A shrinking projection approximant for the split equilibrium problems and
fixed point problems in Hilbert spaces

Abdul Ghaffar1, Zafar Ullah${}^{2},$ Muhammad Aqeel Ahmad Khan3,∗ and Faisal
Mumtaz3

1Department of Mathematical Sciences, Balochistan University of Information
Technology, Engineering and Management Sciences, Quetta 87300, Pakistan

2Department of Mathematics, University of Education Lahore, DG Khan Campus,

DG Khan 32200, Pakistan

3Department of Mathematics, COMSATS Institute of Information Technology
Lahore,

Lahore, 54000, Pakistan

March 06, 2018 ††*Corresponding author
E-mail addresses: (A. Ghaffar<EMAIL_ADDRESS>(Z. Ullah)
<EMAIL_ADDRESS>(M.A.A. Khan<EMAIL_ADDRESS><EMAIL_ADDRESS>(F. Mumtaz<EMAIL_ADDRESS>
Abstract: This work is devoted to establish the strong convergence results of
an iterative algorithm generated by the shrinking projection method in Hilbert
spaces. The proposed approximation sequence is used to find a common element
in the set of solutions of a finite family of split equilibrium problems and
the set of common fixed points of a finite family of total asymptotically
strict pseudo contractions in such setting. The results presented in this
paper improve and extend some recent corresponding results in the literature.
Keywords and Phrases: Split equilibrium problem; fixed point problem; total
asymptotically strict pseudo contraction; inverse strongly monotone mapping;
shrinking projection method; Hilbert space
2010 Mathematics Subject Classification: Primary: 47H05; 47H09; 47J05;
Secondary: 49H05.

##  1\. Introduction and Preliminaries

Throughout this paper, we write $x_{n}\rightarrow x~{}($resp.
$x_{n}\rightharpoonup x)$ to indicate the strong convergence (resp. the weak
convergence) of a sequence $\\{x_{n}\\}_{n=1}^{\infty}$. Let $C$ be a nonempty
subset of a real Hilbert space $H$ and let $T:C\rightarrow C$ be a mapping.
The set of fixed points of the mapping $T$ is defined and denoted:
$F(T)=\\{x\in C:T(x)=x\\}.$ A self-mapping $T$ is said to be: (i) nonexpansive
if $\left\|Tx-Ty\right\|\leq\left\|x-y\right\|;$ (ii) asymptotically
nonexpansive [15] if there exists a sequence
$\\{\lambda_{n}\\}\subset[0,\infty)$ with
$\lim_{n\rightarrow\infty}\lambda_{n}=0$ such that
$\left\|T^{n}x-T^{n}y\right\|\leq(1+\lambda_{n})\left\|x-y\right\|,$ $n\geq
1;$ (iii) Lipschitzian if
$\left\|T^{n}x-T^{n}y\right\|\leq\Theta\left\|x-y\right\|$ for some
$\Theta>0;$ (iv) firmly nonexpansive if

$\left\|Tx-
Ty\right\|^{2}+\left\|(I-T)x-(I-T)y\right\|^{2}\leq\left\|x-y\right\|^{2};$
(1.1)

(v) pseudo-contraction [4], if

$\left\|Tx-
Ty\right\|^{2}\leq\left\|x-y\right\|^{2}+\left\|(I-T)x-(I-T)y\right\|^{2};$
(1.2)

(vi) $k$-strict pseudo contraction [4], if there exists $k\in[0,1)$ such that

$\|Tx-Ty\|^{2}\leq\|x-y\|^{2}+k\|(I-T)x-(I-T)y\|^{2};$ (1.3)

(vii). $(k,\\{\lambda_{n}\\})$-asymptotically strict pseudo contraction [22],
if there exist a constant $k\in[0,1)$ and a sequence
$\\{\lambda_{n}\\}\subset[1,\infty)$ with
$\lim_{n\rightarrow\infty}\lambda_{n}=1$ such that

$\|T^{n}x-T^{n}y\|^{2}\leq\lambda_{n}\|x-y\|^{2}+k\|(I-T^{n})x-(I-T^{n})y)\|^{2};$
(1.4)

(viii). $(\\{\lambda_{n}\\},\\{\mu_{n}\\},\xi)$-total asymptotically
nonexpansive [1] if there exist nonnegative real sequences
$\left\\{\lambda_{n}\right\\}_{n=1}^{\infty}$,
$\left\\{\mu_{n}\right\\}_{n=1}^{\infty}$ with
$\lim_{n\rightarrow\infty}\lambda_{n}=0=\lim_{n\rightarrow\infty}\mu_{n}$ and
a strictly increasing continuous function
$\xi:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ with $\xi(0)=0$ such that

$\|T^{n}x-T^{n}y\|\leq\|x-y\|+\lambda_{n}\xi(\|x-y\|)+\mu_{n};$ (1.5)

(ix). $(k,\\{\lambda_{n}\\},\\{\mu_{n}\\},\xi)$-total asymptotically strictly
pseudo contraction [27], if there exist a constant $k\in[0,1)$ and nonnegative
real sequences $\left\\{\lambda_{n}\right\\}_{n=1}^{\infty}$,
$\left\\{\mu_{n}\right\\}_{n=1}^{\infty}$ with
$\lim_{n\rightarrow\infty}\lambda_{n}=0=\lim_{n\rightarrow\infty}\mu_{n}$ and
a strictly increasing continuous function
$\xi:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ with $\xi(0)=0$ such that

$\|T^{n}x-T^{n}y\|^{2}\leq\|x-y\|^{2}+k\|(I-T^{n})x-(I-T^{n})y)\|^{2}+\lambda_{n}\xi(\|x-y\|)+\mu_{n},$
(1.6)

holds for all $x,y\in C.$
Remark 1.1. It is worth mentioning that the class of nonexpansive mappings
have powerful applications to solve various problems arising in the field of
applied mathematics, such as variational inequality problem, convex
minimization, zeros of a monotone operator, initial value problems of
differential equations, game-theoretic model and image recovery. It is
therefore, natural to extend such powerful results of the class of
nonexpansive mappings to the more general class of mappings. As a consequence,
the notion of nonexpansive mapping has been generalized in several ways. In
1967, Browder and Petryshyn [4] introduced the concept of strict pseudo
contraction as a generalization of nonexpansive mappings. Later on, Alber et
al.[1] introduced the notion of total asymptotically nonexpansive mappings
which is more general in nature and unifies various definitions of mappings
associated with the class of asymptotically nonexpansive mappings. In 2011,
Yang et al. [27] introduced the notion of total asymptotically strict pseudo
contraction which contains properly the class of total asymptotically
nonexpansive mappings and strict pseudo contractions. So, we study this
general class of mappings to contribute in metric fixed point theory.

Let $C$ be a nonempty subset of a real Hilbert space $H_{1}$, $Q$ be a
nonempty subset of a real Hilbert space $H_{2}$ and let $A:H_{1}\rightarrow
H_{2}$ be a bounded linear operator. Let $f:C\times C\rightarrow\mathbb{R}$
and $g:Q\times Q\rightarrow\mathbb{R}$ be two bifunctions. The split
equilibrium problem (SEP) is to find:

$x^{\ast}\in C\text{ \ \ \ such that }f\left(x^{\ast},x\right)\geq 0\text{ for
all }x\in C,$ (1.7)

and

$y^{\ast}=Ax^{\ast}\in Q\text{ \ \ \ such that }g\left(y^{\ast},y\right)\geq
0\text{ for all }y\in Q.$ (1.8)

It is remarked that inequality (1.7) represents the classical equilibrium
problem [12] and its solution set is denoted $EP(f).$ Moreover, inequalities
(1.7) and (1.8) constitute a pair of equilibrium problems which aim to find a
solution $x^{\ast}$ of an equilibrium problem (1.7) such that its image
$y^{\ast}=Ax^{\ast}$ under a given bounded linear operator $A$ also solves
another equilibrium problem (1.8). The set of solutions of SEP (1.7) and (1.8)
is denoted $\Omega=\\{z\in EP(f):Az\in EP(g)\\}.$

Equilibrium problem theory provides a unified approach to address a variety of
mathematical problems arising in various disciplines. In 2012, Censor et al.
[9] proposed the theory of split variational inequality problems (SVIP)
whereas Moudafi [21] generalized the concept of SVIP to that of split monotone
variational inclusions (SMVIP). The split equilibrium problems is a special
case of SMVIP. The SMVIP have already been studied and successfully employed
as a model in intensity-modulated radiation therapy treatment planning, see
[7, 8]. Moreover, this formalism is also at the core of modeling of many
inverse problems arising for phase retrieval and other real-world problems;
for instance, in sensor networks in computerized tomography and data
compression; see, for example, [6, 11]. Some methods have been proposed and
analyzed to solve SEP together with the fixed point problem in Hilbert spaces,
see, for example [16, 17, 18, 24] and the references cited therein.

In 2013, Chang et al. [10] studied the split feasibility problem for a total
asymptotically strict pseudo contraction in infinitely dimensional Hilbert
spaces. In 2015, Ma and Wang [19] established strong convergence results for
the split common fixed point problem of total asymptotically strict pseudo
contractions in Hilbert spaces. Quite recently, some methods have been
proposed and analyzed in [17, 18] for the split equilibrium problem. Inspired
and motivated by the above mentioned results and the ongoing research in this
direction, we aim to employ a hybrid shrinking projection algorithm to find a
common element in the set of solutions of a finite family of split equilibrium
problems and the set of common fixed points of a finite family of total
asymptotically strict pseudo contractions in Hilbert spaces. Our results can
be viewed as a generalization and improvement of various existing results in
the current literature.

##  2\. Preliminaries

This section is devoted to recall some definitions and results required in the
sequel.

Let $C$ be a nonempty closed convex subset of a Hilbert space $H_{1}.$ For
each $x\in H_{1}$, there exists a unique nearest point of $C,$ denoted by
$P_{C}x,$ such that

$\left\|x-P_{C}x\right\|\leq\left\|x-y\right\|\text{ for all }y\in C.$

Such a mapping $P_{C}:H_{1}\rightarrow C$ is known as a metric projection or a
nearest point projection of $H_{1}$ onto $C.$ Moreover, $P_{C}$ satisfies
nonexpansiveness in a Hilbert space and $\left\langle
x-P_{C}x,P_{C}x-y\right\rangle\geq 0$ for all $x,y\in C.$ It is remarked that
$P_{C}$ is firmly nonexpansive mapping from $H_{1}$ onto $C,$ that is,

$\left\|P_{C}x-P_{C}y\right\|^{2}\leq\left\langle
x-y,P_{C}x-P_{C}y~{}\right\rangle,\text{ for all }x,y\in C.$

Recall that a nonlinear mapping $A:C\rightarrow H_{1}$ is $\lambda$-inverse
strongly monotone if it satisfies

$\left\langle x-y~{},Ax-Ay\right\rangle\geq\lambda\left\|Ax-Ay\right\|^{2}.$

Note that, if $A:=I-T$ is a $\lambda$-inverse strongly monotone mapping, then:
(i): $A$ is a $\left(\frac{1}{\lambda}\right)$-Lipschitz continuous mapping;
(ii): if $T$ is a nonexpansive mapping, then $A$ is a
$\left(\frac{1}{2}\right)$-inverse strongly monotone mapping;
(iii): if $\eta\in(0,2\lambda],$ then $I-\eta A$ is a nonexpansive mapping.

The following lemma collects some well-known equations in the context of a
real Hilbert space.
Lemma 2.1. Let $H_{1}$ be a real Hilbert space, then:
(i):
$\left\|x-y\right\|^{2}=\left\|x\right\|^{2}-\left\|y\right\|^{2}-2\left\langle
x-y,y\right\rangle,$ for all $x,y\in H_{1};$
(ii): $\left\|x+y\right\|^{2}\leq\left\|x\right\|^{2}+2\left\langle
x-y,y\right\rangle,$ for all $x,y\in H_{1};$
(iii): $\left\|\alpha
x+(1-\alpha)y\right\|^{2}=\alpha\left\|x\right\|^{2}+(1-\alpha)\left\|y\right\|^{2}-\alpha(1-\alpha)\left\|x-y\right\|^{2}$
for all $x,y\in H_{1}$ and $\alpha\in[0,1]$.
Lemma 2.2 [20]. Let $T:C\rightarrow C$ be a
$(k,\\{\lambda_{n}\\},\\{\mu_{n}\\},\xi)$-total asymptotically strictly pseudo
contraction. If $F(T)\neq\emptyset,$ then for each $p\in F(T)$ and for each
$x\in C,$ the following equivalent inequalities hold:

$\left\|T^{n}x-p\right\|^{2}\leq\left\|x-p\right\|^{2}+k\left\|x-T^{n}x\right\|^{2}+\lambda_{n}\xi(\|x-p\|)+\mu_{n},\newline
$ (2.1) $\left\langle
x-T^{n}x,x-p\right\rangle\geq\frac{1-k}{2}\left\|x-T^{n}x\right\|^{2}-\frac{\lambda_{n}}{2}\xi(\|x-p\|)-\frac{\mu_{n}}{2}\newline
,$ (2.2) $\left\langle
x-T^{n}x,p-T^{n}x\right\rangle\leq\frac{1+k}{2}\left\|x-T^{n}x\right\|^{2}+\frac{\lambda_{n}}{2}\xi(\|x-p\|)+\frac{\mu_{n}}{2}\newline
.$ (2.3)

Lemma 2.3 [20]. Let $C$ be a nonempty subset of a real Hilbert space $H_{1}$
and let $S:C\rightarrow C$ be a uniformly $\Theta$-Lipschitzian and
$(k,\\{\lambda_{n}\\},\\{\mu_{n}\\},\xi)$-total asymptotically strictly pseudo
contraction, then $S$ is demiclosed at origin. That is, if for any sequence
$\\{x_{n}\\}$ in $C$ with $x_{n}\rightharpoonup x$ and
$\left\|x_{n}-Sx_{n}\right\|\rightarrow 0,$ we have $x=Sx.$
Condition 2.4 [3, 12]. Let $f:C\times C\rightarrow\mathbb{R}$ be a bifunction
satisfying the following conditions:
1\. $f(x,x)=0\ $for all$\ x\in C;$
2\. $f$ is monotone, that is, $f(x,y)+f(y,x)\leq 0$ for all$\ x,y\in C;$
3\. $f$ is upper hemicontinuous, that is, for each $x,y,z\in C,$

$\lim_{t\rightarrow 0}f(tz+(1-t)x,y)\leq f(x,y);$

4\. for each $x\in C,$ the function $y\mapsto f(x,y)$ is convex and lower
semi-continuous.
Lemma 2.5 [12]. Let $C$ be a closed convex subset of a real Hilbert space
$H_{1}$ and let $f:C\times C\rightarrow\mathbb{R}$ be a bifunction satisfying
Lemma 2.4. For $r>0$ and $x\in H_{1},$ there exists $z\in C$ such that

$F(z,y)+\frac{1}{r}\langle y-z,z-x\rangle\geq 0,\ \text{for all }y\in C.$

Moreover, define a mapping $T_{r}^{F}:H_{1}\rightarrow C$ by

$T_{r}^{f}(x)=\left\\{z\in C:f(z,y)+\frac{1}{r}\langle y-z,z-x\rangle\geq 0,\
\ \text{for all }y\in C\right\\},$

for all $x\in H_{1}$. Then, the following hold:
(i) $T_{r}^{f}$ is single-valued;
(ii) $T_{r}^{f}$ is firmly nonexpansive, i.e., for every $x,y\in H,$

$\left\|T_{r}^{f}x-T_{r}^{f}y\right\|^{2}\leq\left\langle
T_{r}^{f}x-T_{r}^{f}y,x-y\right\rangle$

(ii) $F(T_{r}^{f})=EP(f);$
(iv) $EP(f)$ is closed and convex.

It is remarked that if $g:Q\times Q\rightarrow\mathbb{R}$ is a bifunction
satisfying Lemma 2.4, then for $s>0$ and $w\in H_{2}$ we can define a mapping:

$T_{s}^{g}(w)=\left\\{d\in C:g(d,e)+\frac{1}{s}\langle e-d,d-w\rangle\geq 0,\
\ \text{for all }e\in Q\right\\},$

which is, nonempty, single-valued and firmly nonexpansive. Moreover, $EP(g)$
is closed and convex, and $F(T_{s}^{g})=EP(g)$.

##  3\. Main results

We now prove our main result of this section.
Theorem 3.1. Let $H_{1}$ and $H_{2}$ be two real Hilbert spaces and let
$C\subseteq H_{1}$ and $Q\subseteq H_{2}$ be nonempty closed convex subsets of
Hilbert spaces $H_{1}$ and $H_{2}$, respectively. Let $f_{i}:C\times
C\rightarrow\mathbb{R}$ and $g_{i}:Q\times Q\rightarrow\mathbb{R}$ be two
finite families of bifunctions satisfying Condition 2.4 such that $g_{i}$ be
upper semicontinuous for each $i\in\\{1,2,3,\cdots,N\\}$. Let
$S_{i}:C\rightarrow C$ be a finite family of uniformly $\Theta$-Lipschitzian
and continuous total asymptotically strict pseudo contractions and let
$A_{i}:H_{1}\rightarrow H_{2}$ be a finite family of bounded linear operators
for each $i\in\\{1,2,3,\cdots,N\\}$. Suppose that
$\mathbb{F}:=\left[\bigcap_{i=1}^{N}F(S_{i})\right]\cap\Omega\neq\emptyset$,
where $\Omega=\left\\{z\in C:z\in\bigcap_{i=1}^{N}EP(f_{i})\text{ and
}A_{i}z\in EP(g_{i})\text{ for }1\leq i\leq N\right\\}.$ Let $\\{x_{n}\\}$ be
a sequence generated by:

$\begin{array}[]{l}x_{1}\in C_{1}=C,\\\
u_{n}=T_{r_{n}}^{f_{n}}\left(x_{n}-\gamma
A_{n(\mathop{\mathrm{m}od}N)}^{\ast}\left(I-T_{s_{n}}^{g_{n}}\right)A_{n(\mathop{\mathrm{m}od}N)}x_{n}\right),\\\
y_{n}=\alpha_{n}u_{n}+\left(1-\alpha_{n}\right)S_{n(\mathop{\mathrm{m}od}N)}^{n}u_{n},\\\
C_{n+1}=\left\\{{z\in
H}_{1}{:}\left\|{y_{n}-z}\right\|{{}^{2}\leq\left\|{x_{n}-z}\right\|^{2}+\theta_{n}}\right\\},\\\
x_{n+1}=P_{C_{n+1}}x_{1},\ \ n\geq 1,\end{array}$ (3.1)

where
$\theta_{n}=(1-\alpha_{n})\left\\{\lambda_{n}\xi_{n}(M_{n})+\lambda_{n}M_{n}^{\ast}D_{n}+\mu_{n}\right\\}$
with $D_{n}=\sup\left\\{\left\|x_{n}-p\right\|:p\ \in\mathbb{F}\right\\}$. Let
$\\{r_{n}\\},\\{s_{n}\\}$ be two positive real sequences and let
$\\{\alpha_{n}\\}$ be in $(0,1).$ Assume that if the following set of
conditions holds:
(C1): $0\leq k<a\leq\alpha_{n}\leq b<1$ and
$\gamma\in\left(0,\frac{1}{L}\right)$ where
$L=\max\left\\{L_{1},L_{2},\cdots,L_{N}\right\\}$ and $L_{i}$ is the spectral
radius of the operator $A_{i}^{\ast}A_{i}$ and $A_{i}^{\ast}$ is the adjoint
of $A_{i}$ for each $i\in\\{1,2,3,\cdots,N\\};$
(C2): $\liminf\limits_{n\rightarrow\infty}r_{n}>0$ and
$\liminf\limits_{n\rightarrow\infty}s_{n}>0;$
(C3): $\sum\limits_{n=1}^{\infty}\lambda_{n}<\infty$ and$\
\sum\limits_{n=1}^{\infty}\mu_{n}<\infty;$
(C4): there exist constants $M_{i},\ M_{i}^{\ast}>0$ such that
$\xi_{i}\left(\lambda_{i}\right)\leq M_{i}^{\ast}\lambda_{i}$ for all
$\lambda_{i}\geq M_{i},i=1,2,3,\cdots,N,$ then the sequence $\\{x_{n}\\}$
generated by (3.1) converges strongly to $P_{\mathbb{F}}x_{1}.$
Proof. For the sake of simplicity, we define
$A_{n}=A_{n(\mathop{\mathrm{m}od}N)}$ and
$S_{n}=S_{n(\mathop{\mathrm{m}od}N)}$ for all $n\geq 1.$ We start our proof to
establish that the sequence $\\{x_{n}\\}$ defined in (3.1) is well defined. In
order to prove this assertion, we first show by mathematical induction that
$\mathbb{F}\subset C_{n}$ for all $n\geq 1.$ Obviously, $\mathbb{F}\subset
C_{1}=C$. Now, assume that $\mathbb{F}\subset C_{i}$ for some $i\geq 1.$ Then
it follows from (3.1) that

$\displaystyle\left\|u_{i}-p\right\|^{2}$ $\displaystyle=$
$\displaystyle\left\|T_{r_{i}}^{f_{i}}\left(x_{i}-\gamma
A_{i}^{\ast}\left(I-T_{s_{i}}^{g_{i}}\right)A_{i}x_{i}\right)-T_{r_{i}}^{f_{i}}p\right\|^{2}$
(3.2) $\displaystyle\leq$ $\displaystyle\left\|x_{i}-\gamma
A_{i}^{\ast}\left(I-T_{s_{i}}^{g_{i}}\right)A_{i}x_{i}-p\right\|^{2}$
$\displaystyle\leq$
$\displaystyle\left\|x_{i}-p\right\|^{2}+\gamma^{2}\left\|A_{i}^{\ast}\left(I-T_{s_{i}}^{g_{i}}\right)A_{i}x_{i}\right\|^{2}+2\gamma\left\langle
p-x_{i},A_{i}^{\ast}\left(I-T_{s_{i}}^{g_{i}}\right)A_{i}x_{i}\right\rangle$
$\displaystyle\leq$
$\displaystyle\left\|x_{i}-p\right\|^{2}+\gamma^{2}\left\langle
A_{i}x_{i}-T_{s_{i}}^{g_{i}}A_{i}x_{i},A_{i}A_{i}^{\ast}\left(I-T_{s_{i}}^{g_{i}}\right)A_{i}x_{i}\right\rangle$
$\displaystyle+2\gamma\left\langle
p-x_{i},A_{i}^{\ast}\left(I-T_{s_{i}}^{g_{i}}\right)A_{i}x_{i}\right\rangle$
$\displaystyle\leq$
$\displaystyle\left\|x_{i}-p\right\|^{2}+L\gamma^{2}\left\langle
A_{i}x_{i}-T_{s_{i}}^{g_{i}}A_{i}x_{i},A_{i}x_{i}-T_{s_{i}}^{g_{i}}A_{i}x_{i}\right\rangle$
$\displaystyle+2\gamma\left\langle
p-x_{i},A_{i}^{\ast}\left(I-T_{s_{i}}^{g_{i}}\right)A_{i}x_{i}\right\rangle$
$\displaystyle=$
$\displaystyle\left\|x_{i}-p\right\|^{2}+L\gamma^{2}\left\|A_{i}x_{i}-T_{s_{i}}^{g_{i}}A_{i}x_{i}\right\|^{2}+2\gamma\left\langle
p-x_{i},A_{i}^{\ast}\left(I-T_{s_{i}}^{g_{i}}\right)A_{i}x_{i}\right\rangle.$

Now letting $\Lambda=2\gamma\left\langle
p-x_{i},A_{i}^{\ast}\left(I-T_{s_{i}}^{g_{i}}\right)A_{i}x_{n}\right\rangle,$
we have

$\displaystyle\Lambda$ $\displaystyle=$ $\displaystyle 2\gamma\left\langle
p-x_{i},A_{i}^{\ast}\left(I-T_{s_{i}}^{g_{i}}\right)A_{i}x_{i}\right\rangle$
$\displaystyle=$ $\displaystyle 2\gamma\left\langle
A_{i}\left(p-x_{i}\right),A_{i}x_{i}-T_{s_{i}}^{g_{i}}A_{i}x_{i}\right\rangle$
$\displaystyle=$ $\displaystyle 2\gamma\left\langle
A_{i}\left(p-x_{i}\right)+\left(A_{i}x_{i}-T_{s_{i}}^{g_{i}}A_{i}x_{i}\right)-\left(A_{i}x_{i}-T_{s_{i}}^{g_{i}}A_{i}x_{i}\right),A_{i}x_{i}-T_{s_{i}}^{g_{i}}A_{i}x_{i}\right\rangle$
$\displaystyle=$ $\displaystyle 2\gamma\left\\{\left\langle
A_{i}p-T_{s_{i}}^{g_{i}}A_{i}x_{i},A_{i}x_{i}-T_{s_{i}}^{g_{i}}A_{i}x_{i}\right\rangle-\left\|A_{i}x_{i}-T_{s_{i}}^{g_{i}}A_{i}x_{i}\right\|^{2}\right\\}$
$\displaystyle\leq$ $\displaystyle
2\gamma\left\\{\frac{1}{2}\left\|A_{i}x_{i}-T_{s_{i}}^{g_{i}}A_{i}x_{i}\right\|^{2}-\left\|A_{i}x_{i}-T_{s_{i}}^{g_{i}}A_{i}x_{i}\right\|^{2}\right\\}$
$\displaystyle=$
$\displaystyle-\gamma\left\|A_{i}x_{i}-T_{s_{i}}^{g_{i}}A_{i}x_{i}\right\|^{2}.$

Using the above simplification of $\Lambda$ in (3.2), we get

$\left\|u_{i}-p\right\|^{2}\leq\left\|x_{i}-p\right\|^{2}+\gamma\left(L\gamma-1\right)\left\|A_{i}x_{i}-T_{s_{n,i}}^{g_{i}}A_{i}x_{i}\right\|^{2}.$
(3.3)

Since $\gamma\in\left(0,\frac{1}{L}\right)$ by condition (C1), the above
estimate then yields

$\left\|u_{i}-p\right\|^{2}\leq\left\|x_{i}-p\right\|^{2}.$ (3.4)

Making use of (3.4), we have the following estimate:

$\displaystyle\left\|y_{i}-p\right\|^{2}$ $\displaystyle=$
$\displaystyle\left\|\alpha_{i}u_{i}+\left(1-\alpha_{i}\right)S_{i}^{i}u_{i}-p\right\|^{2}$
(3.5) $\displaystyle=$
$\displaystyle\alpha_{i}\left\|u_{i}-p\right\|^{2}+\left(1-\alpha_{i}\right)\left\|S_{i}^{i}u_{i}-p\right\|^{2}-\alpha_{i}\left(1-\alpha_{i}\right)\left\|u_{i}-S_{i}^{i}u_{i}\right\|^{2}$
$\displaystyle\leq$
$\displaystyle\alpha_{i}\left\|u_{i}-p\right\|^{2}+\left(1-\alpha_{i}\right)\left\\{\left\|u_{i}-p\right\|^{2}+k\left\|u_{i}-S_{i}^{i}u_{i}\right\|^{2}+\lambda_{i}\xi_{i}(\left\|u_{i}-p\right\|)+\mu_{i}\right\\}$
$\displaystyle-\alpha_{i}\left(1-\alpha_{i}\right)\left\|u_{i}-S_{i}^{i}u_{i}\right\|^{2}$
$\displaystyle\leq$
$\displaystyle\left\|u_{i}-p\right\|^{2}+\left(1-\alpha_{i}\right)\left\\{k\left\|u_{i}-S_{i}^{i}u_{i}\right\|^{2}+\lambda_{i}\xi_{i}\left(M_{i}\right)+\lambda_{i}M_{i}^{\ast}\left\|u_{i}-p\right\|^{2})+\mu_{i}\right\\}$
$\displaystyle-\alpha_{i}\left(1-\alpha_{i}\right)\left\|u_{i}-S_{i}^{i}u_{i}\right\|^{2}$
$\displaystyle\leq$
$\displaystyle\left\|x_{i}-p\right\|^{2}-\left(1-\alpha_{i}\right)\left(\alpha_{i}-k\right)\left\|u_{i}-S_{i}^{i}u_{i}\right\|^{2}$
$\displaystyle+\left(1-\alpha_{i}\right)\left\\{\lambda_{i}\xi_{i}\left(M_{i}\right)+\lambda_{i}M_{i}^{\ast}\left\|x_{i}-p\right\|^{2}+\mu_{i}\right\\}.$

Since $\alpha_{i}-k\geq 0~{}$by condition (C1), so (3.5) implies that

$\left\|y_{i}-p\right\|^{2}\leq\left\|x_{i}-p\right\|^{2}+\theta_{i},$ (3.6)

where
$\theta_{i}=\left(1-\alpha_{i}\right)\left\\{\lambda_{i}\xi_{i}(M_{i})+\lambda_{i}M_{i}^{\ast}D_{i}+\mu_{i}\right\\}$
and $D_{i}=\sup\left\\{\left\|x_{i}-p\right\|^{2}:p\ \in\mathbb{F}\right\\}.$
It now follows from the estimate (3.6) that $p\in C_{i+1}.$ Hence,
$\mathbb{F}\subset C_{n}$ for all $n\geq 1.$ Next, we show that the set
$C_{n}$ is closed and convex for all $n\geq 1.$ Since

$\left\\{{z\in
H_{1}:}\left\|{y_{n}-z}\right\|{{}^{2}\leq\left\|{x_{n}-z}\right\|^{2}}\right\\}=\left\\{{z\in
H_{1}:}\left\|{y_{n}}\right\|{{}^{2}-\left\|{x_{n}}\right\|{{}^{2}}\leq
2}\left\langle{y_{n}}-x_{n},z\right\rangle{+\theta_{n}}\right\\},$

it is closed and convex; hence the sequence $\left\\{x_{n}\right\\}$ defined
in (3.1) is well-defined. Next, from $x_{n}=P_{C_{n}}x_{1},$ we get

$0\leq\left\langle x_{n}-x_{1},x^{\ast}-x_{n}\right\rangle,$

for all $x^{\ast}\in C_{n}.$ Since $\mathbb{F\neq\emptyset},$ then for any
$p\in\mathbb{F\,}$, we get

$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle\left\langle
x_{n}-x_{1},p-x_{n}\right\rangle$ $\displaystyle=$ $\displaystyle\left\langle
x_{n}-x_{1},p+x_{1}-x_{1}-x_{n}\right\rangle$ $\displaystyle=$
$\displaystyle\left\langle x_{n}-x_{1},x_{1}-x_{n}\right\rangle+\left\langle
x_{n}-x_{1},p-x_{1}\right\rangle$ $\displaystyle=$
$\displaystyle-\left\|x_{n}-x_{1}\right\|^{2}+\left\|x_{n}-x_{1}\right\|\left\|p-x_{1}\right\|.$

That is,

$\left\|x_{n}-x_{1}\right\|\leq\left\|p-x_{1}\right\|,\text{ for all
}p\in\mathbb{F}\text{ and }n\geq 1.$

Hence, the sequence $\\{x_{n}\\}$ is bounded, so are $\\{u_{n}\\}$ and
$\\{y_{n}\\}.$ Moreover, from $x_{n}=P_{C_{n}}x_{1}$ and
$x_{n+1}=P_{C_{n+1}}x_{1}\in C_{n+1}\subset C_{n},$ we have

$0\leq\left\langle x_{n}-x_{1},x_{n+1}-x_{n}\right\rangle.$

Similarly, we get the following relation:

$\left\|x_{n}-x_{1}\right\|\leq\left\|x_{n+1}-x_{1}\right\|,\text{ for all
}n\geq 1.$

From the above assertions, we conclude that the sequence
$\\{\left\|x_{n}-x_{1}\right\|\\}$ is bounded and nondecreasing, therefore, we
have

$\lim_{n\rightarrow\infty}\left\|x_{n}-x_{1}\right\|\text{ exists.}$ (3.7)

Further observe that

$\displaystyle\left\|x_{n+1}-x_{n}\right\|^{2}$ $\displaystyle=$
$\displaystyle\left\|x_{n+1}-x_{1}+x_{1}-x_{n}\right\|^{2}$ $\displaystyle=$
$\displaystyle\left\|x_{n+1}-x_{1}\right\|^{2}+\left\|x_{n}-x_{1}\right\|^{2}-2\left\langle
x_{n}-x_{1},x_{n+1}-x_{1}\right\rangle$ $\displaystyle=$
$\displaystyle\left\|x_{n+1}-x_{1}\right\|^{2}+\left\|x_{n}-x_{1}\right\|^{2}-2\left\langle
x_{n}-x_{1},x_{n+1}-x_{n}+x_{n}-x_{1}\right\rangle$ $\displaystyle=$
$\displaystyle\left\|x_{n+1}-x_{1}\right\|^{2}-\left\|x_{n}-x_{1}\right\|^{2}-2\left\langle
x_{n}-x_{1},x_{n+1}-x_{n}\right\rangle$ $\displaystyle\leq$
$\displaystyle\left\|x_{n+1}-x_{1}\right\|^{2}-\left\|x_{n}-x_{1}\right\|^{2}.$

From (3.7), we obtain that
$\left\|x_{n+1}-x_{1}\right\|^{2}-\left\|x_{n}-x_{1}\right\|^{2}\rightarrow 0$
as $n\rightarrow\infty,$ hence the sequence $\\{\left\|x_{n}-x_{1}\right\|\\}$
is Cauchy. That is

$\lim_{n\rightarrow\infty}\left\|x_{n+1}-x_{n}\right\|=0.$ (3.8)

Since $x_{n+1}\in C_{n+1},$ which implies that
$\left\|y_{n}-x_{n+1}\right\|\leq\left\|x_{n}-x_{n+1}\right\|+\theta_{n,i}.$
From (3.8), we conclude that

$\lim_{n\rightarrow\infty}\left\|y_{n}-x_{n+1}\right\|=0,\text{ for all }n\geq
1.$ (3.9)

Now using (3.8), (3.9) and the following triangular inequality, we get

$\left\|y_{n}-x_{n}\right\|\leq\left\|y_{n}-x_{n+1}\right\|+\left\|x_{n+1}-x_{n}\right\|\rightarrow
0$ (3.10)

as $n\rightarrow\infty.$
Consider from (3.1), (3.3) and (3.6), we get

$\displaystyle\gamma\left(1-\gamma
L\right)\left\|A_{n}x_{n}-T_{s_{n}}^{g_{n}}A_{n}x_{n}\right\|^{2}$
$\displaystyle\leq$
$\displaystyle\left\|x_{n}-p\right\|^{2}-\left\|u_{n}-p\right\|^{2}$
$\displaystyle\leq$
$\displaystyle\left\|x_{n}-p\right\|^{2}-\left\|y_{n}-p\right\|^{2}+\theta_{n}$
$\displaystyle\leq$
$\displaystyle\left(\left\|x_{n}-p\right\|+\left\|y_{n}-p\right\|\right)\left\|x_{n}-y_{n}\right\|+\theta_{n}.$

Utilizing the fact that $\gamma\left(1-\gamma L\right)>0$ and the estimate
(3.10), we have

$\lim_{n\rightarrow\infty}\left\|A_{n}x_{n}-T_{s_{n}}^{g_{n}}A_{n}x_{n}\right\|^{2}=0\text{
for all }n\geq 1.$ (3.11)

For any $p\in\mathbb{F}$ and firm nonexpansiveness of $T_{r_{n}}^{f_{n}},$ we
have

$\displaystyle\left\|u_{n}-p\right\|^{2}$ $\displaystyle=$
$\displaystyle\left\|T_{r_{n}}^{f_{n}}\left(x_{n}-\gamma
A_{n}^{\ast}\left(I-T_{s_{n}}^{g_{n}}\right)A_{n}x_{n}\right)-T_{r_{n}}^{f_{n}}p\right\|^{2}$
(3.12) $\displaystyle\leq$ $\displaystyle\left\langle u_{n}-p,x_{n}-\gamma
A_{n}^{\ast}\left(I-T_{s_{n}}^{g_{n}}\right)A_{n}x_{n}-p\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2}\\{\left\|u_{n}-p\right\|^{2}+\left\|x_{n}-\gamma
A_{n}^{\ast}\left(I-T_{s_{n}}^{g_{n}}\right)A_{n}x_{n}-p\right\|^{2}$
$\displaystyle-\left\|u_{n}-x_{n}-\gamma
A_{n}^{\ast}\left(I-T_{s_{n}}^{g_{n}}\right)A_{n}x_{n}\right\|^{2}\\}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\left\\{\left\|u_{n}-p\right\|^{2}+\left\|x_{n}-p\right\|^{2}-\left\|u_{n}-x_{n}-\gamma
A_{n}^{\ast}\left(I-T_{s_{n}}^{g_{n}}\right)A_{n}x_{n}\right\|^{2}\right\\}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\\{\left\|u_{n}-p\right\|^{2}+\left\|x_{n}-p\right\|^{2}-(\left\|u_{n}-x_{n}\right\|^{2}+\gamma^{2}\left\|A_{n}^{\ast}\left(I-T_{s_{n}}^{g_{n}}\right)A_{n}x_{n}\right\|^{2}$
$\displaystyle-2\gamma\left\langle
u_{n}-x_{n},A_{n}^{\ast}\left(I-T_{s_{n}}^{g_{n}}\right)A_{n}x_{n}\right\rangle)\\}$
$\displaystyle\leq$
$\displaystyle\left\|x_{n}-p\right\|^{2}-\left\|u_{n}-x_{n}\right\|^{2}+2\gamma\left\|A\left(u_{n}-x_{n}\right)\right\|\left\|A_{n}x_{n}-T_{s_{n}}^{g_{n}}A_{n}x_{n}\right\|.$

Using the following estimate:

$\left\|y_{n}-p\right\|^{2}\leq\alpha_{n}\left\|x_{n}-p\right\|^{2}+\left(1-\alpha_{n}\right)\left\|u_{n}-p\right\|^{2}+\theta_{n}.$

in (3.12) and re-arranging the terms, we get

$\displaystyle\left(1-\alpha_{n}\right)\left\|u_{n}-x_{n}\right\|^{2}$
$\displaystyle\leq$
$\displaystyle\left(\left\|x_{n}-p\right\|+\left\|y_{n}-p\right\|\right)\left\|x_{n}-y_{n}\right\|$
$\displaystyle+2\gamma\left\|A\left(u_{n}-x_{n}\right)\right\|\left\|A_{n}x_{n}-T_{s_{n}}^{g_{n}}A_{n}x_{n}\right\|+\theta_{n}.$

Letting $n\rightarrow\infty$ and utilizing (3.10) and (3.11), we have

$\lim_{n\rightarrow\infty}\left\|u_{n}-x_{n}\right\|=0\text{ for all }n\geq
1.$ (3.13)

Moreover, from (3.10) and (3.13), we obtain

$\left\|y_{n}-u_{n}\right\|\leq\left\|y_{n}-x_{n}\right\|+\left\|x_{n}-u_{n}\right\|\rightarrow
0$ (3.14)

when $n\rightarrow\infty.$ Observe that
$\left\|y_{n}-u_{n}\right\|=\left(1-\alpha_{n}\right)\left\|S_{n}^{n}u_{n}-u_{n}\right\|.$
Then it follows from condition (C1) and (3.14) that

$\lim_{n\rightarrow\infty}\left\|S_{n}^{n}u_{n}-u_{n}\right\|=0\text{ for all
}n\geq 1.$ (3.15)

Since

$\left\|S_{n}^{n}u_{n}-x_{n}\right\|\leq\left\|S_{n}^{n}u_{n}-u_{n}\right\|+\left\|x_{n}-u_{n}\right\|.$

Therefore from (3.13) and (3.15), we get

$\lim_{n\rightarrow\infty}\left\|S_{n}^{n}u_{n}-x_{n}\right\|=0\text{ for all
}n\geq 1.$ (3.16)

On a similar reasoning, we also obtain

$\lim_{n\rightarrow\infty}\left\|S_{n}^{n}u_{n}-y_{n}\right\|=0\text{ for all
}n\geq 1.$

Observe that each $S_{n}$ is uniformly $\Theta$-Lipschitzian, therefore, we
have

$\displaystyle\left\|S_{n}^{n}x_{n}-x_{n}\right\|$ $\displaystyle\leq$
$\displaystyle\left\|S_{n}^{n}x_{n}-S_{n}^{n}u_{n}\right\|+\left\|S_{n}^{n}u_{n}-x_{n}\right\|$
$\displaystyle\leq$
$\displaystyle\Theta\left\|x_{n}-u_{n}\right\|+\left\|S_{n}^{n}u_{n}-x_{n}\right\|.$

Now, using (3.13), (3.16) and the above estimate, we get

$\lim_{n\rightarrow\infty}\left\|S_{n}^{n}x_{n}-x_{n}\right\|=0\text{ for all
}n\geq 1.$ (3.17)

Moreover, utilizing the uniform continuity of $S_{n}$ and (3.17), the
following estimate:

$\left\|x_{n}-S_{n}x_{n}\right\|\leq\left\|x_{n}-S_{n}^{n}x_{n}\right\|+\left\|S_{n}^{n}x_{n}-S_{n}x_{n}\right\|$

implies that

$\lim_{n\rightarrow\infty}\left\|x_{n}-S_{n}x_{n}\right\|=0\text{ for all
}n\geq 1.$ (3.18)

Similarly, we also have that

$\lim_{n\rightarrow\infty}\left\|u_{n}-S_{n}u_{n}\right\|=0\text{ for all
}n\geq 1.$ (3.19)

Now, we show that $\omega(x_{n})\subset\mathbb{F},$ where $\omega(x_{n})$ is
the set of all weak $\omega$-limits of $\\{x_{n}\\}.$ Since $\\{x_{n}\\}$ is
bounded, therefore $\omega(x_{n})\neq\emptyset.$ Let $q\in\omega(x_{n}),$ then
there exists a subsequence $\\{x_{Nn+i}\\}$ of $\\{x_{n}\\}$ such that
$x_{Nn+i}\rightharpoonup q.$ Using the fact that $S_{Nn+i}=S_{i}$ for all
$n\geq 1$ and the demiclosed principle (Lemma 2.3) for each $S_{i}$, we have
that $x\in F(S_{i})\,$ for each $1\leq i\leq N.$ Next, we show that
$q\in\Omega$, i.e., $q\in\bigcap_{i=1}^{N}EP(f_{i})$ and $A_{i}q\in EP(g_{i})$
for each $1\leq i\leq N.$ In order to show that
$q\in\bigcap_{i=1}^{N}EP(f_{i}),$ that is, $q\in EP(f_{i})$ for each $1\leq
i\leq N,$ we define subsequence $\left\\{n_{j}\right\\}$ of index
$\left\\{n\right\\}$ such that $n_{j}=Nj+i$ for all $n\geq 1.$ As a
consequence, we can write $f_{n_{j}}=f_{i}$ for $1\leq i\leq N.$ From
$u_{n_{j}}=T_{r_{n_{j}}}^{f_{i}}\left(I-\gamma
A_{n_{j}}^{\ast}\left(I-T_{s_{n_{j}}}^{g_{n_{j}}}\right)A_{n_{j}}\right)x_{n_{j}}$
for all $n\geq 1,$ we have

$f_{i}(u_{n_{j}},y)+\frac{1}{r_{n_{j}}}\left\langle
y-u_{n_{j}},u_{n_{j}}-x_{n_{j}}-\gamma
A_{n_{j}}^{\ast}\left(I-T_{s_{n_{j}}}^{g_{n_{j}}}\right)A_{n_{j}}x_{n_{j}}\right\rangle\geq
0,\ \ \text{for all }y\in C.$

This implies that

$f_{i}(u_{n_{j}},y)+\frac{1}{r_{n_{j}}}\left\langle
y-u_{n_{j}},u_{n_{j}}-x_{n_{j}}\right\rangle-\frac{1}{r_{n_{j}}}\left\langle
y-u_{n_{j}},\gamma
A_{n_{j}}^{\ast}\left(I-T_{s_{n_{j}}}^{g_{n_{j}}}\right)A_{n_{j}}x_{n_{j}}\right\rangle\geq
0$

From (A2), we have

$\frac{1}{r_{n_{j}}}\left\langle
y-u_{n_{j}},u_{n_{j}}-x_{n_{j}}\right\rangle-\frac{1}{r_{n_{j}}}\left\langle
y-u_{n_{j}},\gamma
A_{n_{j}}^{\ast}\left(I-T_{s_{n_{j}}}^{g_{n_{j}}}\right)A_{n_{j}}x_{n_{j}}\right\rangle\geq
f_{i}(y,u_{n_{j}}),$

for all $y\in C.$ Since $\liminf_{j\rightarrow\infty}r_{n_{i}}>0$ (by (C2)),
therefore it follows from (3.11) and (3.13) that

$f_{i}(y,q)\leq 0,\ \ \text{for all }y\in C\text{ and for }1\leq i\leq N.$

Let $y_{t}=ty+(1-t)q$ for some $0<t<1$ and $y\in C$. Since $q\in C,$ this
implies that $y_{t}\in C.$ Using (A1) and (A4) from Condition 2.4, the
following estimate:

$0=f_{i}(y_{t},y_{t})\leq tf_{i}(y_{t},y)+(1-t)f_{i}(y_{t},q)\leq
tf_{i}(y_{t},y),$

implies that

$f_{i}(y_{t},y)\geq 0,\text{ for }1\leq i\leq N.$

Letting $t\rightarrow 0,$ we have $f_{i}(q,y)\geq 0$ for all $y\in C.$ Thus,
$q\in EP(f_{i})$ for $1\leq i\leq N.$ That is,
$q\in\bigcap_{i=1}^{N}EP(F_{i}).$ Reasoning as above, we show that $A_{i}q\in
EP(g_{i})$ for each $1\leq i\leq N.$ Since $x_{n_{l}}\longrightarrow q$ and
$A_{n_{l}}$ is a bounded linear operator, therefore
$A_{n_{l}}x_{n_{l}}\longrightarrow A_{n_{l}}q.$ Hence, it follows from (3.11)
that

$T_{s_{n_{l}}}^{g_{n_{l}}}A_{n_{l}}x_{n_{l}}\longrightarrow A_{n_{l}}q\text{ \
\ \ as \ \ }l\rightarrow\infty.$

Now, from Lemma 2.5, we have

$g_{i}\left(T_{s_{n_{l}}}^{g_{n_{l}}}A_{n_{l}}x_{n_{l}},z\right)+\frac{1}{s_{n_{l}}}\left\langle
z-T_{s_{n_{l}}}^{g_{n_{l}}}A_{n_{l}}x_{n_{l}},T_{s_{n_{l}}}^{g_{n_{l}}}A_{n_{l}}x_{n_{l}}-A_{n_{l}}x_{n_{l}}\right\rangle\geq
0,\ \ \text{for all }z\in Q.$

Since $g_{i}$ is upper hemicontinuous in the first argument for each $1\leq
i\leq N$, therefore taking $\limsup$ on both sides of the above estimate as
$l\rightarrow\infty$ and utilizing (C2) and (3.11), we get

$g_{i}\left(A_{n_{l}}x,z\right)\geq 0,\ \ \text{for all }z\in Q\text{ and for
each }1\leq i\leq N.$

Hence $A_{i}q\in EP(g_{i})$ for each $1\leq i\leq N$ and consequently
$q\in\mathbb{F}.$ It remains to show that $x_{n}\rightarrow
q=P_{\mathbb{F}}x_{1}.$ Let $x=P_{\mathbb{F}}x_{1},$ then from
$\left\|x_{n}-x_{1}\right\|\leq\left\|x-x_{1}\right\|,$ therefore, we have

$\displaystyle\left\|x-x_{1}\right\|$ $\displaystyle\leq$
$\displaystyle\left\|q-x_{1}\right\|$ $\displaystyle\leq$
$\displaystyle\liminf_{j\rightarrow\infty}\left\|x_{n_{j}}-x_{1}\right\|$
$\displaystyle\leq$
$\displaystyle\limsup_{j\rightarrow\infty}\left\|x_{n_{j}}-x_{1}\right\|$
$\displaystyle\leq$ $\displaystyle\left\|x-x_{1}\right\|.$

This implies that

$\lim_{j\rightarrow\infty}\left\|x_{n_{j}}-x_{1}\right\|=\left\|q-x_{1}\right\|.$

Hence $x_{n_{j}}\rightarrow q=P_{\mathbb{F}}x_{1}.$ From the arbitrariness of
the subsequence $\left\\{x_{n_{j}}\right\\}$ of $\left\\{x_{n}\right\\},$ we
conclude that $x_{n}\rightarrow x$ as $n\rightarrow\infty.$ It is easy to see
that $y_{n,i}\rightarrow x$ and $u_{n,i}\rightarrow x.$ This completes the
proof. $\square$
Corollary 3.2. Let $H_{1}$ and $H_{2}$ be two real Hilbert spaces and let
$C\subseteq H_{1}$ and $Q\subseteq H_{2}$ be nonempty closed convex subsets of
Hilbert spaces $H_{1}$ and $H_{2}$, respectively. Let $f_{i}:C\times
C\rightarrow\mathbb{R}$ and $g_{i}:Q\times Q\rightarrow\mathbb{R}$ be two
finite families of bifunctions satisfying Condition 2.4 such that $g_{i}$ be
upper semicontinuous for each $i\in\\{1,2,3,\cdots,N\\}$. Let
$S_{i}:C\rightarrow C$ be a finite family nonexpansive mappings and let
$A_{i}:H_{1}\rightarrow H_{2}$ be a finite family of bounded linear operators
for each $i\in\\{1,2,3,\cdots,N\\}$. Suppose that
$\mathbb{F}:=\left[\bigcap_{i=1}^{N}F(S_{i})\right]\cap\Omega\neq\emptyset$,
where $\Omega=\left\\{z\in C:z\in\bigcap_{i=1}^{N}EP(f_{i})\text{ and
}A_{i}z\in EP(g_{i})\text{ for }1\leq i\leq N\right\\}.$ Let $\\{x_{n}\\}$ be
a sequence generated by:

$\begin{array}[]{l}x_{1}\in C_{1}=C,\\\
u_{n}=T_{r_{n}}^{f_{n}}\left(x_{n}-\gamma
A_{n(\mathop{\mathrm{m}od}N)}^{\ast}\left(I-T_{s_{n}}^{g_{n}}\right)A_{n(\mathop{\mathrm{m}od}N)}x_{n}\right),\\\
y_{n}=\alpha_{n}u_{n}+\left(1-\alpha_{n}\right)S_{n(\mathop{\mathrm{m}od}N)}^{n}u_{n},\\\
C_{n+1}=\left\\{{z\in
H}_{1}{:}\left\|{y_{n}-z}\right\|{{}^{2}\leq\left\|{x_{n}-z}\right\|^{2}+\theta_{n}}\right\\},\\\
x_{n+1}=P_{C_{n+1}}x_{1},\ \ n\geq 1,\end{array}$ (3.20)

where
$\theta_{n}=(1-\alpha_{n})\left\\{\lambda_{n}\xi_{n}(M_{n})+\lambda_{n}M_{n}^{\ast}D_{n}+\mu_{n}\right\\}$
with $D_{n}=\sup\left\\{\left\|x_{n}-p\right\|:p\ \in\mathbb{F}\right\\}$. Let
$\\{r_{n}\\},\\{s_{n}\\}$ be two positive real sequences and let
$\\{\alpha_{n}\\}$ be in $(0,1).$ Assume that if the following set of
conditions holds:
(C1): $0\leq k<a\leq\alpha_{n}\leq b<1$ and
$\gamma\in\left(0,\frac{1}{L}\right)$ where
$L=\max\left\\{L_{1},L_{2},\cdots,L_{N}\right\\}$ and $L_{i}$ is the spectral
radius of the operator $A_{i}^{\ast}A_{i}$ and $A_{i}^{\ast}$ is the adjoint
of $A_{i}$ for each $i\in\\{1,2,3,\cdots,N\\};$
(C2): $\liminf\limits_{n\rightarrow\infty}r_{n}>0$ and
$\liminf\limits_{n\rightarrow\infty}s_{n}>0;$
(C3): $\sum\limits_{n=1}^{\infty}\lambda_{n}<\infty$ and$\
\sum\limits_{n=1}^{\infty}\mu_{n}<\infty;$
(C4): there exist constants $M_{i},\ M_{i}^{\ast}>0$ such that
$\xi_{i}\left(\lambda_{i}\right)\leq M_{i}^{\ast}\lambda_{i}$ for all
$\lambda_{i}\geq M_{i},i=1,2,3,\cdots,N,$ then the sequence $\\{x_{n}\\}$
generated by (3.20) converges strongly to $P_{\mathbb{F}}x_{1}.\vskip 6.0pt
plus 2.0pt minus 2.0pt$
Theorem 3.3. Let $H_{1}$ and $H_{2}$ be two real Hilbert spaces and let
$C\subseteq H_{1}$ and $Q\subseteq H_{2}$ be nonempty closed convex subsets of
Hilbert spaces $H_{1}$ and $H_{2}$, respectively. Let $f_{i}:C\times
C\rightarrow\mathbb{R}$ and $g_{i}:Q\times Q\rightarrow\mathbb{R}$ be two
finite families of bifunctions satisfying Condition 2.4 such that $g_{i}$ be
upper semicontinuous for each $i\in\\{1,2,3,\cdots,N\\}$. Let
$S_{i}:C\rightarrow C$ be a finite family of uniformly $\Theta$-Lipschitzian
and continuous total asymptotically strict pseudo contractions and let
$A_{i}:H_{1}\rightarrow H_{2}$ be a finite family of bounded linear operators
for each $i\in\\{1,2,3,\cdots,N\\}$. Suppose that
$\mathbb{F}:=\left[\bigcap_{i=1}^{N}F(S_{i})\right]\cap\Omega\neq\emptyset$,
where $\Omega=\left\\{z\in C:z\in\bigcap_{i=1}^{N}EP(f_{i})\text{ and
}A_{i}z\in EP(g_{i})\text{ for }1\leq i\leq N\right\\}.$ Let $\\{x_{n}\\}$ be
a sequence generated by:

$\begin{array}[]{l}x_{1}\in C_{1}=C,\\\
u_{n}=T_{r_{n}}^{f_{n}}\left(x_{n}-\gamma
A_{n(\mathop{\mathrm{m}od}N)}^{\ast}\left(I-T_{s_{n}}^{g_{n}}\right)A_{n(\mathop{\mathrm{m}od}N)}x_{n}\right),\\\
y_{n}=\alpha_{n}u_{n}+\left(1-\alpha_{n}\right)S_{n(\mathop{\mathrm{m}od}N)}^{n}u_{n},\\\
C_{n+1}=\left\\{{z\in
H}_{1}{:}\left\|{y_{n}-z}\right\|{{}^{2}\leq\left\|{x_{n}-z}\right\|^{2}+\theta_{n}}\right\\},\\\
x_{n+1}=P_{C_{n+1}}x_{1},\ \ n\geq 1,\end{array}$ (3.21)

where
$\theta_{n}=(1-\alpha_{n})\left\\{\lambda_{n}\xi_{n}(M_{n})+\lambda_{n}M_{n}^{\ast}D_{n}+\mu_{n}\right\\}$
with $D_{n}=\sup\left\\{\left\|x_{n}-p\right\|:p\ \in\mathbb{F}\right\\}$. Let
$\\{r_{n}\\},\\{s_{n}\\}$ be two positive real sequences and let
$\\{\alpha_{n}\\}$ be in $(0,1).$ Assume that if the following set of
conditions holds:
(C1): $0\leq k<a\leq\alpha_{n}\leq b<1$ and
$\gamma\in\left(0,\frac{1}{L}\right)$ where
$L=\max\left\\{L_{1},L_{2},\cdots,L_{N}\right\\}$ and $L_{i}$ is the spectral
radius of the operator $A_{i}^{\ast}A_{i}$ and $A_{i}^{\ast}$ is the adjoint
of $A_{i}$ for each $i\in\\{1,2,3,\cdots,N\\};$
(C2): $\liminf\limits_{n\rightarrow\infty}r_{n}>0$ and
$\liminf\limits_{n\rightarrow\infty}s_{n}>0;$
(C3): $\sum\limits_{n=1}^{\infty}\lambda_{n}<\infty$ and$\
\sum\limits_{n=1}^{\infty}\mu_{n}<\infty;$
(C4): there exist constants $M_{i},\ M_{i}^{\ast}>0$ such that
$\xi_{i}\left(\lambda_{i}\right)\leq M_{i}^{\ast}\lambda_{i}$ for all
$\lambda_{i}\geq M_{i},i=1,2,3,\cdots,N,$ then the sequence $\\{x_{n}\\}$
generated by (3.21) converges strongly to $P_{\mathbb{F}}x_{1}.$
Proof. Set $H_{1}=H_{2},~{}C=Q$ and $A=I$(the identity mapping) then the
desired result then follows from Theorem 3.1 immediately. $\square$
Acknowledgment. The author M. A. A. Khan would like to acknowledge the support
provided by the Higher Education Commission of Pakistan for funding this work
through project No. NRPU 5332.

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††thanks: joint first author††thanks: joint first author††thanks:
corresponding author

# Geometric Learning of Knot Topology

Joseph Lahoud Sleiman  School of Physics and Astronomy, University of
Edinburgh  Filippo Conforto  School of Physics and Astronomy, University of
Edinburgh  Yair Augusto Gutierrez Fosado  School of Physics and Astronomy,
University of Edinburgh  Davide Michieletto  School of Physics and Astronomy,
University of Edinburgh  MRC Human Genetics Unit, Institute of Genetics and
Cancer, University of Edinburgh

###### Abstract

Knots are deeply entangled with every branch of science. One of the biggest
open challenges in knot theory is to formalise a knot invariant that can
unambiguously and efficiently distinguish any two knotted curves.
Additionally, the conjecture that the geometrical embedding of a curve encodes
information on its underlying topology is, albeit physically intuitive, far
from proven. Here we attempt to tackle both these outstanding challenges by
proposing a neural network (NN) approach that takes as input a geometric
representation of a knotted curve and tries to make predictions of the curve’s
topology. Intriguingly, we discover that NNs trained with a so-called
geometrical “local writhe” representation of a knot can distinguish curves
that share one or many topological invariants and knot polynomials, such as
mutant and composite knots, and can thus classify knotted curves more
precisely than some knot polynomials. Additionally, we also show that our
approach can be scaled up to classify all prime knots up to 10-crossings with
more than 95% accuracy. Finally, we show that our NNs can also be trained to
solve knot localisation problems on open and closed curves. Our main discovery
is that the pattern of “local writhe” is a potentially unique geometric
signature of the underlying topology of a curve. We hope that our results will
suggest new methods for quantifying generic entanglements in soft matter and
even inform new topological invariants.

##  I Introduction

Knots are fascinating objects that have captured the attention of humans for
centuries. From Incas’ knotted Quipus Adams (1994), and Lord Kelvin’s theory
of elements as knotted ether Thomson (1867), to sailors and climbers whose
lives often rely on the strength of knotted rope, knots are deeply intertwined
with history and art and often carry mystical meaning. The human obsession
with knots brought Peter Guthrie Tait to compile the first knot tabulation of
up to 10 crossings by hand Tait (1885); currently, more than one million
unique knots up to 16 crossings have been tabulated using computer programs
Hoste  _et al._ (1998).

To rigorously prove that the early tabulated knots did not contain duplicates,
so-called topological invariants and knot polynomials were developed, the
first of which was the Alexander polynomial Alexander (1928); Kauffman (1983);
Adams (1994), followed more recently by the Jones and HOMFLY polynomials Hoste
(1986); Adams (1994). Knot polynomials are mathematical constructs that can be
computed on knot diagrams and are invariant under smooth deformations of the
curve, i.e. deformations that preserve the curve topology. However, there are
knots that share many topological invariants and cannot even be distinguished
by knot polynomials. Famously, the 11-crossing Conway knot has the same
Alexander polynomial as the unknot and shares the same Jones polynomial of its
mutant, the Kinoshita–Terasaka (KT) knot Adams (1994). More generally, all
mutants of a knot have the same HOMFLY polynomials and the same hyperbolic
volume Adams (1994), while some composite knots share the same homeomorphic
complements Thurston (1997); Culler  _et al._ ; Gordon and Luecke (1989).

Figure 1: A Examples of equilibrium knotted polymer conformations colour coded
to show the knot contour (red, white, and blue). In this figure we consider
the 5 simplest knots: $0_{1}$, $3_{1}$, $4_{1}$, $5_{1}$ and $5_{2}$. B A
graphical representation of StS writhe $\omega_{StS}(x,y)$ showing a situation
of low and high writhe between two segments. C Examples of patterns for
$\omega_{StA}(x)$ for three different knots. D A graphical representation of
the (feed-forward) network. The input layer contains $N$ (or $3N$) neurons
corresponding to the size of the input feature representation, and the output
layer yields a probability for each knot class. E Accuracy score, tested on
unseen polymer conformations for different input features. The StA writhe
classifies the 5-simplest knots with 99.9% accuracy irrespectively of the
network architecture. F Confusion matrices obtained by training the network
with XYZ and StA writhe input features.

Alongside the development of topological invariants, several attempts were
made to identify a relationship between a specific geometric embedding of a
knot and its underlying topology Stasiak  _et al._ (1998). We note that this
relationship is different from the one sought between so-called geometric and
algebraic invariants Jejjala  _et al._ (2019); Davies  _et al._ (2021), e.g.
between the hyperbolic volume of a knot and its Jones polynomial Murakami  _et
al._ (2002). Perhaps one of the most rigorous results in this direction is the
Fáry-Milnor theorem, stating that the total absolute curvature of non-
trivially knotted curves must be greater than $4\pi$ Milnor (1950).
Unfortunately, this result only imposes a weak constraint on the topology of
the underlying curve, as an unknot can itself have large total curvature due
to, for example, deformations of its contour. In parallel, a large body of
work on so-called “ideal knots” was done with the aim of finding geometric
features that could reflect the underlying knot topology. One impressive
result in this context is that different DNA knots display a spatial
separation when run on a gel electrophoresis that is linearly proportional to
the so-called average crossing number Stasiak  _et al._ (1996a); Michieletto
_et al._ (2015); this result entails that there is an intimate relationship
between the physical shapes assumed by knots and their underlying topology.
Another result that inspired our work is that the total so-called “writhe”
(see below) of an ideal knot is the same (up to a constant that is only a
function of the curve length) as that of a non-ideal, thermally agitated curve
with the same topology Katritch  _et al._ (1997). Though this suggests that
“writhe” may be a good measure that is invariant under thermal fluctuations,
there is no one-to-one relationship between the global writhe of a knot and
its underlying topology; for instance, the global writhe of the $4_{1}$ knot
is 0, the same as the unknot Stasiak  _et al._ (1998).

Thus, the problem of determining a curve topology only based on the geometric
information of its segments (without using any projection and algebraic
invariant) is an open challenge in knot theory that has ramification in many
fields, for instance polymer physics, biophysics and fluid dynamics. In this
paper, we propose to address this open challenge by using the power of
artificial intelligence, and in particular deep learning, at recognising and
classifying patterns in certain knot geometric features. Our main discovery is
that by using a quantity we dub “local writhe”, even simple machine learning
(ML) algorithms can identify the topology of knotted curves undergoing thermal
fluctuations with more than 99% accuracy. We argue that this is an example of
geometric learning as the only quantity we pass to the ML algorithm is a
quantity that can be computed from the Cartesian positions of a curve’s
segments, without the need of computing algebraic invariants such as Alexander
or Jones polynomials. Our method can even distinguish 11-crossing knots that
are otherwise impossible to distinguish using standard invariants (the Conway
and KT knots). Finally, we show how this algorithm can be scaled to classify
all 250 prime knots up to 10 crossings with 95% accuracy and that can even be
employed to solve knot localisation problems. Overall, we argue that local
writhe is an excellent feature – based purely on the 3D position of a curve
segments – that displays patterns that can be easily identified by ML
algorithms. We hope that out results will be applied to other classification
problems such as threading Michieletto and Turner (2016); Staňo  _et al._
(2022) and entanglements Everaers  _et al._ (2004); Smrek  _et al._ (2021) and
also prompt knot theorists to employ local writhe to define new geometric knot
invariants.

##  II Results

Two recent papers by Vandans Vandans  _et al._ (2020) and Braghetto Braghetto
_et al._ (2023) have shown that machine learning is a promising tool to solve
knot classification problems. They mostly considered Cartesian position of the
monomers or adjacent monomer distances and dihedrals to classify the 5
simplest knots. In this work, we set out to test the use of a different type
of geometric features that our group recently considered to identify essential
crossings of a knot and plectoneme-like double folding of ring polymers
Michieletto (2016); Sleiman  _et al._ (2022). More specifically, we focused on
a generalisation of the Gauss linking integral applied to a single closed
curve, often associated with its writhe Dennis and Hannay (2005) and average
crossing number Stasiak  _et al._ (1996b); Michieletto (2016). This choice is
inspired by the intuition that writhe captures the geometrical entanglement of
a curve with itself, and we thus define a generalised local segment-to-segment
(StS) writhe as

$\omega_{StS}(x,y)=\dfrac{(\bm{t}(x)\times\bm{t}(y))\cdot(\bm{r}(x)-\bm{r}(y))}{|\bm{r}(x)-\bm{r}(y)|^{3}}\,,$
(1)

where $\bm{r}(x)$ and $\bm{t}(x)$ are the 3D position of, and the tangent at,
segment $x$, respectively. Intuitively, Eq. (1) captures the magnitude and the
chirality of the entanglement between segment $x$ and segment $y$ (Fig. 1A-B).
The quantity $\omega_{StA}(x)=\oint_{\gamma}\omega_{StS}(x,y)dy$ is the local
segment-to-all (StA) writhe and characterises how geometrically entangled
segment $x$ is with respect to the whole closed curve $\gamma$. In practice,
the calculation of StS and StA writhe are conducted on discrete segments,
taking a finite “window” with length $l_{w}=10$ $\sigma$ to smooth out short
length fluctuations (see SI for details).

The StA writhe, $\omega_{StA}(x)$, is a 1D geometrical representation of a
knot that we hypothesise may display some patterns that are topology-dependent
(Fig. 1A-C). Since complex pattern recognition is a task that naturally lends
itself to being addressed using a machine learning approach, we thus asked
ourselves if a neural network (NN) trained to recognise patterns within
$\omega_{StA}(x)$ was able to solve ambiguous knot classification problems. To
do this, we built feed forward and recurrent (long-short term memory, LSTM)
neural networks (FFNN and RNN, respectively) and trained them using $10^{5}$
statistically uncorrelated and pre-labelled conformations for each knot. To
generate these conformations, we initialised a bead-spring polymer with known
topology, $N=100$ beads, and persistence length $l_{p}=10\sigma$ (other
lengths and $l_{p}$ are reported in the SI) using KnotPlot (knotplot.com) and
subsequently evolved the polymer configurations in LAMMPS Plimpton (1995) via
Langevin dynamics in an implicit solvent and fixed temperature and using a
Kremer-Grest model Kremer and Grest (1990) to preserve polymer topology (see
Methods and SI for more details). The code to generate these conformations are
open access at https://git.ecdf.ed.ac.uk/taplab/mlknotsproject. We confirmed
that the topology was conserved either by computing their Alexander
determinant via KymoKnot (kymoknot.sissa.it) Tubiana  _et al._ (2018) or, when
ambiguous, visually.

Figure 2:  A Snapshots of three knots with identical Alexander polynomial:
square ($3^{l}_{1}\\#3^{r}_{1}$), granny ($3^{l}_{1}\\#3^{l}_{1}$) and
$8_{20}$ knots. B Examples of StA writhe patterns from the three knots. C
Confusion matrices obtained from a 3-class classification problem, training a
FFNN with XYZ (91.7% accuracy) or StA writhe (99.9% accuracy) features. D
Snapshots of Conway (blue) and KT (orange) knots. E Examples of StA writhe
patterns, including the one from the unknot (black). F Confusion matrices
obtained from a 3-class classification problem training a FFNN with XYZ (67%
accuracy) and StA writhe (99.6% accuracy).  Figure 3: A Two example
conformations of $5_{1}$ (blue) and $7_{2}$ (orange) knots. B The XYZ-trained
NN on a 15-class classification problem yields 63.8% accuracy and a rather
non-diagonal confusion matrix. C Examples of $\omega_{StA}(x)$ curves for the
two knots, displaying a degree of similarity between the pattern of maxima and
minima. D The $\omega_{StA}(x)$-trained NN achieves 98% accuracy and the
confusion matrix shows that $5_{1}$ and $7_{2}$ are the knots that are most
confused with each other. E Examples of $\omega_{StS}(x,y)$ geometric feature
for the two knots corresponding to the $\omega_{StA}(x)$ profiles shown in C.
F Confusion matrix for a StS-trained FFNN, achieving 99.8% accuracy. G
Accuracy as a function of number of knot classes to be distinguished, up to
10-crossing (250) prime knots.

The NNs were built with an input layer that was determined according to the
input representation being studied, e.g., the Cartesian (XYZ) coordinate
representation used 3 neurons (one for each dimension) per polymer bead. Other
local input features, such as StA writhe, used one neuron per bead, while the
StS writhe feature requires $N\times N$ input neurons. The optimal number of
hidden layers, hidden units, learning rate and batch size were determined via
an automated hyperparameter tuning method conducted on the Cartesian
representation (KerasTuner O’Malley  _et al._ (2019)). Unless otherwise
stated, our NNs contained 4 hidden layers, with around $4\cdot 10^{5}$
trainable parameters. The output layer consisted of $C$ output neurons,
corresponding to the $C$ knot types being classified, each implemented with a
softmax activation function in order to return the probability that a given
input is a certain knot type. We took the sparse categorical cross-entropy as
the loss function, as the most appropriate for individual class probabilities
and integer target labels, i.e. our knot types (Fig. 1D).

###  II.1 StA writhe yields NNs that are more accurate to classify the
simplest 5 knots than Cartesian features.

We first tackle a 5-knot classification problem with the 5 simplest knots,
which can be satisfactorily solved using NNs trained on center-of-mass-
corrected Cartesian coordinates (XYZ) or adjacent bead input features Vandans
_et al._ (2020); Braghetto  _et al._ (2023). In line with these previous
works, we find that our NNs can accurately predict the topology of unseen
conformations (80.1% accuracy with a FFNN and 86% accuracy with a recurrent NN
architecture, Fig. 1E). These values are lower than the ones reported in Ref.
Vandans  _et al._ (2020) because we use a smaller training dataset and smaller
NNs. We then trained the same NNs using a range of other geometric features,
such as local curvature, density and 1D writhe Sleiman  _et al._ (2022) (see
SI for details), and found that most of them performed more poorly, or at best
equally, with respect to the XYZ representation (Fig. 1E). A similar outcome
was also obtained in Ref. Braghetto  _et al._ (2023). In stark contrast,
models trained using $\omega_{StA}(x)$ outperformed all other models and are
found to achieve 99.9% accuracy, irrespective of the FFNN or RNN architectures
(we also tested random forest algorithms, see SI). Additionally, the networks
reached the early stopping criterion in about 50% fewer epochs or less,
compared to those trained using the XYZ representation (see SI). When plotted
as a confusion matrix, the results clearly indicate that the XYZ input feature
struggles to classify knots with a similar number of crossings, e.g. the
$5_{1}$ and $5_{2}$ knots. In contrast, our local 3D writhe (StA) feature
generated a near-perfect confusion matrix (Fig. 1F).

We found that these results are generally robust for different choices of
dataset splitting, persistence length ($l_{p}=1-10$ $\sigma$), window length
chosen to perform the StA calculation, and length of the chains (see SI).
Nevertheless, they do display a significant reduction in accuracy when tested
on knots generated using a different method (for instance freely jointed
chains), and also when the window length for the StA writhe calculation is
comparable to the full contour of the chain. In this case, the StA writhe is
constant and equals the global writhe of the knot, which is not unique for
different knots Stasiak  _et al._ (1998). This is also in agreement with a
principal component analysis (PCA, see SI), where we see that different knots
are clearly separable in the reduced 2D PCA space, yet the $0_{1}$ and $4_{1}$
cluster together due to the fact that they share the global writhe (zero),
which is related to the mean value of $\omega_{StA}(x)$ along the contour.

###  II.2 NNs trained with StA writhe can distinguish knots with identical
knot polynomials

Given that our NNs distinguished knots with same minimal number of crossings,
i.e., the 5-crossings knots, we ask ourselves if they could also solve more
complicated problems where knots shared algebraic knot polynomials. To this
end, we first considered three knots with identical Alexander polynomial: the
square, granny, and $8_{20}$ knots (see Fig. 2A). The first two knots are
6-crossings knots consisting of trefoil composites with different chirality
(hence they are homeomorphic knot complements), whereas the latter is an
8-crossings knot. Once again, we trained our FFNN using the $\omega_{StA}(x)$
profiles (Fig. 2B) and obtained a striking accuracy of 99.98%, compared with
91.8% obtained by training with COM-shifted XYZ coordinates (Fig. 2C).

We then asked ourselves if our method could also perform just as well in
situations where the knots shared multiple knot polynomials. As mentioned
above, mutant knots share the same hyperbolic volume and several knot
polynomials, including HOMFLY. We therefore performed simulations of the
Conway (K11n34) knot and one of its mutants, the Kinoshita–Terasaka (KT,
K11n42) knot. These 11-crossings knots have a number of identical knot
invariants as they share the same Jones, Alexander, and Conway polynomials
Adams (1994). Intriguingly, the latter two are also shared with the unknot.
Thus, we generated $10^{5}$ statistically uncorrelated conformations of
$N=200$ beads long polymers with the Conway, KT, and unknot topologies (Fig.
2D), and trained our FFNN to classify them either using a COM-subtracted XYZ
or $\omega_{StA}(x)$ (Fig. 2E) representations. When tested on unseen
conformations, we found that the XYZ-trained NN could not distinguish the
Conway and KT knots, but both are accurately distinguished from the unknot
(Fig. 2F). In marked contrast, we discovered that the StA-trained NN perfectly
disentangles the three knots with 99.6% accuracy (Fig. 2F). We therefore
conclude that the StA-trained NN has the ability to convert StA patterns into
a topological knot classification, even for knots sharing multiple knot
polynomials, such as mutants and composites. In turn, we argue that the StA
writhe is a geometric quantity computed on a particular 3D embedding of a
curve that carries high-density information about its underlying topology.
Importantly, we stress that to classify these knots the network does not
compute any knot polynomial, as other standard software do.

Somewhat unsatisfactorily, we cannot fully pinpoint why StA writhe is so
powerful at identifying different topologies. We hypothesise that the 1D
patterns generated by StA writhe, and specifically the sequence, sign and
amplitudes assumed by consecutive maxima and minima, contain information on
the relative orientation and severity of consecutive entanglements. As
mentioned above, the average value of $\omega_{StA}(x)$ is related to the
global writhe of the knot, which itself contains non-unambiguous information
on its topology. Thus, we argue that the NNs can extract additional
information from the full $\omega_{StA}(x)$ patterns that is related to the
chirality of individual entanglements and render the information unique. This
hypothesis is also supported by the fact that considering unsigned StA writhe
(which cannot distinguish chirality) yields in general a lower accuracy (see
Fig. 1E). We thus hypothesise that the information encoded in the pattern of
the StA writhe may be related to the underlying knot’s Dowker code. These
hypotheses will be tested in more detail in future works.

###  II.3 StS writhe outperforms StA writhe on knots with more than 7
crossings

To understand to what extent StA-trained NNs can be used to classify knotted
curves, we trained our NNs on increasingly more complex classification
problems, and generated conformations of all prime knots up to 10-crossings.
Among these 250 prime knots, there are over 30 that share the same Alexander
polynomial (see SI for a table), making them challenging to classify using
standard tools (for instance Kymoknot). We first noticed that XYZ-trained NNs
rapidly declined in accuracy when we included knots with 6 or more crossings
(Fig. 3A-B). In contrast, the confusion matrices from StA-trained NNs retained
relatively high accuracies. However, we noticed that the knots $5_{1}$ and
$7_{2}$ created some confusion even in the StA-trained NNs, causing a drop in
accuracy to 98% (Fig. 3C-D). We argue that this was due to the fact that
$\omega_{StA}(x)$ displayed similar patterns between the two knot types. For
instance, we show two knots that yield particularly similar $\omega_{StA}(x)$
patterns in Fig. 3C. Thus, to further distinguish these (and potentially other
knots with similar $\omega_{StA}(x)$ curves) we decided to consider our
original proposition of using the local StS writhe (Eq. (1)); two examples of
$\omega_{StS}(x,y)$ maps are reported in Fig. 3E, for the same $5_{1}$ and
$7_{2}$ knots configurations used to compute $\omega_{StA}(x)$ in Fig. 3C.
Interestingly, the $\omega_{StS}(x,y)$ maps appear very different, despite
generating very similar StA curves when integrated along $y$ and around the
polymer contour. This is because a given segment $x$ may itself have a certain
sequence of negative and positive entanglements with other segments $y$. Once
integrated along the contour in the $y$ direction, different sequences may
lead to similar overall values. Motivated by this, we trained our FFNNs using
the StS writhe representation of the knots, and discovered we could restore a
very high (99.8%) accuracy for the case of a database containing all knots up
to 7-crossings (Fig. 3F). More specifically, the confusion between $5_{1}$ and
$7_{2}$ knots is now solved thanks to the StS writhe. Ultimately, the StS-
trained NNs produced the most accurate models, achieving 95% for a 250-class
classification task including all knots up to 10 crossings. In comparison, the
XYZ-trained and StA-trained NNs achieved 17% and 72% on the same problem,
respectively (Fig. 3G).

Based on these results, we argue that the StS writhe is therefore the most
scalable and precise geometric feature to employ for knot classification
problems. Most importantly, we stress that the impressive accuracy for a
250-class problem was achieved with a simple feed forward NN with 4 layers
(around 400k parameters). A natural extension going forward will be to employ
more complex architectures and in particular convolutional NNs to classify the
2D StS writhe maps.

Figure 4: A Example of a $4_{1}$ knot where the knotted core is localised
within $\sim$ 80% of the contour.B Signed and unsigned StA writhe profiles for
the conformation shown in A. C Sketch of a LSTM (recurrent) NN, encoding the
sequential information of the segments. D Profile of the knot probability
$P_{k}(x)$ as a function of bead index $x$, as predicted by the RNN with
different geometric features. The ground truth was generated using KymoKnot.

###  II.4 StA-trained NNs can also solve knot localisation problems.

In the final part of this paper we turn our attention to the knot localisation
problem, i.e. determining the shortest knotted arc along the polymer contour.
This task is challenging and particularly important for open curves, such as
linear polymers, DNA, and proteins Dabrowski-Tumanski  _et al._ (2016);
Giulini and Potestio (2019); Klotz  _et al._ (2018); Caraglio  _et al._ (2019,
2020); Soh  _et al._ (2019), which may contain entanglements and knots. In
this context, identifying the shortest portion of a polymer that is knotted is
akin to being able to identify entanglements in chain melts.

We first tackled this problem using the same FFNN architecture as in the knot
classification task, but the accuracies generated were very low. We
hypothesised that this was due to the fact that FFNN do not preserve the
sequential information along the polymer. For this reason, we consider a long-
short term memory (LSTM), also known as recurrent NN (RNN). More specifically,
we employed a sequence-to-sequence LSTM, with an output layer corresponding to
a binary sequence of $N=100$ neurons, equivalent in dimension to the length of
the input polymer. Each output neuron is passed through a sigmoid function,
which converts the output into a probability between 0 and 1 representing the
likelihood that a given monomer is within the knotted segment of the polymer
conformation. The true output labels were generated using KymoKnot Tubiana
_et al._ (2018), which employs a minimally-interfering closure algorithm
followed by a standard Alexander determinant calculation to identify the start
and end monomers of the knot. This data was then transformed into a vector of
100 bits, i.e. a value of 0 or 1, corresponding to whether a certain monomer
was part of the knotted arc.

Unlike normal multi-class classification problems where the classes are
mutually exclusive, here we consider a multi-label classification task, with
mutually non-exclusive class labels (multiple classes per prediction)
Tsoumakas and Katakis (2007). To quantify the error in a multi-label
classification task, we use the binary cross-entropy (BCE) function, suited to
an output layer of sigmoid functions, given by

$BCE=-\frac{1}{N}\sum_{i=1}^{N}y_{i}\log(\hat{y}_{i})+(1-y_{i})\log(1-\hat{y}_{i})$
(2)

where $y_{i}$ is the $i$th element in the true output vector, y, $\hat{y}_{i}$
is the $i$th element in the predicted output vector, $\hat{y}$, and $N$ is the
dimension of the output label, corresponding to the length of the polymer in
our knot localisation task. This error is then used to optimise the model
weights.

Finally, to determine the accuracy of the model we converted the probabilities
generated by the sigmoid function $y_{prob}$ into binary values using an
Heaviside step function (${y_{pred}}=\Theta({y_{prob}}-0.5)$), and compared to
the true binary value obtained using Kymoknot. The final accuracy is given by
the binary accuracy, i.e. $\text{Accuracy}=\text{correct}/\text{total}$.

Overall, we find that the StA-trained RNNs perform extremely well, reaching
above 90% accuracy in localising any knot that we tested: the 5 simplest knot
types, $0_{1}$, $3_{1}$, $4_{1}$, $5_{1}$ and $5_{2}$ (Fig. 4). We argue that
this excellent performance relies on the effectiveness of RNNs in handling
multi-scale sequential data and tracking multi-scale correlations along the
polymer. This capability likely plays a major role in allowing the network to
recognise that nearby monomers are more likely to be in the same knotted arc.
More precisely, we find that the StA writhe representation is superior to all
other descriptors, with a localisation accuracy of 93%, confirming its
potential usefulness as a tool to help in knot localisation tasks. For
instance, in Fig. 4D we report the prediction and ground truth for the $4_{1}$
knot shown in Fig. 4A-B. In this case, the StA writhe perfectly agrees with
the kymoknot ground truth, whereas the XYZ and unsigned StA writhe yield less
accurate localisation predictions.

In the SI (Fig. S9), we also used our StA-trained RNN model to track the
unknotting of a $5_{1}$ knot tied on an open curve. Despite the fact that the
algorithm was not trained on open curves, the results were surprisingly
accurate. The model can be seen to clearly detect the presence of short
knotted arcs even at the final step before complete unknotting. Once again,
also in this case we find that the StA-trained model is largely superior to
the XYZ-trained model.

Overall, our results highlight the power of StA and StS writhe in not only
classifying but also localising knots. We acknowledge that our results are
non-exhaustive and more work will be needed in the future to find the best
architectures and models to optimally solve these tasks.

##  III Conclusions

In conclusion, we have discovered that local “segment-to-all” and “segment-to-
segment” writhe (Eq. (1)) are geometric descriptors of a curve that contain
information about its underlying topology. Our AI-driven approach can
classify, using a single quantity, complex knot topologies that would
otherwise be impossible to disentangle using a single algebraic invariant.
More specifically, we demonstrated, for the first time, that NNs can utilise
the information encoded in StA and StS writhe to classify the curve topology
significantly more accurately than what can be achieved using the Cartesian
coordinates of the curve’s segments or other local geometric quantities (Fig.
1). We hypothesise that our NNs trained on local 3D writhe representations may
numerically encode a new type of geometric topological invariant. This
conjecture is supported by the fact that even a simple FFNN architecture can
distinguish the topology of knot mutants and composites that share several
algebraic knot polynomials (Fig. 2). Finally, we showed that our new proposed
geometric feature (Eq. (1)) is robust to more complex knots than the ones
tackled in the literature so far; indeed, we have managed to classify all 250
prime knots up to 10-crossings with 95% accuracy (Fig. 3). We argue that
deeper NN or convolutional NN may be able to push this result further, to
$>$10 crossing knots.

We stress that this method only requires a snapshot of a knot embedding with a
list of 3D coordinates for each polymer segment and is trained on thermal
conformations under a readily tunable temperature. For this reason, it will
require longer training for longer polymers but should be essentially
insensitive to the number of non-essential crossings, as shown by the
excellent accuracy achieved in spherically confined polymers Braghetto  _et
al._ (2023). This feature is in marked contrast of standard knot topology
algorithms, that take 2D projections and need to compute matrices as big as
the number of crossings in a given projection, irrespectively whether
essential or not Adams (1994). Finally, we show that by deploying recurrent
NNs, our geometric StA descriptor can also solve a knot localisation problem
(Fig. 4). More work will be needed in the future to find optimal NN
architectures.

We note that albeit we do not have a full understanding of how the NNs are
using StA and StS writhe features to identify knots, we hypothesise that they
are classifying the patterns of consecutive maxima and minima, thus capturing
the entanglement of pairs of segments, accounting for their chirality and
magnitude. This argument directly suggests that employing a distance map
between segments or other geometric “unsigned” representations will yield
lower accuracies, due to the fact that they do not capture the chiral nature
of the entanglements between segments. For these reasons, we believe that StS
(or StA) representations are possibly the best ones to connect the geometry a
given curve embedding to its underlying topology. A possible limitation of
this method is that it is restricted to pair-wise entanglement. Generalising
the Gauss linking number to higher-order relations is itself an active field
of research, and it is foreseeable that a local version of the Milnor triple
linking number Polyak (1997) may be used to generate 3D tensors of Brunnian
links, for example.

In conclusion, we established that StS-trained NNs are powerful tools to
accurately classify and localise knots in thermally equilibrated curves.
Importantly, knot classification and localisation are achieved without any
explicit calculation of Alexander or other algebraic invariants. In other
words, we propose that the local writhe – once fed through deep NNs – yields
an accurate map from the configurational space of a curve to its underlying
topology. The approach we reported in our paper naturally lends itself to be
applied to protein folding Dabrowski-Tumanski and Sulkowska (2017); Giulini
and Potestio (2019), DNA Siebert  _et al._ (2017); Goundaroulis  _et al._
(2020) and, in general, entanglements in open curves and complex systems
Dennis  _et al._ (2010); Everaers  _et al._ (2004); Landuzzi  _et al._ (2020);
Staňo  _et al._ (2022); Rosa  _et al._ (2020); Herschberg  _et al._ (2023);
Caraglio  _et al._ (2019). We hope that our results will also inspire
mathematicians and topologists to formulate new topological invariants based
on the geometrical embedding of knotted curves.

## Acknowledgements

DM thanks the Royal Society for support through a University Research
Fellowship. This project has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
program (grant agreement No 947918, TAP). JLS thanks the Physics of Life
network for a student summer bursary in 2021 providing initial funds for this
research. The authors thank Enzo Orlandini, Marco Baiesi, Pawel Dabrowski-
Tumanski, Ken Millett, and Luigi del Debbio for insightful discussions. The
authors also acknowledge the contribution of the COST Action Eutopia, CA17139.

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