International Journal of Analysis and Applications
ISSN 2291-8639
Volume 10, Number 1 (2016), 48-57
http://www.etamaths.com

CONVERGENCE THEOREM FOR GENERALIZED MIXED EQUILIBRIUM

PROBLEM AND COMMON FIXED POINT PROBLEM FOR A FAMILY OF

MULTIVALUED MAPPINGS

J. N. EZEORA∗

Abstract. In this paper, a new hybrid iterative algorithm is constructed using the shrinking projec-

tion method introduced by Takahashi. The sequence of the algorithm is proved to converge strongly
to a common element of the set of solutions of generalized mixed equilibrium problem and the set

of common fixed points of a finite family of multivalued strictly pseudocontractive mappings in real

Hilbert spaces. Furthermore, we apply our main result to convex minimization problem.

1. Introduction

Let H be a real Hilbert space with inner product 〈., .〉 and norm || · ||, and let K be a nonempty closed
convex subset of H. Let B : K → H be a nonlinear mapping, ϕ : K → R ∪{+∞} be a function and
F : K×K → R, be a bifunction where R is the set of real numbers. The generalized mixed equilibrium
problem is defined as follows:

(1.1) find x ∈ K : F(x,y) + ϕ(y) −ϕ(x) + 〈Bx,y −x〉≥ 0 ∀ y ∈ K.

The set of solutions of (1.1) is denoted by GMEP(F,ϕ,B).
If B = 0, problem (1.1) reduces to the following mixed equilibrium problem:

(1.2) find x ∈ K : F(x,y) + ϕ(y) −ϕ(x) ≥ 0 ∀ y ∈ C.

The set of solutions of (1.2) is denoted by MEP(F,ϕ).
If ϕ = 0, problem (1.1) becomes the following generalized equilibrium problem:

(1.3) find x ∈ K : F(x,y) + 〈Bx,y −x〉≥ 0 ∀ y ∈ C.

The set of solutions of (1.3) is denoted by GEP(F,B).
If ϕ = 0 and B = 0, problem (1.1) becomes the following equilibrium problem:

(1.4) Find x ∈ K : F(x,y) ≥ 0 ∀ y ∈ K.

The set of solutions of (1.4) is denoted by EP(F).
If F(x,y) = 0 for all x,y ∈ K, problem (1.1) becomes the following generalized variational inequality
problem:

(1.5) Find x ∈ K : ϕ(y) −ϕ(x) + 〈Bx,y −x〉≥ 0 ∀ y ∈ K.

If ϕ = 0 and F(x,y) = 0 for all x,y ∈ K, problem (1.1) becomes the following variational inequality
problem:

(1.6) Find x ∈ K : 〈Bx,y −x〉≥ 0 ∀ y ∈ K.

If B = 0 and F(x,y) = 0 for all x,y ∈ K, problem (1.1) becomes the following convex minimization
problem:

(1.7) Find x ∈ C : ϕ(y) ≥ ϕ(x) ∀ y ∈ K.

2010 Mathematics Subject Classification. 47H04, 47H06, 47H15, 47H17, 47J25.
Key words and phrases. equilibrium problem; strict pseudo-contractive mappings; multivalued mappings; Hausdorff

metric.

c©2016 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

48



GENERALIZED MIXED EQUILIBRIUM PROBLEM AND COMMON FIXED POINT PROBLEM 49

Let X be a normed space. A subset K of X is called proximinal (see [36]) if for each x ∈ X, there
exists an element k ∈ K such that

d(x,k) = d(x,K),

where d(x,K) = inf{||x−y|| ∀ y ∈ K} is the distance from the point x to the set K.
Let K be a nonempty closed convex subset of X. We denote by CB(K), the family of nonempty closed,
bounded subsets of K, P(K) the family of nonempty proximinal bounded subsets of K. The Hausdorff
metric (see [29]) on CB(X) is defined by

D(A,B) = max
{

sup
x∈A

d(x,B), sup
y∈B

d(y,A)
}
∀A,B ∈ CB(X).

Let T : D(T) ⊂ X → CB(X). An element x ∈ D(T) is called a fixed point of T if x ∈ Tx. The set of
fixed points of T is denoted by F(T).
For multivalued mappings T : K → P(K), the best approximation operator, PT (see [21]) is defined
by PT (x) := {y ∈ T(x) : ||x−y|| = d(x,T(x))} ∀x ∈ K.

A multi-valued mapping T : D(T) ⊆ X → CB(X) is called L-Lipschitzian if there exists L > 0
such that

(1.8) D(Tx,Ty) ≤ L||x−y|| ∀ x,y ∈ D(T).
When L ∈ (0, 1) in (1.8), we say that T is a contraction, and T is called nonexpansive if L = 1.

A multi-valued map T : D(T) ⊂ H → CB(H) is called k-strictly pseudo-contractive (see [10]) if
there exists k ∈ (0, 1) such that for all x,y ∈ D(T),
(1.9) (D(Tx,Ty))2 ≤ ||x−y||2 + k||x−y − (u−v)||2 ∀ u ∈ Tx,v ∈ Ty.
It is well known that the generalized mixed equilibrium problem and indeed equilibrium problem in-
clude variational inequality problem, optimization problem, problems of Nash equilibria, saddle point
problems, fixed point problems and complementarity problems as special cases.( see [5, 19, 20, 28, 33]
and the references therein).
Different iterative algorithms for solving generalized mixed equilibrium problems, mixed equilibrium
problems and equilibrium problems have been developed and studied by many authors. See for in-
stance, [6, 7, 12, 14, 17, 26, 34, 37, 39] and the references therein.
For several years, the study of fixed point theory for multi-valued nonlinear mappings has attracted
the interest of several well known mathematicians (see, for example, Brouwer [4], Chidume et al. [10],
Denavari [13], Kakutani [22], Nash [31, 32] Geanakoplos [18], Nadler [29], Downing and Kirk[15]).
Interest in such studies stems, perhaps, mainly from the usefulness of such fixed point theory in real-
world applications, such as in Game Theory, Market Economy, Non-Smooth Differential Equations
and so on (see e.g., [10, 16]). Game theory is perhaps the most successful area of application of fixed
point theory for multi-valued mappings. However, it has been remarked that the applications of this
theory to equilibrium problems in game theory are mostly static in the sense that while they enhance
the understanding of conditions under which equilibrium may be achieved, they do not indicate how to
construct a process starting from a non-equilibrium point that will converge to an equilibrium solution.
Iterative methods for fixed points of multivalued mappings are designed to address this problem. For
more detail, see [8, 10, 16].

The classical Mann iteration process has been employed successfully to approximate fixed points of
nonlinear mappings (single valued or multi valued).
However, it is known to yield only weak convergence even in Hilbert spaces. To overcome this weak-
ness, Takahashi [40], introduced a method known as the shrinking projection method, and obtained
strong convergence results of the method.

The shrinking projection method has been studied extensively in the literature. See for instance,
Tada and Takahashi [39], Aoyama et al. [2], Yao et al. [41], Kang et. al. [23], Kimura et. al. [25],
Cholamjiak and Suantai [42] and the references contained therein.



50 EZEORA

Motivated by the result of Takahashi [40], Bunyawat and Suantai [6] used the shrinking projection
method and defined a hybrid method for Mixed equilibrium problem and fixed point problem for a
family of nonexpansive multivalued mappings in real Hilbert spaces. Precisely, they proved the follow-
ing result:
Theorem S (Bunyawat and Suantai [6]).
Let D be a nonempty closed and convex subset of a real Hilbert space H. Let F be a bifunction from
D×D to R satisfying (A1) − (A4), and let ϕ be a proper, lower semi continuous and convex function
from D to R∪{+∞} such that D∩domϕ 6= ∅. Let Ti : D → P(D) be multivalued nonexpansive map-
pings for all i ∈ N with Ω :=

⋂∞
i=1 F(Ti)∩MEP(F,ϕ) 6= ∅ such that all PTi are nonexpansive. Assume

that either (B1) or (B2) holds and {αn,i} ⊂ (0, 1) satisfies the condition lim infn→∞αn,iαn,0 > 0 for
all i ∈ N. Define {xn} as follows: x1 ∈ D = C1,

(1.10)




F(un,y) + ϕ(y) −ϕ(un) + 1rn〈y −un,un −xn〉≥ 0 ∀ y ∈ D,
yn = αn,0un +

∑n
i=1 αn,ixn,i, xn,i ∈ PTi (un),

Cn+1 = {z ∈ Cn : ||yn −z|| ≤ ||xn −z||},
xn+1 = PCn+1x0, n ≥ 0.

where PCn is the metric projection of H onto Cn. They proved that {xn} converges strongly to
PΩx0, Ω = ∩∞i=1F(Ti) ∩MEP(F,ϕ).

The class of strictly pseudocontractive mappings was introduced in 1967 by Browder and Petryshyn [3]
as a generalization of the class of nonexpansive mappings. This class of operators have been studied by
several authors under different assumptions. See for instance, [1, 9, 27, 35, 38, 43] and the references
therein.
In 2013, Chidume et. al. [10] introduced and studied the class of multivalued strictly pseudocontrac-
tive mappings as a generalization of the class of multi valued nonexpansive mappings in real Hilbert
spaces.
Recently, Chidume and Ezeora [11] introduced a Krasnoselskii-type sequence and proved that the se-
quence converges strongly to a common fixed point of a finite family of multivalued strictly pseudo
contractive mappings in real Hilbert spaces under some compactness assumption on the operators.

Motivated by the results of Takahashi [40], Bunyawat and Suantai [6], Chidume and Ezeora [11],
it is our purpose in this paper to introduce a new hybrid iterative algorithm based on the shrinking
projection method and prove that the sequence of the scheme converges strongly to a common element
of the set of solution of generalized mixed equilibrium problem and the set of common fixed points of
a finite family of multivalued strictly pseudocontractive mappings in real Hilbert spaces. Our result
extends that of Bunyawat and Suantai [6] from multivalued nonexpansive mappings to the more gen-
eral class of multivalued strictly pseudocontractive mappings and many other important results. In
proving our result, compactness assumption imposed on the operators by Chidume and Ezeora [11]
was dispensed with.

2. preliminaries

Lemma 2.1. [30] Let K be a nonempty closed convex subset of a real Hilbert space H and PK : H → K
be the metric projection from H onto K. Then the following inequality holds:
‖y −PKx‖

2
+ ‖x−PKx‖

2 ≤‖x−y‖2,∀ x ∈ H,y ∈ K.

Lemma 2.2. (see [11]) Let H be a real Hilbert space and {xi}mi=1 ⊂ H. For αi ∈ (0, 1), i = 1, . . . ,m
such that

∑m
i=1 αi = 1, the following identity holds:∣∣∣∣∣

∣∣∣∣∣
m∑
i=1

αixi

∣∣∣∣∣
∣∣∣∣∣
2

=

m∑
i=1

αi ||xi||
2 −

m∑
i,j=1,i6=j

αiαj ||xi −xj||
2

(2.1)

Lemma 2.3. (see [10]) Let X be a reflexive real Banach space and let A,B ∈ CB(X). Assume that
B is weakly closed. Then, for every a ∈ A, there exists b ∈ B such that
(2.1) ||a− b|| ≤ D(A,B)



GENERALIZED MIXED EQUILIBRIUM PROBLEM AND COMMON FIXED POINT PROBLEM 51

Lemma 2.4. (see [10]) Let K be a nonempty subset of a real Hilbert space H and let T : K → CB(K)
be a multivalued k-strictly pseudocontractive mapping. Assume that for every x ∈ K, the set Tx is
weakly closed. Then, T is Lipschitzian. That is

(2.2) D(Tx,Ty) ≤ L||x−y|| ∀ x,y ∈ K.

Lemma 2.5. [24] Let D be a nonempty closed and convex subset of a real Hilbert space H.
Given x,y,z ∈ H and also given a ∈ R, the set
{v ∈ D : ‖y −v‖2 ≤‖x−v‖2 + 〈z,v〉 + a}
is convex and closed.

For solving the generalized mixed equilibrium problem, we assume the bifunction F, ϕ and the set K
satisfy the following conditions:
(A1) F(x,x) = 0 for all x ∈ K;
(A2) F is monotone, that is, F(x,y) + F(y,x) ≤ 0 ∀ x,y ∈ K;
(A3) for each y ∈ K,x 7→ F(x,y) is weakly upper semicontinuous
(A4) for each x ∈ K,y 7→ F(x,y) is convex and lower semicontinuous;
(B1) for each x ∈ H and r > 0, there exist a bounded subset Kx ⊆ K and yx ∈ K ∩domϕ such that
for any z ∈ K \Kx,
F(z,yx) + ϕ(yx) + 〈Bz,yx −z〉 +

1

r
〈yx −z,z −x〉 < ϕ(z);

(B2) K is a bounded set.

Lemma 2.6. [26] Let K be a nonempty closed and convex subset of a real Hilbert space H.
Let F : K×K → R be a bifunction satisfying conditions (A1)-(A4) and ϕ : K → R∪{+∞} be a paper
lower semicontinuous and convex function such that K ∩ domϕ 6= ∅. For r > 0 and x ∈ K, define a
mapping Tr : H → K as follows:
Tr(x) = {z ∈ K : F(z,y) + ϕ(y) + 〈Bz,yx −z〉 +

1

r
〈y −z,z −x〉≥ ϕ(z), ∀y ∈ K}

for all x ∈ H. Assume that either (B1)or (B2) holds. Then the following conditions hold:
(1) for all x ∈ H,Tr(x) 6= ∅;
(2) Tr is single- valued;
(3) Tr is firmly nonexpansive, that is, for any x,y ∈ H,
‖Tr(x) −Tr(y)‖

2 ≤〈Tr(x) −Tr(y),x−y〉;
(4) F(Tr(I −rB)) = GMEP(F,ϕ,B);
(5) GMEP(F,ϕ,B) is closed and convex.

3. Main Result

Let K be a nonempty closed convex subset of a real Hilbert space H. In this section, we denote by
CB(K), the family of nonempty, closed,convex and bounded subsets of K.

Theorem 3.1. Let K be a nonempty closed convex subset of a real Hilbert space H, F be a bi-
function from K × K to R satisfying (A1) − (A4), and let ϕ be a proper lower semicontinuous and
convex function from K to R ∪{+∞} such that K ∩ domϕ 6= ∅ and B an α-inverse strongly mono-
tone mapping from K into H. Let Ti : K → CB(K) be multivalued ki-strictly pseudo-contractive
mappings, ki ∈ (0, 1), i = 1, . . . ,m with Ω := ∩mi=1F(Ti) ∩ GMEP(F,ϕ,B) 6= ∅. Assume that for
p ∈

⋂m
i=1 F(Ti), Tip = {p} and that either (B1) or (B2) holds with {αn,i} ⊂ (k, 1), i = 0, 1, · · · ,m.

Define the sequence {xn} as follows: x1 ∈ K = C1,

(3.1)




F(un,y) + ϕ(y) −ϕ(un) + 〈Bxn,y −un〉
+

1

rn
〈y −un,un −xn〉≥ 0,∀ y ∈ K,

yn = αn,0un +
∑m

i=1 αn,ix
i
n, x

i
n ∈ Tiun,

Cn+1 = {z ∈ Cn : ‖yn −z‖≤‖xn −z‖},
xn+1 = PCn+1x0, n ≥ 0,



52 EZEORA

where the sequence rn ∈ (0,∞) with lim inf
n→∞

rn > 0 and
∑m

i=0 αn,i = 1. Then, the sequence {xn} con-
verges strongly to PΩx0.

Proof. We split the proof into steps.

Step 1. We show that PCn+1x0 is well defined for every x0 ∈ K.
By Lemma 2.6 and the condition on CB(K), we obtain that GMEP (F,ϕ,B) and ∩mi=1F(Ti) are closed
and convex subsets of K. Hence Ω is a closed and convex subset of K. From Lemma 2.5, we have
that Cn+1 is closed and convex for each n ≥ 0. Let p ∈ Ω, then Ti(p) = {p}, i = 1, 2, · · · ,m. Since
un = Trn (xn −rnxn), we have using Lemma 2.6 that

‖un −p‖ = ‖Trn (xn −rnBxn) −Trn (p−rnBp)‖≤‖xn −p‖, ∀ n ≥ 0.

Using Lemma 2.2 and Lemma 2.3, we obtain the following estimates:

||yn −p||2 = αn,0||un −p||2 +
m∑
i=1

αn,i||xin −p||
2

−
m∑
i=1

αn,iαn,0||un −xin||
2 −

m∑
i,j=1,i6=j

αn,iαn,j||xin −x
j
n||

2

≤ αn,0||un −p||2 +
m∑
i=1

αn,i||xin −p||
2 −

m∑
i=1

αn,iαn,0||un −xin||
2

≤ αn,0||un −p||2 +
m∑
i=1

αn,i(D(Tiun,Tip))
2 −

m∑
i=1

αn,iαn,0||un −xin||
2

≤ αn,0||un −p||2 +
m∑
i=1

αn,i||un −p||2 +
m∑
i=1

αn,ik||un −xin||
2

−
m∑
i=1

αn,iαn,0||un −xin||
2

= ||un −p||2 −
m∑
i=1

αn,i(αn,0 −k)||un −xin||
2.(3.2)

Since αn,i ∈ (k, 1), we obtain

||yn −p||2 ≤ ||un −p||2 so that
||yn −p|| ≤ ||un −p|| ≤ ||xn −p||.

This implies that

||yn −p|| ≤ ||xn −p||.(3.3)
Hence p ∈ Cn+1, and so Ω ⊂ Cn+1. Therefore, PCn+1x0 is well defined.

Step 2. We show that lim
n→∞

‖xn −x0‖ exists.

Since Ω is a nonempty closed convex subset of H, there exists a unique v ∈ Ω such that v = PΩx0.
Since xn = PCnx0 and xn+1 ∈ Cn+1 ⊂ Cn, ∀ n ≥ 0, we have ‖xn −x0‖≤‖xn+1 −x0‖, ∀ n ≥ 0.
On the other hand, since v ∈ Ω ⊂ Cn, we obtain

‖xn −x0‖≤‖v −x0‖, ∀ n ≥ 0.

It follows that the sequence {xn} is bounded and {||xn−x0||} is non decreasing and bounded. Therefore,
lim

n→∞
‖xn −x0‖ exists.



GENERALIZED MIXED EQUILIBRIUM PROBLEM AND COMMON FIXED POINT PROBLEM 53

Step 3. We show that lim
n→∞

xn exists in K.

For m > n, by the definition of Cn, we get xm = PCmx0 ∈ Cm ⊂ Cn. By applying Lemma 2.1,
we have

‖xm −xn‖
2 ≤‖xm −x0‖

2 −‖xn −x0‖
2
.

Since lim
n→∞

‖xn − x0‖ exists, it follows that {xn} is a Cauchy sequence. Hence, there exists x∗ ∈ K
such that lim

n→∞
xn = x

∗.

Step 4. We show that ‖xin −xn‖→ 0 as n →∞, i = 1, 2, · · · ,m.

From xn+1 ∈ Cn+1, we have

(3.4) ‖xn −yn‖≤‖xn −xn+1‖ + ‖xn+1 −yn‖≤ 2‖xn −xn+1‖→ 0 as n →∞.

For p ∈ Ω, using inequality (3.2), we get

‖yn −p‖
2 ≤ αn,0||un −p||2 +

m∑
i=1

αn,i||un −p||2 +
m∑
i=1

αn,ik||un −xin||
2

−
m∑
i=1

αn,iαn,0||un −xin||
2

= ||un −p||2 −
m∑
i=1

αn,i(αn,0 −k)||un −xin||
2

≤ ||xn −p||2 −
m∑
i=1

αn,i(αn,0 −k)||un −xin||
2.

Thus,

αn,iαn,0‖xin −un‖
2
≤

m∑
i=1

αn,iαn,0‖xin −un‖
2

≤ ‖xn −p‖
2 −‖yn −p‖

2

≤ M‖xn −yn‖,(3.5)

where M = supn≥0{‖xn −p‖ + ‖yn −p‖}. By the given condition on {αn,i} and (3.5), we get

lim
n→∞

‖xin −un‖ = 0, i = 1, 2, · · · ,m.

By Lemma 2.6, we have

‖un −p‖
2

= ‖Trn (xn −rnBxn) −Trn (p−rnBp)‖
2 ≤〈Trn (xn −rnBxn) −Trn (p−rnBp),xn −p〉

= 〈un −p,xn −p〉

=
1

2
{||un −p||2 + ||xn −p||2 −||xn −un||2}. Hence,

‖un −p‖
2 ≤ ‖xn −p‖

2 −‖xn −un‖
2
.

From inequality (3.2), we get

‖yn −p‖
2 ≤ ‖xn −p‖

2 −‖xn −un‖
2
.

⇒‖xn −un‖
2 ≤ ‖xn −p‖

2 −‖yn −p‖
2

≤ M‖xn −yn‖, where M = sup
n≥0
{‖xn −p‖ + ‖yn −p‖}.

Applying (3.4), we have

‖xn −un‖→ 0,n →∞.



54 EZEORA

Hence,

‖xin −xn‖≤‖x
i
n −un‖ + ‖un −xn‖→ 0 as n →∞, i = 1, 2, · · · ,m.

Step 5. We show that x∗ ∈ Ω.

Using the assumption that lim inf
n→∞

rn > 0, we have

(3.6)
∣∣∣∣xn −un

rn

∣∣∣∣ = 1
rn
‖xn −un‖→ 0,n →∞.

So, since lim
n→∞

xn = x
∗, we obtain

lim
n→∞

un = x
∗.

First, we show that x∗ ∈ GMEP(F,ϕ,B). This proof follows as in the proof of Theorem 3.1 of [26],
we omit the proof. Hence, x∗ ∈ GMEP(F,ϕ,B).

Next, we have to show that x∗ ∈∩mi=1F(Ti). For each i = 1, 2, ...,m, using Lemma 2.4, we have

d(x∗,Tix
∗) ≤ d(x∗,xn) + d(xn,xin) + d(x

i
n,Tix

∗)

≤ d(x∗,xn) + d(xn,xin) + D(Tiun,Tix
∗)

≤ d(x∗,xn) + d(xn,xin) + Li||un −x
∗||

≤ d(x∗,xn) + d(xn,xin) + L||un −x
∗||,

where L = max1≤i≤m{Li}.
Applying Step 3-4, we have d(x∗,Tix

∗) = 0. Hence x∗ ∈ Tix∗ for all i = 1, 2, ...,m. That is,
x∗ ∈∩mi=1F(Ti).

Step 6. We show that x∗ = PΩx0.
Since xn = PCnx0, we get 〈ξ −xn,x0 −xn〉≤ 0,∀ ξ ∈ Cn. Since x∗ ∈ Ω ⊂ Cn, we have

〈ξ −x∗,x0 −x∗〉≤ 0,∀ ξ ∈ Ω.
Thus, x∗ = PΩx0, completing the proof. 2

If for each i = 1, 2, · · · ,m, Ti : K → CB(K) is multivalued nonexpansive mappings, then we have the
following result.

Corollary 3.2. Let K be a nonempty closed and convex subset of a real Hilbert space H, F be a
bi-function from K × K to R satisfying (A1) − (A4), and let ϕ be a proper lower semicontinuous
and convex function from K to R ∪ {+∞} such that K ∩ domϕ 6= ∅. Let Ti : K → CB(K) be
multivalued nonexpansive mappings with Ω := ∩mi=1F(Ti) ∩ GMEP(F,ϕ,B) 6= ∅. Assume that for
p ∈

⋂m
i=1 F(Ti), Tip = {p} and that either (B1) or (B2) holds with {αn,i} ⊂ [0, 1) satisfying the

condition lim inf
n→∞

αn,iαn,0 > 0 ∀ i = 1, 2, · · · ,m. Define the sequence {xn} as follows: x1 ∈ K = C1,

(3.7)



{F(un,y) + ϕ(y) −ϕ(un) +

1

rn
〈y −un,un −xn〉≥ 0,∀ y ∈ K,

yn = αn,0un +
∑m

i=1 αn,ix
i
n, x

i
n ∈ Tiun,

Cn+1 = {z ∈ Cn : ‖yn −z‖≤‖xn −z‖},
xn+1 = PCn+1x0, n ≥ 0,

where the sequence rn ∈ (0,∞) with lim inf
n→∞

rn > 0 and
∑m

i=0 αn,i = 1. Then the sequence {xn} con-
verges strongly to PΩx0.

Setting ϕ ≡ 0 in Theorem 3.1, we have the following result.

Corollary 3.3. Let K be a nonempty closed and convex subset of a real Hilbert space H. Let F be a
bifunction from K × K → R satisfying (A1) − (A4). Let Ti : K → CB(K) be multivalued ki-strictly
pseudo-contractive mappings, ki ∈ (0, 1), i = 1, . . . ,m with Ω := ∩mi=1F(Ti) ∩EP(F,ϕ) 6= ∅. Assume



GENERALIZED MIXED EQUILIBRIUM PROBLEM AND COMMON FIXED POINT PROBLEM 55

that for p ∈
⋂m

i=1 F(Ti), Tip = {p} and that either (B1) or (B2) holds with {αn,i}⊂ [0, 1) satisfying the
condition lim inf

n→∞
αn,iαn,0 > 0 ∀ i = 1, 2, · · · ,m. Define the sequence {xn} as follows: x1 ∈ K = C1,

(3.8)



{F(un,y) +

1

rn
〈y −un,un −xn〉≥ 0,∀ y ∈ K,

yn = αn,0un +
∑
i=1

m
αn,ix

i
n, x

i
n ∈ Tiun,

Cn+1 = {z ∈ Cn : ‖yn −z‖≤‖xn −z‖},
xn+1 = PCn+1x0, n ≥ 0,

where the sequence rn ∈ (0,∞) with lim inf
n→∞

rn > 0 and
∑m

i=0 αn,i = 1. Then the sequence {xn} con-
verges strongly to PΩx0.

Setting F ≡ 0 in Theorem 3.1, we have the following result.

Corollary 3.4. Let K be a nonempty closed and convex subset of a real Hilbert space H. Let F be a
bifunction from K × K → R satisfying (A1) − (A4). Let Ti : K → CB(K) be multivalued ki-strictly
pseudo-contractive mappings, ki ∈ (0, 1), i = 1, . . . ,m with Ω := ∩mi=1F(Ti) ∩CMP(ϕ) 6= ∅. Assume
that for p ∈

⋂m
i=1 F(Ti), Tip = {p} and that either (B1) or (B2) holds with {αn,i}⊂ (k, 1). Define the

sequence {xn} as follows: x1 ∈ K = C1,

(3.9)



{ϕ(y) −ϕ(un) +

1

rn
〈y −un,un −xn〉≥ 0,∀ y ∈ K,

yn = αn,0un +
∑
i=1

m
αn,ix

i
n, x

i
n ∈ Tiun,

Cn+1 = {z ∈ Cn : ‖yn −z‖≤‖xn −z‖},
xn+1 = PCn+1x0, n ≥ 0,

where the sequence rn ∈ (0,∞) with lim inf
n→∞

rn > 0 and
∑m

i=0 αn,i = 1. Then the sequence {xn} con-
verges strongly to PΩx0.

Setting F ≡ 0,ϕ ≡ 0 in Theorem 3.1, we have the following result.

Corollary 3.5. Let K be a nonempty closed and convex subset of a real Hilbert space H. Let F be a
bifunction from K ×K → R satisfying (A1) − (A4). Let Ti : K → CB(K) be multi valued ki-strictly
pseudo-contractive mappings, ki ∈ (0, 1), i = 1, · · · ,m with Ω := ∩mi=1F(Ti) 6= ∅. Assume that for
p ∈

⋂m
i=1 F(Ti), Tip = {p} and that {αn,i}⊂ (k, 1). Define the sequence {xn} as follows: x1 ∈ K = C1,

(3.10)



{yn = αn,0un +

∑
i=1

m
αn,ix

i
n, x

i
n ∈ Tiun,

Cn+1 = {z ∈ Cn : ‖yn −z‖≤‖xn −z‖},
xn+1 = PCn+1x0, n ≥ 0,

where the sequence
∑m

i=0 αn,i = 1. Then the sequence {xn} converges strongly to PΩx0.

Remark 3.6. 1. Theorem 3.1 is a significant improvement on Theorem 3.1 of [11] for the following
reasons;
(a) it solves two major problems, generalized mixed equilibrium problem and common fixed point
problem.
(b) To prove Theorem 3.1 of [11], compactness assumptions were placed on the operators Ti, i =
1, 2, · · · ,N, this is dispensed with in the proof of Theorem 3.1of this paper.

2. Theorem 3.1 of this paper generalizes Theorem 3.1 of [6] from multivalued nonexapnsive map-
pings to multivalued strictly pseudo contractive mapping and from mixed equilibrium problem to
generalized mixed equilibrium problem.



56 EZEORA

3. Corollary 3.4 solves the convex minimization problem (1.7).

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Department of Ind. Mathematics and Statistics, Ebonyi State University, Abakaliki, Nigeria

∗Corresponding author: jerryezeora@yahoo.com