An iterative scheme for split variational inclusion and fixed point problem for demicontractive mappings

Abstract

In this paper, an iterative algorithm is introduced to find a common solution for the split variational inclusion problem and the fixed-point problem of a countable family of demicontractive mappings in Hilbert spaces. As a result, strong convergence theorem concerning the common element along with its applications and numerical examples is obtained.

Keywords

Split variational inclusion / demicontractive mapping / fixed point

1 Introduction

Suppose H₁, H₂ are two real Hilbert spaces. Let A:H_1\to H_2 be a bounded linear operator, and B_1:H_1\to 2^{H_1}and B_2:H_{21}\to 2^{H_2} be two multi-valued maximal monotone mappings. f_1:H_1\to H_1 and f_2:H_2\to H_2 are two given operators. The split Monotone variational inclusion problem (SMVIP) is to find x\in H_1,y=Ax\in H_2, such that

(1.1)0\in f_1\left(x\right)+B_1\left(x\right),0\in f_2\left(y\right)+B_2\left(y\right).

Moudafi [8] studied an iterative method to solve the (SMVIP) (1.1), covering the split zero-point problem, the split variational inequality problem, and the split feasibility problem, etc. These problems have been intensively investigated and used as models for practical applications in image restoration, computerized tomography, and radiotherapy.

If f_1\equiv 0 and f_2\equiv 0, then the problem (SMVIP) (1.1) is simplified to the split variational inclusion problem (SVIP): find x\in H_1,y=Ax\in H_2, such that

(1.2)0\in B_1\left(x\right),0\in B_2\left(y\right).

(SVIP) (1.2) consists of a pair of variational inclusion problems and is related to the bounded linear operator A. The solution set of (SVIP) (1.2) is denoted as \mathrm{\Gamma }=\{x\in H_1:0\in B_1(x),0\in B_2(Ax\left)\right\}.

Many scholars have studied (SVIP) (1.2). In 2012, Byrne et al. [1] investigated the weak and strong convergence of an iterative method for (SVIP) (1.2). Given x_1\in H_1, compute the sequence \left\{x_n\right\} generated by the following iterative procedure

x_{n+1}=J_{\lambda }^{B_1}\left(x_n+\gamma A^{\ast }\left(J_{\lambda }^{B_2}-I\right)A{x}_n\right),

where A* is the conjugate operator of A, L=\parallel A^{\ast }A\parallel ,\gamma \in (0,\frac{2}{L}), \lambda >0.

In 2014, Kazmi and Rizvi [4] studied the strong convergence of an iterative method for (SVIP) (1.2) and the fixed-point problem of the nonexpansive mapping S. The iterative procedure is

\{\begin{array}{l}{u}_n=J_{\lambda }^{B_1}\left(x_n+\gamma A^{\ast }\left(J_{\lambda }^{B_2}-I\right)A{x}_n\right),\\ x_{n+1}={\alpha }_{n}f\left(x_n\right)+(1-{\alpha }_n)S{u}_n,\lambda >0,\end{array}

where L=\parallel A^{\ast }A\parallel ,\gamma \in \left(0,\frac{1}{L}\right),\lambda >0.\left\{{\alpha }_n\right\} is a sequence in (0,1), satisfying the conditions , then the sequence \left\{x_n\right\} converges to the common solution of the two problems.

In 2017, Deepho et al. [3] studied the strong convergence of an iterative method for (SVIP) (1.2) and the fixed-point problem of a finite number of strictly pseudocontractive mappings. The iterative procedure is as follows:

\{\begin{array}{rl}& u_n=J_{\lambda }^{B_1}\left(x_n+\gamma A^{\ast }(J_{\lambda }^{B_2}-I)A{x}_n\right),\\ & y_n={\beta }_n{u}_n+(1-{\beta }_n)\sum _{i=1}^N {\eta }_i^{\left(n\right)}{S}_i{u}_n,\\ & x_{n+1}={\alpha }_n\tau f\left(x_n\right)+(I-{\alpha }_{n}B)y_n,\end{array}

where \lambda >0, \gamma \in \left(0,\frac{1}{L}\right), and A* is the conjugate operator of A, L=\parallel A^{\ast }A\parallel. B is a strongly positive bounded linear operator, and \{S_i{\}}_{i=1}^N is a k_i-strictly pseudocontractive mapping. α_n is the same as that in [4], . When the coefficients satisfy appropriate conditions, the sequence \left\{x_n\right\} converges strongly to q=P_{\mathrm{\Omega }}(I+\tau f-B)q,\mathrm{\Omega }=\bigcap _{i=1}^N F\left(S_i\right)\cap \mathrm{\Gamma }.

If the fixed-point set of a strictly pseudocontractive mapping is non-empty, then the mapping is demicontractive. Consider whether it is possible to relate (SVIP) (1.2) to the fixed-point problem of a demicontractive mapping and find an iterative sequence to approximate the common solution of the two problems.

Based on the above-mentioned work, this paper studies an iterative procedure to approximate a common solution of (SVIP) (1.2) and the fixed-point problem of a countable family of demicontractive mappings in Hilbert spaces. The sequence generated by the iterative method converges strongly to this common solution.

2 Preliminaries

Let H be a real Hilbert space. The sequences x_n\to x and x_n\rightharpoonup x represent that \left\{x_n\right\} converge strongly or weakly to x, respectively.

Call the mapping T: H\to H

(i) α-contractive, if \parallel Tx-Ty\parallel \le \alpha \parallel x-y\parallel ,\alpha \in [0,1),\mathrm{\forall }x,y\in H.

(ii) nonexpansive, if \parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\mathrm{\forall }x,y\in H.

(iii) firmly nonexpansive, if ⟨Tx-Ty,x-y⟩\ge \parallel Tx-Ty{\parallel }^2,\mathrm{\forall }x,y\in H.

(iv) quasi-nonexpansive, if F\left(T\right)\ne \varnothing and \parallel Tx-p\parallel \le \parallel x-p\parallel ,\mathrm{\forall }x\in H,p\in F\left(T\right), where F\left(T\right)=\{x\in H:Tx=x\} is the fixed point set of T.

(v) k-strictly pseudocontractive, if there exists the constant k\in [0,1), such that

\parallel Tx-Ty{\parallel }^2\le \parallel x-y{\parallel }^2+k\parallel x-y-\left(Tx-Ty\right){\parallel }^2,\mathrm{\forall }x,y\in H.

(vi) k-demicontractive, if F\left(T\right)\ne \varnothing and there exists the constant k\in [0,1), such that

\parallel Tx-p{\parallel }^2\le \parallel x-p{\parallel }^2+k\parallel Tx-x{\parallel }^2,\mathrm{\forall }x\in H,p\in F\left(T\right).

Strictly pseudocontractive mappings include nonexpansive mappings and firmly nonexpansive mappings. Demicontractive mappings include quasi-nonexpansive mappings, and the fixed-point set of a demicontractive mapping is a convex set.

A multi-valued mapping M:H\to 2^H is called monotone if \mathrm{\forall }x,y\in H,f\in Mx and g\in My, there is ⟨x-y,f-g⟩\ge 0. A monotone mapping M:H\to 2^H is called maximal if the graph G(M) of M cannot be properly contained in the graph of any other monotone mapping. A monotone mapping is maximal if and only if \mathrm{\forall }\left(x,u\right)\in H\times H, and when \left(y,v\right)\in G\left(M\right) and ⟨x-y,u-v⟩\ge 0, there is u\in Mx. The resolvent J_{\lambda }^M:H\to H is defined by the following formula:

J_{\lambda }^M\left(u\right)=(I+\lambda M{)}^{-1}(u),u\in H,\lambda >0,

where I is the identity mapping. The resolvent is single-valued, nonexpansive, and firmly nonexpansive.

Let C be a non-empty closed convex subset of H. \mathrm{\forall }x\in H, and there exists a nearest point in C denoted as P_{C}x, such that \parallel x-P_{C}x\parallel \le \parallel x-y\parallel ,\mathrm{\forall }y\in C. Then P_C is the metric projection from H onto C. y=P_{C}x is equivalent to

⟨x-y,z-y⟩\le 0,\mathrm{\forall }z\in C.

If there exists a constant r > 0 such that ⟨Bx,x⟩\ge r\parallel x{\parallel }^2,\mathrm{\forall }x\in H, then B is a strongly positive and bounded linear operator on H.

Lemma 2.1 [6]　 Let B be a strongly positive and bounded linear operator on H with coefficient r. If , then \parallel I-\rho B\parallel \le 1-\rho r.

Lemma 2.2 [4]　(SVIP) (1.2) is equivalent to find x\in H_1,y=Ax\in H_2 such that for some λ > 0, there are x=J_{\lambda }^{B_1}x and y=J_{\lambda }^{B_2}y.

Lemma 2.3 [11]　 Let H be a real Hilbert space. Then the following results hold:

(i) \parallel x+y{\parallel }^2\le \parallel x{\parallel }^2+2⟨y,x+y⟩,\mathrm{\forall }x,y\in H;

(ii) 2⟨x,y⟩=\parallel x{\parallel }^2+\parallel y{\parallel }^2-\parallel x-y{\parallel }^2,\mathrm{\forall }x,y\in H;

(iii) for any x,y,z\in H,\alpha ,\beta ,\gamma \in \left[0,1\right] and \alpha +\beta +\gamma =1, there is

\begin{array}{rl}\parallel \alpha x+\beta y+\gamma z{\parallel }^2=& \alpha \parallel x{\parallel }^2+\beta \parallel y{\parallel }^2+\gamma \parallel z{\parallel }^2\\ & -\alpha \beta \parallel x-y{\parallel }^2-\alpha \gamma \parallel x-z{\parallel }^2-\beta \gamma \parallel y-z{\parallel }^2.\end{array}

Lemma 2.4 [10]　 Let the non-negative real-valued sequence \left\{s_n\right\} satisfy s_{n+1}\le \left(1-{\lambda }_n\right)s_n+{\lambda }_n{\beta }_n,\mathrm{\forall }n\ge 0. If the following conditions are established. Then

(i) \left\{{\lambda }_n\right\}\subset \left(0,1\right),\sum _{n=1}^{\mathrm{\infty }}{\lambda }_n=\mathrm{\infty };

(ii) \mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}{\mathrm{p}}_{n\to \mathrm{\infty }}{\beta }_n\le 0 or .

Then {\mathrm{l}\mathrm{i}\mathrm{m}}_{n\to \mathrm{\infty }}{s}_n=0.

Lemma 2.5 [5]　 The sequence \left\{w_n\right\} has a subsequence \left\{w_{n_i}\right\} such that . When n is large enough, define the integer-valued sequence \left\{{\tau }_n\right\} as follows:

When n\to \mathrm{\infty },{\tau }_n\to \mathrm{\infty }, and \mathrm{m}\mathrm{a}\mathrm{x}\{w_{{\tau }_n},w_n\}\le w_{{\tau }_n+1}.

Let S:C\to C be a mapping. I-\mathrm{S} is said to be demi-closed at zero if x\in F\left(S\right) can be deduced from x_n\rightharpoonup x and \underset{n\to \mathrm{\infty }}{{\mathrm{l}\mathrm{i}\mathrm{m}}_{n\to \mathrm{\infty }}} \parallel x_n-S{x}_n\parallel =0 in C.

Lemma 2.6 [7]　 Let C be a non-empty closed convex subset of the real Hilbert space H, and S:C\to C be a k-strictly pseudocontractive mapping. Then I-S is demi-closed at zero.

3 Main conclusions

Theorem 3.1　Let H₁ and H₂ be two real Hilbert spaces. A:H_1\to H_2 is a bounded linear operator, and B_1:H_1\to 2^{H_1} and B_2:H_2\to 2^{H_2} are two maximal monotone operators. S_n:H_1\to H_1,n\in \mathbb{N} is a k-demicontractive mapping and I-S_n is demi-closed at zero. Assume \mathrm{\Omega }=\bigcap _{n=1}^{\mathrm{\infty }} F\left(S_n\right)\cap \mathrm{\Gamma }\ne \varnothing. Let f:H_1\to H_1 be a contractive mapping with contraction coefficient b\in \left(0,1\right) and B be a strongly positive bounded linear operator on H₁ with coefficient r, such that . Take x_1\in H_1, and the sequence \left\{x_n\right\} is generated by the following formula:

(3.1)\{\begin{array}{rl}& u_n=J_{\lambda }^{B_1}\left(x_n+\gamma A^{\ast }\left(J_{\lambda }^{B_2}-I\right)A{x}_n\right),\\ & y_n={\alpha }_n{u}_n+{\beta }_n{x}_n+\sum _{i=1}^n {\gamma }_{n,i}{\tau }_n{S}_i{u}_n,\\ & x_{n+1}={\delta }_n\xi f\left(y_n\right)+(I-{\delta }_{n}B)y_n,\end{array}

where \lambda >0, \gamma \in \left(0,\frac{1}{L}\right), L is the spectral radius of A*A, and A* is the conjugate operator of A. The sequences \left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\tau }_n\right\},\left\{{\delta }_n\right\}, and \{{\gamma }_n,i\} in [0,1] satisfy the following conditions:

(C1) {\alpha }_n+{\beta }_n+{\tau }_n=\sum _{i=1}^n {\gamma }_{n,i}=1,\mathrm{\forall }n\in \mathbb{N};

(C2) \mathrm{l}\mathrm{i}{\mathrm{m}}_{n\to \mathrm{\infty }}{\delta }_n=0,\sum _{n=1}^{\mathrm{\infty }}{\delta }_n=\mathrm{\infty };

(C3) \mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}{\mathrm{f}}_{n\to \mathrm{\infty }}{\beta }_n{\alpha }_n>0,\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}{\mathrm{f}}_{n\to \mathrm{\infty }}\left({\alpha }_n-k\right)>0,{\tau }_n>a>0;

(C4) \mathrm{l}\mathrm{i}\mathrm{m}{\mathrm{i}\mathrm{n}\mathrm{f}}_{n\to \mathrm{\infty }}{\gamma }_{n,i}>0,\mathrm{\forall }i\in \mathbb{N}.

Then the sequence \left\{x_n\right\} generated by formula (2.1) converges strongly to q=P_{\mathrm{\Omega }}\left(I+\xi f-B\right)q.

Proof　 \mathrm{\forall }p\in \mathrm{\Omega }, then p=J_{\lambda }^{B_1}p,Ap=J_{\lambda }^{B_2}Ap, and S_{n}p=p. Through calculation, it is known that

(3.2)\begin{array}{rl}{\Vert u_n-p\Vert }^2& ={\Vert J_{\lambda }^{B_1}\left(x_n+\gamma A^{\ast }\left(J_{\lambda }^{B_2}-I\right)A{x}_n\right)-J_{\lambda }^{B_1}p\Vert }^2\\ & \le {\Vert x_n+\gamma A^{\ast }\left(J_{\lambda }^{B_2}-I\right)A{x}_n-p\Vert }^2\\ & ={\Vert x_n-p\Vert }^2+{\gamma }^2{\Vert A^{\ast }\left(J_{\lambda }^{B_2}-I\right)A{x}_n\Vert }^2+2\gamma ⟨x_n-p,A^{\ast }\left(J_{\lambda }^{B_2}-I\right)A{x}_n⟩.\end{array}

It is noted that

(3.3)\begin{array}{rl}{\gamma }^2\parallel A^{\ast }\left(J_{\lambda }^{B_2}-I\right)A{x}_n{\parallel }^2& ={\gamma }^2⟨A^{\ast }\left(J_{\lambda }^{B_2}-I\right)A{x}_n,A^{\ast }\left(J_{\lambda }^{B_2}-I\right)A{x}_n⟩\\ & ={\gamma }^2⟨\left(J_{\lambda }^{B_2}-I\right)A{x}_n,A{A}^{\ast }\left(J_{\lambda }^{B_2}-I\right)A{x}_n⟩\\ & \le L{\gamma }^2⟨\left(J_{\lambda }^{B_2}-I\right)A{x}_n,\left(J_{\lambda }^{B_2}-I\right)A{x}_n⟩\\ & =L{\gamma }^2\parallel \left(J_{\lambda }^{B_2}-I\right)A{x}_n{\parallel }^2,\end{array}

and

(3.4)\begin{array}{rl}2\gamma & ⟨x_n-p,A^{\ast }(J_{\lambda }^{B_2}-I)A{x}_n⟩\\ & =2\gamma ⟨A{x}_n-Ap+(J_{\lambda }^{B_2}-I)A{x}_n-(J_{\lambda }^{B_2}-I)A{x}_n,(J_{\lambda }^{B_2}-I)A{x}_n⟩\\ & =2\gamma ⟨J_{\lambda }^{B_2}A{x}_n-Ap,(J_{\lambda }^{B_2}-I)A{x}_n⟩-2\gamma \parallel (J_{\lambda }^{B_2}-I)A{x}_n{\parallel }^2\\ & =\gamma \left[\parallel J_{\lambda }^{B_2}A{x}_n-Ap{\parallel }^2+\parallel \left(J_{\lambda }^{B_2}-I\right)A{x}_n{\parallel }^2-\parallel A{x}_n-Ap{\parallel }^2\right]-2\gamma \parallel (J_{\lambda }^{B_2}-I)A{x}_n{\parallel }^2\\ & \le \gamma \parallel (J_{\lambda }^{B_2}-I)A{x}_n{\parallel }^2-2\gamma \parallel (J_{\lambda }^{B_2}-I)A{x}_n{\parallel }^2\\ & =-\gamma \parallel (J_{\lambda }^{B_2}-I)A{x}_n{\parallel }^2.\end{array}

Substitute Eqs. (3.3) and (3.4) into Eq. (3.2), and there is

(3.5)\begin{array}{rl}\parallel u_n-p{\parallel }^2& \le \parallel x_n-p{\parallel }^2-\gamma \left(1-L\gamma \right)\parallel \left(J_{\lambda }^{B_2}-I\right)A{x}_n{\parallel }^2\\ & \le \parallel x_n-p{\parallel }^2.\end{array}

Since S_i is a k-demicontractive mapping, according to Lemma 2.3, there is

(3.6)\begin{array}{rl}\parallel y_n-p{\parallel }^2& ={\Vert {\alpha }_n{u}_n+{\beta }_n{x}_n+\sum _{i=1}^n{\gamma }_{n,i}{\tau }_n{S}_i{u}_n-p\Vert }^2\\ & \le {\alpha }_n\parallel u_n-p{\parallel }^2+{\beta }_n\parallel x_n-p{\parallel }^2+\sum _{i=1}^n{\gamma }_{n,i}{\tau }_n\parallel S_i{u}_n-p{\parallel }^2\\ & -{\alpha }_n{\beta }_n\parallel x_n-u_n{\parallel }^2-\sum _{i=1}^n{\alpha }_n{\tau }_n{\gamma }_{n,i}\parallel S_i{u}_n-u_n{\parallel }^2\\ & \le {\alpha }_n\parallel u_n-p{\parallel }^2+{\beta }_n\parallel x_n-p{\parallel }^2+\sum _{i=1}^n{\gamma }_{n,i}{\tau }_n\parallel \left|u_n-p{\parallel }^2+k\parallel S_i{u}_n-u_n{\parallel }^2\right)\\ & -{\alpha }_n{\beta }_n\parallel x_n-u_n{\parallel }^2-\sum _{i=1}^n{\alpha }_n{\gamma }_{n,i}{\tau }_n\parallel S_i{u}_n-u_n{\parallel }^2\\ & =\parallel x_n-p{\parallel }^2-{\alpha }_n{\beta }_n\parallel x_n-u_n{\parallel }^2-\sum _{i=1}^n{\gamma }_{n,i}{\tau }_n\left({\alpha }_n-k\right)\parallel S_i{u}_n-u_n{\parallel }^2\\ & \le \parallel x_n-p{\parallel }^2.\end{array}

Consequently, \parallel y_n-p\parallel \le \parallel x_n-p\parallel. According to Lemma 2.1, it is known that

\begin{array}{rl}\parallel x_{n+1}-p\parallel & \le {\delta }_n\parallel \xi f\left(y_n\right)-Bp\parallel +\parallel I-{\delta }_{n}B\parallel \parallel y_n-p\parallel \\ & \le {\delta }_n\xi \parallel f\left(y_n\right)-fp\parallel +{\delta }_n\parallel \xi fp-Bp\parallel +\left(1-r{\delta }_n\right)\parallel y_n-p\parallel \\ & \le {\delta }_n\xi b\parallel y_n-p\parallel +{\delta }_n\parallel \xi fp-Bp\parallel +\left(1-r{\delta }_n\right)\parallel y_n-p\parallel \\ & \le \left(1-{\delta }_n\left(r-\xi b\right)\right)\parallel y_n-p\parallel +{\delta }_n\parallel \xi fp-Bp\parallel \\ & \le \left(1-{\delta }_n\left(r-\xi b\right)\right)\parallel x_n-p\parallel +{\delta }_n\parallel \xi fp-Bp\parallel .\end{array}

By induction, it can be known that

\parallel x_n-p\parallel \le \mathrm{m}\mathrm{a}\mathrm{x}\left\{\parallel x_1-p\parallel ,\frac{1}{r-\xi b}\parallel \xi fp-Bp\parallel \right\}, \mathrm{\forall }n\in \mathbb{N}.

Therefore, \left\{x_n\right\} is bounded, and thus \left\{y_n\right\},\left\{u_n\right\},\left\{B{y}_n\right\},\left\{f\left(y_n\right)\right\} are also bounded.

To prove x_n\to q when n\to \mathrm{\infty }, it will be discussed in two cases.

Case 1: Suppose \{\parallel x_n-q\parallel \} is a monotonic sequence. Since \parallel x_n-q\parallel is bounded, it can be deduced that \parallel x_n-q\parallel converges. According to Eq. (3.6), there is

(3.7)\begin{array}{rl}\parallel x_{n+1}-p{\parallel }^2& \le {\delta }_n^2\parallel \xi f\left(y_n\right)-Bp{\parallel }^2+(1-r{\delta }_n{)}^2\parallel y_n-p{\parallel }^2\\ & +2{\delta }_n(1-r{\delta }_n)\parallel \xi f(y_n)-Bp\parallel \parallel y_n-p\parallel \\ & \le {\delta }_{n}M+\parallel x_n-p{\parallel }^2-{\alpha }_n{\beta }_n\parallel x_n-u_n{\parallel }^2\\ & -\sum _{i=1}^n {\gamma }_{n,i}{\tau }_n({\alpha }_n-k)\parallel S_i{u}_n-u_n{\parallel }^2,\end{array}

where M_1={\mathrm{s}\mathrm{u}\mathrm{p}}_n\{\parallel \xi f\left(y_n\right)-Bp{\parallel }^2+2\parallel \xi f\left(y_n\right)-Bp\parallel \parallel y_n-p\parallel \}. For every i=1,2,\dots ,n, there is

(3.8)\begin{array}{rl}{\gamma }_{n,i}{\tau }_n\left({\alpha }_n-k\right)\parallel S_i{u}_n-u_n{\parallel }^2& \le \sum _{i=1}^n{\gamma }_{n,i}{\tau }_n\left({\alpha }_n-k\right)\parallel S_i{u}_n-u_n{\parallel }^2\\ & \le {\delta }_n{M}_1+\parallel x_n-p{\parallel }^2-\parallel x_{n+1}-p{\parallel }^2,\end{array}

(3.9){\alpha }_n{\beta }_n\parallel x_n-u_n{\parallel }^2\le {\delta }_n{M}_1+\parallel x_n-p{\parallel }^2-\parallel x_{n+1}-p{\parallel }^2.

From Eqs. (3.8)‒(3.9) and conditions (C2)‒(C4), it is known that

(3.10)\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}} \parallel S_i{u}_n-u_n\parallel =\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}} \sum _{i=1}^n {\gamma }_{n,i}\parallel S_i{u}_n-u_n{\parallel }^2=0,

(3.11)\begin{array}{c}\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}} \parallel x_n-u_n\parallel =0.\end{array}

Based on Eq. (3.5), there is

\gamma \left(1-L\gamma \right){\Vert \left(J_{\lambda }^{B_2}-I\right)A{x}_n\Vert }^2\le \parallel x_n-u_n\parallel \left(\parallel x_n-p\parallel +\parallel u_n-p\parallel \right).

Based on the boundedness of \left\{x_n\right\},\left\{u_n\right\} and Eq. (3.11), there is

(3.12)\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\Vert \left(J_{\lambda }^{B_2}-I\right)A{x}_n\Vert =0.

From Eq. (3.1), it is known that

\begin{array}{rl}\Vert y_n-u_n{\Vert }^2& ={\Vert {\beta }_n\left(x_n-u_n\right)+\sum _{i=1}^n {\gamma }_{n,i}{\tau }_n\left(S_i{u}_n-u_n\right)\Vert }^2\\ & \le 2\left(\Vert {\beta }_n\left(x_n-u_n\right){\Vert }^2+{\Vert \sum _{i=1}^n {\gamma }_{n,i}{\tau }_n\left(S_i{u}_n-u_n\right)\Vert }^2\right)\\ & \le 2\left(\Vert {\beta }_n\left(x_n-u_n\right){\Vert }^2+\sum _{i=1}^n {\gamma }_{n,i}\Vert S_i{u}_n-u_n{\Vert }^2\right).\end{array}

According to Eqs. (3.10) and (3.11), there is

(3.13)\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\parallel y_n-u_n\parallel =\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\parallel y_n-x_n\parallel =0.

Next illustrate \mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}{\mathrm{p}}_{n\to \mathrm{\infty }}⟨\xi f\left(q\right)-Bq,x_n-q⟩\le 0, where q=P_{\mathrm{\Omega }}(I+\xi f-B)q.

Select a subsequence \left\{x_{n_i}\right\} of \left\{x_n\right\} such that

\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{s}\mathrm{u}\mathrm{p}⟨\xi f\left(q\right)-Bq,x_n-q⟩=\underset{i\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}⟨\xi f\left(q\right)-Bq,x_{n_i}-q⟩.

Since \left\{x_{n_i}\right\} is bounded, without losing generality assume x_{n_i}\rightharpoonup z. From Eq. (3.11), it is known that u_{n_i}\rightharpoonup z. Due to the demi-closedness of I-S_i at zero and Eq. (3.10), there is z\in \bigcap _{i=1}^{\mathrm{\infty }} F\left(S_i\right).

From Eq. (2.1), u_{n_i}=J_{\lambda }^{B_1}\left(x_{n_i}+\gamma A^{\ast }\left(J_{\lambda }^{B_2}-I\right)A{x}_{n_i}\right), so

(3.14)\frac{\left(x_{n_i}-u_{n_i}\right)+\gamma A^{\ast }\left(J_{\lambda }^{B_2}-I\right)A{x}_{n_i}}{\lambda }\in B_1{u}_{n_i}.

Allow i\to \mathrm{\infty } to take the limit in Eq. (3.14), and from Eqs. (3.11) and (3.12) as well as the property that the graph of a maximal monotone mapping is weakly-strongly closed, it is known that 0\in B_{1}z. Since x_{n_i}\rightharpoonup z, A{x}_{n_i}\rightharpoonup Az. From Eq. (3.12), Lemma 2.6, and the nonexpansive property of the resolvent J_{\lambda }^{B_2}, Az=J_{\lambda }^{B_2}Az. From Lemma 2.2, it is known that z\in \mathrm{\Gamma }. Thus,

\mathrm{l}\mathrm{i}\mathrm{m}\underset{n\to \mathrm{\infty }}{\mathrm{s}\mathrm{u}\mathrm{p}} ⟨\xi fq-Bq,x_n-q⟩=⟨\xi fq-Bq,z-q⟩\le 0.

Then, illustrate x_n\to q. Since x_n+1-q={\delta }_n\left(\xi f\left(y_n\right)-Bq\right)+\left(I-{\delta }_{n}B\right)\left(y_n-q\right), through calculation there is

\begin{array}{rl}\parallel x_{n+1}-q{\parallel }^2& \le \parallel \left(I-{\delta }_{n}B\right)\left(y_n-q\right){\parallel }^2+2{\delta }_n⟨\xi f\left(y_n\right)-Bq,x_{n+1}-q⟩\\ & \le {\left(1-r{\delta }_n\right)}^2\parallel y_n-q{\parallel }^2+2\xi {\delta }_n⟨f\left(y_n\right)-fq,x_{n+1}-q⟩\\ & +2{\delta }_n⟨\xi fq-Bq,x_{n+1}-q⟩\\ & \le {\left(1-r{\delta }_n\right)}^2\parallel x_n-q{\parallel }^2+2{\delta }_n\xi b\parallel x_n-q\parallel \parallel x_{n+1}-q\parallel \\ & +2{\delta }_n⟨\xi fq-Bq,x_{n+1}-q⟩\\ & \le {\left(1-r{\delta }_n\right)}^2\parallel x_n-q{\parallel }^2+{\delta }_n\xi b\left(\parallel x_n-q{\parallel }^2+\parallel x_{n+1}-q{\parallel }^2\right)\\ & +2{\delta }_n⟨\xi fq-Bq,x_{n+1}-q⟩\\ & =\left({\left(1-r{\delta }_n\right)}^2+{\delta }_n\xi b\right)\parallel x_n-q{\parallel }^2+{\delta }_n\xi b\parallel x_{n+1}-q{\parallel }^2\\ & +2{\delta }_n⟨\xi fq-Bq,x_{n+1}-q⟩,\end{array}

which can be simplified into

(3.15)\begin{array}{rl}\parallel x_{n+1}-q{\parallel }^2& \le \frac{1-2r{\delta }_n+r^2{\delta }_n^2+{\delta }_n\xi b}{1-{\delta }_n\xi b}\parallel x_n-q{\parallel }^2+\frac{2{\delta }_n}{1-{\delta }_n\xi b}⟨\xi fq-Bq,x_{n+1}-q⟩\\ & =\left(1-\frac{2{\delta }_n(r-\xi b)}{1-{\delta }_n\xi b}\right)\parallel x_n-q{\parallel }^2+\frac{r^2{\delta }_n^2}{1-{\delta }_n\xi b}\parallel x_n-q{\parallel }^2\\ & +\frac{2{\delta }_n}{1-{\delta }_n\xi b}⟨\lambda fq-Bq,x_{n+1}-q⟩\\ & \le \left(1-\frac{2{\delta }_n(r-\xi b)}{1-{\delta }_n\xi b}\right)\parallel x_n-q{\parallel }^2\\ & +\frac{2{\delta }_n(r-\xi b)}{1-{\delta }_n\xi b}\left(\frac{{\delta }_n{r}^2{M}_2}{2(r-\xi b)}+\frac{1}{r-\xi b}⟨\xi fq-Bq,x_{n+1}-q⟩\right)\\ & =(1-{\sigma }_n)\parallel x_n-q{\parallel }^2+{\sigma }_n{\xi }_n,\end{array}

where M_2=\mathrm{s}\mathrm{u}\mathrm{p}\{\parallel x_n-q{\parallel }^2:n\ge 0\},{\sigma }_n=\frac{2{\delta }_n\left(r-\xi b\right)}{1-{\delta }_n\xi b},{\xi }_n=\frac{{\delta }_n{r}^{2}M}{2\left(r-\xi b\right)}+\frac{1}{r-\xi b}⟨\xi fq-Bq,x_{n+1}-q⟩.

It is known that {\sigma }_n\to 0,\sum _{n=1}^{\mathrm{\infty }}{\sigma }_n=\mathrm{\infty }, and \mathrm{l}\mathrm{i}\mathrm{m}{\mathrm{s}\mathrm{u}\mathrm{p}}_{n\to \mathrm{\infty }} {\xi }_n\le 0. According to Eq. (3.15) and Lemma 2.4, there is {\mathrm{l}\mathrm{i}\mathrm{m}}_{n\to \mathrm{\infty }}\parallel x_n-q\parallel =0.

Case 2: Assume \{\parallel x_n-q\parallel \} is not a monotonic sequence. When n is large enough, there exists a subsequence \{\parallel x_{n_i}-q\parallel \} of \{\parallel x_n-q\parallel \} such that \parallel x_{n_i}-q\parallel \le \parallel x_{n_i+1}-q\parallel. Define the integer-valued sequence

and \left\{{\mu }_n\right\} is a non-decreasing sequence. When n\to \mathrm{\infty }, . According to Eq. (2.7), there are {\mathrm{l}\mathrm{i}\mathrm{m}}_{n\to \mathrm{\infty }}\parallel u_{{\mu }_n}-S_i{u}_{{\mu }_n}\parallel =0,{\mathrm{l}\mathrm{i}\mathrm{m}}_{n\to \mathrm{\infty }}\parallel \left(J_{\lambda }^{B_2}-I\right)A{x}_{{\mu }_n}\parallel =0, and {\mathrm{l}\mathrm{i}\mathrm{m}}_{n\to \mathrm{\infty }}\parallel u_{{\mu }_n}-x_{{\mu }_n}\parallel =0. Similar to the discussions in Case 1, there is

\parallel x_{{\mu }_n+1}-q{\parallel }^2\le \left(1-{\sigma }_{{\mu }_n}\right)\parallel x_{{\mu }_n}-q{\parallel }^2+{\sigma }_{{\mu }_n}{\xi }_{{\mu }_n}.

Since {\sigma }_{{\mu }_n}\to 0, \sum _{n=1}^{\mathrm{\infty }}{\sigma }_{{\mu }_n}=\mathrm{\infty }, and \mathrm{l}\mathrm{i}\mathrm{m}{\mathrm{s}\mathrm{u}\mathrm{p}}_{n\to \mathrm{\infty }}{\xi }_{{\mu }_n}\le 0, there are {\mathrm{l}\mathrm{i}\mathrm{m}}_{n\to \mathrm{\infty }}\parallel x_{{\mu }_n}-q\parallel =0,{\mathrm{l}\mathrm{i}\mathrm{m}}_{n\to \mathrm{\infty }}\parallel x_{{\mu }_n+1}-q\parallel =0 by applying Lemma 2.4. Apply Lemma 2.5, and there is

0\le \parallel x_n-q\parallel \le \mathrm{m}\mathrm{a}\mathrm{x}\left\{\parallel x_{{\mu }_n}-q\parallel ,\parallel x_n-q\parallel \right\}\le \parallel x_{{\mu }_n+1}-q\parallel \to 0,n\to \mathrm{\infty }. □

Note 3.1　(1) If S_n is a quasi-nonexpansive mapping, then k = 0.

(2) If S_n is a k-strictly pseudocontractive mapping, the condition that I-S_n is demi-closed at zero can be removed by applying Lemma 2.6.

4 Applications

4.1 Split optimization problem

Let H₁ and H₂ be real Hilbert spaces, and A:H_1\to H_2 be a bounded linear operator. h:H_1\to \mathbb{R} and g:H_2\to \mathbb{R} are two proper convex lower semi-continuous functions. The split optimization problem is to find x^{\ast }\in H_1,A{x}^{\ast }\in H_2 such that

h\left(x^{\ast }\right)=\underset{x\in H_1}{\mathrm{m}\mathrm{i}\mathrm{n}}h\left(x\right), g\left(A{x}^{\ast }\right)=\underset{z\in H_2}{\mathrm{m}\mathrm{i}\mathrm{n}}g\left(z\right).

Let B_1=\mathrm{\partial }h,B_2=\mathrm{\partial }g be the sub-differentials of h and g respectively, as maximal monotone operator. The split optimization problem is equivalent to the following (SVIP) (1.2): find x^{\ast }\in H_1,A{x}^{\ast }\in H_2 such that

0\in \mathrm{\partial }h\left(x^{\ast }\right),0\in \mathrm{\partial }g\left(A{x}^{\ast }\right).

The following conclusion can be obtained from Theorem 3.1.

Theorem 4.1　 Let H₁, H₂, A, B₁, B₂ be the same as the above. Let S_n:H_1\to H_1,n\in \mathbb{N} be a k-demicontractive mapping and I - S_n be demi-closed at zero. Assume \mathrm{\Omega }=\bigcap _{n=1}^{\mathrm{\infty }} F\left(S_n\right)\cap \mathrm{\Gamma }\ne \varnothing. Let f:H_1\to H_1 be a contraction mapping with coefficient b\in \left(0,1\right), and B be a strongly positive bounded linear operator on H₁ with coefficient r such that . Take x_1\in H_1, and the sequence \left\{x_n\right\} is generated by the following formula:

\{\begin{array}{rl}& u_n=J_{\lambda }^{\mathrm{\partial }h}\left(x_n+\gamma A^{\ast }\left(J_{\lambda }^{\mathrm{\partial }g}-I\right)A{x}_n\right),\\ & y_n={\alpha }_n{u}_n+{\beta }_n{x}_n+\sum _{i=1}^n {\gamma }_{n,i}{\tau }_n{S}_i{u}_n,\\ & x_{n+1}={\delta }_n\xi f\left(y_n\right)+(I-{\delta }_{n}B)y_n,\end{array}

where \lambda >0, \gamma \in \left(0,\frac{1}{L}\right), L is the spectral radius of A*A, A* is the conjugate operator of A, and \left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\tau }_n\right\},\left\{{\delta }_n\right\},\left\{{\gamma }_{n,i}\right\} are the same as in Theorem 3.1. Then \left\{x_n\right\} converges strongly to {q=P}_{\mathrm{\Omega }}\left(I+\xi f-B\right)q.

4.2 Split equilibrium problem

Let H₁ and H₂ be real Hilbert spaces and C and Q be non-empty closed convex subsets of H₁ and H₂ respectively. A:H_1\to H_2 is a bounded linear operator. h:C\times C\to \mathbb{R} and g:Q\times Q\to \mathbb{R} are two functionals. The split equilibrium problem is to find x^{\ast }\in C,A{x}^{\ast }\in Q such that h\left(x^{\ast },x\right)\ge 0,\mathrm{\forall }x\in C,g\left(A{x}^{\ast },y\right)\ge 0,\mathrm{\forall }y\in Q.

Let G:D\times D\to \mathbb{R} be a binary function. The equilibrium problem is to find x\in D such that G\left(x,y\right)\ge 0,\mathrm{\forall }y\in D. The solution set is denoted as EP(G). To solve the equilibrium problem, it is assumed that the following conditions are satisfied:

(A1) G\left(x,x\right)=0,\mathrm{\forall }x\in D;

(A2) G is monotone, that is \mathrm{\forall }x,y\in D,G\left(x,y\right)+G\left(y,x\right)\le 0;

(A3) \mathrm{\forall }x,y,z\in D,\mathrm{l}\mathrm{i}\mathrm{m}{\mathrm{s}\mathrm{u}\mathrm{p}}_{t\to 0}G\left(tz+\left(1-t\right)x,y\right)\le G\left(x,y\right);

(A4) \mathrm{\forall }x\in D,y\mapsto G\left(x,y\right) is convex and lower semi-continuous.

Lemma 4.1 [2]　 Let D be a non-empty closed convex subset of the real Hilbert space H, and G:D\times D\to \mathbb{R} satisfy conditions (A1)‒(A4). For r>0, x\in H, define the mapping T_r^G:H\to D as follows:

T_r^{G}x=\left\{z\in D:G\left(z,y\right)+\frac{1}{r}⟨y-z,z-x⟩\ge 0,\mathrm{\forall }y\in D\right\}.

Then the following conclusions hold:

(1) \mathrm{\forall }x\in H,T_r^{G}x\ne \varnothing ,T_r^G is a single-valued mapping;

(2) T_r^G is firmly nonexpansive;

(3) F\left(T_r^G\right)=\mathrm{E}\mathrm{P}\left(G\right);

(4) EP(G) is a closed convex set.

Lemma 4.2 [9]　 Let D be a non-empty closed convex subset of the real Hilbert space H, and G:D\times D\to \mathbb{R} satisfy conditions (A1)‒(A4). The multi-valued mapping is defined as follows:

W_{G}x=\{\begin{array}{ll}\{z\in H:G\left(x,y\right)\ge ⟨y-x,z⟩,\mathrm{\forall }y\in D\},& x\in D;\\ \varnothing ,& x\notin D.\end{array}

Then W_G is maximal monotone, and T_r^{G}x={\left(I+\lambda W_G\right)}^{-1}x,\mathrm{\forall }x\in H,\lambda >0.

Let the function h:C\times C\to \mathbb{R}, and g:Q\times Q\to \mathbb{R} satisfy conditions (A1)‒(A4). Let B_1=W_h, and B_2=W_g. By Lemma 4.2, the two operators are maximal monotone and satisfy the conditions of Theorem 3.1. The split equilibrium problem is equivalent to the following (SVIP) (1.2). Find x^{\ast }\in C,A{x}^{\ast }\in Q such that

0\in W_h\left(x^{\ast }\right), 0\in W_g\left(A{x}^{\ast }\right).

The following conclusion can be obtained from Theorem 3.1 and Lemma 4.1.

Theorem 4.2　 Let H₁, H₂, A, B₁, B₂ be the same as above. Let S_n:H_1\to H_1,n\in \mathbb{N} be a k-demicontractive mapping and I-S_n be demi-closed at zero. Assume \mathrm{\Omega }=\bigcap _{n=1}^{\mathrm{\infty }} F\left(S_n\right)\cap \mathrm{\Gamma }\ne \varnothing. Let f:H_1\to H_1 be a contractive mapping with coefficient b\in \left(0,1\right), and B be a strongly positive bounded linear operator on H₁ with coefficient r such that . Take x_1\in H_1, and the sequence \left\{x_n\right\} is generated by the following formula:

\{\begin{array}{rl}& u_n=T_{\lambda }^h\left(x_n+\gamma A^{\ast }\left(T_{\lambda }^g-I\right)A{x}_n\right),\\ & y_n={\alpha }_n{u}_n+{\beta }_n{x}_n+\sum _{i=1}^n {\gamma }_{n,i}{\tau }_n{S}_i{u}_n,\\ & x_{n+1}={\delta }_n\xi f\left(y_n\right)+(I-{\delta }_{n}B)y_n,\end{array}

where \lambda >0, \gamma \in \left(0,\frac{1}{L}\right), L is the spectral radius of A*A, and A* is the conjugate operator of A, and \left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\tau }_n\right\},\left\{{\delta }_n\right\},\left\{{\gamma }_{n,i}\right\} are the same as in Theorem 3.1. Then \left\{x_n\right\} converges strongly to q=P_{\mathrm{\Omega }}\left(I+\xi f-B\right)q.

5 Examples

Example 5.1　Let the function S\left(x\right)=\left(-a{x}_1,x_2\right),a>1,\parallel x\parallel =\sqrt{x_1^2+x_2^2},\mathrm{\forall }x=\left(x_1,x_2\right). By calculation, there is F\left(S\right)=0\times \mathbb{R}. Let p=\left(0,p_2\right)\in F\left(S\right), then \parallel Sx-x{\parallel }^2={\left(a+1\right)}^2{x}_1^2 and

\begin{array}{rl}\parallel Sx-p{\parallel }^2& =\parallel \left(-a{x}_1,x_2\right)-\left(0,p_2\right){\parallel }^2=x_1^2+{\left(x_2-p_2\right)}^2+\left(a^2-1\right)x_1^2\\ & =\parallel x-p{\parallel }^2+\frac{a-1}{a+1}\parallel Sx-x{\parallel }^2.\end{array}

So S is a \frac{a-1}{a+1}-demicontractive mapping.

Example 4.2　Let H_1=H_2=\left({\mathbb{R}}^2,\parallel \cdot {\parallel }_2\right). The mapping S_i\left(x_1,x_2\right)=\left(-\left(2-\frac{1}{i+1}\right)x_1,x_2\right), so from Example 4.1, it is shown that S_i is a \frac{1}{3}-demicontractive mapping. The mappings B_1\left(x\right)=\left(40x_1,24x_2\right),B_1\left(x\right)=\left(8x_1,2x_2\right), so it is known that B₁ and B₂ are maximal monotone. \mathrm{\forall }\lambda >0, there exist the resolvents J_{\lambda }^{B_1}=(I+\lambda B_1{)}^{-1} and J_{\lambda }^{B_2}=(I+\lambda B_2{)}^{-1} of B₁ and B₂. The bounded linear operator A\left(x\right)=\left(a{x}_1+b{x}_2,c{x}_1+d{x}_2\right),ad-bc\ne 0, and A* is the conjugate operator of A. Let L=\parallel A^{\ast }A{\parallel }_2,\gamma \in \left(0,\frac{1}{L}\right).The contractive mapping f\left(x\right)=\left(m{x}_1,n{x}_2\right),m,n\in \left(0,1\right). The real-valued sequence {\gamma }_{n,i} is defined as

{\mathrm{l}\mathrm{i}\mathrm{m}}_{n\to \mathrm{\infty }}{\gamma }_{n,i}=2^{-i},i\in \mathbb{N}. Select {\alpha }_n=\frac{1}{2},{\beta }_n=\frac{1}{4},{\tau }_n=\frac{1}{4},{\delta }_n=\frac{1}{n},\xi =1, and B = I, which satisfies the conditions of Theorem 2.1. It can be known that \left\{x_n\right\},\left\{u_n\right\}, and \left\{y_n\right\} converge strongly to (0, 0).

6 Conclusions

This paper generalizes and improves the results in Reference [3]:

(1) Generalize the fixed-point problem of a finite number of strictly pseudocontractive mappings to the fixed-point problem of a countable family of demicontractive mappings.

(2) Weaken the conditions of the coefficients in the iterative procedure and remove conditions such as .

(3) There is no need to discuss {\mathrm{l}\mathrm{i}\mathrm{m}}_{n\to \mathrm{\infty }} \parallel x_{n+1}-x_n\parallel =0, which simplifies the calculation process.

References

[1] Byrne C., Censor, Y., Gibali, A. , Reich, S. . The split common null point problem. J. Nonlinear Convex Anal. 2012; 13(4): 759–775

[2] Combettes P.L. , Hirstoaga, S.A. . Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005; 6(1): 117–136

[3] Deepho J., Thounthong, P., Kumam, P. , Phiangsungnoen, S. . A new general iterative scheme for split variational inclusion and fixed point problems of k-strict pseudocontraction mappings with convergence analysis. J. Comput. Appl. Math. 2017; 318: 293–306

[4] Kazmi K.R. , Rizvi, S.H. . An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping. Optim. Lett. 2014; 8(3): 1113–1124

[5] Maingé . Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 2008; 16(7/8): 899–912

[6] Marino G. , Xu, H.K. . A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 2006; 318(1): 43–52

[7] Marino G. , Xu, H.K. . Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 2007; 329(1): 336–346

[8] Moudafi A. . Split monotone variational inclusions. J. Optim. Theory Appl. 2011; 150(2): 275–283

[9] 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(1): 27–41

[10] Xu H.K. . An iterative approach to quadratic optimization. J. Optim. Theory Appl. 2003; 116(3): 659–678

[11] Zegeye H. , Shahzad, N. . Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive mappings. Comput. Math. Appl. 2011; 62(11): 4007–4014

RIGHTS & PERMISSIONS

Higher Education Press 2025