@ Appl. Gen. Topol. 16, no. 2(2015), 99-108doi:10.4995/agt.2015.2988
c© AGT, UPV, 2015

On cyclic relatively nonexpansive mappings in

generalized semimetric spaces

Moosa Gabeleh

Department of Mathematics, Ayatollah Boroujerdi University, Boroujerd, Iran, School of Mathe-

matics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran,

Iran. (gab.moo@gmail.com, Gabeleh@abru.ac.ir)

Abstract

In this article, we prove a fixed point theorem for cyclic relatively non-

expansive mappings in the setting of generalized semimetric spaces by

using a geometric notion of seminormal structure and then we con-

clude a result in uniformly convex Banach spaces. We also discuss on

the stability of seminormal structure in generalized semimetric spaces.

2010 MSC: 47H10; 46B20.

Keywords: Cyclic relatively nonexpansive mapping; seminormal structure;

generalized semimetric space.

1. Introduction

A closed convex subset E of a Banach space X has normal structure in
the sense of Brodskil and Milman ([2]) if for each bounded, closed and convex
subset K of E which contains more than one point, there is a point x ∈ K which
is not a diametral point of K, that is, sup{‖x − y‖ : y ∈ K} < diam(K). In
1965, Kirk proved that if E is a nonempty, weakly compact and convex subset
of a Banach space X with normal structure and T : E → E is a nonexpansive
mapping, that is ‖T x−T y‖ ≤ ‖x−y‖ for all x, y ∈ E, then T has a fixed point
([8]).

As well known, every nonempty, bounded, closed and convex subset of a
uniformly convex Banach space X has normal structure. So, the following
fixed point theorem concludes from the Kirk’s fixed point theorem.

Received 21 May 2014 – Accepted 5 June 2015

http://dx.doi.org/10.4995/agt.2015.2988


M. Gabeleh

Theorem 1.1. Let E be a nonempty, bounded, closed and convex subset of
a uniformly convex Banach space X. Then every nonexpansive mapping T :
E → E has a fixed point.

Now, let (X, d) be a metric space, and let E, F be subsets of X. A mapping
T : E ∪ F → E ∪ F is said to be cyclic provided that T (E) ⊆ F and T (F) ⊆
E. The following interesting theorem is an extension of Banach contraction
principle.

Theorem 1.2 ([10]). Let E and F be nonempty and closed subsets of a com-
plete metric space (X, d). Suppose that T is a cyclic mapping such that

d(T x, T y) ≤ α d(x, y),

for some α ∈ (0, 1) and for all x ∈ E, y ∈ F. Then E ∩ F is nonempty and T
has a unique fixed point in E ∩ F.

If E ∩ F = ∅ then the cyclic mapping T : E ∪ F → E ∪ F cannot have
a fixed point, instead it is interesting to study the existence of best proximity
points, that is, a point p ∈ E ∪ F such that

d(p, T p) = dist(E, F) := inf{d(x, y) : (x, y) ∈ E × F}.

Existence of best proximity points for cyclic relatively nonexpansive mappings
was first studied in [3] (see also [4, 5, 6, 7] for different approaches to the same
problem). We recall that the mapping T : E ∪ F → E ∪ F is called cyclic
relatively nonexpansive provided that T is cyclic on E ∪ F and d(T x, T y) ≤
d(x, y) for all (x, y) ∈ E × F .

Next theorem was established in [3].

Theorem 1.3 (Corollary 2.1 of [3]). Let E and F be two nonempty, bounded,
closed and convex subsets of a uniformly convex Banach space X. Suppose
T : E ∪ F → E ∪ F is a cyclic relatively nonexpansive mapping. Then T has a
best proximity point in E ∪ F.

We mention that Theorem 1.3 is based on the fact that every nonempty,
bounded, closed and convex pair of subsets of a uniformly convex Banach space
X has proximal normal structure (see Proposition 2.1 of [3]).

In this article, motivated by Theorem 1.2, we establish a fixed point theorem
for cyclic relatively nonexpansive mappings in generalized semimetric spaces.
Next we show that if the pair (E, F) considered in Theorem 1.3 has an ap-
propriate geometric condition, then E ∩ F must be nonempty and hence, the
result follows from Theorem 1.1.

2. Preliminaries

Let X be a set and S a linearly ordered set with its order topology having
a smallest element, which denoted by 0. A mapping DS : X × X → S is said
to be a generalized semimetric provided that for each x, y ∈ X
(1) DS(x, y) = 0 ⇔ x = y,
(2) DS(x, y) = DS(y, x).

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On cyclic relatively nonexpansive mappings in generalized semimetric spaces

If S is the set of nonnegative real numbers, then we replace DS with D and
we say that D is a semimetric on X. Also, if DS is a generalized semimetric
on X, then the pair (X, DS) is called generalized semimetric space. An easy
example of a continuous semimetric which is not a metric is given by letting
X = S = [0, 1] and defining D(x, y) := |x − y|2 for all x, y ∈ X.

According to Blumenthal ([1]; p.10), DS generates a topology on X as fol-
lows:
A point p ∈ X is said to be a limit point of a subset E of X if given any α ∈ S
with α 6= 0, there exists a point q ∈ E such that DS(p, q) ∈ (0, α) := {β ∈
S : 0 < β < α}. A set E in X is said to be closed if it contains all of its
limit points and a set U in X is said to be open if X − U is closed. If DS is a
continuous mapping w.r.t. the topology on X induced by DS, then DS is said
to be a continuous generalized semimetric.

Given a generalized semimetric DS, a B-set will be a set like

B(x; α) := {u ∈ X : DS(x, u) ≤ α}.

We say that a set E ⊆ X is spherically bounded if there exists a B-set which
contains E. We also define

cov(E) :=
⋂

{K : K is a B-set containing E}.

Definition 2.1. A subset E of a generalized semimetric space (X, DS) is said
to be admissible if E = cov(E).

The collection of all admissible subsets of a generalized semimetric (X, DS)
will be denoted by A(X). We will say that A(X) is compact provided that any
descending chain of nonempty members of A(X) has nonempty intersection.

The linearly ordered set S is said to have least upper bound property (lub-
property) if each set in S which is bounded above has a smallest upper bound.
Dually, this implies that S has the greatest lower bound property (glb- prop-
erty). We mention that if S is connected relative to its order topology, then S
has the lub- property.

Let (E, F) be a nonempty pair of subsets of a generalized semimetric (X, DS).
We shall adopt the following notations.

dist(E, F) := glb {DS(x, y) : (x, y) ∈ E × F},

δx(E) := lub {DS(x, u) : u ∈ E}, ∀x ∈ X,

δ(E, F) := lub {δx(F) : x ∈ E},

diam(E) := δ(E, E).

E0 := {x ∈ E : DS(x, y) = dist(A, B), for some y ∈ B},

F0 := {y ∈ F : DS(x, y) = dist(A, B), for some x ∈ A}.

Definition 2.2 ([6]). A pair of sets (E, F) in a generalized semimetric space
(X, DS) is said to be a proximal compactness pair provided that every net
{(xα, yα)} of E × F satisfying the condition that DS(xα, yα) → dist(E, F), has
a convergent subnet in E × F .

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M. Gabeleh

3. Seminormal Structure

Throughout this paper, we shall say that a pair (E, F) of subsets of a gen-
eralized semimetric space (X, DS) satisfies a property if both E and F satisfy
that property. For example, (E, F) is admissible if and only if both E and F
are admissible; (E, F) ⊆ (G, H) ⇔ E ⊆ G, and F ⊆ H.

Let (E, F) be a nonempty pair of admissible subsets of X. We say that the
pair (E, F) satisfies the condition (P) if E contained in a B-set centered at a
point of F and the set F contained in a B-set centered at a point of E. Also,
for the pair (E, F) we define

R(E) := {α ∈ S : [
⋂

y∈F

B(y; α)] ∩ E 6= ∅},

R(F) := {β ∈ S : [
⋂

x∈E

B(x; β)] ∩ F 6= ∅}.

Note that the if the pair (E, F) satisfies the condition (P), then (R(E), R(F))
is a nonempty pair of subsets of S. Indeed, if E ⊆ B(v; β) for some v ∈ F
and β ∈ S, then DS(x, v) ≤ β for all x ∈ E and so, v ∈ B(x; β) for all x ∈ E.
Thus v ∈

⋂

x∈E
B(x; β) ∩ F i.e. β ∈ R(F). Similarly, we can see that R(E) is

nonempty.
Furthermore, we set

r(E) := glb R(E), r(F) := glb R(F) and ρ := lub {r(E), r(F)},

and define

CF (E) := {x ∈ E : x ∈
⋂

y∈F

B(y; ρ)},

CE(F) := {y ∈ F : y ∈
⋂

x∈E

B(x; ρ)}.

Next lemma guarantees that (CF (E), CE(F)) is a nonempty pair.

Lemma 3.1. Let (X, DS) be a generalized semimetric space such that A(X) is
compact and S is connected. Let (E, F) be a nonempty and admissible pair of
subsets of X such that (E, F) satisfies the condition (P). Then (CF (E), CE(F))
is a nonempty and admissible pair in X which satisfies the condition (P).

Proof. Let α > ρ and β > ρ be such that the pair (Cα(E), Cβ(F)) is nonempty,
where

Cα(E) := [
⋂

y∈F

B(y; α)] ∩ E & Cβ(F) := [
⋂

x∈E

B(x; β)] ∩ F.

We show that CF (E) =
⋂

α≥ρ
Cα(E) and CE(F) =

⋂

β≥ρ
Cβ(F). Suppose that

u ∈
⋂

α≥ρ
Cα(E). If u is not member of CF (E), then there exists v ∈ F such that

DS(u, v) > ρ. Since S is connected, there exists an element γ ∈ S such that
ρ < γ < DS(u, v). But this is a contradiction by the fact that u ∈ Cγ(E). That

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On cyclic relatively nonexpansive mappings in generalized semimetric spaces

is, u ∈ CF (E) and so,
⋂

α≥ρ
Cα(E) ⊆ CF (E). This implies that CF (E) 6= ∅.

Besides, if u ∈ CF (E), then

u ∈ [
⋂

y∈F

B(y; ρ)] ∩ E ⊆ [
⋂

y∈F

B(y; α)] ∩ E = Cα(E), ∀α ≥ ρ.

Hence, u ∈
⋂

α≥ρ
Cα(E) which deduces that CF (E) =

⋂

α≥ρ
Cα(E). Similar

argument implies that CE(F) =
⋂

β≥ρ
Cα(F). Now, suppose that E ⊆ B(q, γ1)

and F ⊆ B(p, γ2) for some (p, q) ∈ E ×F and γ1, γ2 ∈ S. Put γ := lub {γ1, γ2}.
Then for each α ∈ S with α ≥ ρ, we have Cα(E) ⊆ B(q, γ) which concludes
that

CF (E) =
⋂

α≥ρ

Cα(E) ⊆ B(q, γ).

Similar argument implies that CE(F) ⊆ B(p, γ). That is, the pair (CF (E), CE(F))
satisfies the condition (P). �

Let (E, F) be a nonempty and admissible pair of subsets of a generalized
semimetric space (X, DS) such that (E, F) satisfies the condition (P). In what
follows we set

Σ(E,F ) := {(G, H) ⊆ (E, F) : G, H ∈ A(X) and (G, H) satisfies the condition (P)}.

Here, we introduce the following geometric notion on a nonempty and ad-
missible pair in generalized semimetric spaces.

Definition 3.2. Suppose that (E, F) is a nonempty and admissible pair of
subsets of a generalized semimetric space (X, DS) such that (E, F) satisfies
the condition (P) and A(X) is compact. We say that Σ(E,F ) has seminormal
structure if for each (G, H) ∈ Σ(E,F ), either G ∪ H is singleton or CH(G)  
G, CG(H)  H.

We now state the main result of this paper.

Theorem 3.3. Let (X, DS) be a generalized semimetric space, where S is
connected w.r.t. its order topology and let A(X) be compact. Suppose that
(E, F) is a nonempty and admissible pair of subsets of X which satisfies the
condition (P) and Σ(E,F ) has seminormal structure. If T : E ∪ F → E ∪ F is
a cyclic relatively nonexpansive mapping, then E ∩ F is nonempty and T has
a fixed point in E ∩ F.

Proof. Put

F := {(G, H) : (G, H) ∈ Σ(E,F ) and T is cyclic on G ∪ H}.

By the fact that A(X) is compact and by using Zorn’s lemma, we conclude
that F has a minimal element say (K1, K2) ∈ F. Since T (K1) ⊆ K2 and
K2 ∈ A(X), we deduce that cov(T (K1)) ⊆ K2. Then

T (cov(T (K1))) ⊆ T (K2) ⊆ cov(T (K2)).

Similarly, we can see that T (cov(T (K2))) ⊆ cov(T (K1)), that is, T is cyclic
on cov(T (K2)) ∪ cov(T (K1)). Besides, (cov(T (K2)), cov(T (K1))) satisfies the

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M. Gabeleh

condition (P). Indeed, if K1 ⊆ B(q, α) for some q ∈ K2 and α ∈ S, then for
each x ∈ K1, we have

DS(T x, T q) ≤ DS(x, q) ≤ α,

that is, T x ∈ B(T q, α) for each x ∈ K1. So, T (K1) ⊆ B(T q, α). Thus
cov(T (K1)) ⊆ B(T q, α). Similarly, if K2 ⊆ B(p, β) for some p ∈ K1 and β ∈ S,
then we can see that cov(T (K2)) ⊆ B(T p, β). Hence, (cov(T (K2)), cov(T (K1)))
satisfies the condition (P). Minimality of (K1, K2) implies that

K1 = cov(T (K2)) & K2 = cov(T (K1)).

It follows from Lemma 3.1 that (CK2(K1), CK1(K2)) is a nonempty member of
Σ(E,F ). We show that T is cyclic on CK2(K1) ∪ CK1(K2). Let x ∈ CK2(K1).
Then x ∈ [

⋂

y∈K2
B(y; ρ)] ∩ K1. So, DS(x, y) ≤ ρ for each y ∈ K2. Since T is

cyclic relatively nonexpansive,

DS(T x, T y) ≤ DS(x, y) ≤ ρ, ∀y ∈ K2.

Thus T (K2) ⊆ B(T x; ρ) which implies that

K1 = cov(T (K2)) ⊆ B(T x; ρ).

Hence, T x ∈ [
⋂

u∈K1
B(u; ρ)]∩K2 = CK1(K2). That is, T (CK2(K1)) ⊆ CK1(K2).

Similarly, we can see that T (CK1(K2)) ⊆ CK2(K1). Thereby, T is cyclic on
CK2(K1) ∪ CK1(K2). So, (CK2(K1), CK1(K2)) ∈ F. Again, by the minimality
of (K1, K2) we must have

CK2(K1) = K1 & CK1(K2) = K2.

Since Σ(E,F ) has the seminormal structure, we deduce K1 = K2 = {p} for some
p ∈ X. Therefore, p ∈ E ∩ F is a fixed point of T .

�

Remark 3.4. Note that in Theorem 3.3 we have not the assumption of continu-
ity of DS. We also mention that if the mapping T considered in Theorem 3.3 is
nonexpansive self-mapping, the the main result of [9] is deduces (see Theorem
3 of [9] for more information).

Definition 3.5. Let (E, F) be a nonempty and admissible pair of subsets of
a semimetric space (X, D) such that (E, F) satisfies the condition (P). We say
that (E, F) has the property UC if for each nonempty pair (G, H) ∈ Σ(E,F )
and for any ε > 0, there exists α(ε) > 0 such that for all R > 0 and x1, x2 ∈ G
and y ∈ H with

D(x1, y) ≤ R, D(x2, y) ≤ R and D(x1, x2) ≥ Rε,

there exists u ∈ G such that D(u, y) ≤ R(1 − α(ε)) < R.

We now prove the following existence theorem.

Theorem 3.6. Let (X, D) be a semimetric space such that D is continuous
and A(X) is compact. Suppose (E, F) is a nonempty and admissible pair such
that E0 6= ∅ and (E, F) satisfies the condition (P). Assume that (E, F) is a
proximal compactness pair which has the property UC. If T : E ∪F → E ∪F is

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 104



On cyclic relatively nonexpansive mappings in generalized semimetric spaces

a cyclic relatively nonexpansive mapping, then either E ∩ F is nonempty and
T has a fixed point in E ∩ F, or T has a best proximity point in E ∪ F.

Proof. Let

F′ := {(G, H) ∈ Σ(E,F ) s.t. ∃(x, y) ∈ G × H with D(x, y) = dist(E, F)

and T is cyclic on G ∪ H}.

Since E0 6= ∅, (E, F) ∈ F′. Moreover, if (Gα, Hα) is a descending chain in
F′ and put G :=

⋂

α
Gα and we set H :=

⋂

α
Hα, then by the compactness of

A(X), (G, H) is a nonempty member of Σ(E,F ) and obviously, T is cyclic on
G ∪ H. Now, suppose for each α there exists (xα, yα) ∈ Gα × Hα such that
D(xα, yα) = dist(E, F). Since (E, F) is proximal compactness, {(xα, yα)} has
a convergent subnet say {(xαi, yαi)} such that xαi → x ∈ E and yαi → y ∈ F .
Hence,

D(x, y) = lim
i

D(xαi, yαi) = dist(E, F),

that is, there exists an element (x, y) ∈ G × H such that D(x, y) = dist(E, F).
So, every increasing chain in F′ is bounded above with respect to revers inclu-
sion relation. Using Zorn’s lemma, we obtain a minimal element for F′, say
(K1, K2). If K1 ∪ K2 is singleton, then T has a fixed point in E ∩ F and we
are finished. So, we assume that K1 ∪ K2 is not singleton. Similar argument
of Theorem 3.3 concludes that CK2(K1) = K1 and CK1(K2) = K2. We now
consider the following :
Case 1. If min{diam(K1), diam(K2)} = 0.
We may assume that K1 = {p} for some element p ∈ E. Let q ∈ K2 be such
that D(p, q) = dist(E, F). Since T is cyclic relatively nonexpansive mapping,

D(T p, p) = D(T p, T q) ≤ D(p, q) = dist(E, F),

that is, p is a best proximity point of T and the result follows.
Case 2. If min{diam(K1), diam(K2)} > 0.
Put

R := δ(K1, K2) and r := min{diam(K1), diam(K2)}.

Let x1, x2 ∈ K1 be such that D(x1, x2) ≥
1
2
diam(K1) and let ε > 0 be such

that Rε ≤ r
2
. Now, for each y ∈ K2 we have

D(x1, y) ≤ R, D(x2, y) ≤ R and D(x1, x2) ≥
1

2
r ≥ Rε.

Since (E, F) has the property UC, there exists α(ε) > 0 and u ∈ K1 so that

D(u, y) ≤ R(1 − α(ε)), ∀y ∈ K2.

Then u ∈ [
⋂

y∈K2
B(y; R(1 − α(ε)))] ∩ K1, that is, [

⋂

y∈K2
B(y; R(1 − α(ε)))] ∩

K1 6= ∅. Similarly, we can see that [
⋂

x∈K1
B(x; R(1 − α(ε)))] ∩ K2 6= ∅. Set

r(K1) := inf{s > 0 : [
⋂

y∈K2

B(y; s)] ∩ K1 6= ∅},

r(K2) := inf{s > 0 : [
⋂

x∈K1

B(x; s)] ∩ K2 6= ∅}.

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M. Gabeleh

Note that for ρ := max{r(K1), r(K2)} we have ρ ≤ R(1 − α(ε)). Since
CK2(K1) = K1,

x ∈
⋂

y∈K2

B(y; ρ), ∀x ∈ K1,

which implies that δx(K2) ≤ ρ for all x ∈ K1. Thus

R = δ(K1, K2) = sup
x∈K1

δx(K2) ≤ ρ ≤ R(1 − α(ε)) < R,

which is a contradiction and this completes the proof of Theorem. �

Next corollary is a straightforward consequence of Theorem 3.6 in the setting
of uniformly convex Banach spaces.

Corollary 3.7 (see [3]). Suppose that (E, F) is a nonempty, bounded, closed
and convex pair of subsets of a uniformly convex Banach space X. Let T :
E ∪F → E ∪F be a cyclic relatively nonexpansive mapping. Then either E ∩F
is nonempty and T has a fixed point in E ∩ F or T has a best proximity point
in E ∪ F.

Example 3.8. Let X = R and let A := [−1, 1]. Define the mapping T : A → A
with

T (x) =











−x if x ∈ [−1, 0],

−x if x ∈ [0, 1] ∩ Q,
0 if x ∈ [0, 1] ∩ Qc.

Then T is a self-mapping defined on a nonempty bounded, closed and convex
subset of X. Note that existence of fixed point of T cannot be deduced from
Theorem 1.1, because of the fact T is not continuous (and so is not nonexpan-
sive). Now, Suppose E := [−1, 0] and F := [0, 1] and formulate the mapping
T : E ∪ F → E ∪ F as follows:

T (x) =











−x if x ∈ E,

−x if x ∈ F ∩ Q,
0 if x ∈ F ∩ Qc.

It is easy to see that ‖T x − T y‖ ≤ ‖x − y‖ for all (x, y) ∈ E × F , that is, T is
cyclic relatively nonexpansive mapping on the nonempty, bounded, closed and
convex pair (E, F). Hence, the existence of fixed point for T is concluded from
Corollary 3.7.

4. Stability and Seminormal Structure

We begin our main conclusions of this section with the following notion.

Definition 4.1. Let (X, DS) be a generalized semimetric space and let (E, F)
be a nonempty pair of subsets of X. A mapping T : E ∪ F → E ∪ F is said to
be cyclic relatively h-nonexpansive for some h ∈ S with h > 0 if

DS(T x, T y) ≤ lub {DS(x, y), h},

for all (x, y) ∈ E × F .

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On cyclic relatively nonexpansive mappings in generalized semimetric spaces

Here, we state the following stability result for cyclic relatively h-nonexpansive
mappings.

Theorem 4.2. Let (X, DS) be a generalized semimetric space, where S is
connected w.r.t. its order topology and let A(X) be compact. Suppose that
(E, F) is a nonempty and admissible pair of subsets of X which satisfies the
condition (P) and Σ(E,F ) has seminormal structure. If T : E ∪ F → E ∪ F is a
cyclic relatively h-nonexpansive mapping, then there exists an element p ∈ A∪B
so that DS(p, T p) ≤ h.

Proof. Similar argument of Theorem 3.3 implies that there exists a nonempty
and admissible pair of subsets (K1, K2) ⊆ (E, F) which satisfies the condition
(P) and by minimality,

cov(T (K2)) = K1 and cov(T (K1)) = K2.

If K1 ∪ K2 is singleton, the result follows. So, assume that CK2(K1) $ K1 and
CK1(K2) $ K2. Let u be an arbitrary element of CK2(K1). Suppose ρ < h.
Then DS(u, y) ≤ ρ for all y ∈ K2. Since T is cyclic on K1 ∪ K2, we have
DS(u, T u) ≤ ρ < h and we are finished. We now suppose that h ≤ ρ. Let
y ∈ K2. If DS(u, y) ≥ h, then

DS(T u, T y) ≤ lub{DS(u, y), h} = DS(u, y) ≤ ρ.

Besides, if DS(u, y) < h, then

DS(T u, T y) ≤ lub{DS(u, y), h} = h ≤ ρ,

that is, for each y ∈ K2 we have DS(T u, T y) ≤ ρ which implies that T y ∈
B(T u; ρ) for all y ∈ K2. Hence, T (K2) ⊆ B(T u; ρ). So,

K1 = cov(T (K2)) ⊆ B(T u; ρ),

and then T u ∈ [
⋂

x∈K1
B(x; ρ)]∩K2. Thus T u ∈ CK1(K2). Thereby, T (CK2(K1)) ⊆

CK1(K2). By a similar argument we obtain T (CK1(K2)) ⊆ CK2(K1). Therefore,
T is cyclic on T (CK2(K1)) ∪ CK1(K2). Minimality of (K1, K2) deduces that

K1 = CK2(K1) and K2 = CK1(K2),

which is a contradiction. �

Acknowledgements. This research was in part supported by a grant from
IPM (No. 93470047).

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M. Gabeleh

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