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@ Appl. Gen. Topol. 16, no. 1(2015), 37-44doi:10.4995/agt.2015.2305
c© AGT, UPV, 2015

Nonself KKM maps and corresponding

theorems in Hadamard manifolds

Parin Chaipunya and Poom Kumam
∗

Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thon-

buri, 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand.

(parin.cha@mail.kmutt.ac.th, poom.kum@kmutt.ac.th)

Abstract

In this paper, we consider the KKM maps defined for a nonself map

and the correlated intersection theorems in Hadamard manifolds. We

also study some applications of the intersection results. Our outputs

improved the results of Raj and Somasundaram [17, V. Sankar Raj
and S. Somasundaram, KKM-type theorems for best proximity points,
Appl. Math. Lett., 25(3):496–499, 2012.].

2010 MSC: 53C22; 53C25; 47H04.

Keywords: KKM Maps; Hadamard manifolds; generalized equilibrium

problems; best proximity points.

1. Introduction

The KKM theory, as the term coined by Park [15], is the study of the equiv-
alent formulations, variants, and extensions to the 1929 geometric result due
to Knaster, Kuratowski, and Mazurkiewicz. This result is known nowadays
as the KKM lemma, and it provides a firm foundation to many different ar-
eas of mathematics, e.g., fixed point theory, minimax theory, game theory,
variational inequality, equilibrium theory, and henceforth. This lemma is also
known for being equivalent to both the Brouwer’s fixed point theorem and the
Sperner’s lemma (see [16] for further discussions). One of the most important

∗Corresponding author

Received 19 April 2014 – Accepted 3 October 2014

http://dx.doi.org/10.4995/agt.2015.2305


P. Chaipunya and P. Kumam

enhancement of the KKM lemma is due to Fan [7], whose result is obtained in
a topological vector space.

In [17], the nonself KKM maps have been introduced and studied under the
framework of a normed linear space. As naturally occurs, the best proximity
point theorem is deduced in relation to the nonself KKM lemma.

On the other hand, Colao et al. [6] proved the KKM lemma in a Hadamard
manifold, as an auxiliary tool for proving several results on the existence of
solutions to equilibrium problems. Also, the fixed point, variational inequality
and Nash equilibrium are investigated by the authors.

In this paper, we occupy the nonself KKM lemma in Hadamard manifolds.
The nonself version of the Browder’s fixed point theorem as well as the solv-
ability of a generalized equilibrium problem are studied, as applications of our
KKM lemma.

2. Preliminaries

Recall first that a Hadamard manifold M is a complete simply-connected
smooth Riemannian manifold whose sectional curvature is non-positive. At
each point x ∈ M, we write TxM to represent the tangent plane at x, which is
at the same time a manifold.

With this structure, we can define an exponential map expp : TpM → M by
expp(ν) := γν(1), where γν is a geodesic defined by its position p and velocity
ν at p. Recall that exponential maps are diffeomorphisms.

The exponential maps allow us to characterize the minimal geodesic join-
ing a point p to another point q by the function t 7→ expp(texp

−1
p (q)), with

t ∈ [0,1]. Naturally, a subset K ⊂ M is said to be geodesically convex if
minimal geodesics correspond to each of its elements are contained in K. For
any nonempty subset A ⊂ M, denoted by co(A) the geodesically convex hull
of A, i.e., the smallest geodesically convex set containing A. Note that the
geodesically convex hull of any finite subset is compact. Moreover, the geo-
desic distance d(p,q) between two points p,q ∈ M defined the length of its
minimal geodesic induces the original topology of M.

A real function f : M → R is said to be geodesically convex if the com-
position f ◦ γ is convex (in ordinary sense), provided that γ is the minimal
geodesic joining two arbitrary points in M. In particular, the geodesic distance
is geodesically convex in both of its arguments.

Referring to [18], Hadamard manifolds behave nicely with probability mea-
sures defined on them. Let P(M) be the collection of probability measures
µ on M whose supports are separable and

∫

M
d(x,y)dµ(y) < ∞ for every1

x ∈ M. Then, to each µ ∈ P(M) and y ∈ M, we associate a point z∗ ∈ M
that minimizes the (uniformly) geodesically convex function z 7→

∫

M
[d2(z,x)−

d2(y,x)]dµ(x). Such point z∗ is independent of y ∈ M, so we prefer writing

1The quantifier ‘for some’ is used in some texts. They are however identical as one can
deduce from the triangle inequality.

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Nonself KKM Maps and corresponding theorems in Hadamard manifolds

b(µ) in place of z∗. Moreover, we say that it is the barycenter of µ. If supp(µ) is
contained in some closed geodesically convex set K, it is the case that b(µ) ∈ K.

We can make P(M) into a metric space by endowing it with the Wasserstein
metric given by:

dW (µ,ν) := inf

∫∫

M×M

d(x,y)dλ(x,y), ∀µ,ν ∈ P(M),

where the infimum is taken over λ ∈ P(M × M) whose marginals are µ and
ν. With respect to this metric, the map µ 7→ b(µ) is nonexpansive.

3. Nonself KKM Maps

The pair (A,B) set up by two given nonempty subsets A and B of a metric
space (S,d) is called a proximal pair if to each point (x,y) ∈ A × B, there
corresponds a point (x̄, ȳ) ∈ A × B such that

d(x, ȳ) = d(x̄,y) = dist(A,B),

where dist(A,B) := inf{d(x,y),x ∈ A,y ∈ B}. In addition, if both A and B
are convex, we say that (A,B) is a convex proximal pair.

In the future contents, we assume that M is a Hadamard manifold with
the geodesic distance d. Given a point x ∈ M and two nonempty subsets
A,B ⊂ M, we write d(x,A) := infz∈A d(x,z).

Definition 3.1. Let (A,B) be a proximal pair in a Hadamard manifold M.
A nonself map T : A ⇒ B is said to be KKM if for each finite subset D :=
{x1,x2, · · · ,xm} ⊂ A, there is a subset E := {y1,y2, · · · ,ym} ⊂ B such that
d(xi,yi) = dist(A,B), ∀i ∈ {1,2, · · · ,m}, and

co({yi, i ∈ I}) ⊂ T({xi, i ∈ I})

for every ∅ 6= I ⊂ {1,2, · · · ,m}.

Theorem 3.2. Suppose that (A,B) is a proximal pair in a Hadamard manifold
M and T : A ⇒ B is a KKM map with nonempty closed values. Then, the
family {T(x),x ∈ A} has the finite intersection property.

Proof. Assume to the contrary that there is a finite subset D := {x1,x2, · · · ,xm} ⊂
A such that

⋂

x∈D
T(x) = ∅. Since T is KKM, we can find a subset E :=

{y1,y2, · · · ,ym} ⊂ B so that

co({xj,j ∈ J}) ⊂ T({yj,j ∈ J}),

for every ∅ 6= J ⊂ {1,2, · · · ,m}. Set K := co(E). Define a function λ : K → R
by

λ(y) :=
m
∑

i=1

d(y,K ∩ T(xi)), ∀y ∈ K.

At each y ∈ K, we have λ(y) > 0 since
⋂

x∈D
T(x) = ∅. Then, the map

y ∈ K 7→ µy :=

m
∑

i=1

[

d(y,K ∩ T(xi))δyi
λ(y)

]

,

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P. Chaipunya and P. Kumam

where δu is the Dirac probability measure corresponding to u ∈ K, is continuous
(from K into P(K)). Thus, the composition y 7→ µy 7→ b(µy) is continuous
from K into itself, and it therefore has a fixed point y0 ∈ K (see [11]).

Take J := {j ∈ {1,2, · · · ,m},d(y0,K ∩ T(xj)) > 0}. It is immediate that
y0 6∈

⋃

j∈J
T(xj). As a matter of fact, we have supp(µy0) ⊂ co({xj,j ∈ J})

which implies that y0 = b(µy0) ∈ co({xj,j ∈ J}) ⊂
⋃

j∈J
T(xj), a contradic-

tion. Therefore, the family {T(x),x ∈ A} must possess the finite intersection
property. �

Theorem 3.3. Suppose that (A,B) is a proximal pair in a Hadamard manifold
M and T : A ⇒ B is a KKM map with nonempty closed values. If T(x0) is
compact at some x0 ∈ A, then the intersection

⋂

{T(x),x ∈ A} is nonempty.

Proof. By Theorem 3.2, we know that T(x0) ∩ T(x) is nonempty and closed
for all x ∈ A. Moreover, the family {T(x0) ∩ T(x),x ∈ A} has the f.i.p. The
conclusion follows as T(x0) is compact. �

Remark 3.4. With the same proofs, theorems presented above can also be
extended to CAT(0) spaces, but with an additional assumption that every
continuous map from a compact convex subset of M into itself has a fixed
point. In particular, if A and B are identical, we can obtain KKM results as
of [6, 12]

4. Some Applications

We observe here some applications of our results in the previous section.
Before we go into the main subjects, let us observe the following fact about
convex hulls of finitely many points between a convex proximal pair.

Lemma 4.1. Let (A,B) be a convex proximal pair of a Hadamard manifold
M. Assume that x1,x2, · · · ,xm ∈ A and y1,y2, · · · ,ym ∈ B are points such
that

d(xi,yi) = dist(A,B), ∀i ∈ {1,2, · · · ,m}.

Then, (co({x1,x2, · · · ,xm}),co({y1,y2, · · · ,ym})) is a proximal pair with

(4.1) dist(co({x1,x2, · · · ,xm}),co({y1,y2, · · · ,ym})) = dist(A,B).

Proof. The equity (4.1) is obvious, so let us prove the former part. Let us
write C1 := {x1,x2, · · · ,ym}, and for j ≥ 2, let Cj be the union of minimal
geodesics that join pairs of points in Cj−1. In the same way, we let D1 :=
{y1,y2, · · · ,ym}, and for j ≥ 2, let Dj be the union of minimal geodesics that
join pairs of points in Dj−1. One can simply show, by using mathematical
induction, that (Cj,Dj) is a proximal pair for all j ≥ 1. Now, apply [14,
Proposition 2.5.5] to complete the proof. �

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Nonself KKM Maps and corresponding theorems in Hadamard manifolds

4.1. Generalized Equilibrium Problems. Given a nonempty set Q and
a bifunction ψ : Q × Q → R, the equilibrium problem concerns the existence
(and the determination) of a point x̄ ∈ Q that makes ψ(x̄, ·) into a non-negative
function.

This equilibrium problem is first considered by Fan [7, 8] under Euclidean
spaces. It is then improved and enriched in [3]. As it unifies many problems
in optimization, for examples, minimization problem, variational inequality,
minimax inequality, and Nash equilibrium problem, the equilibrium theory
gained its fame very quickly. Consult [10, 13, 2, 5, 9, 1] for richer details.

In this section, we shall consider the case where the bifunction ψ is defined
on the product P × Q, with P,Q being nonempty and possibly distinct sets.
This leads to a more general aspect of equilibrium problems.

Theorem 4.2. Suppose that (P,Q) is a geodesically convex proximal pair in a
Hadamard manifold M, and ψ : P × Q → R is a bifunction. Assume that the
following conditions hold:

(i) ψ(x,y) ≥ 0 provided x ∈ P, y ∈ Q, and d(x,y) = dist(P,Q),
(ii) ∀x ∈ P, the set {y ∈ Q,ψ(x,y) < 0} is geodesically convex,
(iii) ∀y ∈ Q, the function ψ(·,y) is u.s.c.,
(iv) there exists a nonempty compact set L ⊂ M such that both L ∩ P and

L ∩ Q are nonempty and

ψ(x, ȳ) < 0, ∀x ∈ P \ L,

for some point ȳ ∈ L ∩ Q.

Then, there exists a point x̄ ∈ L ∩ P such that

ψ(x̄,y) ≥ 0, ∀y ∈ Q.

Proof. Define a map G : Q ⇒ P by

G(y) := {x ∈ P,ψ(x,y) ≥ 0}, ∀y ∈ Q.

Since ψ(·,y) is u.s.c., G(y) is closed for each y ∈ Q. From (iv), we have
G(ȳ) ⊂ L and so G(ȳ) is compact.

We shall prove next that G is a KKM map. Suppose that {y1,y2, · · · ,ym} ⊂
Q and {x1,x2, · · · ,xm} ⊂ P such that d(xi,yi) = dist(P,Q), ∀i ∈ {1,2, · · · ,m}.
Let us assume to the contrary that there exists a subset ∅ 6= J ⊂ {1,2, · · · ,m}
and a point x0 ∈ co({xj,j ∈ J}) such that x0 6∈ G({yj,j ∈ J}). Equivalently,
ψ(x0,yj) < 0, ∀j ∈ J. By Lemma 4.1, we can choose a point y0 ∈ co({yj,j ∈
J}) with d(x0,y0) = dist(P,Q). Note that yj ∈ {y ∈ Q,ψ(x0,y) < 0} for each
j ∈ J and {y ∈ Q,ψ(x0,y) < 0} is geodesically convex. Therefore, we have
y0 ∈ co({yj,j ∈ J}) ⊂ {y ∈ Q,ψ(x0,y) < 0}, which contradicts the hypothesis
(i). Hence, G is a KKM map, and the desired result follows immediately from
the construction of G. �

Remark 4.3. If P = Q, then the condition (i) reads as follows:

(i’) ψ(x,x) ≥ 0, ∀x ∈ P,

where it is always assumed in classical equilibrium theory.

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P. Chaipunya and P. Kumam

4.2. Best Proximity Points. Suppose that (S,d) is a metric space and A ⊂ S
is nonempty. Given a map T : A ⇒ S, a point x0 ∈ A is a fixed point of F
if x0 ∈ T(x0). In particular, if T is closed valued, a fixed point is expressed
metrically by d(x0,T(x0)) = 0.

Suppose that B ⊂ S is nonempty. Then, it may be the case that the
map T : A ⇒ B does not have a fixed point. In fact, it is evident that
d(x,T(x)) ≥ dist(A,B) for all x ∈ A. In this case, instead of fixed points,
we can consider the best proximity point x0 ∈ A, i.e., the point such that
d(x0,T(x0)) = dist(A,B).

The notion of best proximity point is stronger than the best approximation.
In details, if T is single-valued and x0 is a best proximity point of T , then
T(x0) is a best approximant to x0 for B.

Now, we state our nonself version of Browder’s fixed point theorem [4] in
the setting of Hadamard manifolds.

Theorem 4.4. Let (A,B) be a proximal pair of a Hadamard manifold M,
where A is assumed to be compact and geodesically convex, and T : A ⇒ B be
a map such that

(i) ∀x ∈ A, T(x) is nonempty and geodesically convex,
(ii) T is an open fibre map, i.e., ∀y ∈ B, the inverse image

T
−1(y) := {x ∈ A,y ∈ T(x)}

is open.

Then, there exists a point x̄ ∈ A such that d(x̄,T(x̄)) = dist(A,B).

Proof. Consider the dual map G : B ⇒ A defined by

G(y) := A \ T −1(y), ∀y ∈ B.

If G(y0) = ∅ for some y0 ∈ B, i.e., T
−1(y0) = A. This means y0 ∈ T(x) for all

x ∈ A. Now, since (A,B) is a proximal pair, we can find a point x̄ ∈ A such
that d(x̄,y0) = dist(A,B). In particular, y0 ∈ T(x̄). Thus, we have

d(x̄,T(x̄)) ≤ d(x̄,y0) = dist(A,B),

yielding the desired result.
On the other hand, suppose that G(y) is nonempty for every y ∈ B. More-

over, G is closed valued. Observe that
⋂

y∈B

G(y) =
⋂

y∈B

(A \ T −1(y)) = A \
⋃

y∈B

T −1(y).

Since {T −1(y),y ∈ B} is an open cover of A, we obtain from the above equality
that

⋂

y∈B
G(y) is empty. We conclude from Theorem 3.3 that G is not a KKM

map. Thus, suppose that {y1,y2, · · · ,ym} ⊂ B and {x1,x2, · · · ,xm} ⊂ A are
sets such that d(xi,yi) = dist(A,B), ∀i ∈ {1,2, · · · ,m} and co({x1,x2, · · · ,xm})
is not contained in G({y1,y2, · · · ,ym}).

In particular, choose x̄ ∈ co({x1,x2, · · · ,xm}) such that x̄ 6∈ G({y1,y2, · · · ,ym}).
Hence, x̄ ∈ T −1(yi) for all i ∈ {1,2, · · · ,m}, or equivalently, yi ∈ T(x̄)
for all i ∈ {1,2, · · · ,m}. According to Lemma 4.1, we can choose a point

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Nonself KKM Maps and corresponding theorems in Hadamard manifolds

z ∈ co({y1,y2, · · · ,ym}) with d(x̄,z) = dist(A,B). Since T(x̄) is convex, we
get z ∈ co({y1,y2, · · · ,ym}) ⊂ T(x̄). We have again

d(x̄,T(x̄)) ≤ d(x̄,z) = dist(A,B),

which leads to the desired result. �

In case A and B are identical, we have the following variant of Browder’s
theorem in the setting of Hadamard manifolds.

Theorem 4.5. Let K be a nonempty, compact, and geodesically convex subset
of a Hadamard manifold M and T : K ⇒ K an open fibre map whose values
are nonempty and geodesically convex. Then, T has a fixed point.

Conclusion

We have proved the intersection theorem for nonself KKM maps in Hadamard
manifolds, which extends the existed result of [17]. We have also provide some
applications of our intersection result towards the existence of an equilibrium
point and a best proximity point.

Acknowledgements. The first author is supported by the Thailand Re-
search Fund through the Royal Golden Jubilee Ph.D. Program (Grant No.
PHD/0045/2555) and the King Mongkut’s University of Technology Thonburi
under the RGJ-Ph.D. scholarship. Moreover, we would like to gratefully thank
the anonymous referees for their suggestions, which improve this paper signifi-
cantly.

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