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An approximate solution to the fixed point problems for an infinite family of asymptotically strictly pseudocontractive mappings in the intermediate sense, cocoercive quasivariational inclusions problems and mixed equilibrium problems in Hilbert spaces

Pattanapong Tianchai

Author Affiliations

Faculty of Science, Maejo University, Chiangmai, 50290, Thailand

Fixed Point Theory and Applications 2012, 2012:214  doi:10.1186/1687-1812-2012-214

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/214


Received:2 May 2012
Accepted:7 November 2012
Published:26 November 2012

© 2012 Tianchai; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 , and , respectively. For a sequence { x n } in H, we denote the strong convergence and the weak convergence of { x n } to x H by x n x and x n x , respectively.

Recall that P C is the metric projection of H onto C; that is, for each x H , there exists the unique point P C x C such that x P C x = min y C x y . A mapping T : C C is called nonexpansive if T x T y x y for all x , y C , and uniformlyL-Lipschitzian if there exists a constant L > 0 such that for each n N , T n x T n y L x y for all x , y C , and a mapping f : C C is called a contraction if there exists a constant α ( 0 , 1 ) such that f ( x ) f ( y ) α x y for all x , y C . A point x C is a fixed point of T provided that T x = x . We denote by F ( T ) the set of fixed points of T; that is, F ( T ) = { x C : T x = x } . If C is a nonempty bounded closed convex subset of H and T is a nonexpansive mapping of C into itself, then F ( T ) is nonempty (see [1]). Recall that a mapping T : C C is said to be

(i) monotone if

T x T y , x y 0 , x , y C , (1.1)

(ii) k-Lipschitz continuous if there exists a constant k > 0 such that

T x T y k x y , x , y C , (1.2)

if k = 1 , then A is nonexpansive,

(iii) α-strongly monotone if there exists a constant α > 0 such that

T x T y , x y α x y 2 , x , y C , (1.3)

(iv) α-inverse-strongly monotone (orα-cocoercive) if there exists a constant α > 0 such that

T x T y , x y α T x T y 2 , x , y C , (1.4)

if α = 1 , then T is called firmly nonexpansive; it is obvious that any α-inverse-strongly monotone mapping T is monotone and ( 1 / α ) -Lipschitz continuous,

(v) κ-strictly pseudocontractive[2] if there exists a constant κ [ 0 , 1 ) such that

T x T y 2 x y 2 + κ ( I T ) x ( I T ) y 2 , x , y C . (1.5)

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) asymptoticallyκ-SPC[3] if there exists a constant κ [ 0 , 1 ) and a sequence { γ n } of nonnegative real numbers with lim n γ n = 0 such that

T n x T n y 2 ( 1 + γ n ) x y 2 + κ ( I T n ) x ( I T n ) y 2 , x , y C , (1.6)

for all n N . If κ = 0 , then T is asymptotically nonexpansive with k n = 1 + γ n for all n N ; that is, T is asymptotically nonexpansive[4] if there exists a sequence { k n } [ 1 , ) with lim n k n = 1 such that

T n x T n y k n x y ( 1 + γ n ) x y , x , y C , (1.7)

for all n N . It is known that the class of 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 ϵ n = γ n K for all n N and for some K > 0 ; that is, T is an asymptotically nonexpansive mapping in the intermediate sense if there exists a sequence { ϵ n } of nonnegative real numbers with lim n ϵ n = 0 such that

T n x T n y x y + ϵ n , x , y C , (1.8)

for all n N ,

(vii) asymptoticallyκ-SPC mapping in the intermediate sense[6] if there exists a constant κ [ 0 , 1 ) and a sequence { γ n } of nonnegative real numbers with lim n γ n = 0 such that

lim sup n sup x , y C ( T n x T n y 2 ( 1 + γ n ) x y 2 κ ( I T n ) x ( I T n ) y 2 ) 0 .

If we define

τ n = max { 0 , sup x , y C ( T n x T n y 2 ( 1 + γ n ) x y 2 κ ( I T n ) x ( I T n ) y 2 ) } ,

then lim n τ n = 0 , and the last inequality is reduced to

T n x T n y 2 ( 1 + γ n ) x y 2 + κ ( I T n ) x ( I T n ) y 2 + τ n , x , y C , (1.9)

for all n N . It is obvious that if τ n = 0 for all n N , then the class of asymptotically κ-SPC mappings in the intermediate sense is reduced to the class of asymptotically κ-SPC mappings; and if τ n = κ = 0 for all n N , then the class of asymptotically κ-SPC mappings in the intermediate sense is reduced to the class of asymptotically nonexpansive mappings; and if γ n = τ n = κ = 0 for all n N , then the class of asymptotically 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 { ϵ n } of nonnegative real numbers such that lim n ϵ n = 0 is reduced to the class of asymptotically κ-SPC mappings in the intermediate sense with τ n = ϵ n K for all n N and for some K > 0 . Some methods have been proposed to solve the fixed point problem of an asymptotically 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 X = R and C = [ 0 , 1 ] . For each x C , we define T : C C by

T x = { κ x , if  x [ 0 , 1 2 ] , 0 , if  x ( 1 2 , 1 ] ,

where κ ( 0 , 1 ) . 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 not continuous. Therefore, T is not an asymptotically κ-SPC and asymptotically nonexpansive mapping.

Example 1.2 (Hu and Cai [7])

Let X = R , C = [ 0 , 1 ] , and κ [ 0 , 1 ) . For each x C , we define T : C C by

T x = { 1 4 1 2 x + 2 2 , if  x [ 0 , 1 2 ] , x , if  x ( 1 2 , 1 ] .

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 X = R and C = [ 0 , 1 ] . For each x C , we define T : C C by

T x = { 2 x + 1 4 1 2 , if  x [ 0 , 1 2 ] , 2 κ x 5 , if  x ( 1 2 , 1 ] ,

where κ [ 0 , 1 ) . 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 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 x = x . The most well-known method is perhaps the Picard successive iteration method when T is a contraction. Picard’s method generates a sequence { x n } successively as x n + 1 = T x n for all n N with x 1 = x chosen arbitrarily, and this sequence converges in norm to the unique fixed point of 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 { x n } recursively by

x n + 1 = α n x n + ( 1 α n ) T x n , n N , (1.10)

where x 1 = x C chosen arbitrarily and the sequence { α n } lies in the interval [ 0 , 1 ] .

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 { x n } generated by (1.10) in a uniformly convex Banach space with a Frechet differentiable norm under the control condition n = 1 α n ( 1 α n ) = . Recently, Marino and Xu [17] developed and extended Reich’s result to a SPC mapping in a Hilbert space setting. More precisely, they proved the weak convergence of Mann’s iteration process (1.10) for a 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 NTLetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetTbe a nonexpansive mapping ofCinto itself such that F ( T ) . Suppose that x 1 = x C chosen arbitrarily and { x n } is the sequence defined by

{ y n = α n x n + ( 1 α n ) T x n , C n = { z C : y n z x n z } , Q n = { z C : x n z , x 1 x n 0 } , x n + 1 = P C n Q n ( x 1 ) , n N ,

where 0 α n α < 1 . Then { x n } converges strongly to P F ( T ) ( x 1 ) .

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

Theorem MXLetCbe a nonempty closed convex subset of a real Hilbert spaceHand let T : C C be aκ-SPC mapping for some 0 κ < 1 . Assume that F ( T ) . Suppose that x 1 = x C chosen arbitrarily and { x n } is the sequence defined by

{ y n = α n x n + ( 1 α n ) T x n , C n = { z C : y n z 2 x n z 2 + ( 1 α n ) ( κ α n ) x n T x n 2 } , Q n = { z C : x n z , x 1 x n 0 } , x n + 1 = P C n Q n ( x 1 ) , n N ,

where 0 α n < 1 . Then the sequence { x n } converges strongly to P F ( T ) ( x 1 ) .

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 KXLetCbe a nonempty closed convex subset of a real Hilbert spaceHand let T : C C be an asymptoticallyκ-SPC mapping for some 0 κ < 1 with a bounded sequence { γ n } [ 0 , ) such that lim n γ n = 0 . Assume that F ( T ) is a nonempty bounded subset ofC. Suppose that x 1 = x C chosen arbitrarily and { x n } is the sequence defined by

{ y n = α n x n + ( 1 α n ) T n x n , C n = { z C : y n z 2 x n z 2 + ( κ α n ( 1 α n ) ) x n T n x n 2 + θ n } , Q n = { z C : x n z , x 1 x n 0 } , x n + 1 = P C n Q n ( x 1 ) , n N ,

where θ n = Δ n 2 ( 1 α n ) γ n 0 as n , Δ n = sup { x n z : z F ( T ) } < and 0 α n < 1 such that lim sup n α n < 1 κ . Then the sequence { x n } converges strongly to P F ( T ) ( x 1 ) .

The domain of the function φ : C R { + } is the set

dom φ = { x C : φ ( x ) < + } .

Let φ : C R { + } be a proper extended real-valued function and let Φ be a bifunction from C × C into ℝ, where ℝ is the set of real numbers. The so-called mixed equilibrium problem is to find x C such that

Φ ( x , y ) + φ ( y ) φ ( x ) 0 , y C . (1.11)

The set of solutions of problem (1.11) is denoted by MEP ( Φ , φ ) , that is,

MEP ( Φ , φ ) = { x C : Φ ( x , y ) + φ ( y ) φ ( x ) 0 , y C } .

It is obvious that if x is a solution of problem (1.11), then x dom φ . If φ = 0 , then problem (1.11) is reduced to finding x C such that

Φ ( x , y ) 0 , y C . (1.12)

We denote by EP ( Φ ) the set of solutions of the 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 Φ : C × C R , the function φ : C R { + } , and the set C satisfy the following conditions:

(A1) Φ ( x , x ) = 0 for all x C ;

(A2) Φ is monotone; that is, Φ ( x , y ) + Φ ( y , x ) 0 for all x , y C ;

(A3) for each x , y , z C ,

lim t 0 Φ ( t z + ( 1 t ) x , y ) Φ ( x , y ) ;

(A4) for each x C , y Φ ( x , y ) is convex and lower semicontinuous;

(A5) for each y C , x Φ ( x , y ) is weakly upper semicontinuous;

(B1) for each x C and r > 0 , there exists a bounded subset D x C and y x C such that for any z C D x ,

Φ ( z , y x ) + φ ( y x ) φ ( z ) + 1 r y x z , z x < 0 ;

(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 B : H H be a single-valued nonlinear mapping and let M : H 2 H be a set-valued mapping. We consider the following quasivariational inclusion problem, which is to find a point x H such that

θ B x + M x , (1.13)

where θ is the zero vector in H. The set of solutions of problem (1.13) is denoted by VI ( H , B , M ) .

A set-valued mapping T : H 2 H is called monotone if for all x , y H , f T x and g T y imply x y , f g 0 . A monotone mapping T : H 2 H is maximal if the graph of G ( T ) of 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 ( x , f ) H × H , x y , f g 0 for all ( y , g ) G ( T ) implies f T x .

Definition 1.4 (see [42])

Let M : H 2 H be a multi-valued maximal monotone mapping. Then the single-valued mapping J M , λ : H H defined by J M , λ ( u ) = ( I + λ M ) 1 ( u ) , for all u H , is called the resolvent operator associated withM, 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 κ i -SPC mappings T i on C. Let x 0 C and { α n } n = 0 be a sequence in [ 0 , 1 ] . Let N 1 be an integer. The sequence { x n } generated in the following way:

(1.14)

is called the explicit iterative scheme of a finite family of asymptotically κ i -SPC mappings T i on C, where i = 1 , 2 , , N . Since, for each n N , it can be written as n = ( h 1 ) N + i , where i = i ( n ) { 1 , 2 , , N } , h = h ( n ) 1 is a positive integer and h ( n ) as n .

For each i = 1 , 2 , , N , let { T i : C C } be a finite family of asymptotically κ i -SPC mappings with the sequence { γ n , i } n = 1 [ 0 , ) such that lim n γ n , i = 0 . One has

T i ( n ) h ( n ) x T i ( n ) h ( n ) y 2 ( 1 + γ h ( n ) , i ( n ) ) x y 2 + κ i ( n ) ( I T i ( n ) h ( n ) ) x ( I T i ( n ) h ( n ) ) y 2 , x , y C , (1.15)

for all n N , and we can rewrite (1.14) in the following compact form:

x n = α n 1 x n 1 + ( 1 α n 1 ) T i ( n ) h ( n ) x n 1 , n N .

To be more precise, they introduced an iterative scheme for finding a common fixed point of a finite family of asymptotically κ i -SPC mappings as the following theorem.

Theorem QCKSLetCbe a nonempty closed convex subset of a real Hilbert spaceH. Let N 1 be an integer. For each i = 1 , 2 , , N , let { T i : C C } be a finite family of asymptotically κ i -SPC mappings defined as in (1.15), when κ i [ 0 , 1 ) with the sequence { γ n , i } [ 0 , ) such that lim n γ n , i = 0 . Let κ = max { κ i : 1 i N } and γ n = max { 1 + γ n , i : 1 i N } . Assume that Ω : = i = 1 N F ( T i ) is a nonempty bounded subset ofC. For x 0 = x C chosen arbitrarily, suppose that { x n } is generated iteratively by

{ y n 1 = α n 1 x n 1 + ( 1 α n 1 ) T i ( n ) h ( n ) x n 1 , C n 1 = { z C : y n 1 z 2 x n 1 z 2 + θ n 1 C n 1 = ( 1 α n 1 ) ( α n 1 κ ) T i ( n ) h ( n ) x n 1 x n 1 2 } , Q n 1 = { z C : x n 1 z , x 0 x n 1 0 } , x n = P C n 1 Q n 1 ( x 0 ) , n N ,

where θ n 1 = ( γ h ( n ) 2 1 ) ( 1 α n 1 ) Δ n 1 2 0 as n such that Δ n 1 = sup { x n 1 z : z Ω } < . If the control sequence { α n } n = 0 is chosen such that 0 α n α < 1 . Then the sequence { x n } converges strongly to P Ω ( x 0 ) .

Recall that a discrete family S = { T n : n 0 } of self-mappings of C is said to be an asymptoticallyκ-SPC semigroup[44] if the following conditions are satisfied:

(1) T 0 = I , where I denotes the identity operator on C;

(2) T n + m x = T n T m x , n , m 0 , x C ;

(3) there exists a constant κ [ 0 , 1 ) and a sequence { γ n } of nonnegative real numbers with lim n γ n = 0 such that

T n x T n y 2 ( 1 + γ n ) x y 2 + κ ( I T n ) x ( I T n ) y 2 , x , y C , (1.16)

for all n 0 . Note that, for a single asymptotically κ-SPC mapping T : C C , (1.16) immediately reduces to (1.6) by taking T n = T n for all n 0 such that T 0 = I .

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 TLetCbe a nonempty closed convex subset of a real Hilbert spaceH, Φ be a bifunction from C × C intosatisfying the conditions (A1)-(A5), and let φ : C R { + } be a proper lower semicontinuous and convex function with the assumption that either (B1) or (B2) holds. Let S = { T n : n 0 } be an asymptoticallyκ-SPC semigroup onCfor some κ [ 0 , 1 ) with the sequence { γ n } [ 0 , ) such that lim n γ n = 0 . Assume that Ω : = F ( S ) MEP ( Φ , φ ) = n = 0 F ( T n ) MEP ( Φ , φ ) is a nonempty bounded subset ofC. For x 0 = x C chosen arbitrarily, suppose that { x n } , { y n } and { u n } are generated iteratively by

{ u n C   such that Φ ( u n , y ) + φ ( y ) φ ( u n ) + 1 r n y u n , u n x n 0 , y C , y n = α n u n + ( 1 α n ) T n u n , C n + 1 = { z C n Q n : y n z 2 x n z 2 C n + 1 = + ( 1 α n ) ( θ n + ( κ α n ) u n T n u n 2 ) } , Q n + 1 = { z C n Q n : x n z , x 0 x n 0 } , C 0 = Q 0 = C , x n + 1 = P C n + 1 Q n + 1 ( x 0 ) , n N { 0 } ,

where θ n = γ n sup { x n z 2 : z Ω } < , { α n } [ a , b ] such that κ < a < b < 1 , { r n } [ r , ) for some r > 0 and n = 0 | r n + 1 r n | < . Then the sequences { x n } , { y n } , and { u n } converge strongly to w = P Ω ( x 0 ) .

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 SXYLetCbe a nonempty closed convex subset of a real Hilbert spaceHand let T : C C be a uniformly continuous and asymptoticallyκ-SPC mapping in the intermediate sense defined as in (1.9), when κ [ 0 , 1 ) , with the sequences { γ n } , { τ n } [ 0 , ) such that lim n γ n = lim n τ n = 0 . Assume that F ( T ) is a nonempty bounded subset ofC. Suppose that x 1 = x C chosen arbitrarily and { x n } is the sequence defined by

{ y n = ( 1 α n ) x n + α n T n x n , C n = { z C : y n z 2 x n z 2 + θ n } , Q n = { z C : x n z , x 1 x n 0 } , x n + 1 = P C n Q n ( x 1 ) , n N ,

where θ n = γ n Δ n + τ n 0 as n , Δ n = sup { x n z : z F ( T ) } < and { α n } ( 0 , 1 ] such that 0 < α α n 1 κ . Then the sequence { x n } converges strongly to P F ( T ) ( x 1 ) .

Let N 1 be an integer. For each i = 0 , 1 , , N 1 , let { T i : C C } be a finite family of asymptotically κ i -SPC mappings in the intermediate sense with the sequences { γ n , i } n = 1 , { τ n , i } n = 1 [ 0 , ) such that lim n γ n , i = lim n τ n , i = 0 . One has

T i ( n ) h ( n ) x T i ( n ) h ( n ) y 2 ( 1 + γ h ( n ) , i ( n ) ) x y 2 + κ i ( n ) ( I T i ( n ) h ( n ) ) x ( I T i ( n ) h ( n ) ) y 2 + τ h ( n ) , i ( n ) , x , y C , (1.17)

for all n N { 0 } such that n = ( h 1 ) N + i , where i = i ( n ) { 0 , 1 , , N 1 } , h = h ( n ) 1 is a positive integer and h ( n ) as n .

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 κ i -SPC mappings in the intermediate sense and the set of solutions of the equilibrium problems (see also Duan and Zhao [8]) as the following theorem.

Theorem HCLetCbe a nonempty closed convex subset of a real Hilbert spaceH, Φ : C × C R be a bifunction satisfying the conditions (A1)-(A4), and let A : C H be aξ-cocoercive mapping. Let N 1 be an integer. For each i = 0 , 1 , , N 1 , let { T i : C C } be a finite family of uniformly continuous and asymptotically κ i -SPC mappings in the intermediate sense defined as in (1.17) when κ i [ 0 , 1 ) with the sequences { γ n , i } , { τ n , i } [ 0 , ) such that lim n γ n , i = lim n τ n , i = 0 . Let κ = max { κ i : 0 i N 1 } , γ n = max { γ n , i : 0 i N 1 } and τ n = max { τ n , i : 0 i N 1 } . Assume that Ω : = i = 0 N 1 F ( T i ) EP ( Φ ) is a nonempty bounded subset ofC. For x 0 = x C chosen arbitrarily, suppose that { x n } and { u n } are generated iteratively by

{ u n C   such that Φ ( u n , y ) + A x n , y u n + 1 r n y u n , u n x n 0 , y C , z n = ( 1 β n ) u n + β n T i ( n ) h ( n ) u n , y n = ( 1 α n ) u n + α n z n , C n = { z H : y n z 2 x n z 2 + θ n } , Q n = { z C : x n z , x 0 x n 0 } , x n + 1 = P C n Q n ( x 0 ) , n N { 0 } ,

where θ n = γ h ( n ) Δ n 2 + τ h ( n ) 0 as n such that Δ n = sup { x n z : z Ω } < . If the control sequences { α n } , { β n } ( 0 , 1 ] , and { r n } [ a , b ] are chosen such that 0 < α α n 1 , 0 < β β n 1 κ , and 0 < a r n b < 2 ξ , then the sequences { x n } and { u n } converge strongly to P Ω ( x 0 ) .

In this paper, we study the sequences { x n , i } n = 1 generated by modifying Mann’s iteration method (1.10) for an infinite family of asymptotically κ i -SPC mappings in the intermediate sense. For each i = 1 , 2 ,  , let { T i : C C } be an infinite family of asymptotically κ i -SPC mappings in the intermediate sense with the sequences { γ n , i } n = 1 , { τ n , i } n = 1 [ 0 , ) such that lim n γ n , i = lim n τ n , i = 0 . One has

T i n x T i n y 2 ( 1 + γ n , i ) x y 2 + κ i ( I T i n ) x ( I T i n ) y 2 + τ n , i , x , y C , (1.18)

for all n N . For each i = 1 , 2 ,  , let x 1 , i C and { α n , i } n = 1 be a sequence in [ 0 , 1 ] , and let the sequences { x n , i } n = 1 be generated in the following way:

x n + 1 , i = ( 1 α n , i ) x n , i + α n , i T i n x n , i , n N .

Quite recently, Ezeora and Shehu [9] introduced an iterative scheme for finding a common fixed point of an infinite family of asymptotically κ i -SPC mappings in the intermediate sense as the following theorem.

Theorem ESLetCbe a nonempty closed convex subset of a real Hilbert spaceH. For each i = 1 , 2 ,  , let { T i : C C } be an infinite family of uniformly continuous and asymptotically κ i -SPC mappings in the intermediate sense defined as in (1.18) when κ i [ 0 , 1 ) with the sequences { γ n , i } n = 1 , { τ n , i } n = 1 [ 0 , ) such that lim n γ n , i = lim n τ n , i = 0 . Assume that Ω : = i = 1 F ( T i ) is a nonempty bounded subset ofC. For x 1 = x C chosen arbitrarily, suppose that { x n } n = 1 is generated iteratively by

{ y n , i = ( 1 α n , i ) x n + α n , i T i n x n , n 1 , C n , i = { z C : y n , i z 2 x n z 2 + θ n , i } , C n = i = 1 C n , i , Q n = { z Q n 1 : x n z , x 1 x n 0 } , n 2 , Q 1 = C , x n + 1 = P C n Q n ( x 1 ) , n N ,

where θ n , i = γ n , i Δ n 2 + τ n , i ( i = 1 , 2 , ), Δ n = sup { x n z : z Ω } < and { α n , i } n = 1 ( 0 , 1 ] ( i = 1 , 2 , ) such that 0 < α i α n , i 1 κ i . Then the sequence { x n } n = 1 converges strongly to P Ω ( x 1 ) .

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 κ i -SPC 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 [9] and many others.

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])

LetCbe a nonempty closed convex subset of a Hilbert spaceH. Then the following inequality holds:

x P C x , P C x y 0 , x H , y C .

Lemma 2.2 (see [46])

LetHbe a Hilbert space. For all x , y , z H and α , β , γ [ 0 , 1 ] such that α + β + γ = 1 , one has

α x + β y + γ z 2 = α x 2 + β y 2 + γ z 2 α β x y 2 α γ x z 2 β γ y z 2 .

Lemma 2.3 (see [25])

LetCbe a nonempty closed convex subset of a real Hilbert spaceH, Φ : C × C R satisfying the conditions (A1)-(A5), and let φ : C R { + } be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r > 0 , define a mapping S r : C C as follows:

S r ( x ) = { z C : Φ ( z , y ) + φ ( y ) φ ( z ) + 1 r y z , z x 0 , y C }

for all x C . Then the following statements hold:

(1) for each x C , S r ( x ) ;

(2) S r is single-valued;

(3) S r is firmly nonexpansive; that is, for any x , y C ,

S r x S r y 2 S r x S r y , x y ;

(4) F ( S r ) = MEP ( Φ , φ ) ;

(5) MEP ( Φ , φ ) is closed and convex.

Lemma 2.4 (see [42])

Let M : H 2 H be a maximal monotone mapping and let B : H H be anα-inverse-strongly monotone mapping. Then, the following statements hold:

(1) Bis a ( 1 / α ) -Lipschitz continuous and monotone mapping.

(2) u H is a solution of quasivariational inclusion (1.13) if and only if u = J M , λ ( u λ B u ) , for all λ > 0 , that is,

VI ( H , B , M ) = F ( J M , λ ( I λ B ) ) , λ > 0 .

(3) If λ ( 0 , 2 α ] , then VI ( H , B , M ) is a closed convex subset inH, and the mapping I λ B is nonexpansive, whereIis the identity mapping onH.

(4) The resolvent operator J M , λ associated withMis single-valued and nonexpansive for all λ > 0 .

(5) The resolvent operator J M , λ is 1-inverse-strongly monotone; that is,

J M , λ ( x ) J M , λ ( y ) 2 x y , J M , λ ( x ) J M , λ ( y ) , x , y H .

Lemma 2.5 (see [47])

Let M : H 2 H be a maximal monotone mapping and let B : H H be a Lipschitz continuous mapping. Then the mapping S = M + B : H 2 H is a maximal monotone mapping.

Lemma 2.6 (see [6])

LetCbe a nonempty closed convex subset of a real Hilbert spaceHand let T : C C be a uniformly continuous and asymptoticallyκ-strictly pseudocontractive mapping in the intermediate sense defined as in (1.9) when κ [ 0 , 1 ) with the sequences { γ n } , { τ n } [ 0 , ) such that lim n γ n = lim n τ n = 0 . Then the following statements hold:

(1) T n x T n y 1 1 κ ( κ x y + ( 1 + ( 1 κ ) γ n ) x y 2 + ( 1 κ ) τ n ) , for all x , y C and n N .

(2) If { x n } is a sequence inCsuch that lim n x n x n + 1 = 0 and lim n x n T n x n = 0 . Then lim n x n T x n = 0 .

(3) I T is demiclosed at zero in the sense that if { x n } is a sequence inCsuch that x n x C as n , and lim sup m lim sup n x n T m x n = 0 , then ( I T ) x = 0 .

(4) F ( T ) is closed and convex.

3 Main results

Theorem 3.1LetHbe a real Hilbert space, Φ be a bifunction from H × H intosatisfying the conditions (A1)-(A5), and let φ : H R { + } be a proper lower semicontinuous and convex function with the assumption that either (B1) or (B2) holds. Let M : H 2 H be a maximal monotone mapping and let B : H H be aξ-cocoercive mapping. For each i = 1 , 2 ,  , let { T i : H H } be an infinite family of uniformly continuous and asymptotically κ i -SPC mappings in the intermediate sense defined as in (1.18) when κ i [ 0 , 1 ) with the sequences { γ n , i } n = 1 , { τ n , i } n = 1 [ 0 , ) such that lim n γ n , i = lim n τ n , i = 0 . Assume that Ω : = i = 1 F ( T i ) VI ( H , B , M ) MEP ( Φ , φ ) is a nonempty bounded subset ofH. For x 1 = x H chosen arbitrarily, suppose that { x n } n = 1 is generated iteratively by

{ u n H   such that Φ ( u n , y ) + φ ( y ) φ ( u n ) + 1 r n y u n , u n x n 0 , y H , y n , i = ( 1 α n , i β n , i ) u n + α n , i T i n u n + β n , i J M , λ ( u n λ B u n ) , C n + 1 , i = { z C n Q n : y n , i z 2 x n z 2 + θ n , i } , C n + 1 = i = 1 C n + 1 , i , Q n + 1 = { z C n Q n : x n z , x 1 x n 0 } , C 1 , i = C 1 = Q 1 = H , x n + 1 = P C n + 1 Q n + 1 ( x 1 ) , n N , (3.1)

where θ n , i = γ n , i Δ n 2 + τ n , i ( i = 1 , 2 , ) and Δ n = sup { x n z : z Ω } < satisfying the following conditions:

(C1) { α n , i } n = 1 [ a 1 , i , b 1 , i ] and { β n , i } n = 1 [ a 2 , i , b 2 , i ] ( i = 1 , 2 , ) such that 0 < a j , i < b j , i < 1 for each j = 1 , 2 , and b 1 , i + b 2 , i < 1 κ i ( i = 1 , 2 , );

(C2) λ ( 0 , 2 ξ ] and { r n } n = 1 [ r , ) for some r > 0 .

Then the sequence { x n } n = 1 converges strongly to w = P Ω ( x 1 ) .

Proof Pick p Ω and fix i = 1 , 2 ,  . From (3.1), by the definition of S r n in Lemma 2.3, we have

u n = S r n x n dom φ , (3.2)

and by T i p = p , Lemmas 2.3(4) and 2.4(2), we have

T i n p = S r n p = J M , λ ( I λ B ) p = p . (3.3)

From Lemma 2.3(3), we know that S r n is nonexpansive. Therefore, by (3.2) and (3.3), we have

u n p = S r n x n S r n p x n p . (3.4)

Let t n = J M , λ ( u n λ B u n ) . From Lemmas 2.4(3) and 2.4(4), we know that J M , λ and I λ B are nonexpansive. Therefore, by (3.3), we have

t n p = J M , λ ( u n λ B u n ) J M , λ ( p λ B p ) u n p . (3.5)

By (3.3), (3.4), (3.5), Lemma 2.2, and the asymptotically κ i -SPC in the intermediate sense of T i , we have

y n , i p 2 = ( 1 α n , i β n , i ) ( u n p ) + α n , i ( T i n u n p ) + β n , i ( t n p ) 2 = ( 1 α n , i β n , i ) u n p 2 + α n , i T i n u n p 2 + β n , i t n p 2 α n , i ( 1 α n , i β n , i ) u n T i n u n 2 β n , i ( 1 α n , i β n , i ) u n t n 2 α n , i β n , i T i n u n t n 2 ( 1 α n , i β n , i ) u n p 2 + α n , i ( ( 1 + γ n , i ) u n p 2 + κ i u n T i n u n 2 + τ n , i ) + β n , i t n p 2 α n , i ( 1 α n , i β n , i ) u n T i n u n 2 β n , i ( 1 α n , i β n , i ) u n t n 2 α n , i β n , i T i n u n t n 2 ( 1 + α n , i γ n , i ) u n p 2 + α n , i τ n , i α n , i ( 1 α n , i β n , i κ i ) u n T i n u n 2 β n , i ( 1 α n , i β n , i ) u n t n 2 α n , i β n , i T i n u n t n 2 ( 1 + γ n , i ) u n p 2 + τ n , i α n , i ( 1 α n , i β n , i κ i ) u n T i n u n 2 x n p 2 + θ n , i , (3.6)

where θ n , i : = γ n , i Δ n 2 + τ n , i and Δ n : = sup { x n z : z Ω } .

Firstly, we show that C n Q n is closed and convex for all n N . It is obvious that C 1 Q 1 is closed and, by mathematical induction, that C n Q n is closed for all n 2 , that is, C n Q n is closed for all n N . Since, for any z H , y n , i z 2 x n z 2 + θ n , i is equivalent to

y n , i x n 2 + 2 y n , i x n , x n z θ n , i 0 (3.7)

for all n N . Therefore, for any z 1 , z 2 C n + 1 Q n + 1 C n Q n and ϵ ( 0 , 1 ) , we have

(3.8)

(3.9)

for all n N . Since C 1 Q 1 is convex, by putting n = 1 in (3.7), (3.8), and (3.9), we have that C 2 Q 2 is convex. Suppose that x k is given and C k Q k is convex for some k 2 . It follows by putting n = k in (3.7), (3.8), and (3.9) that C k + 1 Q k + 1 is convex. Therefore, by mathematical induction, we have that C n Q n is convex for all n 2 , that is, C n Q n is convex for all n N . Hence, we obtain that C n Q n is closed and convex for all n N .

Next, we show that Ω C n Q n for all n N . It is obvious that p Ω H = C 1 Q 1 . Therefore, by (3.1) and (3.6), we have p C 2 , i for all i, and so p C 2 , and note that p H = Q 2 , and so p C 2 Q 2 . Hence, we have Ω C 2 Q 2 . Since C 2 Q 2 is a nonempty closed convex subset of H, there exists a unique element x 2 C 2 Q 2 such that x 2 = P C 2 Q 2 ( x 1 ) . Suppose that x k C k Q k is given such that x k = P C k Q k ( x 1 ) , and p Ω C k Q k for some k 2 . Therefore, by (3.1) and (3.6), we have p C k + 1 , i for all i, and so p C k + 1 . Since x k = P C k Q k ( x 1 ) , therefore, by Lemma 2.1, we have

x k z , x 1 x k 0

for all z C k Q k . Thus, by (3.1), we have p Q k + 1 , and so p C k + 1 Q k + 1 . Hence, we have Ω C k + 1 Q k + 1 . Since C k + 1 Q k + 1 is a nonempty closed convex subset of H, there exists a unique element x k + 1 C k + 1 Q k + 1 such that x k + 1 = P C k + 1 Q k + 1 ( x 1 ) . Therefore, by mathematical induction, we obtain that Ω C n Q n for all n 2 , and so Ω C n Q n for all n N , and we can define x n + 1 = P C n + 1 Q n + 1 ( x 1 ) for all n N . Hence, we obtain that the iteration (3.1) is well defined, and by Lemmas 2.3(5), 2.4(3), and 2.6(4), we also obtain that P Ω ( x 1 ) is well defined.

Next, we show that { x n } is bounded. Since x n = P C n Q n ( x 1 ) for all n N , we have

x n x 1 z x 1 , (3.10)

for all z C n Q n . It follows by p Ω C n Q n that x n x 1 p x 1 for all n N . This implies that { x n } is bounded, and so are { u n } , { t n } , and { y n , i } for each i = 1 , 2 ,  .

Next, we show that x n x n + 1 0 as n . Since x n + 1 = P C n + 1 Q n + 1 ( x 1 ) C n + 1 Q n + 1 C n Q n , therefore, by (3.10), we have x n x 1 x n + 1 x 1 for all n N . This implies that { x n x 1 } is a bounded nondecreasing sequence, there exists the limit of x n x 1 , that is,

lim n x n x 1 = m 1 (3.11)

for some m 1 0 . Since x n + 1 Q n + 1 , therefore, by (3.1), we have

x n x n + 1 , x 1 x n 0 . (3.12)

It follows that

x n x n + 1 2 = x n x 1 2 + 2 x n x 1 , x 1 x n + 2 x n x 1 , x n x n + 1 + x n + 1 x 1 2 x n + 1 x 1 2 x n x 1 2 .

Therefore, by (3.11), we obtain

x n x n + 1 0 as  n . (3.13)

Next, we show that { x n } is a Cauchy sequence. Observe that

C 1 Q 1 C 2 Q 2 C n Q n Ω .

It follows that

x n + m = P C n + m Q n + m ( x 1 ) C n + m Q n + m C n + 1 Q n + 1 Q n + 1

for each m 1 . Therefore, by (3.1), we have

x n x n + m , x 1 x n 0 . (3.14)

Thus, we have

x n + m x n 2 = x n + m x 1 2 + x n x 1 2 2 x n + m x 1 , x n x 1 = x n + m x 1 2 x n x 1 2 2 x n + m x n , x n x 1 x n + m x 1 2 x n x 1 2 . (3.15)

Hence, by (3.11), we obtain that x n + m x n 0 as n , which implies that { x n } is a Cauchy sequence in H, and then there exists a point w H such that x n w as n .

Next, we show that y n , i x n 0 as n . From (3.1), we have x n + 1 = P C n + 1 Q n + 1 ( x 1 ) C n + 1 Q n + 1 C n + 1 C n + 1 , i . Therefore, we have

y n , i x n + 1 2 x n x n + 1 2 + θ n , i . (3.16)

It follows by (3.13) and lim n θ n , i = 0 that

y n , i x n + 1 0 as  n . (3.17)

Since,

y n , i x n y n , i x n + 1 + x n + 1 x n . (3.18)

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

y n , i x n 0 as  n . (3.19)

Next, we show that u n x n 0 and u n + 1 u n 0 as n . By (3.2), (3.3), and the firmly nonexpansiveness of S r n in Lemma 2.3(3), we have

u n p 2 S r n x n S r n p , x n p = u n p , x n p = 1 2 ( u n p 2 + x n p 2 u n x n 2 ) ,

which implies that

u n p 2 x n p 2 u n x n 2 . (3.20)

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

y n , i p 2 ( 1 + γ n , i ) u n p 2 + τ n , i u n p 2 + θ n , i x n p 2 u n x n 2 + θ n , i ,

which implies that

u n x n 2 x n p 2 y n , i p 2 + θ n , i x n y n , i ( x n p + y n , i p ) + θ n , i . (3.21)

Therefore, by (3.19) and lim n θ n , i = 0 , we obtain

u n x n 0 as  n . (3.22)

Since

u n + 1 u n u n + 1 x n + 1 + x n + 1 x n + x n u n , (3.23)

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

u n + 1 u n 0 as  n . (3.24)

Next, we show that w i = 1 F ( T i ) . From (3.4) and (3.6), we have

y n , i p 2 ( 1 + γ n , i ) u n p 2 + τ n , i α n , i ( 1 α n , i β n , i κ i ) u n T i n u n 2 u n p 2 + θ n , i α n , i ( 1 α n , i β n , i κ i )