1. Introduction

Let $H_1$ and $H_2$ be two real Hilbert spaces. Let $S\mathrm{<mo>\; :\; </mo>}{H}_1\to H_1$ and $T\mathrm{<mo>\; :\; </mo>}{H}_2\to H_2$ be two nonlinear mappings. We denote the fixed point sets of S and T by $F\left(S\right)$ and $F\left(T\right)$, respectively. Let $A\mathrm{<mo>\; :\; </mo>}{H}_1\to H_2$ be a bounded linear operator with its adjoint $A^{\mathrm{<mo>\; *\; </mo>}}$. Then, we consider the following split common fixed point problem:

$\text{Finding}x\in H_1\text{such}\text{that}x\in F\left(S\right)\text{and}Ax\in F\left(T\right)\mathrm{.}$ (1.1)

The split common fixed point problem (1.1) is a generalization of the split feasibility problem arising from signal processing and image restoration; see [1] - [7] for instance. It was first introduced and studied by Censor and Segal [8]. Note that solving (1) can be translated to solve the fixed point equation

$x^{*}=S\left(x^{*}-\tau A^{*}\left(I-T\right)A{x}^{*}\right),\mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\tau >0.$

Censor and Segal also proposed the following algorithm for directed mappings.

Algorithm 1.1 Initialization: let $x^{*}\in H_1:={ℝ}_n$ be arbitrary. Iterative step: let

$x_{n+1}=S\left(x_n-\tau A^{*}\left(I-T\right)A{x}_n\right),\mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}n\ge 0,$

where $S\mathrm{<mo>\; :\; </mo>}{ℝ}_n\to {ℝ}_n$ and $T\mathrm{<mo>\; :\; </mo>}{R}_m\to {ℝ}_m$ are two directed mappings and $\tau \in \left(\mathrm{0,}\frac{2}{\lambda }\right)$ with $\lambda$ being the spectral radius of the operator $A^{\mathrm{<mo>\; *\; </mo>}}A$.

Since then, there has been growing interest in the split common fixed point problem; please, see [9] - [15].

Recently, Wang [16] introduced the following new iterative algorithms for the split common fixed point problem of directed mappings.

Algorithm 1.2 Choose an arbitrary initial guess $x_0$. Assume $x_n$ has been constructed. If

$\Vert x_n-S{x}_n+A^{*}\left(I-T\right)A{x}_n\Vert =0,$

then stop; otherwise, continue and construct $x_{n+1}$ via the formula:

$x_{n+1}=x_n-{\tau }_n\left[x_n-S{x}_n+A^{*}\left(I-T\right)A{x}_n\right],\mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\forall n\ge 0,$

where ${\tau }_n$ is chosen self-adaptively as

${\tau }_n=\frac{{\Vert x_n-S{x}_n\Vert }^2+{\Vert \left(I-T\right)A{x}_n\Vert }^2}{{\Vert x_n-S{x}_n+A^{*}\left(I-T\right)A{x}_n\Vert }^2}.$

Algorithm 1.3 Let $u\in H$ and start an initial guess $x_0\in H$. Assume $x_n$ has been constructed. If

$\Vert x_n-S{x}_n+A^{*}\left(I-T\right)A{x}_n\Vert =0,$

then stop; otherwise, continue and construct $x_{n+1}$ via the formula:

$x_{n+1}={\alpha }_{n}u+\left(1-{\alpha }_n\right)\left[x_n-S{x}_n+A^{*}\left(I-T\right)A{x}_n\right],\mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\forall n\ge 0,$

where the stepsize sequence ${\tau }_n$ is chosen self-adaptively as

${\tau }_n=\frac{{\Vert x_n-S{x}_n\Vert }^2+{\Vert \left(I-T\right)A{x}_n\Vert }^2}{{\Vert x_n-S{x}_n+A^{*}\left(I-T\right)A{x}_n\Vert }^2}.$

Wang obtained the weak and strong convergence of Algorithms 1.2 and 1.3, respectively. Inspired by the above work in the literature, Yao, et al. [17] extend Wang’s results in [16] from the directed mappings to the demicontractive mappings. Further, they construct the following two self-adaptive algorithms for solving the split common fixed point problem (1.1).

Algorithm 1.4. Initialization: let $x_0\in H_1$ be arbitrary. For $n\ge 0$, assume the current iterate $x_n$ has been constructed. If

$\Vert x_n-S{x}_n+A^{*}\left(I-T\right)A{x}_n\Vert =0,$

then stop; otherwise, calculate the next iterate $x_{n+1}$ by the following formula

$\{\begin{array}{l}{y}_n=x_n-S{x}_n+A^{*}\left(I-T\right)A{x}_n,\hfill \\ x_{n+1}=x_n-\gamma {\tau }_n{y}_n,\mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\forall n\ge 0,\hfill \end{array}$

where $\gamma \in \left(\mathrm{0,}\min \left\{1-\beta \mathrm{,1}-\mu \right\}\right)$ is a positive constant and ${\tau }_n$ is chosen self-adaptively as

${\tau }_n=\frac{{\Vert x_n-S{x}_n\Vert }^2+{\Vert \left(I-T\right)A{x}_n\Vert }^2}{{\Vert y_n\Vert }^2}.$

Algorithm 1.5. Initialization: Let $u\in H_1$ be a fixed point and let $x_0\in H_1$ be arbitrary. Iterative step: for $n\ge 0$, assume the current iterate $x_n$ has been constructed. If

$\Vert x_n-S{x}_n+A^{*}\left(I-T\right)A{x}_n\Vert =0,$

then stop; otherwise, calculate the next iterate $x_{n+1}$ by the following formula

$\{\begin{array}{l}{y}_n=x_n-S{x}_n+A^{*}\left(I-T\right)A{x}_n,\hfill \\ x_{n+1}={\alpha }_{n}u+\left(1-{\alpha }_n\right)\left(x_n-\gamma {\tau }_n{y}_n\right),\mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\forall n\ge 0,\hfill \end{array}$

where $\gamma \in \left(\mathrm{0,}\min \left\{1-\beta \mathrm{,1}-\mu \right\}\right)$ is a positive constant and ${\tau }_n$ is chosen self-adaptively as

${\tau }_n=\frac{{\Vert x_n-S{x}_n\Vert }^2+{\Vert \left(I-T\right)A{x}_n\Vert }^2}{{\Vert y_n\Vert }^2}.$

They also obtained the weak and strong convergence of Algorithms 1.4 and 1.5, respectively. Motivated and inspired by the work in the literature, the main purpose of this paper is to extend the results of Wang [16] and Yao, et al. [17] from the directed mappings or demicontractive mappings to the demicontractive mappings. We present two self-adaptive algorithms for solving the split common fixed point problem (1.1). Weak and strong convergence theorems are given under some mild assumptions. Our results improve essentially the corresponding results in [16] [17]. Further, some other results are also improved; see [9] - [22].

2. Preliminaries

Let C be a nonempty closed convex subset of a real Hilbert space H.

Definition 2.1. A mapping $T\mathrm{<mo>\; :\; </mo>}C\to C$ is said to be:

1) directed if

${\Vert Tx-x^{\mathrm{<mo>\; *\; </mo>}}\Vert }^2\le {\Vert x-x^{\mathrm{<mo>\; *\; </mo>}}\Vert }^2-{\Vert Tx-x\Vert }^2\mathrm{,\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\forall x\in C\mathrm{,\; <mi>\; </mi>}{x}^{\mathrm{<mo>\; *\; </mo>}}\in F\left(T\right)\mathrm{<mo>\; ;\; </mo>}$

2) β-demicontractive if there exists a constant $\beta \in \left[\mathrm{0,1}\right)$ such that

${\Vert Tx-x^{\mathrm{<mo>\; *\; </mo>}}\Vert }^2\le {\Vert x-x^{\mathrm{<mo>\; *\; </mo>}}\Vert }^2+\beta {\Vert Tx-x\Vert }^2\mathrm{,\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\forall x\in C\mathrm{,\; <mi>\; </mi>}{x}^{\mathrm{<mo>\; *\; </mo>}}\in F\left(T\right)\mathrm{<mo>\; ;\; </mo>}$

3) k-demimetric if there exists a constant $k\in \left(-\infty \mathrm{,1}\right)$ such that

$\langle x-x^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}x-Tx\rangle \ge \frac{1-k}{2}{\Vert x-Tx\Vert }^2\mathrm{,\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\forall x\in C\mathrm{,\; <mi>\; </mi>}{x}^{\mathrm{<mo>\; *\; </mo>}}\in F\left(T\right)\mathrm{.}$ (2.1)

Clearly, (2.1) is equivalent to the following:

${\Vert Tx-x^{\mathrm{<mo>\; *\; </mo>}}\Vert }^2\le {\Vert x-x^{\mathrm{<mo>\; *\; </mo>}}\Vert }^2+k{\Vert Tx-x\Vert }^2\mathrm{,\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\forall x\in C\mathrm{,\; <mi>\; </mi>}{x}^{\mathrm{<mo>\; *\; </mo>}}\in F\left(T\right)\mathrm{.}$

It is obvious that the demimetric mappings include the directed mappings and the demicontractive mappings as special cases. Furthermore, this class mapping also contains the classes of strict pseudo-contractions, firmly-quasinon expansive mappings, 2-generalized hybrid mappings and quasi-non-expansive mappings. The class of demimetric mappings is fundamental because many common types of mappings arising in optimization belong to this class, see for example [23] [24] and references therein.

Definition 2.2 A sequence $\left\{x_n\right\}$ is called Fejér-monotone with respect to a given nonempty set $\Omega$, if for every $x\in \Omega$,

$\Vert x_{n+1}-x\Vert \le \Vert x_n-x\Vert \mathrm{,\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\forall n\ge 0.$

Next we adopt the following notations:

a) $x_n\to x$ and $x_n\rightharpoonup x$ denote the strong and weak convergence of the sequence $\left\{x_n\right\}$, respectively;

b) ${\omega }_w\left(x_n\right)\mathrm{<mo>\; :\; </mo>}=\left\{x\mathrm{<mo>\; :\; </mo>}\exists x_{n_j}\rightharpoonup x\right\}$ is the weak ω-limit set of the sequence $\left\{x_n\right\}$.

Recall that a mapping $f\mathrm{<mo>\; :\; </mo>}C\to C$ is said to be contractive if there exists a constant $v\in \left(\mathrm{0,1}\right)$ such that

$\Vert fx-fy\Vert \le v\Vert x-y\Vert \mathrm{,\; <mi>\; </mi>\; <mi>\; </mi>}\forall x\mathrm{,}y\in C\mathrm{.}$

We use ${\Pi }_C$ to denote the collection of mappings f verifying the above inequality. That is

${\Pi }_C=\left\{f\mathrm{<mo>\; :\; </mo>}C\to H\mathrm{<mo>\; :\; </mo>}f\mathrm{<mi>\; </mi>}\text{is a contraction with constant}\mathrm{<mi>\; </mi>}v\right\}\mathrm{.}$

Let D be a nonempty subset of C. A sequence $\left\{f_n\right\}$ of mappings of C into H is said to be stable on D (see [25]) if $\left\{f_n\left(x\right)\mathrm{<mo>\; :\; </mo>}n\ge 0\right\}$ is a singleton for every $x\in D$. It is clear that if $\left\{f_n\right\}$ is stable on D, then $f_n\left(x\right)=f_0\left(x\right)$ for all $n\ge 0$ and $x\in D$.

Recall that the (nearest point or metric) projection from H onto C, denoted $P_C$, assigns to each $u\in H$, the unique point $P_C\left(u\right)\in C$ with the property

$\Vert u-P_C\left(u\right)\Vert =\inf \left\{\Vert u-v\Vert :v\in C\right\}.$

The metric projection $P_C\left(u\right)$ of H onto C is characterized by

$\langle u-P_C\left(u\right)\mathrm{,}y-P_C\left(u\right)\rangle \le \mathrm{0,\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\forall y\in C\mathrm{,\; <mi>\; </mi>}u\in H\mathrm{.}$

Lemma 2.1 ( [26]) Let $\Omega$ be a nonempty closed convex subset in H. If the sequence $\left\{x_n\right\}$ is Fejér monotone with respect to $\Omega$, then we have the following conclusions:

1) $x_n\rightharpoonup x^{\mathrm{<mo>\; *\; </mo>}}\in \Omega$ iff ${\omega }_w\left(x_n\right)\subset \Omega$ ;

2) the sequence $\left\{P_{\Omega }\left(x_n\right)\right\}$ converges strongly;

3) if $x_n\rightharpoonup x^{\mathrm{<mo>\; *\; </mo>}}\in \Omega$, then $x^{*}={\lim }_{n\to \infty }{P}_{\Omega }\left(x_n\right)$.

Lemma 2.2 ( [27]) Let $\left\{{\alpha }_n\right\}$ be a sequence of nonnegative numbers satisfying the property:

${\alpha }_{n+1}\le \left(1-{\gamma }_n\right){\alpha }_n+{\gamma }_n{c}_n\mathrm{,\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}n\ge \mathrm{0,}$

where $\left\{{\gamma }_n\right\}\mathrm{,}\left\{c_n\right\}$ satisfy the restrictions:

1) ${\displaystyle {\sum }_{n=1}^{\infty }}{\gamma }_n=\infty$ ;

2) $\lim {\sup }_{n\to \infty }{c}_n\le 0$ or ${\displaystyle {\sum }_{n=1}^{\infty }}{c}_n{\gamma }_n<\infty$.

Then, ${\lim }_{n\to \infty }{\alpha }_n=0$.

Lemma 2.3 ( [23] [24]) Let E be a smooth, strictly convex and reflexive Banach space and let k be a real number with $k\in \left(-\infty \mathrm{,1}\right)$. Let U be an k-demimetric mapping of E into itself. Then $F\left(U\right)$ is closed and convex.

3. Main Results

Now we study the split common fixed points problem (1) under the following hypothesis:

Ÿ $H_1$ and $H_2$ are two real Hilbert spaces;

Ÿ $S\mathrm{<mo>\; :\; </mo>}{H}_1\to H_1$ and $T\mathrm{<mo>\; :\; </mo>}{H}_2\to H_2$ are two demimetric mappings with constants $\beta \in \left(-\infty \mathrm{,1}\right)$ and $\mu \in \left(-\infty \mathrm{,1}\right)$, respectively;

Ÿ $A\mathrm{<mo>\; :\; </mo>}{H}_1\to H_2$ is a bounded linear operator with its adjoint operator $A^{\mathrm{<mo>\; *\; </mo>}}$ ;

Ÿ $\left\{f_n\right\}\subset {\Pi }_C$ is stable on $\Omega$, where $\Omega$ denotes the solution set of problem (1.1).

Lemma 3.1 $z^{\mathrm{<mo>\; *\; </mo>}}$ solves problem (1) iff $\Vert z^{*}-S{z}^{*}+A^{*}\left(I-T\right)A{z}^{*}\Vert =0$.

Proof. If $z^{\mathrm{<mo>\; *\; </mo>}}$ solves problem (1), then $z^{*}=S{z}^{*}$ and $\left(I-T\right)A{z}^{*}=0$. Therefore, we get $\Vert z^{*}-S{z}^{*}+A^{*}\left(I-T\right)A{z}^{*}\Vert =0$. To see the converse, suppose that $\Vert z^{*}-S{z}^{*}+A^{*}\left(I-T\right)A{z}^{*}\Vert =0$. Then, we have for any $z\in \Omega$ that

$\begin{array}{c}0=\Vert z^{*}-S{z}^{*}+A^{*}\left(I-T\right)A{z}^{*}\Vert \Vert z^{*}-z\Vert \\ \ge \langle z^{\mathrm{<mo>\; *\; </mo>}}-S{z}^{\mathrm{<mo>\; *\; </mo>}}+A^{\mathrm{<mo>\; *\; </mo>}}\left(I-T\right)A{z}^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}{z}^{\mathrm{<mo>\; *\; </mo>}}-z\rangle \\ \ge \langle z^{\mathrm{<mo>\; *\; </mo>}}-S{z}^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}{z}^{\mathrm{<mo>\; *\; </mo>}}-z\rangle +\langle A^{\mathrm{<mo>\; *\; </mo>}}\left(I-T\right)A{z}^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}{z}^{\mathrm{<mo>\; *\; </mo>}}-z\rangle \\ \ge \langle z^{\mathrm{<mo>\; *\; </mo>}}-S{z}^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}{z}^{\mathrm{<mo>\; *\; </mo>}}-z\rangle +\langle \left(I-T\right)A{z}^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}A{z}^{\mathrm{<mo>\; *\; </mo>}}-Az\rangle \mathrm{.}\end{array}$ (3.1)

Since S and T are demimetric, we have that

$\langle z^{\mathrm{<mo>\; *\; </mo>}}-S{z}^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}{z}^{\mathrm{<mo>\; *\; </mo>}}-z\rangle \ge \frac{1-\beta }{2}{\Vert z^{\mathrm{<mo>\; *\; </mo>}}-S{z}^{\mathrm{<mo>\; *\; </mo>}}\Vert }^2$ (3.2)

and

$\langle \left(I-T\right)A{z}^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}A{z}^{\mathrm{<mo>\; *\; </mo>}}-Az\rangle \ge \frac{1-\mu }{2}{\Vert A{z}^{\mathrm{<mo>\; *\; </mo>}}-TA{z}^{\mathrm{<mo>\; *\; </mo>}}\Vert }^2\mathrm{.}$ (3.3)

Combining (3.1), (3.2) and (3.3), we obtain that

$0\ge \frac{1-\beta }{2}{\Vert z^{\mathrm{<mo>\; *\; </mo>}}-S{z}^{\mathrm{<mo>\; *\; </mo>}}\Vert }^2+\frac{1-\mu }{2}{\Vert A{z}^{\mathrm{<mo>\; *\; </mo>}}-TA{z}^{\mathrm{<mo>\; *\; </mo>}}\Vert }^2\mathrm{.}$ (3.4)

Since $\beta \mathrm{,}\mu \in \left(-\infty \mathrm{,1}\right)$, we infer that $z^{\mathrm{<mo>\; *\; </mo>}}\in F\left(S\right)$ and $A{z}^{\mathrm{<mo>\; *\; </mo>}}\in F\left(T\right)$ by (3.4). Therefore, $z^{\mathrm{<mo>\; *\; </mo>}}$ solves problem (1.1). This completes the proof.

Next we construct the following self-adaptive algorithm to solve problem (1.1).

Algorithm 3.1. Initialization: let $x_0\in H_1$ be arbitrary. For $n\ge 0$, assume the current iterate $x_n$ has been constructed. If

$\Vert x_n-S{x}_n+A^{*}\left(I-T\right)A{x}_n\Vert =0,$

then stop (in this case $x_n$ solves problem (1.1) by Lemma 3.1); otherwise, calculate the next iterate $x_{n+1}$ by the following formula

$\{\begin{array}{l}{y}_n=x_n-S{x}_n+A^{*}\left(I-T\right)A{x}_n,\hfill \\ x_{n+1}=x_n-\gamma {\tau }_n{y}_n,\mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\forall n\ge 0,\hfill \end{array}$ (3.5)

where $\gamma \in \left(\mathrm{0,}\min \left\{1-\beta \mathrm{,1}-\mu \right\}\right)$ is a positive constant and ${\tau }_n$ is chosen self adaptively as

${\tau }_n=\frac{{\Vert x_n-S{x}_n\Vert }^2+{\Vert \left(I-T\right)A{x}_n\Vert }^2}{{\Vert y_n\Vert }^2}.$

We assume that the sequence $\left\{x_n\right\}$ generated by Algorithm 3.1 is infinite. In other words, Algorithm 3.1 does not terminate in a finite number of iterations.

Theorem 3.2. Assume that S and T are demiclosed at zero. If $\Omega \ne \varnothing$, then the sequence $\left\{x_n\right\}$ generated by (3.5) converges weakly to a solution $z^{\mathrm{<mo>\; *\; </mo>}}$ ( $={\lim }_{n\to \infty }{P}_{\Omega }\left(x_n\right)$ ) of problem (1.1).

Proof. Since A is linear and continuous, noticing Lemma 2.3, we see $\Omega$ is closed and convex. Thus we have that $P_{\Omega }$ is well defined.

We next prove that the sequence $\left\{x_n\right\}$ is Fejér-monotone with respect to $\Omega$. Letting $z\in \Omega$, we then obtain that

$\begin{array}{l}\langle y_n\mathrm{,}{x}_n-z\rangle \\ =\langle x_n-S{x}_n+A^{\mathrm{<mo>\; *\; </mo>}}\left(I-T\right)A{x}_n\mathrm{,}{x}_n-z\rangle \\ =\langle x_n-S{x}_n\mathrm{,}{x}_n-z\rangle +\langle A^{\mathrm{<mo>\; *\; </mo>}}\left(I-T\right)A{x}_n\mathrm{,}{x}_n-z\rangle \\ \ge \frac{1-\beta }{2}{\Vert x_n-S{x}_n\Vert }^2+\frac{1-\mu }{2}{\Vert \left(I-T\right)A{x}_n\Vert }^2\\ \ge \frac{1}{2}\min \left\{1-\beta \mathrm{,1}-\mu \right\}\left({\Vert x_n-S{x}_n\Vert }^2+{\Vert A{x}_n-TA{x}_n\Vert }^2\right)\mathrm{.}\end{array}$ (3.6)

In view of Equation (3.5) and Equation (3.6), we deduce

$\begin{array}{l}{\Vert x_{n+1}-z\Vert }^2={\Vert x_n-\gamma {\tau }_n{y}_n-z\Vert }^2\\ ={\Vert x_n-z\Vert }^2-2\gamma {\tau }_n\langle y_n\mathrm{,}{x}_n-z\rangle +{\gamma }^2{\tau }_n^2{\Vert y_n\Vert }^2\\ \le {\Vert x_n-z\Vert }^2+{\gamma }^2\frac{{\left({\Vert x_n-S{x}_n\Vert }^2+{\Vert A{x}_n-TA{x}_n\Vert }^2\right)}^2}{{\Vert y_n\Vert }^2}\\ -\gamma \min \left\{1-\beta \mathrm{,1}-\mu \right\}\frac{{\left({\Vert x_n-S{x}_n\Vert }^2+{\Vert A{x}_n-TA{x}_n\Vert }^2\right)}^2}{{\Vert y_n\Vert }^2}\\ \le {\Vert x_n-z\Vert }^2-\gamma \left(\min \left\{1-\beta \mathrm{,1}-\mu \right\}-\gamma \right)\frac{{\left({\Vert x_n-S{x}_n\Vert }^2+{\Vert A{x}_n-TA{x}_n\Vert }^2\right)}^2}{{\Vert y_n\Vert }^2}\mathrm{.}\end{array}$ (3.7)

This implies that the sequence $\left\{x_n\right\}$ is Fejér monotone.

Next, we show that every weak cluster point of the sequence $\left\{x_n\right\}$ belongs to the solution set of problem (1.1).

From the Fejér-monotonicity of $\left\{x_n\right\}$, it follows that the sequence $\left\{x_n\right\}$ is bounded. Further, we deduce from (3.7) that

$\begin{array}{l}\gamma \left(\min \left\{1-\beta \mathrm{,1}-\mu \right\}-\gamma \right)\frac{{\left({\Vert x_n-S{x}_n\Vert }^2+{\Vert A{x}_n-TA{x}_n\Vert }^2\right)}^2}{{\Vert y_n\Vert }^2}\\ \le {\Vert x_n-z\Vert }^2-{\Vert x_{n+1}-z\Vert }^2\mathrm{.}\end{array}$

An induction induces that

$\gamma \left(\min \left\{1-\beta ,1-\mu \right\}-\gamma \right){\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}\frac{{\left({\Vert x_n-S{x}_n\Vert }^2+{\Vert A{x}_n-TA{x}_n\Vert }^2\right)}^2}{{\Vert y_n\Vert }^2}\le {\Vert x_0-z\Vert }^2<\infty ,$

which implies that

$\underset{n\to \infty }{\lim }\frac{{\left({\Vert x_n-S{x}_n\Vert }^2+{\Vert A{x}_n-TA{x}_n\Vert }^2\right)}^2}{{\Vert y_n\Vert }^2}=0.$

Observe that

$\begin{array}{l}\frac{{\left({\Vert x_n-S{x}_n\Vert }^2+{\Vert A{x}_n-TA{x}_n\Vert }^2\right)}^2}{{\Vert y_n\Vert }^2}\\ =\frac{{\left({\Vert x_n-S{x}_n\Vert }^2+{\Vert A{x}_n-TA{x}_n\Vert }^2\right)}^2}{{\Vert x_n-S{x}_n+A^{*}\left(I-T\right)A{x}_n\Vert }^2}\\ \ge \frac{{\left({\Vert x_n-S{x}_n\Vert }^2+{\Vert A{x}_n-TA{x}_n\Vert }^2\right)}^2}{2\left({\Vert x_n-S{x}_n\Vert }^2+{\Vert A\Vert }^2{\Vert \left(I-T\right)A{x}_n\Vert }^2\right)}\\ \ge \frac{{\left({\Vert x_n-S{x}_n\Vert }^2+{\Vert A{x}_n-TA{x}_n\Vert }^2\right)}^2}{2\max \left\{1,{\Vert A\Vert }^2\right\}\left({\Vert x_n-S{x}_n\Vert }^2+{\Vert \left(I-T\right)A{x}_n\Vert }^2\right)}\\ =\frac{{\Vert x_n-S{x}_n\Vert }^2+{\Vert A{x}_n-TA{x}_n\Vert }^2}{2\max \left\{1,{\Vert A\Vert }^2\right\}}.\end{array}$ (3.8)

By the demiclosedness (at zero) of S and T, we deduce immediately ${\omega }_w\left(x_n\right)\subset \Omega$. To this end, the conditions of Lemma 2.1 are all satisfied. Consequently, $x_n\rightharpoonup z^{\mathrm{<mo>\; *\; </mo>}}={\lim }_{n\to \infty }{P}_{\Omega }\left(x_n\right)$. This completes the proof.

Next, we study an iteration with strong convergence for solving problem (1.1).

Algorithm 3.3 Initialization: Let $x_0\in H_1$ be arbitrary. Iterative step: for $n\ge 0$, assume the current iterate $x_n$ has been constructed. If

$\Vert x_n-S{x}_n+A^{\mathrm{<mo>\; *\; </mo>}}\left(I-T\right)A{x}_n\Vert =\mathrm{0,}$

then stop (in this case $x_n$ solves problem (1.1) by Lemma 3.1); otherwise, calculate the next iterate $x_{n+1}$ by the following formula

$\{\begin{array}{l}{y}_n=x_n-S{x}_n+A^{*}\left(I-T\right)A{x}_n,\hfill \\ x_{n+1}={\alpha }_n{f}_n{x}_n+\left(1-{\alpha }_n\right)\left(x_n-\gamma {\tau }_n{y}_n\right),\mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\forall n\ge 0,\hfill \end{array}$ (3.9)

where $\gamma \in \left(\mathrm{0,}\min \left\{1-\beta \mathrm{,1}-\mu \right\}\right)$ is a positive constant and ${\tau }_n$ is chosen self-adaptively as

${\tau }_n=\frac{{\Vert x_n-S{x}_n\Vert }^2+{\Vert \left(I-T\right)A{x}_n\Vert }^2}{{\Vert y_n\Vert }^2}.$

Theorem 3.4 Assume that:

(C1) $\Omega \ne \varnothing$ ;

(C2) S and T are demiclosed at zero;

(C3) ${\lim }_{n\to \infty }{\alpha }_n=0$ and ${\displaystyle {\sum }_{n=0}^{\infty }}{\alpha }_n=\infty$.

Then the sequence $\left\{x_n\right\}$ generated by (3.9) converges strongly to the solution $z\left(=P_{\Omega }{f}_{0}z\right)$ of problem (1.1).

Proof. Putting $z=P_{\Omega }{f}_{0}z$, we obtain from (3.7) that

$\begin{array}{l}{\Vert x_n-\gamma {\tau }_n{y}_n-z\Vert }^2\\ \le {\Vert x_n-z\Vert }^2-\gamma \left(\min \left\{1-\beta \mathrm{,1}-\mu \right\}-\gamma \right)\frac{{\left({\Vert x_n-S{x}_n\Vert }^2+{\Vert A{x}_n-TA{x}_n\Vert }^2\right)}^2}{{\Vert y_n\Vert }^2}\\ \le {\Vert x_n-z\Vert }^2\mathrm{.}\end{array}$ (3.10)

Next, we show that the sequence $\left\{x_n\right\}$ is bounded. Indeed, we obtain from (3.9) and (3.10) that

$\begin{array}{c}\Vert x_{n+1}-z\Vert =\Vert {\alpha }_n{f}_n{x}_n+\left(1-{\alpha }_n\right)\left(x_n-\gamma {\tau }_n{y}_n\right)-z\Vert \\ \le {\alpha }_n\Vert f_n{x}_n-z\Vert +\left(1-{\alpha }_n\right)\Vert x_n-\gamma {\tau }_n{y}_n-z\Vert \\ \le {\alpha }_n\left(\Vert f_n{x}_n-f_{n}z\Vert +\Vert f_{n}z-z\Vert \right)+\left(1-{\alpha }_n\right)\Vert x_n-z\Vert \\ \le {\alpha }_n\left(v\Vert x_n-z\Vert +\Vert f_{0}z-z\Vert \right)+\left(1-{\alpha }_n\right)\Vert x_n-z\Vert \\ \le {\alpha }_n\Vert f_{0}z-z\Vert +\left(1-{\alpha }_n\left(1-v\right)\right)\Vert x_n-z\Vert .\end{array}$

By induction, we get

$\Vert x_{n+1}-z\Vert \le \max \left\{\frac{\Vert f_{0}z-z\Vert }{1-v}\mathrm{,}\Vert x_0-z\Vert \right\}\mathrm{,}$

which gives that the sequence $\left\{x_n\right\}$ is bounded.

By virtue of (3.9), we deduce

$\begin{array}{c}{\Vert x_{n+1}-z\Vert }^2=\langle {\alpha }_n{f}_n{x}_n+\left(1-{\alpha }_n\right)\left(x_n-\gamma {\tau }_n{y}_n\right)-z,x_{n+1}-z\rangle \\ =\left(1-{\alpha }_n\right)\langle \left(x_n-\gamma {\tau }_n{y}_n\right)-z,x_{n+1}-z\rangle +{\alpha }_n\langle f_n{x}_n-f_{n}z,x_{n+1}-z\rangle \\ +{\alpha }_n\langle f_{0}z-z,x_{n+1}-z\rangle \\ \le \left(1-{\alpha }_n\right)\Vert x_n-\gamma {\tau }_n{y}_n-z\Vert \Vert x_{n+1}-z\Vert +{\alpha }_n\Vert f_n{x}_n-f_{n}z\Vert \Vert x_{n+1}-z\Vert \\ +{\alpha }_n\langle f_{0}z-z,x_{n+1}-z\rangle \end{array}$

$\begin{array}{l}\le \left(1-{\alpha }_n\right)\Vert x_n-\gamma {\tau }_n{y}_n-z\Vert \Vert x_{n+1}-z\Vert +{\alpha }_{n}v\Vert x_n-z\Vert \Vert x_{n+1}-z\Vert \\ +{\alpha }_n\langle f_{0}z-z,x_{n+1}-z\rangle \\ =\left(1-{\alpha }_n\right)\left(\frac{1}{2}{\Vert x_n-\gamma {\tau }_n{y}_n-z\Vert }^2+\frac{1}{2}{\Vert x_{n+1}-z\Vert }^2\right)\\ +{\alpha }_n\left(\frac{1}{2}v{\Vert x_n-z\Vert }^2+\frac{1}{2}{\Vert x_{n+1}-z\Vert }^2\right)+{\alpha }_n\langle f_{0}z-z,x_{n+1}-z\rangle ,\end{array}$

which implies

${\Vert x_{n+1}-z\Vert }^2\le \left(1-{\alpha }_n\right){\Vert x_n-\gamma {\tau }_n{y}_n-z\Vert }^2+{\alpha }_{n}v{\Vert x_n-z\Vert }^2+2{\alpha }_n\langle f_{0}z-z\mathrm{,}{x}_{n+1}-z\rangle \mathrm{.}$

This together with (3.10) implies that

$\begin{array}{l}{\Vert x_{n+1}-z\Vert }^2\\ \le \left(1-{\alpha }_n\left(1-v\right)\right){\Vert x_n-z\Vert }^2+2{\alpha }_n\langle f_{0}z-z,x_{n+1}-z\rangle \\ -\left(1-{\alpha }_n\right)\gamma \left(\min \left\{1-\beta ,1-\mu \right\}-\gamma \right)\frac{{\left({\Vert x_n-S{x}_n\Vert }^2+{\Vert A{x}_n-TA{x}_n\Vert }^2\right)}^2}{{\Vert y_n\Vert }^2}\\ \le \left(1-{\alpha }_n\left(1-v\right)\right){\Vert x_n-z\Vert }^2+{\alpha }_n(2\langle f_{0}z-z,x_{n+1}-z\rangle \begin{array}{c}\\ \\ \end{array}\\ -\frac{\left(1-{\alpha }_n\right)\gamma \left(\min \left\{1-\beta ,1-\mu \right\}-\gamma \right)}{{\alpha }_n}\frac{{\left({\Vert x_n-S{x}_n\Vert }^2+{\Vert A{x}_n-TA{x}_n\Vert }^2\right)}^2}{{\Vert x_n-S{x}_n+A^{*}\left(I-T\right)A{x}_n\Vert }^2}).\end{array}$ (3.11)

Set ${\delta }_n={\Vert x_n-z\Vert }^2$ and

$\begin{array}{c}{\sigma }_n=2\langle f_{0}z-z,x_{n+1}-z\rangle -\frac{\left(1-{\alpha }_n\right)\gamma \left(\min \left\{1-\beta ,1-\mu \right\}-\gamma \right)}{{\alpha }_n}\\ \times \frac{{\left({\Vert x_n-S{x}_n\Vert }^2+{\Vert A{x}_n-TA{x}_n\Vert }^2\right)}^2}{{\Vert x_n-S{x}_n+A^{*}\left(I-T\right)A{x}_n\Vert }^2}\end{array}$ (3.12)

for all $n\ge 0$. Returning to (3.11) to obtain

${\delta }_{n+1}\le \left(1-{\alpha }_n\left(1-v\right)\right){\delta }_n+{\alpha }_n{\sigma }_n\mathrm{,\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\forall n\ge 0.$ (3.13)

From (3.12), we find

${\sigma }_n\le 2\langle f_{0}z-z\mathrm{,}{x}_{n+1}-z\rangle \le 2\Vert f_{0}z-z\Vert \Vert x_{n+1}-z\Vert \mathrm{.}$

It follows that $\lim {\sup }_{n\to \infty }{\sigma }_n<+\infty$.

Next we show that $\lim {\sup }_{n\to \infty }{\sigma }_n\ge -1$.

If $\lim {\sup }_{n\to \infty }{\sigma }_n<-1$, then there exists $n_0$ such that ${\sigma }_n\le -1$ for all $n\ge n_0$. It then follows from (3.13) that

${\delta }_{n+1}\le \left(1-{\alpha }_n\left(1-v\right)\right){\delta }_n-{\alpha }_n\le {\delta }_n-{\alpha }_n\mathrm{.}$

for all $n\ge n_0$. By induction, we have

${\delta }_{n+1}\le {\delta }_{n_0}-{\displaystyle \underset{i=n_0}{\overset{n}{\sum }}}{\alpha }_i.$ (3.14)

By taking $\lim \sup$ as $n\to \infty$ in (3.14), we have

$\underset{n\to \infty }{\lim \sup }{\delta }_n\le {\delta }_{n_0}-\underset{n\to \infty }{\lim }{\displaystyle \underset{i=n_0}{\overset{n}{\sum }}}{\alpha }_i=-\infty ,$

which induces a contradiction. So, $-1\le \lim {\sup }_{n\to \infty }{\sigma }_n<+\infty$. Thus, we can take a subsequence $\left\{n_k\right\}$ such that

$\begin{array}{c}\underset{n\to \infty }{\lim \sup }{\sigma }_n=\underset{k\to \infty }{\lim }{\sigma }_{n_k}\\ =\underset{k\to \infty }{\lim }2\langle f_{0}z-z,x_{n_k+1}-z\rangle -\frac{\left(1-{\alpha }_{n_k}\right)\gamma \left(\min \left\{1-\beta ,1-\mu \right\}-\gamma \right)}{{\alpha }_{n_k}}\\ \times \frac{{\left({\Vert x_{n_k}-S{x}_{n_k}\Vert }^2+{\Vert A{x}_{n_k}-TA{x}_{n_k}\Vert }^2\right)}^2}{{\Vert x_{n_k}-S{x}_{n_k}+A^{*}\left(I-T\right)A{x}_{n_k}\Vert }^2}.\end{array}$ (3.15)

Since $\langle f_{0}z\mathrm{,}{x}_{n_k+1}-z\rangle$ is a bounded real sequence, without loss of generality, we may assume ${\lim }_{k\to \infty }\langle f_{0}z\mathrm{,}{x}_{n_k+1}-z\rangle$ exists. Consequently, from (3.15), the following limit also exists

$\underset{k\to \infty }{\lim }\frac{\left(1-{\alpha }_{n_k}\right)\gamma \left(\min \left\{1-\beta ,1-\mu \right\}-\gamma \right)}{{\alpha }_{n_k}}\frac{{\left({\Vert x_{n_k}-S{x}_{n_k}\Vert }^2+{\Vert A{x}_{n_k}-TA{x}_{n_k}\Vert }^2\right)}^2}{{\Vert x_{n_k}-S{x}_{n_k}+A^{*}\left(I-T\right)A{x}_{n_k}\Vert }^2}.$

It turns out that

$\underset{k\to \infty }{\lim }\frac{{\left({\Vert x_{n_k}-S{x}_{n_k}\Vert }^2+{\Vert A{x}_{n_k}-TA{x}_{n_k}\Vert }^2\right)}^2}{{\Vert x_{n_k}-S{x}_{n_k}+A^{\mathrm{<mo>\; *\; </mo>}}\left(I-T\right)A{x}_{n_k}\Vert }^2}=0.$ (3.16)

Taking into consideration that

$\frac{{\Vert x_{n_k}-S{x}_{n_k}\Vert }^2+{\Vert A{x}_{n_k}-TA{x}_{n_k}\Vert }^2}{2\max \left\{1,{\Vert A\Vert }^2\right\}}\le \frac{{\left({\Vert x_{n_k}-S{x}_{n_k}\Vert }^2+{\Vert A{x}_{n_k}-TA{x}_{n_k}\Vert }^2\right)}^2}{{\Vert x_{n_k}-S{x}_{n_k}+A^{*}\left(I-T\right)A{x}_{n_k}\Vert }^2},$

we then deduce from (3.16) that

$\underset{k\to \infty }{\lim }\Vert x_{n_k}-S{x}_{n_k}\Vert =\underset{k\to \infty }{\lim }\Vert A{x}_{n_k}-TA{x}_{n_k}\Vert =0.$ (3.17)

It follows that any weak cluster point of $\left\{x_{n_k}\right\}$ belongs to $\Omega$. Observe that

$\begin{array}{l}\Vert x_{n+1}-x_n\Vert \\ \le {\alpha }_n\Vert x_n-f_n{x}_n\Vert +\left(1-{\alpha }_n\right)\gamma {\tau }_n\Vert y_n\Vert \\ ={\alpha }_n\Vert x_n-f_n{x}_n\Vert +\left(1-{\alpha }_n\right)\gamma \frac{{\left({\Vert x_n-S{x}_n\Vert }^2+{\Vert A{x}_n-TA{x}_n\Vert }^2\right)}^2}{{\Vert x_n-S{x}_n+A^{*}\left(I-T\right)A{x}_n\Vert }^2}.\end{array}$

By (C3) and (3.16), we derive

$\underset{k\to \infty }{\lim }\Vert x_{n_k+1}-x_{n_k}\Vert =0.$

This means that any weak cluster point of $\left\{x_{n_k+1}\right\}$ also belongs to $\Omega$. Without loss of generality, we assume that $\left\{x_{n_k+1}\right\}$ converges weakly to $\bar{x}\in \Omega$. Hence, we obtain

$\underset{n\to \infty }{\lim \sup }{\sigma }_n\le \underset{k\to \infty }{\lim }2\langle f_{0}z-z\mathrm{,}{x}_{n_k+1}-z\rangle =2\langle f_{0}z-z\mathrm{,}\bar{x}-z\rangle \le 0.$

due to the fact that $z=P_{\Omega }{f}_{0}z$. Rewriting (3.13) as

${\delta }_{n+1}\le \left(1-{\alpha }_n\left(1-v\right)\right){\delta }_n+{\alpha }_n\left(1-v\right)\frac{{\sigma }_n}{1-v}\mathrm{,\; <mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\forall n\ge \mathrm{0,}$

and noticing Lemma 2.2, we get $x_n\to z$ as $n\to \infty$.

Theorem 3.5 Let $S\mathrm{<mo>\; :\; </mo>}{H}_1\to H_1$ and $T\mathrm{<mo>\; :\; </mo>}{H}_2\to H_2$ be two demicontractive mappings with constants $\beta \in \left[\mathrm{0,1}\right)$ and $\mu \in \left[\mathrm{0,1}\right)$, respectively. Then the sequence $\left\{x_n\right\}$ generated by (1.1) converges strongly to the solution $z\left(=P_{\Omega }{f}_{0}z\right)$ of problem (3.9) under the assumption of Theorem 3.4.

4. Conclusion

In this paper, we consider a class of the split common fixed point problems. By extending results in [16] [17] from the directed mappings or the demicontractive mappings to the demimetric mappings, and a fixed point $u\in H_1$ to a sequence mappings $\left\{f_n\right\}\subset \Pi$, we construct two self-adaptive algorithms for solving the split common fixed point problem. Further, we also establish the weak and strong convergence theorems under some certain appropriate assumptions. The results in this paper are the extension and improvement of the recent results in the literature.

Acknowledgements

This research was supported by the Key Scientific Research Projects of Higher Education Institutions in Henan Province (20A110038).