@ Appl. Gen. Topol. 20, no. 1 (2019), 193-210doi:10.4995/agt.2019.10635
c© AGT, UPV, 2019

A viscosity iterative technique for equilibrium

and fixed point problems in a Hadamard space

C. Izuchukwu, K. O. Aremu, A. A. Mebawondu and O. T. Mewomo

School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban,

South Africa. (izuchukwuc@ukzn.ac.za; 218081063@stu.ukzn.ac.za; 216028272@stu.ukzn.ac.za;

mewomoo@ukzn.ac.za)

Communicated by E. A. Sánchez-Pérez

Abstract

The main purpose of this paper is to introduce a viscosity-type proximal
point algorithm, comprising of a nonexpansive mapping and a finite
sum of resolvent operators associated with monotone bifunctions. A
strong convergence of the proposed algorithm to a common solution of
a finite family of equilibrium problems and fixed point problem for a
nonexpansive mapping is established in a Hadamard space. We further
applied our results to solve some optimization problems in Hadamard
spaces.

2010 MSC: 47H09; 47H10; 49J20; 49J40.

Keywords: equilibrium problems; monotone bifunctions; variational in-
equalities; convex feasibility problems; minimization problems;
viscosity iterations; CAT(0) space.

1. Introduction

Optimization theory is one of the most flourishing areas of research in math-
ematics that has received a lot of attention in recent time. One of the most
important problems in optimization theory is the Equilibrium Problem (EP)
since it includes many other optimization and mathematical problems as special
cases; namely, minimization problems, variational inequality problems, comple-
mentarity problems, fixed point problems, convex feasibility problems, among

Received 22 August 2018 – Accepted 11 December 2018

http://dx.doi.org/10.4995/agt.2019.10635


C. Izuchukwu, K. O. Aremu, A. A. Mebawondu and O. T. Mewomo

others (see Section 4, for details). Thus, EPs are of central importance in opti-
mization theory as well as in nonlinear and convex analysis. Given a nonempty
set C and f : C × C → R, the EP is defined as follows:

Find x∗ ∈ C such that f(x∗, y) ≥ 0, ∀y ∈ C.(1.1)

The point x∗ for which (1.1) is satisfied is called an equilibrium point of f.
Throughout this paper, we shall denote the solution set of problem (1.1) by
EP(f, C). EPs have been widely studied in Hilbert, Banach and topological
vector spaces by many authors (see [5, 10, 16, 28]), as well as in Hadamard
manifolds (see [9, 26]). One of the most popular and effective method used for
solving problem (1.1) and other related optimization problems is the Proximal
Point Algorithm (PPA) which was introduced in Hilbert space by Martinet [25]
in 1970 and was further extensively studied in the same space by Rockafellar [30]
in 1976. The PPA and its generalizations have also been studied extensively in
Banach spaces and Hadamard manifolds (see [9, 14, 22, 28] and the references
therein).
Recently, researchers are beginning to extend the study of the PPA and its gen-
eralizations to Hadamard spaces. For instance, Bačák [2] studied the following
PPA for finding minimizers of proper convex and lower semicontinuous func-
tionals in Hadamard spaces: Let X be a Hadamard space, then for arbitrary
point x1 ∈ X, define the sequence {xn} iteratively by

xn+1 = prox
f
µn

(xn),(1.2)

where µn > 0 for all n ≥ 1, and prox
f
µ : X → X is the Moreau-Yosida resolvent

of a proper convex and lower semicontinuous functional f defined by

proxfµ(x) = arg min
v∈X

(

f(v) +
1

2µ
d2(v, x)

)

.(1.3)

Bačák [2] proved that (1.2) ∆-convergence to a minimizer of f. In 2016, Su-
paratulatorn et al [33] extended the results of Bačák [2] by proposing the follow-
ing Halpern-type PPA for approximating a minimizer of a proper convex and
lower semicontinuous functional which is also a fixed point of a nonexpansive
mapping in Hadamard spaces:











u, x1 ∈ X,

yn = prox
f
µn

(xn),

xn+1 = αnu ⊕ (1 − αn)T yn n ≥ 1,

(1.4)

where {αn} ⊂ (0, 1) and µn ≥ λ > 0. They obtained a strong convergence
result under some mild conditions. The PPA was also studied by Khatibzadeh
and Ranjbar in [19] for finding zeroes of monotone operators and in [20] for solv-
ing variational inequality problems in Hadamard spaces. Based on the results of
Suparatulatorn et al [33], Khatibzadeh and Ranjbar [19], Okeke and Izuchukwu
[27] studied the Halpern-type PPA and obtained a strong convergence results
for finding a minimizer of a proper convex and lower semicontinuous functional
which is also a zero of a monotone operator and a fixed point of a nonexpansive

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 194



A viscosity iterative technique for equilibrium and fixed point problems in a Hadamard space

mapping. For more recent important results on PPA in Hadamard spaces and
other general metric spaces, see [1, 17, 36] and the references therein.
Very recently, Kumam and Chaipunya [22] studied EP (1.1) in Hadamard
spaces. First, they established the existence of an equilibrium point of a bi-
function satisfying some convexity, continuity and coercivity assumptions, and
they also established some fundamental properties of the resolvent of the bi-
function. Furthermore, they studied the PPA for finding an equilibrium point
of a monotone bifunction in a Hadamard space. More precisely, they proved
the following theorem.

Theorem 1.1. Let C be a nonempty closed and convex subset of a Hadamard
space X and f : C × C → R be monotone, ∆-upper semicontinuous in the first

variable such that D(J
f
λ
) ⊃ C for all λ > 0 (where D(J

f
λ
) means the domain of

J
f
λ
). Suppose that EP(C, f) 6= ∅ and for an initial guess x0 ∈ C, the sequence

{xn} ⊂ C is generated by

xn := J
f
λn

(xn−1), n ∈ N,

where {λn} is a sequence of positive real numbers bounded away from 0. Then,
{xn} ∆-converges to an element of EP(C, f).

Motivated by the above results of Kumam and Chaipunya [22], we study some
other important properties of the resolvent of monotone bifunctions. We then
introduce a viscosity-type PPA comprising of a nonexpansive mapping and a
finite sum of resolvent operators associated with these bifunctions. We prove
that the sequence generated by our proposed algorithm converges strongly to a
common solution of a finite family of equilibrium problems which is also a fixed
point of a nonexpansive mapping. Furthermore, we applied our results to solve
some optimization problems in Hadamard spaces. Our results serve as a con-
tinuation of the work of Kumam and Chaipunya [22]. They also extend related
results from Hilbert spaces and Hadamard manifolds to Hadamard spaces, and
they complement some recent important results in Hadamard spaces.

2. Preliminaries

In this section, we recall some basic and useful results that will be needed
in establishing our main results. We categorize our study into brief-detailed
subsections.

2.1. Geometry of Hadamard spaces.

Definition 2.1. Let (X, d) be a metric space, x, y ∈ X and I = [0, d(x, y)] be
an interval. A curve c (or simply a geodesic path) joining x to y is an isometry
c : I → X such that c(0) = x, c(d(x, y)) = y and d(c(t), c(t′) = |t − t′| for all
t, t′ ∈ I. The image of a geodesic path is called the geodesic segment, which is
denoted by [x, y] whenever it is unique.

Definition 2.2 ([13]). A metric space (X, d) is called a geodesic space if every
two points of X are joined by a geodesic, and X is said to be uniquely geodesic

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C. Izuchukwu, K. O. Aremu, A. A. Mebawondu and O. T. Mewomo

if every two points of X are joined by exactly one geodesic. A subset C of X
is said to be convex if C includes every geodesic segments joining two of its
points. Let x, y ∈ X and t ∈ [0, 1], we write tx ⊕ (1 − t)y for the unique point
z in the geodesic segment joining from x to y such that

d(x, z) = (1 − t)d(x, y) and d(z, y) = td(x, y).(2.1)

A geodesic triangle ∆(x1, x2, x3) in a geodesic metric space (X, d) consists of
three vertices (points in X) with unparameterized geodesic segment between
each pair of vertices. For any geodesic triangle there is comparison (Alexan-
drov) triangle ∆̄ ⊂ R2 such that d(xi, xj) = dR2(x̄i, x̄j) for i, j ∈ {1, 2, 3}. Let
∆ be a geodesic triangle in X and ∆̄ be a comparison triangle for ∆̄, then ∆
is said to satisfy the CAT(0) inequality if for all points x, y ∈ ∆ and x̄, ȳ ∈ ∆̄,

d(x, y) ≤ dR2(x̄, ȳ).(2.2)

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

2
d2(x, y) +

1

2
d2(x, z) −

1

4
d(y, z).(2.3)

Inequality (2.3) is known as the CN inequality of Bruhat and Titis [7].

Definition 2.3. A geodesic space X is said to be 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.
CAT(0) spaces are examples of uniquely geodesic spaces and complete CAT(0)
spaces are called Hadamard spaces.

Definition 2.4 ([4]). Let X be a CAT(0) space. Denote the pair (a, b) ∈ X×X

by
−→
ab and call it a vector. Then, a mapping 〈., .〉 : (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

is called a quasilinearization mapping.

It is easily to check 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. A geo-

desic space X is said to satisfy the Cauchy-Swartz inequality if 〈
−→
ab,

−→
cd〉 ≤

d(a, b)d(c, d) ∀a, b, c, d ∈ X. It has been established in [4] that a geodesically
connected metric space is a CAT(0) space if and only if it satisfies the Cauchy-
Schwartz inequality. Examples of CAT(0) spaces includes: Euclidean spaces
R

n, Hilbert spaces, simply connected Riemannian manifolds of nonpositive sec-
tional curvature [29], R-trees, Hilbert ball [15], among others.

Lemma 2.5 (see [23, Lemma 7]). Let X be a uniformly convex hyperbolic space
with modulus of uniform convexity η. For any c > 0, ǫ ∈ (0, 2], λ ∈ [0, 1] and
v, x, y ∈ X, we have that d(x, v) ≤ c, d(y, v) ≤ c and d(x, y) ≥ ǫc imply that

d((1 − λ)x ⊕ λy, v) ≤ (1 − 2λ(1 − λ)η(c, ǫ))c.

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A viscosity iterative technique for equilibrium and fixed point problems in a Hadamard space

If X is a CAT(0) space, then X is uniformly convex with modulus of uniform

convexity η(c, ǫ) := ǫ
2

8
(see [23, Proposition 8]).

We end this subsection with the following important lemmas which character-
izes CAT(0) spaces.

Lemma 2.6. Let X be a CAT(0) space, x, y, z ∈ X and t, s ∈ [0, 1]. Then

(i) d(tx ⊕ (1 − t)y, z) ≤ td(x, z) + (1 − t)d(y, z) (see[13]).
(ii) d2(tx ⊕ (1 − t)y, z) ≤ td2(x, z) + (1 − t)d2(y, z) − t(1 − t)d2(x, y) (see

[13]).
(iii) d2(tx⊕(1−t)y, z) ≤ t2d2(x, z)+(1−t)2d2(y, z)+2t(1−t)〈−→xz, −→yz〉 (see

[11]).
(iv) d(tw ⊕ (1 − t)x, ty ⊕ (1 − t)z) ≤ td(w, y) + (1 − t)d(x, z) (see [6]).
(v) z = tx ⊕ (1 − t)y implies 〈−→zy, −→zw〉 ≤ t〈−→xy, −→zx〉, ∀ w ∈ X (see [11]).
(vi) d(tx ⊕ (1 − t)y, sx ⊕ (1 − s)y) ≤ |t − s|d(x, y) (see [8]).

Lemma 2.7 ([37]). Let X be a CAT(0) space. For any t ∈ [0, 1] and u, v ∈ X,
let ut = tu ⊕ (1 − t)v. Then, for all x, y ∈ X,

(1) 〈−→utx,
−→
uty〉 ≤ t〈

−→
ux,

−→
uty〉 + (1 − t)〈

−→
vx,

−→
uty〉;

(2) 〈−→utx,
−→
uy〉 ≤ t〈−→ux, −→uy〉 + (1 − t)〈−→vx, −→ux〉 and

(3) 〈−→utx,
−→
vy〉 ≤ t〈−→ux, −→vy〉 + (1 − t)〈−→vx, −→vy〉.

Lemma 2.8 ([35]). Let X be a CAT(0) space, {xi, i = 1, 2, . . . , N} ⊂ X and

αi ∈ [0, 1], i = 1, 2, . . . , N such that
∑N

i=1 αi = 1. Then,

d

(

N
⊕

i=1

αixi, z

)

≤

N
∑

i=1

αid(xi, z), ∀z ∈ X.

Remark 2.9 (see also [35]). For a CAT(0) space X, if {xi, i = 1, 2, . . . , N} ⊂ X,
and αi ∈ [0, 1], i = 1, 2, . . . , N. Then by induction, we can write

N
⊕

i=1

αixi := (1 − αN)

N−1
⊕

i=1

αi

1 − αN
xi ⊕ αNxN.(2.4)

2.2. The notion of ∆-convergence.

Definition 2.10. Let {xn} be a bounded sequence in a geodesic metric space
X. Then, the asymptotic center A({xn}) of {xn} is defined by

A({xn}) = {v̄ ∈ X : lim sup
n→∞

d(v̄, xn) = inf
v∈X

lim sup
n→∞

d(v, xn).

It is generally known that in a Hadamard space, A({xn}) consists of exactly
one point. A sequence {xn} in X is said to be ∆-convergent to a point v̄ ∈ X if
A({xnk }) = {v̄} for every subsequence {xnk} of {xn}. In this case, we write ∆-
lim
n→∞

xn = v̄ (see [12]). The concept of ∆-convergence in metric spaces was first

introduced and studied by Lim [24]. Kirk and Panyanak [21] later introduced
and studied this concept in CAT(0) spaces, and proved that it is very similar
to the weak convergence in Banach space setting.
The following lemma is very important as regards to ∆-convergence.

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C. Izuchukwu, K. O. Aremu, A. A. Mebawondu and O. T. Mewomo

Lemma 2.11 ([13]). Every bounded sequence in a Hadamard space always have
a △−convergent subsequence.

2.3. Existence of solution of equilibrium problems and resolvent op-
erators.

Theorem 2.12 ([22, Theorem 4.1]). Let C be a nonempty closed and convex
subset of a Hadamard space X and f : C × C → R be a bifunction satisfying
the following:

(A1) f(x, x) ≥ 0 for each x ∈ C,
(A2) for every x ∈ C, the set {y ∈ C : f(x, y) < 0} is convex,
(A3) for every y ∈ C, the function x 7→ f(x, y) is upper semicontinuous,
(A4) there exists a compact subset L ⊂ C containing a point y0 ∈ L such

that f(x, y0) < 0 whenever x ∈ C\L.

Then, problem (1.1) has a solution.

In [22], the authors introduce the resolvent of the bifunction f associated with
the EP (1.1). They defined a perturbed bifunction f̄x̄ : C × C → R (x̄ ∈ X) of
f by

f̄x̄(x, y) := f(x, y) − 〈
−→
xx̄,

−→
xy〉, ∀x, y ∈ C.(2.5)

The perturbed bifunction f̄ has a unique equilibrium called the resolvent op-
erator Jf : X → 2C of the bifunction f (see [22]), defined by

(2.6) Jf (x) := EP(C, f̄x) = {z ∈ C : f(z, y) − 〈
−→
zx,

−→
zy〉 ≥ 0, y ∈ C}, x ∈ X.

It was established in [22] that Jf is well defined.
We now recall the following definitions which will be needed in what follows.

Definition 2.13. Let X be a CAT(0) space. A point x ∈ X is called a fixed
point of a nonlinear mapping T : X → X, if T x = x. We denote the set of
fixed points of T by F(T ). The mapping T is said to be

(i) firmly nonexpansive (see [19]), if

d2(T x, T y) ≤ 〈
−−−→
T xT y,

−→
xy〉 ∀x, y ∈ X,

(ii) nonexpansive, if

d(T x, T y) ≤ d(x, y) ∀x, y ∈ X.

From Cauchy-Schwartz inequality, it is clear that firmly nonexpansive mappings
are nonexpasive.

Definition 2.14. Let X be a CAT(0) space and C be a nonempty closed
and convex subset of X. A function f : C × C → R is called monotone if
f(x, y) + f(y, x) ≤ 0 for all x, y ∈ C.

Definition 2.15. Let X be a CAT(0) space. A function f : D(f) ⊆ X →
(−∞, +∞] is said to be convex if

f(tx ⊕ (1 − t)y) ≤ tf(x) + (1 − t)f(y) ∀x, y ∈ X, t ∈ (0, 1).

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 198



A viscosity iterative technique for equilibrium and fixed point problems in a Hadamard space

f is proper, if D(f) := {x ∈ X : f(x) < +∞} 6= ∅. The function f : D(f) →
(−∞, ∞] is lower semi-continuous at a point x ∈ D(f) if

f(x) ≤ lim inf
n→∞

f(xn),(2.7)

for each sequence {xn} in D(f) such that lim
n→∞

xn = x; f is said to be lower

semicontinuous on D(f) if it is lower semi-continuous at any point in D(f).

Lemma 2.16 ([22, Proposition 5.4]). Suppose that f is monotone and D(Jf ) 6=
∅. Then, the following properties hold.

(i) Jf is single-valued.
(ii) If D(Jf ) ⊃ C, then Jf is nonexpansive restricted to C.
(iii) If D(Jf ) ⊃ C, then F(Jf ) = EP(C, f).

Theorem 2.17 ([22, Theorem 5.2]). Suppose that f has the following proper-
ties

(i) f(x, x) = 0 for all x ∈ C,
(ii) f is monotone,
(iii) for each x ∈ C, y 7→ f(x, y) is convex and lower semicontinuous.
(iv) for each x ∈ C, f(x, y) ≥ lim supt↓0 f((1 − t)x ⊕ tz, y) for all x, z ∈ C.

Then D(Jf ) = X and Jf single-valued.

The following two lemmas will be very useful in establishing our strong con-
vergence theorem.

Lemma 2.18 ([34]). Let {xn} and {yn} be bounded sequences in a metric space
of hyperbolic type X and {βn} be a sequence in [0,1] with lim infn→∞ βn <
lim supn→∞ βn < 1. Suppose that xn+1 = βnxn ⊕ (1 − βn)yn for all n ≥ 0 and
lim supn→∞(d(yn+1, yn) − d(xn+1, xn)) ≤ 0. Then limn→∞ d(yn, xn) = 0.

Lemma 2.19 (Xu, [38]). Let {an} be a sequence of nonnegative real numbers
satisfying the following relation:

an+1 ≤ (1 − αn)an + αnσn + γn, n ≥ 0,

where, (i) {αn} ⊂ [0, 1],
∑

αn = ∞; (ii) lim sup σn ≤ 0; (iii) γn ≥ 0; (n ≥ 0),
∑

γn < ∞. Then, an → 0 as n → ∞.

3. Main results

Lemma 3.1. Let X be a CAT(0) space, {xi, i = 1, 2, . . . , N} ⊂ X, {yi, i =

1, 2, . . . , N} ⊂ X and αi ∈ [0, 1] for each i = 1, 2, . . . , N such that
∑N

i=1 αi = 1.
Then,

d

(

N
⊕

i=1

αixi,

N
⊕

i=1

αiyi

)

≤

N
∑

i=1

αid(xi, yi).(3.1)

Proof. (By induction). For N = 2, the result follows from Lemma 2.6 (iv).
Now, assume that (3.1) holds for N = k, for some k ≥ 2. Then, we prove that

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C. Izuchukwu, K. O. Aremu, A. A. Mebawondu and O. T. Mewomo

(3.1) also holds for N = k + 1. Indeed, by Remark 2.9, Lemma 2.6 (iv) and
our assumption, we obtain that

d

(

k+1
⊕

i=1

αixi,

k+1
⊕

i=1

αiyi

)

≤ (1 − αk+1)d

(

k
⊕

i=1

αi

1 − αk+1
xi,

k
⊕

i=1

αi

1 − αk+1
yi

)

+αk+1d(xk+1, yk+1)

≤

k+1
∑

i=1

αid(xi, yi).

Hence, (3.1) holds for all N ∈ N. �

Remark 3.2. It follows from (2.6) that the resolvent J
f
λ
of the bifunction f and

order λ > 0 is given as

(3.2) J
f
λ
(x) := EP(C, f̄x) = {z ∈ C : f(z, y)+

1

λ
〈−→xz, −→zy〉 ≥ 0, y ∈ C}, x ∈ X,

where f̄ is defined in this case as

f̄x̄(x, y) := f(x, y) +
1

λ
〈
−→
x̄ x,

−→
xy〉, ∀x, y ∈ C, x̄ ∈ X.(3.3)

Lemma 3.3. Let C be a nonempty, closed and convex subset of a Hadamard

space X and f : C × C → R be a monotone bifunction such that C ⊂ D(J
f
λ
)

for λ > 0. Then, the following hold:

(i) J
f
λ
is firmly nonexpansive restricted to C.

(ii) If F(Jλ) 6= ∅, then

d
2(Jλx, x) ≤ d

2(x, v) − d2(J
f
λ
x, v) ∀x ∈ C, v ∈ F(J

f
λ
).

(iii) If 0 < λ ≤ µ, then d(Jfµ x, J
f
λ
x) ≤

√

1 − λ
µ
d(x, Jfµ x), which implies that

d(x, J
f
λ
x) ≤ 2d(x, Jfµ x) ∀x ∈ C.

Proof. (i) Let x, y ∈ C, then by Lemma (2.16) (i) and the definition of J
f
λ
, we

have

f(J
f
λ
x, J

f
λ
y) +

1

λ
〈
−−−→
xJ

f
λ
x,

−−−−−→
J
f
λ
xJ

f
λ
y〉 ≥ 0(3.4)

and

f(J
f
λ
y, J

f
λ
x) +

1

λ
〈
−−−→
yJ

f
λ
y,
−−−−−→
J
f
λ
yJ

f
λ
x〉 ≥ 0.(3.5)

Adding (3.4) and (3.5), and noting that f is monotone, we obtain

1

λ

(

〈
−−−→
xJ

f
λ
x,

−−−−−→
J
f
λ
xJ

f
λ
y〉 + 〈

−−−→
yJ

f
λ
y,
−−−−−→
J
f
λ
yJ

f
λ
x〉

)

≥ 0,

which implies that

〈−→xy,
−−−−−→
J
f
λ
xJ

f
λ
y〉 ≥ 〈

−−−−−→
J
f
λ
xJ

f
λ
y,
−−−−−→
J
f
λ
xJ

f
λ
y〉.

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 200



A viscosity iterative technique for equilibrium and fixed point problems in a Hadamard space

That is,

〈−→xy,
−−−−−→
J
f
λ
xJ

f
λ
y〉 ≥ d2(J

f
λ
x, J

f
λ
y).(3.6)

(ii) It follows from (3.6) and the definition of quasilinearization that

d
2(x, J

f
λ
x) ≤ d2(x, v) − d2(v, J

f
λ
x) ∀x ∈ C, v ∈ F(J

f
λ
).

(iii) Let x ∈ C and 0 < λ ≤ µ, then we have that

f(J
f
λ
x, Jfµ x) +

1

λ
〈
−−−→
xJ

f
λ
x,

−−−−−→
J
f
λ
xJfµ x〉 ≥ 0(3.7)

and

f(Jfµ x, J
f
λ
x) +

1

µ
〈
−−−→
xJ

f
µ x,

−−−−−→
J
f
µ xJ

f
λ
x〉 ≥ 0.(3.8)

Adding (3.7) and (3.8), and by the monotonicity of f, we obtain that

〈
−−−→
J
f
λ
xx,

−−−−−→
Jfµ xJ

f
λ
x〉 ≥

λ

µ
〈
−−−→
Jfµ xx,

−−−−−→
Jfµ xJ

f
λ
x〉.

By the definition of quasilinearization, we obtain that
(

λ

µ
+ 1

)

d2(Jfµ x, J
f
λ
x) ≤

(

1 −
λ

µ

)

d2(x, Jµx) +

(

λ

µ
− 1

)

d2(x, J
f
λ
x).

Since λ
µ
≤ 1, we obtain that

(

λ

µ
+ 1

)

d
2(Jfµ x, J

f
λ
x) ≤

(

1 −
λ

µ

)

d
2(x, Jfµ x),

which implies

d(Jfµ x, J
f
λ
x) ≤

√

1 −
λ

µ
d(x, Jfµ x).(3.9)

Furthermore, by triangle inequality and (3.9), we obtain

d(x, J
f
λ
x) ≤ 2d(x, Jfµ x).

�

Remark 3.4. We note here that, if the bifunction f satisfies assumption (i)-(iv)
of Theorem 2.17, the conclusions of Lemma 3.3 hold in the whole space X.

Lemma 3.5. Let C be a nonempty, closed and convex subset of a Hadamard
space X and T be a nonexpansive mapping on C. Let fi : C × C → R, i =

1, 2, . . . , N be a finite family of monotone bifunctions such that C ⊂ D(J
fi
λ
)

for λ > 0. Then, for βi ∈ (0, 1) with
∑N

i=0 βi = 1, the mapping Sλ : C → C

defined by Sλx := β0x ⊕ β1J
f1
λ
x ⊕ β2J

f2
λ
x ⊕ · · · ⊕ βNJ

fN
λ

x for all x ∈ C, is

nonexpansive and F(T ◦ Sλ2) ⊆ ∩
N
i=1F(J

fi
λ1
) ∩ F(T ) for 0 < λ1 ≤ λ2, where

Sλ2 : C → C is defined by Sλ2x := β0x ⊕ β1J
f1
λ2
x ⊕ β2J

f2
λ2
x ⊕ · · · ⊕ βNJ

fN
λ2

x for
all x ∈ C.

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 201



C. Izuchukwu, K. O. Aremu, A. A. Mebawondu and O. T. Mewomo

Proof. Since f is monotone, it follows from Lemma 3.3 (i) (or Lemma 2.16 (ii))

that J
fi
λ

is nonexpansive for λ > 0, i = 1, 2, . . . , N. Thus, by Lemma 3.1, we
obtain

d(Sλx, Sλy) ≤ β0d(x, y) + β1d(J
f1
λ
x, J

f1
λ
y) + · · · + βNd(J

fN
λ

x, J
fN
λ

y)

≤

N
∑

i=0

βid(x, y)

= d(x, y).

Hence, Sλ is nonexpansive.

Now, let x ∈ F(T ◦ Sλ2) and v ∈ ∩
N
i=1F(J

fi
λ2
) ∩ F(T ). Then, by Lemma 3.1,

we obtain

d(x, v) ≤ d(Sλ2x, v)

≤ β0d(x, v) + β1d(J
f1
λ2
x, v) + · · · + βNd(J

fN
λ2

x, v)

≤

N−1
∑

i=0

βid(x, v) + βNd(J
fN
λ2

x, v)(3.10)

≤ d(x, v).

From (3.10), we obtain that

d(x, v) =

N−1
∑

i=0

βid(x, v) + βNd(J
fN
λ2

x, v) = (1 − βN)d(x, v) + βNd(J
fN
λ2

x, v),

which implies that d(x, v) = d(J
fN
λ2

x, v) = d(Sλ2x, v) = d(β0x ⊕ β1J
f1
λ2
x ⊕

β2J
f2
λ2
x ⊕ · · · ⊕ βNJ

fN
λ2

x, v). Similarly, we obtain

d(x, v) = d(J
fN−1
λ2

x, v) = · · · = d(J
f2
λ2
x, v) = d(J

f1
λ2
x, v).

Thus,
(3.11)

d(x, v) = d(J
fN
λ2

x, v) = · · · = d(J
f1
λ2
x, v) = d(β0x⊕β1J

f1
λ2
x⊕β2J

f2
λ2
x⊕· · ·⊕βNJ

fN
λ2

x, v).

Now, let d(x, v) = c. If c > 0, and there exist ǫ > 0 and i, j ∈ {0, 1, 2, . . ., N},

i 6= j such that d(J
fi
λ2
x, J

fj
λ2
x) ≥ ǫc (where J

f0
λ2

= I), then by Lemma 2.5, we
obtain that

d(β0x ⊕ β1J
f1
λ2
x ⊕ β2J

f2
λ2
x ⊕ · · · ⊕ βNJ

fN
λ2

x, v) < c = d(x, v),

and this contradicts (3.11). Hence, c = 0. This implies that x = v, hence

x ∈ ∩Ni=1F(J
fi
λ2
) ∩ F(T ).(3.12)

Since 0 < λ1 ≤ λ2, we obtain from Lemma 3.3 (iii) and (3.12) that

d(x, J
fi
λ1
x) ≤ 2d(x, J

fi
λ2
x) = 0, i = 1, 2, . . . , N.

Hence, x ∈ ∩Ni=1F(J
fi
λ1
) ∩ F(T ). Therefore, we conclude that F(T ◦ Sλ2) ⊆

∩Ni=1F(J
fi
λ1
) ∩ F(T ). �

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 202



A viscosity iterative technique for equilibrium and fixed point problems in a Hadamard space

We now present our strong convergence theorem.

Theorem 3.6. Let C be a nonempty closed and convex subset of a Hadamard
space X and fi : C × C → R, i = 1, 2, . . . , N be a finite family of monotone

and upper semicontinuous bifunctions such that C ⊂ D(J
fi
λ
) for λ > 0. Let

T : C → C be a nonexpansive mapping and g : C → C be a contraction mapping
with coefficient τ ∈ (0, 1). Suppose that Γ := ∩Ni=1EP(fi, C) ∩ F(T ) 6= ∅ and
for arbitrary x1 ∈ C, the sequence {xn} is generated by

(3.13)

{

yn = Sλnxn := β0xn ⊕ β1J
f1
λn

xn ⊕ β2J
f2
λn

xn ⊕ · · · ⊕ βNJ
fN
λn

xn,

xn+1 = αng(xn) ⊕ βnxn ⊕ γnT yn, n ≥ 1,

where {αn}, {βn} and {γn} are sequences in (0, 1), and {λn} is a sequence of
positive real numbers satisfying the following conditions:

(i) lim
n→∞

αn = 0 and
∑∞

n=1 αn = ∞,

(ii) 0 < lim inf
n→∞

βn ≤ lim sup
n→∞

βn < 1, αn + βn + γn = 1 ∀n ≥ 1,

(iii) 0 < λ ≤ λn ∀n ≥ 1 and lim
n→∞

λn = λ,

(iv) βi ∈ (0, 1) with
∑N

i=0 βi = 1.

Then, {xn} converges strongly to z̄ ∈ Γ.

Proof. Step 1: We show that {xn} is bounded. Let u ∈ Γ, then by Lemma
2.8, we obtain that

d(xn+1, u) ≤ αnd(g(xn), u) + βnd(xn, u) + γnd(T yn, u)

≤ αnτd(xn, u) + αnd(g(u), u) + βnd(xn, u) + γnd(yn, u)

≤ αnτd(xn, u) + (αn + βn)d(xn, u) + αnd(g(u), u)

= (1 − αn(1 − τ))d(xn, u) + αnd(g(u), u)

≤ max

{

d(xn, u) +
d(g(u), u)

1 − τ

}

...

≤ max

{

d(x1, u) +
d(g(u), u)

1 − τ

}

.

Hence, {xn} is bounded. Consequently, {yn}, {g(xn)} and {T (yn)} are all
bounded.
Step 2: We show that lim

n→∞
d(xn+1, xn) = 0. Observe from Remark 2.9, that

(3.13) can be rewritten as











yn = Sλn xn := β0xn ⊕ β1J
f1
λn

xn ⊕ · · · ⊕ βNJ
fN
λn

xn,

wn =
αn

1−βn
g(xn) ⊕

γn
1−βn

T yn,

xn+1 = βnxn ⊕ (1 − βn)wn, n ≥ 1.

(3.14)

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 203



C. Izuchukwu, K. O. Aremu, A. A. Mebawondu and O. T. Mewomo

Now, from (3.14), Lemma 2.6 (iv),(vi) and the nonexpansivity of T, we obtain
that

d(wn+1, wn) = d
(

αn+1

1 − βn+1
g(xn+1) ⊕

γn+1

1 − βn+1
T yn+1,

αn

1 − βn
g(xn) ⊕

γn

1 − βn
T yn

)

≤d
(

αn+1

1 − βn+1
g(xn+1) ⊕

(

1 −
αn+1

1 − βn+1

)

T yn+1,
αn+1

1 − βn+1
g(xn) ⊕

(

1 −
αn+1

1 − βn+1

)

T yn

)

+d
(

αn+1

1 − βn+1
g(xn) ⊕

(

1 −
αn+1

1 − βn+1

)

T yn,
αn

1 − βn
g(xn) ⊕

(

1 −
αn

1 − βn

)

T yn

)

≤
αn+1

1 − βn+1
τd(xn+1, xn) +

(

1 −
αn+1

1 − βn+1

)

d(yn+1, yn) + |
αn+1

1 − βn+1
−

αn

1 − βn
|d(g(xn), T yn)

(3.15)

Without loss of generality, we may assume that 0 < λn+1 ≤ λn ∀n ≥ 1. Thus,
from (3.14), condition (iv), Lemma 3.1 and Lemma 3.3 (iii), we obtain

d(yn+1, yn) =d
(

β0xn+1 ⊕ β1J
f1
λn+1

xn+1 ⊕ · · · ⊕ βNJ
fN
λn+1

xn+1, β0xn ⊕ β1J
f1
λn

xn ⊕ · · · ⊕ βNJ
fN
λn

xn
)

≤β0d(xn+1, xn) +

N
∑

i=1

βid(J
fi
λn+1

xn+1, J
fi
λn

xn)

≤β0d(xn+1, xn) +

N
∑

i=1

βid(J
fi
λn+1

xn+1, J
fi
λn+1

xn) +

N
∑

i=1

βid(J
fi
λn+1

xn, J
fi
λn

xn)

≤d(xn+1, xn) +
(

√

1 −
λn+1

λn

)

N
∑

i=1

βid(J
fi
λn+1

xn, xn)

≤d(xn+1, xn) +
(

√

1 −
λn+1

λn

)

M̄,

(3.16)

where M̄ := sup
n≥1

{
∑N

i=1 βid(J
fi
λn+1

xn, xn)
}

. Substituting (3.16) into (3.15), we

obtain that

d(wn+1, wn) ≤
αn+1

1 − βn+1
τd(xn+1, xn) +

(

1 −
αn+1

1 − βn+1

)

d(xn+1, xn)

+
(

√

1 −
λn+1

λn

)(

1 −
αn+1

1 − βn+1

)

M +
∣

∣

∣

αn+1

1 − βn+1
−

αn

1 − βn

∣

∣

∣
d(g(xn), T yn)

=
[

1 −
αn+1

1 − βn+1
(1 − τ)

]

d(xn+1, xn) +
(

√

1 −
λn+1

λn

)(

1 −
αn+1

1 − βn+1

)

M

+
∣

∣

∣

αn+1

1 − βn+1
−

αn

1 − βn

∣

∣

∣
d(g(xn), T yn).

Since lim
n→∞

αn = 0, lim
n→∞

λn = λ and {g(xn)}, {T yn} are bounded, we obtain

that

lim sup
n→∞

(

d(wn+1, wn) − d(xn+1, xn)
)

≤ 0.

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 204



A viscosity iterative technique for equilibrium and fixed point problems in a Hadamard space

Thus, by Lemma 2.18 and condition (ii), we obtain that

lim
n→∞

d(wn, xn) = 0.(3.17)

Hence, by Lemma 2.6 we obtain that

d(xn+1, xn) ≤ (1 − βn)d(wn, xn) → 0, as n → ∞.(3.18)

Step 3: We show that lim
n→∞

d(xn, T (Sλn)xn) = 0 = lim
n→∞

d(wn, T (Sλn)wn).

Notice also that (3.13) can be rewritten as

xn+1 = αng(xn) ⊕ (1 − αn)
(

βnxn ⊕ γnT yn
(1 − αn)

)

, yn = Sλn xn.

Thus, by Lemma 2.6, we obtain that
(3.19)

d
(

xn+1,
βnxn ⊕ γnT yn

(1 − αn)

)

≤ αnd
(

g(xn),
βnxn ⊕ γnT yn

(1 − αn)

)

→ 0, as n → ∞.

Also, from (2.1), we obtain

d
(

xn,
βnxn ⊕ γnT yn

(1 − αn)

)

=
γn

1 − αn
d(xn, T yn),

which implies from (3.18) and (3.19) that

γn

1 − αn
d(xn, T yn) ≤ d(xn, xn+1)+d

(

xn+1,
βnxn ⊕ γnT yn

(1 − αn)

)

→ 0, as n → ∞.

Hence,

lim
n→∞

d(xn, T yn) = lim
n→∞

d(xn, T (Sλn)xn) = 0.(3.20)

Now, since {xn} is bounded and X is a complete CAT(0) space, then from
Lemma 2.11, there exists a subsequence {xnk } of {xn} such that ∆− lim

k→∞
xnk =

z̄. Again, since T ◦ Sλn is nonexpansive (and every nonexpansive mapping is
demiclosed), it follows from (3.20), condition (iii), Lemma 3.5 and Lemma 2.16

(iii) that z̄ ∈ F(T ◦ Sλn) ⊆ ∩
N
i=1F(J

fi
λ
) ∩ F(T ) = Γ.

d(wn, T (Sλn)wn) ≤ d(wn, xn) + d(xn, T (Sλn)xn) + d(T (Sλn)xn, T (Sλn)wn)

≤ 2d(wn, xn) + d(xn, T (Sλnxn) → 0, as n → ∞.(3.21)

Step 4: We show that lim sup
n→∞

〈
−−−→
g(z̄)z̄,

−−→
xnz̄〉 ≤ 0.

Now, define Tnx = βnx ⊕ (1 − βn)w, where w =
αn

(1−βn)
g(x) ⊕ γn

(1−βn)
T (Sλn)x,

then Tn is a contractive mapping for each n ≥ 1. Thus, there exists a unique
fixed point zn of Tn ∀n ≥ 1. That is,
zm = βmzm ⊕ (1 − βm)wm, where wm =

αm
(1−βm)

g(zm) ⊕
γm

(1−βm)
T (Sλm)zm.

Moreover, lim
m→∞

zm = z ∈ Γ (see [31]).

Thus, we obtain that

d(zm, wn) = d(βmzm ⊕ (1 − βm)wm, wn)

≤ βmd(zm, wn) + (1 − βm)d(wm, wn),

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 205



C. Izuchukwu, K. O. Aremu, A. A. Mebawondu and O. T. Mewomo

which implies that

d(zm, wn) ≤ d(wm, wn).(3.22)

From (3.22) and Lemma 2.6(v), we obtain that

d2(wm, wn) = 〈
−−−−→
wmwn,

−−−−→
wmwn〉

= 〈
−−−−−−−−−−→
wmT (Sλm)zm,

−−−−→
wmwn〉 + 〈

−−−−−−−−−→
T (Sλm)zmwn,

−−−−→
wmwn〉

≤
αm

(1 − βm)
〈
−−−−−−−−−−−→
g(zm)T (Sλm)zm,

−−−−→
wmwn〉 + 〈

−−−−−−−−−→
T (Sλmzm)wn,

−−−−→
wmwn〉

=
αm

(1 − βm)
〈
−−−−−−−−−−−→
g(zm)T (Sλmzm),

−−−−→
wmzm〉 +

αm

(1 − βm)
〈
−−−−−→
g(zm)wn,

−−−→
zmwn〉

+
αm

(1 − βm)
〈
−−−−−−−−−→
wnT (Sλmzm),

−−−−→
zmwm〉 + 〈

−−−−−−−−−−−−−−−→
T (Sλmzm)T (Sλmwn),

−−−−→
wmwm〉

+ 〈
−−−−−−−→
T (Sλmwm),

−−−−→
wmwn〉

≤
αm

(1 − βm)
d(g(zm), T (Sλmzm))d(wm, zm) +

αm

(1 − βm)
〈
−−−−−→
g(zm)zm,

−−−→
zmwn〉

+
αm

(1 − βm)
〈
−−−−−−−−−→
zmT (Sλmzm),

−−−→
zmwn〉 + d(T (Sλmzm), T (Sλmwn))d(wm, wn)

+ d(T (Sλmwn), wn)d(wm, wn)

≤
αm

(1 − βm)
d(g(zm), T (Sλmzm))d(wn, zm) +

αm

(1 − βm)
〈
−−−−−→
g(zm)zm,

−−−→
zmwn〉

+
αm

(1 − βm)
〈
−−−−−−−−−→
zmT (Sλmzm),

−−−→
zmwn〉 + d(zm, wm)d(wm, wn) + d(T (Sλmwn), wn)d(wn, wm)

≤
αm

(1 − βm)
d(g(zm), T (Sλmzm))d(wn, zm) +

αm

(1 − βm)
〈
−−−−−→
g(zm)zm,

−−−→
zmwn〉

+
αm

(1 − βm)
d(zm, T (Sλmzm))d(wm, zm) + d(wm, wn) + d(T (Sλmwn), wn)d(wn, wm),

which implies that

〈
−−−−−→
g(zm)zm,

−−−→
wnzm〉 ≤ d(g(zm), T (Sλm)zm)d(wn, zm) + d(zm, T (Sλm)zm)d(zm, wm)

+
(1 − βm)

αm
d(T (Sλn)wn, wn)d(wm, wm).

Thus, taking lim sup as n → ∞ first, then as m → ∞, it follows from (3.17),(3.20)
and (3.21) that

lim sup
m→∞

lim sup
n→∞

〈
−−−−−→
g(zm)zm,

−−−→
wnzm〉 ≤ 0.(3.23)

Furthermore,

〈
−−−→
g(z)z̄,

−−→
xnz̄〉 = 〈

−−−−−−→
g(z̄)g(zm),

−−→
xnz〉 + 〈

−−−−−→
g(zm)zm,

−−−→
xnwn〉 + 〈

−−−−−→
g(zm)zm,

−−−→
wnzm〉 + 〈

−−−−−→
g(zm)zm,

−−→
zmz〉 + 〈

−−→
zmz̄,

−−→
xnz̄〉

≤ d(g(z), g(zm))d(xn, z) + d(g(zm), zm)d(xn, wn) + 〈
−−−−−→
g(zm)zm,

−−−→
wnzm〉

+ d(g(zm), zm)d(zm, z) + d(zm, z̄)d(xn, z̄)

≤ (1 + τ)d(zm, z)d(xn, z̄) + 〈
−−−−−→
g(zm)zm,

−−−→
wnzm〉 + [d(xn, wn) + d(zm, z)]d(g(zm), zm),

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 206



A viscosity iterative technique for equilibrium and fixed point problems in a Hadamard space

which implies from (3.17), (3.23) and the fact that lim
m→∞

zm = z, that

lim sup
n→∞

〈
−−−→
g(z)z,

−−→
xnz̄〉 = lim sup

m→∞
lim sup
n→∞

〈
−−−→
g(z̄)z̄,

−−→
xnz̄〉

≤ lim sup
m→∞

lim sup
n→∞

〈
−−−−−→
g(zm)zm,

−−−→
wnzm〉 ≤ 0.(3.24)

Step 5: Lastly, we show that {xn} converges strongly to z ∈ Γ.
From Lemma 2.7, we obtain that

〈
−−→
wnz,

−−→
xnz̄〉 ≤

αn

(1 − βn)
〈
−−−−→
g(xn)z,

−−→
xnz〉 +

γn

(1 − βn)
〈
−−−−−−−→
T (Sλn)xnz,

−−→
xnz〉

≤
αn

(1 − βn)
〈
−−−−−−→
g(xn)g(z),

−−→
xnz〉 +

αn

(1 − βn)
〈
−−−→
g(z)z,

−−→
xnz〉 +

γn

(1 − βn)
d(T (Sλn)xn, z)d(xn, z)

≤
αn

(1 − βn)
τd

2(xn, z) +
αn

(1 − βn)
〈
−−−→
g(z)z,

−−→
xnz〉 + (1 −

αn

1 − βn
)d2(xn, z)

=

[

αn

(1 − βn)
τ + (1 −

αn

1 − βn
)

]

d2(xn, z) +
αn

(1 − βn)
〈
−−−→
g(z)z,

−−→
xnz〉.

Thus, from Lemma 2.6, we have

d2(xn+1, z) ≤ βnd
2(xn, z) + (1 − βn)d

2(wn, z)

= βnd
2(xn, z) + (1 − βn)〈

−−→
wnz,

−−→
wnz〉

= βnd
2(xn, z) + (1 − βn)[〈

−−→
wnz,

−−−→
wnxn〉 + 〈

−−→
wnz,

−−→
xnz〉]

≤ [βn + αnτ + γn]d
2(xn, z) + (1 − βn)〈

−−→
wnz,

−−−→
wnxn〉 + αn〈g(z)z, xnz〉

≤ (1 − αn(1 − τ))d
2(xn, z̄) + αn(1 − τ)

[

1

1 − τ
〈
−−−→
g(z̄)z̄,

−−→
xnz̄〉

]

+ (1 − βn)d(wn, xn)M.

(3.25)

By (3.17) and applying Lemma 2.19 to (3.25), we obtain that {xn} converges
strongly to z̄. �

Corollary 3.7. Let C be a nonempty closed and convex subset of a Hadamard
space X and fi : C × C → R, i = 1, 2, . . . , N be a finite family of monotone

and upper semicontinuous bifunctions such that C ⊂ D(J
fi
λ
) for λ > 0. Let

g : C → C be a contraction mapping with coefficient τ ∈ (0, 1). Suppose
that Γ := ∩Ni=1EP(fi, C) 6= ∅ and for arbitrary x1 ∈ C, the sequence {xn} is
generated by

(3.26)

{

yn = Sλnxn := β0xn ⊕ β1J
f1
λn

xn ⊕ β2J
f2
λn

xn ⊕ · · · ⊕ βNJ
fN
λn

xn,

xn+1 = αng(xn) ⊕ βnxn ⊕ γnyn, n ≥ 1,

where {αn}, {βn} and {γn} are sequences in (0, 1), and {λn} is a sequence of
positive real numbers satisfying the following conditions:

(i) lim
n→∞

αn = 0 and
∑∞

n=1 αn = ∞,

(ii) 0 < lim inf
n→∞

βn ≤ lim sup
n→∞

βn < 1, αn + βn + γn = 1 ∀n ≥ 1,

(iii) 0 < λ ≤ λn ∀n ≥ 1 and lim
n→∞

λn = λ,

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 207



C. Izuchukwu, K. O. Aremu, A. A. Mebawondu and O. T. Mewomo

(iv) βi ∈ (0, 1) with
∑N

i=0 βi = 1.

Then, {xn} converges strongly to z̄ ∈ Γ.

Corollary 3.8. Let C be a nonempty closed and convex subset of a Hadamard
space X and f : C×C → R be a monotone and upper semicontinuous bifunction

such that C ⊂ D(J
f
λ
) for λ > 0. Let T : C → C be a nonexpansive mapping

and g : C → C be a contraction mapping with coefficient τ ∈ (0, 1). Suppose
that Γ := EP(f, C) ∩ F(T ) 6= ∅ and for arbitrary x1 ∈ C, the sequence {xn}
is generated by

{

yn = J
f
λn

xn,

xn+1 = αng(xn) ⊕ βnxn ⊕ γnT yn, n ≥ 1,
(3.27)

where {αn}, {βn} and {γn} are sequences in (0, 1), and {λn} is a sequence of
positive real numbers satisfying the following conditions:

(i) lim
n→∞

αn = 0 and
∑∞

n=1 αn = ∞,

(ii) 0 < lim inf
n→∞

βn ≤ lim sup
n→∞

βn < 1, αn + βn + γn = 1 ∀n ≥ 1,

(iii) 0 < λ ≤ λn ∀n ≥ 1 and lim
n→∞

λn = λ.

Then, {xn} converges strongly to z̄ ∈ Γ.

4. Application to optimization problems

In this section, we give application of our results to solve some optimiza-
tion problems. Throughout this section, X is a Hadamard space and C is
a nonempty closed and convex subset of X.

4.1. Minimization problem. Let h : X → R be a proper convex and lower
semicontimnuous function. Consider the bifunction fh : C × C → R defined by

fh(x, y) = h(y) − h(x), ∀x, y ∈ C.

Then, fh is monotone and upper semicontinuous (see [22]). Moreover, EP(C, fh) =
arg minC h, J

fh = proxh and D(proxh) = X (see [22]). Now, consider the fol-
lowing finite family of minimization problem and fixed point problem:

Find x ∈ F(T ) such that hi(x) ≤ hi(y), ∀y ∈ C, i = 1, 2 . . . , N,(4.1)

where T is a nonexpansive mapping. Thus, by setting J
fi
λn

= proxhi
λn

in Algo-

rithm (3.13), we can apply Theorem 3.6 to approximate solutions of problem
(4.1).

4.2. Variational inequality problem. Let S : C → C be a nonexpansive

mapping. Now define the bifunction fS : C × C → R by fS(x, y) = 〈
−−→
Sxx,

−→
xy〉.

Then, fS is monotone and J
fS = JS (see [20, 3]). Consider the following finite

family of variational inequality and fixed point problems:

Find x ∈ F(T ) such that 〈
−−→
Sixx,

−→
xy〉 ≥ 0, ∀y ∈ C, i = 1, 2 . . . , N,(4.2)

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 208



A viscosity iterative technique for equilibrium and fixed point problems in a Hadamard space

where T is a nonexpansive mapping on C. Thus, by setting J
fi
λn

= JSi
λn

in Algo-

rithm (3.13), we can apply Theorem 3.6 to approximate solutions of problem
(4.2).

4.3. Convex feasibility problem. Let Ci, i = 1, 2, . . . , N be a finite family
of nonempty closed and convex subsets of C such that ∩Ni=1Ci 6= ∅. Now,
consider the following convex feasibility problem:

Find x ∈ F(T ) such that x ∈ ∩Ni=1Ci.(4.3)

We know that the indicator function δC : X → R defined by

δC(x) =

{

0, if x ∈ C,

+∞, otherwise

is a proper convex and lower semicontinuous function. By letting δC = h and
following similar argument as in Subsection 4.1, we obtain that fδC is monotone
and upper semicontinuous, and JfδC = proxδC = PC. Therefore, by setting
Jfi = PCi, i = 1, 2, . . . , N in Algorithm (3.13), we can apply Theorem 3.6 to
approximate solutions of (4.3).

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