Iterative algorithms for common elements in fixed point sets and zero point sets with applications

Abstract

In this study, Mann-type iterative process is considered for finding a common element in the fixed point set of strict pseudocontractions and in the zero point set of the operator which is the sum of inverse strongly- monotone operators and maximal monotone operators. Weak convergence theorems of common elements are established in the framework of Hilbert spaces. Some applications of main results are also provided.

AMS Subject Classification: 47H05; 47H09; 47J25; 90C33.

1 Introduction and preliminaries

Throughout this article, we always assume that H is a real Hilbert space with the inner product 〈 , 〉, and the norm ||·|| and that C is a nonempty closed convex subset of H.

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

$$⟨Ax-Ay,x-y⟩\ge 0,\forall x,y\in C.$$

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

$$⟨Ax-Ay,x-y⟩\ge \alpha {∥Ax-Ay∥}^2,\forall x,y\in C.$$

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

Let M : H → 2^Hbe a set-valued mapping. The set D(M) defined by D(M) = {x ∈ H: Mx ≠ ∅} is said to be the domain of M. The set R(M) defined by $R\left(M\right)={\bigcup }_{x\in H}Mx$ is said to be the range of M. The set G(M) defined by G(M) = {(x, y) ∈ H × H: x ∈ D(M), y ∈ R(M)} is said to be the graph of M.

Recall that M is said to be monotone if

$$⟨x-y,f-g⟩\ge 0,\forall \left(x,f\right),\left(y,g\right)\in G\left(M\right).$$

M is said to be maximal monotone if it is not properly contained in any other monotone operator. Equivalently, M is maximal monotone if R(I + rM) = H for all r > 0. The class of monotone mappings is one of the most important classes of mappings. Within the past several decades, many authors have been devoting to the studies on the existence and convergence of zero points for maximal monotone mappings, see [1–15] and the references therein. For a maximal monotone operator M on H and r > 0, we may define the single-valued resolvent J_r = (I + rM)^-1 : H → D(M). It is known that J _r is firmly nonexpansive and M^-1 (0) = F(J _r ), where F (J _r ) denotes the fixed point set of J _r .

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

$$∥Sx-Sy∥\le ∥x-y∥,\forall x,y\in C.$$

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

$${∥Sx-Sy∥}^2\le {∥x-y∥}^2+\kappa {∥\left(x-Sx\right)-\left(y-Sy\right)∥}^2,\forall x,y\in C.$$

The class of strictly pseudocontractive mappings was introduced by Browder and Petryshyn [16]. If κ = 0, the class of strictly pseudocontractive mappings is reduced to the class of nonexpansive mappings. In case that κ = 1, we call S a pseudocontractive mapping. Marino and Xu [17] proved that fixed point sets of strictly pseudocontractive mappings are closed and convex. They also proved that I - S is demi-closed at zero. To be more precise, if {x _n } is a sequence in C with x _n ⇀ x and x _n - Sx _n → 0, then x ∈ F (S).

Let A : C → H be an inverse strongly-monotone mapping. Recall that the classical variational inequality problem is to find x ∈ C such that

$$⟨Ax,y-x⟩\ge 0,\forall y\in C.$$

(1.1)

Denote by V I(C, A) of the solution set of (1.1). It is known that x ∈ C is a solution to (1.1) if and only if x is a fixed point of the mapping P _C (I - λA), where λ > 0 is a constant and I is the identity mapping. In [3], Iiduka and Takahashi showed that if λ ∈ [0, 2α], then I - λA is nonexpansive.

Let F be a bifunction from C × C to ℝ, where ℝ denotes the set of real numbers. Recall the following equilibrium problem.

$$\mathsf{\text{Find}}x\in C\mathsf{\text{such}}\mathsf{\text{that}}F\left(x,y\right)\ge 0,\forall y\in C.$$

(1.2)

To study the equilibrium problems (1.2), we may assume that F satisfies the following conditions:

(A 1) F (x, x) = 0 for all x ∈ C;

(A 2) F is monotone, i.e., F (x, y) + F (y, x) ≤ 0 for all x, y ∈ C;

(A 3) for each x, y, z ∈ C,

$$\underset{t\downarrow 0}{\mathsf{\text{lim}}\mathsf{\text{sup}}}F\left(tz+\left(1-t\right)x,y\right)\le F\left(x,y\right);$$

(A 4) for each x ∈ C, y ↦ F (x, y) is convex and lower semi-continuous.

Putting F (x, y) = 〈Ax, y - x〉 for every x, y ∈ C, we see that the equilibrium problem (1.2) is reduced to the variational inequality (1.1).

Recently, many authors considered the convergence of iterative sequences for the variational inequality (1.1), the equilibrium problem (1.2) and fixed point problems of nonlinear mappings; see, for example, [2, 4–7, 11–15, 18–26].

In 2003, Takahashi and Toyoda [13] proved the following weak convergence theorem.

Theorem 1.1. Let C be a closed convex subset of a real Hilbert space H. Let A be an α-inverse strongly-monotone mapping from C into H and S be a non-expansive mapping from C into itself such that F(S) ∩ VI (C, A) ≠ ∅. Let {x _n } be a sequence generated by

$$x_0\in C,x_{n+1}={\alpha }_n{x}_n+\left(1-{\alpha }_n\right)S{P}_C\left(x_n-{\lambda }_{n}A{x}_n\right),\forall n\ge 0,$$

where λ _n ∈ [a, b] for some a, b ∈ (0, 2α) and α _n ∈ [c, d] for some c, d ∈ (0, 1). Then, {x _n } converges weakly to z ∈ F(S) ∩ VI (C, A), where z = lim_n→∞P_{F(S)∩VI (C, A)}x _n .

In 2007, Tada and Takahashi [11] obtained the following weak convergence theorem.

Theorem 1.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C × C to ℝ satisfying (A 1)-(A 4) and S be a nonexpansive mapping from C into H such that F (S) ∩ EP (F) ≠ ∅. Let {x _n } and {u _n } be sequences generated by x ₁ = x ∈ H and let

$$\left\{\begin{array}{cc}\hfill u_n\in CsuchthatF\left(u_n,u\right)+\frac{1}{r_n}⟨u-u_n,u_n-x_n⟩\ge 0,\hfill & \hfill \forall u\in C,\hfill \\ x_{n+1}={\alpha }_n{x}_n+\left(1-{\alpha }_n\right)S{u}_n\hfill \end{array}\right.$$

for each n ≥ 1, where {α _n } ⊂ [a, b] for some a, b ∈ (0, 1) and {r _n } ⊂ (0, ∞) satisfies lim inf_n→∞r _n > 0. Then, {x _n } converges weakly to w ∈ F(S)∩EP(F) where w = lim_n→∞P_F(S)∩EP(F)x _n .

A very common problem in diverse areas of mathematics and physical sciences consists of trying to find a point in the intersection of convex sets. This problem is referred to as the convex feasibility problem; its precise mathematical formulation is as follows. Find an $x\in {\bigcap }_{m=1}^N{C}_r$, where N ≥ 1 is an integer and each C _m is a nonempty closed convex subset of H. There is a considerable investigation on the convex feasibility problem in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration, computer tomography, and radiation therapy treatment planning.

Let K be an integer, S : C → C a strict pseudocontraction, A _m : C → H be an α _m -inverse strongly-monotone mapping and M _m : H → 2^Hbe a maximal monotone operator such that D(M _m ) ⊂ C, where D(M _m ) is the domain of M _m , where m ∈ {1, 2, ..., K}. In this article, motivated by Theorems 1.1 and 1.2, we consider the problem of finding a common element in the following set: $F\left(S\right)\cap {\bigcap }_{m=1}^K{\left(A_m+M_m\right)}^{-1}\left(0\right)$, where F (S) is the fixed point set of S and (A _m + M _m )^{- 1}(0) is the zero point set of A _m + M _m . Weak convergence theorems of common elements are established in real Hilbert spaces. The results presented in this article improve and extend the corresponding results announced by Tada and Takahashi [11] and Takahshi and Toyoda [13].

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

Lemma 1.3. [16] Let C be a nonempty closed convex subset of a real Hilbert space H and S : C → C be a κ-strict pseudo-contraction with a fixed point. Define S : C → C by S _a x = ax + (1 - a)Sx for each x ∈ C. If a ∈ [κ, 1), then S _a is nonexpansive with F (S _a ) = F (S).

Lemma 1.4. [17] Let C be a nonempty closed convex subset of a real Hilbert space H and S : C → C be a κ-strict pseudocontraction. Then

• (a) S is $\frac{1+\kappa }{1-\kappa }$-Lipschitz;

• (b) I - S is demi-closed, this is, if {x _n } is a sequence in C with x _n ⇀ x and x _n - Sx _n → 0, then x ∈ F (S).

Lemma 1.5. [27] Let H be a real Hilbert space and 0 < p ≤ t _n ≤ q < 1 for all n ≥ 1. Suppose that {x _n } and {y _n } are sequences in H such that

$$\underset{n\to \infty }{\mathsf{\text{lim}}\mathsf{\text{sup}}}∥x_n∥\le r,\underset{n\to \infty }{\mathsf{\text{lim}}\mathsf{\text{sup}}}∥y_n∥\le r$$

and

$$\underset{n\to \infty }{\mathsf{\text{lim}}}∥t_n{x}_n+\left(1-t_n\right)y_n∥=r$$

hold for some r ≥ 0. Then lim_n→∞||x _n -y _n || = 0.

Lemma 1.6. [13] Let C be a nonempty closed convex subset of a real Hilbert space H and P _C be the metric projection from H onto C. Let {x _n } be a sequence in H. Suppose that, for all y ∈ C

$$∥x_{n+1}-y∥\le ∥x_n-y∥,\forall n\ge 1.$$

Then {P _C x _n } converges strongly to some z ∈ C.

Lemma 1.7. [28] Let C be a nonempty closed convex subset of a real Hilbert space H, A : C → H be a mapping and M : H → 2^H be a maximal monotone mapping. Then

$$F\left(J_r\left(I-rA\right)\right)={\left(A+M\right)}^{-1}\left(0\right),\forall r>0.$$

Lemma 1.8. [29] Let H be a Hilbert space and suppose {x _n } converges weakly to x. Then

$$\underset{n\to \infty }{\mathsf{\text{lim}}\mathsf{\text{inf}}}∥x_n-x∥<\underset{n\to \infty }{\mathsf{\text{lim}}\mathsf{\text{inf}}}∥x_n-y∥$$

for all y ∈ H with x ≠ y.

2 Main results

Theorem 2.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let S : C → C be a κ-strict pseudocontraction, A : C → H be an α-inverse strongly monotone mapping and B : C → H be a β-inverse strongly monotone mapping. Let M : H → 2^H and W : H → 2^H be maximal monotone operators such that D(M) ⊂ C and D(W) ⊂ C. Assume that $\mathcal{F}:=F\left(S\right)\cap {\left(A+M\right)}^{-1}\left(0\right)\cap {\left(B+W\right)}^{-1}\left(0\right)\ne \varnothing$. Let {x _n } be a sequence generated in the following manner:

$$\left\{\begin{array}{c}{x}_0\in C,\hfill \\ y_n={\gamma }_n{J}_{r_n}\left(x_n-r_{n}A{x}_n\right)+\left(1-{\gamma }_n\right)J_{s_n}\left(x_n-s_{n}B{x}_n\right),\hfill \\ x_{n+1}={\alpha }_n{x}_n+\left(1-{\alpha }_n\right)\left({\beta }_n{y}_n+\left(1-{\beta }_n\right)S{y}_n\right),n\ge 0,\hfill \end{array}\right.$$

where $J_{r_n}={\left(I+r_{n}M\right)}^{-1}$, $J_{s_n}={\left(I+s_{n}W\right)}^{-1}$, {r _n } is a sequence in (0, 2α), {s _n } is a sequence in (0, 2β) and {α _n }, {β _n }, and {γ _n } are sequences in (0, 1). Assume that the following restrictions are satisfied

• (a) 0 < a ≤ r _n ≤ b < 2α and 0 < c ≤ s _n ≤ d < 2β;

• (b) 0 ≤ κ ≤ β _n < e < 1, 0 < h ≤ α _n ≤ i < 1 and 0 < j ≤ γ _n ≤ k < 1,

where a, b, c, d, e, h, i, j, k are real numbers. Then the sequence {x _n } converges weakly to $\bar{x}\in \mathcal{F}$, where $\bar{x}={\mathsf{\text{lim}}}_{n\to \infty }{P}_{\mathcal{F}}{x}_n$.

Proof. Note that (I - r _n A) and (I - s _n B) are nonexpansive for each fixed n ≥ 0.

Indeed, for any x, y, ∈ C, we see from the restriction (a) that

$$\begin{array}{cc}\hfill {∥\left(I-r_{n}A\right)x-\left(I-r_{n}A\right)y∥}^2& ={∥x-y∥}^2-2r_n⟨x-y,Ax-Ay⟩+r_n^2{∥Ax-Ay∥}^2\hfill \\ \le {∥x-y∥}^2-r_n\left(2\alpha -r_n\right){∥Ax-Ay∥}^2\hfill \\ \le {∥x-y∥}^2.\hfill \end{array}$$

This shows that (I - r _n A) is nonexpansive for each fixed n ≥ 0, so is (I - s _n B).

Put

$$S_{n}x={\beta }_{n}x+\left(1-{\beta }_n\right)Sx,\forall x\in C.$$

In the restriction (b), we obtain from Lemma 1.3 that S _n is nonexpansive for each fixed n ≥ 0. Fixing $p\in \mathcal{F}$ and since $J_{r_n}$, $J_{s_n}$, I - r _n A, and I - s _n B are nonexpansive, we see that

$$\begin{array}{cc}\hfill ∥y_n-p∥& \le {\gamma }_n∥J_{r_n}\left(x_n-r_{n}A{x}_n\right)-p∥+\left(1-{\gamma }_n\right)∥J_{s_n}\left(x_n-s_{n}B{x}_n\right)-p∥\hfill \\ \le ∥x_n-p∥.\hfill \end{array}$$

Since S _n is nonexpansive, we see that

$$\begin{array}{cc}\hfill ∥x_{n+1}-p∥|& \le {\alpha }_n∥x_n-p∥+\left(1-{\alpha }_n\right)∥S_n{y}_n-p∥\hfill \\ \le {\alpha }_n∥x_n-p∥+\left(1-{\alpha }_n\right)∥y_n-p∥\hfill \\ \le ∥x_n-p∥.\hfill \end{array}$$

(2.1)

Hence, the limit of the sequence {||x _n - p||} exists. This shows that the sequence {x _n } is bounded, so is {y _n }. Without loss of generality, we may assume that lim_n→∞||x _n -p|| = d > 0. Notice that

$$\begin{array}{cc}\hfill {∥y_n-p∥}^2& \le {\gamma }_n{∥J_{r_n}\left(x_n-r_{n}A{x}_n\right)-p∥}^2+\left(1-{\gamma }_n\right){∥J_{s_n}\left(x_n-s_{n}B{x}_n\right)-p∥}^2\hfill \\ \le {\gamma }_n{∥\left(x_n-r_{n}A{x}_n\right)-p∥}^2+\left(1-{\gamma }_n\right){∥\left(x_n-s_{n}B{x}_n\right)-p∥}^2\hfill \\ \le {\gamma }_n\left({∥x_n-p∥}^2-r_n\left(2\alpha -r_n\right){∥A{x}_n-Ap∥}^2\right)\hfill \\ +\left(1-{\gamma }_n\right)\left({∥x_n-p∥}^2-s_n\left(2\beta -s_n\right){∥B{x}_n-Bp∥}^2\right)\hfill \\ \le {∥x_n-p∥}^2-r_n{\gamma }_n\left(2\alpha -r_n\right){∥A{x}_n-Ap∥}^2\hfill \\ -s_n\left(1-{\gamma }_n\right)\left(2\beta -s_n\right){∥B{x}_n-Bp∥}^2.\hfill \end{array}$$

This in turn implies that

$$\begin{array}{cc}\hfill {∥x_{n+1}-p∥}^2& \le {\alpha }_n{∥x_n-p∥}^2+\left(1-{\alpha }_n\right){∥S_n{y}_n-p∥}^2\hfill \\ \le {\alpha }_n{∥x_n-p∥}^2+\left(1-{\alpha }_n\right){∥y_n-p∥}^2\hfill \\ \le {∥x_n-p∥}^2-\left(1-{\alpha }_n\right)r_n{\gamma }_n\left(2\alpha -r_n\right){∥A{x}_n-Ap∥}^2\hfill \\ -\left(1-{\alpha }_n\right)s_n\left(1-{\gamma }_n\right)\left(2\beta -s_n\right){∥B{x}_n-Bp∥}^2.\hfill \end{array}$$

(2.2)

It follows from the restrictions (a) and (b) that

$$\left(1-i\right)aj\left(2\alpha -b\right){∥A{x}_n-Ap∥}^2\le {∥x_n-p∥}^2-{∥x_{n+1}-p∥}^2.$$

Since lim_n→∞||x _n -p|| = d, we see that

$$\underset{n\to \infty }{\mathsf{\text{lim}}}∥A{x}_n-Ap∥=0.$$

(2.3)

In view of (2.2), we see from the restrictions (a) and (b) that

$$\left(1-i\right)c\left(1-k\right)\left(2\beta -d\right){∥B{x}_n-Bp∥}^2\le {∥x_n-p∥}^2-{∥x_{n+1}-p∥}^2.$$

Since lim_n→∞||x _n -p|| = d, we see that

$$\underset{n\to \infty }{\mathsf{\text{lim}}}∥B{x}_n-Bp∥=0.$$

(2.4)

Notice that $J_{r_n}$ is firmly nonexpansive. Putting $u_n=J_{r_n}\left(x_n-r_{n}A{x}_n\right)$ and $v_n=J_{s_n}\left(x_n-s_{n}B{x}_n\right)$, we see that

$$\begin{array}{cc}\hfill {∥u_n-p∥}^2& ={∥J_{r_n}\left(x_n-r_{n}A{x}_n\right)-J_{r_n}\left(p-r_{n}Ap\right)∥}^2\hfill \\ \le ⟨u_n-p,\left(x_n-r_{n}A{x}_n\right)-\left(p-r_{n}Ap\right)⟩\hfill \\ =\frac{1}{2}\left({∥u_n-p∥}^2+{∥\left(x_n-r_{n}A{x}_n\right)-\left(p-r_{n}Ap\right)∥}^2\right.\hfill \\ \left.-{∥\left(u_n-p\right)-\left(\left(x_n-r_{n}A{x}_n\right)-\left(p-r_{n}Ap\right)\right)∥}^2\right)\hfill \\ \le \frac{1}{2}\left({∥u_n-p∥}^2+{∥x_n-p∥}^2-{∥u_n-x_n+r_n\left(A{x}_n-Ap\right)∥}^2\right)\hfill \\ =\frac{1}{2}{\left({∥u_n-p∥}^2+{∥x_n-p∥}^2-{∥u_n-x_n∥}^2-r_n^2∥A{x}_n-Ap∥\right.}^2\hfill \\ \left.-2r_n⟨u_n-x_n,A{x}_n-Ap⟩\right)\hfill \\ \le \frac{1}{2}\left({∥u_n-p∥}^2+{∥x_n-p∥}^2-{∥u_n-x_n∥}^2+2r_n∥u_n-x_n∥∥A{x}_n-Ap∥\right).\hfill \end{array}$$

This in turn implies that

$${∥u_n-p∥}^2\le {∥x_n-p∥}^2-{∥u_n-x_n∥}^2+2r_n∥u_n-x_n∥∥A{x}_n-Ap∥.$$

(2.5)

In a similar way, we can obtain that

$${∥v_n-p∥}^2\le {∥x_n-p∥}^2-{∥v_n-x_n∥}^2+2s_n∥v_n-x_n∥∥B{x}_n-Bp∥.$$

(2.6)

Combining (2.5) with (2.6) yields that

$$\begin{array}{cc}\hfill {∥x_{n+1}-p∥}^2& \le {\alpha }_n{∥x_n-p∥}^2+\left(1-{\alpha }_n\right){∥S_n{y}_n-p∥}^2\hfill \\ \le {\alpha }_n{∥x_n-p∥}^2+\left(1-{\alpha }_n\right){∥y_n-p∥}^2\hfill \\ \le {\alpha }_n{∥x_n-p∥}^2+\left(1-{\alpha }_n\right)\left({\gamma }_n{∥u_n-p∥}^2+\left(1-{\gamma }_n\right){∥v_n-p∥}^2\right)\hfill \\ \le {∥x_n-p∥}^2-\left(1-{\alpha }_n\right){\gamma }_n{∥u_n-x_n∥}^2+2r_n∥u_n-x_n∥∥A{x}_n-Ap∥\hfill \\ -\left(1-{\alpha }_n\right)\left(1-{\gamma }_n\right){∥v_n-x_n∥}^2+2s_n∥v_n-x_n∥∥B{x}_n-Bp∥.\hfill \end{array}$$

(2.7)

It follows that

$$\begin{array}{cc}\hfill \left(1-{\alpha }_n\right){\gamma }_n{∥u_n-x_n∥}^2& \le {∥x_n-p∥}^2-{∥x_{n+1}-p∥}^2+2r_n∥u_n-x_n∥∥A{x}_n-Ap∥\hfill \\ +2s_n∥v_n-x_n∥∥B{x}_n-Bp∥.\hfill \end{array}$$

In view of (2.3) and (2.4), we see from the restrictions (a) and (b) that

$$\underset{n\to \infty }{\mathsf{\text{lim}}}∥u_n-x_n∥=0.$$

(2.8)

It also follows from (2.7) that

$$\begin{array}{cc}\hfill \left(1-{\alpha }_n\right)\left(1-{\gamma }_n\right){∥v_n-x_n∥}^2& \le {∥x_n-p∥}^2-{∥x_{n+1}-p∥}^2+2r_n∥u_n-x_n∥∥A{x}_n-Ap∥\hfill \\ +2s_n∥v_n-x_n∥∥B{x}_n-Bp∥.\hfill \end{array}$$

In view of (2.3) and (2.4), we see from the restrictions (a) and (b) that

$$\underset{n\to \infty }{\mathsf{\text{lim}}}∥v_n-x_n∥=0$$

(2.9)

Notice that

$$∥y_n-x_n∥\le ∥u_n-x_n∥+∥v_n-x_n∥.$$

It follows from (2.8) and (2.9) that

$$\underset{n\to \infty }{\mathsf{\text{lim}}}∥y_n-x_n∥=0.$$

(2.10)

On the other hand, we have

$$\underset{n\to \infty }{\mathsf{\text{lim}}}∥{\alpha }_n\left(x_n-p\right)+\left(1-{\alpha }_n\right)\left(S_n{y}_n-p\right)∥=d.$$

Notice that

$$∥S_n{y}_n-p∥\le ∥y_n-p∥\le ∥x_n-p∥\mathsf{\text{.}}$$

This implies that

$$\underset{n\to \infty }{\mathsf{\text{lim}}\mathsf{\text{sup}}}∥S_n{y}_n-p∥\le d.$$

In view of Lemma 1.5, we arrive at

$$\underset{n\to \infty }{\mathsf{\text{lim}}}∥S_n{y}_n-x_n∥=0.$$

(2.11)

Note that

$$S{y}_n-x_n=\frac{S_n{y}_n-x_n}{1-{\beta }_n}+\frac{{\beta }_n\left(x_n-y_n\right)}{1-{\beta }_n}.$$

From (2.10), (2.11) and the restriction (b), we get that

$$\underset{n\to \infty }{\mathsf{\text{lim}}}∥S{y}_n-x_n∥=0.$$

(2.12)

On the other hand, we see from Lemma 1.4 that

$$\begin{array}{cc}\hfill ∥S{x}_n-x_n∥& \le ∥S{x}_n-S{y}_n∥+∥S{y}_n-x_n∥\hfill \\ \le \frac{1+\kappa }{1-\kappa }∥x_n-y_n∥+∥S{y}_n-x_n∥.\hfill \end{array}$$

It follows from (2.10) and (2.12) that

$$\underset{n\to \infty }{\mathsf{\text{lim}}}∥S{x}_n-x_n∥=0.$$

(2.13)

Since {x _n } is bounded, we see that there exists a subsequence $\left\{x_{n_i}\right\}$ of {x _n } which converges weakly to $\bar{x}$. By virtue of Lemma 1.4, we obtain that $\bar{x}\in F\left(S\right)$. Next, we show that $\bar{x}\in {\left(A+M\right)}^{-1}\left(0\right).$ Notice that

$$x_n-r_{n}A{x}_n\in u_n+r_{n}M{u}_n.$$

Let μ ∈ Mν. Since M is monotone, we have

$$〈\frac{x_n-u_n}{r_n}-A{x}_n-\mu ,u_n-\nu 〉\ge 0.$$

In view of the restriction (a), we see from (2.8) that

$$⟨-A\bar{x}-\mu ,\bar{x}-\nu ⟩\ge 0.$$

This implies that $-A\bar{x}\in M\bar{x}$, that is, $\bar{x}\in {\left(A+M\right)}^{-1}\left(0\right)$. In a similar way, we can obtain that $\bar{x}\in {\left(B+W\right)}^{-1}\left(0\right)$. This proves that $\bar{x}\in \mathcal{F}$.

Assume that there exists another subsequence $\left\{x_{n_j}\right\}$ of {x _n } such that $\left\{x_{n_j}\right\}$ converges weakly to x'. By the above proof, we also have $x^{\prime }\in \mathcal{F}$. If $\bar{x}\ne x^{\prime }$, we get from Lemma 1.8 that

$$\begin{array}{cc}\hfill \underset{n\to \infty }{\mathsf{\text{lim}}}∥x_n-\bar{x}∥& =\underset{i\to \infty }{\mathsf{\text{lim}}\mathsf{\text{inf}}}∥x_{n_i}-\bar{x}∥<\underset{i\to \infty }{\mathsf{\text{lim}}\mathsf{\text{inf}}}∥x_{n_i}-x^{\prime }∥\hfill \\ =\underset{n\to \infty }{\mathsf{\text{lim}}}∥x_n-x^{\prime }∥=\underset{j\to \infty }{\mathsf{\text{lim}}\mathsf{\text{inf}}}∥x_{n_j}-x^{\prime }∥\hfill \\ <\underset{j\to \infty }{\mathsf{\text{lim}}\mathsf{\text{inf}}}∥x_{n_j}-\bar{x}∥=\underset{n\to \infty }{\mathsf{\text{lim}}}∥x_n-\bar{x}∥.\hfill \end{array}$$

This derives a contradiction. Hence, we have $\bar{x}=x^{\prime }$. This implies that $x_n\rightharpoonup \bar{x}\in \mathcal{F}$. Let $e_n=P_{\mathcal{F}}{x}_n$. In view of (2.1), we obtain from Lemma 1.6 that {e _n } converges strongly to some $e\in \mathcal{F}$. On the other hand, we see from $\bar{x}\in \mathcal{F}$ that $⟨x_n-e_n,e_n-\bar{x}⟩\ge 0$. Note that {x _n } converges weakly to $\bar{x}$. It follows that

$$⟨\bar{x}-e,e-\bar{x}⟩\ge 0.$$

This implies that $\bar{x}=e={\mathsf{\text{lim}}}_{n\to \infty }{P}_{\mathcal{F}}{x}_n$. The proof is completed. □

From Theorem 2.1, we can obtain the following immediately.

Theorem 2.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let S : C → C be a κ-strict pseudocontraction, A _m : C → H be an α _m -inverse strongly monotone mapping and M _m : H → 2^H be a maximal monotone operator such that D(M _m ) ⊂ C, where m ∈ {1, 2, ..., K}. Assume that $\mathcal{F}:=F\left(S\right)\cap {\bigcap }_{m=1}^N{\left(A_m+M_m\right)}^{-1}\left(0\right)\ne \varnothing$. Let {x _n } be a sequence generated in the following manner:

$$\left\{\begin{array}{c}{x}_0\in C,\hfill \\ y_n={\sum }_{m=1}^K{\gamma }_{n,m}{J}_{r_n,m}\left(x_n-r_{n,m}{A}_m{x}_n\right),\hfill \\ \hfill x_{n+1}={\alpha }_n{x}_n+\left(1-{\alpha }_n\right)\left({\beta }_n{y}_n+\left(1-{\beta }_n\right)S{y}_n\right),\hfill & \hfill n\ge 0,\hfill \end{array}\right.$$

where $J_{r_n,m}={\left(I+r_{n,m}{M}_m\right)}^{-1}$, {r_n,m} is a sequence in (0, 2α _m ), {α _n }, {β _n }, and {γ_n,m} are sequences in (0, 1). Assume that the following restrictions are satisfied

• (a) 0 < a _m ≤ r _n,m ≤ b _m < 2α _m for each m ∈ {1, 2, ..., K};

• (b) ${\sum }_{m=1}^K{\gamma }_{n,m}=1;$

• (c) 0 ≤ k ≤ β _n < c < 1, 0 < d ≤ α _n ≤ e < 1 and 0 < h _m ≤ γ _n,m ≤ i _m < 1, where a₁, a₂, ..., a _K , b₁, b₂, ..., b _K , c, d, e, h₁, h₂, ..., h _K , i₁, i₂, ..., i _K are real numbers. Then the sequence {x _n } converges weakly to $\bar{x}\in \mathcal{F}$, where $\bar{x}={\mathsf{\text{lim}}}_{n\to \infty }{P}_{\mathcal{F}}{x}_n$.

If S = I, where I denotes the identity, then Theorem 2.2 is reduced to the following.

Corollary 2.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A _m : C → H be an α _m -inverse strongly monotone mapping and M _m : H → 2^H be a maximal monotone operator such that D(M _m ) ⊂ C, where m ∈ {1, 2, ..., K}. Assume that $\mathcal{F}:={\bigcap }_{m=1}^N{\left(A_m+M_m\right)}^{-1}\left(0\right)\ne \varnothing$. Let {x _n } be a sequence generated in the following manner:

$$x_0\in C,x_{n+1}={\alpha }_n{x}_n+\left(1-{\alpha }_n\right)\sum _{m=1}^K{\gamma }_{n,m}{J}_{r_n,m}\left(x_n-r_{n,m}{A}_m{x}_n\right),n\ge 0,$$

where $J_{r_n,m}={\left(I+r_{n,m}{M}_m\right)}^{-1}$, {r _n,m } is a sequence in (0, 2α _m ) and {α _n }, {β _n } and {γ _n,m } are sequences in (0, 1). Assume that the following restrictions are satisfied

• (a) 0 < a _m ≤ r _n,m ≤ b _m < 2α _m for each m ∈ {1, 2, ..., K};

• (b) ${\sum }_{m=1}^K{\gamma }_{n,m}=1;$

• (c) 0 < c ≤ α _n < d < 1 and 0 < h _m ≤ γ _n,m ≤ i _m < 1,

where a₁, a₂, ..., a _K , b₁, b₂, ..., b _K , c, d, h₁, h₂, ..., h _K , i₁, i₂, ..., i _K are real numbers. Then the sequence {x _n } converges weakly to $\bar{x}\in \mathcal{F}$, where $\bar{x}={\mathsf{\text{lim}}}_{n\to \infty }{P}_{\mathcal{F}}{x}_n$.

3 Applications

Let H be a Hilbert space and f : H → (-∞, +∞] a proper convex lower semicontinuous function. Then the subdifferential ∂f of f is defined as follows:

$$\partial f\left(x\right)=\left\{y\in H:f\left(z\right)\ge f\left(x\right)+⟨z-x,y⟩,z\in H\right\},\forall x\in H.$$

From Rockafellar [9, 30], we know that ∂f is maximal monotone. It is easy to verify that 0 ∈ ∂f(x) if and only if f(x) = min_y∈Hf(y).

First, we consider the problem of finding common minimizers of proper convex lower semicontinuous functions.

Theorem 3.1. Let H be a real Hilbert space. Let f : H → (-∞, +∞] and g : H → (-∞, +∞] be proper convex lower semi-continuous functions. Assume that $\mathcal{F}:={\left(\partial f\right)}^{-1}\left(0\right)\cap {\left(\partial g\right)}^{-1}\left(0\right)\ne \varnothing$. Let {x _n } be a sequence generated in the following manner:

$$\left\{\begin{array}{c}{x}_0\in H,\hfill \\ z_n=\mathsf{\text{arg}}{\mathsf{\text{min}}}_{z\in H}\left\{g\left(z\right)+\frac{{∥z-x_n∥}^2}{2s_n}\right\},\hfill \\ y_n=\mathsf{\text{arg}}{\mathsf{\text{min}}}_{z\in H}\left\{f\left(z\right)+\frac{{∥z-x_n∥}^2}{2r_n}\right\},\hfill \\ x_{n+1}={\alpha }_n{x}_n+\left(1-{\alpha }_n\right)\left({\gamma }_n{y}_n+\left(1-{\gamma }_n\right)z_n\right),\hfill & \hfill n\ge 0,\hfill \end{array}\right.$$

where {α _n }, {β _n }, and {γ _n } are sequences in (0, 1). Assume that the following restrictions are satisfied

• (a) 0 < a ≤ r _n ≤ b < ∞ and 0 < c ≤ s _n ≤ d < ∞;

• (b) 0 < h ≤ α _n ≤ i < 1 and 0 < j ≤ γ _n ≤ k < 1,

where a, b, c, d, h, i, j, k are real numbers. Then the sequence {x _n } converges weakly to $\bar{x}\in \mathcal{F}$, where $\bar{x}={\mathsf{\text{lim}}}_{n\to \infty }{P}_{\mathcal{F}}{x}_n$.

Proof. Putting A = B = 0 and S = I, the identity mapping, we can conclude from Theorem 2.1 the desired conclusion immediately. □

Let I _C be the indicator function of C, i.e.,

$$I_C\left(x\right)=\left\{\begin{array}{cc}\hfill 0,\hfill & \hfill x\in C,\hfill \\ \hfill +\infty ,\hfill & \hfill x\notin C.\hfill \end{array}\right.$$

(3.1)

Since I _C is a proper lower semicontinuous convex function on H, we see that the subdifferential ∂I _C of I _C is a maximal monotone operator.

Lemma 3.2. [12] Let C be a nonempty closed convex subset of a real Hilbert space H. Let P _C be the metric projection from H onto C, ∂I _C be the subdifferential of I _C , where I _C is as defined in (3.1) and J _r = (I + r∂I _C )^{- 1}. Then

$$y=J_{r}x\iff y=P_{C}x,x\in H,y\in C.$$

Second, we consider the variation inequality (1.1).

Theorem 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H and P _C be the metric projection from H onto C. Let S : C → C be a κ-strict pseudocontraction, A : C → H be an α-inverse strongly monotone mapping and B : C → H be a β-inverse strongly monotone mapping. Assume that $\mathcal{F}:=F\left(S\right)\cap VI\left(C,A\right)\cap VI\left(C,B\right)\ne \varnothing$. Let {x _n } be a sequence generated in the following manner:

$$\left\{\begin{array}{c}{x}_0\in C,\hfill \\ y_n={\gamma }_n{P}_C\left(x_n-r_{n}A{x}_n\right)+\left(1-{\gamma }_n\right)P_C\left(x_n-s_{n}B{x}_n\right),\hfill \\ x_{n+1}={\alpha }_n{x}_n+\left(1-{\alpha }_n\right)\left({\beta }_n{y}_n+\left(1-{\beta }_n\right)S{y}_n\right),n\ge 0,\hfill \end{array}\right.$$

where {r _n } is a sequence in (0, 2α), {s _n } is a sequence in (0, 2β) and {α _n }, {β _n } and {γ _n } are sequences in (0, 1). Assume that the following restrictions are satisfied

• (a) 0 < a ≤ r _n ≤ b < 2α and 0 < c ≤ s _n ≤ d < 2β;

• (b) 0 ≤ κ ≤ β _n < e < 1, 0 < h ≤ α _n ≤ i < 1 and 0 < j ≤ γ _n ≤ k < 1,

where a, b, c, d, e, h, i, j, k are real numbers. Then the sequence {x _n } converges weakly to $\bar{x}\in \mathcal{F}$, where $\bar{x}={\mathsf{\text{lim}}}_{n\to \infty }{P}_{\mathcal{F}}{x}_n$.

Proof. Put M = W = ∂I _C . Next, we show that V I(C, A) = (A + ∂I _C )^{- 1}(0) and VI(C, B) = (B + ∂I _C )^{- 1}(0), respectively. Notice that

$$\begin{array}{cc}\hfill x\in {\left(A+\partial I_C\right)}^{-1}\left(0\right)& \iff 0\in Ax+\partial I_{C}x\hfill \\ \iff -Ax\in \partial I_{C}x\hfill \\ \iff ⟨Ax,y-x⟩\ge 0\hfill \\ \iff x\in VI\left(C,A\right)\hfill \end{array}$$

In the same way, we can obtain that x ∈ (B + ∂I _C )^{- 1}⇔ (0) x ∈ V I(C, B). From Lemma 3.2, we can conclude the desired conclusion immediately. □

Remark 3.1. Let S be a nonexpansive mapping, A = B, M = W and β _n = 0 in Theorem 3.3. Then Theorem 3.3 is reduced to Theorem 1.1 in Section 1.

Third, we consider the problem of finding common fixed points of three strict pseudocontractions.

Theorem 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H. Let S : C → C be a κ-strict pseudocontraction, T : C → C be an α-strict pseudocontraction and R : C → C be a β-strict pseudocontraction. Assume that $\mathcal{F}:=F\left(R\right)\cap F\left(S\right)\cap F\left(T\right)\ne \varnothing$. Let {x _n } be a sequence generated in the following manner:

$$\left\{\begin{array}{c}{x}_0\in C,\hfill \\ y_n={\gamma }_n\left(\left(1-r_n\right)x_n+r_{n}T{x}_n\right)+\left(1-{\gamma }_n\right)\left(\left(1-s_n\right)x_n+s_{n}R{x}_n\right),\hfill \\ x_{n+1}={\alpha }_n{x}_n+\left(1-{\alpha }_n\right)\left({\beta }_n{y}_n+\left(1-{\beta }_n\right)S{y}_n\right),n\ge 0,\hfill \end{array}\right.$$

where {r _n } is a sequence in (0, 1 - α), {s _n } is a sequence in (0, 1 - β) and {α _n }, {β _n } and {γ _n } are sequences in (0, 1). Assume that the following restrictions are satisfied

• (a) 0 < a ≤ r _n ≤ b < 1 - α and 0 < c ≤ s _n ≤ d < 1 - β;

• (b) 0 ≤ κ ≤ β _n < e < 1, 0 < h ≤ α _n ≤ i < 1 and 0 < j ≤ γ _n ≤ k < 1,

where a, b, c, d, e, h, i, j, k are real numbers. Then the sequence {x _n } converges weakly to $\bar{x}\in \mathcal{F}$, where $\bar{x}={\mathsf{\text{lim}}}_{n\to \infty }{P}_{\mathcal{F}}{x}_n$.

Proof. Putting A = I - T, we see that A is $\frac{1-\alpha }{2}$-inverse-strongly monotone. We also have F (T) = V I(C, A) and P _C (x _n - r _n Ax _n ) = (1 - r _n )x _n + r _n Tx _n . Putting B = I - R, we see that B is $\frac{1-\beta }{2}$-inverse-strongly monotone. We also have F(R) = V I(C, B) and P _C (x _n - s _n Bx _n ) = (1 - s _n )x _n + s _n Ru _n . In view of Theorem 3.2, we can obtain the desired result immediately. □

The following lemma can be found in [31, 32].

Lemma 3.5. Let C be a nonempty closed convex subset of a real Hilbert space H and let F be a bifunction from C × C to ℝ which satisfies (A 1)-(A 4). Then, for any r > 0 and x∈ H, there exists z∈ C such that

$$F\left(z,y\right)+\frac{1}{r}⟨y-z,z-x⟩\ge 0,\forall y\in C.$$

Further, define

$$T_{r}x=\left\{z\in C:F\left(z,y\right)+\frac{1}{r}⟨y-z,z-x⟩\ge 0,\forall y\in C\right\}$$

(3.2)

for all r > 0 and x ∈ H. Then, the following hold:

• (a) T _r is single-valued;

• (b) T _r is firmly nonexpansive, i.e., for any x, y ∈ H.,

  $${∥T_{r}x-T_{r}y∥}^2\le ⟨T_{r}x-T_{r}y,x-y⟩;$$

• (c) F (T _r ) = EP (F);

• (d) EP (F) is closed and convex.

Lemma 3.6. [12] Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C × C to ℝ which satisfies (A 1)-(A 4) and A _F be a multivalued mapping from H into itself defined by

$$A_{F}x=\left\{\begin{array}{cc}\left\{z\in H:F\left(x,y\right)\ge ⟨y-x,z⟩,\forall y\in C\right\},\hfill & x\in C,\hfill \\ \varnothing ,\hfill & x\notin C.\hfill \end{array}\right.$$

(3.3)

Then A _F is a maximal monotone operator with the domain $D\left(A_F\right)\subset C,EP\left(F\right)=A_F^{-1}\left(0\right)$ and

$$T_{r}x={\left(I+r{A}_F\right)}^{-1}x,\forall x\in H,r>0,$$

where T _r is defined as in (3.2).

Finally, we consider the problem of finding common elements in solution set of equilibrium problems and in the fixed point set of strict pseudocontractions.

Theorem 3.7. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C × C to ℝ which satisfies (A 1)-(A 4), G be a bifunction from C × C to ℝ which satisfies (A 1)-(A 4) and S : C → C be a κ-strict pseudocontraction. Assume that $\mathcal{F}:=F\left(S\right)\cap EP\left(F\right)\cap EP\left(G\right)\ne \varnothing$. Let {r _n } and {s _n } be two positive sequences and {α _n }, {β _n }, and {γ _n } sequences in (0, 1). Let {x _n } be a sequence generated in the following manner:

$$\left\{\begin{array}{c}{x}_0\in C,\hfill \\ y_n={\gamma }_n{u}_n+\left(1-{\gamma }_n\right)v_n,\hfill \\ x_{n+1}={\alpha }_n{x}_n+\left(1-{\alpha }_n\right)\left({\beta }_n{y}_n+\left(1-{\beta }_n\right)S{y}_n\right),\hfill & \hfill n\ge 0,\hfill \end{array}\right.$$

where u _n is such that

$$F\left(u_n,u\right)+\frac{1}{r_n}⟨u-u_n,u_n-x_n⟩\ge 0,\forall u\in C$$

and v _n is such that

$$G\left(v_n,v\right)+\frac{1}{s_n}⟨v-v_n,v_n-x_n⟩\ge 0,\forall v\in C.$$

Assume that the following restrictions are satisfied

• (a) 0 < a ≤ r _n ≤ b < ∞ and 0 < c ≤ s _n ≤ d < ∞;

• (b) 0 ≤ κ ≤ β _n < e < 1, 0 < h ≤ α _n ≤ i < 1 and 0 < j ≤ γ _n ≤ k < 1,

where a, b, c, d, e, h, i, j, k are real numbers. Then the sequence {x _n } converges weakly to $\bar{x}\in \mathcal{F}$, where $\bar{x}={\mathsf{\text{lim}}}_{n\to \infty }{P}_{\mathcal{F}}{x}_n$.

Proof. Putting A = B = 0, we can conclude from Lemma 3.6 the desired conclusion immediately. □

Remark 3.2. Let S be a nonexpansive mapping, F = G and β _n = 0 in Theorem 3.7. Then Theorem 3.7 is reduced to Theorem 1.2 in Section 1.