Strong and weak convergence of Ishikawa iterations for best proximity pairs

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 : A ∪ B → A ∪ B 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 : A ∪ B → A ∪ B be a noncyclic relatively nonexpansive mapping. Let x₀ ∈ A₀ and define xn+1=xn+Txn2. Then ∥x_n − Tx_n∥ → 0. Moreover, if T(A) is compact, then {x_n} 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.

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, p ∈ X, R > 0 and r ∈ (0, 2R]:

   ∥x−p∥≤R,∥y−p∥≤R,∥x−y∥≥r⇒∥x+y2−p∥≤(1−δ(rR))R;

2. strictly convex if the following implication holds x, y, p ∈ X and R > 0:

   ∥x−p∥≤R,∥y−p∥≤R,x≠y⇒∥x+y2−p∥<R.

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

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

for all λ ∈ [0, 1] and all x, y ∈ X such that ∥x∥ ≤ r and ∥y∥ ≤ 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 {r_n} in [0, ∞) satisfies lim_n→∞ ϕ(r_n) = 0, then lim_n→∞ r_n = 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 {x_n} and {z_n} be sequences in A and {y_n} be a sequence in B such that lim_n→∞ ∥x_n − y_n∥ = dist(A, B) and lim_n→∞ ∥z_n − y_n∥ = dist(A, B), then we have lim_n→∞ ∥x_n − z_n∥ = 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) ⇔ A ⊆ C, and B ⊆ D. We shall also adopt the notation

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 p ∈ X with radius r > 0.

Given (A, B) a pair of nonempty subsets of a Banach space, if ∥x − y∥ = dist(A, B), x ∈ A, y ∈ B 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

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 : A ∪ B → A ∪ B the set of all best proximity pairs of T will be denoted by Prox_A×B(T).

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

for any x, y ∈ A and 0 < λ < 1. Also, a mapping T : A ∪ B → A ∪ B is called affine provided that both T∣_A and T∣_B 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 𝓟_A : X → 2^A is defined as

where 2^A 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 𝓟_A 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:A₀ ∪ B₀ → A₀ ∪ B₀ as

Then the following statements hold:

1. ∥x − 𝓟x∥ = dist(A, B) for any x ∈ A₀ ∪ B₀ and 𝓟(A₀) ⊆ B₀, 𝓟(B₀) ⊆ A₀.

2. 𝓟 is an isometry, that is, ∥𝓟x − 𝓟y∥ = ∥x − y∥ for 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

whenever x₁, x₂ ∈ A₀ and y₁, y₂ ∈ B₀.

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 {x_n} in X with x_n →^w x we have

It is well known that finite dimensional Banach spaces, Hilbert spaces and l^p 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 {x_n} in A(respectively in B) with x_n →^w u ∈ A(respectively x_n →^w u ∈ B) we have

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 {e_n} and let

Then dist(A, B) = 1. Also, the closed and convex pair (A, B) is not proximinal. Notice that e2n→w0. Let u = 0 and y=14e1+78e3. Then 𝓟u = e₃ and we have

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

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 {x_n} be a sequence in A such that xn→wu and y ≠ 𝓟u in B. By Pythagorean property

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

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 l^p (1 < p < ∞) spaces such that (A, B) has the Pythagorean property. Then the pair (A₀, B₀) has the proximal Opiaľs condition.

Theorem 3.1

Let (A, B) be a nonempty, bounded, closed and convex pair in a uniformly convex Banach space and T : A ∪ B → A ∪ B be a noncyclic relatively nonexpansive mapping such that T(A) is compact. Let x₀ ∈ A₀ and y₀ ∈ B₀ be a unique proximal point of x₀. Define

Then ∥x_n − Tx_n∥ → 0, ∥y_n − Ty_n∥ → 0 and the sequence {(x_n,y_n)} converges to a best proximity pair of T.

Proof

It follows from the proof of Theorem 1.2 that ∥x_n − Tx_n∥ → 0 and that the sequence {x_n} converges to a fixed point of T in A, say p ∈ A. Besides,

Equivalently, and by induction we obtain ∥x_n − y_n∥ = dist(A, B) for all n ∈ ℕ. Also for each n, ∥Tx_n − Ty_n∥ = dist(A, B) and ∥Tx_n − y_n∥ → dist(A, B). Using Lemma 2.4, ∥y_n − Ty_n∥ → 0. Since B is bounded and X is reflexive, there exists a subsequence {y_nk} of the sequence {y_n} so that y_nk →^w q ∈ B. Thus x_nk −y_nk →^w p−q. It now follows from the lower semi-continuity of norm that

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

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 x₁ ∈ A₀, put x1′ := 𝓟(x₁) ∈ B₀. Define the sequence pair {(xn,xn′)} as follows:

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

(C) 0 < ϵ ≤ α_n(1 − α_n) and β_n → 0 as n → ∞,

(D) 0 < ϵ ≤ α_n ≤ 1 and 0 < ϵ ≤ β_n(1 − β_n).

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 : A ∪ B → A ∪ B be a noncyclic relatively nonexpansive mapping. Suppose {x_n} and {xn′} are given by (1). Then lim_n→∞∥ x_n − q ∥ exists for all q ∈ Fix(T) ∩ B₀ and lim_n→∞ ∥xn′−p∥ 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

This implies {∥ x_n − q ∥} is non-increasing and hence lim_n→∞ ∥ x_n − q∥ exists. Similarly, we can show that lim_n→∞ ∥xn′−p∥ 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 : A ∪ B → A ∪ B be a noncyclic relatively nonexpansive mapping. For x₁ ∈ A₀ let x1′ ∈ B₀ be a unique proximal point of x₁. Assume {x_n} and {xn′} are given by (1) where {α_n}, {β_n} satisfy either (C) or (D). Then

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

From the above, we can deduce the following inequalities:

We consider two following cases.

1. Suppose α_n and β_n satisfy (C). From (2) we conclude that

   ∑n=1mαn(1−αn)φ(∥Tyn−xn∥)≤∥x1−p∥2−xm+1−p2.

   Letting m → ∞, we obtain ∑n=1∞ α_n (1 − α_n)φ(∥ Ty_n − x_n ∥) < ∞. In view of the fact that α_n(1 − α_n) ≥ ϵ, it results that lim_n φ(∥ Ty_n − x_n ∥) = 0. Therefore, ∥ Ty_n − x_n ∥ → 0. Since 𝓟 is affine and isometry and 𝓟T = T𝓟 on A₀ ∪ B₀,

   ∥Txn−Pxn∥≤∥Txn−PTyn∥+∥PTyn−Pxn∥=∥Txn−T(Pyn)∥+∥Tyn−xn∥≤∥xn−Pyn∥+∥Tyn−xn∥≤∥xn−P{(1−βn)xn+βnTxn}∥+∥Tyn−xn∥=∥xn−Pxn+βn(Pxn−PTxn)∥+∥Tyn−xn∥≤∥xn−Pxn∥+βn∥Pxn−PTxn∥+∥Tyn−xn∥=∥xn−Pxn∥+βn∥xn−Txn∥+∥Tyn−xn∥.

   Thus,

   ∥Txn−Pxn∥≤∥xn−Pxn∥+βn∥xn−Txn∥+∥Tyn−xn∥. (4)

   If n → ∞ in above relation, we obtain

   limn∥Txn−Pxn∥≤limn∥xn−Pxn∥=dist(A,B).

   Lemma 2.4 implies that lim_n→∞ ∥Tx_n − x_n ∥ → 0.

2. Suppose the sequences {α_n} and {β_n} satisfy (D). By (3) we deduce that

   ∑n=1mαnβn(1−βn)φ(∥Txn−xn∥)≤∥x1−p∥2−xm+1−p2.

   As m → ∞, we get ∑n=1∞ α_n β_n (1 − β_n)φ(∥ Tx_n − x_n∥) < ∞. On account of the facts that α_nβ_n (1 − β_n) ≥ ϵ², one can be sure that lim supn→∞ φ(∥ Tx_n − x_n ∥) = 0 which means lim supn→∞ ∥Tx_n − x_n∥ = 0.

   Therefore, ∥ x_n − Tx_n ∥ → 0 in both the cases.

   Now, suppose that T(A) lies in a compact subset. Then {Tx_n} has a convergent subsequence {Tx_nk}, converging to some point u ∈ A₀. Since lim_n→∞∥ Tx_nk-x_nk ∥ = 0, x_nk → u. Besides, from T(𝓟u) = 𝓟(Tu) we have

   ∥|Txnk−P(Tu)∥=∥Txnk−T(Pu)∥≤∥xnk−Pu∥→dist(A,B),

   which deduces that Tx_nk → Tu. Thus Tu = u and so T(𝓟u) = 𝓟u. Therefore by Lemma 3.2, limn ∥x_n −𝓟u∥ exists and

   limn∥xn−Pu∥=limk∥xnk−Pu∥=∥u−Pu∥=dist(A,B),

   which implies x_n → u ∈ Fix(T) ∩ A. Besides, by the assumption ∥x1−x1′∥ = dist(A, B). We now have

   ∥x2−x2′∥=∥(1−α1)x1+α1Ty1−((1−α1)x1′+α1Ty1′)∥≤(1−α1)∥x1−x1′∥+α1∥Ty1−Ty1′∥≤(1−α1)∥x1−x1′∥+α1∥y1−y1′∥=(1−α1)∥x1−x1′∥+α1∥(1−β1)x1+β1Tx1−((1−β1)x1′+β1Tx1′)∥≤(1−α1)∥x1−x1′∥+α1∥x1−x1′∥=∥x1−x1′∥=dist(A,B).

   Continuing this process and by induction we obtain ∥xn−xn′∥ = dist(A, B) for each n ∈ ℕ. By a similar argument of aforesaid discussion we conclude that

   ∥xn′−Txn′∥→0andxn′→v∈Fix(T)∩B.

   On the other hand,

   ∥u−v∥=limn→∞∥xn−xn′∥=dist(A,B),

   which deduces that (u, v) ∈ Prox_A×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 : A ∪ B → A ∪ B be a noncyclic relatively nonexpansive mapping. Then I − T is demi-closed at zero, i.e., for each sequence pair {(xn,xn′)} in (A₀, B₀) with ∥xn−xn′∥ = dist(A, B) for each n ∈ ℕ if {(xn,xn′)} converges weakly to (x, y) and {(Txn−xn,Txn′−xn′)} converges to (0, 0) then (x, y) ∈ Prox_A×B(T).

Proof

Let {x_n} weakly converge to x in A₀ and x_n − Tx_n → 0. Since 𝓟Tx = T(𝓟x) for all x ∈ A₀,

Thereby,

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,

that is, (x, y) ∈ Prox_A×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 : A ∪ B → A ∪ B be a noncyclic relatively nonexpansive mapping. Let x₁ ∈ A₀ and x1′ ∈ B₀ be a unique proximal point of x₁. Suppose {x_n} and {xn′} are given by (1) where {α_n}, {β_n} satisfy either condition (C) or (D). Then {(xn,xn′)} converges weakly to a best proximity pair of T.

Proof

We assert that {x_n} weakly converges to a member of Fix(T) ∩ A₀. Let {x_nk} and {x_mk} be subsequences of {x_n} converging weakly to u and v respectively, such that u ≠ v. Since ∥ x_nk−Tx_nk ∥ and ∥ x_mk−Tx_mk ∥ converge to 0, by Theorem 3.4 Tu = u and Tv = v. From Lemma 3.2, lim_n→∞∥ x_n − 𝓟u∥ and lim_n→∞∥ x_n − 𝓟v∥ exist. It now follows from the proximal Opiaľs condition that

which is a contradiction. Thus u = v and so {x_n} 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, ∥xn−xn′∥ = dist(A, B) for each n ∈ ℕ and that lim_n→∞ ∥ x_n − Tx_n ∥ = 0 and lim_n→∞ ∥xn′−Txn′∥ = 0. Therefore, from Theorem 3.4, (u, u′) ∈ Prox_A×B(T) and this completes the proof of theorem.□

Theorem 4.1

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

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 x ∈ X, 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 : A ∪ B → A ∪ B is a noncyclic contraction mapping, that is, T is noncyclic on A ∪ B and

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

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

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 : A ∪ B → A ∪ B is said to be a generalized noncyclic relatively nonexpansive mapping provided that T is noncyclic on A ∪ B and

and

for some α ∈ [0, 1] and for all (x, y) ∈ A × B with ∥x − y∥ > 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 : A ∪ B → A ∪ B 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 : A ∪ B → A ∪ B is said to be a noncyclic contraction type mapping if there exists α ∈ [0, 1) such that

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

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

Set u := e₁ + e₃, v := e₂ + e₃ and w := e₃ + e₄. Then u, v, w are tree distinct elements in B and we have ∥x − u∥ = ∥x − v∥ = 2 . Also, for each y = (y₁, y₂, 1, y₄, …) ∈ B we have ∥y∥₂ ≤ 2 which implies that ∑_i≠3 ∣y_i∣² ≤ 1 and since ∥y∥_∞ ≤ 1, we conclude that ∣y_i∣ ≤ 1 for each i ∈ ℕ. Thus, for all y ∈ B we have ∥x − y∥ ≥ 2 which deduces that dist(A, B) = 2 . Let T : A ∪ B → A ∪ B be a mapping defined as follows

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

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

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

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 : A ∪ B → A ∪ B be a noncyclic contraction type mapping. Assume that T∣_A, T∣_B 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 E ∪ F. 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 T − invariant. Note that (conv(T(K₁)),conv(T(K₂))) ⊆ (K₁, K₂) is a nonempty, bounded, closed and convex pair in X. Further,

and also,

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

We now assert that

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 u ∈ T[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 α_iTx_i, where 0 ≤ α_i ≤ 1 and ∑i=1n α_i = 1 and x_i ∈ K₁ for i = 1, 2, …, n. Since T∣_A is affine, we obtain

which implies that u ∈ conv(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 x ∈ K₁. Then for each y ∈ K₂ we have

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

Hence, ∥y − Tx∥ ≤ αδ_x(K₂) + (1 − α)dist(A, B) for each y ∈ K₂ and

This implies that

It now follows from Lemma 4.6 that