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Hybrid algorithms for equilibrium and common fixed point problems with applications

Abstract

In this paper, hybrid algorithms are investigated for equilibrium and common fixed point problems. Strong convergence of the algorithms is obtained in the framework of reflexive Banach spaces.

1 Introduction

Iterative algorithms have emerged as an effective and powerful tool for studying a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity and optimization; see [111] and the references therein. The computation of solutions of nonlinear operator equations (inequalities) is important in the study of many real-world problems. Recently, the study of the convergence of various iterative algorithms for solving various nonlinear mathematical models has formed a major part of numerical mathematics. Among these iterative algorithms, the Krasnoselski-Mann iterative algorithm and the Ishikawa iterative algorithm are popular and much discussed. It is well known that both the Krasnoselski-Mann iterative algorithm and the Ishikawa iterative algorithm only have weak convergence even for nonexpansive mappings in infinite-dimensional Hilbert spaces. In many disciplines, including economics, image recovery, quantum physics, and control theory, problems arise in infinite-dimensional spaces. In such problems, strong convergence is often much more desirable than weak convergence. To improve the weak convergence of the Krasnoselski-Mann iterative algorithm and the Ishikawa iterative algorithm, so-called projection algorithms have been investigated in different frameworks of spaces; see [1226] and the references therein.

The aim of this article is to investigate solutions of equilibrium and common fixed point problems in Hilbert spaces. In Section 2, we provide some necessary preliminaries. In Section 3, a projection algorithm is investigated. Strong convergence of the algorithm is obtained in a uniformly smooth and strictly convex Banach space which also has the Kadec-Klee property. In Section 4, applications are provided to support the main results of this paper.

2 Preliminaries

Let E be a real Banach space with the dual E . Recall that the normalized duality mapping J from E to 2 E is defined by Jx={ f E :x, f = x 2 = f 2 }, where , denotes the generalized duality pairing. Let U E ={xE:x=1} be the unit sphere of E. E is said to be smooth iff lim t 0 x + t y x t exists for each x,y U E . It is also said to be uniformly smooth iff the above limit is attained uniformly for x,y U E . E is said to be strictly convex iff x + y 2 <1 for all x,yE with x=y=1 and xy. It is said to be uniformly convex iff lim n x n y n =0 for any two sequences { x n } and { y n } in E such that x n = y n =1 and lim n x n + y n 2 =1. It is well known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. It is also well known that if E is (uniformly) smooth if and only if E is (uniformly) convex.

In what follows, we use and → to stand for the weak and strong convergence, respectively. Recall that a space E has the Kadec-Klee property iff for any sequence { x n }E, xE with x n x, and x n x, we have x n x0 as n. It is known that if E is a uniformly convex Banach spaces, then E has the Kadec-Klee property.

Let E be a smooth Banach space. Let us consider the functional defined by ϕ(x,y)= x 2 2x,Jy+ y 2 , x,yE. Observe that, in a Hilbert space H, the equality is reduced to ϕ(x,y)= x y 2 , x,yH. As we know, if C is a nonempty, closed, and convex subset of a Hilbert space H and P C :HC is the metric projection of H onto C, then P C is nonexpansive. This fact actually characterizes Hilbert spaces, and consequently it is not available in more general Banach spaces. In this connection, Alber [27] recently introduced a generalized projection operator Π C in a Banach space E which is an analog of the metric projection P C in Hilbert spaces. Recall that the generalized projection Π C :EC is a map that assigns to an arbitrary point xE the minimum point of the functional ϕ(x,y), that is, Π C x= x ¯ , where x ¯ is the solution to the minimization problem ϕ( x ¯ ,x)= min y C ϕ(y,x). Existence and uniqueness of the operator Π C follow from the properties of the functional ϕ(x,y) and strict monotonicity of the mapping J. If E is a reflexive, strictly convex, and smooth Banach space, then ϕ(x,y)=0 if and only if x=y; for more details, see [27] and the references therein. In Hilbert spaces, Π C = P C . From the definition of the function ϕ, we also have

( x y ) 2 ϕ(x,y) ( y + x ) 2 ,x,yE
(2.1)

and

( x y ) 2 ϕ(x,y) ( y + x ) 2 ,x,yE.
(2.2)

Let F be a bifunction from C×C to , where denotes the set of real numbers. Recall the following equilibrium problem. Find pC such that F(p,y)0, yC. We use EP(F) to denote the solution set of the equilibrium problem. Numerous problems in physics, optimization and economics reduce to finding a solution of the equilibrium problem.

Next, we give the following assumptions:

(A1) F(x,x)=0, xC;

(A2) F is monotone, i.e., F(x,y)+F(y,x)0, x,yC;

(A3) lim sup t 0 F(tz+(1t)x,y)F(x,y), x,y,zC;

(A4) for each xC, yF(x,y) is convex and weakly lower semi-continuous.

Let C be a nonempty subset of E and let T:CC be a mapping. In this paper, we use F(T) to stand for the fixed point set of T. Recall that T is said to be asymptotically regular on C iff for any bounded subset K of C, lim sup n { T n + 1 x T n x:xK}=0. Recall that T is said to be closed iff for any sequence { x n }C such that lim n x n = x 0 and lim n T x n = y 0 , then T x 0 = y 0 . Recall that a point p in C is said to be an asymptotic fixed point of T iff C contains a sequence { x n } which converges weakly to p such that lim n x n T x n =0. The set of asymptotic fixed points of T will be denoted by F ˜ (T). T is said to be relatively nonexpansive iff F ˜ (T)=F(T) and

ϕ(p,Tx)ϕ(p,x),xC,pF(T).

T is said to be relatively asymptotically nonexpansive iff F ˜ (T)=F(T) and

ϕ ( p , T n x ) (1+ μ n )ϕ(p,x),xC,pF(T),n1,

where { μ n }[0,) is a sequence such that μ n 0 as n.

Remark 2.1 The class of relatively asymptotically nonexpansive mappings, which is an extension of the class of relatively nonexpansive mappings, was first introduced in [28].

Recall that T is said to be quasi-ϕ-nonexpansive iff F(T) and

ϕ(p,Tx)ϕ(p,x),xC,pF(T).

Recall that T is said to be asymptotically quasi-ϕ-nonexpansive iff there exists a sequence { μ n }[0,) with μ n 0 as n such that

F(T),ϕ ( p , T n x ) (1+ μ n )ϕ(p,x),xC,pF(T),n1.

Remark 2.2 The class of asymptotically quasi-ϕ-nonexpansive mappings [14, 29] which is an extension of the class of quasi-ϕ-nonexpansive mappings [8]. The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings. Quasi-ϕ-nonexpansive mappings and asymptotically quasi-ϕ-nonexpansive ones do not require the restriction F(T)= F ˜ (T).

In order to prove our main results, we need the following lemmas.

Lemma 2.3 [30]

Let E be a smooth and uniformly convex Banach space and let r>0. Then there exists a strictly increasing, continuous, and convex function g:[0,2r]R such that g(0)=0 and

i = 1 ( α i x i ) 2 i = 1 ( α i x i 2 ) α i α j g ( x i x j ) ,

for all x 1 , x 2 ,, x N , B r :={xE:xr} and α 1 , α 2 ,, α N ,[0,1] such that i = 1 α i =1.

Lemma 2.4 [27]

Let E be a reflexive, strictly convex, and smooth Banach space. Let C be a nonempty, closed, and convex subset of E and let xE. Then

ϕ(y, Π C x)+ϕ( Π C x,x)ϕ(y,x),yC.

Lemma 2.5 [27]

Let C be a nonempty, closed, and convex subset of a smooth Banach space E and let xE. Then x 0 = Π C x if and only if

x 0 y,JxJ x 0 0yC.

Lemma 2.6 [31]

Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property and let C be a nonempty, closed, and convex subset of E. Let T:CC be an asymptotically quasi-ϕ-nonexpansive mapping. Then F(T) is closed and convex.

Lemma 2.7 [8, 32]

Let C be a closed and convex subset of a smooth, strictly convex, and reflexive Banach space E. Let F be a bifunction from C×C to satisfying the assumptions (A1)-(A4). Let r>0 and let xE. Then there exists zC such that F(z,y)+ 1 r yz,JzJx0, yC. Define a mapping T r :EC by

S r x= { z C : f ( z , y ) + 1 r y z , J z J x , y C } .

Then the following conclusions hold:

  1. (1)

    S r is a single-valued firmly nonexpansive-type mapping, i.e., for all x,yE, S r x S r y,J S r xJ S r y S r x S r y,JxJy;

  2. (2)

    F( S r )=EP(F) is closed and convex;

  3. (3)

    S r is quasi-ϕ-nonexpansive;

  4. (4)

    ϕ(q, S r x)+ϕ( S r x,x)ϕ(q,x), qF( S r ).

3 Main results

Theorem 3.1 Let E be a uniformly smooth and strictly convex Banach space which also has the Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E and let Δ be an index set. Let F j be a bifunction from C×C to satisfying (A1)-(A4) for every jΔ. Let T i :CC an asymptotically quasi-ϕ-nonexpansive mapping for every i1. Assume that T i is closed asymptotically regular on C and i = 1 F( T i ) j Δ EP( F j ) is nonempty and bounded. Let { x n } be a sequence generated in the following manner:

{ x 0 E , chosen arbitrarily, C 1 , j = C , C 1 = j Δ C 1 , j , x 1 = Π C 1 x 0 , y n = J 1 ( α n , 0 J x n + i = 1 α n , i J T i n x n ) , u n , j C  such that  F j ( u n , j , y ) + 1 r n , j y u n , j , J u n , j J y n 0 , y C , C n + 1 , j = { z C n : ϕ ( z , u n , j ) ϕ ( z , x n ) + i = 1 μ n , i M n } , C n + 1 = j Δ C n + 1 , j , x n + 1 = Π C n + 1 x 0 ,

where { α n , i } is a real number sequence in (0,1) for every i1, { r n , j } is a real number sequence in [r,), where r is some positive real number and M n :=sup{ϕ(z, x n ):z i = 1 F( T i ) j Δ EP( F j )}. Assume that i = 0 α n , i =1 and lim inf n α n , 0 α n , i >0 for every i1. Then the sequence { x n } converges strongly to Π i = 1 F ( T i ) j Δ E P ( F j ) x 0 , where Π i = 1 F ( T i ) j Δ E P ( F j ) is the generalized projection from E onto i = 1 F( T i ) j Δ EP( F j ).

Proof Using Lemma 2.6 and Lemma 2.7, we find that i = 1 F( T i ) j Δ EP( F j ) is closed and convex so that Π i = 1 F ( T i ) j Δ E P ( F j ) x 0 is well defined. By induction, we easily find that the sets C n are convex and closed.

Next, we show that i = 1 F( T i ) j Δ EF( F j ) C n . It suffices to claim that

i = 1 F( T i ) j Δ EF( F j ) C n , j ,

for every jΔ. It is clear that i = 1 F( T i ) j Δ EF( F j ) C 1 , j =C. Now, we assume that i = 1 F( T i ) j Δ EF( F j ) C m , j for some m and for every jΔ. Since z i = 1 F( T i ) j Δ EF( F j ) C m , j , one finds that

ϕ ( z , u m , j ) = ϕ ( z , S r m , j y m ) ϕ ( z , y m ) = ϕ ( z , J 1 ( α m , 0 J x m + i = 1 α m , i J T i m x m ) ) = z 2 2 z , α m , 0 J x m + i = 1 α m , i J T i m x m + α m , 0 J x m + i = 1 α m , i J T i m x m 2 z 2 2 α m , 0 z , J x m 2 i = 1 α m , i z , J T i m x m + α m , 0 x m 2 + i = 1 α m , i T i m x m 2 = α m , 0 ϕ ( z , x m ) + i = 1 α m , i ϕ ( z , T i m x m ) α m , 0 ϕ ( z , x m ) + i = 1 α m , i ϕ ( z , x m ) + i = 1 α m , i μ m , i ϕ ( z , x m ) ϕ ( z , x m ) + i = 1 μ m , i ϕ ( z , x m ) ϕ ( z , x m ) + i = 1 μ m , i M m ,
(3.1)

which yields z C m + 1 , j , that is, i = 1 F( T i ) j Δ EF( F j ) C n .

Next, we prove that x n p, where p i = 1 F( T i ) j Δ EF( F j ). It follows from Lemma 2.4 that ϕ( x n , x 0 )ϕ(w, x 0 )ϕ(w, x n )ϕ(w, x 0 ), for w i = 1 N F( T i ) j Δ EF( F j ). This shows that the sequence ϕ( x n , x 0 ) is bounded. Hence { x n } is also bounded. Since the space is reflexive, we may, without loss of generality, assume that x n p, where p C n . Using ϕ( x n , x 0 )ϕ(p, x 0 ), we find that

ϕ(p, x 0 ) lim inf n ϕ( x n , x 0 ) lim sup n ϕ( x n , x 0 )ϕ(p, x 0 ).

It follows that lim n ϕ( x n , x 0 )=ϕ(p, x 0 ). Hence, we have lim n x n =p. Since E has the Kadec-Klee property, one sees that x n p as n. Next, we prove p j Δ EF( F j ). In view of C n + 1 C n and x n + 1 = Π C n + 1 x 0 C n , we have ϕ( x n + 1 , x n )ϕ( x n + 1 , x 0 )ϕ( x n , x 0 ). Letting n, we obtain ϕ( x n + 1 , x n )0. Since x n + 1 C n + 1 , we see that ϕ( x n + 1 , u n , j )ϕ( x n + 1 , x n )+ i = 1 μ n , i M n . We, therefore, obtain lim n ϕ( x n + 1 , u n , j )=0. Hence lim n u n , j =p. It follows that lim n J u n , j =Jp. This shows that {J u n , j } is a bounded sequence. Since E is reflexive ( E is also reflexive), we may assume that J u n , j u , j E . Using the reflexivity of E, we also see that J(E)= E . This shows that there exists an u j E such that J u j = u , j . It follows that ϕ( x n + 1 , u n )= x n + 1 2 2 x n + 1 ,J u n + J u n 2 . Taking lim inf n in both sides of the equality above yields

0 p 2 2 p , u , j + u , j 2 = p 2 2 p , J u j + J u j 2 = ϕ ( p , u j ) ,

that is, p= u j . This yields Jp= u , j . It follows that J u n , j Jp E . Since E has the Kadec-Klee property, we obtain J u n , j Jp0 as n. Since J 1 : E E is demicontinuous. It follows that u n , j p. Since the space E has the Kadec-Klee property, one finds that u n , j p as n. In view of x n u n , j x n p+p u n , j , we have lim n x n u n , j =0. It follows that lim n J x n J u n , j =0. Notice that

ϕ ( z , x n ) ϕ ( z , u n , j ) = x n 2 u n , j 2 2 z , J x n J u n , j x n u n , j ( x n + u n , j ) + 2 z J x n J u n , j .

It follows that lim n ϕ(z, x n )ϕ(z, u n , j )=0. By virtue of (3.1), we find that ϕ(z, y n )ϕ(z, x n )+ i = 1 N μ n , j M n , where z i = 1 F( T i ) j Δ EF( F j ). In view of u n , j = S r n , j y n , we find from Lemma 2.7 that

ϕ ( u n , j , y n ) = ϕ ( S r n , j y n , y n ) ϕ ( z , y n ) ϕ ( z , S r n , j y n ) ϕ ( z , x n ) ϕ ( z , S r n , j y n ) + i = 1 μ n , j M n = ϕ ( z , x n ) ϕ ( z , u n , j ) + i = 1 μ n , j M n .

It follows that lim n ϕ( u n , j , y n )=0. This in turn yields that u n , j y n 0 as n. Since u n , j p as n, one finds that lim n y n =p. It follows that lim n J y n =Jp. Since E is also reflexive, we may assume that J y n y E . In view of J(E)= E , we see that there exists yE such that Jy= y . It follows that

ϕ( u n , j , y n )= u n , j 2 2 u n , j ,J y n + J y n 2 .

Taking lim inf n in both sides of the equality above yields

0 p 2 2 p , y + y 2 = p 2 2 p , J y + J y 2 = ϕ ( p , y ) .

That is, p=y, which in turn implies that y =Jp. It follows that J y n Jp E . Since the space E has the Kadec-Klee property, we arrive at J y n Jp0 as n. Note that J 1 : E E is demicontinuous. It follows that y n p. Since E has the Kadec-Klee property, we obtain y n p as n. In view of u n , j y n u n , j p+p y n , we find that lim n u n , i y n =0. Since J is uniformly norm-to-norm continuous on any bounded sets, we have lim n J u n , j J y n =0. From the assumption r n , j r j , we see that lim n J u n , j J y n r n , j =0. Notice that

F j ( u n , j ,y)+ 1 r n , j y u n , j ,J u n , j J y n 0,yC.

From the condition (A2), we find that

y u n , j J u n , j J y n r n , j 1 r n , j y u n , j ,J u n , j J y n F j (y, u n , j ),yC.

Taking the limit as n, we find that F j (y,p)0, yC. For 0< t j <1 and yC, define y t j = t j y+(1 t j )p. It follows that y t , j C, which yields F j ( y t , j ,p)0. It follows from the conditions (A1) and (A4) that

0= F j ( y t , j , y t , j ) t j F j ( y t , j ,y)+(1 t j ) F j ( y t , j ,p) t j F j ( y t , j ,y).

This yields F j ( y t , j ,y)0. Letting t j 0, we find from the condition (A3) that F j (p,y)0, yC. This implies that pEP( F j ) for every jΔ.

Now, we are in a position to show that p i = 1 F( T i ). It follows from Lemma 2.3 that

ϕ ( z , u n , j ) ϕ ( z , y n ) = ϕ ( z , J 1 ( α n , 0 J x n + i = 1 α n , i J T i n x n ) ) = z 2 2 z , α n , 0 J x n + i = 1 α n , i J T i n x n + α n , 0 J x n + i = 1 α n , i J T i n x n 2 z 2 2 α n , 0 z , J x n 2 i = 1 α n , i z , J T i n x n + α n , 0 x n 2 + i = 1 α n , i T i n x n 2 α n , 0 ( 1 α n , i ) g ( J x n J T i n x n ) = α n , 0 ϕ ( z , x m ) + i = 1 α n , i ϕ ( z , T i n x n ) α n , 0 ( 1 α n , i ) g ( J x n J T i n x n ) α n , 0 ϕ ( z , x m ) + i = 1 α n , i ϕ ( z , x m ) + i = 1 α n , i μ n , i ϕ ( z , x n ) α n , 0 ( 1 α n , i ) g ( J x n J T i n x n ) ϕ ( z , x n ) + i = 1 μ n , i ϕ ( z , x m ) α n , 0 ( 1 α n , i ) g ( J x n J T i n x n ) ϕ ( z , x n ) + i = 1 μ n , i M n α n , 0 ( 1 α n , i ) g ( J x n J T i n x n ) .

From lim inf n α n , 0 (1 α n , i )>0, we find that lim n g(J x n J T i n x n )=0. Hence, we have

lim n J x n J T i n x n =0.
(3.2)

Since x n p as n and J:E E is demicontinuous, we obtain J x n Jp E . Note that |J x n Jp|=| x n p| x n p. This implies that J x n Jp as n. Since E has the Kadec-Klee property, we see that

lim n J x n Jp=0.
(3.3)

On the other hand, we have J T i n x n JpJ T i n x n J x n +J x n Jp. Combining (3.2) with (3.3), one obtains that lim n J T i n x n Jp=0. Since J 1 : E E is demicontinuous, one sees that T i n x n p. Notice that

| T i n x n p | = | J T i n x n J p | J T i n x n J p .

This yields lim n T i n x n =p. Since the space E has the Kadec-Klee property, we obtain lim n T i n x n p=0. Since T n + 1 x n p T n + 1 x n T n x n + T n x n p and T is asymptotically regular, we find that lim n T i n + 1 x n p=0. That is, T i T i n x n p0 as n. It follows from the closedness of T i that T i p=p for every i1.

Finally, we prove that p= Π i = 1 F ( T i ) j Δ E F ( F j ) x 0 . Since x n = Π C n x 0 , we see that

x n z,J x 0 J x n 0,z C n .

Since i = 1 F( T i ) j Δ EP( F j ) C n , we find that

x n w,J x 0 J x n 0,w i = 1 N F( T i ) j Δ EF( F j ).

Letting n, we arrive at pw,J x 0 Jp0, w i = 1 F( T i ) j Δ EP( F j ). In view of Lemma 2.5, we find that p= Π i = 1 F ( T i ) j Δ E F ( F j ) x 0 . This completes the proof. □

Remark 3.2 Theorem 3.1 mainly improves the corresponding results in Kim [11] and Qin et al. [33].

If T is quasi-ϕ-nonexpansive, we have the following result.

Corollary 3.3 Let E be a uniformly smooth and strictly convex Banach space which also has the Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E and let Δ be an index set. Let F j be a bifunction from C×C to satisfying (A1)-(A4) for every jΔ. Let T i :CC a quasi-ϕ-nonexpansive mapping for every i1. Assume that T i is closed on C and i = 1 F( T i ) j Δ EP( F j ) is nonempty. Let { x n } be a sequence generated in the following manner:

{ x 0 E , chosen arbitrarily, C 1 , j = C , C 1 = j Δ C 1 , j , x 1 = Π C 1 x 0 , y n = J 1 ( α n , 0 J x n + i = 1 α n , i J T i x n ) , u n , j C  such that  F j ( u n , j , y ) + 1 r n , j y u n , j , J u n , j J y n 0 , y C , C n + 1 , j = { z C n : ϕ ( z , u n , j ) ϕ ( z , x n ) } , C n + 1 = j Δ C n + 1 , j , x n + 1 = Π C n + 1 x 0 ,

where { α n , i } is a real number sequence in (0,1) for every i1, { r n , j } is a real number sequence in [r,), where r is some positive real number. Assume that i = 0 N α n , i =1 and lim inf n α n , 0 α n , i >0 for every i1. Then the sequence { x n } converges strongly to Π i = 1 F ( T i ) j Δ E P ( F j ) x 0 , where Π i = 1 F ( T i ) j Δ E P ( F j ) is the generalized projection from E onto i = 1 F( T i ) j Δ EP( F j ).

Remark 3.4 Corollary 3.3 mainly improves the corresponding results in Qin et al. [8]. Indeed, the space is extended from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space; the mapping is extended from a pari of mappings to infinite many mappings. We also remark here that the framework of the space in this paper can be applicable to L p , p1.

Finally for a single mapping and bifunction, we have the following result.

Corollary 3.5 Let E be a uniformly smooth and strictly convex Banach space which also has the Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E and let F be a bifunction from C×C to satisfying (A1)-(A4). Let T:CC an asymptotically quasi-ϕ-nonexpansive mapping. Assume that T is closed asymptotically regular on C and F(T)EP(F) is nonempty and bounded. Let { x n } be a sequence generated in the following manner:

{ x 0 E , chosen arbitrarily, C 1 = C , x 1 = Π C 1 x 0 , y n = J 1 ( α n J x n + ( 1 α n ) J T n x n ) , u n C  such that  F ( u n , y ) + 1 r n y u n , J u n J y n 0 , y C , C n + 1 = { z C n : ϕ ( z , u n ) ϕ ( z , x n ) + μ n M n } , x n + 1 = Π C n + 1 x 0 ,

where { α n } is a real number sequence in (0,1), { r n } is a real number sequence in [r,), where r is some positive real number and M n :=sup{ϕ(z, x n ):zF(T)EP(F)}. Assume lim inf n α n (1 α n )>0. Then the sequence { x n } converges strongly to Π F ( T ) E P ( F ) x 0 , where Π F ( T ) E P ( F ) is the generalized projection from E onto F(T)EP(F).

4 Applications

First, we give the corresponding results in the framework of Hilbert spaces.

Theorem 4.1 Let E be a Hilbert space. Let C be a nonempty, closed, and convex subset of E and let Δ be an index set. Let F j be a bifunction from C×C to satisfying (A1)-(A4) for every jΔ. Let T i :CC an asymptotically quasi-nonexpansive mapping for every i1. Assume that T i is closed asymptotically regular on C and i = 1 F( T i ) j Δ EP( F j ) is nonempty and bounded. Let { x n } be a sequence generated in the following manner:

{ x 0 E , chosen arbitrarily, C 1 , j = C , C 1 = j Δ C 1 , j , x 1 = Proj C 1 x 0 , y n = α n , 0 x n + i = 1 α n , i T i n x n , u n , j C such that F j ( u n , j , y ) + 1 r n , j y u n , j , u n , j y n 0 , y C , C n + 1 , j = { z C n : z u n , j 2 z x n 2 + i = 1 μ n , i M n } , C n + 1 = j Δ C n + 1 , j , x n + 1 = Proj C n + 1 x 0 ,

where { α n , i } is a real number sequence in (0,1) for every i1, { r n , j } is a real number sequence in [r,), where r is some positive real number and M n :=sup{ z x n 2 :z i = 1 F( T i ) j Δ EP( F j )}. Assume that i = 0 α n , i =1 and lim inf n α n , 0 α n , i >0 for every i1. Then the sequence { x n } converges strongly to Proj i = 1 F ( T i ) j Δ E P ( F j ) x 0 , where Proj i = 1 F ( T i ) j Δ E P ( F j ) is the metric projection from E onto i = 1 F( T i ) j Δ EP( F j ).

Proof Note that ϕ(x,y)= x y 2 , J=I, the identity mapping, and the generalized projection is reduced to the metric projection. In the framework of Hilbert spaces, the class of asymptotically quasi-ϕ-nonexpansive mappings is reduced to the class of asymptotically quasi-nonexpansive mappings. Using Theorem 3.1, we easily find the desired conclusion. □

Next, we give a results on common solutions of a family of equilibrium problems.

Corollary 4.2 Let E be a Hilbert space. Let C be a nonempty, closed, and convex subset of E and let Δ be an index set. Let F j be a bifunction from C×C to satisfying (A1)-(A4) for every jΔ. Assume that j Δ EP( F j ) is nonempty. Let { x n } be a sequence generated in the following manner:

{ x 0 E , chosen arbitrarily, C 1 , j = C , C 1 = j Δ C 1 , j , x 1 = Proj C 1 x 0 , u n , j C  such that  F j ( u n , j , y ) + 1 r n , j y u n , j , u n , j x n 0 , y C , C n + 1 , j = { z C n : z u n , j z x n } , C n + 1 = j Δ C n + 1 , j , x n + 1 = Proj C n + 1 x 0 ,

where { r n , j } is a real number sequence in [r,), where r is some positive real number. Then the sequence { x n } converges strongly to Proj j Δ E P ( F j ) x 0 , where Proj j Δ E P ( F j ) is the metric projection from E onto j Δ EP( F j ).

If F j (x,y)=0, then we find from Theorem 4.1 the following result.

Corollary 4.3 Let E be a Hilbert space and let C be a nonempty, closed, and convex subset of E. Let T i :CC an asymptotically quasi-nonexpansive mapping for every i1. Assume that T i is closed asymptotically regular on C and i = 1 F( T i ) is nonempty and bounded. Let { x n } be a sequence generated in the following manner:

{ x 0 E , chosen arbitrarily, C 1 = C , x 1 = Proj C 1 x 0 , y n = α n , 0 x n + i = 1 α n , i T i n x n , C n + 1 = { z C n : z y n 2 z x n 2 + i = 1 μ n , i M n } , x n + 1 = Proj C n + 1 x 0 ,

where { α n , i } is a real number sequence in (0,1) for every i1, and M n :=sup{ z x n 2 :z i = 1 F( T i )}. Assume that i = 0 α n , i =1 and lim inf n α n , 0 α n , i >0 for every i1. Then the sequence { x n } converges strongly to Proj i = 1 F ( T i ) x 0 , where Proj i = 1 F ( T i ) is the metric projection from E onto i = 1 F( T i ).

Remark 4.4 Comparing with Theorem 2.2 in Kim and Xu [34], we have the following: (1) one mapping is extended to an infinite family of mappings; (2) the set Q n is relaxed; (3) the mapping is extended from asymptotically nonexpansive mappings to asymptotically quasi-nonexpansive mappings, which may not be continuous.

From Corollary 4.3, the following result is not hard to derive.

Corollary 4.5 Let E be a Hilbert space. Let C be a nonempty, closed, and convex subset of E. Let T:CC be a closed quasi-nonexpansive mapping. Let { x n } be a sequence generated in the following manner:

{ x 0 E , chosen arbitrarily, C 1 = C , x 1 = Proj C 1 x 0 , y n = α n x n + ( 1 α n ) T x n , C n + 1 = { z C n : z y n z x n } , x n + 1 = Proj C n + 1 x 0 ,

where { α n } is a real number sequence in (0,1) such that lim inf n α n (1 α n )>0. Then the sequence { x n } converges strongly to Proj F ( T ) x 0 , where Proj F ( T ) is the metric projection from E onto F(T).

Remark 4.6 Corollary 4.5 is a shrinking version of the corresponding results in Nakajo and Takahashi [35]. It deserves mentioning that the mapping in our result is quasi-nonexpansive. The restriction of the demiclosed principal is relaxed.

References

  1. Vanderluge A: Optical Signal Processing. Wiley, New York; 2005.

    Google Scholar 

  2. Byrne C: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 2008, 20: 103–120.

    MathSciNet  Article  MATH  Google Scholar 

  3. Combettes PL: The convex feasibility problem in image recovery. 95. In Advanced in Imaging and Electron Physcis. Edited by: Hawkes P. Academic Press, New York; 1996:155–270.

    Google Scholar 

  4. Chen JH: Iterations for equilibrium and fixed point problems. J. Nonlinear Funct. Anal. 2013., 2013: Article ID 4

    Google Scholar 

  5. Cho SY, Qin X, Kang SM: Iterative processes for common fixed points of two different families of mappings with applications. J. Glob. Optim. 2013, 57: 1429–1446. 10.1007/s10898-012-0017-y

    MathSciNet  Article  MATH  Google Scholar 

  6. Park S: A review of the KKM theory on ϕ A -space or GFC-spaces. Adv. Fixed Point Theory 2013, 3: 355–382.

    Google Scholar 

  7. Noor MA, Noor KI, Waseem M: Decomposition method for solving system of linear equations. Eng. Math. Lett. 2013, 2: 34–41.

    Google Scholar 

  8. Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 2009, 225: 20–30. 10.1016/j.cam.2008.06.011

    MathSciNet  Article  MATH  Google Scholar 

  9. Al-Bayati AY, Al-Kawaz RZ: A new hybrid WC-FR conjugate gradient-algorithm with modified secant condition for unconstrained optimization. J. Math. Comput. Sci. 2012, 2: 937–966.

    MathSciNet  Google Scholar 

  10. Cho SY, Kang SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci. 2012, 32: 1607–1618. 10.1016/S0252-9602(12)60127-1

    Article  MathSciNet  MATH  Google Scholar 

  11. Tanaka Y: Constructive proof of the existence of Nash equilibrium in a strategic game with sequentially locally non-constant payoff functions. Adv. Fixed Point Theory 2012, 2: 398–416.

    Google Scholar 

  12. Kim JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi- ϕ -nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 10

    Google Scholar 

  13. Yang L, Zhao F, Kim JK: Hybrid projection method for generalized mixed equilibrium problem and fixed point problem of infinite family of asymptotically quasi- ϕ -nonexpansive mappings in Banach spaces. Appl. Math. Comput. 2012, 218: 6072–6082. 10.1016/j.amc.2011.11.091

    MathSciNet  Article  MATH  Google Scholar 

  14. Qin X, Cho SY, Kang SM: On hybrid projection methods for asymptotically quasi- ϕ -nonexpansive mappings. Appl. Math. Comput. 2010, 215: 3874–3883. 10.1016/j.amc.2009.11.031

    MathSciNet  Article  MATH  Google Scholar 

  15. Qin X, Agarwal RP, Cho SY, Kang SM: Convergence of algorithms for fixed points of generalized asymptotically quasi- ϕ -nonexpansive mappings with applications. Fixed Point Theory Appl. 2012., 2012: Article ID 58

    Google Scholar 

  16. Qin X, Agarwal RP: Shrinking projection methods for a pair of asymptotically quasi- ϕ -nonexpansive mappings. Numer. Funct. Anal. Optim. 2010, 31: 1072–1089. 10.1080/01630563.2010.501643

    MathSciNet  Article  MATH  Google Scholar 

  17. Chang SS, Lee HWJ, Chan CK, Liu JA: Strong convergence theorems for countable families of asymptotically relatively nonexpansive mappings with applications. Appl. Math. Comput. 2011, 218: 3187–3198. 10.1016/j.amc.2011.08.055

    MathSciNet  Article  MATH  Google Scholar 

  18. Chang SS, Lee HWJ, Chan CK, Yang L: Approximation theorems for total quasi- ϕ -asymptotically nonexpansive mappings with applications. Appl. Math. Comput. 2011, 218: 2921–2931. 10.1016/j.amc.2011.08.036

    MathSciNet  Article  MATH  Google Scholar 

  19. Cho SY, Qin X, Kang SM: Hybrid projection algorithms for treating common fixed points of a family of demicontinuous pseudocontractions. Appl. Math. Lett. 2012, 25: 854–857. 10.1016/j.aml.2011.10.031

    MathSciNet  Article  MATH  Google Scholar 

  20. Li X, Huang NJ, Regan DO: Strong convergence theorems for relatively nonexpansive mappings in Banach spaces with applications. Comput. Math. Appl. 2010, 60: 1322–1331. 10.1016/j.camwa.2010.06.013

    MathSciNet  Article  MATH  Google Scholar 

  21. Cho SY, Qin X: On the strong convergence of an iterative process for asymptotically strict pseudocontractions and equilibrium problems. Appl. Math. Comput. 2014, 235: 430–438.

    MathSciNet  Article  Google Scholar 

  22. Ge CS: A hybrid algorithm with variable coefficients for asymptotically pseudocontractive mappings in the intermediate sense on unbounded domains. Nonlinear Anal. 2012, 75: 2859–2866. 10.1016/j.na.2011.11.026

    MathSciNet  Article  MATH  Google Scholar 

  23. Qin X, Cho SY, Kim JK: Convergence theorems on asymptotically pseudocontractive mappings in the intermediate sense. Fixed Point Theory Appl. 2010., 2010: Article ID 186874

    Google Scholar 

  24. Hao Y: Some results on a modified Mann iterative scheme in a reflexive Banach space. Fixed Point Theory Appl. 2013., 2013: Article ID 227

    Google Scholar 

  25. Hao Y, Cho SY: Fixed point iterations of a pair of hemirelatively nonexpansive mappings. Fixed Point Theory Appl. 2010., 2010: Article ID 270150

    Google Scholar 

  26. Wu C, Lv S: Bregman projection methods for zeros of monotone operators. J. Fixed Point Theory 2013., 2013: Article ID 7

    Google Scholar 

  27. Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartsatos AG. Dekker, New York; 1996.

    Google Scholar 

  28. Agarwal RP, Cho YJ, Qin X: Generalized projection algorithms for nonlinear operators. Numer. Funct. Anal. Optim. 2007, 28: 1197–1215. 10.1080/01630560701766627

    MathSciNet  Article  MATH  Google Scholar 

  29. Zhou H, Gao G, Tan B: Convergence theorems of a modified hybrid algorithm for a family of quasi- ϕ -asymptotically nonexpansive mappings. J. Appl. Math. Comput. 2010, 32: 453–464. 10.1007/s12190-009-0263-4

    MathSciNet  Article  MATH  Google Scholar 

  30. Hao Y: Some weak convergence theorems for a family of asymptotically nonexpansive nonself mappings. Fixed Point Theory Appl. 2010., 2010: Article ID 218573

    Google Scholar 

  31. Qin X, Huang S, Wang T: On the convergence of hybrid projection algorithms for asymptotically quasi-image-nonexpansive mappings. Comput. Math. Appl. 2011, 61: 851–859. 10.1016/j.camwa.2010.12.033

    MathSciNet  Article  MATH  Google Scholar 

  32. Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 2009, 70: 45–57. 10.1016/j.na.2007.11.031

    MathSciNet  Article  MATH  Google Scholar 

  33. Qin X, Cho SY, Kang SM: Strong convergence of shrinking projection methods for quasi- ϕ -nonexpansive mappings and equilibrium problems. J. Comput. Appl. Math. 2010, 234: 750–760. 10.1016/j.cam.2010.01.015

    MathSciNet  Article  MATH  Google Scholar 

  34. Kim TH, Xu HK: Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal. 2006, 64: 1140–1152. 10.1016/j.na.2005.05.059

    MathSciNet  Article  MATH  Google Scholar 

  35. Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 2003, 279: 372–379. 10.1016/S0022-247X(02)00458-4

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

This paper is dedicated to Professor Chang with respect and admiration. The authors are grateful to the editor and the three anonymous reviewers for useful suggestions which improved the contents of the article.

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Zhang, Q., Wu, H. Hybrid algorithms for equilibrium and common fixed point problems with applications. J Inequal Appl 2014, 221 (2014). https://doi.org/10.1186/1029-242X-2014-221

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Keywords

  • asymptotically quasi-ϕ-nonexpansive mapping
  • quasi-ϕ-nonexpansive mapping
  • algorithm
  • equilibrium problem
  • fixed point