Approximation of the fixed point of the product of two operators in Banach
algebras with applications to some functional equations

Khaled Ben Amara1, Maria Isabel Berenguer2,∗ Aref Jeribi3

(1,3) Department of Mathematics. Faculty of Sciences of Sfax. University of
Sfax. Road Soukra Km $3.5B.P.1171,3000,$ Sfax, Tunisia.
(2) Department of Applied Mathematics and Institute of Mathematics (IEMath-
GR), E.T.S. de Ingenieria de Edificación, University of Granada, Granada,
Spain.
(∗) Corresponding author.

e-mails: (1<EMAIL_ADDRESS>(2<EMAIL_ADDRESS>(3)
<EMAIL_ADDRESS>
Abstract. In this paper, the existence and uniqueness of the fixed point for
the product of two nonlinear operator in Banach algebra is discussed. In
addition, an approximation method of the fixed point of hybrid nonlinear
equations in Banach algebras is established. This method is applied to two
interesting different types of functional equations. In addition, to
illustrate the applicability of our results we give some numerical examples.

Keywords: Banach algebras, Fixed point theory, Integro-differential operators,
Schauder Bases.

AMS Classification: $65$L$03$, $65$R$20$, $47$H$10$, $47$G$20.$

##  1 Introduction

Many phenomena in physics, chemistry, mechanics, electricity, and so as, can
be formulated by using the following nonlinear differential equations with a
nonlocal initial condition of the form:

$\left\\{\begin{array}[]{rl}\displaystyle\frac{d}{dt}\left(\frac{x(t)}{f(t,x(t))}\right)&=g(t,x(t)),t\in
J,\\\ \\\ x(0)&=\Gamma(x),\end{array}\right.$ (P1)

where $\Gamma$ is a mapping from $C(J)$ into $\mathbb{R}$ which represents the
nonlocal initial condition of the considered problem, see [12, 19]. The
nonlocal condition $x(0)=\Gamma(x)$ can be more descriptive in physics with
better effect than the classical initial condition $x(0)=x_{0},$ (see, e.g.
[10, 11, 13, 19]). In the last case, i.e. $x(0)=x_{0},$ the problem (P1) has
been studied by Dhage [17] and O’Regan [26] for the existence of solutions.
Therefore it is of interest to discuss and to approximate the solution of (P1)
with a nonlocal initial condition for various aspects of its solution under
some suitable conditions.
Similarly another class of nonlinear equations is used frequently to describe
many phenomen a in various fields of applied sciences such as physics, control
theory, chemistry, biology, and so forth (see [3], [15], [23] and [24]). This
class is generated by the equations of the form:

$x(t)=f(t,x(\sigma(t)))\cdot\left[q(t)+\displaystyle\int_{0}^{\eta(t)}K(t,s,x(\tau(s)))ds\right],t\in
J.$ (P2)

Both, (P1) and (P2), can be interpreted as fixed point problems in which the
equation involved is a hybrid equation of the type

$x=Ax\cdot Bx.$ (P3)

A hybrid fixed point theorem to (P3) was proved by Dhage in [14] and since
then, several extensions and generalizations of this fixed point result have
been proved. See [16, 18] and the references therein. These results can be
used to establish the existence and uniqueness of solutions. Although the
explicit calculation of the fixed point is only possible in some simple cases,
these results are regarded as one of the most powerful tools to approximate
this fixed point by a computational method and to develop numerical methods
that allow us to approximate the solution of these equations.

In Banach spaces, several works deals to develop numerical techniques to
approximate the solutions of some systems of integro-differential equations,
by using different methods such as the Chebyshev polynomial method [1], the
parameterization method [20], the wavelet methods [22], the secant-like
methods [Argyros], a collocation method in combination with operational
matrices of Berstein polynomials [25], the variational iteration method [27],
etc. A combination method of a fixed point result and Schauder’s basis in a
Banach space have been used in [4, 5, 6, 7] to solve numerically systems of
integral and integro-differential equations. The advantages of this method
over other numerical methods is its simplicity to implement it in a computer
and the approximating functions are the sum of integrals of piecewise
univariate and bivariate polynomials with coefficients easy to calculate.

Since the Banach algebras represents a practical framework for equations such
as (P1) and (P2), and in general (P3), the purposes of this paper are twofold.
Firstly, to present, under suitable conditions, a method to approximate the
fixed point of a hybrid equation of type (P3), by means of the product and
composition of operators defined in a Banach algebra. Secondly, to develop and
apply the method presented to obtain an approximation of the solutions of (P1)
and (P2).

The structure of this paper is as follows: in section 2 we present some
definitions and auxiliary results which will be needed in the sequel; in
section 3 we derive an approximation method for the fixed point of the hybrid
equation (P3); in sections 4 and 5, we apply our results to prove the
existence and the uniqueness of solution of (P1) and (P2), we give an
approximation method for these solutions and moreover, we establish some
numerical examples to illustrate the applicability of our results.

##  2 Analytical tools

In this section, we provide some concepts and results that we will need in the
following sections. The first analytical tool to be used comes from the theory
of the fixed point.

###### Definition 2.1

A mapping $A:X\longrightarrow X$ is said to be $\mathcal{D}$-Lipschitzian, if
there exists a continuous nondecreasing function
$\phi:\mathbb{R}_{+}\longrightarrow\mathbb{R}_{+}$ such that

$\|Ax-Ay\|\leq\phi(\|x-y\|)$

for all $x,y\in X$, with $\phi(0)=0$. The mapping $\phi$ is called the
$\mathcal{D}$-function associate to $A$. If $\phi(r)<r$ for $r>0,$ the mapping
$A$ is called a nonlinear contraction on $X$. $\hfill\diamondsuit$

###### Remark 2.1

The class of $\mathcal{D}$-Lipschitzian mapping on $X$ contains the class of
Lipschitzian mapping on $X$, indeed if $\phi(r)=\alpha\,r$, for some
$\alpha>0$, then $A$ is called Lipschitzian mapping with Lipschitz constant
$\alpha$ or an $\alpha$-Lipschitzian mapping. When $0\leq\alpha<1,$ we say
that $A$ is a contraction. $\hfill\diamondsuit$

The Banach fixed point theorem ensures that every contraction operator $A$ on
a complete metric space $X$ has a unique fixed point $\tilde{x}\in X,$ and the
sequence $\\{A^{n}x,n\in\mathbb{N}\\}$ converges to $\tilde{x}.$ One of the
more useful generalizations of the Banach fixed point principal is the
following result due to Boyd and Wong in [8].

###### Theorem 2.1

Let $(X,d)$ be a complete metric space, and let $A:X\to X.$ Assume that there
exists a continuous function $\varphi:[0,\infty)\to[0,\infty)$ such that
$\varphi(r)<r$ if $r>0,$ and

$d(A(x),A(y))\leq\varphi(d(x,y)),\ \ \forall x,y\in X.$

Then $A$ has a unique fixed point $\tilde{x}\in X.$ Moreover, for any $x\in
X,$ the sequence $x_{n}=A^{n}(x)$ converges to $\tilde{x}.$

###### Remark 2.2

The operator $A(x)=x-x^{2}$ mapping $X=[0,1]$ into itself, and possessing the
unique fixed point $x=0,$ does not satisfy the assumptions of the contraction
principal (the smallest possible $k$ equals $1$), whereas it satisfies those
of Theorem 2.1 with $\varphi(r)=r-r^{2}.$

On the other hand, Schauder bases will constitute the second essential tool.

A biorthogonal system in a Banach space $E$ is a system
$\\{(\tau_{n},\xi_{n}),n\geq 1\\}$ of $E\times E^{*},$ where $E^{*}$ denotes
the topological dual space of $E.$ Moreover, $\\{(\tau_{n},\xi_{n}),n\geq
1\\}$ said to be a fundamental biorthogonal system if
$\overline{span}\\{\tau_{n}\\}=E.$ Now, a sequence
$\\{\tau_{n},n\in\mathbb{N}\\}\subset E$ defines a Schauder basis of $E$ if,
for every $x\in E,$ there is a unique sequence $(a_{n})_{n}\subset\mathbb{R}$
such that

$x=\sum_{n\geq 1}a_{n}\tau_{n}.$

This produces the concept of the canonical sequence of finite dimensional
projections $P_{n}:E\to E,$ defined by the formula

$P_{n}\left(\sum_{k\geq 1}a_{k}\tau_{k}\right)=\sum_{k=1}^{n}a_{k}\tau_{k},$

and the associated sequence of coordinate functionals $\tau_{n}^{*}\in E^{*}$
defined by the formula

$\tau^{*}_{n}\left(\sum_{k\geq 1}a_{k}\tau_{k}\right)=a_{n}.$

Note that a Schauder basis is always a fundamental biorthogonal system, under
the interpretation of the coordinate functionals as biorthogonal functionals.
Moreover, in view of the Baire category theorem [9], that for all $n\geq 1,$
$\tau_{n}^{*}$ and $P_{n}$ are continuous. This yields, in particular, that

$\lim_{n\rightarrow\infty}\|P_{n}(x)-x\|=0.$

The above mentioned notions play an important role to approximating the
solution of different integral and integro-differential equations (see [4, 5,
6].)

##  3 The existence, uniqueness and approximation of a fixed point of the
product of two operators in Banach algebras.

Based on the Boyd-Wong theorem, we establish the following fixed point result
for the product of two nonlinear operators in Banach algebras.

###### Theorem 3.1

Let $X$ be a nonempty closed convex subset of a Banach algebra $E.$ Let
$A,B:X\to E$ be two operators satisfying the following conditions:
$(i)$ $A$ and $B$ are $\mathcal{D}$-lipschitzian with $\mathcal{D}$-functions
$\varphi$ and $\psi$ respectively,
$(ii)$ $A(X)$ and $B(X)$ are bounded,
$(iii)$ $Ax\cdot Bx\in X,$ for all $x\in X.$
Then, there is a unique point $\tilde{x}\in X$ such that $A\tilde{x}\cdot
B\tilde{x}=\tilde{x}$ and the sequence $\\{(A\cdot B)^{n}x\\},x\in X,$
converges to $\tilde{x}$ provided that
$\|A(X)\|\psi(r)+\|B(X)\|\varphi(r)<r,r>0.$ $\hfill\diamondsuit$

Proof. Let $x,y\in X.$ we have

$\begin{array}[]{rcl}\displaystyle\|Ax\cdot Bx-Ay\cdot By\|&\leq&\|Ax\cdot(Bx-
By)\|+\|(Ax-Ay)\cdot By\|\\\ \\\ &\leq&\|Ax\|\,\|Bx-By\|+\|By\|\,\|Ax-Ay\|\\\
\\\ &\leq&\|A(X)\|\,\psi(\|x-y\|)+\|B(X)\|\,\varphi(\|x-y\|).\end{array}$

This implies that $A\cdot B$ defines a nonlinear contraction with
$\mathcal{D}$-function

$\phi(r)=\|A(X)\|\,\psi(r)+\|B(X)\|\,\varphi(r),\ r>0.$

Applying the Boyd-Wong fixed point theorem [8], we obtain the desired result.
$\Box$

Boyd-Wong’s fixed-point theorem allows us to express the fixed point of
$A\cdot B$ as the limit of the sequence of functions $\\{(A\cdot
B)^{n}(x),n\in\mathbb{N}\\},$ with $x\in X.$ Obviously, if it were possible to
explicitly calculate, for each iteration, the expression $(A\cdot B)^{n}(x),$
then for each $n$ we would have an approximation of the fixed point. But, as a
practical matter, such an explicit calculation is only possible in very
particular cases. For this reason, we need to construct another approximation
of the fixed point which is simple to calculate in practice. Therefore, we
need the following Lemmas. The proofs of these Lemmas are similar to those of
Lemma 1 and Lemma 2 in [7].

###### Lemma 3.1

Let $X$ be a nonempty closed convex subset of a Banach algebra $E$ and let
$A,B:X\to E$ and $\phi:\mathbb{R}^{+}\to\mathbb{R}^{+}$ be a continuous
nondecreasing function such that for all $n\geq 1,$

$\left\|(A\cdot B)^{n}x-(A\cdot B)^{n}y\right\|\leq\phi^{n}(\|x-y\|).$

Let $x\in X$ and $T_{0},T_{1},\ldots,T_{m}:E\to E,$ with $T_{0}\equiv I.$ Then

$\begin{array}[]{rcl}\left\|(A\cdot B)^{m}x-T_{m}\circ\ldots\circ
T_{1}x\right\|&\leq&\displaystyle\sum_{p=1}^{m-1}\phi^{m-p}\left(\left\|A\cdot
B\circ T_{p-1}\circ\ldots\circ T_{1}x-T_{p}\circ\ldots\circ
T_{1}x\right\|\right)\\\ \\\ &&+\left\|A\cdot B\circ T_{m-1}\circ\ldots\circ
T_{1}x-T_{m}\circ\ldots\circ T_{1}x\right\|.\end{array}$

$\hfill\diamondsuit$

###### Lemma 3.2

Let $X$ be a nonempty closed convex subset of a Banach algebra $E.$ Let
$A,B:X\to E$ be two $\mathcal{D}$-Lipschitzian operators with
$\mathcal{D}$-functions $\varphi$ and $\psi,$ respectively, and $A\cdot B$
maps $X$ into $X.$ Moreover, suppose that

$\phi(r):=\|A(X)\|\psi(r)+\|B(X)\|\varphi(r)<r,r>0.$

Let $\tilde{x}$ be the unique fixed point of $A\cdot B,$ $x\in X,$
$\varepsilon>0,$ $n\in\mathbb{N}$ such that

$\left\|(A\cdot B)^{n}x-T_{n}\circ\ldots\circ
T_{1}x\right\|\leq\frac{\varepsilon}{2},$

then

$\left\|\tilde{x}-T_{n}\circ\ldots\circ T_{1}x\right\|\leq\varepsilon.$

$\hfill\diamondsuit$

Taking into account the above Lemmas, so as to approximate the solutions of
the problems (P1) and (P2), we will begin with an initial function $x_{0}\in
X$ and we will construct a sequence of operators
$\left\\{S_{n},n\in\mathbb{N}\right\\}$ in order to obtain successive
$T_{n}\circ\ldots\circ T_{1}(x_{0})$ approximations of the fixed point
$\tilde{x}$ of the product $A\cdot B$ following the scheme:

$\begin{array}[]{ccc}x_{0}&&\\\ \downarrow&&\\\ (A\cdot
B)(x_{0})&\approx&T_{1}(x_{0})=A(x_{0})\cdot S_{1}(x_{0})\\\
\downarrow&&\downarrow\\\ (A\cdot B)^{2}(x_{0})&\approx&T_{2}\circ
T_{1}x_{0}=(A\cdot S_{2})\circ T_{1}(x_{0})\\\ \vdots&\vdots&\vdots\\\ \\\
\vdots&\vdots&\vdots\\\ \downarrow&&\downarrow\\\ (A\cdot
B)^{n}(x_{0})&\approx&T_{n}\circ\ldots\circ T_{1}(x_{0})=(A\cdot S_{n})\circ
T_{n-1}\circ\ldots\circ T_{1}(x_{0})\approx\tilde{x}\end{array}$

##  4 Nonlinear differential problems (P1)

In this section we focus our attention in the following nonlinear differential
equation with a nonlocal initial condition:

$\left\\{\begin{array}[]{rl}\displaystyle\frac{d}{dt}\left(\frac{x(t)}{f(t,x(t))}\right)&=g(t,x(t)),t\in
J,\\\ \\\ x(0)&=\Gamma(x),\end{array}\right.$ (P1)

where $J:=[0,\rho],$ $f:J\times\mathbb{R}\to\mathbb{R}\setminus\\{0\\},$
$g:J\times\mathbb{R}\to\mathbb{R}$ and $\Gamma:C(J)\to\mathbb{R}.$ Here $C(J)$
is the space of all continuous functions from $J$ into $\mathbb{R}$ endowed
with the norm $\|\cdot\|_{\infty}=\sup_{t\in J}|x(t)|.$
This equation will be studied under the following assumptions:
$(i)$ The partial mappings $t\mapsto f(t,x),$ $t\mapsto g(t,x)$ are continuous
and the mapping $\Gamma$ is $L_{\Gamma}$-Lipschitzian.
$(ii)$ There exist $r>0$ and two nondecreasing, continuous functions
$\varphi,\psi:\mathbb{R}_{+}\longrightarrow\mathbb{R}_{+}$ such that

$\left|f(t,x)-f(t,{y})\right|\leq\alpha(t)\varphi(|x-y|),t\in J,\hbox{ and
}x,y\in\mathbb{R}\hbox{ with }|x|,|y|\leq r,$

and

$\left|g(t,x)-g(t,{y})\right|\leq\gamma(t)\psi(|x-y|),t\in J\text{ and
}x,y\in\mathbb{R}\hbox{ with }|x|,|y|\leq r.$

$(iii)$ There is a constant $\delta>0$ such that $\sup_{x\in\mathbb{R},|x|\leq
r}|f(0,x)|^{-1}\leq\delta.$

###  4.1 The existence and uniqueness of a solution to problem (P1).

In this subsection, we prove the existence and the uniqueness of a solution to
the functional differential problem (P1).

###### Theorem 4.1

Assume that the assumptions $(i)$-$(iii)$ hold. If

$\displaystyle\displaystyle\left\\{\begin{array}[]{lll}\displaystyle
M_{F}\delta
L_{\Gamma}t+\left(M_{F}\delta^{2}\alpha(0)\left(L_{\Gamma}r+\Gamma(0)\right)+M_{G}\|\alpha\|_{\infty}\right)\varphi(t)+M_{F}\|\gamma(\cdot)\|_{L^{1}}\psi(t)<t,\
t>0,\\\ \\\ \displaystyle M_{F}M_{G}\leq r,\end{array}\right.$

where $r$ is defined in the assumption $(ii),$ then the nonlinear differential
problem (P1) has a unique solution in ${B}_{r}.$

Proof. Let

$\Omega:=\\{x\in C(J);\|x\|\leq r\\}.$

Here the constant $r$ is defined in $(ii).$ Observe that $\Omega$ is a non-
empty, closed, convex and bounded subset of $C(J),$ and the problem of the
existence of a solution to (P1) can be formulated in the following fixed point
problem $Fx\cdot Gx=x,$ where $F,G$ are given for $x\in C(J)$ by

$\displaystyle\left\\{\begin{array}[]{ll}(Fx)(t)&=\displaystyle f(t,x(t))\\\
\\\
(Gx)(t)&=\left[\displaystyle\frac{1}{f(0,x(0))}\Gamma(x)+\displaystyle\int_{0}^{t}g(s,x(s))ds\right],t\in
J.\end{array}\right.$ (4.5)

Let $x\in\Omega$ and $t,t^{\prime}\in J.$ Since $f$ is
$\mathcal{D}$-lipschitzian with respect to the second variable and is
continuous with respect to the first variable, then by using the inequality

$\begin{array}[]{rcl}\displaystyle|f(t,x(t))-f(t^{\prime},x(t^{\prime}))|&\leq&\displaystyle|f(t,x(t))-f(t^{\prime},x(t))|+|f(t^{\prime},x(t))-f(t^{\prime},x(t^{\prime}))|,\end{array}$

we can show that $F$ maps $\Omega$ into $C(J).$
Now, let us claim that $G$ maps $\Omega$ into $C(J).$ In fact, let
$x\in\Omega$ and $t,t^{\prime}\in J$ be arbitrary. Taking into account that
$t\mapsto g(t,x)$ is a continuous mapping, it follows from assumption $(ii)$
that

$\begin{array}[]{rcl}\displaystyle|G(x)(t)-G(x)(t^{\prime})|&\leq&\displaystyle\int_{t^{\prime}}^{t}|g(s,x(s))-g(s,0)|ds+(t-t^{\prime})\|g(\cdot,0)\|_{\infty}\\\
\\\
&\leq&\displaystyle(t-t^{\prime})\left(\|\gamma\|_{\infty}\psi(r)+\|g(\cdot,0)\|_{\infty}\right).\end{array}$

This proves the claim. Our strategy is to apply Theorem 3.1 to show the
existence and the uniqueness of a fixed point for the product $F\cdot G$ in
$\Omega$ which in turn is a continuous solution for problem (P1).

For this purpose, we will claim, first, that $F$ and $G$ are
$\mathcal{D}$-lipschitzian mappings on $\Omega.$ The claim regarding $F$ is
clear in view of assumption $(ii),$ that is $F$ is $\mathcal{D}$-lipschitzian
with $\mathcal{D}$-function $\Phi$ such that

$\Phi(t)=\|\alpha\|_{\infty}\varphi(t),t\in J.$

We corroborate now the claim for $G.$ Let $x,y\in\Omega,$ and let $t\in J.$ By
using our assumptions, we obtain

$\begin{array}[]{rcl}\displaystyle\left|G(x)(t)-G(y)(t)\right|&=&\left|\displaystyle\frac{1}{f(0,x(0))}\Gamma(x)-\frac{1}{f(0,y(0))}\Gamma(y)+\int_{0}^{t}g(s,x(s))-g(s,y(s))ds\right|\\\
\\\
&\leq&\displaystyle\frac{L_{\Gamma}}{|f(0,x(0))|}\|x-y\|+\frac{\alpha(0)}{|f(0,x(0))f(0,y(0))|}\left(L_{\Gamma}r+\Gamma(0)\right)\varphi(\|x-y\|)\\\
\\\ &&+\displaystyle\int_{0}^{t}|\gamma(s)|\psi(|x(s)-y(s))|ds\\\ \\\
&\leq&\delta
L_{\Gamma}\|x-y\|+\delta^{2}\alpha(0)\left(L_{\Gamma}r+\Gamma(0)\right)\varphi(\|x-y\|)+\|\gamma(\cdot)\|_{L^{1}}\psi(\|x-y\|).\end{array}$

Taking the supremum over $t,$ we obtain that $G$ is $\mathcal{D}$-lipschitzian
with $\mathcal{D}$-function $\Psi$ such that

$\Psi(t)=\delta
L_{\Gamma}t+\delta^{2}\alpha(0)\left(L_{\Gamma}r+\Gamma(0)\right)\varphi(t)+\|\gamma(\cdot)\|_{L^{1}}\psi(t),t\in
J.$

On the other hand, bearing in mind assumption $(i),$ by using the above
discussion we can see that $F(\Omega)$ and $G(\Omega)$ are bounded with bounds

$M_{F}:=\|\alpha\|_{\infty}\varphi(r)+\|f(\cdot,0)\|_{\infty}\hbox{ and
}M_{G}:=\delta(L_{\Gamma}r+|\Gamma(0)|)+\|\gamma\|_{\infty}\rho\psi(r)+\rho\|g(\cdot,0)\|_{\infty}.$

Taking into account the estimate $M_{F}M_{G}\leq r,$ we obtain that $F\cdot G$
maps $\Omega$ into $\Omega.$
Now, applying Theorem 3.1, we infer that (P1) has one and only one solution
$\tilde{x}$ in $\Omega,$ and for each $x\in\Omega$ we have

$\displaystyle\lim_{n\rightarrow\infty}(F\cdot G)^{n}x=\tilde{x}.$

$\Box$

Notice that by induction argument we can show that

$\displaystyle\displaystyle\|(F\cdot G)^{n}x-(F\cdot
G)^{n}y\|\leq\Theta^{n}(\|x-y\|),$ (4.6)

where $\Theta(t):=M_{F}\Psi(t)+M_{G}\Phi(t),t\geq 0.$

###  4.2 Numerical method to approximate the solution of (P1).

In this subsection we find a numerical approximation of the solution to the
nonlinear equation (P1) using a Schauder basis in $C(J).$

First, let us consider a Schauder basis $\\{\tau_{n}\\}_{n\geq 1}$ in $C(J)$
and the sequence of associated projections $\\{\xi_{n}\\}_{n\geq 1}.$ Let

$\left\\{\begin{array}[]{ll}T_{p}:C(J)\longrightarrow C(J)\\\ \\\
x\mapsto\displaystyle
T_{p}(x)(t)=F(x)(t)\left(\displaystyle\frac{1}{f(0,x(0))}\Gamma(x)+\displaystyle\int_{0}^{t}\xi_{n_{p}}(U_{0}(x))(s)ds\right),\end{array}\right.$

where $F:C(J)\longrightarrow C(J)$ such that

$F(x)(t)=f(t,x(t))$

and $U_{0}:C(J)\longrightarrow C(J)$ such that

$U_{0}(x)(s)=g(s,x(s)).$

###### Remark 4.1

$(i)$ For all fixed $p\geq 1,$ the mapping $T_{p}$ maps $\Omega$ into
$\Omega.$
In fact, let $x\in\Omega,$ we have

$\begin{array}[]{rcl}\left|T_{p}(x)(t)\right|&=&\left|F(x)(t)\left(\displaystyle\frac{1}{f(0,x(0))}\Gamma(x)+\displaystyle\int_{0}^{t}\xi_{n_{p}}(U_{0}(x))(s)ds\right|\right)\\\
\\\
&\leq&\left|f(t,x(t))\right|\left(\displaystyle\delta|\Gamma(x)|+\int_{0}^{t}\left|\xi_{n_{p}}(U_{0}(x))(s)\right|ds\right).\end{array}$

Proceeding essentially as in the above subsection and using the fact that
$\xi_{n_{p}}$ is a bounded linear operator on $C(J),$ we get

$\begin{array}[]{rcl}\left|T_{p}(x)(t)\right|&\leq&M_{F}\left[\displaystyle\delta|\Gamma(x)|+\rho\left\|\xi_{n_{p}}\left(U_{0}(x)\right)\right\|\right]\\\
\\\
&\leq&M_{F}\displaystyle\left[\displaystyle\delta(L_{\Gamma}r+|\Gamma(0)|)+\rho\sup_{s\in
J}|g(s,x(s))|\right]\\\ \\\ &\leq&M_{F}M_{G}.\end{array}$

In view of assumption $(iii),$ we infer that $T_{p}$ maps $\Omega$ into
$\Omega.$

$(ii)$ Item $(i)$ means, in particular, that for all fixed $p\geq 1,$ the
operator $T_{p}\circ\ldots\circ T_{1}$ maps $\Omega$ into
$\Omega.$$\hfill\diamondsuit$

Our objective is to justify that we can choose $n_{1},\ldots,n_{m},$ so that
the operators $T_{1},\ldots,T_{m}$ can be used to obtain an approximation to
the unique solution of equation (P1).

###### Theorem 4.2

Let $\tilde{x}$ be the unique solution to the nonlinear problem (P1). Let
$x\in\Omega$ and $\varepsilon>0,$ then there exists $n\in\mathbb{N}$ such that

$\left\|\tilde{x}-T_{n}\circ\ldots\circ T_{1}x\right\|\leq\varepsilon.$

$\hfill\diamondsuit$

Proof. Let $x\in\Omega$ and $\varepsilon>0.$ For $p\in\\{1,\ldots,m\\},$ we
define $U_{p}:C(J)\to C(J)$ by

$U_{p}(x)(s):=g(s,T_{p}\circ\ldots\circ T_{1}(x)(s)),\ s\in J,x\in C(J)$

and $F_{p}:C(J)\to C(J)$ by

$F_{p}(x)(s):=F\left(s,T_{p}\circ\ldots\circ T_{1}(x)(s)\right),\ s\in J,x\in
C(J).$

According to inequality (4.6), in view of Lemma 3.1, we get

$\left\|(F\cdot G)^{m}x-T_{m}\circ\ldots\circ T_{1}x\right\|\leq$
$\displaystyle\sum_{p=1}^{m-1}\Theta^{m-p}\left(\left\|(F\cdot G)\circ
T_{p-1}\circ\ldots\circ T_{1}x-T_{p}\circ\ldots\circ
T_{1}x\right\|\right)+\left\|(F\cdot G)\circ T_{m-1}\circ\ldots\circ
T_{1}x-T_{m}\circ\ldots\circ T_{1}x\right\|.$

Taking into account (4.1)-$(i)$ and using similar arguments as in the above
section, we infer that $\left\|F_{p-1}(x)\right\|$ is bounded, and
consequently we get

$\begin{array}[]{rcl}&&\displaystyle\left|(F\cdot G)\circ
T_{p-1}\circ\ldots\circ T_{1}(x)(t)-T_{p}\circ T_{p-1}\circ\ldots\circ
T_{1}(x)(t)\right|\\\ \\\
&\leq&\left|F_{p-1}(x)(t)\left(\displaystyle\int_{0}^{t}g\left(s,T_{p-1}\circ\ldots\circ
T_{1}(x)(s)\right)\,ds-\displaystyle\int_{0}^{t}\xi_{n_{p}}(U_{p-1}(x))(s)\,ds\right)\right|\\\
\\\
&\leq&\left|F_{p-1}(x)(t)\right|\,\displaystyle\int_{0}^{t}\left|\left(\xi_{n_{p}}(U_{p-1}(x))-U_{p-1}(x)\right)(s)\right|\,ds\\\
\\\
&\leq&\rho\left\|F_{p-1}(x)\right\|\,\left\|\xi_{n_{p}}(U_{p-1})(x)-U_{p-1}(x)\right\|.\end{array}$

Then, we obtain

$\left\|(F\cdot G)^{m}x-T_{m}\circ\ldots\circ
T_{1}x\right\|\leq\displaystyle\sum_{p=1}^{m-1}\Theta^{m-p}\left(\rho
M_{F}\,\left\|\xi_{n_{p}}(U_{p-1}(x))-U_{p-1}(x)\right\|\right)+\rho
M_{F}\,\left\|\xi_{n_{m}}(U_{m-1}(x))-U_{m-1}(x)\right\|.$

In view of the convergence property of the Projection operators associated to
the Schauder basis and the continuity of $\Theta,$ we can find
$n_{1},\ldots,n_{m}\geq 1$ and therefore $T_{1},\ldots,T_{m},$ such that

$\|(F\cdot G)^{m}x-T_{m}\circ\ldots\circ T_{1}x\|\leq$
$\displaystyle\sum_{p=1}^{m-1}\Theta^{m-p}\Big{(}\rho
M_{F}\left\|\xi_{n_{p}}(U_{p-1}(x))-U_{p-1}(x)\right\|\Big{)}+\rho
M_{F}\left\|\xi_{n_{m}}(U_{m-1}(x))-U_{m-1}(x)\right\|\leq\displaystyle\frac{\varepsilon}{2}.$

Now apply Lemma 3.2, in order to get

$\left\|\tilde{x}-T_{m}\circ\ldots\circ T_{1}(x)\right\|<\varepsilon.$

$\Box$

###  4.3 Numerical experiments.

This section is devoted to providing some examples and their numerical results
to illustrate the theorems of the above sections. We will consider $J=[0,1]$
and the classical Faber-Schauder system in $C(J)$ where the nodes are the
naturally ordered dyadic numbers (see [21, 28] for details).

###### Example 4.3.1

Consider the nonlinear differential equation with a nonlocal initial condition

$\left\\{\begin{array}[]{ll}&\displaystyle\frac{d}{dt}\left(\frac{x(t)}{f(t,x(t))}\right)=\displaystyle
ae^{-x(t)},\ \ t\in J,\\\ \\\ &\displaystyle x(0)=b(\sup_{t\in
J}|x(t)|+3/4)\end{array}\right.$ (4.7)

where $f(t,x)=\displaystyle\frac{b}{1+ae^{-b}t},$ $g(t,x)=\displaystyle
ae^{-x},$ $\Gamma(u)=b\left(\sup_{t\in J}|u(t)|+3/4\right),$ with
$a<1/\log(2).$
Let $R$ be small enough such that $a(\log(2)+R)<1.$ Let $x,y\in[-R,R],$ by an
elementary calculus we can show that

$\begin{array}[]{rcl}\left|f(t,x)-f(t,y)\right|&\leq&\displaystyle\alpha(t)\varphi(|x-y|)\end{array}$

and

$\begin{array}[]{rcl}\left|g(t,x)-g(t,y)\right|&\leq&\displaystyle\gamma(t)\psi(|x-y|)\end{array}$

where $\displaystyle\alpha(t)=\varphi(t)=0,$ $\gamma(t)=ae^{R}(1-e^{-t}),$ and
$\psi(t)=t.$
On the other hand, we have that $\Gamma$ is Lipschizian with a Lipschiz
constant $L_{\Gamma}=b,$ and

$\displaystyle\sup_{x,|x|\leq R}[f(0,x)]^{-1}\leq\delta=\frac{1}{b}.$

Applying Theorem 4.1, we obtain that (4.7) has a unique solution in
$\Omega=\left\\{x\in C([0,1]);\|x\|\leq 3/4\right\\},$ when $a$ is small
enough. In fact the solution is $x(t)=b.$ We apply the numerical method for
$a=0.1$ and the initial $x_{0}(t)=\frac{1}{4}\left(\sqrt{bt}+1\right).$

Table 1. Numerical results for (4.7) with initial
$x_{0}(t)=\frac{1}{4}\left(\sqrt{bt}+1\right)$.

|  | $n_{1}=\dots=n_{m}=9$ | $n_{1}=\dots=n_{m}=33$
---|---|---|---
$t$ | $x^{*}(t)$ | $m=2$ | $m=4$ | $m=2$ | $m=4$
$0.1$ | $0.25$ | 0.25964068562641207 | 0.2526360625738145 | 0.25779577744548676 | 0.25062384017038686
$0.2$ | $0.25$ | 0.2581608132685836 | 0.2512245431325148 | 0.2576778369861067 | 0.2506151528771704
$0.3$ | $0.25$ | 0.25785705013803817 | 0.25102089532293176 | 0.2575623161293616 | 0.25060665510642743
$0.4$ | $0.25$ | 0.25774159101031774 | 0.25100874582984495 | 0.25744919245075903 | 0.2505983412941664
$0.5$ | $0.25$ | 0.2576285346430475 | 0.2509968386936278 | 0.25733843642963494 | 0.2505902060799007
$0.6$ | $0.25$ | 0.25751784650251447 | 0.2509851672563384 | 0.2572300145734221 | 0.2505822442972077
$0.7$ | $0.25$ | 0.2574094899054699 | 0.25097372508850474 | 0.2571238909612659 | 0.2505744509661791
$0.8$ | $0.25$ | 0.25730342712624404 | 0.2509625059364119 | 0.25702002815700664 | 0.25056682128612107
$0.9$ | $0.25$ | 0.25719961932300417 | 0.25095150376429876 | 0.2569183878122034 | 0.25055935062726176
$1$ | $0.25$ | 0.2570980270442474 | 0.2509407127451644 | 0.25681893107062653 | 0.2505520345235613

| $n_{1}=\dots=n_{m}=9$ | $n_{1}=\dots=n_{m}=33$
---|---|---
| $m=2$ | $m=4$ | $m=2$ | $m=4$
$\|x^{*}-\tilde{x}\|_{\infty}$ | $9.90603\times 10^{-3}$ | $2.86369\times 10^{-3}$ | $8.33966\times 10^{-3}$ | $1.0862\times 10^{-3}$

###### Example 4.3.2

Consider the nonlinear differential equation with a nonlocal initial condition

$\left\\{\begin{array}[]{ll}&\displaystyle\frac{d}{dt}\left(\frac{x(t)}{f(t,x(t))}\right)=\displaystyle
a(x(t))^{2},\ \ t\in J,\\\ \\\ &\displaystyle x(0)=1/(4b)\sup_{t\in
J}|x(t)|^{2},\end{array}\right.$ (4.8)

where $f(t,x)=\displaystyle\frac{b(t+1)}{1+\frac{ab^{2}}{3}(x^{3}/b^{3}-1)}$
and $g(t,x)=ax^{2},$ with $ab^{2}<3.$
Let $R>0$ such that $2b\leq R$ and $\frac{a}{3b}(b^{3}+R^{3})<1.$ Let
$x,y\in[-R,R].$ By an elementary calculus we can show that

$\begin{array}[]{rcl}\left|f(t,x)-f(t,y)\right|&\leq&\displaystyle\alpha(t)\varphi(|x-y|)\end{array}$

and

$\begin{array}[]{rcl}\left|g(t,x)-g(t,y)\right|&\leq&\displaystyle\gamma(t)\psi(|x-y|)\end{array}$

where
$\displaystyle\alpha(t)=\frac{a(t+1)R^{2}}{\left(1-\frac{a}{3b}(R^{3}+b^{3})\right)^{2}},$
$\gamma(t)=2aR,$ and $\varphi(t)=\psi(t)=t.$
On the other hand, we have that

$\begin{array}[]{rcl}\displaystyle|\Gamma(u)-\Gamma(v)|&\leq&\displaystyle\frac{R}{2b}\|u-v\|.\end{array}$

Consequently, $\Gamma$ is Lipschizian with aLipschiz constant
$L_{\Gamma}=\frac{R}{2b}.$ It is easy to prove that

$\sup_{x\in\mathbb{R},|x|\leq R}[f(0,x)]^{-1}\leq\delta=aR^{3}/(3b^{2})+1/b.$

Now, applying Theorem 4.1, in order to obtain that (4.8), with $a$ is small
enough, has a unique solution in $\Omega=\left\\{x\in C([0,1]);\|x\|\leq
1/2\right\\}.$ We can check that the solution is $x(t)=b(t+1).$

The following table shows the numerical results of the proposed method for
$a=0.05$, $b=1/4$ and $x_{0}(t)=\frac{1}{2}t.$

Table 2. Numerical results for (4.8) with initial $x_{0}(t)=\frac{1}{2}t$.

|  | $n_{1}=\dots=n_{m}=9$ | $n_{1}=\dots=n_{m}=33$
---|---|---|---
$t$ | $x^{*}(t)$ | $m=2$ | $m=4$ | $m=2$ | $m=4$
$0.1$ | $0.275$ | 0.27417118067351837 | 0.27151545133640886 | 0.2740819659311709 | 0.27145329704728827
$0.2$ | $0.3$ | 0.2990105059004299 | 0.29611673530305527 | 0.2989981055587191 | 0.29613324654650613
$0.3$ | $0.325$ | 0.32391591487977106 | 0.3207837845940706 | 0.3239149335622202 | 0.3208140511167786
$0.4$ | $0.35$ | 0.3488342313971621 | 0.34546352791535867 | 0.3488329357145739 | 0.34549585475483185
$0.5$ | $0.375$ | 0.3737541821287371 | 0.3701445199310059 | 0.3737524860894689 | 0.3701788114857308
$0.6$ | $0.40$ | 0.39867580493805804 | 0.3948268789541488 | 0.39867366683899425 | 0.39486308640853285
$0.7$ | $0.425$ | 0.42359867388137695 | 0.4195107187398104 | 0.4235961144223516 | 0.41954885401447617
$0.8$ | $0.45$ | 0.4485219829977168 | 0.4441962543294659 | 0.448518998226605 | 0.44423629583080837
$0.9$ | $0.475$ | 0.47344474190449987 | 0.4688837174935067 | 0.47344130014978747 | 0.468925600958778
$1$ | $0.5$ | 0.498366567564148 | 0.49357335586512446 | 0.49836264112050677 | 0.4936169655580174

| $n_{1}=\dots=n_{m}=9$ | $n_{1}=\dots=n_{m}=33$
---|---|---
| $m=2$ | $m=4$ | $m=2$ | $m=4$
$\|x^{*}-\tilde{x}\|_{\infty}$ | $1.63343\times 10^{-3}$ | $6.42664\times 10^{-3}$ | $1.63736\times 10^{-3}$ | $6.38303\times 10^{-3}$

##  5 Nonlinear integral equations of type (P2).

This section deals with the following nonlinear integral equation:

$x(t)=f(t,x(\sigma(t)))\cdot\left[q(t)+\displaystyle\int_{0}^{\eta(t)}K(t,s,x(\tau(s)))ds\right],t\in
J.$ (P2)

where $\sigma,\tau,\eta:J\to J,$ $f:J\times\mathbb{R}\to\mathbb{R},q\in C(J)$
and $K:J\times J\times\mathbb{R}\to\mathbb{R}.$
More precisely, we prove the existence and the uniqueness of a solution to
equation (P2), and then we provide an approximation method of this solution.
In our consideration, we need the following hypotheses:
$(i)$ The partial mappings $t\mapsto f(t,x)$ and $(t,s)\mapsto K(t,s,x)$ are
continuous.
$(ii)$ There exist $r>0$ and two nondecreasing, continuous functions
$\varphi,\psi:\mathbb{R}_{+}\longrightarrow\mathbb{R}_{+}$ such that

$\left|f(t,x)-f(t,{y})\right|\leq\alpha(t)\varphi(|x-y|),t\in J,\hbox{ and
}x,y\in\mathbb{R}\hbox{ with }|x|,|y|\leq r,$

and

$\left|K(t,s,x)-K(t,s,{y})\right|\leq\gamma(t,s)\psi(|x-y|),t,s\in J\text{ and
}x,y\in\mathbb{R}\hbox{ with }|x|,|y|\leq r.$

###  5.1 The existence and uniqueness of a solution to Eq. (P2)

To allow the abstract formulation of equation (P2), we define the following
operators by

$\displaystyle\left\\{\begin{array}[]{ll}(Fx)(t)&=\displaystyle
f(t,x(\sigma(t)))\\\ \\\ (Gx)(t)&=\left[\displaystyle
q(t)+\displaystyle\int_{0}^{\eta(t)}K(t,s,x(\tau(s)))ds\right],t\in
J.\end{array}\right.$ (5.4)

First, we will establish the following result which shows the existence and
uniqueness of a solution.

###### Theorem 5.1

Assume that the assumptions $(i)$-$(ii)$ hold. If

$\displaystyle\displaystyle\left\\{\begin{array}[]{ll}\displaystyle
M_{F}\rho\|\gamma\|_{\infty}\psi(t)+M_{G}\|\alpha\|_{\infty}\varphi(t)<t,\
t>0\\\ \\\ \displaystyle M_{F}M_{G}\leq r,\end{array}\right.$ (5.8)

where

$M_{F}=\|\alpha\|_{\infty}\varphi(r)+\|f(\cdot,\theta)\|_{\infty}\hbox{ and
}M_{G}=\displaystyle\|q(\cdot)\|_{\infty}+\rho\left(\|k(\cdot,\cdot,0)\|_{\infty}+\|\gamma\|_{\infty}\psi(r)\right),$

then the nonlinear integral equation (P2) has a unique solution in ${B}_{r}.$

Proof. Let $\Omega:=\\{x\in C(J);\|x\|_{\infty}\leq r\\}.$ By using similar
arguments to those in the above section, we can show that $F$ and $G$ define
$\mathcal{D}$-lipschitzian mappings from $\Omega$ into $C(J),$ with
$\mathcal{D}$-functions $\|\alpha\|_{\infty}\varphi$ and
$\rho\|\gamma\|_{\infty}\psi,$ respectively. Also it is easy to see that
$F(\Omega)$ and $G(\Omega)$ are bounded with bounds, respectively,

$M_{F}=\|\alpha\|_{\infty}\varphi(r)+\|f(\cdot,0)\|_{\infty}\hbox{ and
}M_{G}=\|q\|_{\infty}+\rho\left(\|\gamma\|_{\infty}\psi(r)+\|K(\cdot,\cdot,0)\|_{\infty}\right).$

Taking into account our assumptions, we deduce that $F\cdot G$ maps $\Omega$
into $\Omega.$ Now, an application of Theorem 3.1 yields that (P2) has one and
only one solution $\tilde{x}$ in $\Omega,$ and for each $x\in\Omega$ we have
$\displaystyle\lim_{n\rightarrow\infty}(F\cdot G)^{n}x=\tilde{x}.$ $\Box$

Moreover, by induction argument we can obtain

$\displaystyle\displaystyle\|(F\cdot G)^{n}x-(F\cdot
G)^{n}y\|\leq\Theta^{n}(\|x-y\|),$ (5.9)

where
$\Theta(t):=\rho\|\gamma\|_{\infty}M_{F}\psi(t)+\|\alpha\|_{\infty}M_{G}\varphi(t),t\geq
0.$

###  5.2 A Numerical method to approximate the solution to (P2).

In this subsection we provide a numerical approximation of the solution to the
nonlinear equation (P2). Now let us consider a Schauder basis
$\\{\tau_{n}\\}_{n\geq 1}$ in $C(J\times J)$ and the sequence of associated
projections $\\{\xi_{n}\\}_{n\geq 1}.$ Let

$\left\\{\begin{array}[]{ll}T_{p}:C(J)\longrightarrow C(J)\\\ \\\
x\mapsto\displaystyle
T_{p}(x)(t)=F(x)(t)\left({q}(t)+\displaystyle\int_{0}^{\eta(t)}\xi_{n_{p}}(U_{0}(x))(t,s)ds\right),\end{array}\right.$

where $F:C(J)\longrightarrow C(J)$ such that

$F(x)(t)=f(t,x(\sigma(t)))$

and $U_{0}:C(J)\longrightarrow C(J\times J)$ such that

$U_{0}(x)(t,s)=K(t,s,x(\tau(s))).$

###### Remark 5.1

$(i)$ For all fixed $p\geq 1,$ the mapping $T_{p}$ maps $\Omega$ into
$\Omega.$
In fact, let $x\in\Omega,$ we have

$\left|T_{p}(x)(t)\right|=\left|F(x)(t)\left(\displaystyle{q}(t)+\displaystyle\int_{0}^{\eta(t)}\xi_{n_{p}}(U_{0}(x))(t,s)ds\right)\right|\\\
\leq\left|f(t,x(\sigma(t)))\right|\,\left(\displaystyle\left|{q}(t)\right|+\displaystyle\int_{0}^{\eta(t)}\left|\xi_{n_{p}}(U_{0}(x))(t,s)\right|ds\right).$

Proceeding essentially as in the above section and using the fact that
$\xi_{n_{p}}$ is a bounded linear operator on $C(J\times J),$ we get

$\left|T_{p}(x)(t)\right|\leq
M_{F}\left(\left|q(t)\right|+\displaystyle\rho\left\|\xi_{n_{p}}\left(U_{0}(x)\right)\right\|\right)\leq
M_{F}\left(\left\|q\right\|_{\infty}+\displaystyle\rho\sup_{t,s\in
J}|k(t,s,x(\tau(s)))|\right)\\\ \\\ \leq M_{F}M_{G}.$

In view of our assumptions, we infer that $T_{p}$ maps $\Omega$ into $\Omega.$

$(ii)$ Item $(i)$ means, in particular, that for all fixed $p\geq 1,$ the
operator $T_{p}\circ\ldots\circ T_{1}$ maps $\Omega$ into $\Omega.$

Again, our objective is to justify that we can choose $n_{1},\ldots,n_{m},$ so
that the operators $T_{1},\ldots,T_{m}$ can be used to obtain an approximation
of the unique solution to equation (P2).

###### Theorem 5.2

Let $\tilde{x}$ be the unique solution to the nonlinear equation (P2). Let
$x\in\Omega$ and $\varepsilon>0,$ then there exists $n\in\mathbb{N}$ such that

$\left\|\tilde{x}-T_{n}\circ\ldots\circ T_{1}x\right\|\leq\varepsilon.$

$\hfill\diamondsuit$

Proof. Let $x\in\Omega$ and $\varepsilon>0.$ For $p\in\\{1,\ldots,m\\},$ we
define $U_{p}:C(J)\to C(J\times J)$ by

$U_{p}(x)(t,s):=K(t,s,T_{p}\circ\ldots\circ T_{1}(x)(s)),\ t,s\in J,x\in C(J)$

and $F_{p}:C(J)\to C(J)$ by

$F_{p}(x)(s):=f\left(s,T_{p}\circ\ldots\circ T_{1}(x)(s)\right),\ s\in J,x\in
C(J).$

According to Lemma 3.1, we get

$\left\|(F\cdot G)^{m}x-T_{m}\circ\ldots\circ T_{1}x\right\|\leq$
$\displaystyle\sum_{p=1}^{m-1}\Theta^{m-p}\left(\left\|(F\cdot G)\circ
T_{p-1}\circ\ldots\circ T_{1}x-T_{p}\circ\ldots\circ
T_{1}x\right\|\right)+\left\|(F\cdot G)\circ T_{m-1}\circ\ldots\circ
T_{1}x-T_{m}\circ\ldots\circ T_{1}x\right\|.$

Taking into account Remark 5.1, we infer that $\left\|F_{p-1}(x)\right\|$ is
bounded. Proceeding essentially, as in the above section, we get

$\begin{array}[]{rcl}\displaystyle\left|(F\cdot G)\circ
T_{p-1}\circ\ldots\circ T_{1}(x)(t)-T_{p}\circ T_{p-1}\circ\ldots\circ
T_{1}(x)(t)\right|\leq\rho\left\|F_{p-1}(x)\right\|\,\left\|\xi_{n_{p}}(U_{p-1})(x)-U_{p-1}(x)\right\|,\end{array}$

which implies that

$\left\|(F\cdot G)^{m}x-T_{m}\circ\ldots\circ T_{1}x\right\|\leq$
$\displaystyle\sum_{p=1}^{m-1}\Theta^{m-p}\left(\rho
M_{F}\,\left\|\xi_{n_{p}}(U_{p-1})(x)-U_{p-1}(x)\right\|\right)+\rho
M_{F}\,\left\|\xi_{n_{m}}(U_{m-1})(x)-U_{m-1}(x)\right\|.$

In view of the convergence property of the Projection operators associated to
the Schauder basis, we can find $n_{1},\ldots,n_{m}\geq 1$ and therefore
$T_{1},\ldots,T_{m},$ such that

$\|(F\cdot G)^{m}x-T_{m}\circ\ldots\circ T_{1}x\|\leq$
$\displaystyle\sum_{p=1}^{m-1}\Theta^{m-p}\Big{(}\rho
M_{F}\left\|\xi_{n_{p}}(U_{p-1})(x)-U_{p-1}(x)\right\|\Big{)}+\rho
M_{F}\left\|\xi_{n_{m}}(U_{m-1})(x)-U_{m-1}(x)\right\|\\\ \\\
\leq\displaystyle\frac{\varepsilon}{2}.$

Now apply Lemma 3.2, in order to get

$\|\tilde{x}-T_{m}\circ\ldots\circ T_{1}(x)\|<\varepsilon.$

$\Box$

###  5.3 Numerical experiments.

This section is devoted to give some examples and their numerical results to
illustrate the previous results using the usual Schauder basis in
$C([0,1]^{2})$ with the well know square ordering (see for example [21]).

###### Example 5.3.1

Consider the nonlinear differential equation

$\displaystyle x(t)=\displaystyle
a(t+1)\left[\frac{b}{a}-\frac{b^{2}}{3}\left((t+1)^{3}-1\right)+\int_{0}^{t}(x(s))^{2}ds\right],\
\ \ t\in J.$ (5.10)

Proceeding essentially as in subsection 5.1, equation (5.10) can be written as
a fixed point problem

$x=F(x)\cdot G(x),$

where $F$ and $G$ are defined in (5.4), with $f(t,x)=a(t+1),$
$q(t)=b/a-\frac{b^{2}}{3}\left((t+1)^{3}-1\right)$ and $k(t,s,x)=x^{2}.$
Let $x,y\in[-R,R],$ we have that

$\left|k(t,s,x)-k(t,s,y)\right|\leq\gamma(t,s)\psi(|x-y|)$

where $\displaystyle\gamma(t,s)=2R,$ and $\psi(t)=t.$
An application of Theorem 5.1, yields that (4.8) has a unique solution in
$\Omega=\left\\{x\in C([0,1]);\|x\|\leq 3\right\\}$, in fact the solution is
$x(t)=b(t+1).$
Using the proposed method with $a=0.1,$ $b=0.1$ and $x_{0}(t)=t^{2},$ we
obtain the following table:

Table 3. Numerical results for the (5.10) with initial $x_{0}(t)=t^{2}$.
$n_{1}=\dots=n_{m}=9$ $n_{1}=\dots=n_{m}=33$ $t$ $x^{*}(t)$ $m=2$ $m=4$ $m=2$
$m=4$ $0.1$ $0.11$ 0.10994463333333335 0.10994463333333335 0.10995952813954675
0.10995955765685321 $0.2$ $0.12$ 0.11981787538985864 0.11981791805770493
0.11994695097301926 0.11994727822516114 $0.3$ $0.13$ 0.12975090261315447
0.12975116990203317 0.12993149120265635 0.1299327014013851 $0.4$ $0.14$
0.1396858063225927 0.13968664031615474 0.13991267664284926 0.13991561146443787
$0.5$ $0.15$ 0.14960946152701812 0.1496116012197044 0.14989041600508018
0.14989578496520412 $0.6$ $0.16$ 0.15952079194994145 0.1595251486759711
0.15986592032169128 0.15987299132148372 $0.7$ $0.17$ 0.16941936215524034
0.1694262809122463 0.1698435692932222 0.1698469898893412 $0.8$ $0.18$
0.17930673541360537 0.1793140741901599 0.17983433359550066 0.17981752625254807
$0.9$ $0.19$ 0.18918899833227526 0.18918756887790722 0.18986179093331174
0.1897843325246908 $1$ $0.2$ 0.19908194969111695 0.1990457618518603
0.1999725602822185 0.1997471266515799

| $n_{1}=\dots=n_{m}=9$ | $n_{1}=\dots=n_{m}=33$
---|---|---
| $m=2$ | $m=4$ | $m=2$ | $m=4$
$\|x^{*}-\tilde{x}\|_{\infty}$ | $9.1805\times 10^{-4}$ | $9.544238\times 10^{-4}$ | $1.65588\times 10^{-4}$ | $2.52873\times 10^{-4}$

###### Example 5.3.2

Consider the nonlinear differential equation

$\displaystyle
x(t)=\displaystyle\left(ae^{-x(t)}+b\right)\left[\frac{t}{ae^{-t}+b}+\frac{1}{1-c}\log(\cos(1-c)t)+\int_{0}^{t}\tan(1-c)x(s)ds\right].$
(5.11)

Similarly to that above, (5.11) can be written as a fixed point problem

$x=F(x)\cdot G(x),$

with the same notations in (5.4).
Let $R>0$ and let $x,y\in[-R,R].$ By an elementary calculus we can show that

$|f(t,x)-f(t,y)|\leq\alpha(t)\varphi(|x-y|)$

and

$|k(t,s,x)-k(t,s,y)|\leq\gamma(t)\psi(|x-y|)$

where $\alpha(t)=ae^{R},$ $\displaystyle\gamma(t)=(1+\tan^{2}(1-c)R),$ and
$\varphi(t)=(1-e^{-t})$ and $\psi(t)=\tan(1-c)t.$
Apply Theorem 5.1, (5.11), with $a$ small enough and $c=1-a,$ has a unique
solution in $\Omega=\left\\{x\in C([0,1]);\|x\|\leq 3\right\\}$, in fact the
solution is $x(t)=t.$

The following table show the numerical results of the proposed method for
$a=0.01,b=1,R=3,$ and $x_{0}(t)=\sin(t).$

Table 4. Numerical results for (5.11) with initial $x_{0}(t)=\sin(t).$

|  | $n_{1}=\dots=n_{m}=9$ | $n_{1}=\dots=n_{m}=33$
---|---|---|---
$t$ | $x^{*}(t)$ | $m=2$ | $m=4$ | $m=2$ | $m=4$
$0.1$ | $0.1$ | 0.09994959265924196 | 0.09994959275258121 | 0.09997341307990056 | 0.09997341318295203
$0.2$ | $0.2$ | 0.19982697772113361 | 0.19982698062053245 | 0.19994196511596454 | 0.19994196762406424
$0.3$ | $0.3$ | 0.29970145954436234 | 0.2997014781005956 | 0.2999105557764353 | 0.29991056948622924
$0.4$ | $0.4$ | 0.39957605185527684 | 0.3995761128223367 | 0.39987915932823637 | 0.3998792008487213
$0.5$ | $0.5$ | 0.49945067663057896 | 0.49945081633085925 | 0.4998477565970089 | 0.49984784689621164
$0.6$ | $0.6$ | 0.5993252823989378 | 0.5993255387084228 | 0.5998163380147611 | 0.5998164954408373
$0.7$ | $0.7$ | 0.699199836738067 | 0.6992002390137386 | 0.699784904433895 | 0.6997851365136741
$0.8$ | $0.8$ | 0.799074326433192 | 0.7990748839377436 | 0.7997534658105931 | 0.7997537620153589
$0.9$ | $0.9$ | 0.8989487530236073 | 0.8989494465775325 | 0.8997220384049086 | 0.8997223654190059
$1$ | $1$ | 0.9988231278772514 | 0.9988239054111422 | 0.9996906411587515 | 0.9996909415162489

| $n_{1}=\dots=n_{m}=9$ | $n_{1}=\dots=n_{m}=33$
---|---|---
| $m=2$ | $m=4$ | $m=2$ | $m=4$
$\|x^{*}-\tilde{x}\|_{\infty}$ | $1.17687\times 10^{-3}$ | $1.17609\times 10^{-3}$ | $3.09359\times 10^{-4}$ | $3.09058\times 10^{-4}$

###### Example 5.3.3

Consider the nonlinear differential equation

$\displaystyle x(t)=\displaystyle
at\left[(b+t)^{2}+\frac{t}{(t+1)}\int_{0}^{t}\left(1-e^{-(t+1)(as+1)}\right)ds\right]^{-1}\left[(b+t)^{2}+\int_{0}^{t}\int_{0}^{x(s)+1}e^{-(t+1)u}duds\right].$
(5.12)

According to the above discussion, (5.12) can be written as a fixed point
problem

$x=F(x)\cdot G(x),$

where $F$ and $G$ are defined in (5.4), with $f(t,x)=\displaystyle
at\left[(b+t)^{2}+\frac{t}{(t+1)}\int_{0}^{t}\left(1-e^{-(t+1)(as+1)}\right)ds\right]^{-1}$
and $k(t,s,x)=\int_{0}^{x+1}e^{-(t+1)u}du.$

Let $0<R<1$ and let $x,y\in[-R,R].$ By an elementary calculus, we can show
that

$|f(t,x)-f(t,y)|\leq\alpha(t)\varphi(|x-y|)$

and

$|k(t,s,x)-k(t,s,y)|\leq\gamma(t)\psi(|x-y|)$

where $\alpha(t)=\varphi(t)=0,$ $\psi(t)=\int_{0}^{2t}e^{-s}ds,$ and
$\gamma(t,s)=\frac{1}{t+1}e^{(t+1)(R-1)}.$
Taking $a=0.1,b=1,$ and applying Theorem 5.1, (5.12) has a unique solution in
$\Omega=\left\\{x\in C([0,1]);\|x\|\leq R\right\\}$. In fact the solution is
$at.$

Table 5. Numerical results for (5.12) with initial $x_{0}(t)=1/2cos(10\pi t).$

|  | $n_{1}=\dots=n_{m}=9$ | $n_{1}=\dots=n_{m}=33$
---|---|---|---
$t$ | $x^{*}(t)$ | $m=2$ | $m=4$ | $m=2$ | $m=4$
$0.1$ | $0.01$ | 0.009807889768197995 | 0.009807889768197995 | 0.009850176149181912 | 0.009850173620253975
$0.2$ | $0.02$ | 0.01913347555848587 | 0.01913346934141619 | 0.019763982715543943 | 0.01976400675926519
$0.3$ | $0.03$ | 0.028858901595188682 | 0.028858870390823594 | 0.029713698642337805 | 0.029713848529122386
$0.4$ | $0.04$ | 0.038745648738524936 | 0.038745618536895766 | 0.039685125981635705 | 0.039685476825051164
$0.5$ | $0.05$ | 0.04868660113266646 | 0.0486866179763731 | 0.04967020617859288 | 0.04967087311797989
$0.6$ | $0.06$ | 0.05866572236367756 | 0.05866579674631664 | 0.05966446431697861 | 0.05966546941999516
$0.7$ | $0.06$ | 0.06866841551194598 | 0.06866853944486338 | 0.06966463823770887 | 0.06966603029961263
$0.8$ | $0.08$ | 0.07868630321311545 | 0.07868650513410157 | 0.07966879607286238 | 0.07967053753105567
$0.9$ | $0.09$ | 0.08871376330117915 | 0.08871405879246874 | 0.08967554656783944 | 0.08967762811140002
$1$ | $0.09$ | 0.0987469779423797 | 0.0987473453913395 | 0.09968401150389958 | 0.09968636366339986

| $n_{1}=\dots=n_{m}=9$ | $n_{1}=\dots=n_{m}=33$
---|---|---
| $m=2$ | $m=4$ | $m=2$ | $m=4$
$\|x^{*}-\tilde{x}\|_{\infty}$ | $1.33714\times 10^{-3}$ | $1.33705\times 10^{-3}$ | $3.35272\times 10^{-4}$ | $3.34982\times 10^{-4}$

##  6 Conclusions

In this paper we have presented a numerical method, based on the use of
Schauder’s bases, to solve hybrid nonlinear equations in Banach algebras. To
do this, we have used Boyd-Wong’s theorem to establish the existence and
uniqueness of a fixed point for the product of two nonlinear operators in
Banach algebra (Theorem 3.1). The method is applied to a wide class of
nonlinear integro-differential equations such as the ones we have illustrated
by means of several numerical examples.

The possibility of applying this process or a similar idea to other types of
hybrid equations or systems of such equations is open and we hope to discuss
this in the near future.

## Acknowledgements

The research of Aref Jeribi and Khaled Ben Amara has been partially supported
by the University of Sfax.

The research of Maria Isabel Berenguer has been partially supported by Junta
de Andalucia, Project FQM359 and by the “Maria de Maeztu” Excellence Unit
IMAG, reference CEX2020-001105-M, funded by MCIN/AEI/10.13039/501100011033/ .

## References

  * [1] Akyüz-Daşcioǧlu, A.; Sezer, M. Chebyshev polynomial solutions of systems of higher-order linear Fredholm-Volterra integro-differential equations. J. Franklin Inst. 342 (2005), 688–701.
  * [2] Argyros, I. L.; Ezquerro, J. A.; Henandez, M. A.; Hilout, S.; Domero, N.; Velasco, A. I. Expanding the applicability of secant like methods for solving nonlinear equations. Carphatian J. Math. 31 (2015), 11–30.
  * [3] Ben Amar, A.; Chouayekh, S.; Jeribi, A. Fixed point theory in a new class of Banach algebras and application. Afr. Mat. 24 (2013), 705–724.
  * [4] Berenguer, M. I.; Garralda-Guillem, A. I.; Ruiz Galán, M. Biorthogonal systems approximating the solution of the nonlinear Volterra integro-differential equation. Fixed Point Theory Appl. 2010, Art. ID 470149, 9 pp.
  * [5] Berenguer, M.I.; Gámez, D.; López Linares, A.J. Fixed point techniques and Schauder bases to approximate the solution of the first order nonlinear mixed Fredholm-Volterra integro-differential equation. J. Comput. Appl. Math. 252 (2013), 52–61.
  * [6] Berenguer, M.I.; Gámez, D., A computational method for solving a class of two dimensional Volterra integral equations. J. Comput. Appl. Math. 318 (2017), 403–410.
  * [7] Berenguer, M. I.; Gámez, D. Projected iterations of fixed-point type to solve nonlinear partial Volterra integro-differential equations. Bull. Malays. Math. Sci. Soc. 43 (2020), no. 6, 4431–4442.
  * [8] Boyd, D. W.; Wong, J. S. W.On nonlinear contractions. Proc. Amer. Math. Soc. 20 (1969), 458–464.
  * [9] Brezis, H. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. xiv+599 pp. ISBN: 978-0-387-70913-0
  * [10] Byszewski, L. Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162 (1991), no. 2, 494–505.
  * [11] Byszewski, L. Existence and uniqueness of mild and classical solutions of semilinear functional-differential evolution nonlocal Cauchy problem. Selected problems of mathematics, 25–33, 50th Anniv. Cracow Univ. Technol. Anniv. Issue, 6, Cracow Univ. Technol., Krakow, 1995
  * [12] Deimling, K. Nonlinear functional analysis. Springer-Verlag, Berlin, 1985. xiv+450 pp. ISBN: 3-540-13928-1
  * [13] Deng, K.Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions. J. Math. Anal. Appl. 179 (1993), no. 2, 630–637.
  * [14] Dhage, B.C. On some variants of Schauder’s fixed point principle and applications to nonlinear integral equations. J. Math. Phy. Sci., 25 (1988), 603-611.
  * [15] Dhage, B.C. On some nonlinear alternatives of Leray-Schauder type and functional integral equations. Arch. Math. (Brno), 42 (2006), 11–23.
  * [16] Dhage, B.C. On a fixed point theorem in Banach algebras with aplications. Appl. Math. Lett., 18 (2005),273–280.
  * [17] Dhage, B.C. Multi-valued mappings and fixed points I. Nonlinear Functional Anal. Appl., 10(3) (2005), 359-378.
  * [18] Dhage, B.C. A hybrid fixed point theorem in Banach algebras with applications. Comm. Appl. Nonlinear Anal., 13 (2006), 71–84.
  * [19] Djebali, S.; Sahnoun, Z. Nonlinear alternatives of Schauder and Krasnosel’skii types with applications to Hammerstein integral equations in $L^{1}$-spaces. J. Differ. Equ. 249 (2010), 2061–2075.
  * [20] Dzhumabaev, D.S.On one approach to solve the linear boundary value problems for Fredholm integro differential equations. J. Comput. Appl. Math. 294 (2016), 342–357.
  * [21] Gelbaum, B. R.; Gil de Lamadrid, J.Bases of tensor products of Banach spaces. Pacific J. Math. 11 (1961), 1281–1286.
  * [22] Heydari, M.H.; Hooshmandasl, M.R.; Mohammadi, F.; Cattani, C.Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations. Commun. Nonlinear Sci. 19 (2014), 37–48.
  * [23] Jeribi, A.; Kaddachi, N.; Krichen, B. Existence results for a system of nonlinear integral equations in Banach algebras under weak topology. Fixed Point Theory 18 (2017), no. 1, 247–267
  * [24] Jeribi, A.; Krichen, B. Nonlinear functional analysis in Banach spaces and Banach algebras: Fixed point theory under weak topology for nonlinear operators and block operator matrices with applications. (Monographs and Research Notes in Mathematics), CRC Press/ Taylor and Francis, (2015).
  * [25] Maleknejad, K.; Basirat, B.; Hashemizadeh, E. A Berstein operational matrix approach for solving a system of high order linear Volterra-Fredholm integro-differential equations. Math. Comput. Model. 55 (2012), 1363–1372
  * [26] O’Regan, D. New fixed point results for 1-set contractive set-valued maps. Computes Math. Appl., 35(4) (1998), 27-34.
  * [27] Saberi-Nadjafi, J.; Tamamgar, M. The variational iteration method: A highly promising method for solving the system of integro-differential equations. Comput. Math. Appl. 56 (2008), 346-351.
  * [28] Semadeni, Z. Product Schauder bases and approximationwith nodes in spaces of continuous functions. Bull. Acad. Polon. Sci., 11 (1963), 387-391.


* Kel [00] , _An ergodic theoretic approach to mean field coupled maps_ , Fractal geometry and stochastics II, Springer, 2000, pp. 183–208.
  * KK [92] Gerhard Keller and Martin Künzle, _Transfer operators for coupled map lattices_ , Ergodic Theory and Dynamical Systems 12 (1992), no. 2, 297–318.
  * KKN [92] Gerhard Keller, Martin Künzle, and Tomasz Nowicki, _Some phase transitions in coupled map lattices_ , Physica D: Nonlinear Phenomena 59 (1992), no. 1-3, 39–51.
  * KL [06] Gerhard Keller and Carlangelo Liverani, _Uniqueness of the srb measure for piecewise expanding weakly coupled map lattices in any dimension_ , Communications in Mathematical Physics 262 (2006), no. 1, 33–50.
  * KL [09] , _Map lattices coupled by collisions_ , Communications in Mathematical Physics 291 (2009), no. 2, 591–597.
  * KP [95] Jürgen Kurths and Arkady S Pikovsky, _Symmetry breaking in distributed systems and modulational spatio-temporal intermittency_ , Chaos, Solitons & Fractals 5 (1995), no. 10, 1893–1899.
  * KT [01] Kunihiko Kaneko and Ichiro Tsuda, _Complex systems: Chaos and beyond: Chaos and beyond: A constructive approach with applications in life sciences_ , Springer Science & Business Media, 2001.
  * Kur [84] Yoshiki Kuramoto, _Chemical turbulence_ , Chemical oscillations, waves, and turbulence, Springer, 1984, pp. 111–140.
  * KY [10] José Koiller and Lai-Sang Young, _Coupled map networks_ , Nonlinearity 23 (2010), no. 5, 1121.
  * KYY+ [14] Kiyoshi Kotani, Ikuhiro Yamaguchi, Lui Yoshida, Yasuhiko Jimbo, and G Bard Ermentrout, _Population dynamics of the modified theta model: macroscopic phase reduction and bifurcation analysis link microscopic neuronal interactions to macroscopic gamma oscillation_ , Journal of The Royal Society Interface 11 (2014), no. 95, 20140058.
  * KZ [01] Gerhard Keller and Roland Zweimüller, _Unidirectionally coupled interval maps: between dynamics and statistical mechanics_ , Nonlinearity 15 (2001), no. 1, 1.
  * Lai [18] Carlo R Laing, _The dynamics of networks of identical theta neurons_ , The Journal of Mathematical Neuroscience 8 (2018), no. 1, 1–24.
  * Liv [95] Carlangelo Liverani, _Decay of correlations_ , Annals of Mathematics (1995), 239–301.
  * LL [94] André Lambert and Ricardo Lima, _Stability of wavelengths and spatiotemporal intermittency in coupled map lattices_ , Physica D: Nonlinear Phenomena 71 (1994), no. 4, 390–411.
  * LL [07] Jean-Michel Lasry and Pierre-Louis Lions, _Mean field games_ , Japanese journal of mathematics 2 (2007), no. 1, 229–260.
  * LMM [95] Jérôme Losson, John Milton, and Michael C Mackey, _Phase transitions in networks of chaotic elements with short and long range interactions_ , Physica D: Nonlinear Phenomena 81 (1995), no. 1-2, 177–203.
  * Luc [18] Valerio Lucarini, _Revising and extending the linear response theory for statistical mechanical systems: Evaluating observables as predictors and predictands_ , Journal of Statistical Physics 173 (2018), no. 6, 1698–1721.
  * Mor [97] Satoru Morita, _Lyapunov analysis of collective behavior in a network of chaotic elements_ , Physics Letters A 226 (1997), no. 3-4, 172–178.
  * MPA [16] Erik A Martens, Mark J Panaggio, and Daniel M Abrams, _Basins of attraction for chimera states_ , New Journal of Physics 18 (2016), no. 2, 022002.
  * MVM [97] Christian Maes and A Van Moffaert, _Stochastic stability of weakly coupled lattice maps_ , Nonlinearity 10 (1997), no. 3, 715.
  * NK [98] Naoko Nakagawa and Teruhisa S Komatsu, _Collective motion occurs inevitably in a class of populations of globally coupled chaotic elements_ , Physical Review E 57 (1998), no. 2, 1570.
  * NK [99] , _Confined chaotic behavior in collective motion for populations of globally coupled chaotic elements_ , Physical Review E 59 (1999), no. 2, 1675.
  * NOEE+ [22] Eddie Nijholt, Jorge Luis Ocampo-Espindola, Deniz Eroglu, István Z Kiss, and Tiago Pereira, _Emergent hypernetworks in weakly coupled oscillators_ , Nature communications 13 (2022), no. 1, 1–8.
  * NT [00] Duane Q Nykamp and Daniel Tranchina, _A population density approach that facilitates large-scale modeling of neural networks: Analysis and an application to orientation tuning_ , Journal of computational neuroscience 8 (2000), no. 1, 19–50.
  * Ome [18] Oleh E Omel'chenko, _The mathematics behind chimera states_ , Nonlinearity 31 (2018), no. 5, R121.
  * PA [15] Mark J Panaggio and Daniel M Abrams, _Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators_ , Nonlinearity 28 (2015), no. 3, R67.
  * PC [92] Gabriel Perez and Hilda A Cerdeira, _Instabilities and nonstatistical behavior in globally coupled systems_ , Physical Review A 46 (1992), no. 12, 7492.
  * PG [16] Mason A Porter and James P Gleeson, _Dynamical systems on networks_ , Frontiers in Applied Dynamical Systems: Reviews and Tutorials 4 (2016).
  * PK [94] Arkady S Pikovsky and Jürgen Kurths, _Do globally coupled maps really violate the law of large numbers?_ , Physical review letters 72 (1994), no. 11, 1644.
  * PR [15] Arkady Pikovsky and Michael Rosenblum, _Dynamics of globally coupled oscillators: Progress and perspectives_ , Chaos: An Interdisciplinary Journal of Nonlinear Science 25 (2015), no. 9, 097616.
  * PRK [02] Arkady Pikovsky, Michael Rosenblum, and Jürgen Kurths, _Synchronization: a universal concept in nonlinear science_ , 2002.
  * PS [88] Ya B Pesin and Ya G Sinai, _Space-time chaos in the system of weakly interacting hyperbolic systems_ , Journal of Geometry and Physics 5 (1988), no. 3, 483–492.
  * PvST [20] Tiago Pereira, Sebastian van Strien, and Matteo Tanzi, _Heterogeneously coupled maps: hub dynamics and emergence across connectivity layers_ , Journal of the European Mathematical Society 22 (2020), no. 7, 2183–2252.
  * RPJK [16] Francisco A Rodrigues, Thomas K DM Peron, Peng Ji, and Jürgen Kurths, _The kuramoto model in complex networks_ , Physics Reports 610 (2016), 1–98.
  * RS [14] Bob Rink and Jan Sanders, _Coupled cell networks and their hidden symmetries_ , SIAM Journal on Mathematical Analysis 46 (2014), no. 2, 1577–1609.
  * RS [15] , _Coupled cell networks: semigroups, lie algebras and normal forms_ , Transactions of the American Mathematical Society 367 (2015), no. 5, 3509–3548.
  * Rue [09] David Ruelle, _A review of linear response theory for general differentiable dynamical systems_ , Nonlinearity 22 (2009), no. 4, 855\.
  * Rug [02] Hans Henrik Rugh, _Coupled maps and analytic function spaces_ , Annales Scientifiques de l'École Normale Supérieure, vol. 35, Elsevier, 2002, pp. 489–535.
  * Rul [01] Nikolai F Rulkov, _Regularization of synchronized chaotic bursts_ , Physical Review Letters 86 (2001), no. 1, 183.
  * Rul [02] , _Modeling of spiking-bursting neural behavior using two-dimensional map_ , Physical Review E 65 (2002), no. 4, 041922.
  * Say [14] Ali H Sayed, _Adaptive networks_ , Proceedings of the IEEE 102 (2014), no. 4, 460–497.
  * SB [81] Richard S Sutton and Andrew G Barto, _Toward a modern theory of adaptive networks: expectation and prediction._ , Psychological review 88 (1981), no. 2, 135.
  * SB [16] Fanni Sélley and Péter Bálint, _Mean-field coupling of identical expanding circle maps_ , Journal of Statistical Physics 164 (2016), no. 4, 858–889.
  * SBAL [92] Sudeshna Sinha, D Biswas, M Azam, and SV Lawande, _Nonstatistical behavior of higher-dimensional coupled systems_ , Physical Review A 46 (1992), no. 6, 3193.
  * Sch [04] Matthias Schmitt, _Spectral theory for nonanalytic coupled map lattices_ , Nonlinearity 17 (2004), no. 2, 671.
  * Sél [18] Fanni M Sélley, _Symmetry breaking in a globally coupled map of four sites_ , Discrete & Continuous Dynamical Systems 38 (2018), no. 8, 3707\.
  * Sél [21] , _A self-consistent dynamical system with multiple absolutely continuous invariant measures_ , Journal of Computational Dynamics 8 (2021), no. 1, 9.
  * Ser [20] Sylvia Serfaty, _Mean field limit for coulomb-type flows_ , Duke Mathematical Journal 169 (2020), no. 15, 2887–2935.
  * SJ [01] Frank Schmüser and Wolfram Just, _Non-equilibrium behaviour in unidirectionally coupled map lattices_ , Journal of Statistical Physics 105 (2001), no. 3, 525–559.
  * SK [97] Tatsuo Shibata and Kunihiko Kaneko, _Heterogeneity-induced order in globally coupled chaotic systems_ , EPL (Europhysics Letters) 38 (1997), no. 6, 417.
  * [172] , _Collective chaos_ , Physical review letters 81 (1998), no. 19, 4116.
  * [173] , _Tongue-like bifurcation structures of the mean-field dynamics in a network of chaotic elements_ , Physica D: Nonlinear Phenomena 124 (1998), no. 1-3, 177–200.
  * SM [91] Steven H Strogatz and Renato E Mirollo, _Stability of incoherence in a population of coupled oscillators_ , Journal of Statistical Physics 63 (1991), no. 3, 613–635.
  * SPMS [17] Tomislav Stankovski, Tiago Pereira, Peter VE McClintock, and Aneta Stefanovska, _Coupling functions: universal insights into dynamical interaction mechanisms_ , Reviews of Modern Physics 89 (2017), no. 4, 045001.
  * Spo [12] Herbert Spohn, _Large scale dynamics of interacting particles_ , Springer Science & Business Media, 2012.
  * ST [00] T Shimada and S Tsukada, _A resolution of the puzzle of the posi-nega switch mechanism in the globally coupled map lattice_ , arXiv preprint nlin/0012036 (2000).
  * ST [21] Fanni M Sélley and Matteo Tanzi, _Linear response for a family of self-consistent transfer operators_ , Communications in Mathematical Physics 382 (2021), no. 3, 1601–1624.
  * ST [22] , _Synchronization for networks of globally coupled maps in the thermodynamic limit_ , Journal of Statistical Physics 189 (2022), no. 1, 1–26.
  * Str [00] Steven H Strogatz, _From kuramoto to crawford: exploring the onset of synchronization in populations of coupled oscillators_ , Physica D: Nonlinear Phenomena 143 (2000), no. 1-4, 1–20.
  * Szn [91] Alain-Sol Sznitman, _Topics in propagation of chaos_ , Ecole d'été de probabilités de Saint-Flour XIX—1989, Springer, 1991, pp. 165–251.
  * Tan [22] Matteo Tanzi, _Uniformly expanding coupled maps: Self-consistent transfer operators and propagation of chaos_ , arXiv preprint arXiv:2209.13571 (2022).
  * Vol [94] VA Volevich, _Construction of an analogue of bowen-ruelle-sinaĭ measure for a multidimensional lattice of interacting hyperbolic mappings_ , Sbornik: Mathematics 79 (1994), no. 2, 347.
  * WG [18] Caroline L Wormell and Georg A Gottwald, _On the validity of linear response theory in high-dimensional deterministic dynamical systems_ , Journal of Statistical Physics 172 (2018), no. 6, 1479–1498.
  * WG [19] , _Linear response for macroscopic observables in high-dimensional systems_ , Chaos: An Interdisciplinary Journal of Nonlinear Science 29 (2019), no. 11, 113127.
  * WH [89] Kurt Wiesenfeld and Peter Hadley, _Attractor crowding in oscillator arrays_ , Physical Review Letters 62 (1989), no. 12, 1335.
  * Win [67] Arthur T Winfree, _Biological rhythms and the behavior of populations of coupled oscillators_ , Journal of theoretical biology 16 (1967), no. 1, 15–42.
  * WO [11] Matthias Wolfrum and E Omel'chenko, _Chimera states are chaotic transients_ , Physical Review E 84 (2011), no. 1, 015201.
  * Wor [22] Caroline L Wormell, _Non-hyperbolicity at large scales of a high-dimensional chaotic system_ , Proceedings of the Royal Society A 478 (2022), no. 2261, 20210808.
  * YZ [16] Nan Yao and Zhigang Zheng, _Chimera states in spatiotemporal systems: Theory and applications_ , International Journal of Modern Physics B 30 (2016), no. 7, 1630002.


# Comment on “New physics constraints from atomic parity violation in 133Cs”

B. M. Roberts<EMAIL_ADDRESS>J. S. M. Ginges<EMAIL_ADDRESS>School
of Mathematics and Physics, The University of Queensland, Brisbane QLD 4072,
Australia

###### Abstract

In a recent Letter, B. K. Sahoo, B. P. Das, and H. Spiesberger, Phys. Rev. D
103, L111303 (2021) Sahoo _et al._ (2021), a calculation of the parity
violating $6S-7S$ E1 amplitude in Cs is reported, claiming an uncertainty of
just 0.3%. In this Comment, we point out that key contributions have been
omitted, and the theoretical uncertainty has been significantly
underestimated. In particular, the contribution of missed QED radiative
corrections amounts to several times the claimed uncertainty.

The $6S$–$7S$ atomic parity violation (APV) amplitude in Cs may be expressed
as $\langle\widetilde{7S}|D_{z}|\widetilde{6S}\rangle$, where $D_{z}$ is the
$z$ component of the electric dipole (E1) operator, and
$|\widetilde{6S}\rangle$ and $|\widetilde{7S}\rangle$ are weak-interaction-
perturbed atomic states; the source of this interaction is $Z$-boson exchange
between the electrons and the nucleus. In the lowest-order single-particle
picture, it may be written

$\displaystyle{E}_{\rm PV}$ $\displaystyle=\sum_{n}\left[\frac{\langle
7s|h_{w}|n\rangle\langle
n|d_{z}|6s\rangle}{\varepsilon_{7s}-\varepsilon_{n}}+\frac{\langle
7s|d_{z}|n\rangle\langle
n|h_{w}|6s\rangle}{\varepsilon_{6s}-\varepsilon_{n}}\right],$ (1)

where $d_{z}$ is the single-particle E1 operator,
$h_{w}=-\frac{G_{F}}{2\sqrt{2}}Q_{w}\rho(r)\gamma_{5}$ is the parity-violating
weak interaction operator, with $G_{F}$ the Fermi constant, $Q_{w}$ the
nuclear weak charge, $\rho$ the nuclear density, and $\gamma_{5}$ the Dirac
matrix, and $n$ runs over all $p_{1/2}$ states including the (occupied) core;
see Ref. Ginges and Flambaum (2004). The accuracy of the calculation is
determined by account of many-body effects and smaller corrections including
higher-order relativistic effects. Evaluation of $E_{\rm PV}$ in Cs with an
accuracy matching or exceeding that of the measurement Wood _et al._ (1997)
remains a formidable challenge. There is a rich history connected to this
spanning more than 20 years as the theoretical accuracy has reached the
fraction-of-a-percent level; see, e.g., reviews Ginges and Flambaum (2004);
Roberts _et al._ (2015); Safronova _et al._ (2018) and Ref. Toh _et al._
(2019). A major development over this time, following the realization of the
significance of the Breit contribution Derevianko (2000, 2001); Kozlov _et
al._ (2001); Dzuba _et al._ (2001a), was the recognition of the importance of
quantum electrodynamics (QED) radiative corrections and the formulation of
methods to account for them in precision calculations for heavy atoms Sushkov
(2001); Johnson _et al._ (2001); Dzuba _et al._ (2002); Milstein _et al._
(2002); *Milstein2002a; *Milstein2003; Kuchiev and Flambaum (2002);
*Kuchiev2002b; *KuchievQED03; Sapirstein _et al._ (2003); Shabaev _et al._
(2005a); Flambaum and Ginges (2005) (see also Sapirstein and Cheng (2005);
Shabaev _et al._ (2013); Ginges and Berengut (2016a, b)).

We have identified a number of shortcomings in the theoretical evaluation of
$E_{\rm PV}$ in the Letter Sahoo _et al._ (2021), some of which are detailed
below. Most notably, the treatment of QED radiative corrections omits
important contributions to $E_{\rm PV}$, which amount to several times the
theoretical uncertainty claimed in Ref. Sahoo _et al._ (2021).

##  I QED correction to $E_{\rm PV}$

QED radiative corrections in the strong Coulomb field of the nucleus make a
significant contribution to $E_{\rm PV}$, $\lesssim$ 1%. These have been
calculated before Johnson _et al._ (2001); Dzuba _et al._ (2002); Milstein _et
al._ (2002); *Milstein2002a; *Milstein2003; Kuchiev and Flambaum (2002);
Kuchiev (2002); *KuchievQED03; Sapirstein _et al._ (2003); Shabaev _et al._
(2005a, b); Flambaum and Ginges (2005); Roberts _et al._ (2013a) and are well
established. It is said in the Letter Sahoo _et al._ (2021) that one of the
key improvements is the treatment of these QED corrections. However, details
of the QED calculation are not presented in the Letter, and the reader is
directed to the unpublished manuscript Sahoo and Das (2020) for explanation
111Note that the reference to Sahoo and Das (2020) in the Letter Sahoo _et
al._ (2021) is incorrect, linking to an unrelated arXiv paper. There it is
said that the self-energy QED correction to $E_{\rm PV}$ (and to other atomic
properties) is accounted for by including the radiative potential Flambaum and
Ginges (2005); Ginges and Berengut (2016a) into the Hamiltonian from the start
222Vacuum polarization is included using the standard Uehling potential, and a
simplified form of the Wichmann-Kroll potential from Ref. Dzuba _et al._
(2002); Flambaum and Ginges (2005)., which the authors claim to be a more
rigorous approach compared to previous calculations.

Figure 1: Feynman diagrams for self-energy corrections to matrix elements.
Dashed line with triangle represents the external field (e.g., E1, weak,
hyperfine), wavy line the photon propagator, and double line the bound
electron wavefunction and propagator. Middle diagram is vertex correction.

The radiative potential method Flambaum and Ginges (2005) enables the accurate
inclusion of self-energy corrections to the energies and wavefunctions of
many-electron atoms. It may also be used to account for QED corrections to
matrix elements of external fields whose operators act at radial distances
much larger than the electron Compton wavelength, $r\gg e\hbar/(m_{e}c)$,
e.g., the E1 field. However, this is not the case for operators that act at
small distances, including the weak and hyperfine interactions. We illustrate
this in Fig. 1. For the E1 interaction, the dominant contribution is given by
the left and right diagrams, which may be accounted for by using the radiative
potential method. However, for the weak and hyperfine interactions, other
contributions are important. In particular, the middle vertex diagram – where
the external field is locked inside the photon loop – simply cannot be
accounted for using this method. We refer the reader to the original Flambaum
and Ginges (2005) and subsequent Roberts _et al._ (2013a) works for details on
the applicability of the radiative potential method.

The QED correction to the full Cs APV amplitude (involving both E1 and weak
interactions) was determined in Refs. Flambaum and Ginges (2005); Shabaev _et
al._ (2005a). In Ref. Flambaum and Ginges (2005), the radiative potential
method was used to calculate corrections to the E1 matrix elements and energy
denominators in the sum (1), with QED corrections to weak matrix elements
$\langle s|h_{w}|p_{1/2}\rangle$ taken from previous works Milstein _et al._
(2002); *Milstein2002a; *Milstein2003; Kuchiev and Flambaum (2002); Kuchiev
(2002); *KuchievQED03; Sapirstein _et al._ (2003). In Ref. Shabaev _et al._
(2005a), Shabaev et al. calculated the total correction by applying a rigorous
QED formalism. The results of Refs. Shabaev _et al._ (2005a) and Flambaum and
Ginges (2005) are in excellent agreement, $-0.27(3)$% and $-0.32(3)$%,
respectively 333The QED results of both Refs. Shabaev _et al._ (2005a) and
Flambaum and Ginges (2005) were misattributed in Table III of the Letter Sahoo
_et al._ (2021); these papers were not cited in Ref. Sahoo _et al._ (2021)..

It is unclear how the authors of Sahoo _et al._ (2021) arrive at a QED
correction of $-0.4\%$ for the weak matrix elements and $-0.3\%$ for $E_{\rm
PV}$, in agreement with existing calculations Milstein _et al._ (2002);
*Milstein2002a; *Milstein2003; Kuchiev and Flambaum (2002); Kuchiev (2002);
*KuchievQED03; Sapirstein _et al._ (2003); Flambaum and Ginges (2005); Shabaev
_et al._ (2005a, b); Roberts _et al._ (2013a), given the important short-range
effects, including the vertex contribution, have been omitted. In an attempt
to reproduce the results of Ref. Sahoo _et al._ (2021), we calculate the
radiative potential value for the QED correction to weak matrix elements,
including vacuum polarization. The result is $-2.1\%$, too large by a factor
of five compared to the correct calculations, demonstrating the importance of
the missed short-range effects. This difference amounts to a change in $E_{\rm
PV}$ that is nearly six times the atomic theory uncertainty claimed in Ref.
Sahoo _et al._ (2021).

##  II Hyperfine constants

In the Letter Sahoo _et al._ (2021), calculations of hyperfine constants are
performed to test the accuracy of the wavefunctions in the nuclear region,
crucial for assessing the accuracy of APV calculations (see Refs. Ginges _et
al._ (2017); Ginges and Volotka (2018); Roberts and Ginges (2020, 2021) for
recent studies of the nuclear magnetization distribution for Cs). By
demonstrating excellent agreement with experiment, the authors conclude the
accuracy of their wavefunctions is high, and so estimate a tremendously small
uncertainty for the APV calculation. However, it appears that serious
omissions have been made in the hyperfine calculations.

As for $E_{\rm PV}$, the vertex and short-range contributions to QED
corrections to hyperfine constants are important Sapirstein and Cheng (2003);
Ginges _et al._ (2017) (see also Blundell _et al._ (1997); Sunnergren _et al._
(1998); Artemyev _et al._ (2001); Volotka _et al._ (2008, 2012)). Moreover,
the magnetic loop vacuum polarization correction also gives a significant
contribution Sapirstein and Cheng (2003); Ginges _et al._ (2017). In the
Letter Sahoo _et al._ (2021), the radiative potential method is employed, with
no account for these contributions. Given this, it is unclear how the authors
of Sahoo _et al._ (2021); Sahoo and Das (2020) arrive at a correction of
$-0.3\%$ to the hyperfine constants for $s$ states of Cs, in good agreement
with existing calculations Sapirstein and Cheng (2003); Ginges _et al._
(2017). To investigate this result, we again use the radiative potential
method and find it gives a correction of $-1.2\%$, three times too large
compared to rigorous QED calculations Sapirstein and Cheng (2003); Ginges _et
al._ (2017), confirming the importance of the omitted effects. This difference
amounts to two times the uncertainty of the hyperfine calculations (0.4%)
claimed in the Letter Sahoo _et al._ (2021).

##  III Core contribution

The contribution to $E_{\rm PV}$ coming from the (occupied) $n$ = 2–5 terms in
Eq. (1) is called the “core” (or autoionization) contribution. In the Letter
Sahoo _et al._ (2021), it is said that the main difference in the $E_{\rm PV}$
result compared to the previous calculation of Dzuba et al. Dzuba _et al._
(2012) stems from the opposite sign of the core contribution. The difference
in core contribution between Refs. Sahoo _et al._ (2021) and Dzuba _et al._
(2012) is larger than the theoretical uncertainty claimed in the Letter Sahoo
_et al._ (2021) and should be investigated thoroughly.

In Ref. Dzuba _et al._ (2012), Dzuba et al. showed that many-body effects
(core polarization and correlations) have a significant impact on the core
contribution, changing its sign compared to the lowest-order Hartree-Fock
value; see also Ref. Roberts _et al._ (2015). The authors of Ref. Sahoo _et
al._ (2021) claim their result confirms the core calculation of Ref. Porsev
_et al._ (2009); *Porsev2010 and agrees with the result of Ref. Blundell _et
al._ (1990); *Blundell1992. However, in both of those works, the core
contribution was evaluated in the lowest-order approximation.

Here, we re-examine the core contribution in detail in an attempt to elucidate
the source of this discrepancy. We include core polarization using the time-
dependent Hartree-Fock (TDHF) method Dzuba _et al._ (1984), in which the
single-particle operators are modified: $d_{z}\to{\tilde{d}_{z}}=d_{z}+\delta
V_{d}$, and ${h_{w}}\to{\tilde{h}_{w}}={h}_{w}+\delta V_{w}$. The $\delta V$
corrections are found by solving the set of TDHF equations for all electrons
in the core Dzuba _et al._ (1984). We obtain the corrections to lowest-order
in the Coulomb interaction by solving the set of equations once, and to all-
orders by iterating the equations until self-consistency is reached Dzuba _et
al._ (1984) (equivalent to the random-phase approximation with exchange, RPA
Johnson _et al._ (1980)). The equations for $\delta V_{d}$ are solved at the
frequency of the $6S$–$7S$ transition (see Roberts _et al._ (2013b) for a
numerical study). We account for correlation corrections using the second-
order Dzuba _et al._ (1987) and all-orders Dzuba _et al._ (1988);
*DzubaCPM1989plaEn; *DzubaCPM1989plaE1 correlation potential methods (see also
Dzuba _et al._ (2002)).

The core contribution arises as the sum of two terms, due to the weak-
perturbation of $6s$ and $7s$ states, respectively. These have similar
magnitude though opposite sign, and strongly cancel, meaning numerical error
may be significant. We test the numerical accuracy in a number of ways.
Firstly, we vary the number of radial grid points used for solving the
differential equations, and vary the number of basis states used in any
expansions. We find numerical errors stemming from grid/basis choices can
easily be made insignificant. More importantly, we have three physically
equivalent, but numerically distinct, ways to compute $E_{\rm PV}$:

$\displaystyle\sum_{n}\left[\frac{\langle 7s|\tilde{h}_{w}|n\rangle\langle
n|{\tilde{d}_{z}}|6s\rangle}{\varepsilon_{7s}-\varepsilon_{n}}+\frac{\langle
7s|{\tilde{d}_{z}}|n\rangle\langle
n|\tilde{h}_{w}|6s\rangle}{\varepsilon_{6s}-\varepsilon_{n}}\right]$ (2)
$\displaystyle=\langle\delta\psi_{7s}|{\tilde{d}_{z}}|6s\rangle+\langle
7s|{\tilde{d}_{z}}|\delta\psi_{6s}\rangle$ (3) $\displaystyle=\langle
7s|\tilde{h}_{w}|\Delta\psi_{6s}\rangle+\langle\Delta\psi_{7s}|\tilde{h}_{w}|6s\rangle,$
(4)

where $\delta\psi$ and $\Delta\psi$ are corrections to the valence
wavefunctions ($\psi$) due to the time-independent weak interaction, and the
time-dependent E1 interaction, respectively. These are called the sum-over-
states (2), weak-mixed-states (3), and E1-mixed-states (4) methods 444These
formulas exclude the double-core-polarization effect, which is very small for
Cs, and has been studied in detail in Ref. Roberts _et al._ (2013b). The sign
change of the core cannot be explained by the double-core-polarization
correction, which even if entirely assigned to the core contribution is a
factor of two too small to account for the difference Roberts _et al._
(2013b)..

In the sum-over-states method, a B-spline basis (e.g., Johnson _et al._
(1988); Beloy and Derevianko (2008)) is used to sum over the set of
intermediate states. In contrast, the mixed-states approach does not require a
basis at all; the $\delta$ and $\Delta$ corrections are found by solving the
differential equations Dalgarno and Lewis (1955):

$\displaystyle(h-\varepsilon)\delta\psi$ $\displaystyle=-{\tilde{h}_{w}}\psi$
(5) $\displaystyle(h-\varepsilon-\nu)\Delta\psi$
$\displaystyle=-{\tilde{d}_{z}}\psi,$ (6)

where $h$ is the single-particle atomic Hamiltonian, and $\nu$ is the
$6S$–$7S$ transition frequency. In the mixed-states approach, the core
contribution is found by projecting the corrections $\delta\psi$ and
$\Delta\psi$ onto the core states, while in the sum-over-states method it is
found by restricting the sum to include only core states. Note that the
numerics involved in solving each of the above equations is significantly
different, and the coincidence of results is indicative of high numerical
accuracy. Even with moderate choice for the radial grid, we find the results
of the two mixed-states methods agree to parts in $10^{8}$, and the mixed-
states and sum-over-states methods agree to parts in $10^{7}$, demonstrating
excellent numerical precision and completeness of the basis.

Our calculations of the core term are summarized in Table 1. The sign change
in the core contribution is mostly due to polarization of the core by the
external E1 field. This is sensitive to the frequency of the E1 field. While
correlations beyond core polarization are important, they affect both terms in
roughly the same manner; the core term and its sign are robust to the
treatment of correlations. We also performed calculations for the
$7S$-$6D_{3/2}$ $E_{\rm PV}$ for 223Ra+ to test against previous calculations;
at the RPA level, we find the core contribution to be 6.81 [in units
$-10^{-11}i(-Q_{w}/N)\,|e|a_{B}$], in excellent agreement with the result 6.83
of Ref. Pal _et al._ (2009) (see also Dzuba _et al._ (2001b); Wansbeek _et
al._ (2008)). It is unclear why the sign of the result of Ref. Sahoo _et al._
(2021) remains the same as the Hartree-Fock value, however, we note that it
may not be straight forward to compare individual contributions across
different methods as discussed in Refs. Wieman and Derevianko (2019);
Safronova _et al._ (2018).

##  IV Conclusion

For the above reasons, we are not convinced the result presented in the Letter
Sahoo _et al._ (2021) is an improved value for the Cs $E_{\rm PV}$. We
conclude that the most reliable and accurate values that have been obtained to
date are: $E_{\rm PV}=0.898(5)$ Dzuba _et al._ (2002); Flambaum and Ginges
(2005) and $E_{\rm PV}=0.8977(40)$ Porsev _et al._ (2009); Dzuba _et al._
(2012), in units $-10^{-11}i(-Q_{w}/N)\,|e|a_{B}$, which agree precisely and
were obtained using different approaches. These results are also in excellent
agreement with previous calculations Dzuba _et al._ (1989c); Blundell _et al._
(1990); *Blundell1992; Kozlov _et al._ (2001); Shabaev _et al._ (2005b),
though in disagreement with the result of the Letter Sahoo _et al._ (2021).

Table 1: Core contribution to 133Cs 6S-7S $E_{\rm PV}$ in different
approximations, in units $-10^{-11}i(-Q_{w}/N)\,|e|a_{B}$, where $N=78$ is the
number of neutrons.a Here, HF denotes relativistic Hartree-Fock, $\delta
V^{(1)}$ and $\delta V^{(\infty)}$ denote lowest-order and all-orders core-
polarization, respectively, with subscripts $w$ and $d$ indicating
polarization by the weak or E1 fields, $\Sigma^{(2)}$ and $\Sigma^{(\infty)}$
denote second- and all-orders correlations, respectively, and $\lambda$
indicates correlations have been re-scaled to reproduce the lowest
experimental binding energies.

Method | $\langle\delta\psi_{7s}|{\tilde{d}_{z}}|6s\rangle$ | $\langle 7s|{\tilde{d}_{z}}|\delta\psi_{6s}\rangle$ | Sum
---|---|---|---
HF | $-0.02645$ | $0.02472$ | $-0.00174$
HF+$\delta V_{w}^{(1)}$ | $-0.03747$ | $0.03539$ | $-0.00208$
HF+$\delta V_{w}^{(\infty)}$ | $-0.04319$ | $0.04119$ | $-0.00201$
E1 TDHF equations solved at HF frequency:
HF+$\delta V_{w}^{(\infty)}$+$\delta V_{d}^{(1)}$ | $-0.05506$ | $0.05442$ | $-0.00063$
HF+$\delta V_{w}^{(\infty)}$+$\delta V_{d}^{(\infty)}$222HF+$\delta V_{w}^{(\infty)}$+$\delta V_{d}^{(\infty)}$ is commonly called RPA level. | $-0.05822$ | $0.05992$ | $0.00170$
E1 TDHF equations solved at experimental frequency:
HF+$\delta V_{w}^{(\infty)}$+$\delta V_{d}^{(1)}$ | $-0.05468$ | $0.05466$ | $-0.00002$
HF+$\delta V_{w}^{(\infty)}$+$\delta V_{d}^{(\infty)}$222HF+$\delta V_{w}^{(\infty)}$+$\delta V_{d}^{(\infty)}$ is commonly called RPA level. | $-0.05784$ | $0.06043$ | $0.00259$
Including correlation corrections (and $\delta V_{w}^{(\infty)}+\delta
V_{d}^{(\infty)}$):
$\Sigma^{(2)}$ | $-0.06739$ | $0.06924$ | $0.00184$
$\lambda\Sigma^{(2)}$ | $-0.06547$ | $0.06732$ | $0.00184$
$\Sigma^{(\infty)}$ | $-0.06514$ | $0.06695$ | $0.00181$
$\lambda\Sigma^{(\infty)}$ | $-0.06516$ | $0.06696$ | $0.00181$
Other calculations:
HF Blundell _et al._ (1990); *Blundell1992 |  |  | $-0.002$
HF Porsev _et al._ (2009); *Porsev2010 |  |  | $-0.002$
$\Sigma^{(\infty)}$+RPA Dzuba _et al._ (2012) |  |  | $0.00182$
Values from the Letter Sahoo _et al._ (2021):
HF Sahoo _et al._ (2021) |  |  | $-0.0017$
RCCSD Sahoo _et al._ (2021) |  |  | $-0.0019$
RCCSDT Sahoo _et al._ (2021) |  |  | $-0.0018$

11footnotetext: To avoid possible ambiguity in the sign, we note that the
total amplitude is positive in these units; at the HF level it is $0.7395$.

Acknowledgments— We thank V. A. Dzuba and V. V. Flambaum for useful
discussions. This work was supported by the Australian Government through ARC
DECRA Fellowship DE210101026 and ARC Future Fellowship FT170100452.

## References

  * Sahoo _et al._ (2021) B. K. Sahoo, B. P. Das, and H. Spiesberger, Phys. Rev. D 103, L111303 (2021), arXiv:2101.10095 .
  * Ginges and Flambaum (2004) J. S. M. Ginges and V. V. Flambaum, Phys. Rep. 397, 63 (2004).
  * Wood _et al._ (1997) C. S. Wood, S. C. Bennett, D. Cho, B. P. Masterson, J. L. Roberts, C. E. Tanner, and C. E. Wieman, Science 275, 1759 (1997).
  * Roberts _et al._ (2015) B. M. Roberts, V. A. Dzuba, and V. V. Flambaum, Annu. Rev. Nucl. Part. Sci. 65, 63 (2015), arXiv:1412.6644 .
  * Safronova _et al._ (2018) M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark, Rev. Mod. Phys. 90, 025008 (2018).
  * Toh _et al._ (2019) G. Toh, A. Damitz, C. E. Tanner, W. R. Johnson, and D. S. Elliott, Phys. Rev. Lett. 123, 073002 (2019), arXiv:1905.02768 .
  * Derevianko (2000) A. Derevianko, Phys. Rev. Lett. 85, 1618 (2000).
  * Derevianko (2001) A. Derevianko, Phys. Rev. A 65, 012106 (2001).
  * Kozlov _et al._ (2001) M. G. Kozlov, S. G. Porsev, and I. I. Tupitsyn, Phys. Rev. Lett. 86, 3260 (2001).
  * Dzuba _et al._ (2001a) V. A. Dzuba, C. Harabati, W. R. Johnson, and M. S. Safronova, Phys. Rev. A 63, 044103 (2001a).
  * Sushkov (2001) O. P. Sushkov, Phys. Rev. A 63, 042504 (2001).
  * Johnson _et al._ (2001) W. R. Johnson, I. Bednyakov, and G. Soff, Phys. Rev. Lett. 87, 233001 (2001).
  * Dzuba _et al._ (2002) V. A. Dzuba, V. V. Flambaum, and J. S. M. Ginges, Phys. Rev. D 66, 076013 (2002).
  * Milstein _et al._ (2002) A. I. Milstein, O. P. Sushkov, and I. S. Terekhov, Phys. Rev. Lett. 89, 283003 (2002).
  * Milstein and Sushkov (2002) A. I. Milstein and O. P. Sushkov, Phys. Rev. A 66, 022108 (2002).
  * Milstein _et al._ (2003) A. I. Milstein, O. P. Sushkov, and I. S. Terekhov, Phys. Rev. A 67, 062103 (2003).
  * Kuchiev and Flambaum (2002) M. Y. Kuchiev and V. V. Flambaum, Phys. Rev. Lett. 89, 283002 (2002).
  * Kuchiev (2002) M. Y. Kuchiev, J. Phys. B 35, L503 (2002).
  * Kuchiev and Flambaum (2003) M. Y. Kuchiev and V. V. Flambaum, J. Phys. B 36, R191 (2003).
  * Sapirstein _et al._ (2003) J. Sapirstein, K. Pachucki, A. Veitia, and K. T. Cheng, Phys. Rev. A 67, 052110 (2003).
  * Shabaev _et al._ (2005a) V. M. Shabaev, K. Pachucki, I. I. Tupitsyn, and V. A. Yerokhin, Phys. Rev. Lett. 94, 213002 (2005a).
  * Flambaum and Ginges (2005) V. V. Flambaum and J. S. M. Ginges, Phys. Rev. A 72, 052115 (2005).
  * Sapirstein and Cheng (2005) J. Sapirstein and K. T. Cheng, Phys. Rev. A 71, 022503 (2005).
  * Shabaev _et al._ (2013) V. M. Shabaev, I. I. Tupitsyn, and V. A. Yerokhin, Phys. Rev. A 88, 012513 (2013), arXiv:1305.6333 .
  * Ginges and Berengut (2016a) J. S. M. Ginges and J. C. Berengut, Phys. Rev. A 93, 052509 (2016a), arXiv:1603.09116 .
  * Ginges and Berengut (2016b) J. S. M. Ginges and J. C. Berengut, J. Phys. B 49, 095001 (2016b).
  * Shabaev _et al._ (2005b) V. M. Shabaev, I. I. Tupitsyn, K. Pachucki, G. Plunien, and V. A. Yerokhin, Phys. Rev. A 72, 062105 (2005b).
  * Roberts _et al._ (2013a) B. M. Roberts, V. A. Dzuba, and V. V. Flambaum, Phys. Rev. A 87, 054502 (2013a), arXiv:1302.0593 .
  * Sahoo and Das (2020) B. K. Sahoo and B. P. Das,  (2020), arXiv:2008.08941 .
  * Note (1) Note that the reference to Sahoo and Das (2020) in the Letter Sahoo _et al._ (2021) is incorrect, linking to an unrelated arXiv paper.
  * Note (2) Vacuum polarization is included using the standard Uehling potential, and a simplified form of the Wichmann-Kroll potential from Ref. Dzuba _et al._ (2002); Flambaum and Ginges (2005).
  * Note (3) The QED results of both Refs. Shabaev _et al._ (2005a) and Flambaum and Ginges (2005) were misattributed in Table III of the Letter Sahoo _et al._ (2021); these papers were not cited in Ref. Sahoo _et al._ (2021).
  * Ginges _et al._ (2017) J. S. M. Ginges, A. V. Volotka, and S. Fritzsche, Phys. Rev. A 96, 062502 (2017), arXiv:1709.07725 .
  * Ginges and Volotka (2018) J. S. M. Ginges and A. V. Volotka, Phys. Rev. A 98, 032504 (2018), arXiv:1707.00551 .
  * Roberts and Ginges (2020) B. M. Roberts and J. S. M. Ginges, Phys. Rev. Lett. 125, 063002 (2020), arXiv:2001.01907 .
  * Roberts and Ginges (2021) B. M. Roberts and J. S. M. Ginges, Phys. Rev. A 104, 022823 (2021), arXiv:2101.09924 .
  * Sapirstein and Cheng (2003) J. Sapirstein and K. T. Cheng, Phys. Rev. A 67, 022512 (2003).
  * Blundell _et al._ (1997) S. A. Blundell, K. T. Cheng, and J. Sapirstein, Phys. Rev. A 55, 1857 (1997).
  * Sunnergren _et al._ (1998) P. Sunnergren, H. Persson, S. Salomonson, S. M. Schneider, I. Lindgren, and G. Soff, Phys. Rev. A 58, 1055 (1998).
  * Artemyev _et al._ (2001) A. N. Artemyev, V. M. Shabaev, G. Plunien, G. Soff, and V. A. Yerokhin, Phys. Rev. A 63, 062504 (2001).
  * Volotka _et al._ (2008) A. V. Volotka, D. A. Glazov, I. I. Tupitsyn, N. S. Oreshkina, G. Plunien, and V. M. Shabaev, Phys. Rev. A 78, 062507 (2008), arXiv:0806.1121 .
  * Volotka _et al._ (2012) A. V. Volotka, D. A. Glazov, O. V. Andreev, V. M. Shabaev, I. I. Tupitsyn, and G. Plunien, Phys. Rev. Lett. 108, 073001 (2012).
  * Dzuba _et al._ (2012) V. A. Dzuba, J. C. Berengut, V. V. Flambaum, and B. M. Roberts, Phys. Rev. Lett. 109, 203003 (2012), arXiv:1207.5864 .
  * Porsev _et al._ (2009) S. G. Porsev, K. Beloy, and A. Derevianko, Phys. Rev. Lett. 102, 181601 (2009).
  * Porsev _et al._ (2010) S. G. Porsev, K. Beloy, and A. Derevianko, Phys. Rev. D 82, 036008 (2010).
  * Blundell _et al._ (1990) S. A. Blundell, W. R. Johnson, and J. Sapirstein, Phys. Rev. Lett. 65, 1411 (1990).
  * Blundell _et al._ (1992) S. A. Blundell, J. Sapirstein, and W. R. Johnson, Phys. Rev. D 45, 1602 (1992).
  * Dzuba _et al._ (1984) V. A. Dzuba, V. V. Flambaum, and O. P. Sushkov, J. Phys. B 17, 1953 (1984).
  * Johnson _et al._ (1980) W. R. Johnson, C. D. Lin, K. T. Cheng, and C. M. Lee, Phys. Scr. 21, 409 (1980).
  * Roberts _et al._ (2013b) B. M. Roberts, V. A. Dzuba, and V. V. Flambaum, Phys. Rev. A 88, 042507 (2013b), arXiv:1309.3371 .
  * Dzuba _et al._ (1987) V. A. Dzuba, V. V. Flambaum, P. G. Silvestrov, and O. P. Sushkov, J. Phys. B 20, 1399 (1987).
  * Dzuba _et al._ (1988) V. A. Dzuba, V. V. Flambaum, P. G. Silvestrov, and O. P. Sushkov, Phys. Lett. A 131, 461 (1988).
  * Dzuba _et al._ (1989a) V. A. Dzuba, V. V. Flambaum, and O. P. Sushkov, Phys. Lett. A 140, 493 (1989a).
  * Dzuba _et al._ (1989b) V. A. Dzuba, V. V. Flambaum, A. Y. Kraftmakher, and O. P. Sushkov, Phys. Lett. A 142, 373 (1989b).
  * Note (4) These formulas exclude the double-core-polarization effect, which is very small for Cs, and has been studied in detail in Ref. Roberts _et al._ (2013b). The sign change of the core cannot be explained by the double-core-polarization correction, which even if entirely assigned to the core contribution is a factor of two too small to account for the difference Roberts _et al._ (2013b).
  * Johnson _et al._ (1988) W. R. Johnson, S. A. Blundell, and J. Sapirstein, Phys. Rev. A 37, 307 (1988).
  * Beloy and Derevianko (2008) K. Beloy and A. Derevianko, Comput. Phys. Commun. 179, 310 (2008), arXiv:0710.3142 .
  * Dalgarno and Lewis (1955) A. Dalgarno and J. T. Lewis, Proceedings of the Royal Society A 233, 70 (1955).
  * Pal _et al._ (2009) R. Pal, D. Jiang, M. S. Safronova, and U. I. Safronova, Phys. Rev. A 79, 062505 (2009).
  * Dzuba _et al._ (2001b) V. A. Dzuba, V. V. Flambaum, and J. S. M. Ginges, Phys. Rev. A 63, 062101 (2001b).
  * Wansbeek _et al._ (2008) L. W. Wansbeek, B. K. Sahoo, R. G. E. Timmermans, K. Jungmann, B. P. Das, and D. Mukherjee, Phys. Rev. A 78, 050501 (2008).
  * Wieman and Derevianko (2019) C. E. Wieman and A. Derevianko,  (2019), arXiv:1904.00281 .
  * Dzuba _et al._ (1989c) V. A. Dzuba, V. V. Flambaum, and O. P. Sushkov, Phys. Lett. A 141, 147 (1989c).