### Abstract

We introduce a new iterative scheme by modifying Mann’s iteration method to find a common element for the set of common fixed points of an infinite family of asymptotically strictly pseudocontractive mappings in the intermediate sense, the set of solutions of the cocoercive quasivariational inclusions problems, and the set of solutions of the mixed equilibrium problems in Hilbert spaces. The strong convergence theorem of the iterative scheme to a common element of the three aforementioned sets is obtained based on the shrinking projection method which extends and improves that of Ezeora and Shehu (Thai J. Math. 9(2):399-409, 2011) and many others.

**MSC: **
46C05, 47H09, 47H10, 49J30, 49J40.

##### Keywords:

fixed point; asymptotically strictly pseudocontractive in the intermediate sense; variational inequalities; mixed equilibrium; strong convergence; Hilbert space### 1 Introduction

Throughout this paper, we always assume that *C* is a nonempty closed convex subset of a real Hilbert space *H* with inner product and norm denoted by
*H*, we denote the strong convergence and the weak convergence of

Recall that
*metric projection* of *H* onto *C*; that is, for each
*nonexpansive* if
*uniformly**L-Lipschitzian* if there exists a constant
*contraction* if there exists a constant
*fixed point* of *T* provided that
*T*; that is,
*C* is a nonempty bounded closed convex subset of *H* and *T* is a nonexpansive mapping of *C* into itself, then

(i) *monotone* if

(ii) *k-Lipschitz continuous* if there exists a constant

if
*A* is nonexpansive,

(iii) *α-strongly monotone* if there exists a constant

(iv) *α-inverse-strongly monotone* (*or**α-cocoercive*) if there exists a constant

if
*T* is called *firmly nonexpansive*; it is obvious that any *α*-inverse-strongly monotone mapping *T* is monotone and

(v) *κ-strictly pseudocontractive*[2] if there exists a constant

In brief, we use *κ*-SPC to denote *κ*-strictly pseudocontractive. It is obvious that *T* is nonexpansive if and only if *T* is 0-SPC,

(vi) *asymptotically**κ-SPC*[3] if there exists a constant

for all
*T* is asymptotically nonexpansive with
*T* is *asymptotically nonexpansive*[4] if there exists a sequence

for all
*κ*-SPC mappings and the class of asymptotically *κ*-SPC mappings are independent (see [5]). And the class of asymptotically nonexpansive mappings is reduced to the class of
asymptotically nonexpansive mappings in the intermediate sense with
*T* is an *asymptotically nonexpansive mapping in the intermediate sense* if there exists a sequence

for all

(vii) *asymptotically**κ-SPC mapping in the intermediate sense*[6] if there exists a constant

If we define

then

for all
*κ*-SPC mappings in the intermediate sense is reduced to the class of asymptotically
*κ*-SPC mappings; and if
*κ*-SPC mappings in the intermediate sense is reduced to the class of asymptotically
nonexpansive mappings; and if
*κ*-SPC mappings in the intermediate sense is reduced to the class of nonexpansive mappings;
and the class of asymptotically nonexpansive mappings in the intermediate sense with
*κ*-SPC mappings in the intermediate sense with
*κ*-SPC mapping in the intermediate sense (1.9); related work can also be found in [6-13] and the references therein.

**Example 1.1** (Sahu *et al.*[6])

Let

where

(1) *T* is an asymptotically nonexpansive mapping in the intermediate sense. Therefore, *T* is an asymptotically *κ*-SPC mapping in the intermediate sense.

(2) *T* is not continuous. Therefore, *T* is not an asymptotically *κ*-SPC and asymptotically nonexpansive mapping.

**Example 1.2** (Hu and Cai [7])

Let

Then

(1) *T* is an asymptotically nonexpansive mapping in the intermediate sense. Therefore, *T* is an asymptotically *κ*-SPC mapping in the intermediate sense.

(2) *T* is continuous but not uniformly *L*-Lipschitzian. Therefore, *T* is not an asymptotically *κ*-SPC mapping.

**Example 1.3** Let

where

(1) *T* is an asymptotically nonexpansive mapping in the intermediate sense. Therefore, *T* is an asymptotically *κ*-SPC mapping in the intermediate sense.

(2) *T* is not continuous. Therefore, *T* is not an asymptotically *κ*-SPC and asymptotically nonexpansive mapping.

Iterative methods are often used to solve the fixed point equation
*T* is a contraction. Picard’s method generates a sequence
*T*. However, if *T* is not a contraction (for instance, if *T* is nonexpansive), then Picard’s successive iteration fails, in general, to converge.
Instead, Mann’s iteration method for a nonexpansive mapping *T* (see [14]) prevails, generates a sequence

where

Mann’s algorithm for nonexpansive mappings has been extensively investigated (see
[2,15,16] and the references therein). One of the well-known results is proven by Reich [16] for a nonexpansive mapping *T* on *C*, which asserts the weak convergence of the sequence
*κ*-SPC mapping *T* on *C*, and subsequently, this result was improved and carried over the class of asymptotically
*κ*-SPC mappings by Kim and Xu [18].

It is known that Mann’s iteration (1.10) is in general not strongly convergent (see [19]). The way to guarantee strong convergence has been proposed by Nakajo and Takahashi [20]. They modified Mann’s iteration method (1.10), which is to find a fixed point of a nonexpansive mapping by the hybrid method, called the shrinking projection method (or the CQ method), as the following theorem.

**Theorem NT***Let**C**be a nonempty closed convex subset of a real Hilbert space**H*. *Let**T**be a nonexpansive mapping of**C**into itself such that*
*Suppose that*
*chosen arbitrarily and*
*is the sequence defined by*

*where*
*Then*
*converges strongly to*

Subsequently, Marino and Xu [21] introduced an iterative scheme for finding a fixed point of a *κ*-SPC mapping as the following theorem.

**Theorem MX***Let**C**be a nonempty closed convex subset of a real Hilbert space**H**and let*
*be a**κ*-*SPC mapping for some*
*Assume that*
*Suppose that*
*chosen arbitrarily and*
*is the sequence defined by*

*where*
*Then the sequence*
*converges strongly to*

Quite recently, Kim and Xu [18] have improved and carried Theorem MX over a wider class of asymptotically *κ*-SPC mappings as the following theorem.

**Theorem KX***Let**C**be a nonempty closed convex subset of a real Hilbert space**H**and let*
*be an asymptotically**κ*-*SPC mapping for some*
*with a bounded sequence*
*such that*
*Assume that*
*is a nonempty bounded subset of**C*. *Suppose that*
*chosen arbitrarily and*
*is the sequence defined by*

*where*
*as*
*and*
*such that*
*Then the sequence*
*converges strongly to*

The *domain* of the function

Let
*mixed equilibrium problem* is to find

The set of solutions of problem (1.11) is denoted by

It is obvious that if *x* is a solution of problem (1.11), then

We denote by
*equilibrium problem*. The theory of equilibrium problems has played an important role in the study of
a wide class of problems arising in economics, finance, transportation, network and
structural analysis, elasticity, and optimization and has numerous applications, including
but not limited to problems in economics, game theory, finance, traffic analysis,
circuit network analysis, and mechanics. The ideas and techniques of this theory are
being used in a variety of diverse areas and have proved to be productive and innovative.
Problem (1.12) is very general in the sense that it includes, as special cases, optimization
problems, variational inequalities, minimax problems, the Nash equilibrium problem
in noncooperative games, and others; see, for instance, [22,23] and the references therein. Some methods have been proposed to solve equilibrium
problem (1.12); related work can also be found in [7,12,24-34].

For solving the mixed equilibrium problem, let us assume that the bifunction
*C* satisfy the following conditions:

(A1)

(A2) Φ is monotone; that is,

(A3) for each

(A4) for each

(A5) for each

(B1) for each

(B2) *C* is a bounded set.

Variational inequality theory provides us with a simple, natural, general, and unified
framework for studying a wide class of unrelated problems arising in elasticity, structural
analysis, economics, optimization, oceanography, and regional, physical, and engineering
sciences, *etc.* (see [35-41] and the references therein). In recent years, variational inequalities have been
extended and generalized in different directions, using novel and innovative techniques,
both for their own sake and for their applications. A useful and important generalization
of variational inequalities is a variational inclusion.

Let
*quasivariational inclusion problem*, which is to find a point

where *θ* is the zero vector in *H*. The set of solutions of problem (1.13) is denoted by

A set-valued mapping
*monotone* if for all
*T* is not properly contained in the graph of any other monotone mappings. It is known
that a monotone mapping *T* is maximal if and only if for

**Definition 1.4** (see [42])

Let
*the resolvent operator associated with**M*, where *λ* is any positive number and *I* is the identity mapping.

Recently, Qin *et al.*[43] introduced the following algorithm for a finite family of asymptotically
*C*. Let

is called the explicit iterative scheme of a finite family of asymptotically
*C*, where

For each

for all

To be more precise, they introduced an iterative scheme for finding a common fixed
point of a finite family of asymptotically

**Theorem QCKS***Let**C**be a nonempty closed convex subset of a real Hilbert space**H*. *Let*
*be an integer*. *For each*
*let*
*be a finite family of asymptotically*
*SPC mappings defined as in* (1.15), *when*
*with the sequence*
*such that*
*Let*
*and*
*Assume that*
*is a nonempty bounded subset of**C*. *For*
*chosen arbitrarily*, *suppose that*
*is generated iteratively by*

*where*
*as*
*such that*
*If the control sequence*
*is chosen such that*
*Then the sequence*
*converges strongly to*

Recall that a discrete family
*C* is said to be an *asymptotically**κ-SPC semigroup*[44] if the following conditions are satisfied:

(1)
*I* denotes the identity operator on *C*;

(2)

(3) there exists a constant

for all
*κ*-SPC mapping

On the other hand, Tianchai [24] introduced an iterative scheme for finding a common element of the set of solutions
of the mixed equilibrium problems and the set of common fixed points for a discrete
asymptotically *κ*-SPC semigroup which is a subclass of the class of infinite families for the asymptotically
*κ*-SPC mapping as the following theorem.

**Theorem T***Let**C**be a nonempty closed convex subset of a real Hilbert space**H*, Φ *be a bifunction from*
*into* ℝ *satisfying the conditions* (A1)-(A5), *and let*
*be a proper lower semicontinuous and convex function with the assumption that either* (B1) *or* (B2) *holds*. *Let*
*be an asymptotically**κ*-*SPC semigroup on**C**for some*
*with the sequence*
*such that*
*Assume that*
*is a nonempty bounded subset of**C*. *For*
*chosen arbitrarily*, *suppose that*
*and*
*are generated iteratively by*

*where*
*such that*
*for some*
*and*
*Then the sequences*
*and*
*converge strongly to*

Recently, Saha *et al.*[6] modified Mann’s iteration method (1.10) for finding a fixed point of the asymptotically
*κ*-SPC mapping in the intermediate sense which is not necessarily uniformly Lipschitzian
(see, *e.g.*, [6,7]) as the following theorem.

**Theorem SXY***Let**C**be a nonempty closed convex subset of a real Hilbert space**H**and let*
*be a uniformly continuous and asymptotically**κ*-*SPC mapping in the intermediate sense defined as in* (1.9), *when*
*with the sequences*
*such that*
*Assume that*
*is a nonempty bounded subset of**C*. *Suppose that*
*chosen arbitrarily and*
*is the sequence defined by*

*where*
*as*
*and*
*such that*
*Then the sequence*
*converges strongly to*

Let

for all

Subsequently, Hu and Cai [7] modified Ishikawa’s iteration method (see [45]) for finding a common element of the set of common fixed points for a finite family
of asymptotically

**Theorem HC***Let**C**be a nonempty closed convex subset of a real Hilbert space**H*,
*be a bifunction satisfying the conditions* (A1)-(A4), *and let*
*be a**ξ*-*cocoercive mapping*. *Let*
*be an integer*. *For each*
*let*
*be a finite family of uniformly continuous and asymptotically*
*SPC mappings in the intermediate sense defined as in* (1.17) *when*
*with the sequences*
*such that*
*Let*
*and*
*Assume that*
*is a nonempty bounded subset of**C*. *For*
*chosen arbitrarily*, *suppose that*
*and*
*are generated iteratively by*

*where*
*as*
*such that*
*If the control sequences*
*and*
*are chosen such that*
*and*
*then the sequences*
*and*
*converge strongly to*

In this paper, we study the sequences

for all

Quite recently, Ezeora and Shehu [9] introduced an iterative scheme for finding a common fixed point of an infinite family
of asymptotically

**Theorem ES***Let**C**be a nonempty closed convex subset of a real Hilbert space**H*. *For each*
*let*
*be an infinite family of uniformly continuous and asymptotically*
*SPC mappings in the intermediate sense defined as in* (1.18) *when*
*with the sequences*
*such that*
*Assume that*
*is a nonempty bounded subset of**C*. *For*
*chosen arbitrarily*, *suppose that*
*is generated iteratively by*

*where*
*and*
*such that*
*Then the sequence*
*converges strongly to*

Inspired and motivated by the works mentioned above, in this paper, we introduce a
new iterative scheme (3.1) below by modifying Mann’s iteration method (1.10) to find
a common element for the set of common fixed points of an infinite family of asymptotically

### 2 Preliminaries

We collect the following lemmas which be used in the proof of the main results in the next section.

**Lemma 2.1** (see [1])

*Let**C**be a nonempty closed convex subset of a Hilbert space**H*. *Then the following inequality holds*:

**Lemma 2.2** (see [46])

*Let**H**be a Hilbert space*. *For all*
*and*
*such that*
*one has*

**Lemma 2.3** (see [25])

*Let**C**be a nonempty closed convex subset of a real Hilbert space**H*,
*satisfying the conditions* (A1)-(A5), *and let*
*be a proper lower semicontinuous and convex function*. *Assume that either* (B1) *or* (B2) *holds*. *For*
*define a mapping*
*as follows*:

*for all*
*Then the following statements hold*:

(1) *for each*

(2)
*is single*-*valued*;

(3)
*is firmly nonexpansive*; *that is*, *for any*

(4)

(5)
*is closed and convex*.

**Lemma 2.4** (see [42])

*Let*
*be a maximal monotone mapping and let*
*be an**α*-*inverse*-*strongly monotone mapping*. *Then*, *the following statements hold*:

(1) *B**is a*
*Lipschitz continuous and monotone mapping*.

(2)
*is a solution of quasivariational inclusion* (1.13) *if and only if*
*for all*
*that is*,

(3) *If*
*then*
*is a closed convex subset in**H*, *and the mapping*
*is nonexpansive*, *where**I**is the identity mapping on**H*.

(4) *The resolvent operator*
*associated with**M**is single*-*valued and nonexpansive for all*

(5) *The resolvent operator*
*is* 1-*inverse*-*strongly monotone*; *that is*,

**Lemma 2.5** (see [47])

*Let*
*be a maximal monotone mapping and let*
*be a Lipschitz continuous mapping*. *Then the mapping*
*is a maximal monotone mapping*.

**Lemma 2.6** (see [6])

*Let**C**be a nonempty closed convex subset of a real Hilbert space**H**and let*
*be a uniformly continuous and asymptotically**κ*-*strictly pseudocontractive mapping in the intermediate sense defined as in* (1.9) *when*
*with the sequences*
*such that*
*Then the following statements hold*:

(1)
*for all*
*and*

(2) *If*
*is a sequence in**C**such that*
*and*
*Then*

(3)
*is demiclosed at zero in the sense that if*
*is a sequence in**C**such that*
*as*
*and*
*then*

(4)
*is closed and convex*.

### 3 Main results

**Theorem 3.1***Let**H**be a real Hilbert space*, Φ *be a bifunction from*
*into* ℝ *satisfying the conditions* (A1)-(A5), *and let*
*be a proper lower semicontinuous and convex function with the assumption that either* (B1) *or* (B2) *holds*. *Let*
*be a maximal monotone mapping and let*
*be a**ξ*-*cocoercive mapping*. *For each*
*let*
*be an infinite family of uniformly continuous and asymptotically*
*SPC mappings in the intermediate sense defined as in* (1.18) *when*
*with the sequences*
*such that*
*Assume that*
*is a nonempty bounded subset of**H*. *For*
*chosen arbitrarily*, *suppose that*
*is generated iteratively by*

*where*
*and*
*satisfying the following conditions*:

(C1)
*and*
*such that*
*for each*
*and*

(C2)
*and*
*for some*

*Then the sequence*
*converges strongly to*

*Proof* Pick

and by

From Lemma 2.3(3), we know that

Let

By (3.3), (3.4), (3.5), Lemma 2.2, and the asymptotically

where

Firstly, we show that

for all

for all

Next, we show that
*i*, and so
*H*, there exists a unique element
*i*, and so

for all
*H*, there exists a unique element

Next, we show that

for all

Next, we show that

for some

It follows that

Therefore, by (3.11), we obtain

Next, we show that

It follows that

for each

Thus, we have

Hence, by (3.11), we obtain that
*H*, and then there exists a point

Next, we show that

It follows by (3.13) and

Since,

Therefore, by (3.13) and (3.17), we obtain

Next, we show that

which implies that

By (3.4), (3.6), and (3.20), we have

which implies that

Therefore, by (3.19) and

Since

therefore, by (3.13) and (3.22), we obtain

Next, we show that