and the weight $(-2)$–representation of $\operatorname{U}(1)$ in the
representation ring of $\operatorname{U}(1)$. The coefficient before each of
them, is given by $\mu_{t}(D(p))$ and $\mu_{n}(D(p))$ respectively. Then the
theorem follows from checking the formula (1.2) term-wisely with the formula
in [boden1998the, Theorem 1]. Note that our definition of the equivariant
spectral flow cancels out the factor $\frac{1}{2}$ in _loc.cit._ and our
convention of spectral flow adds the term $[\ker d_{B}]$. Although the formula
of Boden-Herald is written down using a holonomy perturbation which is not of
the preferred form, the identification between (1.2) and
$\lambda_{\operatorname{SU}(3)}$ results from the independence of
$\lambda_{\operatorname{SU}(3)}$ on holonomy perturbations, established in
[bai2020equivariant] and Section 2. Therefore the theorem is proved. ∎

## References


# New averaged type algorithms for solving split common fixed-point problem
for demicontractive mappings

Vasile Berinde  Department of Mathematics and Computer Science Technical
University of Cluj-Napoca, North University Centre at Baia Mare, Victoriei 76,
430122 Baia Mare, Romania Academy of Romanian Scientists, 3 Ilfov, 050044,
Bucharest, Romania  Khairul Saleh  Department of Mathematics, King Fahd
University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

###### Abstract

In this paper we propose new averaged iterative algorithms designed for
solving a split common fixed-point problem in the class of demicontractive
mappings. The algorithms are obtained by inserting an averaged term into the
algorithms used in [Li, R. and He, Z., A new iterative algorithm for split
solution problems of quasi-nonexpansive mappings J. Inequal. Appl. 131 (2015),
1–12.] for solving the same problem but in the class of quasi-nonexpansive
mappings, which is a subclass of demicontractive mappings. Basically, our
investigation is based on the embedding of demicontractive operators in the
class of quasi-nonexpansive operators by means of averaged mappings. For the
considered algorithms we prove weak and strong convergence theorems in the
setting of a real Hilbert space and also provide examples to show that our
results are effective generalizations of existing results in literature.

Keywords: Hilbert space, demicontractive mapping, split common fixed point,
weak convergence
$2010$ Mathematics Subject Classification: 47H10; 47H09.

##  1 Introduction

Let ${\cal H}$ be a real Hilbert space with norm $\|\cdot\|$ and inner product
$\langle\cdot,\cdot\rangle.$ Let $D\subset{\cal H}$ be a closed convex set,
and consider the self-mapping $G:D\to D.$ Throughout this paper the set of all
fixed point of G in $D$ is denoted by

$Fix(G)=\\{u\in D~{}:~{}Gu=u\\}.$

The mapping $G$ is said to be

  1. (a)

_nonexpansive_ if

$\|Gu-Gv\|\leq\|u-v\|,\quad\text{for all}~{}u,v\in D;$ (1.1)

  2. (b)

_quasi-nonexpansive_ if $Fix(G)\neq\emptyset$ and

$\|Gu-v\|\leq\|u-v\|,\quad\text{for all}~{}u\in D~{}\text{and}~{}v\in Fix(G);$
(1.2)

  3. (c)

_$\beta$ -demicontractive_ if $Fix(G)\neq\emptyset$ and there exists a
positive number $\beta<1$ such that

$\|Gu-v\|^{2}\leq\|u-v\|^{2}+\beta\|u-Gu\|^{2},$ (1.3)

for all $u\in D$ and $v\in Fix(G)$.

By the previous definitions, it is obvious that any nonexpansive mapping $G$
with $Fix(G)\neq\emptyset$ is demicontractive and that any quasi-nonexpansive
mapping is demicontractive, too, but the reverses are no more true, as
illustrated by the next example.

######  Example 1.1 ([2])

Let ${\cal H}$ be the real line with the usual norm and $D=[0,1].$ Define $F$
on $D$ as

$F(u)=\begin{cases}7/8,&\mbox{if }0\leq u<1\\\
1/4,&\text{if}~{}u=1.\end{cases}$ (1.4)

Then $F$ is demicontractive but $F$ is neither quasi-nonexpasive nor
nonexpansive.

Let ${\cal H}_{1}$ and ${\cal H}_{2}$ be two real Hilbert spaces and $F:{\cal
H}_{1}\to{\cal H}_{1},~{}G:{\cal H}_{2}\to{\cal H}_{2}$ be two nonlinear
mappings with $Fix(F)\neq\emptyset$ and $Fix(G)\neq\emptyset.$ The split
common fixed point problem for $F$ and $G$ is to

$\text{find an element}~{}u\in Fix(F)~{}\text{such that}~{}Au\in Fix(G).$
(1.5)

where $A:{\cal H}_{1}\to{\cal H}_{2}$ is a bounded linear operator. We denote
the set of all solutions of Problem (1.5) by

$\Gamma=\\{u\in Fix(F):Au\in Fix(G)\\}.$

This problem was first introduced by Censor and Segal [3] in 2009. It was
mentioned in [3] that the split common fixed point problem is a generalization
of the split and the convex feasibility problems that are very useful in image
recovery, convex optimization and other areas of applied mathematics. Some
convergence theorems for the split common fixed point problem have been
analyzed, see [5, 6] and the references therein. For instance, Moudafi [6]
proved weak convergence of the following iteration method for two quasi-
nonexpasive mappings $F$ and $G,$ assuming that $F-I$ and $G-I$ are demiclosed
at zero.

Let $u_{1}\in{\cal H}_{1}.$ For all $p\in\mathbb{N},$ define

$\begin{cases}y_{p}=u_{p}+\gamma\mu A^{*}(G-I)Au_{p},\\\
u_{p+1}=(1-\alpha_{p})y_{p}+\alpha_{p}F(y_{p}),\end{cases}$ (1.6)

where $\mu\in(0,1),~{}\\{\alpha_{p}\\}\subset(\delta,1-\delta)$ for a small
enough $\delta>0$ and $\gamma\in(0,\frac{1}{\lambda\mu}),$ with $\lambda$
being the spectral radius of $A^{*}A.$ Here $A^{*}$ denotes the adjoint of
$A.$

It was established in Moudafi [6] that the sequence $\\{u_{p}\\}$ given by
(1.6) converges weakly to a split common fixed point of $F$ and $G$.

A few years later, Li and He [5] introduced a new iteration scheme with
certain conditions that was shown to converge strongly to a split common fixed
point of two quasi-nonexpansive mappings.

In this paper, our aim is to solve the split common fixed point problem in the
setting Hilbert spaces for the case of the larger class of demicontractive
mappings, thus extending the main results in Li and He [5]. Our results are
obtained by considering new averaged iterative algorithms and prove weak and
strong convergence theorems.

##  2 Preliminaries

We recall some important lemmas used in the proofs of our main results. The
following two lemmas are due to Berinde [1].

###### Lemma 2.1

[1] Let ${\cal H}$ be a real Hilbert space and $D\subset{\cal H}$ a closed and
convex set. If $G:D\to D$ is $\beta$-demicontractive, then the Krasnoselskij
perturbation $G_{\nu}=(1-\nu)I+\nu G$ of $G$ is
$(1+\beta/\nu-1/\nu)$-demicontractive.

###### Lemma 2.2

[1] Let ${\cal H}$ be a real Hilbert space and $D\subset{\cal H}$ a closed and
convex set. If $G:D\to D$ is $\beta$-demicontractive, then for any
$\nu\in(0,1-\beta)$

$G_{\nu}=(1-\nu)I+\nu G$

is quasi-nonexpansive.

###### Lemma 2.3

Let ${\cal H}$ be a real Hilbert space, $D\subset{\cal H}$ a closed and convex
set and $F:D\to D$ a mapping. Then, for any $\nu\in(0,1),$ we have
$Fix(F_{\nu})=Fix(F).$

The next lemma is known as Opial’s lemma.

######  Lemma 2.4 ([7])

Let ${\cal H}$ be a Hilbert space and $\\{u_{p}\\}\subset{\cal H}$ a sequence
such that there exists a nonempty set $C\subset{\cal H}$ satisfying the
following.

  1. (a)

$\lim_{p\to\infty}\|u_{p}-v\|$ exists $\forall v\in C.$

  2. (b)

Any weak-cluster point of the sequence $\\{u_{p}\\}$ belongs to $C.$

Then there exists $\bar{u}\in C$ such that $\\{u_{p}\\}$ weakly converges to
$\bar{u}.$

The next lemma is due to Li and He [5].

###### Lemma 2.5

[5] Let $F_{1},\cdots,F_{n}:{\cal H}_{1}\to{\cal H}_{1}$ be quasi-nonexpansive
mappings and set $T=\sum_{i=1}^{n}b_{i}F_{a_{i}},$ where $b_{i}\in(0,1)$ with
$\sum_{i=1}^{n}b_{i}=1,$ and $F_{a_{i}}=(1-a_{i})I+a_{i}F_{i}$ with
$a_{i}\in(0,1),i=1,2,\cdots,n.$ Then $T$ is quasi-nonexpansive and

$Fix(T)=\bigcap_{i=1}^{n}Fix(F_{i})=\bigcap_{i=1}^{n}Fix(F_{a_{i}}).$

###### Lemma 2.6

[6] Let $S$ be a quasi-nonexpansive mapping and set $S_{a}:=(1-a)I+aS$ for
$a\in(0,1].$ The following properties hold for all $u\in{\cal H}$ and $v\in
Fix(S).$

  1. (1)

$\langle u-Su,u-v\rangle\geq\frac{1}{2}\|u-Su\|^{2}$ and $\langle
u-Su,v-Su\rangle\leq\frac{1}{2}\|u-Su\|$;

  2. (2)

$\|S_{a}u-v\|^{2}\leq\|u-v\|^{2}-a(1-a)\|u-Su\|$;

  3. (3)

$\langle u-S_{a}u,u-v\rangle\geq\frac{a}{2}\|u-Su\|^{2}.$

##  3 Split Common Fixed Points for Demicontractive mappings

In this section we state and prove our main results: a weak convergence
theorem (Theorem 3.1) and a strong convergence theorem (Theorem 3.2) for the
averaged type algorithm (3.1) used to approximate a split common fixed point
of two demicontractive mappings.

We also obtain a common fixed point result for two demicontrative mappings as
a special case of the main convergence theorem (Theorem 3.2).

###### Definition 3.1

[6] A sequence $\\{u_{p}\\}$ is said to be Fejér-monotone with respect to
$\Gamma$ if for every $v\in\Gamma,$

$\|u_{p+1}-v\|\leq\|u_{p}-v\|,\quad\forall p\in\mathbb{N}.$

###### Theorem 3.1

Let ${\cal H}_{1}$ and ${\cal H}_{2}$ be two real Hilbert spaces. Given a
bounded linear operator $A:{\cal H}_{1}\to{\cal H}_{2},$ let $F:{\cal
H}_{1}\to{\cal H}_{1},~{}G:{\cal H}_{2}\to{\cal H}_{2}$ be two
$\beta$-demicontractive operators. Assume that $F-I$ and $G-I$ are demiclosed
at zero. Let $\\{u_{p}\\}$ be a sequence generated by

$\begin{cases}u_{1}\in{\cal H}_{1},&\\\ y_{p}=u_{p}+\gamma\mu
A^{*}(b(G-I))Au_{p},\quad b\in(0,1-\beta),&\\\
u_{p+1}=(1-\alpha_{p})y_{p}+\alpha_{p}F(y_{p}),~{}~{}\forall
p\in\mathbb{N},\end{cases}$ (3.1)

where $\mu\in(0,1),~{}\\{\alpha_{p}\\}\subset(\delta,1-\delta)$ for a small
enough $\delta>0$ and $\gamma\in(0,\frac{1}{\lambda\mu})$ with $\lambda$ being
the spectral radius of $A^{*}A$ and $A^{*}$ being the adjoint of $A.$ Then
$\\{u_{p}\\}$ converges weakly to a split common fixed point $u^{*}\in\Gamma.$

Proof. Since $F$ and $G$ are $\beta$-demicontractive mappings, in view of
Lemma 2.2 the averaged mappings

$\displaystyle F_{a}=(1-a)I+aF\quad\text{and}\quad G_{b}=(1-b)I+bG$

are quasi-nonexpansive for $\\{a,b\\}\subset(0,1-\beta).$ Clearly, $F_{a}-I$
and $G_{b}-I$ are demiclosed at zero.

For any sequence $\\{t_{p}\\}\subset(\delta,1-\delta)$ with a small enough
$\delta>0,$ consider the iteration $\\{u_{p}\\}$ generated by

$\begin{cases}u_{1}\in{\cal H}_{1},&\\\ y_{p}=u_{p}+\gamma\mu
A^{*}(G_{b}-I)Au_{p},&\\\ u_{p+1}=(1-t_{p})y_{p}+t_{p}F_{a}y_{p},~{}~{}\forall
p\in\mathbb{N}.\end{cases}$

Take $v\in\Gamma.$ By Property (2) in Lemma 2.6, we obtain

$\|u_{p+1}-v\|^{2}\leq\|y_{k}-v\|^{2}-t_{p}(1-t_{p})\|G_{b}(y_{p})-y_{p}\|^{2}.$
(3.2)

We also have

$\displaystyle\|y_{p}-v\|^{2}$ $\displaystyle=\|u_{p}+\gamma\mu
A^{*}(G_{b}-I)(Au_{p})-v\|^{2}$
$\displaystyle=\|u_{p}-v\|^{2}+\gamma^{2}\mu^{2}\|(G_{b}-I)(Au_{p})\|^{2}+2\gamma\mu\left\langle
u_{p}-v,A^{*}(G_{b}-I)(Au_{p})\right\rangle$
$\displaystyle=\|u_{p}-v\|^{2}+\gamma^{2}\mu^{2}\left\langle(G_{b}-I)(Au_{p}),AA^{*}(G_{b}-I)(Au_{p})\right\rangle$
$\displaystyle\quad+2\gamma\mu\left\langle
u_{p}-v,A^{*}(G_{b}-I)(Au_{p})\right\rangle.$

Let $\Delta:=2\gamma\mu\langle u_{p}-v,A^{*}(G_{p}-I)(Au_{p})\rangle.$ Propery
(1) of Lemma 2.6 yields

$\displaystyle\Delta$ $\displaystyle=2\gamma\mu\langle
A(u_{p}-v),(G_{p}-I)(Au_{p})\rangle$ $\displaystyle=2\gamma\mu\langle
A(u_{p}-v)+(G_{p}-I)(Au_{p})-(G_{p}-I)(Au_{p}),(G_{p}-I)(Au_{p})\rangle$
$\displaystyle=2\gamma\mu\left(\langle
T(Au_{p})-A(v),(G_{p}-I)(Au_{p})\rangle-\|(G_{p}-I)(Au_{p})\|^{2}\right)$
$\displaystyle\leq
2\gamma\mu\left(\frac{1}{2}\|(G_{p}-I)(Au_{p})\|^{2}-\|(G_{p}-I)(Au_{p})\|^{2}\right)$
$\displaystyle\leq-\gamma\mu\|(G_{p}-I)(Au_{p})\|^{2}.$

From the definition of $\lambda,$ we have

$\displaystyle\gamma^{2}\mu^{2}\langle(G_{b}-I)(Au_{p}),AA^{*}(G_{b}-I)(Au_{p})\rangle$
$\displaystyle\leq\lambda\gamma^{2}\mu^{2}\langle(G_{b}-I)(Au_{p}),(G_{b}-I)(Au_{p})\rangle$
$\displaystyle=\lambda\gamma^{2}\mu^{2}\|(G_{b}-I)(Au_{p})\|^{2}.$

From 3.2, we obtain

$\|u_{p+1}-v\|^{2}\leq\|u_{p}-v\|^{2}-\gamma\mu(1-\lambda\gamma\mu)\|(G_{b}-I)(Au_{p})\|^{2}-t_{p}(1-t_{p})\|F_{a}(y_{p})-y_{p}\|^{2}.$

Thus, $\|u_{p+1}-v\|\leq\|u_{p}-v\|,$ that is, the sequence $\\{u_{p}\\}$ is
Fejér-monotone.

From the assumption that $\\{\alpha_{p}\\}\subset(\delta,1-\delta)$ and
$\gamma\in(0,\frac{1}{\lambda\mu}),$ we have

$\|u_{p+1}-v\|^{2}\leq\|u_{p}-v\|^{2}-\gamma\mu(1-\lambda\gamma\mu)\|(G_{b}-I)(Au_{p})\|^{2}-\delta^{2}\|F_{a}(y_{p})-y_{p}\|^{2}$
(3.3)

for any $v\in\Gamma.$ This implies that the sequence $\\{\|u_{p}-v\|\\}$ is
monotone decreasing, and thus converges to a positive number. Hence,

$\lim_{p\to\infty}\|(G_{b}-I)(Au_{p})\|=0\quad\text{and}\quad\lim_{p\to\infty}\|F_{a}(y_{p})-y_{p}\|=0.$
(3.4)

Fejér monotonicity of $\\{u_{p}\\}$ implies that the sequence $\\{u_{p}\\}$ is
bounded. Let $u^{*}$ be a weak-cluster point of $\\{u_{p}\\}.$ Demiclosedness
of $F_{a}-I$ and $G_{b}-I$ at zero, (3.4) and the weak convergence of a
subsequence $\\{u_{p_{j}}\\}$ of $\\{u_{p}\\}$ to $u^{*}$ gives

$F_{a}(u^{*})=u^{*}\quad\text{and}\quad G_{b}(Au^{*})=Au^{*}.$

Thus, $u^{*}\in\\{u\in Fix(F_{a}):Au\in Fix(G_{b})\\}$ and $Au^{*}\in
Fix(G_{b}).$ Lemma 2.4 implies that $\\{u_{p}\\}$ converges weakly to the
split common fixed point $u^{*}.$

By Lemma 2.3, $Fix(F_{a})=Fix(F),$ $Fix(G_{b})=Fix(G)$ and

$(1-t_{p})y_{p}+t_{p}F_{a}y_{p}=(1-at_{p})y_{p}+at_{p}Fy_{p}.$

Since $G_{b}-I=b(G-I),$ putting $\alpha_{p}=at_{p},$ shows that $\\{u_{p}\\}$
generated by $(\ref{Alg1})$ converges weakly to a split common fixed point
$u^{*}\in\Gamma.$ Note that since $0<a<1-\beta$ and
$\\{t_{n}\\}\subset(\delta,1-\delta),$ it follows that
$\\{\alpha_{p}\\}\subset(\delta,1-\delta).$ $\Box$

By considering a projection version of Algorithm (3.1), it is possible to
obtain a strong convergence result as shown by the following theorem.

###### Theorem 3.2

Let ${\cal H}_{1}$ and ${\cal H}_{2}$ be two real Hilbert spaces, $D$ a
nonempty closed convex subset of ${\cal H}_{1}$ and $E$ a nonempty closed
convex subset of ${\cal H}_{2}.$ Given a bounded linear operator $A:{\cal
H}_{1}\to{\cal H}_{2}$, let $F:D\to{\cal H}_{1},~{}G:E\to{\cal H}_{2}$ be two
$\beta$-demicontractive mappings with $Fix(F)\neq\emptyset$ and
$Fix(G)\neq\emptyset$ and both $F-I$ and $G-I$ are demiclosed at $0$. Let
$\\{u_{n}\\}$ be a sequence generated in the following manner:

$\begin{cases}u_{0}\in D,\quad D_{0}=D,&\\\ z_{p}=P_{D}(u_{p}+\lambda
A^{*}(b(G-I))Au_{p}),\quad b\in(0,1-\beta)&\\\
y_{p}=\alpha_{p}z_{p}+(1-\alpha_{p})Fz_{p},\quad\\{\alpha_{p}\\}\subset(0,\eta)\subset(0,1)&\\\
D_{p+1}=\\{u\in D_{p}:\|y_{p}-u\|\leq\|z_{p}-u\|\leq\|u_{p}-u\|\\},&\\\
u_{p+1}=P_{D_{p+1}}(u_{0}),\qquad\forall
p\in\mathbb{N}\cup\\{0\\},\end{cases}$ (3.5)

where $P$ is a projection operator and $A^{*}$ denotes the adjoint of $A,$
$\lambda\in(0,1/\|A^{*}\|).$ If $\Gamma=\\{u\in Fix(F):Au\in
Fix(G)\\}\neq\emptyset,$ then

$u_{p}\to u^{*}\in\Gamma~{}\text{and}~{}Au_{p}\to Au^{*}\in Fix(G).$

Proof. Since $F$ and $G$ are $\beta$-demicontractive mappings, it follows from
Lemma 2.2 that the averaged mappings

$\displaystyle F_{a}u=(1-a)u+aFu\quad\text{and}\quad G_{b}u=(1-b)u+bGu$

are quasi-nonexpansive for $\\{a,b\\}\subset(0,1-\beta).$ Clearly, $F_{a}-I$
and $G_{b}-I$ are also demiclosed at zero.

Now consider the sequence $\\{u_{p}\\}$ generated by (3.5) It is
straightforward to see that $D_{p}$ is closed and convex for any
$p\in\mathbb{N}\cup\\{0\\}.$ Let $v\in\hat{\Gamma}=\\{u\in Fix(F_{a}):Au\in
Fix(G_{b})\\}.$ We have

$\displaystyle 2\lambda$ $\displaystyle\langle
u_{p}-v,A^{*}(G_{b}Au_{p}-Au_{p})\rangle$ $\displaystyle=$
$\displaystyle~{}2\lambda\langle
A(u_{p}-v)+(G_{b}Au_{p}-Au_{p})-(G_{b}Au_{p}-Au_{p}),G_{b}Au_{p}-Au_{p}\rangle$
$\displaystyle=$ $\displaystyle~{}2\lambda(\langle
G_{b}Au_{p}-Av,G_{b}Au_{p}-Au_{p}\rangle-\|G_{b}Au_{p}-Au_{p}\|^{2}$
$\displaystyle=$
$\displaystyle~{}2\lambda\left(\frac{1}{2}\|G_{b}Au_{p}-Av\|^{2}+\frac{1}{2}\|G_{b}Au_{p}-Au_{p}\|^{2}\right.$
$\displaystyle\qquad\left.-\frac{1}{2}\|Au_{p}-Av\|^{2}-\|G_{b}Au_{p}-Au_{p}\|^{2}\right)$
$\displaystyle\leq$
$\displaystyle~{}2\lambda\left(\frac{1}{2}\|G_{b}Au_{p}-Au_{p}\|^{2}-\|G_{b}Au_{p}-Au_{p}\|^{2}\|\right)$
$\displaystyle=$ $\displaystyle-\lambda\|G_{b}Au_{p}-Au_{p}\|^{2}.$

Moreover,

$\displaystyle\|z_{p}-v\|^{2}$ $\displaystyle=\|P_{D}(u_{p}+\lambda
A^{*}(G_{b}Au_{p}-P_{D}(v)\|^{2}$ $\displaystyle\leq\|u_{p}+\lambda
A^{*}(G_{b}Au_{p}-Au_{p})-v\|^{2}$ $\displaystyle=$
$\displaystyle\|u_{p}-v\|^{2}+\|\lambda
A^{*}(G_{b}Au_{p}-Au_{p}\|^{2}+2\lambda\langle
u_{p}-v,A^{*}(G_{b}Au_{p}-Au_{p})\rangle$
$\displaystyle\leq\|u_{p}-v\|^{2}+\lambda^{2}\|A^{*}\|^{2}\|(G_{b}Au_{p}-Au_{p}\|^{2}-\lambda\|G_{b}Au_{p}-Au_{p}\|$
$\displaystyle=\|u_{p}-v\|^{2}+\lambda(1-\lambda\|A^{*}\|^{2})\|(G_{b}Au_{p}-Au_{p}\|^{2}\|$
(3.6)

Hence we have

$\|y_{p}-v\|\leq\|z_{p}-v\|\leq\|u_{p}-v\|$

This implies that $v\in D_{p}$ and $\hat{\Gamma}\subset D_{p}$ for each
$p\in\mathbb{N}\cup\\{0\\}.$ For all $p\in\mathbb{N}$ we have

$\|u_{p+1}-u_{0}\|\leq\|v-u_{0}\|.$

Thus, $\\{u_{p}\\}$ is bounded and $\\{\|u_{p}-u_{0}\|\\}$ is nondecreasing.
Therefore, $\\{u_{p}\\}$ is Cauchy. Let $u*=\lim_{p\to\infty}u_{p}.$ We obtain

$\displaystyle\|z_{p}-u_{p}\|$ $\displaystyle\leq 2\|u_{p+1}-u_{p}\|\to 0,$
$\displaystyle\|y_{p}-u_{p}\|$ $\displaystyle\leq 2\|u_{p+1}-u_{p}\|\to 0,$
$\displaystyle\|y_{p}-z_{p}\|$
$\displaystyle\leq\|y_{p}-u_{p}\|+\|u_{p}-z_{p}\|\to
0~{}\text{as}~{}p\to\infty.$

Since $\lambda(1-\lambda\|A^{*}\|^{2})>0,$ from (3.6) we obtain

$\|G_{b}Au_{p}-Au_{p}\|^{2}\leq\frac{1}{\lambda(1-\lambda\|A^{*}\|^{2})}\|u_{p}-z_{p}\|\\{\|u_{p}-v\|+\|z_{n}-v\|\\}\to
0,$

which yields $Au^{*}\in Fix(G_{b}).$ Hence, $u^{*}\in\hat{\Gamma}$ and
$\\{u_{p}\\}$ generated by (3.5) converges strongly to $u^{*}.$ By Lemma 2.3,
for any $a,b\in(0,1),$ we have $Fix(F_{a})=Fix(F)$ and $Fix(G_{b})=Fix(G).$
Let $k_{p}=1-t_{p},$ we have

$t_{p}z_{p}+(1-t_{p})F_{a}z_{p}=(1-k_{p})z_{p}+k_{p}F_{a}z_{p}=(1-ak_{p})z_{p}+ak_{p}Fz_{p}.$

Since $G_{b}-I=b(G-I),$ putting $\alpha_{p}=1-ak_{p}$ shows that for the
sequence $\\{u_{p}\\}$ generated by (3.5), we have

$u_{p}\to u^{*}\in\Gamma~{}\text{and}~{}Au_{p}\to Au^{*}\in Fix(G).$

Note that since $0<a,b<1-\beta$ and $\\{t_{p}\\}\subset(0,\eta)\subset(0,1),$
it follows that

$\\{\alpha_{p}\\}\subset(0,\eta)\subset(0,1).$

$\Box$

In the following theorem, for $i=1,\cdots,n$ and $j=1,\cdots,k$ we denote

$\hat{F}_{i}=(1-a_{i})I+a_{i}F_{i}\text{ and
}\hat{G}_{j}=(1-b_{j})I+b_{j}G_{j},\quad a_{i},b_{j}\in(0,1).$

###### Theorem 3.3

Let ${\cal H}_{1}$ and ${\cal H}_{2}$ be two real Hilbert spaces, $D$ a
nonempty closed convex subset of ${\cal H}_{1}$ and $E$ a nonempty closed
convex subset of ${\cal H}_{2}.$ Given a bounded linear operator $A:{\cal
H}_{1}\to{\cal H}_{2}$, let $F_{1},\cdots,F_{n}:D\to{\cal H}_{1}$ be
$\beta$-demicontractive mappings with
$\bigcap_{i=1}^{n}Fix(F_{i})\neq\emptyset,$ and let $G_{1},\cdots,G_{k}:{\cal
H}_{2}\to{\cal H}_{2}$ be $\beta$-demicontractive mappings with
$\bigcap_{i=1}^{k}Fix(G_{i})\neq\emptyset.$ Suppose

$(F_{i}-I),~{}~{}(i=1,\cdots,n)\text{ and }(G_{i}-I),~{}~{}(i=1,\cdots,k)$

are demiclosed at zero. Let $\\{u_{p}\\}$ be a sequence generated in the
following manner:

$\begin{cases}u_{0}\in D,\quad D_{0}=D,&\\\ z_{p}=P_{D}\left(u_{p}+\lambda
A^{*}\left(\sum_{j=1}^{k}d_{j}{G}_{\varphi_{j}}-I\right)Au_{p}\right),\quad
d_{j},\varphi_{j}\in(0,1),\quad\sum_{j=1}^{k}d_{j}=1,&\\\
y_{p}=\alpha_{p}z_{p}+(1-\alpha_{p})\sum_{i=1}^{n}c_{i}{F}_{\vartheta_{i}}z_{p},\quad
c_{i},\vartheta_{i}\in(0,1),\quad\sum_{i=1}^{n}c_{i}=1,&\\\
D_{p+1}=\left\\{u\in
D_{p}:\|y_{p}-u\|\leq\|z_{p}-u\|\leq\|u_{p}-u\|\right\\},&\\\
u_{p+1}=P_{D_{p+1}}(u_{0}),\qquad\forall
p\in\mathbb{N}\cup\\{0\\},\end{cases}$ (3.7)

where $P$ is a projection operator and $A^{*}$ denotes the adjoint of $A,$

$\lambda\in\left(0,\frac{1}{\|A^{*}\|^{2}}\right)\text{ and
}\\{\alpha_{p}\\}\subset(0,\eta)\subset(0,1).$

For $i=1,\cdots,n,$
$F_{\vartheta_{i}}:=(1-\vartheta_{i})I+\vartheta_{i}\hat{F}_{i}$ and for
$j=1,\cdots,k,$ let $G_{\varphi_{j}}=(1-\varphi_{j})I+\varphi_{j}\hat{G}_{j},$
and

$\Omega=\left\\{u\in\bigcap_{i=1}^{n}Fix(F_{i}):Au\in\bigcap_{j=1}^{k}Fix(G_{j})\right\\}.$

Then the sequence $\\{u_{p}\\}$ converges strongly to $u^{*}\in\Omega.$

Proof. Let
$T_{1}=\sum_{i=1}^{n}c_{i}{F}_{\vartheta_{i}},T_{2}=\sum_{j=1}^{k}d_{j}G_{\varphi_{j}}.$
Lemma 2.5 implies that $T_{1}$ and $T_{2}$ are quasi-nonexpansive.
Furthermore,

$Fix(T_{1})=\bigcap_{i=1}^{n}Fix(F_{i})\neq\emptyset\text{ and
}Fix(T_{2})=\bigcap_{j=1}^{k}Fix(G_{j})\neq\emptyset.$

It is straightforward to verify that $T_{1}-I$ and $T_{2}-I$ are demiclosed at
the origin. The rest of the proof is the same as the proof of Theorem 3.2.
$\Box$

As a direct consequence of Theorem 3.2 we obtain strong convergence of the
iteration to a common fixed point of $F$ and $G,$ considering ${\cal
H}_{1}={\cal H}_{2}={\cal H}$ and $A$ as an identity mapping.

###### Corollary 3.1

Let ${\cal H}$ be a real Hilbert space, $D$ a nonempty closed convex subset of
${\cal H}.$ Let $F:D\to{\cal H}$ and $G:{\cal H}\to{\cal H}$ be two
$\beta$-demicontractive mappings with $Fix(F)\cap Fix(G)\neq\emptyset.$ Assume
that $F-I$ and $G-I$ are demiclosed at zero. Let $\\{u_{p}\\}$ be a sequence
defined as follows:

$\begin{cases}u_{0}\in D,\quad D_{0}=D,&\\\
z_{p}=P_{D}((1-\lambda)u_{p}+\lambda((1-a)I+G)u_{p}),\quad
a\in(0,1-\beta),~{}\lambda\in(0,1),&\\\
y_{p}=\alpha_{p}z_{p}+(1-\alpha_{p})Fz_{p},\quad\\{\alpha_{p}\\}\subset(0,\eta)\subset(0,1),&\\\
D_{p+1}=\\{u\in D_{p}:\|y_{p}-u\|\leq\|z_{p}-u\|\leq\|u_{p}-u\|\\},&\\\
u_{p+1}=P_{D_{p+1}}(u_{0}),\qquad\forall
p\in\mathbb{N}\cup\\{0\\},\end{cases}$

where $P$ is a projection operator. Then $u_{p}\to u^{*}\in Fix(F)\cap
Fix(G).$

The next theorem is a generalization of Theorem 3.2 in which the operators $F$
and $G$ are demicontractive with different constants. Its proof is similar to
that of Theorem 3.2.

###### Theorem 3.4

Let ${\cal H}_{1}$ and ${\cal H}_{2}$ be two real Hilbert spaces, $D$ a
nonempty closed convex subset of ${\cal H}_{1}$ and $E$ a nonempty closed
convex subset of ${\cal H}_{2}.$ Given a bounded linear operator $A:{\cal
H}_{1}\to{\cal H}_{2}$, let $F:D\to{\cal H}_{1}$ be an
$\alpha$-demicontractive mapping and $G:E\to{\cal H}_{2}$ be a
$\beta$-demicontractive mapping and both $F-I$ and $G-I$ are demiclosed at
$0$. Let $\\{u_{n}\\}$ be a sequence generated in the following manner:

$\begin{cases}u_{0}\in D,\quad D_{0}=D,&\\\ z_{p}=P_{D}(u_{p}+\lambda
A^{*}(b(G-I))Au_{p}),\quad b\in(0,1-\beta)&\\\
y_{p}=\alpha_{p}z_{p}+(1-\alpha_{p})Fz_{p},\quad\\{\alpha_{p}\\}\subset(0,\eta)\subset(0,1)&\\\
D_{p+1}=\\{u\in D_{p}:\|y_{p}-u\|\leq\|z_{p}-u\|\leq\|u_{p}-u\|\\},&\\\
u_{p+1}=P_{D_{p+1}}(u_{0}),\qquad\forall
p\in\mathbb{N}\cup\\{0\\},\end{cases}$ (3.8)

where $P$ is a projection operator and $A^{*}$ denotes the adjoint of $A,$
$\lambda\in(0,1/\|A^{*}\|).$ If $\Gamma=\\{u\in Fix(F):Au\in
Fix(G)\\}\neq\emptyset,$ then

$u_{p}\to u^{*}\in\Gamma~{}\text{and}~{}Au_{p}\to Au^{*}\in Fix(G).$

##  4 Conclusions

1\. We have proven a weak convergence theorem for an iteration scheme used to
approximate split common fixed point of two demicontractive mappings in
Hilbert spaces derived from an associated weak convergence theorem in the
class of quasi-nonexpansive operators.

2\. We also have shown a strong convergence theorem for an iteration scheme
used to approximate split common fixed point of two demicontractive mappings
in Hilbert spaces derived from a corresponding strong convergence theorem in
the class of quasi-nonexpansive operators.

3\. Our investigation is based on an embedding technique by means of an
average mappings: if $F$ is $\beta$-demicontractive, then for any
$a\in(0,1-\beta),$ $F_{a}=(1-a)I+aF$ is quasi-nonexpansive.

4\. For other related works that allow similar developments, we refer to
Kingkam and Nantadilok [4], Padcharoen et al. [8], Sharma and Chandok [9], Shi
et al. [10], Tiammee and Tiamme [11],…

## Acknowledgements

The first draft of this paper was carried out during the first author’s short
visit (December 2023) at the Department of Mathematics, King Fahd University
of Petroleum and Minerals, Dhahran, Saudi Arabia. He is grateful to Professor
Monther Alfuraidan, the Chairman of Department of Mathematics, for the
invitation and for providing excellent facilities during his visit.

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