishikawa iterates for nonexpansive maps, Proceedings of the World Congress on Engineering and Computer Science (October 20-22, 2010, San Francisco, USA), 20–22.

  • [15]

    Kok-Keong Tan and Hong Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993), 301–308.

    • Crossref
    • Export Citation
  • [16]

    Rafael Espínola and Moosa Gabeleh, On the structure of minimal sets of relatively nonexpansive mappings, Numer. Funct. Anal. Optim. 34 (2013), no. 8, 845–860.

  • [17]

    Moosa Gabeleh, Noncyclic mappings and best proximity pair results in Hilbert and uniformly convex Banach spaces, Quaestiones Mathematicae, 39 (2016), no. 2, 203–2015.

    • Crossref
    • Export Citation
  • [18]

    Ali Abkar and Moosa Gabeleh, Proximal quasi-normal structure and a best proximity point theorem, J. Nonlin. Convex Anal. 14 (2013), no. 4, 653–659.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    William A. Kirk, A fixed point theorem for mapping which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006.

    • Crossref
    • Export Citation
  • [2]

    A. Anthony Eldred, William A. Kirk, and P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Math. 171 (2005), no. 3, 283–293.

    • Crossref
    • Export Citation
  • [3]

    Dan Butnariu, Simeon Reich, and Alexander J. Zaslavski, Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Applied Anal. 7 (2001), no. 2, 151–174.

  • [4]

    William A. Kirk, Simeon Reich, and P. Veeramani, Proximinal retracts and best proximity pair theorems, Numer. Funct. Anal. Optim. 24 (2003), no. 7-8, 851–862.

    • Crossref
    • Export Citation
  • [5]

    Vaclav E. Zizler, On some rotundity and smoothness properties of Banach spaces, Dissertationes Mat. (Rozprawy Mat.) 87 (1971), 33 pages.

  • [6]

    Hong-Kun Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991), no. 12, 1127–1138.

    • Crossref
    • Export Citation
  • [7]

    A. Anthony Eldred and P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2006), no. 2, 1001–1006.

    • Crossref
    • Export Citation
  • [8]

    Moosa Gabeleh, Common best proximity pairs in strictly convex Banach spaces, Georgian Math. J. 24 (2017), no. 3, 363–372.

    • Crossref
    • Export Citation
  • [9]

    V. Sankar Raj, A best proximity point theorem for weakly contractive non-self-mappings, Nonlinear Anal. 74 (2011), no. 14, 4804–4808.

    • Crossref
    • Export Citation
  • [10]

    Ali Abkar and Moosa Gabeleh, Global optimal solutions of noncyclic mappings in metric spaces, J. Optim. Theory Appl. 153 (2012), no. 2, 298–305.

    • Crossref
    • Export Citation
  • [11]

    V. Sankar Raj and A. Anthony Eldred, Characterizaton of strictly convex spaces and applications, J. Optim. Theory Appl. 160 (2014), no. 2, 703–710.

    • Crossref
    • Export Citation
  • [12]

    Zdzislaw Opial, Weak convergence of the sequences of approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), no. 4, 591–597.

    • Crossref
    • Export Citation
  • [13]

    Rafael Espínola, G. Sankara Raju Kosuru, and P. Veeramani, Pythagorean property and best proximity point theorems, J. Optim. Theory Appl. 164 (2015), 534–550.

    • Crossref
    • Export Citation
  • [14]

    Abdul Rahim Khan and Hafiz Fukhar-ud-din, Weak convergence of Ishikawa iterates for nonexpansive maps, Proceedings of the World Congress on Engineering and Computer Science (October 20-22, 2010, San Francisco, USA), 20–22.

  • [15]

    Kok-Keong Tan and Hong Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993), 301–308.

    • Crossref
    • Export Citation
  • [16]

    Rafael Espínola and Moosa Gabeleh, On the structure of minimal sets of relatively nonexpansive mappings, Numer. Funct. Anal. Optim. 34 (2013), no. 8, 845–860.

  • [17]

    Moosa Gabeleh, Noncyclic mappings and best proximity pair results in Hilbert and uniformly convex Banach spaces, Quaestiones Mathematicae, 39 (2016), no. 2, 203–2015.

    • Crossref
    • Export Citation
  • [18]

    Ali Abkar and Moosa Gabeleh, Proximal quasi-normal structure and a best proximity point theorem, J. Nonlin. Convex Anal. 14 (2013), no. 4, 653–659.

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