Nonlinear Analysis 70 (2009) 3332–3341

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

Best proximity points for cyclic and noncyclic set-valued relativelyquasi-asymptotic contractions in uniform spacesKazimierz Włodarczyk ∗, Robert Plebaniak, Artur BanachDepartment of Nonlinear Analysis, Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, 90-238 Łódź, Poland

a r t i c l e i n f o

Article history:Received 19 February 2008Accepted 24 April 2008

MSC:54C6047H1054E1546A0354E50

Keywords:Cyclic and noncyclic set-valued dynamicsystemsRelatively quasi-asymptotic contractionBest proximity pointFamily of generalized pseudodistancesUniform spaceLocally convex spaceMetric spaceClosed mapUpper semicontinuous mapGeneralized sequence of iterationsDynamic process

a b s t r a c t

Given a uniform space X and nonempty subsets A and B of X, we introduce the conceptsof some families V of generalized pseudodistances on X, of set-valued dynamic systemsof relatively quasi-asymptotic contractions T : A ∪ B → 2A∪B with respect to V and bestproximity points for T in A ∪ B, and we describe the methods which we use to establishthe conditions guaranteeing the existence of best proximity points for T when T is cyclic(i.e. T : A → 2B and T : B → 2A) or when T is noncyclic (i.e. T : A → 2A andT : B → 2B). Moreover, we establish conditions guaranteeing that for each starting pointeach generalized sequence of iterations of these contractions (in particular, each dynamicprocess) converges and the limit is a best proximity point for T in A∪B. These best proximitypoints for T are determined by unique endpoints in A ∪ B for a map T[2] when T is cyclicand for a map T when T is noncyclic. The results and the methods are new for set-valuedand single-valued dynamic systems in uniform, locally convex, metric and Banach spaces.Various examples illustrating the ideas of our definitions and results, and fundamentaldifferences between our results and the well-known ones are given.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Let A and B be nonempty subsets of a metric space (X, d) and let dist(A, B) = inf{d(x, y) : x ∈ A, y ∈ B}. The single-valuedmap T : A ∪ B → A ∪ B is called cyclic if T(A) ⊂ B and T(B) ⊂ A. Recall that if T is cyclic, then a point w ∈ A ∪ B is called abest proximity point for T if d(w, T(w)) = dist(A, B). The single-valued map T : A ∪ B → A ∪ B is called noncyclic if T(A) ⊂ Aand T(B) ⊂ B. If T is noncyclic, then a point (u, v) ∈ A × B is called a best proximity point for T if T(u) = u, T(v) = v andd(u, v) = dist(A, B). For details, see [1,3–6].

The results concerning the existence of best proximity points were established by: Eldred, Kirk and Veeramani [3] forrelatively nonexpansive cyclic and noncyclic maps T : A ∪ B → A ∪ B in uniformly convex Banach spaces and in Banachspaces such that the pair (A, B) has a proximal normal structure; A.A. Eldred and P. Veeramani [4] for cyclic contractionT : A ∪ B→ A ∪ B of the Banach type in metric and uniformly convex Banach spaces X; and Di Bari, Suzuki and Vetro [1] forcyclic contractions T : A∪B→ A∪B of the Meir–Keeler type in uniformly convex Banach spaces. Additionally, in papers [4,1],

∗ Corresponding author.E-mail addresses: wlkzxa@math.uni.lodz.pl (K. Włodarczyk), robpleb@math.uni.lodz.pl (R. Plebaniak).

0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2008.04.037

K. Włodarczyk et al. / Nonlinear Analysis 70 (2009) 3332–3341 3333

the uniqueness of best proximity points, the convergence to these best proximity points of every sequence {w2m} or {w2m+1

}

where wm= T[m](w0) for m ∈ {0} ∪ N and w0

∈ A ∪ B, and relations between best proximity points of T in A ∪ B and fixedpoints of T[2] in A ∪ B were proved.

It is natural to ask whether there are some results of the above type concerning set-valued dynamic systems in uniformspaces. The main aim of this paper is to show that the answer is affirmative.

For a given uniform space X and nonempty subsets A and B of X, we introduce the concepts of some families V ofgeneralized pseudodistances on X, the set-valued dynamic systems of relatively quasi-asymptotic contractions T : A ∪ B→2A∪B with respect to V and best proximity points for T in A ∪ B, and we describe the methods which we use to establish theconditions guaranteeing the existence of best proximity points for T when T is cyclic (i.e. T : A → 2B and T : B → 2A) orwhen T is noncyclic (i.e. T : A→ 2A and T : B→ 2B). Moreover, we establish conditions guaranteeing that for each startingpoint each generalized sequence of iterations of these contractions (in particular, each dynamic process) converges and thelimit is a best proximity point for T in A ∪ B. These best proximity points for T are determined by unique endpoints in A ∪ Bfor a map T[2] when T is cyclic and for a map T when T is noncyclic. The results and the methods are new for set-valuedand single-valued dynamic systems in uniform, locally convex, metric and Banach spaces. Various examples illustrating theideas of our definitions and results, and a fundamental differences between our results and the well-known ones are given.

2. Definitions, notations and statement of results

To describe our results we need some definitions and notations.Assume once and for all that X is a Hausdorff uniform space with uniformity defined by a saturated family {dα : α ∈ A}

of pseudometrics dα, α ∈ A, uniformly continuous on X2. Recall that a set-valued dynamic system is defined as a pair (X, T),where X is a certain space and T is a set-valued map T : X → 2X; in particular, a set-valued dynamic system includes theusual dynamic system where T is a single-valued map. For T : E→ 2X , E ⊂ X, let T(E) =

⋃x∈E T(x). Here 2X denotes the family

of all nonempty subsets of a space X.

Definition 2.1. Let X be a Hausdorff uniform space and let A and B be nonempty subsets of X. (a) (A ∪ B, T) is called a cyclicset-valued dynamic system on A ∪ B if T : A→ 2B and T : B→ 2A. (b) (A ∪ B, T) is called a noncyclic set-valued dynamic systemon A ∪ B if T : A→ 2A and T : B→ 2B.

Definition 2.2. Let X be a Hausdorff uniform space. The family V = {Vα : 2X→ [0,∞],α ∈ A} is said to be a V-semifamily

of generalized pseudodistances on X (V-semifamily, for short) if the following two conditions hold:

(V1) ∀α∈A∀E1,E2∈2X {E1 ⊂ E2 ⇒ Vα(E1) ≤ Vα(E2)}; and(V2) ∃α0∈A{Vα0(X) > 0}.

Let V = {Vα : 2X→ [0,∞],α ∈ A} be a V-semifamily and, for each α ∈ A, let DV;α(E1, E2) = inf{Vα({x, y}) : x ∈ E1, y ∈

E2}, E1, E2 ∈ 2X .A point w ∈ X is said to be an endpoint (or a stationary point) of T if w is a fixed point of T (i.e., w ∈ T(w)) and T(w) = {w}.

Now let us introduce the notion of best proximity points for cyclic and noncyclic set-valued dynamic systems in uniformspaces.

Definition 2.3. Let X be a Hausdorff uniform space, let V = {Vα : 2X→ [0,∞],α ∈ A} be a V-semifamily and let A and B

be nonempty subsets of X.

(a) Let (A∪ B, T) be a cyclic set-valued dynamic system on A∪ B. A point w ∈ A∪ B is called a best proximity point for T if T(w)is a singleton (i.e. T(w) = {T(w)}) and, for each α ∈ A, Vα({w, T(w)}) = DV; α(A, B).

(b) Let (A∪ B, T) be a noncyclic set-valued dynamic system on A∪ B. A point (u, v) ∈ A× B is called a best proximity point forT if u and v are endpoints of T and, for each α ∈ A, Vα({u, v}) = DV;α(A, B).

It is natural to ask the following:

Question 2.1. Let (A ∪ B, T) be a cyclic or noncyclic set-valued dynamic system in uniform space X. Are there any conditionsguaranteeing the existence of best proximity points for T such that the convergence property holds?

The following concept of relatively quasi-asymptotic contractions is needed to present our results which are the answerto Question 2.1.

Definition 2.4. Let X be a Hausdorff uniform space and let V = {Vα : 2X→ [0,∞],α ∈ A} be a V-semifamily on X. Let A and

B be nonempty subsets of X and let (A∪ B, T) be a set-valued dynamic system on A∪ B. We say that (A∪ B, T) is a V-relativelyquasi-asymptotic contraction on A ∪ B (V-RQAC on A ∪ B, for short) if the following two conditions hold:

(A1) ∀α∈A{Vα(A ∪ B) > DV;α(A, B)⇒ ∃m∈N{Vα(T[m](A ∪ B)) < Vα(A ∪ B)}}; and(A2) ∀α∈A∀ε>0∃η>0∀n∈N{Vα(T[n](A ∪ B)) < DV;α(A, B)+ ε+ η⇒ ∃m∈N{Vα(T[m+n](A ∪ B)) ≤ DV;α(A, B)+ ε}}.

We now state the first main result.

3334 K. Włodarczyk et al. / Nonlinear Analysis 70 (2009) 3332–3341

Theorem 2.1. Assume that: (a) X is a Hausdorff uniform space; (b) The family V = {Vα : 2X→ [0,∞],α ∈ A} is a V-semifamily

on X; (c) A and B are nonempty subsets of X; and (d) (A ∪ B, T) is V-RQAC on A ∪ B.

(i) If (A ∪ B, T) is cyclic on A ∪ B, then each sequence {wm}, where wm

∈ T[m](w0) for m ∈ N and w0∈ A ∪ B, satisfies

∀α∈A{limm Vα({wm,wm+1}) = DV;α(A, B)}.

(ii) If (A ∪ B, T) is noncyclic on A ∪ B, then every two sequences {um} and {vm}, where (um, vm) ∈ T[m](u0) × T[m](v0) for m ∈ N

and (u0, v0) ∈ A× B, satisfy ∀α∈A{limm Vα({um, vm}) = DV;α(A, B)}.

In [7] we introduced the notion of a V-family.

Definition 2.5 ([7, Definition 2.1]). Let X be a Hausdorff uniform space. The family V = {Vα : 2X→ [0,∞],α ∈ A} is said to

be a V-family of generalized pseudodistances on X (V-family, for short) if it is a V-semifamily and, additionally, the followingtwo conditions hold:

(V3) ∀α∈A∀x,y,z∈X{Vα({x, z}) ≤ Vα({x, y})+ Vα({y, z})}; and(V4) For any sequence {xm} in X such that

∀α∈A{limn

supm>n

Vα({xn, xm}) = 0}, (2.1)

if there exists a sequence {ym} in X satisfying

∀α∈A{limm

Vα({xm, ym}) = 0}, (2.2)

then

∀α∈A{limm

dα(xm, ym) = 0}. (2.3)

Using the definition of V-family we introduced in [8] the following concept of V-quasi-asymptotic contraction.

Definition 2.6 ([8, Definition 2.2]). Let X be a Hausdorff uniform space, let V = {Vα : 2X→ [0,∞],α ∈ A} be a V-family on

X and let (X, T) be a set-valued dynamic system. We say that (X, T) is V-quasi-asymptotic contraction on X (V-QAC on X, forshort) if the following two conditions hold:

(C1) ∀α∈A{Vα(X) > 0⇒ ∃m∈N{Vα(T[m](X)) < Vα(X)}}; and(C2) ∀α∈A∀ε>0∃η>0∀n∈N{Vα(T[n](X)) < ε+ η⇒ ∃m∈N{Vα(T[m+n](X)) ≤ ε}}.

Remark 2.1. The condition (C2) is

∀α∈A∀ε>0∃η>0∀n∈N{ε < Vα(T[n](X)) < ε+ η⇒ ∃m∈N{Vα(T

[m+n](X)) ≤ ε}} (2.4)

in [8, Definition 2.2]. However, both are equivalent. Indeed, it is clear that (C2) implies (2.4). Assuming now that (2.4) issatisfied, we conclude that (C2) holds. In fact, if Vα0(T

[n0](X)) ≤ ε0 for some α0 ∈ A, n0 ∈ N and ε0 > 0, then, using (V1) andproperty ∀m∈N{T[m+n0](X) ⊂ T[n0](X)}, we get that ∀m∈N{Vα0(T

[m+n0](X)) ≤ Vα0(T[n0](X)) ≤ ε0}.

The graph of T is⋃{(x, T(x)) : x ∈ X} ⊂ X × X. Let (D,≥) denote a directed set whose elements will be indicated by

β, n,m, . . ..The following are equivalent: (a) T is closed, i.e. the graph of T is closed in X × X; (b) Whenever {xβ : β ∈ D} is a net

converging to x and {yβ : β ∈ D} is a net converging to y such that yβ ∈ T(xβ) for all β ∈ D, then y ∈ T(x). It is well-knownthat every upper semicontinuous map is closed [2, Theorem 6, p. 112] and, if X is a compact space, then the map is closed ifand only if it is upper semicontinuous [2, Corollary, p. 112].

In [8] we proved the following endpoint theorem.

Theorem 2.2 ([8, Theorem 2.2]). Assume that: (a) X is a Hausdorff complete uniform space; (b) V = {Vα : 2X→ [0,∞],α ∈ A}

is a V-family on X; (c) (X, T) is V-QAC on X; and (d) T[p] is closed in X for some p ∈ N. Then: (i) T has a unique endpoint w in X;and (ii) Each sequence {wm

}, where wm∈ T[m](w0) for m ∈ N and w0

∈ X, converges to w.

Using Theorems 2.1 and 2.2, we show that the answer concerning Question 2.1 is affirmative.

Theorem 2.3. Assume that: (a) X is a Hausdorff complete uniform space; (b) V = {Vα : 2X→ [0,∞],α ∈ A} is a V-family on

X; (c) A and B are nonempty closed subsets of X; (d) (A ∪ B, T) is cyclic on A ∪ B; (e) (A ∪ B, T) is V-RQAC on A ∪ B and T[2s−1]

is closed in A ∪ B for some s ∈ N; (f) (A, T[2]) is V-QAC on A and T[2p] is closed in A for some p ∈ N; and (g) (B, T[2]) is V-QACon B and T[2q] is closed in B for some q ∈ N. Then there exists w ∈ A such that: (i) w is a unique endpoint of T[2] in A; (ii) T(w)is a singleton and T(w) is a unique endpoint of T[2] in B; (iii) Each sequence {wm

}, where wm∈ T[m](w0) for m ∈ N and w0

∈ A,satisfies limm w2m

= w and limm w2m+1= T(w); and (iv) Each sequence {wm

}, where wm∈ T[m](w0) for m ∈ N and w0

∈ A ∪ B,satisfies ∀α∈A{limm Vα({wm,wm+1

}) = Vα({w, T(w)}) = DV;α(A, B)}.

K. Włodarczyk et al. / Nonlinear Analysis 70 (2009) 3332–3341 3335

Theorem 2.4. Assume that: (a) X is a Hausdorff complete uniform space; (b) V = {Vα : 2X→ [0,∞],α ∈ A} is a V-family on

X; (c) A and B are nonempty closed subsets of X; (d) (A ∪ B, T) is noncyclic on A ∪ B; (e) (A ∪ B, T) is V-RQAC on A ∪ B; (f) (A, T)is V-QAC on A and T[p] is closed in A for some p ∈ N; and (g) (B, T) is V-QAC on B and T[q] is closed in B for some q ∈ N. Thenthere exists (u, v) ∈ A× B such that: (i) u is a unique endpoint of T in A; (ii) v is a unique endpoint of T in B; and (iii) Every twosequences {um

} and {vm}, where (um, vm) ∈ T[m](u0)×T[m](v0) for m ∈ N and (u0, v0) ∈ A×B, satisfy limm um= u and limm vm = v

and ∀α∈A{limm Vα({um, vm}) = Vα({u, v}) = DV;α(A, B)}.

3. Proofs of Theorems 2.1, 2.3 and 2.4

Proof of Theorem 2.1. The proof will be broken into nine steps.Step I. We see that

∀α∈A∀m∈{0}∪N{DV;α(A, B) ≤ DV;α(T[m](A), T[m](B)) ≤ Vα(T

[m](A ∪ B))} (3.1)

and

∀α∈A∀m∈{0}∪N{Vα(T[m+1](A ∪ B)) ≤ Vα(T

[m](A ∪ B))}. (3.2)

Since

∀m∈{0}∪N{T[m+1](A ∪ B) ⊂ T[m](A ∪ B)} (3.3)

holds and (A ∪ B, T)is cyclic or noncyclic on A ∪ B, we obtain that (3.1) and (3.2) are a consequence of (V1) and (3.3).Step II. We show that, for each α ∈ A, there exists mα ∈ {0} ∪ N such that the set

H(mα) = {Vα(T[m](A ∪ B))− DV;α(A, B) : m ≥ mα} (3.4)

is bounded. First, assume that Vα(A∪ B) = DV;α(A, B) for some α ∈ A. Then, by (3.1)–(3.3), H(mα) = {0}where mα = 0. Next,if Vα(A ∪ B) > DV;α(A, B) for some α ∈ A, then, by (A1), ∃mα∈N{Vα(T[mα](A ∪ B)) < Vα(A ∪ B)} which, by (3.1)–(3.3), gives∀m≥mα {Vα(T

[m](A ∪ B))− DV;α(A, B) ≤ Vα(T[mα](A ∪ B))− DV;α(A, B) <∞}. This proves that the set H(mα) is bounded.Step III. We see that

∀α∈A∀n∈N{0 < Vα(T[n](A ∪ B))− DV;α(A, B) <∞⇒ ∃m∈N{Vα(T

[m+n](A ∪ B))− DV;α(A, B)

< Vα(T[n](A ∪ B))− DV;α(A, B)}}. (3.5)

Let 0 < Vα0(T[n0](A ∪ B)) − DV;α0(A, B) < ∞ for some α0 ∈ A and n0 ∈ N. This implies that, for some ε0 > 0 and η0 > 0,

ε0 < Vα0(T[n0](A ∪ B)) − DV;α0(A, B) < ε0 + η0. Hence, by (A2), we get ∃m∈N{Vα0(T

[m+n0](A ∪ B)) − DV;α0(A, B) ≤ ε0 <Vα0(T

[n0](A ∪ B))− DV;α0(A, B)}. This shows that (3.5) holds.Step IV. We show that

∀α∈A{Vα(A ∪ B) = DV;α(A, B)⇒ limm

Vα(T[m](A ∪ B)) = DV;α(A, B)}. (3.6)

By (3.1)–(3.3), since ∀α∈A{Vα(A ∪ B) = DV;α(A, B)⇒ ∀m∈NVα(T[m](A ∪ B))} = DV;α(A, B), we obtain (3.6).Step V. We show that

∀α∈A{Vα(A ∪ B) > DV;α(A, B)⇒ limm

Vα(T[m](A ∪ B)) = DV;α(A ∪ B)}. (3.7)

Suppose α0 ∈ A and Vα0(A ∪ B) > DV;α0(A, B). By (3.2), the sequence {Vα0(T[m](A ∪ B))} is decreasing and, by Step II, there

exists mα0 ∈ {0} ∪ N such that the set H(mα0) is bounded which, by (3.4), implies that there exists the limit

limm

Vα0(T[m](A ∪ B)) = DV;α0(A, B)+ γα0 (3.8)

where

0 ≤ γα0 <∞. (3.9)

It is clear from (3.2) and (3.8) that

∀m≥mα0{γα0 ≤ Vα0(T

[m](A ∪ B))− DV;α0(A, B) ≤ Vα0(T[mα0 ](A ∪ B))− DV;α0(A, B) <∞}. (3.10)

On the other hand, if

γα0 6= 0, (3.11)

then

∀m≥mα0{γα0 < Vα0(T

[m](A ∪ B))− DV;α0(A, B) ≤ Vα0(T[mα0 ](A ∪ B))− DV;α0(A, B) <∞}. (3.12)

3336 K. Włodarczyk et al. / Nonlinear Analysis 70 (2009) 3332–3341

Indeed, otherwise, by property (3.10), there exists m0 ≥ mα0 such that γα0 = Vα0(T[m0](A∪ B))−DV;α0(A, B) which, by (3.10)

and (3.2), implies that

∀n≥m0 {γα0 = Vα0(T[n](A ∪ B))− DV;α0(A, B)}. (3.13)

Next, if n0 ≥ m0 is arbitrary and fixed, then, by (3.11), (3.10), (3.5) and (3.13), we get ∃m∈N{γα0 = Vα0(T[m+n0](A ∪ B)) −

DV;α0(A, B) < Vα0(T[n0](A ∪ B))− DV;α0(A, B) = γα0 }which is impossible. Therefore, (3.12) holds if (3.11) is satisfied.

Now we observe that

γα0 = 0. (3.14)

The assertion (3.7) follows by contradiction. Suppose that

γα0 > 0 (3.15)

and let η > 0 be arbitrary and fixed. Then, due to (3.12) and (3.8), we have that there exists k ∈ N such that k ≥ mα0 and∀m≥k{γα0 < Vα0(T

[m](A∪ B))−DV;α0(A, B) < γα0 + η}. Hence, by (A2), ∀m≥k∃j∈N{Vα0(T[m+j](A∪ B))−DV;α0(A, B) ≤ γα0 }which,

by (3.15) and (3.12), is impossible. Therefore (3.14) holds. From (3.8) and (3.14) we obtain (3.7).Step VI. The following holds

∀α∈A{limm

Vα(T[m](A ∪ B)) = DV;α(A, B)}. (3.16)

This is a consequence of (3.1), (3.6) and (3.7).Step VII. We see that

∀α∈A∀E⊂A∪B, E∩A 6=∅, E∩B6=∅{limm

Vα(T[m](E)) = DV;α(A, B)}. (3.17)

Indeed, then ∀m∈{0}∪N{T[m](E) ∩ A 6= ∅ ∧ T[m](E) ∩ B 6= ∅}. Hence, by (V1), we have ∀α∈A∀m∈{0}∪N{DV;α(A, B) ≤ Vα(T[m](E)) ≤Vα(T[m](A ∪ B))}which, using (3.16), gives (3.17).

Step VIII. Let (A∪ B, T) be cyclic on A∪ B. Let w0∈ A∪ B and {wm

}, wm∈ T[m](w0) for m ∈ N, be arbitrary and fixed. The result

is that

∀α∈A{limm

Vα({wm,wm+1

}) = DV;α(A, B)}. (3.18)

Indeed, we see that ∀m∈N{{wm,wm+1} ⊂ T[m](E1)} where E1 = {w0

} ∪ T(w0), E1 ⊂ A ∪ B, E1 ∩ A 6= ∅ and E1 ∩ B 6= ∅.Clearly, ∀m∈{0}∪N{{wm,wm+1

} ⊂ A ∪ B ∧ {wm,wm+1} ∩ A 6= ∅ ∧ {wm,wm+1

} ∩ B 6= ∅}. Hence, by (V1), we deduce that∀α∈A∀m∈N{DV;α(A, B) ≤ Vα({wm,wm+1

}) ≤ Vα(T[m](E1))}which, by (3.17), gives (3.18).Step IX. Let (A ∪ B, T) be noncyclic on A ∪ B. Let (u0, v0) ∈ A× B, {um

} and {vm}, (um, vm) ∈ T[m](u0)× T[m](v0) for m ∈ N, bearbitrary and fixed. The result is that

∀α∈A{limm

Vα({um, vm}) = DV;α(A, B)}. (3.19)

Indeed, we see that ∀m∈N{{um, vm} ⊂ T[m](E2)} where E2 = {u0, v0}, E2 ⊂ A ∪ B, E2 ∩ A 6= ∅ and E2 ∩ B 6= ∅. Clearly,

∀m∈{0}∪N{{um, vm} ⊂ A ∪ B ∧ {um, vm} ∩ A 6= ∅ ∧ {um, vm} ∩ B 6= ∅}. Hence, by (V1), we deduce that ∀α∈A∀m∈N{DV;α(A, B) ≤Vα({um, vm}) ≤ Vα(T[m](E2))}which, by (3.17), gives (3.19). �

Proof of Theorem 2.3. By assumptions (c), (f) and (g) and Theorem 2.2(i), the map T[2] : A→ 2A has a unique endpoint w inA and the map T[2] : B→ 2B has a unique endpoint v in B; i.e. w and v satisfy

T[2](w) = {w} and T[2](v) = {v}. (3.20)

Thus, using (3.20) we show that

∀m∈N{T[2m](w) = {w} ∧ T[2m](v) = {v}}. (3.21)

Moreover, from Theorem 2.2(ii) it follows, in particular, that each sequence {wm}, where wm

∈ T[m](w0) for m ∈ N andw0∈ A, satisfies

limm

w2m= w (3.22)

and

limm

w2m+1= v. (3.23)

Also note that Theorem 2.1 implies that

∀α∈A{limm

Vα({wm,wm+1

}) = DV;α(A, B)} (3.24)

since a V-family is a V-semifamily.

K. Włodarczyk et al. / Nonlinear Analysis 70 (2009) 3332–3341 3337

Additionally, since w2m∈ T[2m](w0) = T[2s−1](T[2(m−s)+1](w0)), m > s, there exists v2(m−s)+1

∈ T[2(m−s)+1](w0), m > s, suchthat

w2m∈ T[2s−1](v2(m−s)+1), m > s, (3.25)

and, by (3.23),

limm

v2(m−s)+1= v. (3.26)

However, T[2s−1] is closed in A∪ B and, for each m > s, {w2m, v2(m−s)+1} ⊂ A∪ B. Therefore, using (3.22) and (3.26), from (3.25)

we obtain w ∈ T[2s−1](v) which implies, when we use (3.21), that T(w) ⊂ T[2s](v) = {v}, i.e.

v = T(w). (3.27)

Finally, we see, by (3.21) and (3.27), that

∀m∈N{T[2m](w) = {w} ∧ T[2m+1](w) = {T(w)}}. (3.28)

We now define

um= T[m](w), m ∈ N. (3.29)

Hence we get that {um} is a generalized sequence of iterations starting at u0

= w. Applying this sequence to (3.24) and using(3.28) we conclude that

∀α∈A{limm

Vα({um, um+1

}) = Vα({w, T(w)}) = DV;α(A, B)}. (3.30)

Therefore, the assertions (i)–(iv) follow directly from (3.20)–(3.24) and (3.27)–(3.30). �

Proof of Theorem 2.4. By Theorem 2.2(i) and assumptions (c), (f) and (g), the map T : A→ 2A has a unique endpoint u in Aand the map T : B→ 2B has a unique endpoint v in B. Moreover, from Theorems 2.2(ii) and 2.1(ii) it follows that every twosequences {um

} and {vm}, where (um, vm) ∈ T[m](u0)× T[m](v0) for m ∈ N and (u0, v0) ∈ A× B, satisfy limm um= u, limm vm = v

and ∀α∈A{limm Vα({um, vm}) = DV;α(A, B)}. In particular, defining (um, vm) = T[m](u) × T[m](v) = (u, v) for m ∈ N, we obtainthat ∀α∈A{limm Vα({um, vm}) = Vα({u, v}) = DV;α(A, B)}. This shows that assertions (i)–(iii) hold. �

4. Examples

In this section, we will discuss various examples. Let X be a Hausdorff uniform space with uniformity defined by asaturated family {dα : α ∈ A} of pseudometrics dα, α ∈ A, uniformly continuous on X2. Let δα(E) = sup{dα(x, y) : x, y ∈E}, E ∈ 2X,α ∈ A. First, we present some examples of V-families.

Example 4.1. Let X be a Hausdorff uniform space containing at least two different points, let a, b ∈ X be arbitrary and fixedand such that a 6= b and, for each α ∈ A, let cα > 0 be arbitrary and fixed. We show that the family V = {Vα : 2X

→

[0,∞] α ∈ A}, where

Vα(E) ={

0 if E = {a} or E = {b}cα if E 6= {a} and E 6= {b},

E ⊂ X,α ∈ A, (4.1)

is a V-family on X. In fact, condition (V1) is satisfied since for E1 ⊂ E2, E1, E2 ∈ 2X , the following two cases hold:Case 1. If E2 = {a} or E2 = {b}, then E1 = E2 and ∀α∈A{Vα(E1) = Vα(E2) = 0};Case 2. If E2 6= {a} and E2 6= {b}, then ∀α∈A{Vα(E2) = cα} and, for each α ∈ A, Vα(E1) = 0 or Vα(E1) = cα which gives

∀α∈A{Vα(E1) ≤ Vα(E2)}.Condition (V2) holds, since, by (4.1) and property (V1), we get ∀α∈A{0 < cα = Vα({a, b}) ≤ Vα(X)}. Since, for every

x, y, z ∈ X and α ∈ A,

Vα({x, y}) = cα

≤

cα + cα = Vα({x, z})+ Vα({z, y}) if x 6= y ∧ z 6∈ {a, b}cα + 0 = Vα({x, z})+ Vα({z, y}) if x 6= y = z ∈ {a, b}0+ cα = Vα({x, z})+ Vα({z, y}) if y 6= x = z ∈ {a, b}cα + cα = Vα({x, z})+ Vα({z, y}) if x = y 6∈ {a, b}

and Vα({x, y}) = 0 ≤ Vα({x, z})+ Vα({z, y}) if x = y ∈ {a, b} ∧ z ∈ X, (V3) holds.By proving that (V4) holds we see that if the sequences {xm} and {ym} in X satisfy (2.1) and (2.2), then, in particular, (2.2)

yields

∀α∈A∀0<εα<cα∃m0=m0(α,εα)∈N∀m≥m0 {Vα({xm, ym}) < εα}. (4.2)

3338 K. Włodarczyk et al. / Nonlinear Analysis 70 (2009) 3332–3341

Now, by (4.1), denoting m′ = min{m0(α, εα) : α ∈ A}, we conclude that condition (4.2) gives

∀m≥m′ {(xm= ym = a) ∨ (xm = ym = b)}. (4.3)

Using (4.2) and (4.3), we get ∀α∈A∀0<εα<cα∃m0=m0(α,εα)∈N∀m≥m0 {dα(x

m, ym) = 0 < εα}. The result is that the sequences {xm} and{ym} satisfy (2.3). Hence we find (V4).

Example 4.2 ([7, Example 6.2]). Let X be a Hausdorff uniform space, let E0 be a closed and bounded subset of X, which containsat least two points and let cα ∈ (0,∞) be constants such that cα ≥ δα(E0), α ∈ A. The family V = {Vα : 2X

→ [0,∞], α ∈ A}where

Vα(E) ={δα(E) if E ∩ E0 = Ecα if E ∩ E0 6= E,

E ⊂ X,α ∈ A, (4.4)

is a V-family on X.

Example 4.3. The family V = {Vα : 2X→ [0,∞], α ∈ A}, ∀α∈A∀E∈2X {Vα(E) = δα(E)}, is a V-family on X.

Now we illustrate Theorems 2.1, 2.3 and 2.4 in the case of single-valued maps.

Example 4.4. Let X = [0, 1/2]. Defining

V(E) ={

0 if E = {0} or E = {1/2}2 if E 6= {0} and E 6= {1/2}, E ∈ 2X,

we see, by Example 4.1, that the family V = {V : 2X→ [0,∞]} is a V-family on X. Let A = [0, 1/4] ∪ {1/2}, B = {0, 1/2} and

let T : A ∪ B→ A ∪ B be a not closed map of the form

T(x) ={

0 for x = 1/41/2 for x ∈ [0, 1/4) ∪ {1/2}.

We prove that all assumptions of Theorem 2.3 hold. Because T(A) = {0, 1/2} = B and T(B) = {1/2} ⊂ A, we obtainthat condition (d) is satisfied. We show that (A ∪ B, T) is V-RQAC on A ∪ B and T[3] is closed in A ∪ B. Indeed, condition(A1) holds since V(A ∪ B) = 2 > 0 = V({0, 0}) = V({1/2, 1/2}) = DV(A, B) and there exists m (e.g. m = 2) such thatV(T[m](A ∪ B)) = V({1/2}) = 0 < 2 = V(A ∪ B). Obviously, (A2) holds since T[m](A ∪ B) = {1/2} for m ≥ 2. Therefore,assumption (e) holds (for s = 2). We see also that assumptions (f) (for p = 1) and (g) (for q = 1) are satisfied. Thus, allassumptions of Theorem 2.3 hold, w = 1/2 and T(w) = w (w is a unique fixed point of T in A ∪ B). All assertions (i)–(iv) aresatisfied and V({w, T(w)}) = V({1/2, 1/2}) = 0 = DV(A, B).

Example 4.5. Let X = [0, 3] and E0 = {1, 3} be a closed and bounded subset of X. By Example 4.2, the family V = {V : 2X→

[0,∞]},

V(E) =

0 if E = {1} or E = {3}2 for E = E03 for E ∩ E0 6= E,

E ∈ 2X,

is a V-family on X. Let A = [0, 1] and B = [2, 3].(I) Let T : A ∪ B→ A ∪ B be a not closed map of the form

T(x) =

3 for x ∈ A \ {1/2}2 for x = 1/21 for x ∈ B.

We prove that all assumptions of Theorem 2.3 hold. Since T(A) = {2, 3} ⊂ B and T(B) = {1} ⊂ A, (d) holds. (A ∪ B, T) isV-RQAC on A ∪ B. In fact, it is obvious that V(A ∪ B) = 3 > 2 = V({1, 3}) = DV(A, B). Next, we can calculate that, for eachn ∈ N,

T[2n](x) ={

1 for x ∈ A3 for x ∈ B,

T[2n−1](x) ={

3 for x ∈ A1 for x ∈ B.

Hence, for m = 2, V(T[m](A ∪ B)) = V({1, 3}) = 2 < 3 = V(A ∪ B), so (A1) holds. Further, for each m ≥ 2,V(T[m](A ∪ B)) = V({1, 3}) = 2. Since DV(A, B) = 2, for each ε > 0, we get V(T[m+n](A ∪ B)) ≤ DV(A, B) + ε; thus (A2)holds. In consequence, (A ∪ B, T) is V-RQAC on A ∪ B. Moreover, assumptions (f) (for p = 2) and (g) (for q = 1) are satisfied.Thus, all assumptions of Theorem 2.3 hold, T[2] has a unique fixed point w = 1 in A, T[2] has a unique fixed point w = 3 in Band V({w, T(w)}) = V({1, 3}) = 2 = DV(A, B).

K. Włodarczyk et al. / Nonlinear Analysis 70 (2009) 3332–3341 3339

(II) Let T : A ∪ B→ A ∪ B be a not closed map of the form

T(x) =

1 for x ∈ A \ {1/2}0 for x = 1/23 for x ∈ B.

We prove that all assumptions of Theorem 2.4 hold. Since T(A) = {0, 1} ⊂ A and T(B) = {3} ⊂ B, T is noncyclic. Moreover, itis easy to calculate that: for each m ≥ 2, V(T[m](A ∪ B)) = V({1, 3}) = 2; V(A ∪ B) = 3; DV(A, B) = 2. Thus all assumptions ofTheorem 2.4 hold. We see that T(1) = 1, T(3) = 3 and (1, 3) ∈ A× B is a best proximity point of T.

Example 4.6. Let (X, d) be a metric space where X = [0, 1] and d(x, y) = |x − y| for all x, y ∈ X. Of course, the familyV = {V : 2X

→ [0,∞]}, where V(E) = δ(E) = sup{d(x, y) : x, y ∈ E}, E ∈ 2X , is a V-family on X; see Example 4.3. Let A 6= ∅,A ⊂ [0, 1/4], B 6= ∅ and B ⊂ [3/4, 1]. Let T : A ∪ B→ A ∪ B be of the form

T(x) =

−(25/96)x+ 15/16 if x ∈ [0, 6/25] ∩ A−(25/2)x+ 31/8 if x ∈ [6/25, 1/4] ∩ A−(25/2)x+ 77/8 if x ∈ [3/4, 76/100] ∩ B−(25/96)x+ 31/96 if x ∈ [76/100, 1] ∩ B.

We consider the following two cases:Case 1. A = [0, 1/4] and B = [3/4, 1]. Then condition (d) is satisfied since T(A) = [3/4, 15/16] ⊂ B and T(B) =

[1/16, 1/4] ⊂ A. Moreover, condition (A1) holds since V(A ∪ B) = 1 > 1/2 = DV(A, B) and V(T(A ∪ B)) = V([1/16, 1/4] ∪[3/4, 15/16]) = 7/8 < 1 = V(A ∪ B); thus m = 1. Obviously (A2) holds since if ε > 0 is arbitrary and η = 1/2, then, for alln ∈ N, we have V(T[n](A ∪ B)) < DV(A, B)+ ε+ η and there exists m ∈ N such that V(T[n+m](A ∪ B)) ≤ DV(A, B)+ ε becauselimm V(T[n+m](A ∪ B)) = DV(A, B) = 1/2. We also see that T[2s−1] is closed in A ∪ B for some s ∈ N. Thus, (e) holds. Nextwe show that (A, T[2]) is a V-QAC on A. The condition (C1) holds since V(A) = 1/4 > 0 and V(T[2](A)) < V(A). Moreover,condition (C2) holds since if ε is arbitrary and fixed and η = 1/2, then, for all n ∈ N, we have V(T[2n](A)) < ε+ η and thereexists m ∈ N such that V(T[2(m+n)](A)) ≤ ε because limm Vα(T[2m](A)) = 0. It is clear that T[2p] is closed in A for each p ∈ N.Thus, (f) is satisfied. Analogously we can show that (B, T[2]) is V-QAC on B and T[2q] is closed in B for each q ∈ N, i.e. (g) holds.Therefore, assumptions of Theorem 2.3 are satisfied, T[2] has a unique fixed point w = 1/4 in A, ∀x∈A{limm T[2m](x) = 1/4}, T[2]has a unique fixed point w = 3/4 in B and ∀x∈B{limm T[2m](x) = 3/4}. Moreover, for each w ∈ {1/4, 3/4}, T(w) ∈ {1/4, 3/4}and V({w, T(w)}) = DV(A, B).

Case 2. A ⊂ [0, 1/4), B ⊂ (3/4, 1], 1/4 ∈ A′ and 3/4 ∈ B′ where A′ and B′ denote the derivatives of the sets of A and B,respectively. In this case it is clear that all assumptions of Theorem 2.1 are satisfied.

Let A and B be nonempty subsets in a Banach space (X, ‖·‖) and let dist(A, B) = inf{‖x− y‖ : x ∈ A, y ∈ B}. A.A. Eldred,W.A. Kirk and P. Veeramani established among others the following results:

Theorem 4.1 ([3, Theorem 2.1 and Corollary 2.1]). Let A and B be nonempty subsets in a Banach space (X, ‖·‖). Assume that oneof the two conditions holds: (a′) A and B are weakly compact and convex and (A, B) has a proximal normal structure; (a′′) A andB are bounded, closed and convex and X is uniformly convex. Additionally, assume that: (b) T : A ∪ B→ A ∪ B; (c) T(A) ⊂ B andT(B) ⊂ A; and (d) T is relatively nonexpansive, i.e. ∀(x,y)∈A×B{‖T(x)− T(y)‖ ≤ ‖x− y‖}. Then there exists (x, y) ∈ A× B such that‖x− T(x)‖ = ‖y− T(y)‖ = dist(A, B).

Theorem 4.2 ([3, Theorem 2.2 and Corollary 2.2]). Let A and B be nonempty subsets in a Banach space (X, ‖·‖). Assume that onefrom the two conditions holds: (a′) A and B are weakly compact and convex, X is strictly convex and (A, B) has a proximal normalstructure; (a′′) A and B are bounded closed and convex and X is uniformly convex. Additionally, assume that: (b) T : A ∪ B →A∪ B; (c) T(A) ⊂ A and T(B) ⊂ B; and (d) T is relatively nonexpansive, i.e. ∀(x,y)∈A×B{‖T(x)− T(y)‖ ≤ ‖x− y‖}. Then there exists(x, y) ∈ A× B such that T(x) = x, T(y) = y and ‖x− y‖ = dist(A, B).

The A.A. Eldred and P. Veeramani theorem can be read as follows.

Theorem 4.3 ([4, Theorem 3.10]). Let (X, ‖·‖) be a uniformly convex Banach space and let A and B be nonempty closed andconvex subsets of X. Let T : A ∪ B → A ∪ B be a cyclic contraction, that is, T(A) ⊂ B, T(B) ⊂ A and there exists k ∈ (0, 1)such that ∀x∈A, y∈B{‖T(x)− T(y)‖ ≤ k ‖x− y‖ + (1− k)dist(A, B)}. Then there exists a unique best proximity point w ∈ A, that is‖w− T(w)‖ = dist(A, B). Further, for each x ∈ A, {T[2n](x)} converges to w.

C. Di Bari, T. Suzuki and C. Vetro find a more general result.

Theorem 4.4 ([1, Theorem 2]). Let (X, ‖·‖) be a uniformly convex Banach space and let A and B be nonempty closed and convexsubsets of X. Let T : A ∪ B → A ∪ B be a cyclic Meir–Keeler contraction, that is, T(A) ⊂ B, T(B) ⊂ A and for each ε > 0, thereexists η > 0 such that ∀x∈A, y∈B{‖x− y‖ < dist(A, B)+ ε+ η⇒ ‖T(x)− T(y)‖ < dist(A, B)+ ε}. Then there exists a unique bestproximity point w ∈ A, that is ‖w− T(w)‖ = dist(A, B). Further, for each x ∈ A, {T[2n](x)} converges to w.

3340 K. Włodarczyk et al. / Nonlinear Analysis 70 (2009) 3332–3341

We show that, in general, our V-RQAC on A ∪ B, even for single-valued maps on metric spaces X, are not relativelynonexpansive [3], not cyclic contractions [4] and not cyclic Meir–Keeler contractions [1] and we present three exampleswhich illustrate the fundamental differences between Theorems 2.1, 2.3, 2.4 and 4.1–4.4.

Example 4.7. Let X, A, B and the map T : A ∪ B → A ∪ B be such as in Example 4.4. We see that dist(A, B) = inf{|x− y| :x ∈ A , y ∈ B} = 0. We obtain: (a) For (x, y) = (1/4, 0) we obtain that |T(x)− T(y)| = |0− 1/2| = 1/2 > 1/4 =|1/4− 0| = |x− y|. Hence T is not relatively nonexpansive; (b) Assuming (x, y) = (1/4, 0) we get that, for each k ∈ (0, 1),|T(x)− T(y)| = |0− 1/2| = 1/2 > (1/4)k = k |1/4− 0| = k |x− y| + (1 − k)dist(A, B). Therefore, T is not a cycliccontraction; (c) There exists ε0 = 1/4 such that for all η > 0 there exist x = 1/4 ∈ A and y = 0 ∈ B such that|x− y| = |1/4− 0| = 1/4 < 0 + 1/4 + η = dist(A, B) + ε0 + η and |T(x)− T(y)| = 1/2 > 0 + 1/4 = dist(A, B) + ε0.Consequently, T is not a cyclic Meir–Keeler contraction.

Example 4.8. Let X, A, B and the map T : A ∪ B→ A ∪ B be defined as in Example 4.5. Then dist(A, B) = 1. We have: (a) Forx = 1 ∈ A and y = 2 ∈ B we get that |T(x)− T(y)| = 2 > 1 = |x− y|. Hence T is not relatively nonexpansive; (b) For eachk ∈ (0, 1) there exist x = 1 ∈ A and y = 2 ∈ B such that |T(x)− T(y)| = 2 > k+ (1− k) = k |x− y| + (1− k)dist(A, B). HenceT is not a cyclic contraction; (c) There exists ε0 = 1/4 such that for all η > 0 there exist x = 1 ∈ A and y = 2 ∈ B such that|x− y| = 1 < 1+ 1/4+ η = dist(A, B)+ ε0 + η and |T(x)− T(y)| = 2 > 1+ 1/4 = dist(A, B)+ ε0. Consequently, T is not acyclic Meir–Keeler contraction.

Example 4.9. Let X, A, B and the map T : A∪ B→ A∪ B be defined as in Case 1 of Example 4.6. Then dist(A, B) = 1/2. We get:(a) For x = 1/4− 1/100 = 24/100 ∈ A and y = 3/4+ 1/100 = 76/100 ∈ B we have that |T(x)− T(y)| = |14/16− 2/16| =3/4 > 52/100 = |x− y|. Hence T is not relatively nonexpansive; (b) For each k ∈ (0, 1) there exist x = 24/100 ∈ Aand y = 76/100 ∈ B such that |T(x)− T(y)| = |14/16− 2/16| = 3/4 > 1/2 + (1/50)k = (52/100)k + (1 − k)/2 =k |x− y| + (1 − k)dist(A, B). Hence T is not a cyclic contraction; (c) There exists ε0 = 2/100 such that for all η > 0 thereexist x = 24/100 ∈ A and y = 76/100 ∈ B such that |x− y| = 1/2 + 2/100 = dist(A, B) + ε0 < dist(A, B) + ε0 + η and|T(x)− T(y)| = 3/4 > 1/2+ 2/100 = dist(A, B)+ ε0. Consequently, T is not a cyclic Meir–Keeler contraction.

Now, we illustrate Theorem 2.3 in the case when T is a set-valued map.

Example 4.10. Let X = [0, 3], let E0 = {1, 3} and let V = {V : 2X→ [0,∞]} be such as in Example 4.5. For A = [0, 1] and

B = [2, 3] let T : A ∪ B→ 2A∪B be a not closed map of the form

T(x) =

{3} for x ∈ A \ {1/2}{2} for x = 1/2{0, 1} for x ∈ B \ {3}{1} for x = 3.

Since T(A) = {2, 3} ⊂ [2, 3] = B and T(B) = {0, 1} ⊂ [0, 1] = A, (d) holds.We show that (A ∪ B, T) is V-RQAC on A ∪ B. We see that V(A ∪ B) = 3 > 2 = V({1, 3}) = DV(A, B). By the definition of T,

we have

T[2](x) =

{1} for x ∈ A \ {1/2}{0, 1} for x = 1/2{3} for x ∈ [2, 3]

and, by induction, for n ≥ 2,

T[2n](x) ={{1} for x ∈ A{3} for x ∈ B

, T[2n−1](x) ={{3} for x ∈ A{1} for x ∈ B.

Hence, for m = 3, V(T[m](A ∪ B)) = V({1, 3}) = 2 < 3 = V(A ∪ B), so (A1) holds. It is easy to see that, for each m ≥ 2,V(T[m](A ∪ B)) = V({1, 3}) = 2. Since DV(A, B) = V({1, 3}) = 2, for each ε > 0 we get V(T[m+n](A ∪ B)) ≤ DV(A, B) + ε,thus (A2) holds. The next considerations and conclusion are similar as in Example 4.5(I) and will be omitted. Therefore, theassumptions of Theorem 2.3 hold, w = 1 ∈ A is a unique endpoint of T[2] in A, T(w) = {3/4} ∈ B is a unique endpoint of T[2]in B and V({w, T(w)}) = 2 = DV(A, B).

Example 4.11. Let (X, d), d and V = {V : 2X→ [0,∞]} be such as in Example 4.6. Let A = [0, 1/4], B = [3/4, 1] and let

T : A ∪ B→ 2A∪B be a cyclic closed map of the form

T(x) =

[3/4,−(25/96)x+ 15/16] if x ∈ [0, 6/25][3/4,−(25/2)x+ 31/8] if x ∈ [6/25, 1/4][−(25/2)x+ 77/8, 1/4] if x ∈ [3/4, 76/100][−(25/96)x+ 31/96, 1/4] if x ∈ [76/100, 1].

Similarly as in Case 1 of Example 4.6 we can deduce that assumptions of Theorem 2.3 are satisfied, w = 1/4 is a uniqueendpoint of T[2] in A, T(w) = {3/4} is a unique endpoint of T[2] in B and V({w, T(w)}) = 1/2 = DV(A, B).

K. Włodarczyk et al. / Nonlinear Analysis 70 (2009) 3332–3341 3341

We close this section with an example which shows that the assumption in Theorems 2.1, 2.3 and 2.4 concerning theexistence of V-families on X which are different from the V-family on X defined in Example 4.3, is essential.

Example 4.12. If in Example 4.5 (Case (I) or (II)) we use the V-family on X defined in Example 4.3, then we obtain that,for each m ≥ 2, V(T[m](A ∪ B)) = V({1, 3}) = 2. However, DV(A, B) = 1. Hence, for 0 < ε < 1 and m, n ∈ N,V(T[m+n](A ∪ B)) = 2 > 1+ ε = DV(A, B)+ ε, from which it follows that (A2) does not hold.

References

[1] C. Di Bari, T. Suzuki, C. Vetro, Best proximity for cyclic Meir–Keeler contractions, Nonlinear Anal. (2007), in press (doi:10.1016/j.na.2007.10.014).[2] C. Berge, Topological Spaces, Oliver & Boyd, Edinburgh, 1963.[3] A.A. Eldred, W.A. Kirk, P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Math. 171 (2005) 283–293.[4] A.A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2006) 1001–1006.[5] W.A. Kirk, S. Reich, P. Veeramani, Proximal retracts and best proximity pair theorems, Numer. Funct. Anal. Optim. 24 (2003) 851–862.[6] W.A. Kirk, P.S. Srinivasan, P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory 4 (2003) 79–89.[7] K. Włodarczyk, R. Plebaniak, Endpoint theory for set-valued nonlinear asymptotic contractions with respect to generalized pseudodistances in uniform

spaces, J. Math. Anal. Appl. 339 (2008) 344–358.[8] K. Włodarczyk, R. Plebaniak, Quasi asymptotic contractions, set-valued dynamic systems, uniqueness of endpoints and generalized pseudodistances in

uniform spaces, Nonlinear Anal. (2008), in press (doi:10.1016/j.na.2008.01.032).