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.
We know that the metric projections in Banach and Hadamard spaces have the same property.
Lemma 2.4.3. [16] Let C be a nonempty, closed and convex subset of a reflexive, strictly convex and smooth Banach space X. If x∈ X and q ∈ C, then
q = ΠCx ⇐⇒ ⟨y−q, J x−J q⟩ ≤0, ∀ y∈ C (2.4.2) and
ϕ(y, ΠCx) +ϕ(ΠCx, x)≤ϕ(y, x), ∀ y∈ C, x∈ X. (2.4.3) We are now in the position to present some examples of the metric projection.
Example 2.4.4. If C = {y ∈ H : ∥y−s∥ ≤ α} is a closed ball centered at s ∈ H with radius α≥0,
PCx=
(s+α(x−s)∥x−s∥, if x /∈ C x, ifx∈ C.
Example 2.4.5. Let C = [a, b] be a closed rectangle in Rn, where a = (a1, a2,· · · , an)T and b = (b1, b2,· · · , bn)T,then for 1≤i≤n
(PCx)i =
ai, xi < ai, xi, xi ∈[ai, bi], bi, xi > bi
(2.4.4)
is the metric projection with the ith coordinate.
Example 2.4.6. Let C = {y ∈ H : ⟨s, y⟩ ≤ α} be a closed half space, with s ̸= 0 and α ∈R, then
PCx=
(x− ⟨s,x⟩−α∥s∥2 s, if ⟨s, x⟩> α,
x, if ⟨s, x⟩ ≤α (2.4.5)
is the metric projection onto C.
Example 2.4.7. Let C = {y ∈ H : ⟨s, y⟩ = α} be a hyperplane, with s ̸= 0 and α ∈ R, then
PCx=x−⟨s, x⟩ −α
∥s∥2 s (2.4.6)
is the metric projection onto C.
Example 2.4.8. If C is the range of a m ×n matrix A with full column rank, then PCx=A(A∗A)−1A∗X, whereA∗ is the adjoint of A.
2.4.1 Convex functions
Definition 2.4.9. A function c : H → R is said to be Gˆateaux differential at x ∈ H, if there exists an element denoted by c′(x)∈ H such that
h→0lim
c(x+hv)−c(x)
h =⟨v, c′(x)⟩, ∀v ∈ H,
where c′(x) is called the Gˆateaux differential of c at x. We say that if for each x∈ H, c is Gˆateaux differentiable at x, then c is Gˆateaux differentiable on H.
Definition 2.4.10. A functionc:H → Ris called convex, if for allt ∈[0,1]andx, y ∈ H, c(tx+ (1−t)y)≤tc(x) + (1−t)c(y).
Remark 2.4.11. We note that in an Hadamard space X, the convex function is defined by
c(tx⊕(1−t)y)≤tc(x) + (1−t)c(y)∀x, y ∈X, t∈(0,1);
Definition 2.4.12. Let X be a geodesic metric space. The function f : D(f) ⊆ X → R∪ {+∞} is said to be uniformly convex (see [77]), if there exists a strictly increasing function ϕ :R+ →R+ such that
f 1
2x⊕ 1 2y
≤ 1
2f(x) + 1
2f(y)−ϕ(d(x, y)).
Now, we present an example of a convex function in an Hadamard space.
Example 2.4.13. [35] Let X be an Hadamard space. For a finite number of points a1, a2, . . . , aN and (w1, w2, . . . , wN)∈S (whereS is the convex hull of the canonical basis e1, e2, . . . , eN ∈ RN), the function f :X →R defined by f(x) =
N
P
n=1
wnd2(x, an) is convex and continuous.
Definition 2.4.14. A convex function c:H → Ris said to be subdifferentiable at a point x∈ H if the set
∂c(x) ={u∈ H | c(y)≥c(x) +⟨u, y−x⟩, ∀y∈ H} (2.4.7) is nonempty, where each element in ∂c(x) is called a subgradient of cat x, ∂c(x) is called the subdifferential of c at x and the inequality in (2.4.7) is called the subdifferential in- equality of c at x. We say that c is subdifferentiable on H if c is subdifferentiable at each x∈ H.
We also note that if c is Gˆateaux differentiable at x, then c is subdifferentiable at x, and
∂c(x) ={c′(x)}.
Remark 2.4.15. We note that in a Banach spaceX with a dual spaceX∗, the subdiffer- ential inequality (2.4.7) is given by
∂c(x) = {u∈ X∗| c(y)≥c(x) +⟨u, y−x⟩, ∀y∈ X }. (2.4.8)
Definition 2.4.16. LetHbe a Hilbert space. The domain of a functionf :H →R∪{+∞}
is defined by D(f) ={x∈ H:f(x)<+∞}.
The function f :D(f)⊆ H →R∪ {+∞} is said to be (a) proper, if D(f)̸=∅;
(b) lower semicontinuous at a point x∈D(f), if f(x)≤lim inf
xn→x f(xn), (c) weakly lower semicontinuous at a point x∈D(f), if
lim inf
n→∞ f(xn)≥f(x),
holds for an arbitrary sequence {xn}∞n=0 in H satisfying xn ⇀ x;
(d) weakly upper semicontinuous at a point x∈D(f), if f(x)≥lim sup
n→∞
f(xn),
holds for an arbitrary sequence {xn}∞n=0 in H satisfying xn ⇀ x;
Definition 2.4.17. A bifunction f :C × C →R is called
(i) strongly monotone on C if there exists a constant β >0 such that f(x, y) +f(y, x)≤ −β∥x−y∥2, ∀x, y ∈ C;
(ii) monotone on C if
f(x, y) +f(y, x)≤0, ∀x, y ∈ C;
(iii) pseudomonotone on C if
f(x, y)≥0 =⇒ f(y, x)≤0 ∀ x, y ∈ C.
It is easy to see that (i) =⇒ (ii) =⇒ (iii) but the converses are not generally true.
Definition 2.4.18. A function f : C → R is said to be hemicontinuous at y ∈ C, if and only if
lim
t→0+f(tx+ (1−t)y) =f(y), for all x∈ C.
Nonlinear Multivalued mappings
In this section, we shall denote by P(C), CB(C) and 2C, the family of all nonempty prox- iminal subsets of C, the family of all nonempty, closed and bounded subsets of C and the family of all nonempty subsets of C respectively. Let H denote the Hausdorff metric on CB(C), then for all A, B ∈CB(C),
H(A, B) = max{sup
a∈A
d(a, B), sup
b∈B
d(b, A)}, (2.4.9)
whered(a, B) = inf
b∈B∥a−b∥is the distance from the pointato the subsetB. LetT :C →2C be a multivalued mapping. A point x ∈ C is called a fixed point of T, if x ∈ T x while x∈ C is called a strict fixed point of T, if T x={x}.
Definition 2.4.19. The mapping T :C → 2C is called
L-Lipschitz, if there exists L >0 such that
H(T x, T y)≤L∥x−y∥ ∀ x, y ∈ C,
if L= 1, then T is called nonexpansive, while T is called a contraction if L∈[0,1);
quasinonexpansive, if F(T)̸=∅ and
H(T x, p)≤ ∥x−p∥ ∀ x∈ C and p∈F(T).
Definition 2.4.20. Let X be a Banach space and B a mapping of X into 2X∗. The effective domain of B denoted by dom(B) is given as dom(B) = {x ∈ X : Bx ̸= ∅}. Let B :X → 2X∗ be a multivalued operator on X. Then
(i) The graph G(B) is defined by
G(B) := {(x, u)∈ X × X :u∈B(x)},
(ii) the operatorB is said to be monotone if⟨x−y, u∗−v∗⟩ ≥0for allx, y ∈dom(A), u∗ ∈ Ax and v∗ ∈Ay.
(iii) A monotone operator B on X is said to be maximal if its graph is not properly contained in the graph of any other monotone operator on X.
LetX be a uniformly convex and smooth Banach space with a Gˆateaux differential norm and let B :X →2X∗ be a maximal monotone operator. We consider the metric resolvent of B,
QBµ = I+µJX−1B−1
, µ >0.
It is known that the operatorQBµ is firmly nonexpansive and the fixed points of the operator QBµ are the null points ofB (see [152, 153]). The resolvent plays an important role in the approximation theory for zero points of maximal monotone operators in Banach spaces.
The following are the properties of the resolvent (see [21])
⟨QBµx−x∗, J(x−QBµx)⟩ ≥0, x∈ X, x∗ ∈B−1(0), (2.4.10) in particular, if X is a real Hilbert space, then
⟨JµBx−x∗, x−JµBx⟩ ≥0, x∈ X, x∗ ∈B−1(0),
where JµB = (I+µB)−1 is the general resolvent,B−1(0) = {z ∈ X : 0∈Bz} is the set of null points of B. Also, we know that B−1(0) is closed and convex (see [237]).
Lemma 2.4.21. [44] Let B :H →2H be a maximal monotone mapping, and A:H → H be a Lipschitz continuous and monotone mapping. Then the mapping A+B is a maximal monotone.