Viscosity splitting methods for variational inclusion and fixed point problems in Hilbert spaces

Research Article

Viscosity splitting methods for variational inclusion and fixed point problems in Hilbert spaces

Xiaolong Qin^a, B. A. Bin Dehaish^b, Sun Young Cho^c,∗

aInstitute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Sichuan, China.

bDepartment of mathematics, Faculty of Science, AL Faisaliah Campus, King Abdulaziz University, Jeddah, Saudi Arabia.

cDepartment of Mathematics, Gyeongsang National University, Jinju 660-701, Korea.

Communicated by Y. Yao

Abstract

In this paper, a viscosity splitting method is investigated for treating variational inclusion and fixed point problems. Strong convergence theorems of common solutions are established in the framework of Hilbert spaces. Applications are also provided to support the main results. c2016 All rights reserved.

Keywords: Convex feasibility problem, fixed point, iterative process, monotone operator, splitting algorithm.

2010 MSC: 47H05, 47H09.

1. Introduction and Preliminaries

Let H be a real Hilbert space with inner product hx, yi and induced normkxk =p

hx, xi forx, y ∈H.

LetC be a nonempty convex and closed subset of H.

Let T :C → C be a mapping. In this paper, we use F ix(T) to stand for the set of fixed points of T. Recall thatT is said to be anα-contractive mapping iff there exists a constantα with 0< α <1 such that

kT x−T yk ≤αkx−yk, ∀x, y∈C.

T is said to be nonexpansive iff

kT x−T yk ≤ kx−yk, ∀x, y∈C.

∗Corresponding author

Email addresses: [email protected](Xiaolong Qin),[email protected](B. A. Bin Dehaish),[email protected] (Sun Young Cho)

Received 2015-12-27

IfCis also bounded, then the set of fixed points ofS is not empty; see [5] and the references therein. In the real world, many important problems have reformulations which require finding fixed points of nonexpansive mapping. Mann iteration is powerful to study fixed points of nonexpansive mappings. However it is only weakly convergent. Recently, many authors studied the problem of modifying Mann iteration so that strong convergence is guaranteed without any compactness assumption; see [7, 8, 12, 13, 14, 16, 24, 25, 26, 27] and the references therein.

T is said to be a λ-strict pseudocontraction iff there exists a constantλwith 0≤λ <1 such that kT x−T yk² ≤ kx−yk+λkx−T x−y+T yk², ∀x, y∈C.

The class of λ-strict pseudocontractions was introduced by Browder and Petryshyn [6] in 1967. It is clear that the class ofλ-strict pseudocontractions strictly include the class of nonexpansive mappings as a special cases. It is also known that everyλ-strict pseudocontraction is Lipschitz continuous; see [6] and the references therein.

Let A:C →H be a mapping. Recall thatA is said to be monotone iff hAx−Ay, x−yi ≥0, ∀x, y∈C.

A is said to beλ-strongly monotone iff there exists a positive constantλ >0 such that hAx−Ay, x−yi ≥λkx−yk², ∀x, y∈C.

A is said to be inverseλ-strongly monotone iff there exists a positive constantλ >0 such that hAx−Ay, x−yi ≥λkAx−Ayk², ∀x, y∈C.

From the above, we see thatA is inverseλ-strongly monotone iffA⁻¹ is strongly monotone. Ais said to be L-Lipschitz continuous iff there exists a positive constantL >0 such that

kAx−Ayk ≤Lkx−yk, ∀x, y∈C.

It is obvious that Ais inverse λ-strongly monotone, thenAis also monotone and ¹_λ-Lipschitz continuous.

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

hAx, y−xi ≥0, ∀y∈C. (1.1)

The solution set of variational inequality (1.1) is denoted by V I(C, A). Projection methods have been recently investigated for solving variational inequality (1.1). Let P roj_C be the metric projection from H onto C and I the identity on H. It is known that x is a solution to (1.1) iff x is a fixed point of mappingP roj_C(I−rA). If Ais inverseλ-strongly monotone, then P roj_C(I−rA) is nonexpansive. IfC is bounded, closed and convex, then the existence of solutions of variational inequality (1.1) is guaranteed by the nonexpansivity of mapping P rojC(I−rA).

Recall that an operator B : H ⇒ H is said to be monotone iff, for all x, y ∈ H, f ∈ Bx and g ∈ By imply hx−y, f −gi ≥ 0. In this paper, we use B⁻¹(0) to stand for the zero point of B. A monotone mapping B : H ⇒ H is maximal iff the graph G(B) of B is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping B is maximal if and only if, for any (x, f)∈H×H,hx−y, f −gi ≥0, for all (y, g)∈G(B) implies f ∈Bx. For a maximal monotone operator B on H, and r > 0, we may define the single-valued resolvent J_r^B = (I +rB)⁻¹, where D(B) denote the domain of B. It is known that J_r^B :H → D(B) is firmly nonexpansive, and B⁻¹(0) =F ix(J_r^B). The property of the resolvent ensures that the Picard iterative algorithm x_n+1 = J_r^Bx_n converge weakly to a fixed point ofJ_r^B, which is necessarily a zero point ofB. Rockafellar introduced this iteration method and call it the proximal point algorithm (PPA); for more detail, see [20, 23] and the references therein. The PPA and its dual version in the context of convex programming, the method of multipliers of Hesteness and

Powell, have been extensively studied and are known to yield as special cases decomposition methods such as the method of partial inverses [22], the Douglas-Rachford splitting method, and the alternating direction method of multipliers [10]. In the case of B = B1 +B2, where B1 and B2 are maximal monotone on H, the forward-backward splitting method x_n+1 = (I +r_nB₁)⁻¹(I −r_nB₂)x_n, n = 0,1,· · · , where r_n > 0, was proposed by Lions and Mercier [15], and in a dual form for convex programming, by Han and Lou [11]. In the case whereB1 =NC, this method reduces to a projection method proposed by Sibony [21] for monotone variational inequalities (1.1). Recently, many authors have studied the splitting algorithm; see [2, 3, 4, 9, 17, 18, 19] and the references therein.

In this paper, a viscosity splitting method is investigated for treating a inclusion problem with two monotone operators and a fixed point problem ofλ-strict pseudocontractions. Strong convergence theorems of common solutions are established in the framework of Hilbert spaces. Applications are also provided to support the main results.

The following lemmas are essential to prove our main results.

Lemma 1.1 ([3]). Let C be a nonempty convex and closed subset of a real Hilbert space H.Let A:C→H be a mapping, and B :H⇒H a maximal monotone operator. Then F(Jr(I−rA)) = (A+B)⁻¹(0).

Lemma 1.2 ([25]). Let {a_n} be a sequence of nonnegative numbers satisfying the condition an+1 ≤ (1− t_n)a_n+t_nb_n+c_n,∀n≥0,where {t_n} is a number sequence in(0,1)such thatlimn→∞t_n= 0 andP∞

n=0t_n=

∞, {b_n} is a number sequence such that lim sup_n→∞bn ≤0, and {c_n} is a positive number sequence such thatP∞

n=0cn<∞. Then limn→∞an= 0.

Lemma 1.3([1]). Let H be a Hilbert space, andA an maximal monotone operator. For λ >0, µ >0, and x∈E, we have J_λx=J_µ

µ λx+

1−^µ_λ J_λx

, where J_λ = (I +λA)⁻¹ and J_µ= (I+µA)⁻¹.

Lemma 1.4 ([6]). Let C be a nonempty convex and closed subset of a real Hilbert space H.Let T :C→C be a λ-strict pseudocontraction. Define a mapping S by S = βI + (1−β)T. If beta ∈ [λ,1), then S is nonexpansive andF ix(T) =F ix(S).

Lemma 1.5 ([6]). Let C be a nonempty convex and closed subset of a real Hilbert space H.Let T :C→C be a λ-strict pseudocontraction. ThenT is Lipschitz continuous and I−T is demiclosed at zero.

2. Main results

Theorem 2.1. Let C be a nonempty convex closed subset of a real Hilbert space H. Let A:C→H be an inverseκ-strongly monotone mapping and letB be a maximal monotone operator onH. Letf :C→C be a fixedα-contraction and letT :C→C be a λ-strict pseudocontraction. Assume that (A+B)⁻¹(0)∩F ix(T) is not empty. Let {α_n}, {β_n} be real number sequences in (0,1)and let {r_n} be a real number sequence in (0,2κ). Let{x_n} be a sequence inC generated in the following process: x0∈C,xn+1 =βnyn+ (1−βn)T yn,

∀n≥0,where{y_n} is a sequence inC such that ky_n−(I+r_nB)⁻¹ α_nf(x_n) + (1−α_n)x_n−r_nA α_nf(x_n) + (1−αn)xn

k ≤ en. Assume that the control sequences satisfy the following restrictions: limn→∞αn = 0, P∞

n=0αn = ∞, P∞

n=1|α_n−αn−1|< ∞, 0 < a ≤ rn ≤ a⁰ <2κ, P∞

n=1|r_n−rn−1| <∞, P∞

n=0ke_nk <∞, P∞

n=1|β_n −βn−1| < ∞, and λ ≤ β_n ≤ a⁰⁰ < 1, where a, a⁰ and a⁰⁰ are three real numbers. Then {x_n} converges strongly to a point x¯ ∈F ix(T)∩(A+B)⁻¹(0), where x¯ =P roj_{F ix(T)∩(A+B)}⁻¹₍₀₎f(¯x), that is, x¯ solves the following variational inequality hf(¯x)−x,¯ x¯−xi ≥0, ∀x∈F ix(T)∩(A+B)⁻¹(0).

Proof. Since A is inverseκ-strongly monotone, one has

k(I−r_nA)x−(I−r_nA)yk² =kx−yk²−2r_nhx−y, Ax−Ayi+r_n²kAx−Ayk²

≤ kx−yk²−rn(2κ−rn)kAx−Ayk².

From the restriction imposed on {r_n}, one has I −r_nA is nonexpansive. Setting T_n = β_nI + (1−β_n)T,

whereI is the identity, one sees from Lemma 1.4 thatT_n is nonexpansive with F ix(T_n) =F ix(T). Fixing p ∈ F ix(T) ∩(A+B)⁻¹(0), one has from Lemma 1.1 that p = Tnp = (I +rnB)⁻¹(p−rnAp). Setting zn=αnf(xn) + (1−αn)xn, one has

kz_n−pk ≤αnkf(xn)−pk+ (1−αn)kx_n−pk

≤ 1−αn(1−α)

kx_n−pk+αnkf(p)−pk.

Hence, one has

kx_n+1−pk ≤ ky_n−pk

≤ ky_n−(I+r_nB)⁻¹ z_n−r_nAz_n

k+k z_n−r_nAz_n

− p−r_nAp k

≤ kz_n−pk+en

≤ 1−αn(1−α)

kx_n−pk+αnkf(p)−pk+en

≤max{kx_n−pk,kf(p)−pk

1−α }+e_n.

By mathematical induction, one finds that sequence{x_n} is bounded, so are{y_n} and {z_n}. Notice that kz_n−zn−1k ≤ |α_n−αn−1|kx_n−1−f(xn−1)k+ 1−αn(1−α)

kx_n−1−xnk. (2.1) Puttingw_n=z_n−r_nAz_n, we find from (2.1) that

kw_n−wn−1k ≤ kz_n−zn−1k+kr_n−rn−1kkAz_n−1k

≤ |α_n−αn−1|kx_n−1−f(xn−1)k+ 1−αn(1−α)

kx_n−1−xnk +|r_n−rn−1|kAz_n−1k.

(2.2)

SetJ_r^B_n = (I+r_nB)⁻¹.Using Lemma 1.3, one has kx_n−xn+1k=kT_n−1yn−1−Tnynk

≤ kyn−1−ynk+|β_n−βn−1|ky_n−T ynk

≤ kJ_r^B_nwn−J_r^B_n−1wn−1k+|β_n−βn−1|ky_n−T ynk+en−1+en

≤ k(1−rn−1

rn

)(J_r^B_nwn−wn−1) +rn−1

rn

(wn−wn−1)k +|β_n−βn−1|ky_n−T ynk+en−1+en

≤ k(1−rn−1

rn

)(J_r^B_nwn−wn) + (wn−wn−1)k +|β_n−βn−1|ky_n−T ynk+en−1+en

≤ |r_n−rn−1|

r_n kw_n−J_r^B_nw_nk+kw_n−1−w_nk +|β_n−βn−1|ky_n−T ynk+en−1+en.

(2.3)

Combining (2.2) with (2.3), one has

kx_n−xn+1k ≤ |r_n−rn−1|

r_n kw_n−J_r^B_nwnk+|α_n−αn−1|kx_n−1−f(xn−1)k + 1−αn(1−α)

kx_n−1−xnk+|r_n−rn−1|kAz_n−1k +|β_n−βn−1|ky_n−T y_nk+en−1+e_n.

Using the restrictions imposed on{r_n},{e_n},{α_n}and{β_n}and Lemma 1.1, we find limn→∞kx_n−x_n+1k= 0.Since αn→0 as n→ ∞,we find

n→∞lim kx_n−z_nk= 0. (2.4)

Since k · k² is convex, we have

kz_n−pk²≤αnkf(x_n)−pk²+ (1−αn)kx_n−pk²

≤ kx_n−pk²+α_nkf(x_n)−pk². This in turn implies

kx_n+1−pk²≤ ky_n−J_r^B_n(zn−rnAzn)k²+kJ_r^B_n(zn−rnAzn)−pk² + 2kJ_r^B_n(zn−rnAzn)−pkky_n−J_r^B_n(zn−rnAzn)k

≤ k(z_n−rnAzn)−(p−rnAp)k²+ 2enkJ_r^B_n(zn−rnAzn)−pk+e²_n

≤ kz_n−pk²−r_n(2κ−r_n)kAz_n−Apk²+ 2e_nkz_n−pk+e²_n

≤ kx_n−pk²+αnkf(xn)−pk²−rn(2κ−rn)kAz_n−Apk²+ 2enkz_n−pk+e²_n. It follows that

rn(2κ−rn)kAz_n−Apk² ≤ kx_n−pk²+αnkf(xn)−pk²− kx_n+1−pk²+ (2kz_n−pk+en)en. Therefore, one finds

n→∞lim kAp−Az_nk= 0. (2.5)

Since J_r^B_n is firmly nonexpansive, one has kJ_r^B

n(z_n−r_nAz_n)−pk² ≤ h(z_n−r_nAz_n)−(p−r_nAp), J_r^B_n(z_n−r_nAz_n)−pi

≤ 1 2

k(z_n−rnAzn)−(p−rnAp)k²+kJ_r^B

n(zn−rnAzn)−pk²

− kz_n−J_r^B_n(zn−rnAzn)−rn(Azn−Ap)k² . It follows that

kJ_r^B_n(zn−rnAzn)−pk² ≤ kz_n−pk²− kz_n−J_r^B_n(zn−rnAzn)k²

−rnkAz_n−Apk²+ 2rnkz_n−J_r^B_n(zn−rnAzn)kkAz_n−Apk.

Hence, one has

kx_n+1−pk²≤ ky_n−J_r^B_n(z_n−r_nAz_n)k²+kJ_r^B

n(z_n−r_nAz_n)−pk² + 2kJ_r^B

n(z_n−r_nAz_n)−pkky_n−J_r^B_n(z_n−r_nAz_n)k

≤ kJ_r^B

n(z_n−r_nAz_n)−pk²+e_n(2kJ_r^B

n(z_n−r_nAz_n)−pk+e_n)

≤ kz_n−pk²− kz_n−J_r^B_n(z_n−r_nAz_n)k² + 2r_nkz_n−J_r^B_n(z_n−r_nAz_n)kkAz_n−Apk +e_n(2kJ_r^B

n(z_n−r_nAz_n)−pk+e_n)

≤ kx_n−pk²+α_nkf(x_n)−pk²− kz_n−J_r^B_n(z_n−r_nAz_n)k² + 2r_nkz_n−J_r^B_n(z_n−r_nAz_n)kkAz_n−Apk

+e_n(2kJ_r^B

n(z_n−r_nAz_n)−pk+e_n), which further implies from (2.5)

n→∞lim kz_n−J_r^B_n(z_n−r_nAz_n)k= 0. (2.6)

Since T_n is nonexpansive, one has

kβ_nx_n+ (1−β_n)T x_n−x_nk ≤ kβ_nx_n+ (1−β_n)T x_n−β_ny_n−(1−β_n)T y_nk +kx_n−βnyn−(1−βn)T ynk

≤ kx_n−ynk+kx_n−xn+1k

≤ kx_n−znk+kz_n−J_r^B_n(zn−rnAzn)k+kx_n−xn+1k+en. In view of (2.4) and (2.6), one has limn→∞kβ_nx_n+ (1−β_n)T x_n−x_nk= 0.Note that

kT x_n−x_nk ≤ kT x_n−β_nx_n−(1−β_n)T x_nk+kβ_nx_n+ (1−β_n)T x_n−x_nk

≤βnkx_n−T xnk+kβ_nxn+ (1−βn)T xn−xnk.

From the restriction imposed on sequence{β_n}, one finds that limn→∞kT x_n−xnk= 0.

Next, we show that

lim sup

n→∞

hf(¯x)−x, z¯ _n−xi ≤¯ 0, (2.7)

where ¯xis the unique fixed point of the mappingP roj_(A+B)⁻¹_{(0)∩F ix(T)}f.To show this inequality, we choose a subsequence{z_ni}of {z_n} such that

lim sup

n→∞

hf(¯x)−x, z¯ _n−xi¯ = lim

i→∞hf(¯x)−x, z¯ _ni−xi ≤¯ 0.

Since{z_ni}is bounded, we find that there exists a subsequence{z_nij}of{z_ni} which converges weakly to ˆx.

Without loss of generality, we assume thatzni *x.ˆ Puttingµn=J_r^B_n(zn−rnAzn),we find that µni *x.ˆ Next, we show ˆx∈(A+B)⁻¹(0). Notice thatz_n−r_nAz_n∈µ_n+r_nBµ_n; that is,

zn−rnAzn−µn

r_n ∈Bµn. Letµ∈Bν. Since B is maximal monotone, we find

zn−µn

rn

−Azn−µ, µn−ν

≥0.

It follows that h−Aˆx−µ,xˆ−νi ≥0.This in turn implies that −Aˆx∈Bx, that is, ˆˆ x∈(A+B)⁻¹(0).

Now, we are in a position to show that ˆxis also inF ix(T).Sincex_ni *x, we find from Lemma 1.5 thatˆ ˆ

x∈F ix(T) immediately. This proves that (2.7) holds.

Finally, we show that {x_n} converges strongly to ¯x, where ¯x is the unique fixed point of mapping P roj_(A+B)⁻¹_{(0)∩F ix(T)}f.

Note that

kz_n−xk¯ ² ≤α_nhf(¯x)−x, z¯ _n−xi¯ + 1−α_n(1−α)

kz_n−xkkx¯ _n−xk.¯ It follows that kz_n−xk¯ ² ≤2α_nhf(¯x)−x, z¯ _n−xi¯ + 1−α_n(1−α)

kx_n−xk¯ ².Hence, one has kx_n+1−xk¯ ² ≤ ky_n−xk¯ ²

≤ kJ_r^B_n(zn−rnAzn)−xk¯ ²+en(2kJ_r^B_n(zn−rnAzn)−xk¯ +en)

≤ kz_n−xk¯ ²+en(2kJ_r^B_n(zn−rnAzn)−xk¯ +en)

≤2α_nhf(¯x)−x, z¯ _n−xi¯ + 1−α_n(1−α)

kx_n−xk¯ ² +e_n(2kJ_r^B

n(z_n−r_nAz_n)−xk¯ +e_n).

An application of Lemma 1.2 to the above inequality yields that limn→∞kx_n−xk¯ = 0.This completes the proof.

From Theorem 2.1, the following results are not hard to derive.

Corollary 2.2. Let C be a nonempty convex closed subset of a real Hilbert space H. Let A:C →H be an inverse κ-strongly monotone mapping and let B be a maximal monotone operator onH. Let f :C →C be a fixed α-contraction and let T :C →C be a nonexpansive mapping. Assume that (A+B)⁻¹(0)∩F ix(T) is not empty. Let {α_n}, {β_n} be real number sequences in (0,1)and let {r_n} be a real number sequence in (0,2κ). Let{x_n} be a sequence inC generated in the following process: x₀∈C,x_n+1 =β_ny_n+ (1−β_n)T y_n,

∀n≥0,where{y_n} is a sequence inC such that ky_n−(I+rnB)⁻¹ αnf(xn) + (1−αn)xn−rnA αnf(xn) + (1−αn)xn

k ≤ en. Assume that the control sequences satisfy the following restrictions: limn→∞αn = 0, P∞

n=0α_n = ∞, P∞

n=1|α_n−αn−1|< ∞, 0 < a ≤ r_n ≤ a⁰ <2κ, P∞

n=1|r_n−rn−1| <∞, P∞

n=0ke_nk <∞, P∞

n=1|β_n −βn−1| < ∞, and 0 ≤ βn ≤ a⁰⁰ < 1, where a, a⁰ and a⁰⁰ are three real numbers. Then {x_n} converges strongly to a point x¯ ∈F ix(T)∩(A+B)⁻¹(0), where x¯ =P roj_{F ix(T)∩(A+B)}⁻¹₍₀₎f(¯x), that is, x¯ solves the following variational inequality hf(¯x)−x,¯ x¯−xi ≥0, ∀x∈F ix(T)∩(A+B)⁻¹(0).

Corollary 2.3. Let C be a nonempty convex closed subset of a real Hilbert space H. Let A:C →H be an inverse κ-strongly monotone mapping and let B be a maximal monotone operator onH. Let f :C →C be a fixed α-contraction. Assume that (A+B)⁻¹(0) is not empty. Let {α_n}, {β_n} be real number sequences in (0,1) and let {r_n} be a real number sequence in (0,2κ). Let x₀ ∈C and {x_n} be a sequence in C such that kx_n+1−(I +rnB)⁻¹ αnf(xn) + (1−αn)xn−rnA αnf(xn) + (1−αn)xn

k ≤ en. Assume that the control sequences satisfy the following restrictions: limn→∞α_n= 0,P∞

n=0α_n=∞,P∞

n=1|α_n−αn−1|<∞, 0 < a ≤ r_n ≤ a⁰ < 2κ, P∞

n=1|r_n−rn−1| < ∞, P∞

n=0ke_nk < ∞, where a and a⁰ are three real numbers.

Then{x_n} converges strongly to a point x¯∈(A+B)⁻¹(0), where x¯=P roj_(A+B)⁻¹₍₀₎f(¯x),that is, x¯ solves the following variational inequality hf(¯x)−x,¯ x¯−xi ≥0, ∀x∈(A+B)⁻¹(0).

Let C be a nonempty closed and convex subset of a Hilbert spaceH. Let iC be the indicator function ofC, that is,

i_C(x) =

(0, x∈C,

∞, x /∈C.

SinceiC is a proper lower and semicontinuous convex function onH, the subdifferential∂iC ofiC is maximal monotone. So, we can define the resolventJ_r^∂iCof∂iC forr >0, i.e.,J_r^∂iC := (I+r∂iC)⁻¹. Lettingx=J_r^∂iCy, we find that

y∈x+r∂iCx⇐⇒y∈x+rNCx

⇐⇒ hy−x, v−xi ≤0,∀v∈C

⇐⇒x=P rojCy,

whereP roj_C is the metric projection from H onto C and N_Cx:={e∈H :he, v−xi,∀v∈C}.

From Theorem 2.1, we have the following results on variational inequality (1.1).

Corollary 2.4. Let C be a nonempty convex closed subset of a real Hilbert space H. Let A : C → H be an inverse κ-strongly monotone mapping. Let f : C → C be a fixed α-contraction and let T : C → C be a λ-strict pseudocontraction. Assume that V I(C, A)∩F ix(T) is not empty. Let {α_n}, {β_n} be real number sequences in (0,1) and let {r_n} be a real number sequence in (0,2κ). Let {x_n} be a sequence in C generated in the following process: x0 ∈C, xn+1 =βnyn+ (1−βn)T yn, ∀n≥0,where {y_n} is a sequence in C such thatky_n−P roj_C α_nf(x_n) + (1−α_n)x_n−r_nA α_nf(x_n) + (1−α_n)x_n

k ≤e_n.Assume that the control sequences satisfy the following restrictions: limn→∞αn= 0,P∞

n=0αn=∞,P∞

n=1|α_n−αn−1|<∞, 0< a≤rn≤a⁰<2κ,P∞

n=1|r_n−r_n−1|<∞,P∞

n=0ke_nk<∞,P∞

n=1|β_n−β_n−1|<∞,andλ≤βn≤a⁰⁰<1, where a, a⁰ and a⁰⁰ are three real numbers. Then{x_n} converges strongly to a point x¯∈F ix(T)∩V I(C, A), wherex¯=P roj_{F ix(T})∩V I(C,A)f(¯x),that is, x¯ solves the following variational inequalityhf(¯x)−x,¯ x¯−xi ≥0,

∀x∈F ix(T)∩V I(C, A).

Acknowledgements

This article was supported by the National Natural Science Foundation of China under grant No.11401152.

References

[1] V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Translated from the Romanian, Editura Academiei Republicii Socialiste Romˆania, Bucharest; Noordhoff International Publishing, Leiden (1976).

1.3

[2] H. H. Bauschke,A note on the paper by Eckstein and Svaiter on general projective splitting methods for sums of maximal monotone operators,SIAM J. Control Optim.,48(2009), 2513–2515. 1

[3] B. A. Bin Dehaish, A. Latif, H. O. Bakodah, X. Qin, A viscosity splitting algorithm for solving inclusion and equilibrium problems,J. Inequal. Appl.,2015(2015), 14 pages. 1, 1.1

[4] B. A. Bin Dehaish, X. Qin, A. Latif, H. O. Bakodah,Weak and strong convergence of algorithms for the sum of two accretive operators with applications,J. Nonlinear Convex Anal.,16(2015), 1321–1336. 1

[5] F. E. Browder,Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A,54 (1965), 1041–1044. 1

[6] F. E. Browder, W. V. Petryshyn,Construction of fixed points of nonlinear mappings in Hilbert space,J. Math.

Anal. Appl.,20(1967), 197–228. 1, 1.4, 1.5

[7] S. Y. Cho,Generalized mixed equilibrium and fixed point problems in a Banach space,J. Nonlinear Sci. Appl.,9 (2016), 1083–1092. 1

[8] S. Y. Cho, X. Qin,On the strong convergence of an iterative process for asymptotically strict pseudocontractions and equilibrium problems,Appl. Math. Comput.,235(2014), 430–438. 1

[9] S. Y. Cho, X. Qin, L. Wang,Strong convergence of a splitting algorithm for treating monotone operators,Fixed Point Theory Appl.,2014(2014), 15 pages. 1

[10] J. Eckstein, D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators,Math. Programming,55(1992), 293–318. 1

[11] S. P. Han, G. Lou, A parallel algorithm for a class of convex programs,SIAM J. Control Optim., 26 (1988), 345–355. 1

[12] J. K. Kim, S. Y. Cho, X. Qin,Some results on generalized equilibrium problems involving strictly pseudocontractive mappings,Acta Math. Sci. Ser. B Engl. Ed.,31(2011), 2041–2057. 1

[13] J. K. Kim, G. S. Saluja,Convergence of composite implicit iterative process with errors for asymptotically non- expansive mappings in Banach spaces,Nonlinear Funct. Anal. Appl.,18(2013), 145–162. 1

[14] Y. Kimura,Shrinking projection methods for a family of maximal operators,Nonlinear Funct. Anal. Appl., 16 (2011), 481–489. 1

[15] P. L. Lions, B. Mercier,Splitting algorithms for the sum of two nonlinear operators,SIAM J. Numer. Anal.,16 (1979), 964–979. 1

[16] B. Liu, C. Zhang, Strong convergence theorems for equilibrium problems and quasi-φ-nonexpansive mappings, Nonlinear Funct. Anal. Appl.,16(2011), 365–385. 1

[17] X. Qin, S. Y. Cho, L. Wang,Iterative algorithms with errors for zero points ofm-accretive operators,Fixed Point Theory Appl.,2013(2013), 17 pages. 1

[18] X. Qin, S. Y. Cho, L. Wnag, A regularization method for treating zero points of the sum of two monotone operators,Fixed Point Theory Appl.,2014(2014), 10 pages. 1

[19] X. Qin, S. Y. Cho, L. Wang, Convergence of splitting algorithms for the sum of two accretive operators with applications,Fixed Point Theory Appl.,2014(2014), 12 pages. 1

[20] R. T. Rockfellar,Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res.,1(1976), 97–116. 1

[21] M. Sibony,M´ethodes it´eratives pour les ´equations et in´equations aux d´eriv´ees partielles nonlin´eares de type mono- tone,(French), Calcolo,7(1970), 65–183. 1

[22] J. E. Spingarn, Applications of the method of partial inverses to convex programming: decomposition, Math.

Programming,32(1985), 199–223. 1

[23] M. V. Solodov, B. F. Svaiter,Forcing strong convergence of proximal point iterations in a Hilbert space,Math.

Program.,87(2000), 189–202. 1

[24] G. Wang, S. Sun,Hybrid projection algorithms for fixed point and equilibrium problems in a Banach space,Adv.

Fixed Point Theory,3(2013), 578–594. 1

[25] Z. Q. Xue, H. Y. Zhou, Y. J. Cho,Iterative solutions of nonlinear equations for m-accretive operators in Banach spaces,J. Nonlinear Convex Anal.,1(2000), 313–320. 1, 1.2

[26] Y. Yao, Y. J. Cho, Y. C. Liou,Iterative algorithms for variational inclusions, mixed equilibrium problems and fixed point problems approach to optimization problems,Central European J. Math.,9(2011), 640–656. 1

[27] Y. Yao, M. A. Noor, S. Zainab, Y. C. Liou, Mixed equilibrium problems and optimization problems, J. Math.

Anal. Appl.,354(2009), 319–329. 1