Skip to main content

On solutions of inclusion problems and fixed point problems

Abstract

An inclusion problem and a fixed point problem are investigated based on a hybrid projection method. The strong convergence of the hybrid projection method is obtained in the framework of Hilbert spaces. Variational inequalities and fixed point problems of quasi-nonexpansive mappings are also considered as applications of the main results.

MSC:47H05, 47H09, 47J25.

1 Introduction and preliminaries

Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decomposed as the sum of two nonlinear operators. Splitting methods for linear equations were introduced by Peaceman and Rachford [1] and Douglas and Rachford [2]. Extensions to nonlinear equations in Hilbert spaces were carried out by Kellogg [3] and Lions and Mercier [4]. The central problem is to iteratively find a zero of the sum of two monotone operators A and B in a Hilbert space H. In this paper, we consider the problem of finding a solution to the following problem: find an x in the fixed point set of the mapping S such that

x∈ ( A + B ) − 1 (0),

where A and B are two monotone operators. The problem has been addressed by many authors in view of the applications in image recovery and signal processing; see, for example, [5–9] and the references therein.

Throughout this paper, we always assume that H is a real Hilbert space with the inner product 〈⋅,⋅〉 and norm ∥⋅∥, respectively. Let C be a nonempty closed convex subset of H and P C be the metric projection from H onto C. Let S:C→C be a mapping. In this paper, we use F(S) to denote the fixed point set of S; that is, F(S):={x∈C:x=Sx}.

Recall that S is said to be nonexpansive iff

∥Sx−Sy∥≤∥x−y∥,∀x,y∈C.

If C is a bounded, closed, and convex subset of H, then F(S) is not empty, closed, and convex; see [10].

S is said to be quasi-nonexpansive iff F(S)≠∅ and

∥Sx−y∥≤∥x−y∥,∀x∈C,y∈F(S).

It is easy to see that nonexpansive mappings are Lipschitz continuous; however, the quasi-nonexpansive mapping is discontinuous on its domain generally. Indeed, the quasi-nonexpansive mapping is only continuous in its fixed point set.

Let A:C→H be a mapping. Recall that A is said to be monotone iff

〈Ax−Ay,x−y〉≥0,∀x,y∈C.

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

〈Ax−Ay,x−y〉≥α ∥ x − y ∥ 2 ,∀x,y∈C.

For such a case, A is also said to be α-strongly monotone. A is said to be inverse-strongly monotone iff there exists a constant α>0 such that

〈Ax−Ay,x−y〉≥α ∥ A x − A y ∥ 2 ,∀x,y∈C.

For such a case, A is also said to be α-inverse-strongly monotone. Notice that

α ∥ A x − A y ∥ 2 ≤〈Ax−Ay,x−y〉≤∥Ax−Ay∥∥x−y∥

clearly shows that A is 1 α -Lipschitz continuous.

Recall that the classical variational inequality is to find an x∈C such that

〈Ax,y−x〉≥0,∀y∈C.
(1.1)

In this paper, we use VI(C,A) to denote the solution set of (1.1). It is known that x ∗ ∈C is a solution to (1.1) iff x ∗ is a fixed point of the mapping P C (I−λA), where λ>0 is a constant, I stands for the identity mapping, and P C stands for the metric projection from H onto C.

A multivalued operator T:H→ 2 H with the domain D(T)={x∈H:Tx≠∅} and the range R(T)={Tx:x∈D(T)} is said to be monotone if for x 1 ∈D(T), x 2 ∈D(T), y 1 ∈T x 1 , and y 2 ∈T x 2 , we have 〈 x 1 − x 2 , y 1 − y 2 〉≥0. A monotone operator T is said to be maximal if its graph G(T)={(x,y):y∈Tx} is not properly contained in the graph of any other monotone operator. Let I denote the identity operator on H and T:H→ 2 H be a maximal monotone operator. Then we can define, for each λ>0, a nonexpansive single-valued mapping J λ :H→H by J λ = ( I + λ T ) − 1 . It is called the resolvent of T. We know that T − 1 0=F( J λ ) for all λ>0 and J λ is firmly nonexpansive.

The Mann iterative algorithm is efficient to study fixed point problems of nonlinear operators. Recently, many authors have studied the common solution problem, that is, find a point in a solution set and a fixed point (zero) point set of some nonlinear problems; see, for example, [11–30] and the references therein.

In [11], Kamimura and Takahashi investigated the problem of finding zero points of a maximal monotone operator by considering the following iterative algorithm:

x 0 ∈H, x n + 1 = α n x n +(1− α n ) J λ n x n ,n=0,1,2,…,
(1.2)

where { α n } is a sequence in (0,1), { λ n } is a positive sequence, T:H→ 2 H is a maximal monotone, and J λ n = ( I + λ n T ) − 1 . They showed that the sequence { x n } generated in (1.2) converges weakly to some z∈ T − 1 (0) provided that the control sequence satisfies some restrictions. Further, using this result, they also investigated the case that T=∂f, where f:H→(−∞,∞] is a proper lower semicontinuous convex function. Convergence theorems are established in the framework of real Hilbert spaces.

In [12], Takahashi an Toyoda investigated the problem of finding a common solution of the variational inequality problem (1.1) and a fixed point problem involving nonexpansive mappings by considering the following iterative algorithm:

x 0 ∈C, x n + 1 = α n x n +(1− α n )S P C ( x n − λ n A x n ),∀n≥0,
(1.3)

where { α n } is a sequence in (0,1), { λ n } is a positive sequence, S:C→C is a nonexpansive mapping, and A:C→H is an inverse-strongly monotone mapping. They showed that the sequence { x n } generated in (1.3) converges weakly to some z∈VI(C,A)∩F(S) provided that the control sequence satisfies some restrictions.

The above convergence theorems are weak. In this paper, motivated by the above results, we consider the problem of finding a common solution to the zero point problems and fixed point problems based on hybrid iterative methods with errors. Strong convergence theorems are established in the framework of Hilbert spaces.

To obtain our main results in this paper, we need the following lemmas and definitions.

Let C be a nonempty, closed, and convex subset of H. Let S:C→C be a mapping. Then the mapping I−S is demiclosed at zero, that is, if { x n } is a sequence in C such that x n ⇀ x ¯ and x n −S x n →0, then x ¯ ∈F(S).

Lemma [9]

Let C be a nonempty, closed, and convex subset of H, A:C→H be a mapping, and B:H⇉H be a maximal monotone operator. Then F( J r (I−λA))= ( A + B ) − 1 (0).

2 Main results

Theorem 2.1 Let C be a nonempty closed convex subset of a real Hilbert space H, A:C→H be an α-inverse-strongly monotone mapping, S:C→C be a quasi-nonexpansive mapping such that I−S is demiclosed at zero, and B be a maximal monotone operator on H such that the domain of B is included in C. Assume that F=F(S)∩ ( A + B ) − 1 (0)≠∅. Let { λ n } be a positive real number sequence. Let { α n } be a real number sequence in [0,1]. Let { x n } be a sequence in C generated in the following iterative process:

{ x 1 ∈ C , C 1 = C , y n = α n x n + ( 1 − α n ) S J λ n ( x n − λ n A x n ) , C n + 1 = { z ∈ C n : ∥ y n − z ∥ ≤ ∥ x n − z ∥ } , x n + 1 = P C n + 1 x 1 , n ≥ 1 ,

where J λ n = ( I + λ n B ) − 1 . Suppose that the sequences { α n } and { λ n } satisfy the following restrictions:

  1. (a)

    0≤ α n ≤a<1;

  2. (b)

    0<b≤ λ n ≤c<2α.

Then the sequence { x n } converges strongly to P F x 1 .

Proof First, we show that C n is closed and convex. Notice that C 1 =C is closed and convex. Suppose that C i is closed and convex for some i≥1. We show that C i + 1 is closed and convex for the same i. Indeed, for any v∈ C i , we see that

∥ y i −z∥≤∥ x i −z∥

is equivalent to

∥ y i ∥ 2 − ∥ x i ∥ 2 −2〈z, y i − x i 〉≥0.

Thus C i + 1 is closed and convex. This shows that C n is closed and convex.

Next, we prove that I− λ n A is a nonexpansive mapping. Indeed, we have

In view of the restriction (b), we obtain that I− λ n A is nonexpansive. Next, we show that F⊂ C n for each n≥1. From the assumption, we see that F⊂C= C 1 . Assume that F⊂ C i for some i≥1. For any z∈F⊂ C i , we find from Lemma that

z=Sz= J λ i (z− λ i Az).

Put z n = J λ n ( x n − λ n A x n ). Since J λ n and I− λ n A are nonexpansive, we have

∥ z n − p ∥ ≤ ∥ ( x n − λ n A x n ) − ( p − λ n A p ) ∥ ≤ ∥ x n − p ∥ .
(2.1)

It follows from (2.1) that

∥ y i − z ∥ = ∥ α i x i + ( 1 − α i ) S z i − z ∥ ≤ α i ∥ x i − z ∥ + ( 1 − α i ) ∥ z i − z ∥ ≤ ∥ x i − z ∥ .

This shows that z∈ C i + 1 . This proves that F⊂ C n . Notice that x n = P C n x 1 . For every z∈F⊂ C n , we have

∥ x 1 − x n ∥≤∥ x 1 −z∥.

In particular, we have

∥ x 1 − x n ∥≤∥ x 1 − P F x 1 ∥.

This implies that { x n } is bounded. Since x n = P C n x 1 and x n + 1 = P C n + 1 x 1 ∈ C n + 1 ⊂ C n , we arrive at

0 ≤ 〈 x 1 − x n , x n − x n + 1 〉 ≤ − ∥ x 1 − x n ∥ 2 + ∥ x 1 − x n ∥ ∥ x 1 − x n + 1 ∥ .

It follows that

∥ x n − x 1 ∥≤∥ x n + 1 − x 1 ∥.

This implies that lim n → ∞ ∥ x n − x 1 ∥ exists. On the other hand, we have

It follows that

lim n → ∞ ∥ x n − x n + 1 ∥=0.
(2.2)

Notice that x n + 1 = P C n + 1 x 1 ∈ C n + 1 . It follows that

∥ y n − x n + 1 ∥≤∥ x n − x n + 1 ∥.

This in turn implies that

∥ y n − x n ∥≤∥ y n − x n + 1 ∥+∥ x n − x n + 1 ∥≤2∥ x n − x n + 1 ∥.

In view of (2.2), we obtain that

lim n → ∞ ∥ x n − y n ∥=0.
(2.3)

On the other hand, we have

∥ x n − y n ∥=(1− α n )∥ x n −S z n ∥.

It follows from (2.3) that

lim n → ∞ ∥ x n −S z n ∥=0.
(2.4)

For any p∈F, we see that

∥ z n − p ∥ 2 = ∥ J λ n ( x n − λ n A x n ) − J λ n ( p − λ n A p ) ∥ 2 ≤ ∥ x n − p ∥ 2 − 2 〈 x n − p , A x n − A p 〉 + λ n 2 ∥ A x n − A p ∥ 2 ≤ ∥ x n − p ∥ 2 − λ n ( 2 α − λ n ) ∥ A x n − A p ∥ 2 .
(2.5)

Notice that

∥ y n − p ∥ 2 ≤ α n ∥ x n − p ∥ 2 + ( 1 − α n ) ∥ S z n − p ∥ 2 ≤ α n ∥ x n − p ∥ 2 + ( 1 − α n ) ∥ z n − p ∥ 2 .
(2.6)

Substituting (2.5) into (2.6), we see that

∥ y n − p ∥ 2 ≤ ∥ x n − p ∥ 2 −(1− α n ) λ n (2α− λ n ) ∥ A x n − A p ∥ 2 .

It follows that

( 1 − α n ) λ n ( 2 α − λ n ) ∥ A x n − A p ∥ 2 ≤ ∥ x n − p ∥ 2 − ∥ y n − p ∥ 2 ≤ ( ∥ x n − p ∥ + ∥ y n − p ∥ ) ∥ x n − y n ∥ .

This implies from (2.3) that

lim n → ∞ ∥A x n −Ap∥=0.
(2.7)

On the other hand, we have

∥ z n − p ∥ 2 = ∥ J λ n ( x n − λ n A x n ) − J λ n ( p − λ n A p ) ∥ 2 ≤ 〈 ( x n − λ n A x n ) − ( p − λ n A p ) , z n − p 〉 = 1 2 ( ∥ ( x n − λ n A x n ) − ( p − λ n A p ) ∥ 2 + ∥ z n − p ∥ 2 − ∥ ( x n − λ n A x n ) − ( p − λ n A p ) − ( z n − p ) ∥ 2 ) ≤ 1 2 ( ∥ x n − p ∥ 2 + ∥ z n − p ∥ 2 − ∥ x n − z n − λ n ( A x n − A p ) ∥ 2 ) ≤ 1 2 ( ∥ x n − p ∥ 2 + ∥ z n − p ∥ 2 − ∥ x n − z n ∥ 2 − λ n 2 ∥ A x n − A p ∥ 2 + 2 λ n ∥ x n − z n ∥ ∥ A x n − A p ∥ ) ≤ 1 2 ( ∥ x n − p ∥ 2 + ∥ z n − p ∥ 2 − ∥ x n − z n ∥ 2 + 2 λ n ∥ x n − z n ∥ ∥ A x n − A p ∥ ) .

It follows that

∥ z n − p ∥ 2 ≤ ∥ x n − p ∥ 2 − ∥ x n − z n ∥ 2 +2 λ n ∥ x n − z n ∥∥A x n −Ap∥.
(2.8)

Substituting (2.8) into (2.6), we see that

∥ y n − p ∥ 2 ≤ ∥ x n − p ∥ 2 −(1− α n ) ∥ x n − z n ∥ 2 +2(1− α n ) λ n ∥ x n − z n ∥∥A x n −Ap∥.

It follows that

In view of the restriction (a), we obtain from (2.7) that

lim n → ∞ ∥ x n − z n ∥=0.
(2.9)

Since { x n } is bounded, we may assume that there is a subsequence { x n i } of { x n } converging weakly to some point x ∗ . It follows from (2.9) that z n i converges weakly to x ∗ . Notice that

∥S z n − z n ∥≤∥S z n − x n ∥+∥ x n − z n ∥.

It follows from (2.4) and (2.9) that

lim n → ∞ ∥S z n − z n ∥=0.

In view of the assumption that S is demiclosed at zero, we see that x ∗ ∈F(S).

Next, we show that x ∗ ∈ ( A + B ) − 1 (0). Notice that z n = J λ n ( x n − λ n A x n ). This implies that

x n − λ n A x n ∈(I+ λ n B) z n .

That is,

x n − z n λ n −A x n ∈B z n .

Since B is monotone, we get for any (u,v)∈B, that

〈 z n − u , x n − z n λ n − A x n − v 〉 ≥0.
(2.10)

Replacing n by n i and letting i→∞, we obtain from (2.10) that

〈ω−u,−Aω−v〉≤0.

This means −Aω∈Bω, that is, 0∈(A+B)(ω). Hence, we get ω∈ ( A + B ) − 1 (0). This completes the proof that x ∗ ∈F.

Notice that P F x 1 ⊂ C n + 1 and x n + 1 = P C n + 1 x 1 , we have

∥ x 1 − x n + 1 ∥≤∥ x 1 − P F x 1 ∥.

On the other hand, we have

∥ x 1 − P F x 1 ∥ ≤ ∥ x 1 − x ∗ ∥ ≤ lim inf i → ∞ ∥ x 1 − x n i ∥ ≤ lim sup i → ∞ ∥ x 1 − x n i ∥ ≤ ∥ x 1 − P F x 1 ∥ .

We, therefore, obtain that

∥ x 1 − x ∗ ∥ = lim i → ∞ ∥ x 1 − x n i ∥=∥ x 1 − P F x 1 ∥.

This implies x n i → x ∗ = P F x 1 . Since { x n i } is an arbitrary subsequence of { x n }, we obtain that x n → P F x 1 as n→∞. This completes the proof. □

From Theorem 2.1, we have the following results immediately.

Corollary 2.2 Let C be a nonempty closed convex subset of a real Hilbert space H, A:C→H be an α-inverse-strongly monotone mapping, and B be a maximal monotone operator on H such that the domain of B is included in C. Assume that ( A + B ) − 1 (0)≠∅. Let { λ n } be a positive real number sequence. Let { α n } be a real number sequence in [0,1]. Let { x n } be a sequence in C generated in the following iterative process:

{ x 1 ∈ C , C 1 = C , y n = α n x n + ( 1 − α n ) J λ n ( x n − λ n A x n ) , C n + 1 = { z ∈ C n : ∥ y n − z ∥ ≤ ∥ x n − z ∥ } , x n + 1 = P C n + 1 x 1 , n ≥ 1 ,

where J λ n = ( I + λ n B ) − 1 . Suppose that the sequences { α n } and { λ n } satisfy the following restrictions:

  1. (a)

    0≤ α n ≤a<1;

  2. (b)

    0<b≤ λ n ≤c<2α.

Then the sequence { x n } converges strongly to P ( A + B ) − 1 ( 0 ) x 1 .

Let f:H→(−∞,∞] be a proper lower semicontinuous convex function. Define the subdifferential

∂f(x)= { z ∈ H : f ( x ) + 〈 y − x , z 〉 ≤ f ( y ) , ∀ y ∈ H }

for all x∈H. Then ∂f is a maximal monotone operator of H into itself; see [23] for more details. Let C be a nonempty closed convex subset of H and i C be the indicator function of C, that is,

i C x={ 0 , x ∈ C , ∞ , x ∉ C .

Furthermore, we define the normal cone N C (v) of C at v as follows:

N C v= { z ∈ H : 〈 z , y − v 〉 ≤ 0 , ∀ y ∈ H }

for any v∈C. Then i C :H→(−∞,∞] is a proper lower semicontinuous convex function on H and ∂ i C is a maximal monotone operator. Let J λ x= ( I + λ ∂ i C ) − 1 x for any λ>0 and x∈H. From ∂ i C x= N C x and x∈C, we get

v = J λ x ⇔ x ∈ v + λ N C v ⇔ 〈 x − v , y − v 〉 ≤ 0 , ∀ y ∈ C , ⇔ v = P C x ,

where P C is the metric projection from H into C. Similarly, we can get that x∈ ( A + ∂ i C ) − 1 (0)⇔x∈VI(A,C). Putting B=∂ i C in Theorem 2.1, we can see J λ n = P C . The following is not hard to derive.

Corollary 2.3 Let C be a nonempty closed convex subset of a real Hilbert space H, A:C→H be an α-inverse-strongly monotone mapping, and S:C→C be a quasi-nonexpansive mapping such that I−S is demiclosed at zero. Assume that F=F(S)∩VI(C,A)≠∅. Let { λ n } be a positive real number sequence. Let { α n } be a real number sequence in [0,1]. Let { x n } be a sequence in C generated in the following iterative process:

{ x 1 ∈ C , C 1 = C , y n = α n x n + ( 1 − α n ) S P C ( x n − λ n A x n ) , C n + 1 = { z ∈ C n : ∥ y n − z ∥ ≤ ∥ x n − z ∥ } , x n + 1 = P C n + 1 x 1 , n ≥ 1 .

Suppose that the sequences { α n } and { λ n } satisfy the following restrictions:

  1. (a)

    0≤ α n ≤a<1;

  2. (b)

    0<b≤ λ n ≤c<2α.

Then the sequence { x n } converges strongly to P F x 1 .

In view of Corollary 2.3, we have the following corollary on variational inequalities.

Corollary 2.4 Let C be a nonempty closed convex subset of a real Hilbert space H and A:C→H be an α-inverse-strongly monotone mapping. Assume that F=VI(C,A)≠∅. Let { λ n } be a positive real number sequence. Let { α n } be a real number sequence in [0,1]. Let { x n } be a sequence in C generated in the following iterative process:

{ x 1 ∈ C , C 1 = C , y n = α n x n + ( 1 − α n ) P C ( x n − λ n A x n ) , C n + 1 = { z ∈ C n : ∥ y n − z ∥ ≤ ∥ x n − z ∥ } , x n + 1 = P C n + 1 x 1 , n ≥ 1 .

Suppose that the sequences { α n } and { λ n } satisfy the following restrictions:

  1. (a)

    0≤ α n ≤a<1;

  2. (b)

    0<b≤ λ n ≤c<2α.

Then the sequence { x n } converges strongly to P VI ( C , A ) x 1 .

References

  1. Peaceman DH, Rachford HH: The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 1995, 3: 28–415.

    Article  MathSciNet  Google Scholar 

  2. Douglas J, Rachford HH: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 1956, 82: 421–439. 10.1090/S0002-9947-1956-0084194-4

    Article  MathSciNet  Google Scholar 

  3. Kellogg RB: Nonlinear alternating direction algorithm. Math. Comput. 1969, 23: 23–28. 10.1090/S0025-5718-1969-0238507-3

    Article  MathSciNet  Google Scholar 

  4. Lions PL, Mercier B: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 1979, 16: 964–979. 10.1137/0716071

    Article  MathSciNet  Google Scholar 

  5. Qin X, Kang JI, Cho YJ: On quasi-variational inclusions and asymptotically strict pseudo-contractions. J. Nonlinear Convex Anal. 2010, 11: 441–453.

    MathSciNet  Google Scholar 

  6. Zhang M: Iterative algorithms for common elements in fixed point sets and zero point sets with applications. Fixed Point Theory Appl. 2012, 2012: 21. 10.1186/1687-1812-2012-21

    Article  Google Scholar 

  7. Takahashi S, Takahashi W, Toyoda M: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl. 2010, 147: 27–41. 10.1007/s10957-010-9713-2

    Article  MathSciNet  Google Scholar 

  8. Kamimura S, Takahashi W: Weak and strong convergence of solutions to accretive operator inclusions and applications. Set-Valued Anal. 2010, 8: 361–374.

    Article  MathSciNet  Google Scholar 

  9. Aoyama K, Kimura Y, Takahashi W, Toyoda M: On a strongly nonexpansive sequence in Hilbert spaces. J. Nonlinear Convex Anal. 2007, 8: 471–489.

    MathSciNet  Google Scholar 

  10. Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. Pure Math. 1976, 18: 78–81.

    Google Scholar 

  11. Kamimura S, Takahashi W: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 2000, 106: 226–240. 10.1006/jath.2000.3493

    Article  MathSciNet  Google Scholar 

  12. Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 2003, 118: 417–428. 10.1023/A:1025407607560

    Article  MathSciNet  Google Scholar 

  13. Ye J, Huang J: Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces. J. Math. Comput. Sci. 2011, 1: 1–18.

    Article  MathSciNet  Google Scholar 

  14. Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett. 2011, 24: 224–228. 10.1016/j.aml.2010.09.008

    Article  MathSciNet  Google Scholar 

  15. Zegeye H, Shahzad N: Strong convergence theorem for a common point of solution of variational inequality and fixed point problem. Adv. Fixed Point Theory 2012, 2: 374–397.

    Google Scholar 

  16. 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

    Article  MathSciNet  Google Scholar 

  17. Lu H, Wang Y: Iterative approximation for the common solutions of a infinite variational inequality system for inverse-strongly accretive mappings. J. Math. Comput. Sci. 2012, 2(6):1660–1670.

    MathSciNet  Google Scholar 

  18. Husain S, Gupta S: A resolvent operator technique for solving generalized system of nonlinear relaxed cocoercive mixed variational inequalities. Adv. Fixed Point Theory 2012, 2: 18–28.

    Google Scholar 

  19. Noor MA, Huang Z: Some resolvent iterative methods for variational inclusions and nonexpansive mappings. Appl. Math. Comput. 2007, 194: 267–275. 10.1016/j.amc.2007.04.037

    Article  MathSciNet  Google Scholar 

  20. 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

    Article  MathSciNet  Google Scholar 

  21. Kim JK, Tuyen TM: Regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2011, 2011: 52. 10.1186/1687-1812-2011-52

    Article  MathSciNet  Google Scholar 

  22. Wei Z, Shi G: Convergence of a proximal point algorithm for maximal monotone operators in Hilbert spaces. J. Inequal. Appl. 2012, 2012: 137. 10.1186/1029-242X-2012-137

    Article  MathSciNet  Google Scholar 

  23. Qin X, Chang SS, Cho YJ: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. 2010, 11: 2963–2972. 10.1016/j.nonrwa.2009.10.017

    Article  MathSciNet  Google Scholar 

  24. Qin X, Shang M, Su Y: Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems. Math. Comput. Model. 2008, 48: 1033–1046. 10.1016/j.mcm.2007.12.008

    Article  MathSciNet  Google Scholar 

  25. He XF, Xu YC, He Z: Iterative approximation for a zero of accretive operator and fixed points problems in Banach space. Appl. Math. Comput. 2011, 217: 4620–4626. 10.1016/j.amc.2010.11.014

    Article  MathSciNet  Google Scholar 

  26. Wu C, Liu A: Strong convergence of a hybrid projection iterative algorithm for common solutions of operator equations and of inclusion problems. Fixed Point Theory Appl. 2012., 2012: Article ID 90

    Google Scholar 

  27. Qin X, Su Y: Approximation of a zero point of accretive operator in Banach spaces. J. Math. Anal. Appl. 2007, 329: 415–424. 10.1016/j.jmaa.2006.06.067

    Article  MathSciNet  Google Scholar 

  28. Abdel-Salam HS, Al-Khaled K: Variational iteration method for solving optimization problems. J. Math. Comput. Sci. 2012, 2: 1457–1497.

    MathSciNet  Google Scholar 

  29. 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

    Article  MathSciNet  Google Scholar 

  30. Zegeye H, Shahzad N, Alghamdi M: Strong convergence theorems for a common point of solution of variational inequality, solutions of equilibrium and fixed point problems. Fixed Point Theory Appl. 2012., 2012: Article ID 119

    Google Scholar 

Download references

Acknowledgements

The author is grateful to the reviewers’ suggestions which improved the contents of the article.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yuan Hecai.

Additional information

Competing interests

The author declares that they have no competing interests.

Rights and permissions

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

Reprints and permissions

About this article

Cite this article

Hecai, Y. On solutions of inclusion problems and fixed point problems. Fixed Point Theory Appl 2013, 11 (2013). https://doi.org/10.1186/1687-1812-2013-11

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-1812-2013-11

Keywords