Definition 2.3.5. Let X be a geometric metric space. A subset C of X is said to be convex in C if it includes every geodesic segment joining two of its point. This means that C is convex if x, y ∈C, we have that tx⊕(1−t)y∈C.
Remark 2.3.6. [45]
A geodesic segment in a space that is not uniquely geodesic may not necessarily be convex.
(a)
(b) A subset of a uniquely geodesic metric space which is endowed with the induced metric is geodesic if and only if it is convex.
2.3.2 Geodesic triangles
Definition 2.3.7. A geodesic triangle △(x1, x2, x3) in a geodesic space X consists of three points x1, x2, x3 ∈X (which are also called the vertices of △) and a geodesic segment between each pair of vertices (which are also known as edges of △).
Definition 2.3.8. A comparison triangle for the geodesic triangle △(x1, x2, x3) ∈ X is a triangle △(x¯ 1, x2, x3) := △( ¯x1,x¯2,x¯3) in Euclidean space R2 such that d(xi, xj) = dR2( ¯xi,x¯j) ∀ i, j ∈ {1,2,3}.
Definition 2.3.9. Let {xn} be a bounded sequence in a geodesic metric space X. Then, the asymptotic centerA({xn})of {xn}is defined by A({xn}) ={¯v ∈X : lim sup
n→∞
d(¯v, xn) =
v∈Xinf lim sup
n→∞
d(v, xn)}.
The sequence {xn} in X is said to be ∆-convergent to a point v¯∈ X if A({xnk}) = {¯v}
for every subsequence {xnk} of {xn}. In this case, we write ∆-lim
n→∞xn= ¯v and we say that
¯
v is the ∆-limit of {xn}.
2.3.3 Geometric properties of Hadamard spaces
The Hadamard spaces which were named after Jacques Hadamard are known as complete uniquely geodesic metric spaces of nonpositive curvatures and they include Hilbert spaces, Euclidean spaces (Rn),R-trees, hyperbolic spaces, Hilbert ball, among others. The geom- etry of Hadamard spaces can be seen as the nonlinearization of the geometry of Hilbert spaces. Karl Menger [176] introduced the notion of a geodesic in metric spaces and he gen- eralized classical results in geometry to his new metric space with geodesic. To improve the work of Menger [176], Wald [258] introduced the notion of two-dimensional curva- ture in metric spaces. Alexandror [17] also contributed by discovering some interesting characteristics of the spaces. Over the years, the Hadamard spaces has shown to be an appropriate framework for the study of optimization problems which has applications in diverse fields such as economics, engineering and science.
Now we are in the position to present some characterizations and conditions required for geodesic spaces to be CAT(0) spaces.
Characterizations of CAT(0) spaces
Gromov [114] coined the term CAT(0) from the initials of three mathematicians where C stands for Cartan, A stands for Alexandrov and T stands for Toponogov. Precisely, CAT(0) spaces are spaces of non-positive curvature bounded above by 0.
Definition 2.3.10. A geodesic space is called a CAT(0) space if all geodesic triangles satisfy the comparison axiom.
Definition 2.3.11. Let△be a geodesic triangle in X and let △¯ be its comparison triangle in R2. Then, △ is said to satisfy the CAT(0) inequality, if for all x, y ∈ △ and all comparison points x,¯ y¯∈△,¯
d(x, y)≤dR2(¯x,y).¯
Let x, y, z be points in X and y0 be the midpoint of the segment [y, z], then the CAT(0) inequality implies
d2(x, y0)≤ 1
2d2(x, y) + 1
2d2(x, z)−1
4d2(y, z). (2.3.2) Thus, a geodesic space is a CAT(0) space if and only if it satisfies (2.3.2). It is generally known that a CAT(0) space is a uniquely geodesic space. Inequality (2.3.2) is known as the CN inequality of Bruhat and Titis [48].
Definition 2.3.12. A geodesic metric space X is called a CAT(0) space if all geodesic triangles satisfy the CAT(0) inequality. Equivalently, X is called a CAT(0) space if and only if it satisfies the CN inequality.
Theorem 2.3.13. (see [33, Theorem 1.3.2]). Let X be a complete metric space. Then the following are equivalent.
(i) The space X is a CAT(0) space.
(ii) For every pair of points x, y ∈X, there exists m∈ X such that for each z ∈X, we have that
d2(m, z)≤ 1
2d2(x, z) + 1
2d2(y, z)− 1
4d2(x, y).
(iii) For every pair of points x, y ∈X and ϵ >0, there exists m ∈X such that for each z ∈X, we have that
d2(m, z)≤ 1
2d2(x, z) + 1
2d2(y, z)−1
4d2(x, y) +ϵ.
Next, we present the following Theorem which gives equivalent conditions for a geodesic space to be a CAT(0) space.
Theorem 2.3.14. (see [33, Theorem 1.3.3]). Let X be a geodesic metric space. Then the following are equivalent:
(i) The space X is a CAT(0) space.
(ii) For every pair of points x, y, z∈X, we have d2(m, x)≤ 1
2d2(x, y) + 1
2d2(x, z)− 1
4d2(x, z), where m is the midpoint of [y, z].
(iii) For every geodesic x: [0,1]→X and every point p∈X, we have d2(p, x1)≤(1−t)d2(p, x0) +td2(p, x1)−t(1−t)d2(x0, x1).
(iv) For every x, y, u, v ∈X, we have
d2(x, u) +d2(y, v)≤d2(x, y) +d2(u, v) + 2d(x, v)d(y, u).
(v) For every x, y, u, v ∈X, we have
d2(x, u) +d2(y, v)≤d2(x, y) +d2(y, u) +d2(u, v) +d2(v, x).
Definition 2.3.15. A complete CAT(0) space is an Hadamard space.
2.3.4 Examples of Hadamard spaces
Now, we present some examples of Hadamard spaces;
Example 2.3.16. (Hilbert space)[33]. Hilbert spaces are Hadamard. The geodesics are the line segments. It is also known that a Banach space is CAT(0) if and only if it is Hilbert.
Example 2.3.17. (R- trees)[33]. A metric space (X, d) is an R-tree if it is uniquely geodesic and for every x, y, z ∈X, we have [x, z] = [x, y]∪[y, z] whenever [x, y]∩[y, z] = {y}. Also, all triangles in an R-tree are trivial.
Example 2.3.18. All simply connected Riemannian manifold with non-positive sectional curvature induced with the Riemannian metric.
2.3.5 Quasilinearization mapping and dual space
The concept of quasilinearization in Hadamard spaces was introduced by Berg and Niko- laev [41]. They denoted a pair (a, b) ∈ X ×X by −→
ab and called it a vector. Using this concept, they defined the quasilinearization as a map ⟨·,·⟩ : (X ×X)×(X ×X) → R defined by
⟨−→ ab,−→
cd⟩= 1
2 d2(a, d) +d2(b, c)−d2(a, c)−d2(b, d)
, ∀a, b, c, d∈X. (2.3.3) It is easy to verify that ⟨−→
ab,−→
ab⟩=d2(a, b), ⟨−→ ba,−→
cd⟩ =−⟨−→ ab,−→
cd⟩, ⟨−→ ab,−→
cd⟩=⟨−→ae,−→ cd⟩+
⟨−→ eb,−→
cd⟩ and ⟨−→ ab,−→
cd⟩=⟨−→ cd,−→
ab⟩, for all a, b, c, d, e∈X.
Definition 2.3.19. The space X is said to satisfy the Cauchy Schwartz inequality, if
⟨−→ ab,−→
cd⟩ ≤d(a, b)d(c, d) ∀a, b, c, d∈X.
Moreover, a geodesic space is a CAT(0) space if and only if it satisfies the Cauchy-Schwartz inequality (see [138]).
Using the idea of quasilinearization mapping, Kakavandi and Amini [138] introduced the concept of dual space of an Hadamard space X as follows:
Consider the map θ:R×X×X →C(X,R) defined by θ(t, a, b)(x) =t⟨−→
ab,−ax⟩→ (t∈R, a, b, x ∈X),
whereC(X,R) denotes the space of all continuous real valued functions onX.The Cauchy- Schwartz inequality implies thatθ(t, a, b) is a Lipschitz function with Lipschitz semi-norm L(θ(t, a, b)) = |t|d(a, b) (t∈R, a, b∈X), where
L(φ) = sup
nφ(x)−φ(y)
d(x, y) :x, y ∈X, x̸=y o
is the Lipschitz semi-norm for any function φ:X →R.
Definition 2.3.20. A pseudometric D on R×X×X is defined by
D((t, a, b),(s, c, d)) = L(θ(t, a, b)−θ(s, c, d)), (t, s∈R, a, b, c, d∈X).
In an Hadamard space (X, d), the pseudometric space (R×X×X, D) can be considered as a subset of the pseudometric space of all real valued Lipschitz functions (Lip(X,R), L) (see [93, 211, 255]).
It is shown in [138] thatD((t, a, b),(s, c, d)) = 0 if and only ift⟨−→
ab,−xy⟩→ =s⟨−→
cd,−xy⟩→ for allx, y ∈ X. Thus,Dinduces an equivalence relation on R×X×X,where the equivalence class of (t, a, b) is defined as
[−→
tab] :={−→
scd:D((t, a, b),(s, c, d)) = 0}.
Thus, the set X∗ = {[−→
tab] : (t, a, b) ∈ R × X × X} is a metric space with metric D([−→
tab],[−→
scd]) :=D((t, a, b),(s, c, d)).
Definition 2.3.21. Let (X, d) be an Hadamard space. Then, the pair (X∗, D) is called the dual space of (X, d).
Throughout this dissertation, we shall write X∗ for the dual space of an Hadamard space X.
Remark 2.3.22. [138] The dual of a closed and convex subset of a Hilbert space H with nonempty interior is an Hadamard space and t(b−a)≡[−→
tab] for all t∈R, a, b ∈H.We also note that X∗ acts on X×X by
⟨x∗,−xy⟩→ =t⟨−→
ab,−xy⟩,→ (x∗ = [−→
tab]∈X∗, x, y∈X and t ∈R).
2.3.6 Some inequalities that characterize Hadamard spaces
Lemma 2.3.23. [194] Let X be a CAT(0) space, x, y, z,∈X and t∈[0,1]. Then (a) d(tx⊕(1−t)y, z)≤td(x, z) + (1−t)d(y, z).
(b) d2(tx⊕(1−t)y, z)≤td2(y, z) + (1−t)d2(y, z)−t(1−t)d2(x, y).
(c) d2(tx⊕(1−t)y, z)≤t2d2(x, z) + (1−t)2d2(y, z) + 2t(1−t)⟨−xz,→ −→yz⟩.
2.3.7 Some nonlinear single-valued mappings in Hadamard spaces
Let X be an Hadamard space and C be a nonempty closed and convex subset of X. A mapping T :C →C is said to be
(i) L-Lipschitzian, if there exists L >0 such that
d(T x, T y)≤Ld(x, y), ∀x, y ∈C, (ii) nonexpansive, if
d(T x, T y)≤d(x, y) ∀ x, y ∈C, (iii) quasi-nonexpansive, ifF(T)̸=∅ and
d(T x, y)≤d(x, y), ∀x∈C and y∈F(T), (iv) demicontractive, ifF(T)̸=∅ and there exists k∈(0,1) such that
d2(T x, y)≤d2(x, y) +kd2(x, T x), ∀ x∈C, ∀ y∈F(T).
(v) quasi-pseudocontractive if F(T)̸=∅ and
d2(T x, y)≤d2(x, y) +d2(x, T x), ∀ x∈C, ∀ y∈F(T). (2.3.4) Remark 2.3.24. The class of quasi-pseudocontractive mappings includes some nonlinear mappings like nonexpansive mappings (with nonempty fixed points set), quasi-nonexpansive mappings and demicontractive mappings.
Example 2.3.25. [186] LetCbe the closed interval [0,1] with the absolute value as norm.
Define T :C →C by
T x= (1
2, if x∈[0,12] 0, if x∈(12,1].
T is quasi-pseudocontractive but not demicontractive.
2.3.8 Monotone operators and its resolvents in Hadamard spaces
One of the most important area of nonlinear and convex analysis is the monotone operator theory. This is due to the role it plays in optimization theory and related mathematical problems. In this section, we study the concept of monotone operators and its resolvent in Hadamard spaces.
Definition 2.3.26. LetX be an Hadamard space andX∗ be its dual space. A multivalued operator A:X →2X∗ is monotone if and only if for allx, y ∈D(A), x∗ ∈Ax, y∗ ∈Ay, we have
⟨x∗−y∗,−yx⟩ ≥→ 0.
A monotone operator A is called a maximal monotone operator if the graph G(A) of A defined by
G(A) := {(x, x∗)∈X×X∗ :x∗ ∈A(x)}, is not properly contained in the graph of any other monotone operator.
Definition 2.3.27. The resolvent of a monotone operator A of order λ >0 is the multi- valued mapping JλA:X →2X∗ defined by
JλAx:=
z ∈X
1 λ
−→ zx
∈Az
. (2.3.5)
The operator A satisfies the range condition if for every λ >0, D(JλA) = X.
The resolvent of monotone operators plays a crucial role in the approximation of solutions of MIPs. We present some lemmas that relate the fixed points of a resolvent of a monotone operator and the set of solutions of the MIP (1.2.16).
Lemma 2.3.28. [150] Let X be a CAT(0) space and JλA be the resolvent of the operator A of orderλ. Then we have that
(a) For any λ > 0, R(JλA)⊂D(A) and F(JλA) = A−1(0), where R(JλA) is the range of JλA.
(b) If A is monotone, then JλA is a single-valued and firmly nonexpansive mapping.
(c) If A is monotone and 0< λ≤µ, then d2(JλAx, JµAx)≤ µ−λµ+λd2(x, JµAx), which implies that d(x, JλAx)≤2d(x, JµAx).
Lemma 2.3.29. [255] Let X be an Hadamard space and A : X → 2X∗ be a monotone mapping. Then,
d2(u, JλAx) +d2(JλAx, x)≤d2(u, x), (2.3.6) for all u∈F(JλA), x∈X and λ >0.
The Moreau-Yosida resolvent Jλf : X →X of a proper convex and lower semicontinuous function f in X is defined by
Jλf(x) = arg min
y∈X
f(y) + 1
2λd2(y, x)
∀x∈X, λ >0. (2.3.7) Definition 2.3.30. Let C be a nonempty closed and convex subset of an Hadamard space X. A mapping T : C →C is said to be ∆-demiclosed, if for any bounded sequence {xn} in C such that ∆-lim
n→∞xn=x and lim
n→∞d(xn, T xn) = 0, then x=T x.