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Weak convergence theorems of a hybrid algorithm in Hilbert spaces

Abstract

In this paper, a hybrid algorithm is investigated for solving common solutions of a generalized equilibrium problem, a variational inequality, and fixed point problems of an asymptotically strict pseudocontraction. Weak convergence theorems are established in the framework of real Hilbert spaces.

1 Introduction

Monotone variational inequalities recently have been investigated as an effective and powerful tool for studying a wide class of real world problems which arise in economics, finance, image reconstruction, ecology, transportation, and network; see [19] and the references therein. Monotone variational inequalities, which include many important problems in nonlinear analysis and optimization, such as the Nash equilibrium problem, complementarity problems, fixed point problems, saddle point problems, and game theory recently have been extensively studied based on projection methods. Many well-known problems can be studied by using methods which are iterative in their nature. As an example, in computer tomography with limited data, each piece of information implies the existence of a convex set in which the required solution lies. The problem of finding a point in the intersection of these convex subsets is then of crucial interest, and it cannot be usually solved directly. Therefore, an iterative algorithm must be used to approximate such a point. Krasnoselskii-Mann iteration, which is also known as a one-step iteration, is a classic algorithm to study fixed points of nonlinear operators. However, Krasnoselskii-Mann iteration only enjoys weak convergence for nonexpansive mappings; see [10] and the references therein.

The purposes of this paper is to study common solutions of a generalized equilibrium problem, a variational inequality, and fixed point problems of an asymptotically strict pseudocontraction based on a hybrid algorithm. Weak convergence theorems are established in the framework of real Hilbert spaces. The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a hybrid algorithm is introduced and the convergence analysis is given. Weak convergence theorems are established in a real Hilbert space.

2 Preliminaries

From now on, we always assume that H is a real Hilbert space with the inner product , and the norm , C is a nonempty closed convex subset of H and P C denotes the metric projection from H onto C.

Let A:CH be a mapping. Recall that A is said to be monotone if

AxAy,xy0,x,yC.

A is said to be inverse-strongly monotone if there exists a constant α>0 such that

AxAy,xyα A x A y 2 ,x,yC.

For such a case, we also call it an α-inverse-strongly monotone mapping.

A set-valued mapping T:H 2 H is said to be monotone if for all x,yH, fTx and gTy imply xy,fg>0. A monotone mapping T:H 2 H is maximal if the graph G(T) of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if, for any (x,f)H×H, xy,fg0 for all (y,g)G(T) implies fTx. Let A be a monotone mapping of C into H and N C v be the normal cone to C at vC, i.e.,

N C v= { w H : v u , w 0 , u C }

and define a mapping T on C by

Tv={ A v + N C v , v C , , v C .

Then T is maximal monotone and 0Tv if and only if Av,uv0 for all uC; see [6] and the references therein.

Recall that the classical variational inequality problem is to find xC such that

Ax,yx0,yC.
(2.1)

It is known that xC is a solution to (2.1) if and only if x is a fixed point of the mapping P C (IλA), where λ>0 is a constant and I is the identity mapping. Projection methods recently have been studied for variational inequality (2.1); see [1122] and the references therein.

Let S:CC be a nonlinear mapping. In this paper, we use F(S) to denote the fixed point set of S. Recall that S is said to be nonexpansive if

SxSyxy,x,yC.

S is said to be asymptotically nonexpansive if there exists a sequence { k n }[1,) with lim n k n =1 such that

SxSy k n xy,x,yC.

S is said to be κ-strictly pseudocontractive if there exists a constant k[0,1) such that

S x S y 2 x y 2 +κ ( x S x ) ( y S y ) 2 ,x,yC.

The class of strict pseudocontractions was introduced by Browder and Petryshyn [23]. It is clear that every nonexpansive mapping is a 0-strict pseudocontraction.

T is said to be an asymptotically κ-strict pseudocontraction if there exists a sequence { k n }[1,) with lim n k n =1 and a constant κ[0,1) such that

T n x T n y 2 k n x y 2 +κ ( I T n ) x ( I T n ) y 2 ,x,yC,n1.

The class of asymptotically strict pseudocontractions was introduced by Qihou [24]. It is clear that every asymptotically nonexpansive mapping is an asymptotically 0-strict pseudocontraction.

Let F be a bifunction of C×C into , where denotes the set of real numbers and A:CH is an inverse-strongly monotone mapping. In this paper, we consider the following generalized equilibrium problem:

Find xC such that F(x,y)+Ax,yx0,yC.
(2.2)

In this paper, the set of such xC is denoted by EP(F,A), i.e.,

EP(F,A)= { x C : F ( x , y ) + A x , y x 0 , y C } .

To study the generalized equilibrium problem (2.2), we may assume that F satisfies the following conditions:

  1. (A1)

    F(x,x)=0 for all xC;

  2. (A2)

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

  3. (A3)

    for each x,y,zC,

    lim sup t 0 F ( t z + ( 1 t ) x , y ) F(x,y);
  4. (A4)

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

If A0, then the generalized equilibrium problem (2.2) is reduced to the following equilibrium problem:

Find xC such that F(x,y)0,yC.
(2.3)

In this paper, the set of such xC is denoted by EP(F), i.e.,

EP(F)= { x C : F ( x , y ) 0 , y C } .

If F0, then the generalized equilibrium problem (2.2) is reduced to the classical variational inequality (2.1).

Recently, equilibrium problems (2.2) and (2.3) have been investigated by many authors; see [2531] and the references therein. Motivated by the research going on in this direction, we study a hybrid algorithm for solving common solutions of variational inequality (2.1), generalized equilibrium problem (2.2), and fixed points of an asymptotically strict pseudocontraction. Possible computation errors are taken into account. Weak convergence theorems are established in the framework of real Hilbert spaces.

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

Lemma 2.1 [32]

Let C be a nonempty closed convex subset of H, and let F:C×CR be a bifunction satisfying (A1)-(A4). Then, for any r>0 and xH, there exists zC such that

F(z,y)+ 1 r yz,zx0,yC.

Further, define

T r x= { z C : F ( z , y ) + 1 r y z , z x 0 , y C }

for all r>0 and xH. Then the following hold:

  1. (a)

    T r is single-valued;

  2. (b)

    T r is firmly nonexpansive, i.e., for any x,yH,

    T r x T r y 2 T r x T r y,xy;
  3. (c)

    F( T r )=EP(F);

  4. (d)

    EP(F) is closed and convex.

Lemma 2.2 [24]

Let C be a nonempty closed convex subset of a Hilbert space H and S:CC be an asymptotically strict pseudocontraction. Then IS is demi-closed, that is, if { x n } is a sequence in C with x n x and x n S x n 0, then xF(S).

Lemma 2.3 [33]

Let H be a Hilbert space and 0<p t n q<1 for all n1. Suppose that { x n } and { y n } are sequences in H such that

lim sup n x n d, lim sup n y n d

and

lim n t n x n + ( 1 t n ) y n =d

hold for some d0. Then lim n x n y n =0.

Lemma 2.4 [34]

Let { a n }, { b n }, and { c n } be three nonnegative sequences satisfying the following condition:

a n + 1 (1+ b n ) a n + c n ,n n 0 ,

where n 0 is some nonnegative integer, n = 1 b n < and n = 1 c n <. Then the limit lim n a n exists.

3 Main results

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C×C to which satisfies (A1)-(A4). Let A:CH be an α-inverse-strongly monotone mapping, and let B:CH be a β-inverse-strongly monotone mapping. Let S:CC be an asymptotically κ-strict pseudocontraction with the sequence { k n } such that n = 1 ( k n 1)<. Assume that Ω=F(S)VI(C,B)EP(F,A) is not empty. Let { α n }, { α n }, { α n }, and { β n } be real number sequences in (0,1). Let { r n } and { s n } be two positive real number sequences. Let { x n } be a sequence generated in the following process:

{ x 1 C , F ( z n , z ) + A x n , z z n + 1 r n z z n , z n x n 0 , z C , y n = P C ( z n s n B z n ) , x n + 1 = α n x n + α n ( β n y n + ( 1 β n ) S n y n ) + α n e n ,

where { e n } is a bounded sequence in C. Assume that the control sequences satisfy the following restrictions:

  1. (a)

    α n + α n + α n =1;

  2. (b)

    0<p α n q<1 and n = 1 α n <;

  3. (c)

    0<κ< β n b<1;

  4. (d)

    0<s s n s <2β and 0<r r n r <2α,

where p, q, b, s, s , r, r are real constants. Then { x n } converges weakly to some point in Ω.

Proof First, we show that the sequences { x n }, { y n }, and { z n } are bounded. Let pΩ be fixed arbitrarily. For any x,yC, we see that

( I r n A ) x ( I r n A ) y 2 = ( x y ) r n ( A x A y ) 2 = x y 2 2 r n x y , A x A y + r n 2 A x A y 2 x y 2 r n ( 2 α r n ) A x A y 2 .
(3.1)

Using the restriction (d), we see that (I r n A)x(I r n A)yxy. This implies that I r n A is nonexpansive. In the same way, we find that I s n B is also nonexpansive. Using the restriction (c), we obtain that

β n y n + ( 1 β n ) S n y n p 2 = β n y n p 2 + ( 1 β n ) S n y n S n p 2 β n ( 1 β n ) ( y n p ) ( S n y n S n p ) 2 β n y n p 2 + ( 1 β n ) ( k n y n p 2 + κ ( y n p ) ( S n y n S n p ) 2 ) β n ( 1 β n ) ( y n p ) ( S n y n S n p ) 2 = k n y n p 2 ( 1 β n ) ( β n κ ) ( y n p ) ( S n y n S n p ) 2 k n y n p 2 .
(3.2)

It follows that

x n + 1 p 2 α n x n p 2 + α n β n y n + ( 1 β n ) S n y n p 2 + α n e n p 2 α n x n p 2 + α n k n y n p 2 + α n e n p 2 = α n x n p 2 + α n k n P C ( I s n B ) z n p 2 + α n e n p 2 α n x n p 2 + α n k n T r n ( I r n A ) x n p 2 + α n e n p 2 k n x n p 2 + α n e n p 2 .

This implies from Lemma 2.4 that lim n x n p exists. This shows that { x n } is bounded, so are { y n } and { z n }. From (3.2), we have

x n + 1 p 2 α n x n p 2 + α n k n y n p 2 + α n e n p 2 α n x n p 2 + α n k n ( I s n B ) z n p 2 + α n e n p 2 α n x n p 2 + α n k n ( z n p 2 s n ( 2 β s n ) B z n B p 2 ) + α n e n p 2 k n x n p 2 s n k n α n ( 2 β s n ) B z n B p 2 + α n e n p 2 .

It follows that

s n k n α n (2β s n ) B z n B p 2 k n x n p 2 x n + 1 p 2 + α n e n p 2 .

With the aid of the restrictions (b) and (d), we find that

lim n B z n Bp=0.
(3.3)

Since P C is firmly nonexpansive, we have

y n p 2 = P C ( I s n B ) z n P C ( I s n B ) p 2 ( I s n B ) z n ( I s n B ) p , y n p = 1 2 { ( I s n B ) z n ( I s n B ) p 2 + y n p 2 ( I s n B ) z n ( I s n B ) p ( y n p ) 2 } 1 2 { z n p 2 + y n p 2 z n y n s n ( B z n B p ) 2 } = 1 2 { z n p 2 + y n p 2 z n y n 2 + 2 s n z n y n , B z n B p s n 2 B z n B p 2 } 1 2 { x n p 2 + y n p 2 z n y n 2 + 2 s n z n y n , B z n B p s n 2 B z n B p 2 } ,

which implies that

y n p 2 x n p 2 z n y n 2 +2 s n z n y n B z n Bp.

Hence, we find from (3.2) that

x n + 1 p 2 α n x n p 2 + α n k n y n p 2 + α n e n p 2 k n x n p 2 α n k n z n y n 2 + 2 α n s n k n z n y n B z n B p + α n e n p 2 .

Therefore, we obtain that

α n k n z n y n 2 k n x n p 2 x n + 1 p 2 + 2 s n k n z n y n B z n B p + α n e n p 2 .

From the restrictions (b) and (d), we find from (3.3) that

lim n z n y n =0.
(3.4)

It follows from (3.1) that

z n p 2 = T r n ( I r n A ) x n p 2 x n p 2 r n ( 2 α r n ) A x n A p 2 .

Hence, we have

x n + 1 p 2 α n x n p 2 + α n k n y n p 2 + α n e n p 2 α n x n p 2 + α n k n z n p 2 + α n e n p 2 k n x n p 2 α n r n ( 2 α r n ) k n A x n A p 2 + α n e n p 2 .

This implies that

α n r n (2α r n ) k n A x n A p 2 k n x n p 2 x n + 1 p 2 + α n e n p 2 .

Using the restrictions (b) and (d), we obtain that

lim n A x n Ap=0.
(3.5)

Since T r n is firmly nonexpansive, we find that

z n p 2 = T r n ( I r n A ) x n T r n ( I r n A ) p 2 ( I r n A ) x n ( I r n A ) p , z n p = 1 2 ( ( I r n A ) x n ( I r n A ) p 2 + z n p 2 ( I r n A ) x n ( I r n A ) p ( z n p ) 2 ) 1 2 ( x n p 2 + z n p 2 x n z n r n ( A x n A p ) 2 ) = 1 2 ( x n p 2 + z n p 2 ( x n z n 2 2 r n x n z n , A x n A p r n 2 A x n A p 2 ) ) ,

which implies that

z n p 2 x n p 2 x n z n 2 +2 r n x n z n A x n Ap.

It follows that

x n + 1 p 2 α n x n p 2 + α n k n y n p 2 + α n e n p 2 α n x n p 2 + α n k n z n p 2 + α n e n p 2 k n x n p 2 α n k n x n z n 2 + 2 r n α n k n x n z n A x n A p + α n e n p 2 ,

which yields that

α n k n x n z n 2 k n x n p 2 x n + 1 p 2 + 2 r n α n x n z n A x n A p + α n e n p 2 .

Using the restrictions (b) and (d), we find from (3.5) that

lim n x n z n =0.
(3.6)

It follows from (3.4) and (3.6) that

lim n x n y n =0.
(3.7)

Since { x n } is bounded, we see that there exists a subsequence { x n i } of { x n } which converges weakly to ξ. Let T be a maximal monotone mapping defined by

Tx={ B x + N C x , x C , , x C .

For any given (x,y)Graph(T), we have yBx N C x. Since y n C, by the definition of N C , we have x y n ,yBx0. Since y n = P C (I s n B) z n , we see that x y n , y n (I s n B) z n 0 and hence

x y n , y n z n s n + B z n 0.

It follows that

x y n i , y x y n i , B x x y n i , B x x y n i , y n i z n i s n i + B z n i = x y n i , B x B y n i + x y n i , B y n i B z n i x y n i , y n i z n i s n i x y n i , B y n i B z n i x y n i , y n i z n i s n i .

Since y n i converges weakly to ξ and B is 1 β -Lipschitz continuous, we see that xξ,y0. Notice that T is maximal monotone and hence 0Tξ. This shows that ξVI(C,B). From (3.6), we see that z n i converges weakly to ξ. It follows that

F( z n ,z)+A x n ,z z n + 1 r n z z n , z n x n 0,zC.

From condition (A2), we see that

A x n ,z z n + 1 r n z z n , z n x n F(z, z n ),zC.

Replacing n by n i , we arrive at

A x n i ,z z n i + z z n i , z n i x n i r n i F(z, z n i ),zC.
(3.8)

For t with 0<t1 and zC, let u t =tz+(1t)ξ. Since zC and ξC, we have u t C. In view of (3.8), we find that

u t z n i , A u t u t z n i , A u t A x n i , u t z n i u t z n i , z n i x n i r n i + F ( u t , z n i ) = u t z n i , A u t A z n i + u t z n i , A z n i A x n i u t z n i , z n i x n i r n i + F ( u t , z n i ) .

Using (3.6), we have lim i A z n i A x n i =0. Since A is monotone, we see that u t z n i ,A u t A z n i 0. It follows from condition (A4) that

u t ξ,A u t F( u t ,ξ).
(3.9)

Using conditions (A1) and (A4), we see from (3.9) that

0 = F ( u t , u t ) t F ( u t , z ) + ( 1 t ) F ( u t , ξ ) t F ( u t , z ) + ( 1 t ) u t ξ , A u t = t F ( u t , u ) + ( 1 t ) t z ξ , A u t ,

which yields that

F( u t ,z)+(1t)zξ,A u t 0.

Letting t0, we find

F(ξ,z)+zξ,Aξ0,

which implies that ξEP(F,A).

Now, we are in a position to show ξF(S). Since lim n x n p exists, we may assume that lim n x n p=d>0. Put λ n = β n y n +(1 β n ) S n y n . It follows from (3.2) that lim sup n x n p+ α n ( e n λ n )d and lim sup n λ n p+ α n ( e n λ n )d. On the other hand, we have

lim n x n + 1 p= lim n α n ( ( x n p ) + α n ( e n λ n ) ) + ( 1 α n ) ( λ n p + α n ( e n λ n ) ) =d.

Using Lemma 2.3, we obtain that lim n λ n x n =0. Note that

S n y n x n = λ n x n 1 β n + β n ( x n y n ) 1 β n .

Hence, we have lim n S n y n x n =0. Note that S n x n x n S n x n S n y n + S n y n x n . Since S is Lipschitz continuous, we have lim n S n x n x n =0. Further, we find that lim n S x n x n =0. Using Lemma 2.2, we see that ξF(S). This proves that ηΩ.

Finally, we show that the sequence { x n } converges weakly to ξ. Assume that there exists another subsequence { x n j } of { x n } such that { x n j } converges weakly to η. In the same way, we find ηΩ. If ηξ, we see from the Opial condition [35] that

lim n x n ξ = lim inf i x n i ξ < lim inf i x n i η = lim inf n x n η = lim inf j x n j η < lim inf j x n j ξ = lim n x n ξ .

This derives a contradiction. Hence, we have η=ξ. This implies that x n ξΩ. This completes the proof. □

Remark 3.2 The key of the weak convergence of the algorithm is due to the fact that A is inverse-strongly monotone, which yields that I r n A is nonexpansive. The nonexpansivity of the mapping I r n A plays an important role in this theorem. Therefore, it is of interest to relax the monotonicity of A such that the algorithm is still weakly convergent.

Next, we give some subresults of Theorem 3.1. If S is asymptotically nonexpansive, we find the following result.

Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C×C to which satisfies (A1)-(A4). Let A:CH be an α-inverse-strongly monotone mapping, and let B:CH be a β-inverse-strongly monotone mapping. Let S:CC be an asymptotically nonexpansive mapping with the sequence { k n } such that n = 1 ( k n 1)<. Assume that Ω=F(S)VI(C,B)EP(F,A) is not empty. Let { α n }, { α n }, and { α n } be real number sequences in (0,1). Let { r n } and { s n } be two positive real number sequences. Let { x n } be a sequence generated in the following process:

{ x 1 C , F ( z n , z ) + A x n , z z n + 1 r n z z n , z n x n 0 , z C , y n = P C ( z n s n B z n ) , x n + 1 = α n x n + α n S n y n + α n e n ,

where { e n } is a bounded sequence in C. Assume that the control sequences satisfy the following restrictions:

  1. (a)

    α n + α n + α n =1;

  2. (b)

    0<p α n p<1 and n = 1 α n <;

  3. (c)

    0<s s n s <2β and 0<r r n r <2α,

where p, q, s, s , r, r are real constants. Then { x n } converges weakly to some point in Ω.

Further, if S is an identity mapping, we have the following result.

Corollary 3.4 Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C×C to which satisfies (A1)-(A4). Let A:CH be an α-inverse-strongly monotone mapping, and let B:CH be a β-inverse-strongly monotone mapping. Assume that Ω=VI(C,B)EP(F,A) is not empty. Let { α n }, { α n }, and { α n } be real number sequences in (0,1). Let { r n } and { s n } be two positive real number sequences. Let { x n } be a sequence generated in the following process:

{ x 1 C , F ( y n , z ) + A x n , z y n + 1 r n z y n , y n x n 0 , x n + 1 = α n x n + α n P C ( y n s n B y n ) + α n e n ,

where { e n } is a bounded sequence in C. Assume that the control sequences satisfy the following restrictions:

  1. (a)

    α n + α n + α n =1;

  2. (b)

    0<p α n q<1 and n = 1 α n <;

  3. (c)

    0<s s n s <2β and 0<r r n r <2α,

where p, q, s, s , r, r are real constants. Then { x n } converges weakly to some point in Ω.

Next, we give a result on variational inequality (2.1).

Corollary 3.5 Let C be a nonempty closed convex subset of a real Hilbert space H. Let A:CH be an α-inverse-strongly monotone mapping, and let B:CH be a β-inverse-strongly monotone mapping. Assume that Ω=VI(C,B)VI(C,A) is not empty. Let { α n }, { α n }, and { α n } be real number sequences in (0,1). Let { r n } and { s n } be two positive real number sequences. Let { x n } be a sequence generated in the following process:

{ x 1 C , z n = P C ( x n s n A x n ) , x n + 1 = α n x n + α n P C ( z n s n B z n ) + α n e n ,

where { e n } is a bounded sequence in C. Assume that the control sequences satisfy the following restrictions:

  1. (a)

    α n + α n + α n =1;

  2. (b)

    0<p α n q<1 and n = 1 α n <;

  3. (c)

    0<s s n s <2β and 0<r r n r <2α,

where p, q, s, s , r, r are real constants. Then { x n } converges weakly to some point in Ω.

Proof Putting F0, we see that

A x n ,z z n + 1 r n z z n , z n x n 0,zC

is equivalent to

x n r n A x n z n , z n z0,zC.

This implies that z n = P C ( x n r n A x n ). Let β n =0 and S be the identity. Then we can obtain from Theorem 3.1 the desired results immediately. □

Finally, we consider solving common fixed points of a pair of strict pseudocontractions.

Corollary 3.6 Let C be a nonempty closed convex subset of a real Hilbert space H. Let T 1 :CC be an α-strict pseudocontraction, and let T 2 :CC be a β-strict pseudocontraction. Assume that Ω=F( T 1 )F( T 2 ) is not empty. Let { α n }, { α n }, and { α n } be real number sequences in (0,1). Let { r n } and { s n } be two positive real number sequences. Let { x n } be a sequence generated in the following process:

{ x 1 C , z n = ( 1 r n ) x n + r n T 2 x n , y n = ( 1 s n ) x n + s n T 1 x n , x n + 1 = α n x n + α n y n + α n e n ,

where { e n } is a bounded sequence in C. Assume that the control sequences satisfy the following restrictions:

  1. (a)

    α n + α n + α n =1;

  2. (b)

    0<p α n q<1 and n = 1 α n <;

  3. (c)

    0<s s n s <1α and 0<r r n r <1β,

where p, q, s, s , r, r are real constants. Then { x n } converges weakly to some point in Ω.

Proof Put F0, A=I T 2 and B=I T 1 . It follows that A is 1 α 2 -inverse-strongly monotone and B is 1 β 2 -inverse-strongly monotone. We also have F( T 1 )=VI(C,B) and F( T 2 )=VI(C,A). In view of Theorem 3.1, we find the desired result immediately. □

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Hao, Y. Weak convergence theorems of a hybrid algorithm in Hilbert spaces. J Inequal Appl 2014, 378 (2014). https://doi.org/10.1186/1029-242X-2014-378

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Keywords

  • equilibrium problem
  • variational inequality
  • nonexpansive mapping
  • common solution