A minimax inequality and its applications to fixed point theorems in CAT(0) spaces

Open Access
Research
Part of the following topical collections:
  1. S. Park's Contribution to the Development of Fixed Point Theory and KKM Theory

Abstract

In this paper, a CAT(0) version of famous Fan's minimax inequality is established and as its application, we obtain some fixed point theorems and best approximation theorems in CAT(0) spaces.

2000 Mathematics Subject Classification: 47H10.

Keywords

CAT(0) space minimax inequality fixed point 

1 Introduction

A metric space is a CAT(0) space if it is geodesically connected and if every geodesic triangle in this space is at least as thin as its comparison triangle in Euclidean plane. CAT(0) spaces play fundamental role in various areas of mathematics [1]. Moreover, there are applications in biology and computer science as well [2, 3].

Fixed point theory in a CAT(0) space was first studied by Kirk [4]. Since then, the fixed point theory for single valued and multivalued mappings in CAT(0) spaces has been developed [5, 6, 7, 8].

The famous Knaster-Kuratowski-Mazurkiewicz theorem (in short, KKM theorem) and its generalization have a fundamental importance in modern nonlinear analysis [9, 10]. Recently, Niculescu and Roventa established the KKM mapping principle for CAT(0) spaces [11].

In this paper, a minimax inequality in CAT(0) spaces is established and as its application, some fixed point and best approximation theorems in CAT(0) spaces are proved.

2 Preliminaries

Let (X, d) be a metric space. A geodesic path joining xX to yY (briefly, a geodesic from x to y) is a map c from a closed interval [0, l] ⊆ ℝ to X such that c(0) = x, c(l) = y and d(c(t), c(t')) = |t - t'| for all t, t' ∈ [0, l]. In particular, c is an isometry and d(x, y) = l. The image of c is called a geodesic segment joining x and y. When it is unique, this geodesic is denoted by [x, y].

The metric space (X, d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, yX. A subset Y of X is said to be convex if Y includes every geodesic segment joining any of two its points.

A geodesic triangle △(x1, x2, x3) in a geodesic space consists of three points x1, x2, x3 in X (the vertices of △) and a geodesic segment between each pair of vertices (the edges of △). A comparison triangle for geodesic triangle △ (x1, x2, x3) in (X, d) is a triangle ̄ ( x 1 , x 2 , x 3 ) : = ( x ̄ 1 , x ̄ 2 , x ̄ 3 ) Open image in new window in the Euclidian plane E 2 Open image in new window such that
d E 2 ( x ̄ i , x ̄ j ) = d ( x i , x j ) , Open image in new window

for i, j ∈ {1, 2, 3}.

A geodesic space is called a CAT(0) space if all geodesic triangles satisfy the CAT(0) inequality:

For every geodesic triangle, △ in X and its comparison triangle ̄ Open image in new window in E 2 Open image in new window, if x, y ∈ △, and x ̄ , ȳ Open image in new window are comparison points in ̄ Open image in new window, then
d ( x , y ) d E 2 ( x ̄ , ȳ ) . Open image in new window

We now collect some elementary facts about CAT(0) spaces which will be used in the proofs of our main results.

Lemma 2.1[1]Every CAT(0) space (X, d) is uniquely geodesic, and the balls in (X, d) are convex.

Lemma 2.2[12]Let (X, d) be a CAT(0) space. Then,

1. for each x, yX such that xy then d(x, z) + d(z, y) = d(x, y) if and only if z ∈ [x, y],

2. for each x, yX and t ∈ [0, 1], there exists a unique point z ∈ [x, y] such that d(x, z) = td(x, y) and d(y, z) = (1 - t)d(x, y).

Recall that we say a topological space K has the fixed point property if every continuous map f : KK has a fixed point. Let X and Y be topological Hausdorff spaces, BY and T : XY be a multivalued map with nonempty values. Define
T - ( B ) = { x X : T ( x ) B } , Open image in new window

and let int(B), ∂B and F ( B ) Open image in new window denote the interior, boundary and the set of all nonempty finite subsets of B.

Let E be a CAT(0) space and FE. Recall that the notion of a convex hull is introduced via the formula
c o ( F ) = n = 0 F n , Open image in new window

where F0 = F and for n ≥ 1, the set F n consists of all points in E which lie on geodesics which start and end in Fn-1. The convex hull of a finite subset is not necessarily closed, but in any locally convex Hausdorff space, if K1,..., K n are compact convex subsets, then the convex hull of their union is compact too [13].

Definition 2.1[11] Let C be a nonempty subset of a CAT(0) space E. A multivalued mapping G : C → 2 E is said a KKM mapping if
c o ( F ) x F G ( x ) , Open image in new window

for every nonempty finite set F F ( C ) Open image in new window.

Example 2.1 Let C be a convex subset of CAT(0) space E and f : C → ℝ be such that for each x1,..., x n X if xco({x1,..., x n }), then there exist a1,..., a n ∈ ℝ with i = 1 n a i = 1 Open image in new window such that f ( x ) i = 1 n a i f ( x i ) Open image in new window. For each xC define
ϕ ( x ) = { y C : f ( y ) f ( x ) } . Open image in new window

We show that ϕ is a KKM map. By contradiction, suppose yco({x1,..., x n }) and y ∉ ∪ i ϕ(x i ). Therefore, there exist a1,..., a n ∈ ℝ with i = 1 n a i = 1 Open image in new window such that f ( y ) i = 1 n a i f ( x i ) Open image in new window. Since f (x i ) < f (y) for each i = 1,..., n, so we have a contradiction. Thus, ϕ is a KKM map.

Definition 2.2 We say that a CAT(0) space X has the convex hull finite property if the closed convex hull of every nonempty finite family of points of X has the fixed point property.

Example 2.2[14] In a locally compact CAT(0) space, the closed convex hull of each finite family of points has the fixed point property. So, every locally compact CAT(0) space has the convex hull finite property.

The following important result is established in [11].

Theorem 2.1 (KKM mapping principle) Suppose that E is a complete CAT(0) space with the convex hull finite property and X is a nonempty subset of E. Furthermore, suppose M : X → 2 X is a KKM mapping with closed values. Then, if M(z) is compact for some zX, then
x X M ( x ) . Open image in new window

3 Main results

The following theorem is a direct application of KKM mapping principle.

Theorem 3.1 Suppose X is a compact subset of a complete CAT(0) space E with convex hull finite property and F : XE is continuous. Then, there exists y0X such that
d ( y 0 , F ( y 0 ) ) = inf x X d ( x , F ( y 0 ) ) . Open image in new window
Proof. Consider the map G : X → 2 E defined by
G ( x ) = { y X : d ( y , F ( y ) ) d ( x , F ( y ) ) } . Open image in new window
Since F is continuous, so G(x) is closed for every xX. We claim that
c o ( A ) x A G ( x ) , Open image in new window
for all finite set AX. On the contrary, there exists {x1,..., x n } ⊆ X and yco({x1,..., x n }) such that y i G ( x i ) Open image in new window. This clearly implies
d ( x i , F ( y ) ) < d ( y , F ( y ) ) , Open image in new window
for i = 1,..., n. Hence, x i B(F (y), d(y, F (y))) for i = 1,..., n. Therefore, we have
c o ( { x 1 , , x n } ) B ( F ( y ) , d ( y , F ( y ) ) ) , Open image in new window

which implies that yB(F(y), d(y, F(y))). Clearly, this gets a contradiction.

By compactness of X, we deduce that G(x) is compact for every xX. Therefore, there exists y0 ∈ ∩xXG(x). This clearly implies d(y0, F (y0)) ≤ d(x, F(y0)) for every xX which implies
d ( y 0 , F ( y 0 ) ) = inf x X d ( x , F ( y 0 ) ) , Open image in new window

and the proof is complete. □

Theorem 3.2 Suppose X is a compact subset of a complete CAT(0) space E with convex hull finite property and F : XE is a continuous map such that for every cX, with cF(c), there exists α ∈ (0, 1) such that
X B ( F ( c ) , ( 1 - α ) d ( c , F ( c ) ) ) . Open image in new window

Then, F has a fixed point.

Proof. By Theorem 3.1, there exists y0X such that
d ( y 0 , F ( y 0 ) ) = inf x X d ( x , F ( y 0 ) ) . Open image in new window
We claim that y0 is a fixed point of F. Indeed, assume not, i.e., y0F(y0). Then, our assumption on X implies the existence of α ∈ (0, 1) such that
X B ( F ( y 0 ) , ( 1 - α ) d ( y 0 , F ( y 0 ) ) ) . Open image in new window
Let xXB(F(y0), (1 - α)d(y0, F (y0))). Clearly, xy0, and we have
d ( x , F ( y 0 ) ) < ( 1 - α ) d ( y 0 , F ( y 0 ) ) . Open image in new window

Since d(y0, F(y0)) ≤ d(x, F(y0)), we clearly get a contradiction and this completes the proof. □

Definition 3.1 Let E be a CAT(0) space, and C be a convex subset of E. A function f : C → ℝ is said to be metrically quasi-concave (resp., metrically quasi-convex) if for each λ ∈ ℝ, the set {xC : f(x) > λ} (resp., {xC : f(x) < λ}) is convex.

Example 3.1 Consider Hilbert space λ2 consisting of all complex sequences with the norm x = ( i = 1 ξ i 2 ) 1 2 Open image in new window, where x = (ξ j ) ∈ λ2. Define the functions f, g : λ2 → ℝ defined by
f ( x ) = 0 x = 0 1 | | x | | x 0 , Open image in new window

and g(x) = ||x||. It is easy to see that f is metrically quasi-concave and is not quasi-convex, and g is metrically quasi-convex and not metrically quasi-concave.

Lemma 3.1 Let C be a convex subset of a CAT(0) space X, and the function f : C × C → ℝ satisfies the following conditions.

1. for each xC, the function f(·, x) : C → ℝ is metrically quasi-concave (resp., metrically quasi-convex),

2. there exists γ ∈ ℝ such that f(x, x) ≤ γ (resp., f(x, x) ≥ ) for each xC.

Then, the mapping G : C → 2 X , which is defined by
G ( x ) = { y C : f ( x , y ) γ } ( r e s p . , G ( x ) = { y C : f ( x , y ) γ } ) , Open image in new window

is a KKM mapping.

Proof. The conclusion is proved for the concave case, the convex case is completely similar. On the contrary assume that G is not a KKM mapping. Suppose that there exists a finite subset A = {x1,..., x n } of C and a point x0co(A) such that x0G(x i ) for each i = 1,..., n. By setting
λ = min { f ( x i , x 0 ) : i = 1 , , n } > γ , Open image in new window
and
B = { z C : f ( z , x 0 ) > λ 0 } , Open image in new window

where λ > λ0> γ. For each i, we have x i B. According to hypothesis 1, B is convex and hence co(A) ⊆ B. So, x0B, and we have f (x0, x0) > λ0> γ which is a contradiction by (2). Thus, G is a KKM mapping. □

Definition 3.2 Let X, Y be CAT(0) spaces. A map F : X → 2 Y is said to be

  • upper semicontinuous if for each closed set BY , F-(B) is closed in X.

  • lower semicontinuous if for each open set BY , F-(B) is open in X.

It is well known that if F(x) is compact for each xX, then F is upper semicontinuous if and only if for each xX and ε > 0, there exist δ > 0 such that for each x' ∈ B(x, δ), we have F(x') ⊆ B(F(x), ε).

The following is a CAT(0) version of the Fan's minimax inequality [15].

Theorem 3.3 Suppose C is a compact and convex subset of a complete CAT(0) space E with convex hull finite property and f : C×C → ℝ satisfies the following,

1. for each xC, the function f(x,·) : C → ℝ is lower semicontinuous (resp., upper semicontinuous),

2. for each yC, the function f(·, y) : C → ℝ is metrically quasi-concave (resp., metrically quasi-convex),

3. there exists γ ∈ ℝ such that f(x, x) ≤ γ (resp., f(x, x) ≥ γ) for each xC.

Then, there exists a y0C such that f (x, y0) ≤ γ (resp., f (x, y0) ≥ γ) for all xC and hence
sup x C f ( x , y 0 ) sup x C f ( x , x ) ( r e s p . , inf x C f ( x , y 0 ) inf x C f ( x , x ) ) . Open image in new window
Proof. By hypothesis 3, λ = supxCf (x, x) < ∞. For each xC, define the mapping G : C → 2 C by
G ( x ) = { y C : f ( x , y ) λ } , Open image in new window
which is closed by hypothesis (1). By Lemma 3.1, G is a KKM mapping. By Theorem 2.1,
x C G ( x ) . Open image in new window

Therefore, there exists a y 0 x C G ( x ) Open image in new window. Thus, f (x, y0) ≤ λ for every xC.

Hence,
sup x C f ( x , y 0 ) sup x C f ( x , x ) . Open image in new window

This completes the proof. □

Definition 3.3 Let X be a CAT(0) space and DX. The map G : D → 2 X is called quasi-convex if the set G-(C) is convex for each convex subset C of Y.

Theorem 3.4 Suppose X is a compact subset of a complete CAT(0) space E with convex hull finite property and F, G : X → 2 E are upper semicontinuous maps with nonempty compact convex values and G is quasi-convex. Then, there exists x0X such that
d ( G ( x 0 ) , F ( x 0 ) ) = inf x X d ( G ( x ) , F ( x 0 ) ) . Open image in new window
Proof. Let H : X → 2 X be defined by
H ( y ) = { x X : d ( G ( x ) , F ( x ) ) d ( G ( y ) , F ( x ) ) } . Open image in new window

For each yX, since yH(y), so H(y) ≠ ∅.

We claim that H(y) is closed for each yX. Suppose that {y n } be a sequence in H(y) such that y n y*. We show that y* ∈ H(y). Let ε > 0 be arbitrary. Since F is upper semicontinuous with compact values, so there exists N1 such that for each nN1, we have
F ( y n ) B ̄ ( F ( y * ) , ε ) . Open image in new window
Similarly, we can prove there exists N1 such that for each nN2, we have
G ( y n ) B ̄ ( G ( y * ) , ε ) . Open image in new window
Let N = max{N1, N2}. Then, we have
d ( G ( y * ) , F ( y * ) ) d ( G ( y * ) , G ( y n ) ) + d ( G ( y n ) , F ( y n ) ) + d ( F ( y n ) , F ( y * ) ) 2 ε + d ( G ( y n ) , F ( y n ) ) 2 ε + d ( G ( y ) , F ( y n ) ) 2 ε + d ( G ( y ) , F ( y * ) ) + d ( F ( y * ) , F ( y n ) ) 3 ε + d ( G ( y ) , F ( y * ) ) . Open image in new window
Since ε was arbitrary, so
d ( G ( y * ) , F ( y * ) ) d ( G ( y ) , F ( y * ) ) , Open image in new window

and this proves our claim.

Now, we show that for each A F ( X ) Open image in new window, co(A) ⊆ H(A). On the contrary, suppose co(A) ⊄ H(A) for some A F ( X ) Open image in new window. Then, there exists yco(A) such that yH(a) for every aA. Therefore,
d ( G ( a ) , F ( y ) ) < d ( G ( y ) , F ( y ) ) , Open image in new window
(1)
for some aA. For each aA, we have
G ( a ) y F ( y ) B ( y , max b A d ( G ( a ) , F ( y ) ) ) . Open image in new window
Since F(y) is convex, so
y F ( y ) B ( y , max b A d ( G ( b ) , F ( y ) ) ) ) Open image in new window
is convex. This shows that
G ( y ) y F ( y ) B ( y , max b A d ( G ( b ) , F ( y ) ) ) , Open image in new window
because G is quasi-convex. Therefore, by (1), we have
d ( G ( y ) , F ( y ) ) max b A d ( G ( b ) , F ( y ) ) < d ( G ( y ) , F ( y ) ) . Open image in new window
This is a contradiction. Now, by Theorem 2.1, it follow that there exists x0X such that
x 0 y X H ( y ) . Open image in new window
Hence,
d ( G ( x 0 ) , F ( x 0 ) ) = inf x X d ( G ( x ) , F ( x 0 ) ) . Open image in new window

This completes the proof. □

Corollary 3.1 Suppose X is a compact subset of a complete CAT(0) space E with convex hull finite property and G : X → 2 X is an onto, quasi-convex and upper semicontinuous map with nonempty compact convex values and f : XX is a continuous single valued map. Then, there exists x0X such that f(x0) ∈ G(x0).

Corollary 3.2 Suppose X is a compact subset of a complete CAT(0) space E with convex hull finite property and G : X → 2 X is a quasi-convex and an upper semicontinuous map with nonempty compact convex values. Then, there exists x0X such that
d ( x 0 , G ( x 0 ) ) = inf x X d ( G ( x ) , x 0 ) . Open image in new window
Corollary 3.3 Suppose X is a compact subset of a complete CAT(0) space E with convex hull finite property and G : X → 2 X is an upper semicontinuous map with nonempty compact convex values. Then, there exists x0X such that
d ( G ( x 0 ) , x 0 ) = inf x X d ( G ( x 0 ) , x ) . Open image in new window

Moreover, if x0G(x0) then x0 ∈ ∂X.

Proof. By Theorem 3.4, clearly there exists x0X such that
d ( G ( x 0 ) , x 0 ) = inf x X d ( G ( x 0 ) , x ) . Open image in new window
(2)
Suppose x0G(x0). Since G has compact values, so d(x0, G(x0)) = r > 0. We prove that x0 ∈ ∂X. Assume, it is not. Then, x0int(X). Therefore, there exists an ε ∈ (0, r) such that B(x0, ε) ⊆ X. Take z0G(x0) such that d ( x 0 , z 0 ) < r + ε 2 Open image in new window. By Lemma 2.2(2), there exists y0 ∈ [x0, z0] such that d ( x 0 , y 0 ) = ε 2 Open image in new window. Again by Lemma 2.2(1), we have
d ( y 0 , G ( x 0 ) ) d ( y 0 , z 0 ) = d ( x 0 , z 0 ) d ( x 0 , y 0 ) < r = d ( x 0 , G ( x 0 ) , Open image in new window

which is a contradiction by (2). Therefore, x0 ∈ ∂X. □

Corollary 3.4 Suppose X is a compact subset of a complete CAT(0) space E with convex hull finite property and G : X → 2 X is an upper semicontinuous map with nonempty compact convex values. If G(x) ∩ X = ∅ for all x ∈ ∂X, then G has a fixed point.

Proof. On the contrary, assume that G does not have a fixed point. Therefore, by Theorem 3.4, there exists x0 ∈ ∂X such that
0 < d ( x 0 , G x 0 ) d ( x , G x 0 ) , Open image in new window
(3)

for all xX. Since x0 ∈ ∂X, we have Gx0X ≠ ∅. This is a contradiction by (3). □

If in Theorem 3.4, G is single valued, then it reduces to the following analog of Fan's best approximation to single-valued mappings in CAT(0) spaces.

Corollary 3.5 Suppose X is a compact subset of E and G : XE is a continuous map. Then, there exists x0X such that
d ( G x 0 , x 0 ) d ( G x 0 , x ) , Open image in new window

for all xX.

The following is an analog of Fan's fixed point theorem in CAT(0) spaces [16].

Theorem 3.5 Suppose X is a compact subset of a complete CAT(0) space E with convex hull finite property and G : XE is a continuous map and for every xX with xGx,
( x , G x ] = [ x , G x ] \ { x } , Open image in new window

contains at least one point of X, then G has a fixed point.

Proof. By the Corollary 3.5, there exists x0X such that
d ( x 0 , G x 0 ) d ( x , G x 0 ) , Open image in new window
(4)

for all xX.

We claim that x0 is a fixed point of T. On the contrary, assume that x0Gx0. Then, by assumptions, there exists zX such that z ∈ (x0, Gx0].

Therefore,
d ( z , G x 0 ) = d ( x 0 , G x 0 ) - d ( x 0 , z ) < d ( x 0 , G x 0 ) , Open image in new window

which by (4) it is a contradiction. □

Notes

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© Shabanian and Vaezpour; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceAmirkabir University of TechnologyTehranIran

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