# On the Local convergence of two-step Newton type Method in Banach Spaces
under generalized Lipschitz Conditions

Akanksha Saxena1, J. P. Jaiswal2

1 Department of Mathematics, Maulana Azad National Institute of Technology,
Bhopal, M.P., India-462003, Email<EMAIL_ADDRESS>
2 Department of Mathematics, Guru Ghasidas Vishwavidyalaya (A Central
University), Bilaspur, C.G., India-495009, Email<EMAIL_ADDRESS>
Abstract. The motive of this paper is to discuss the local convergence of a
two-step Newton type method of convergence rate three for solving nonlinear
equations in Banach spaces. It is assumed that the first order derivative of
nonlinear operator satisfies the generalized Lipschitz i.e. $L$-average
condition. Also, some results on convergence of the same method in Banach
spaces are established under the assumption that the derivative of the
operators satisfies the radius or center Lipschitz condition with a weak
$L$-average particularly it is assumed that $L$ is positive integrable
function but not necessarily non-decreasing.

Mathematics Subject Classification (2000). 65H10

Keywords and Phrases. Banach space, Nonlinear problem, Local convergence,
Lipschitz condition, $L$-average, Convergence ball.

##  1\. Introduction

Consider a nonlinear operator $t:\Omega\subseteq X\to Y$ such that $X$ and $Y$
are two Banach spaces, $\Omega$ is a non-empty open convex subset and $t$ is
Fréchet differentiable nonlinear operator. Nonlinear problems has so many
applications in the field of chemical engineering, transportation, operational
research etc. which can be seen in the form of

$\displaystyle t(x)=0.$ (1.1)

To find the solution of equation $(\ref{eqn:11})$ , Newton’s method defined as

$\displaystyle x_{k+1}$ $\displaystyle=$ $\displaystyle
x_{k}-[t^{\prime}(x_{k})]^{-1}t(x_{k}),\ k\geq 0,$ (1.2)

is being preferred though its speed of convergence is low. Newton’s method
[9], is a well known iterative method which converges quadratically, has been
initially studied by Kantorovich [5] and then scrutinized by Rall [10].

Some Newton-type methods with third-order convergence that do not require the
computation of second order derivatives have been developed in the refs.([4],
[7], [8] and [14]). While the methods of higher $R$-order of convergence are
generally not executed frequently despite having fast speed of convergence
because its operational cost is high. But the method of higher $R$-order of
convergence can be used in the problems of stiff system where fast convergence
is required.

In the numerical point of view, the convergence domain plays a crucial role
for the stable behavior of an iterative scheme. Research about the convergence
study of Newton methods involves two types: semilocal and local convergence
analysis. The semilocal convergence study is based on the information around
an initial point to give criteria ensuring the convergence of iterative
methods; meanwhile, the local one is, based on the information around a
solution, to find estimates for the radii of the convergence balls. Numerous
researchers studied the local convergence analysis for Newton-type, Jarratt-
type, Weerakoon-type, etc. in Banach space setting in the articles ([1], [2],
[6] and [13]).

Here, we discuss the local convergence of the classical third-order
modification of two-step Newton’s method [11] under the $L$-average condition
which is expressed as:

$\displaystyle y_{k}$ $\displaystyle=$ $\displaystyle
x_{k}-[t^{\prime}(x_{k})]^{-1}t(x_{k}),$ $\displaystyle x_{k+1}$
$\displaystyle=$ $\displaystyle y_{k}-[t^{\prime}(x_{k})]^{-1}t(y_{k}),\ k\geq
0.$ (1.3)

The important characteristic of the method $(\ref{eqn:12})$ is that: it is
simplest and efficient third-order iterative method, per iteration it requires
two evaluations of the function $t$, one of the first derivative $t^{\prime}$
and no evaluations of the second derivative $t^{\prime\prime}$ hence makes it
computationally efficient. We find, in the literature, several studies on the
weakness and/or extension of the hypotheses made on the underlying operators.

For re-investigating the local convergence of Newton’s method, generalized
Lipschitz conditions was constructed by Wang [15], in which a non-decreasing
positive integrable function was used instead of usual Lipschitz constant.
Further, Wang and Li [16] derived some results on convergence of Newton’s
method in Banach spaces when derivative of the operators satisfies the radius
or center Lipschitz condition but with a weak $L$-average. Shakhno [12] have
studied the local convergence of the two step Secant-type method when the
first-order divided differences satisfy the generalized Lipschitz conditions.

Now, the intriguing question strikes out that whether the radius Lipschitz
condition with $L$-average and the non-decreasing of $L$ are necessary for the
convergence the third-order modification of Newton’s method. Motivated and
inspired by the above mentioned research works in this direction in the
present paper, we derived some theorems for scheme $(\ref{eqn:12})$. In the
first result generalized Lipschitz conditions has been used to study the local
convergence which is important to enlarge the convergence region without
additional hypotheses along with an error estimate. In the second theorem, the
domain of uniqueness of solution has been derived under center Lipschitz
condition. In the last two theorems, weak $L$-average has been used to derive
the convergence result of the considered third-order scheme. Also, few
corollaries are stated.

The rest part of the paper is structured as follows: section $2$ contains the
definitions related to $L$-average conditions. The local convergence and its
domain of uniqueness is mentioned in section $3$ and $4$, respectively.
Section $5$ deals with the improvement in assumption that the derivative of
$t$ satisfies the radius and center Lipschitz condition with weak $L$-average
namely $L$ is assumed to belong to some family of positive integrable
functions that are not necessarily non-decreasing for convergence theorems. An
example is also presented to justify the significance of the last result.

##  2\. Generalized Lipschitz conditions

Here, we denote by $V(x^{*},r)=\\{x:||x-x^{*}||<r\\}$ a ball with radius $r$
and center $x^{*}$. The condition imposed on the function $t$

$||t(x)-t(y^{\tau})||\leq L(1-\tau)(||x-x^{*}||+||y-x^{*}||),\forall\ x,\ y\in
V(x^{*},r),$ (2.1)

where $y^{\tau}=x^{*}+\tau(y-x^{*}),0\leq\tau\leq 1,$ is usually called radius
Lipschitz condition in the ball $V(x^{*},r)$ with constant $L$. Sometimes, if
it is only required to satisfy

$\displaystyle||t(x)-t(x^{*})||\leq 2L||x-x^{*}||,\forall\ x\in V(x^{*},r),$
(2.2)

we call it the center Lipschitz condition in the ball $V(x^{*},r)$ with
constant $L$. Furthermore, $L$ in the Lipschitz conditions does not
necessarily have to be constant but can be a positive integrable function. In
this case, conditions $(2.1)-(2.2)$ are respectively, replaced by

$\displaystyle||t(x)-t(y^{\tau})||\leq\int_{\tau(\rho(x)+\rho(y))}^{\rho(x)+\rho(y)}L(u)du,\forall\
x,\ y\in V(x^{*},r),0\leq\tau\leq 1,$

and

$\displaystyle||t(x)-t(x^{*})||\leq\int_{0}^{2\rho(x)}L(u)du,\forall\ x\in
V(x^{*},r),$ (2.4)

where $\rho(x)=||x-x^{*}||$. At the same time, the corresponding ’ Lipschitz
conditions’ is referred as to as having the $L$-average or generalized
Lipschitz conditions. Now, we start with the following lemmas, which will be
used later in the main theorems.

######  Lemma 2.1.

Suppose that $t$ has a continuous derivative in $V(x^{*},r)$ and
$[t^{\prime}(x^{*})]^{-1}$ exists.
(i) If $[t^{\prime}(x^{*})]^{-1}t^{\prime}$ satisfies the radius Lipschitz
condition with the $L$-average:

$\displaystyle||[t^{\prime}(x^{*})]^{-1}(t^{\prime}(x)-t^{\prime}(y^{\tau}))||\leq\int_{\tau(\rho(x)+\rho(y))}^{\rho(x)+\rho(y)}L(u)du,\forall\
x,\ y\in V(x^{*},r),0\leq\tau\leq 1,$

where $y^{\tau}=x^{*}+\tau(y-x^{*})$, $\rho(x)=||x-x^{*}||$ and $L$ is non-
decreasing, then we have

$\int_{0}^{1}||[t^{\prime}(x^{*})]^{-1}(t^{\prime}(x)-t^{\prime}(y^{\tau}))||\rho(y)d\tau\leq\int_{0}^{\rho(x)+\rho(y)}L(u)\frac{u}{\rho(x)+\rho(y)}\rho(y)du.$
(2.6)

(ii) If $[t^{\prime}(x^{*})]^{-1}t^{\prime}$ satisfies the center Lipschitz
condition with the $L$-average:

$||[t^{\prime}(x^{*})]^{-1}(t^{\prime}(x^{\tau})-t^{\prime}(x^{*}))||\leq\int_{0}^{2\tau\rho(x)}L(u)du,\forall\
x,\ y\in V(x^{*},r),0\leq\tau\leq 1,$ (2.7)

where $\rho(x)=||x-x^{*}||$ and $L$ is non-decreasing, then we have

$\int_{0}^{1}||[t^{\prime}(x^{*})]^{-1}(t^{\prime}(x^{\tau})-t^{\prime}(x^{*}))||\rho(x)d\tau\leq\int_{0}^{2\rho(x)}L(u)\left(\rho(x)-\frac{u}{2}\right)du.$
(2.8)

###### Proof.

The Lipschitz conditions $(\ref{eqn:25})$ and $(\ref{eqn:27})$, respectively,
imply that

$\displaystyle\int_{0}^{1}||[t^{\prime}(x^{*})]^{-1}(t^{\prime}(x)-t^{\prime}(y^{\tau}))||\rho(y)d\tau$
$\displaystyle\leq$
$\displaystyle\int_{0}^{1}\int_{\tau(\rho(x)+\rho(y))}^{\rho(x)+\rho(y)}L(u)du\rho(y)d\tau$
$\displaystyle=$
$\displaystyle\int_{0}^{\rho(x)+\rho(y)}L(u)\frac{u}{\rho(x)+\rho(y)}\rho(y)du,$
$\displaystyle\int_{0}^{1}||[t^{\prime}(x^{*})]^{-1}(t^{\prime}(x^{\tau})-t^{\prime}(x^{*}))||\rho(x)d\tau$
$\displaystyle\leq$
$\displaystyle\int_{0}^{1}\int_{0}^{2\tau\rho(x)}L(u)du\rho(x)d\tau$
$\displaystyle=$
$\displaystyle\int_{0}^{2\rho(x)}L(u)\left(\rho(x)-\frac{u}{2}\right)du.$

where $x^{\tau}=x^{*}+\tau(x-x^{*})$ and $y^{\tau}=x^{*}+\tau(y-x^{*})$. ∎

######  Lemma 2.2.

[16] Suppose that $L$ is positive integrable. Assume that the function $L_{a}$
defined by relation $(\ref{eqn:56})$ is non-decreasing for some $a$ with
$0\leq a\leq 1$. Then, $for\ each\ b\geq 0$, the function $\varphi_{b,a}$
defined by

$\varphi_{b,a}(f)=\frac{1}{f^{a+b}}\int_{0}^{f}u^{b}L(u)du,$ (2.9)

is also non-decreasing.

##  3\. Local convergence of Newton type method $(\ref{eqn:12})$

In this section, we state existence theorem under radius Lipschitz condition
for Newton type method $(\ref{eqn:12})$.

######  Theorem 3.1.

Suppose that $t(x^{*})=0$, $t$ has a continuous derivative in $V(x^{*},r)$,
$[t^{\prime}(x^{*})]^{-1}$ exists and $[t^{\prime}(x^{*})]^{-1}t^{\prime}$
satisfies the radius Lipschitz condition with the $L$-average:

$||[t^{\prime}(x^{*})]^{-1}(t^{\prime}(x)-t^{\prime}(y^{\tau}))||\leq\int_{\tau(\rho(x)+\rho(y))}^{\rho(x)+\rho(y)}L(u)du,\forall\
x,\ y\in V(x^{*},r),0\leq\tau\leq 1,$ (3.1)

where $y^{\tau}=x^{*}+\tau(y-x^{*})$, $\rho(x)=||x-x^{*}||$ and $L$ is non-
decreasing. Let $r$ satisfies the relation

$\displaystyle\frac{\int_{0}^{2r}L(u)udu}{2r(1-\int_{0}^{2r}L(u)du)}\leq 1.$
(3.2)

Then two step Newton type method $(\ref{eqn:12})$ is convergent for all
$x_{0}\in V(x^{*},r)$ and

$\displaystyle||y_{n}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{\int_{0}^{2\rho(x_{n})}L(u)udu}{2(1-\int_{0}^{2\rho(x_{n})}L(u)du)}\leq\frac{q_{1}}{\rho(x_{0})}\rho(x_{n})^{2},$
(3.3) $\displaystyle||x_{n+1}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{\int_{0}^{\rho(x_{n})+\rho(y_{n})}L(u)udu}{(\rho(x_{n})+\rho(y_{n}))(1-\int_{0}^{2\rho(x_{n})}L(u)du)}\rho(y_{n})\leq\frac{q_{2}q_{1}}{\rho(x_{0})\rho(y_{0})}\rho(x_{n})^{3},$

where the quantities

$\displaystyle
q_{1}=\frac{\int_{0}^{2\rho(x_{0})}L(u)udu}{2\rho(x_{0})(1-\int_{0}^{2\rho(x_{0})}L(u)du)},\
q_{2}=\frac{\int_{0}^{\rho(x_{0})+\rho(y_{0})}L(u)udu}{(\rho(x_{0})+\rho(y_{0}))(1-\int_{0}^{2\rho(x_{0})}L(u)du)},$

are less than $1$. Furthermore,

$\displaystyle||x_{n}-x^{*}||\leq C^{3^{n}-1}||x_{0}-x^{*}||,\
n=1,2,\cdots,C=q_{1}\frac{\rho(x_{0})}{\rho(y_{0})}.$ (3.6)

###### Proof.

On arbitrarily choosing $x_{0}\in V(x^{*},r)$, where $r$ satisfies the
relation $(\ref{eqn:32})$, $q_{1}$ and $q_{2}$, defined according to
inequality $(\ref{eqn:36})$ are less than 1. Indeed, since $L$ is monotone, we
get

$\displaystyle\left(\frac{1}{t_{2}^{2}}\int_{0}^{t_{2}}-\frac{1}{t_{1}^{2}}\int_{0}^{t_{1}}\right)L(u)udu$
$\displaystyle=\left(\frac{1}{t_{2}^{2}}\int_{t_{1}}^{t_{2}}+\left(\frac{1}{t_{2}^{2}}-\frac{1}{t_{1}^{2}}\right)\int_{0}^{t_{1}}\right)L(u)udu$
$\displaystyle\geq
L(t_{1})\left(\frac{1}{t_{2}^{2}}\int_{t_{1}}^{t_{2}}+\left(\frac{1}{t_{2}^{2}}-\frac{1}{t_{1}^{2}}\right)\int_{0}^{t_{1}}\right)udu$
$\displaystyle=L(t_{1})\left(\frac{1}{t_{2}^{2}}\int_{0}^{t_{2}}-\frac{1}{t_{1}^{2}}\int_{0}^{t_{1}}\right)udu=0,$

for $0<t_{1}<t_{2}.$ Thus, $\frac{1}{t^{2}}\int_{0}^{t}L(u)udu$ is non-
decreasing with respect to $t$. Now, we have

$\displaystyle q_{1}$ $\displaystyle=$
$\displaystyle\frac{\int_{0}^{2\rho(x_{0})}L(u)udu}{2\rho(x_{0})^{2}(1-\int_{0}^{2\rho(x_{0})}L(u)du)}\rho(x_{0})$
$\displaystyle\leq$
$\displaystyle\frac{\int_{0}^{2r}L(u)udu}{2r^{2}(1-\int_{0}^{2r}L(u)du)}\rho(x_{0})\leq\frac{||x_{0}-x^{*}||}{r}<1,$
$\displaystyle q_{2}$ $\displaystyle=$
$\displaystyle\frac{\int_{0}^{\rho(x_{0})+\rho(y_{0})}L(u)udu}{(\rho(x_{0})+\rho(y_{0}))^{2}(1-\int_{0}^{2\rho(x_{0})}L(u)du)}(\rho(x_{0}+\rho(y_{0}))$
$\displaystyle\leq$
$\displaystyle\frac{\int_{0}^{2r}L(u)udu}{2r^{2}(1-\int_{0}^{2r}L(u)du)}(\rho(x_{0})+\rho(y_{0}))\leq\frac{||x_{0}-x^{*}||+||y_{0}-x^{*}||}{2r}<1.$

Obviously, if $x\in V(x^{*},r)$, then using center Lipschitz condition with
the $L$-average and the relation $(\ref{eqn:32})$, we have

$\displaystyle||[t^{\prime}(x^{*})]^{-1}[t^{\prime}(x)-t^{\prime}(x^{*})]||\leq\int_{0}^{2\rho(x)}L(u)du\leq
1.$ (3.7)

Taking into account the Banach Lemma and the below equation

$\displaystyle||I-([t^{\prime}(x^{*})]^{-1}t^{\prime}(x)-I)||^{-1}=||[t^{\prime}(x)]^{-1}t^{\prime}(x^{*})||,$

we come to following inequality by using the relation $(\ref{eqn:37})$

$\displaystyle||[t^{\prime}(x)]^{-1}t^{\prime}(x^{*})||$ $\displaystyle\leq$
$\displaystyle\frac{1}{1-\int_{0}^{2\rho(x)}L(u)du}.$ (3.8)

Now, if $x_{n}\in V(x^{*},r)$ then we may write from expression
$(\ref{eqn:12})$

$\displaystyle||y_{n}-x^{*}||$ $\displaystyle=$
$\displaystyle||x_{n}-x^{*}-[t^{\prime}(x_{n})]^{-1}t(x_{n})||$ (3.9)
$\displaystyle=$
$\displaystyle||[t^{\prime}(x_{n})]^{-1}[t^{\prime}(x_{n})(x_{n}-x^{*})-t(x_{n})+t(x^{*})]||.$

Expanding $t(x_{n})$ along $x^{*}$ from Taylor’s Expansion, we attain

$\displaystyle
t(x^{*})-t(x_{n})+t^{\prime}(x_{n})(x_{n}-x^{*})=t^{\prime}(x^{*})\int_{0}^{1}[t^{\prime}(x^{*})]^{-1}[t^{\prime}(x_{n})-t^{\prime}(x^{\tau})]d\tau(x_{n}-x^{*}).$

Also, from the expression $(\ref{eqn:31})$ and combining the equations
$(\ref{eqn:39})$ and $(\ref{eqn:310})$, it can written as

$\displaystyle||y_{n}-x^{*}||$ $\displaystyle\leq$
$\displaystyle||[t^{\prime}(x_{n})]^{-1}t^{\prime}(x^{*})||.||\int_{0}^{1}[t^{\prime}(x^{*})]^{-1}[t^{\prime}(x_{n})-t^{\prime}(x^{\tau})]d\tau||.||(x_{n}-x^{*})||$
(3.11) $\displaystyle\leq$
$\displaystyle\frac{1}{\int_{0}^{2\rho(x_{n})}L(u)du}\int_{0}^{1}\int_{2\tau\rho(x_{n})}^{2\rho(x_{n})}L(u)du\rho(x_{n})d\tau.$

In view of Lemma $(\ref{lm:21})$ and the above inequality, we can obtain the
first inequality of expression $(\ref{eqn:34})$. By similar analogy and using
the last sub-step of the scheme $(\ref{eqn:12})$, we can write

$\displaystyle||x_{n+1}-x^{*}||$ $\displaystyle\leq$
$\displaystyle||[t^{\prime}(x_{n})]^{-1}t^{\prime}(x^{*})||.||\int_{0}^{1}[t^{\prime}(x^{*})]^{-1}[t^{\prime}(x_{n})-t^{\prime}(y^{\tau})]d\tau||.||(y_{n}-x^{*})||$
(3.12) $\displaystyle\leq$
$\displaystyle\frac{1}{\int_{0}^{2\rho(x_{n})}L(u)du}\int_{0}^{1}\int_{\tau(\rho(x_{n})+\rho(y_{n}))}^{\rho(x_{n})+\rho(y_{n})}L(u)du\rho(y_{n})d\tau.$

Using Lemma $(\ref{lm:21})$ and above expression, we can get the first
inequality of expression $(\ref{eqn:34a})$. Furthermore, $\rho(x_{n})$ and
$\rho(y_{n})$ are decreasing monotonically, therefore for all $n=0,1,...$, we
have

$\displaystyle||y_{n}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{\int_{0}^{2\rho(x_{n})}L(u)udu}{2(1-\int_{0}^{2\rho(x_{n})}L(u)du)}$
$\displaystyle\leq$
$\displaystyle\frac{\int_{0}^{2\rho(x_{0})}L(u)udu}{2\rho(x_{0})^{2}(1-\int_{0}^{2\rho(x_{n})}L(u)du)}2\rho(x_{n})^{2}\leq\frac{q_{1}}{\rho(x_{0})}\rho(x_{n})^{2},$

Also, by using second inequality of expression $(\ref{eqn:34})$, we have

$\displaystyle||x_{n+1}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{\int_{0}^{\rho(x_{n})+\rho(y_{n})}L(u)udu}{(\rho(x_{n})+\rho(y_{n}))^{2}(1-\int_{0}^{2\rho(x_{n})}L(u)du)}\rho(y_{n}).[\rho(x_{n})+\rho(y_{n})]$
$\displaystyle\leq$
$\displaystyle\frac{q_{2}}{\rho(x_{0})+\rho(y_{0})}[\rho(x_{0})\rho(y_{n})+\rho(y_{n})^{2}]\leq\frac{q_{2}q_{1}}{\rho(x_{0})\rho(y_{0})}\rho(x_{n})^{3},$

Hence, we have the complete inequalities of expressions $(\ref{eqn:34})$ and
$(\ref{eqn:34a})$. Also, it can be seen that inequality $(\ref{eqn:33})$ may
be easily derived from the expression $(\ref{eqn:313})$. ∎

##  4\. The uniqueness ball for the solution of equations

Here, we derived uniqueness theorem under center Lipschitz condition for
Newton type method $(\ref{eqn:12})$.

######  Theorem 4.1.

Suppose that $t(x^{*})=0$, $t$ has a continuous derivative in $V(x^{*},r)$,
$[t^{\prime}(x^{*})]^{-1}$ exists and $[t^{\prime}(x^{*})]^{-1}t^{\prime}$
satisfies the center Lipschitz condition with the $L$-average:

$\displaystyle|||[t^{\prime}(x^{*})]^{-1}(t^{\prime}(x)-t^{\prime}(x^{*}))||\leq\int_{0}^{2\rho(x)}L(u)du,\forall\
x\in V(x^{*},r),$ (4.1)

where $\rho(x)=||x-x^{*}||$ and $L$ is positive integrable function. Let $r$
satisfies the relation

$\displaystyle\frac{\int_{0}^{2r}L(u)(2r-u)du}{2r}\leq 1.$ (4.2)

Then the equation $t(x)=0$ has a unique solution $x^{*}$ in $V(x^{*},r)$.

###### Proof.

On arbitrarily choosing $y^{*}\in V(x^{*},r)$, $y^{*}\neq x^{*}$ and
considering the iteration

$\displaystyle||y^{*}-x^{*}||$ $\displaystyle=$
$\displaystyle||y^{*}-x^{*}-[t^{\prime}(x^{*})]^{-1}t(y^{*})||.$ (4.3)
$\displaystyle=$
$\displaystyle||[t^{\prime}(x^{*})]^{-1}[t^{\prime}(x^{*})(y^{*}-x^{*})-t(y^{*})+t(x^{*})]||.$

Expanding $t(y^{*})$ along $\ x^{*}$ from Taylor’s expansion, we have

$\displaystyle
t(x^{*})-t(y^{*})+t^{\prime}(x^{*})(y^{*}-x^{*})=\int_{0}^{1}[t^{\prime}(x^{*})]^{-1}[t^{\prime}(y^{*})^{\tau}-t^{\prime}(x^{*})]d\tau(y^{*}-x^{*})$

Following the expression $(\ref{eqn:41})$ and combining the inequalities
$(\ref{eqn:43})$ and $(\ref{eqn:44})$, we can write

$\displaystyle||y^{*}-x^{*}||$ $\displaystyle\leq$
$\displaystyle||[t^{\prime}(x^{*})]^{-1}t^{\prime}(x^{*})||.||\int_{0}^{1}[t^{\prime}(x^{*})]^{-1}[t^{\prime}(y^{*})^{\tau}-t^{\prime}(x^{*})]d\tau||.||(y^{*}-x^{*})||$
(4.5) $\displaystyle\leq$
$\displaystyle\int_{0}^{1}\int_{0}^{2\tau\rho(y^{*})}L(u)du\rho(y^{*})d\tau.$

In view of Lemma $(\ref{lm:21})$ and expression $(\ref{eqn:45})$, we obtain

$\displaystyle||y^{*}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{1}{2\rho(y^{*})}\int_{0}^{2\rho(y^{*})}L(u)[2\rho(y^{*})-u]du(y^{*}-x^{*})$
(4.6) $\displaystyle\leq$
$\displaystyle\frac{\int_{0}^{2r}L(u)(2r-u)du}{2r}\rho(y^{*})\leq||y^{*}-x^{*}||.$

but this contradicts our assumption. Thus, we see that $y^{*}=x^{*}$. This
completes the proof of the theorem. ∎

In particular, assuming that $L$ is a constant, we obtain the following
corollaries $(\ref{cr:51})$ and $(\ref{cr:52})$ from theorems $(\ref{th:31})$
and $(\ref{th:41})$, respectively.

######  Corollary 4.1.

Suppose that $x^{*}$ satisfies $t(x^{*})=0$, $t$ has a continuous derivative
in $V(x^{*},r)$, $[t^{\prime}(x^{*})]^{-1}$ exists and
$[t^{\prime}(x^{*})]^{-1}t^{\prime}$ satisfies the radius Lipschitz condition
with the $L$-average:

$\displaystyle||[t^{\prime}(x^{*})]^{-1}(t^{\prime}(x)-t^{\prime}(y^{\tau}))||$
$\displaystyle\leq$ $\displaystyle L(1-\tau)(||x-x^{*}||+||y-x^{*}||),$ (4.7)
$\displaystyle\forall\ x,\ y\in V(x^{*},r),0\leq\tau\leq 1,$

where $y^{\tau}=x^{*}+\tau(y-x^{*})$, $\rho(x)=||x-x^{*}||$ and $L$ is a
positive number. Let $r$ satisfies the relation

$\displaystyle r$ $\displaystyle=$ $\displaystyle\frac{1}{3L}.$ (4.8)

Then two-step Newton type method $(\ref{eqn:12})$ is convergent for all
$x_{0}\in V(x^{*},r)$ and

$\displaystyle||y_{n}-x^{*}||\leq\frac{q_{1}}{\rho(x_{0})}\rho(x_{n})^{2},$
(4.9)
$\displaystyle||x_{n+1}-x^{*}||\leq\frac{q_{2}q_{1}}{\rho(x_{0})\rho(y_{0})}\rho(x_{n})^{3},$
(4.10)

where the quantities

$\displaystyle q_{1}=\frac{L\rho(x_{0})}{[1-2L\rho(x_{0})]},\
q_{2}=\frac{L(\rho(x_{0})+\rho(y_{0}))}{2(1-2L\rho(x_{0}))},$ (4.11)

are less than $1$. Moreover

$\displaystyle||x_{n}-x^{*}||\leq C^{3^{n}-1}||x_{0}-x^{*}||,\
n=1,2,...;C=q_{1}\frac{\rho(x_{0})}{\rho(y_{0})}.$ (4.12)

######  Corollary 4.2.

Suppose that $x^{*}$ satisfies $t(x^{*})=0$, $t$ has a continuous derivative
in $V(x^{*},r)$, $[t^{\prime}(x^{*})]^{-1}$ exists and
$[t^{\prime}(x^{*})]^{-1}t^{\prime}$ satisfies the center Lipschitz condition
with the $L$-average:

$\displaystyle||[t^{\prime}(x^{*})]^{-1}(t^{\prime}(x)-t^{\prime}(x^{*}))||\leq
2L||x-x^{*}||,\forall\ x\in V(x^{*},r),$ (4.13)

where $\rho(x)=||x-x^{*}||$ and $L$ is a positive number. Let $r$ fulfills the
condition

$\displaystyle r$ $\displaystyle=$ $\displaystyle\frac{1}{L}.$ (4.14)

Then the equation $t(x)=0$ has a unique solution $x^{*}$ in $V(x^{*},r)$.
Moreover, the ball radius $r$ depends only on $L$.

Now, we will apply our main theorems to some special function $L$ and
immediately obtain the following corollaries.

######  Corollary 4.3.

Suppose that $x^{*}$ satisfies $t(x^{*})=0$, $t$ has a continuous derivative
in $V(x^{*},r)$, $[t^{\prime}(x^{*})]^{-1}$ exists and
$[t^{\prime}(x^{*})]^{-1}t^{\prime}$ satisfies the radius Lipschitz condition
with the $L$-average where given fixed positive constants $\gamma$ and $L>0$
with $L(u)=\gamma+Lu$ i.e.:

$\displaystyle||[t^{\prime}(x^{*})]^{-1}(t^{\prime}(x)-t^{\prime}(y^{\tau}))||$
$\displaystyle\leq$ $\displaystyle\gamma(1-\tau)(||x-x^{*}||+||y-x^{*}||)$
(4.15) $\displaystyle+$
$\displaystyle\frac{L}{2}(1-\tau^{2})(||x-x^{*}||+||y-x^{*}||)^{2},$
$\displaystyle\forall\ x,\ y\in V(x^{*},r),0\leq\tau\leq 1,$

where $y^{\tau}=x^{*}+\tau(y-x^{*})$, $\rho(x)=||x-x^{*}||$. Let $r$ satisfies
the relation

$\displaystyle r$ $\displaystyle=$
$\displaystyle\frac{-3\gamma+\sqrt{9\gamma^{2}+40/3L}}{7L}.$ (4.16)

Then two-step Newton type method $(\ref{eqn:12})$ is convergent for all
$x_{0}\in V(x^{*},r)$ and

$\displaystyle||y_{n}-x^{*}||\leq\frac{q_{1}}{\rho(x_{0})}\rho(x_{n})^{2},$
(4.17)
$\displaystyle||x_{n+1}-x^{*}||\leq\frac{q_{2}q_{1}}{\rho(x_{0})\rho(y_{0})}\rho(x_{n})^{3},$
(4.18)

where the quantities

$\displaystyle q_{1}$ $\displaystyle=$
$\displaystyle\frac{\rho(x_{0})[\gamma+4/3\rho(x_{0})]}{[1-2\gamma\rho(x_{0})-2L\rho(x_{0})^{2}]},$
(4.19) $\displaystyle q_{2}$ $\displaystyle=$
$\displaystyle\frac{\rho(x_{0})+\rho(y_{0})[\gamma/2+L/3(\rho(x_{0})+\rho(y_{0})]}{[1-2\gamma\rho(x_{0})-2L\rho(x_{0})^{2}]},$

are less than $1$. Moreover

$\displaystyle||x_{n}-x^{*}||\leq C^{3^{n}-1}||x_{0}-x^{*}||,\
n=1,2,...;C=q_{1}\frac{\rho(x_{0})}{\rho(y_{0})}.$ (4.21)

######  Corollary 4.4.

Suppose that $x^{*}$ satisfies $t(x^{*})=0$, $t$ has a continuous derivative
in $V(x^{*},r)$, $[t^{\prime}(x^{*})]^{-1}$ exists and
$[t^{\prime}(x^{*})]^{-1}t^{\prime}$ satisfies the center Lipschitz condition
with the $L$-average where given fixed positive constants $\gamma$ and $L>0$
with $L(u)=\gamma+Lu$ i.e.:

$\displaystyle||[t^{\prime}(x^{*})]^{-1}(t^{\prime}(x)-t^{\prime}(x^{*}))||\leq
2||x-x^{*}||(\gamma+L||x-x^{*}||),\forall\ x\in V(x^{*},r),$

where $\rho(x)=||x-x^{*}||$. Let $r$ satisfies the relation

$\displaystyle r$ $\displaystyle=$
$\displaystyle\frac{2\gamma-\sqrt{4\gamma^{2}-16/3L}}{8/3L}.$ (4.23)

Then the equation $t(x)=0$ has a unique solution $x^{*}$ in $V(x^{*},r)$.
Moreover, the ball radius $r$ depends only on $L$ and $\gamma$.

##  5\. Convergence under weak $L$-average

This section contains the results on re-investigation of the conditions and
radius of convergence of considered scheme already presented in the first
theorem but $L$ is not taken as non-decreasing function. It has been noticed
that the convergence order decreases. The second theorem of this section gives
a similar result to theorem $(\ref{th:31})$ but under the assumption of center
Lipschitz condition.

######  Theorem 5.1.

Suppose that $t(x^{*})=0$, $t$ has a continuous derivative in $V(x^{*},r)$,
$[t^{\prime}(x^{*})]^{-1}$ exists and $[t^{\prime}(x^{*})]^{-1}t^{\prime}$
satisfies the radius Lipschitz condition with the $L$-average:

$\displaystyle||[t^{\prime}(x^{*})]^{-1}(t^{\prime}(x)-t^{\prime}(y^{\tau}))||\leq\int_{\tau(\rho(x)+\rho(y))}^{\rho(x)+\rho(y)}L(u)du,\forall\
x,\ y\in V(x^{*},r),$

$0\leq\tau\leq 1$, where $y^{\tau}=x^{*}+\tau(y-x^{*})$, $\rho(x)=||x-x^{*}||$
and $L$ is positive integrable. Let $r$ satisfies

$\displaystyle\int_{0}^{2r}L(u)du\leq\frac{1}{2}.$ (5.2)

Then two-step Newton type method $(\ref{eqn:12})$ is convergent for all
$x_{0}\in V(x^{*},r)$ and

$\displaystyle||y_{n}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{\int_{0}^{2\rho(x_{n})}L(u)udu}{2(1-\int_{0}^{2\rho(x_{n})}L(u)du)}\leq
q_{1}\rho(x_{n}),$ (5.3) $\displaystyle||x_{n+1}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{\int_{0}^{\rho(x_{n})+\rho(y_{n})}L(u)udu}{(\rho(x_{n})+\rho(y_{n}))(1-\int_{0}^{2\rho(x_{n})}L(u)du)}\rho(y_{n})\leq
q_{2}q_{1}\rho(x_{n}),$

where the quantities

$\displaystyle
q_{1}=\frac{\int_{0}^{2\rho(x_{0})}L(u)du}{(1-\int_{0}^{2\rho(x_{0})}L(u)du)},\
q_{2}=\frac{\int_{0}^{\rho(x_{0})+\rho(y_{0})}L(u)du}{(1-\int_{0}^{2\rho(x_{0})}L(u)du)},$
(5.5)

are less than $1$. Moreover,

$\displaystyle||x_{n}-x^{*}||\leq(q_{1}q_{2})^{n}||x_{0}-x^{*}||,\ n=1,2,....$
(5.6)

Furthermore, suppose that the function $L_{a}$ is defined by

$\displaystyle L_{a}(f)=f^{1-a},$ (5.7)

is non-decreasing for some $a$ with $0\leq a\leq 1$ and $r$ satisfies

$\frac{1}{2r}\int_{0}^{2r}(2r+u)L(u)du\leq 1.$ (5.8)

Then the two-step Newton type method $(\ref{eqn:12})$ is convergent for all
$x_{0}\in V(x^{*},r)$ and

$\displaystyle||x_{n}-x^{*}||\leq C^{(1+2a)^{n}-1}||x_{0}-x^{*}||,\
n=1,2,\cdots,C=Q_{1}\frac{\rho(x_{0})}{\rho(y_{0})},$ (5.9)

where the quantity

$\displaystyle
Q_{1}=\frac{\int_{0}^{2\rho(x_{0})}L(u)udu}{2\rho(x_{0})(1-\int_{0}^{2\rho(x_{0})}L(u)du)},$
(5.10)

is less than $1$.

###### Proof.

On arbitrarily choosing $x_{0}\in V(x^{*},r)$, where $r$ satisfies the
relation $(\ref{eqn:52})$, the quantities $q_{1}$ and $q_{2}$ defined by the
equation $(\ref{eqn:55})$ are less than 1. Indeed, since $L$ is positive
integrable, we can get

$\displaystyle q_{1}$ $\displaystyle=$
$\displaystyle\frac{\int_{0}^{2\rho(x_{0})}L(u)du}{(1-\int_{0}^{2\rho(x_{0})}L(u)du)}\leq\frac{\int_{0}^{2r}L(u)du}{(1-\int_{0}^{2r}L(u)du)}<1,$
$\displaystyle q_{2}$ $\displaystyle=$
$\displaystyle\frac{\int_{0}^{\rho(x_{0})+\rho(y_{0})}L(u)du}{((1-\int_{0}^{2\rho(x_{0})}L(u)du)}\leq\frac{\int_{0}^{2r}L(u)du}{(1-\int_{0}^{2r}L(u)du)}<1.$

Obviously, if $x\in V(x^{*},r)$, then using center Lipschitz condition with
the $L$ average and the relation $(\ref{eqn:52})$, we have

$\displaystyle||[t^{\prime}(x^{*})]^{-1}[t^{\prime}(x)-t^{\prime}(x^{*})]||\leq\int_{0}^{2\rho(x)}L(u)du\forall\
x\in V(x^{*},r)\leq 1.$ (5.11)

Taking into account the Banach Lemma and the below equation

$\displaystyle||I-([t^{\prime}(x^{*})]^{-1}t^{\prime}(x)-I)||^{-1}=||[t^{\prime}(x)]^{-1}t^{\prime}(x^{*})||,$

we come to following inequality using the relation $(\ref{eqn:510})$

$\displaystyle||[t^{\prime}(x)]^{-1}t^{\prime}(x^{*})||$ $\displaystyle\leq$
$\displaystyle\frac{1}{1-\int_{0}^{2\rho(x)}L(u)du}.$ (5.12)

Now, if $x_{n}\in V(x^{*},r)$, then we may write from first sub-step of scheme
$(\ref{eqn:12})$

$\displaystyle||y_{n}-x^{*}||$ $\displaystyle=$
$\displaystyle||x_{n}-x^{*}-[t^{\prime}(x_{n})]^{-1}t(x_{n})||$ (5.13)
$\displaystyle=$
$\displaystyle||[t^{\prime}(x_{n})]^{-1}[t^{\prime}(x_{n})(x_{n}-x^{*})-t(x_{n})+t(x^{*})]||$

Expanding $t(x_{n})$ around $x^{*}$ from Taylor’s Expansion, it can written as

$\displaystyle
t(x^{*})-t(x_{n})+t^{\prime}(x_{n})(x_{n}-x^{*})=t^{\prime}(x^{*})\int_{0}^{1}[t^{\prime}(x^{*})]^{-1}[t^{\prime}(x_{n})-t^{\prime}(x^{\tau})]d\tau(x_{n}-x^{*}).$

Following the hypothesis $(\ref{eqn:51})$ and combining the inequalities
$(\ref{eqn:512})$ and $(\ref{eqn:513})$, we may write

$\displaystyle||y_{n}-x^{*}||$ $\displaystyle\leq$
$\displaystyle||[t^{\prime}(x_{n})]^{-1}t^{\prime}(x^{*})||.||\int_{0}^{1}[t^{\prime}(x^{*})]^{-1}[t^{\prime}(x_{n})-t^{\prime}(x^{\tau})]d\tau||.||(x_{n}-x^{*})||$
(5.15) $\displaystyle\leq$
$\displaystyle\frac{1}{\int_{0}^{2\rho(x_{n})}L(u)du}\int_{0}^{1}\int_{2\tau\rho(x_{n})}^{2\rho(x_{n})}L(u)du\rho(x_{n})d\tau.$

Using the results of Lemma $(\ref{lm:21})$ and the inequality
$(\ref{eqn:511})$ in the above expression we can obtain the first inequality
of $(\ref{eqn:54})$. By similar analogy for the last sub-step of the scheme
$(\ref{eqn:12})$, we can write

$\displaystyle||x_{n+1}-x^{*}||$ $\displaystyle\leq$
$\displaystyle||[t^{\prime}(x_{n})]^{-1}t^{\prime}(x^{*})||.||\int_{0}^{1}[t^{\prime}(x^{*})]^{-1}[t^{\prime}(x_{n})-t^{\prime}(y^{\tau})]d\tau||.||(y_{n}-x^{*})||$
(5.16) $\displaystyle\leq$
$\displaystyle\frac{1}{\int_{0}^{2\rho(x_{n})}L(u)du}\int_{0}^{1}\int_{\tau(\rho(x_{n})+\rho(y_{n}))}^{\rho(x_{n})+\rho(y_{n})}L(u)du\rho(y_{n})d\tau.$

Using Lemma $(\ref{lm:21})$ in the above expression, we can get the first
inequality of $(\ref{eqn:54a})$. Furthermore, $\rho(x_{n})$ and $\rho(y_{n})$
are decreasing monotonically, therefore for all $n=0,1,...,$ we have

$\displaystyle||y_{n}-x^{*}||\leq\frac{\int_{0}^{2\rho(x_{n})}L(u)udu}{2(1-\int_{0}^{2\rho(x_{n})}L(u)du)}\leq\frac{\int_{0}^{2\rho(x_{0})}L(u)du}{(1-\int_{0}^{2\rho(x_{n})}L(u)du)}\rho(x_{n})\leq
q_{1}\rho(x_{n}).$

Using the second inequality of expression $(\ref{eqn:54})$, we arrive at

$\displaystyle||x_{n+1}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{\int_{0}^{\rho(x_{n})+\rho(y_{n})}L(u)udu}{(\rho(x_{n})+\rho(y_{n}))(1-\int_{0}^{2\rho(x_{n})}L(u)du)}\rho(y_{n})$
(5.17) $\displaystyle\leq$
$\displaystyle\frac{\int_{0}^{\rho(x_{0})+\rho(y_{0})}L(u)du}{(1-\int_{0}^{2\rho(x_{0})}L(u)du)}\rho(y_{n})\leq
q_{2}q_{1}\rho(x_{n}).$

Also, the inequality $(\ref{eqn:53})$ may be easily derived from the
expression $(\ref{eqn:517})$. Furthermore, if the function $L_{a}$ defined by
the relation $(\ref{eqn:56})$ is non-decreasing for some $a$ with $0\leq a\leq
1$ and $r$ is determined by inequality $(\ref{eqn:57})$, it follows from the
first inequality of expression $(\ref{eqn:54})$ and Lemma $(\ref{lm:22})$ that

$\displaystyle||y_{n}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{\varphi_{1,a}(2\rho(x_{n}))2^{a}}{(1-\int_{0}^{2\rho(x_{n})}L(u)du)}\rho(x_{n})^{a+1}$
$\displaystyle\leq$
$\displaystyle\frac{\varphi_{1,a}(2\rho(x_{0}))2^{a}}{(1-\int_{0}^{2\rho(x_{n})}L(u)du)}\rho(x_{n})^{a+1}=\frac{Q_{1}}{\rho(x_{0})^{a}}\rho(x_{n})^{a+1}.$

Moreover, from the first inequality of $(\ref{eqn:54a})$ and Lemma
$(\ref{lm:22})$, we can write

$\displaystyle||x_{n+1}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{\varphi_{1,a}(\rho(x_{n})+\rho(y_{n}))(\rho(x_{n})+\rho(y_{n}))^{a}}{(1-\int_{0}^{2\rho(x_{n})}L(u)du)}\rho(y_{n}),$
$\displaystyle\leq$
$\displaystyle\frac{\varphi_{1,a}(\rho(x_{0})+\rho(y_{0}))(\rho(x_{n})+\rho(y_{n}))^{a}}{(1-\int_{0}^{2\rho(x_{n})}L(u)du)}\rho(y_{n}),$
$\displaystyle=$
$\displaystyle\frac{Q_{2}Q_{1}}{\rho(x_{0})^{a}\rho(y_{0})^{a}}\rho(x_{n})^{2a+1},$

where $Q_{1}<1$ and $Q_{2}<1$ are determined by the expression
$(\ref{eqn:59})$. Also, the inequality $(\ref{eqn:58})$ may be easily derived
and hence $x_{n}$ converges to $x^{*}$. Thus the proof is completed. ∎

######  Theorem 5.2.

Suppose that $t(x^{*})=0$, $t$ has a continuous derivative in $V(x^{*},r)$,
$[t^{\prime}(x^{*})]^{-1}$ exists and $[t^{\prime}(x^{*})]^{-1}t^{\prime}$
satisfies the center Lipschitz condition with the $L$-average:

$\displaystyle||[t^{\prime}(x^{*})]^{-1}(t^{\prime}(x)-t^{\prime}(x^{*}))||\leq\int_{0}^{2\rho(x)}L(u)du,\forall\
x\in V(x^{*},r),$ (5.18)

where $\rho(x)=||x-x^{*}||$ and $L$ is positive integrable function. Let $r$
satisfies

$\displaystyle\int_{0}^{2r}L(u)du\leq\frac{1}{3}.$ (5.19)

Then the two-step Newton type method $(\ref{eqn:12})$ is convergent for all
$x_{0}\in V(x^{*},r)$ and

$\displaystyle||y_{n}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{2\int_{0}^{2\rho(x_{n})}L(u)du}{1-\int_{0}^{2\rho(x_{n})}L(u)du}\rho(x_{n})\leq
q_{1}\rho(x_{n}),$ $\displaystyle||x_{n+1}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{\int_{0}^{2\rho(x_{n})}L(u)du+\int_{0}^{2\rho(y_{n})}L(u)du}{1-\int_{0}^{2\rho(x_{n})}L(u)du}\rho(y_{n})\leq
q_{2}q_{1}\rho(x_{n}),$

where the quantities

$\displaystyle
q_{1}=\frac{2\int_{0}^{2\rho(x_{0})}L(u)du}{(1-\int_{0}^{2\rho(x_{0})}L(u)du)},\
q_{2}=\frac{\int_{0}^{2\rho(x_{0})}L(u)du+\int_{0}^{2\rho(y_{0})}L(u)du}{(1-\int_{0}^{2\rho(x_{0})}L(u)du)}.$
(5.21)

are less than $1$. Moreover,

$\displaystyle||x_{n}-x^{*}||\leq(q_{1}q_{2})^{n}||x_{0}-x^{*}||,\ n=1,2,...,$
(5.22)

Furthermore, suppose that the function $L_{a}$ defined by the relation
$(\ref{eqn:56})$ is non-decreasing for some $a$ with $0\leq a\leq 1$, then

$\displaystyle||x_{n}-x^{*}||\leq C^{(1+2a)^{n}-1}||x_{0}-x^{*}||,\
n=1,2,\cdots,C=q_{1}\frac{\rho(x_{0})}{\rho(y_{0})}.$ (5.23)

and $q_{1}$ is given by the first expression of the equation
$(\ref{eqn:524})$.

###### Proof.

Let $x_{0}\in V(x^{*},r)$ and ${x_{n}}$ be the sequence generated by two-step
Newton type method given in $(\ref{eqn:12})$. Let $r$, $q_{1}$ and $q_{2}$ be
determined by the expressions $(\ref{eqn:522})$ and $(\ref{eqn:524})$,
respectively. Assume that $x_{n}\in V(x^{*},r)$. Then

$\displaystyle||y_{n}-x^{*}||$ $\displaystyle=$
$\displaystyle||y_{n}-x^{*}-[t^{\prime}(x_{n})]^{-1}t(x_{n})||$
$\displaystyle=$
$\displaystyle||[t^{\prime}(x_{n})]^{-1}[t^{\prime}(x_{n})(x_{n}-x^{*})-t(x_{n})+t(x^{*})]||.$

Expanding $t(x_{n})$ along $\ x^{*}$ from Taylor’s Expansion, we have

$\displaystyle
t(x^{*})-t(x_{n})+t^{\prime}(x_{n})(x_{n}-x^{*})=t^{\prime}(x^{*})\int_{0}^{1}[t^{\prime}(x^{*})]^{-1}[t^{\prime}(x_{n})-t^{\prime}(x)^{\tau}]d\tau(x_{n}-x^{*}).$

Following the hypothesis $(\ref{eqn:521})$ of the theorem and using the
equations $(\ref{eqn:526})$ and $(\ref{eqn:527})$, it can be written as

$\displaystyle||y_{n}-x^{*}||$ $\displaystyle\leq$
$\displaystyle||[t^{\prime}(x_{n})]^{-1}t^{\prime}(x^{*})||.||\int_{0}^{1}[t^{\prime}(x^{*})]^{-1}[t^{\prime}(x_{n})-t^{\prime}(x^{*})+t^{\prime}(x^{*})-t^{\prime}(x)^{\tau}]d\tau||$
(5.26) $\displaystyle.||(x_{n}-x^{*})||$ $\displaystyle\leq$
$\displaystyle\frac{1}{(1-\int_{0}^{2\rho(x_{n})}L(u)du}\left(\int_{0}^{1}\int_{0}^{2\tau\rho(x_{n})}L(u)du\rho(x_{n})d\tau\right)$
$\displaystyle+\left(\int_{0}^{1}\int_{0}^{2\rho(x_{n})}L(u)du\rho(x_{n})d\tau\right).$

In view of Lemma $(\ref{lm:21})$, the above inequality becomes

$\displaystyle||y_{n}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{2\int_{0}^{2\rho(x_{n})}L(u)du\rho(x_{n})-\frac{1}{2}\int_{0}^{2\rho(x_{n})}L(u)udu}{1-\int_{0}^{2\rho(x_{n})}L(u)du}$
$\displaystyle\leq$
$\displaystyle\frac{2\int_{0}^{2\rho(x_{n})}L(u)du}{1-\int_{0}^{2\rho(x_{n})}L(u)du}\rho(x_{n})=q_{1}\rho(x_{n}),$

which is same as first inequality of $(\ref{eqn:520a})$. By similar analogy
and form the final sub-step of the scheme $(\ref{eqn:12})$, we can write

$\displaystyle||x_{n+1}-x^{*}||$ $\displaystyle\leq$
$\displaystyle||[t^{\prime}(x_{n})]^{-1}t^{\prime}(x^{*})||\left(||\int_{0}^{1}[t^{\prime}(x^{*})]^{-1}[t^{\prime}(x_{n})-t^{\prime}(x^{*})]d\tau||.||(y_{n}-x^{*})||\right)$
(5.27)
$\displaystyle+\left(||\int_{0}^{1}[t^{\prime}(x^{*})]^{-1}[t^{\prime}(x^{*})-t^{\prime}(y^{\tau})]d\tau||.||(y_{n}-x^{*})||\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{(1-\int_{0}^{2\rho(x_{n})}L(u)du}\left(\int_{0}^{1}\int_{0}^{2\tau\rho(y_{n})}L(u)du\rho(y_{n})d\tau\right)$
$\displaystyle+\left(\int_{0}^{1}\int_{0}^{2\rho(x_{n})}L(u)du\rho(y_{n})d\tau\right).$

By virtue of Lemma $(\ref{lm:21})$, the above expression becomes

$\displaystyle||x_{n+1}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{\int_{0}^{2\rho(x_{n})}L(u)du\rho(y_{n})+\int_{0}^{2\rho(y_{n})}L(u)du\rho(y_{n})-\frac{1}{2}\int_{0}^{2\rho(y_{n})}L(u)udu}{1-\int_{0}^{2\rho(x_{n})}L(u)du}$
$\displaystyle\leq$
$\displaystyle\frac{\int_{0}^{2\rho(x_{n})}L(u)du\rho(y_{n})+\int_{0}^{2\rho(y_{n})}L(u)du\rho(y_{n})}{1-\int_{0}^{2\rho(x_{n})}L(u)du}$
$\displaystyle=$ $\displaystyle q_{2}q_{1}\rho(x_{n}),$

where $q_{1}<1$ and $q_{2}<1$ are determined by the relation
$(\ref{eqn:524})$. Also, it can be seen that inequality $(\ref{eqn:523})$ may
be easily derived from the second expression $(\ref{eqn:520a})$ and hence
$x_{n}$ converges to $x^{*}$.

Furthermore, if the function $L_{a}$ defined by the relation $(\ref{eqn:56})$
is non-decreasing for some $a$ with $0\leq a\leq 1$ and $r$ is determined by
the inequality $(\ref{eqn:522})$, it follows from the first inequality of the
expression $(\ref{eqn:520a})$ and Lemma $(\ref{lm:22})$ that

$\displaystyle||y_{n}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{2\varphi_{0,a}(2\rho(x_{n}))2^{a}}{(1-\int_{0}^{2\rho(x_{n})}L(u)du)}\rho(x_{n})^{a+1},$
$\displaystyle\leq$
$\displaystyle\frac{2\varphi_{0,a}(2\rho(x_{0}))2^{a}}{(1-\int_{0}^{2\rho(x_{0})}L(u)du)}\rho(x_{n})^{a+1}=\frac{q_{1}}{\rho(x_{0})^{a}}\rho(x_{n})^{a+1}.$

Moreover, from the second inequality of expression $(\ref{eqn:520a})$ and
Lemma $(\ref{lm:22})$, we get

$\displaystyle||x_{n+1}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{\varphi_{0,a}(2\rho(x_{n}))+\varphi_{0,a}(2\rho(y_{n})).(2\rho(x_{n}))^{a}.\rho(y_{n})}{(1-\int_{0}^{2\rho(x_{n})}L(u)du)}\rho(y_{n}),$
$\displaystyle\leq$
$\displaystyle\frac{\varphi_{0,a}(2\rho(x_{0}))+\varphi_{0,a}(2\rho(y_{0})).(2\rho(x_{n}))^{a}.\rho(y_{n})}{(1-\int_{0}^{2\rho(x_{n})}L(u)du)}\rho(y_{n}),$
$\displaystyle=$
$\displaystyle\frac{q_{2}.q_{1}}{\rho(x_{0})^{a}\rho(y_{0})^{a}}\rho(x_{n})^{2a+1}.$

Hence, it can be seen that inequality $(\ref{eqn:525})$ may be easily derived
and hence $x_{n}$ converges to $x^{*}$. ∎

Now, we wil apply our newly improved theorems to some special functions $L$
and results from Theorems $(\ref{th:51})$ and $(\ref{th:52})$ are recaptured.

######  Corollary 5.1.

Suppose that $x^{*}$ satisfies $t(x^{*})=0$, $t$ has a continuous derivative
in $V(x^{*},r)$, $[t^{\prime}(x^{*})]^{-1}$ exists and
$[t^{\prime}(x^{*})]^{-1}t^{\prime}$ satisfies the radius Lipschitz condition
with the $L$-average with $L(u)=cau^{a-1}$i.e. :

$||[t^{\prime}(x^{*})]^{-1}(t^{\prime}(x)-t^{\prime}(y^{\tau}))||\leq
c.(1-\tau^{a})(||x-x^{*}||+||y-x^{*}||)^{a},$ (5.28)

$\forall\ x,\ y\in V(x^{*},r),0\leq\tau\leq 1$, where
$y^{\tau}=x^{*}+\tau(y-x^{*})$, $\rho(x)=||x-x^{*}||$,$0<a<1$ and $c>0$. Let
$r$ satisfies

$\displaystyle r$ $\displaystyle=$
$\displaystyle\left(\frac{a+1}{c2^{a}(1+2a)}\right)^{\frac{1}{a}}.$ (5.29)

Then two-step Newton type method $(\ref{eqn:12})$ is convergent for all
$x_{0}\in V(x^{*},r)$ and

$\displaystyle||y_{n}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{\int_{0}^{2\rho(x_{n})}L(u)udu}{2(1-\int_{0}^{2\rho(x_{n})}L(u)du)}\leq
q_{1}\rho(x_{n}),$ (5.30) $\displaystyle||x_{n+1}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{\int_{0}^{\rho(x_{n})+\rho(y_{n})}L(u)udu}{(\rho(x_{n})+\rho(y_{n}))(1-\int_{0}^{2\rho(x_{n})}L(u)du)}\rho(y_{n})\leq
q_{2}q_{1}\rho(x_{n}),$

where the quantities

$\displaystyle
q_{1}=\frac{ca2^{a}\rho(x_{0})^{a}}{(1+a)[1-2^{a}c\rho(x_{0})^{a}]},\
q_{2}=\frac{ca(\rho(x_{0})+\rho(y_{0}))^{a}}{(a+1)(1-2^{a}c\rho(x_{0})^{a})},$
(5.32)

are less than $1$. Furthermore,

$\displaystyle||x_{n}-x^{*}||\leq C^{3^{n}-1}||x_{0}-x^{*}||,\
n=1,2,...,C=q_{1}\frac{\rho(x_{0})}{\rho(y_{0})}.$ (5.33)

######  Corollary 5.2.

Suppose that $x^{*}$ satisfies $t(x^{*})=0$, $t$ has a continuous derivative
in $V(x^{*},r)$, $[t^{\prime}(x^{*})]^{-1}$ exists and
$[t^{\prime}(x^{*})]^{-1}t^{\prime}$ satisfies the center Lipschitz condition
with the $L$-average with $L(u)=cau^{a-1}$ i.e.:

$\displaystyle||[t^{\prime}(x^{*})]^{-1}(t^{\prime}(x)-t^{\prime}(x^{*}))||\leq
c2^{a}||x-x^{*}||^{a},\forall\ x\in V(x^{*},r),$ (5.34)

where $\rho(x)=||x-x^{*}||$, $0<a<1$ and $c>0$. Let $r$ satisfies

$\displaystyle r$ $\displaystyle=$
$\displaystyle\left(\frac{1}{3c2^{a}}\right)^{\frac{1}{a}}.$ (5.35)

Then two-step Newton type method $(\ref{eqn:12})$ is convergent for all
$x_{0}\in V(x^{*},r)$ and

$\displaystyle||y_{n}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{2\int_{0}^{2\rho(x_{n})}L(u)du}{1-\int_{0}^{2\rho(x_{n})}L(u)du}\rho(x_{n})\leq
q_{1}\rho(x_{n}),$ $\displaystyle||x_{n+1}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{\int_{0}^{2\rho(x_{n})}L(u)du+\int_{0}^{2\rho(y_{n})}L(u)du}{1-\int_{0}^{2\rho(x_{n})}L(u)du}\rho(y_{n})\leq
q_{2}q_{1}\rho(x_{n}),$

where the quantities

$\displaystyle
q_{1}=\frac{c2^{a+1}\rho(x_{0})^{a}}{[1-2^{a}c\rho(x_{0})^{a}]},\
q_{2}=\frac{c2^{a}(\rho(x_{0})^{a}+\rho(y_{0})^{a})}{(1-2^{a}c\rho(x_{0})^{a})},$
(5.37)

are less than 1. Furthermore,

$\displaystyle||x_{n}-x^{*}||\leq C^{3^{n}-1}||x_{0}-x^{*}||,\
n=1,2,...,C=q_{1}\frac{\rho(x_{0})}{\rho(y_{0})}$ (5.38)

######  Corollary 5.3.

Suppose that $x^{*}$ satisfies $t(x^{*})=0$, $t$ has a continuous derivative
in $V(x^{*},r)$, $[t^{\prime}(x^{*})]^{-1}$ exists and
$[t^{\prime}(x^{*})]^{-1}t^{\prime}$ satisfies the center Lipschitz condition
with the $L$-average with $L(u)=\frac{2\gamma c}{(1-\gamma u)^{3}}$ i.e.:

$\displaystyle||[t^{\prime}(x^{*})]^{-1}(t^{\prime}(x)-t^{\prime}(x^{*}))||\leq\frac{c}{(1-2\gamma\rho(x))^{2}}-c,\forall\
x\in V(x^{*},r)$ (5.39)

where $\rho(x)=||x-x^{*}||$, $\gamma>0$ and $c>0$. Let $r$ satisfies

$\displaystyle r$ $\displaystyle=$
$\displaystyle\frac{3c+1-\sqrt{3c(3c+1)}}{2\gamma(3c+1)}.$ (5.40)

Then two-step Newton type method $(\ref{eqn:12})$ is convergent for all
$x_{0}\in V(x^{*},r)$ and

$\displaystyle||y_{n}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{2\int_{0}^{2\rho(x_{n})}L(u)du}{1-\int_{0}^{2\rho(x_{n})}L(u)du}\rho(x_{n})\leq
q_{1}\rho(x_{n}),$ $\displaystyle||x_{n+1}-x^{*}||$ $\displaystyle\leq$
$\displaystyle\frac{\int_{0}^{2\rho(x_{n})}L(u)du+\int_{0}^{2\rho(y_{n})}L(u)du}{1-\int_{0}^{2\rho(x_{n})}L(u)du}\rho(y_{n})\leq
q_{2}q_{1}\rho(x_{n}),$

where the quantities

$\displaystyle q_{1}$ $\displaystyle=$
$\displaystyle\frac{2c-2c(1-2\gamma\rho(x_{0}))^{2}}{[1-2\gamma\rho(x_{0})]^{2}(1+c)-c},$
(5.42) $\displaystyle q_{2}$ $\displaystyle=$
$\displaystyle\frac{[c-c(1-2\gamma\rho(x_{0}))^{2}](1-2\gamma\rho(y_{0}))^{2})+[c-c(1-2\gamma\rho(y_{0}))^{2}](1-2\gamma\rho(x_{0}))^{2})}{([1-2\gamma\rho(x_{0})]^{2}(1+c)-c)(1-2\gamma\rho(y_{0}))^{2})},$

are less than 1. Furthermore,

$\displaystyle||x_{n}-x^{*}||\leq C^{3^{n}-1}||x_{0}-x^{*}||,\
n=1,2,...,C=q_{1}\frac{\rho(x_{0})}{\rho(y_{0})}.$ (5.44)

Example 5.1 Let $X=Y=R$, the reals. Define

$\displaystyle t(x)=\int_{0}^{x}\left(1+2x\sin\frac{\pi}{x}\right)dx,\ \forall
x\in R.$

Then

$t^{\prime}(x)=\begin{cases}1+2x\sin\frac{\pi}{x},&x\neq 0,\\\
1,&x=0,\end{cases}$

Obviously, $x^{*}=0$ is a zero of $t$ and $t^{\prime}$ satisfies that

$\displaystyle||[t^{\prime}(x^{*})]^{-1}(t^{\prime}(x)-t^{\prime}(x^{*}))||=\left|2x\sin\frac{\pi}{x}\right|\leq
2|x-x^{*}|,\forall\ x\in R.$

It follows from Theorem $(\ref{th:52})$ that for any $x_{0}\in V(x^{*},1/6)$

$\displaystyle||x_{n}-x^{*}||\leq C^{3^{n}-1}||x_{0}-x^{*}||,\
n=1,2,\cdots,C=\left(\frac{4|x_{0}|^{2}}{1-2|x_{0}|.|y_{0}|}\right).$

However, there is no positive integrable function $L$ such that the
inequality $(\ref{eqn:51})$ is satisfied. In fact, notice that

$||[t^{\prime}(x^{*})]^{-1}(t^{\prime}(x)-t^{\prime}(y^{\tau}))||=\left|2x\sin\frac{\pi}{x}-2y\tau\sin\frac{\pi}{y\tau}\right|=\frac{4}{2k+1},$

for $x=1/k,y=1/k,\tau=\frac{2k}{2k+1}$ and $k=1,2,\cdots$ Thus, if there was a
positive integrable function $L$ such that the inequality $(\ref{eqn:51})$
holds on $V(x^{*},r)$ for some $r>0$, it follows that there exists some
$n_{0}>1$ such that

$\displaystyle\int_{0}^{2r}L(u)du\geq\sum_{k=n_{0}}^{+\infty}\int_{\frac{4}{2k+1}}^{\frac{2}{k}}L(u)du\geq\sum_{k=n_{0}}^{+\infty}\frac{4}{2k+1}=+\infty,$

which is a contradiction. This example shows that Theorem $(\ref{th:52})$ is a
crucial improvement of Theorem $(\ref{th:51})$ if the radius of the
convergence ball is ignored.

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##  Appendix A Figure Content Description

In this section, we include content descriptions for figure images.

###  A.1. Figure 1

Left to right: (1) Text: COVID-19 Financial Relief. While Congress did nothing
for 6 months, here’s what the rest of the world is doing: Money distributed by
countries during the pandemic: Japan: Up to 100% of your salary. Norway: Up to
90% of your salary. Germany: Up to 87% of your salary. France: Up to 84% of
your salary. U.K.: Up to 80% of your salary. Italy: Up to 80% of your salary.
Canada: $2000 a month since April. U.S.A.: $700 one-time payment, first relief
in 6 months. Visuals: Teal and yellow table with a row for each country. (2)
Text: Build Black. Buy Black. Bank Black. Visuals: Black background and white
text, once in bold, filled text and once in white-outlined text. Accompanied
by a yellow shopping card icon. (3) Text: The fight for trans Black Lives is
not over. Let’s continue to rise up! Visuals:  Text is white and outlined in
black. The background is comprised of pink, blue, and white wavy rows. (4)
Text:  The Black Panther Party. Founded October 1966. What did they give us?
Visuals:  The background includes a picture of a black panther in a white
circle over a yellow-tinted image of a Black Panther protest. (5) Text:  Black
Women & Femmes Are Divine: A Reclamation in the Name of Breonna Taylor.
Saturday, 10/3 12PM PST. Visuals:  A pink-tinted image of four women/femmes in
white T-shirts. (6) Text:  BREATHE Day. IG Take Over. From the Streets to
Congress: A BREATHE Day conversation with Jamaal Bowman. Monifa Bandele,
Jamaal Bowman (jamaalbowmanny). September 29th 11:00AM ET. The BREATHE Act.
M4BL. Visuals:  Orange, cyan, and purple patterns in the background. Images of
Bowman and Bandele in the center. (7) Text:  Seven Years of Growth: BLM’s
Founder and incoming Executive Director reflects on the movement. Visuals:
The background is a purple, red, and yellow tinted collage of posters and
protesting individuals, including a Black Lives Matter Sign and a BLM sign
with a yellow fist. (8) Text:  Here is how we #DefundThePolice. First. Demand
that lawmakers support reparations for all families of those killed and
survivors of police violence. Second. Demand that State, city and mincipality
spend less on law enforcement and incarceration. Period. Finally. Demand
investment into Black communities. It is not enough to defund the police, we
need to put in place systems to uplift and protect. That means divesting from:
police in schools, criminalizing mental health, military weapons against
citizens and investing in teachers & counselors, mental health & restorative
services, community-led harm reduction. Visuals:  Yellow and black background
with black, white, yellow text throughout. Headers have brushstroke-like
highlighting. (9) Text:  Call to Action. Continue sharing her story, posting
her face, and saying her name. PLEASE USE #BreonnaTaylor & #JusticeForBre. Tag
all responsible parties. [Twitter:] LMPD, MayorGregFisher, GovAndyBeshear.
[Instagram:] LMPD.KY, MayorGregFisher, GovAndyBeshear. Make calls & send
emails for Breonna to the investigative agencies, institutions and individuals
in charge and make the demands known!. FightForBreonna.org. Visuals:  Yellow
box outline with white and yellow text. The background is a black-and-white
image of a protest. (10) Text:  Crisis Response: Sign the Petitions. We need
more. We demand more. Visuals:  The text is in bold white and ”Sign the
Petitions” has a blue highlight. The background is black with red coronavirus
speckled throughout. (11) Text:  We celebrate the Life and Legacy of Martin
Luther King Jr. We will continue to fight. Visuals:  Gray grainy background
with a yellow circle. Two black and white pictures that appear to be taped on.
The left one is of MLK at a gathering and the right one is of a protest. (12)
Text:  Black Futures Month. Global Pop Up: Torontoa, Atlanta, Los Angeles.
February 17-March 3. Featuring Karen Miranda Augustine, Rodney Diverlus,
Jasmyn Fyffe, Kim Ninkuru, Kike Otuije, Pete Owusu-Ansah, Gloria Swain,
Diseiye Thompson, Kyisha Williams, Ravyn Wngz. February 17, 2019 7-10 PM
Opening Reception ASL Provided. Access Info: There are three steps to the
entrance. ASL provided for opening event. 76 Geary Ave.
infoblacklivesmatter.ca. Visuals:  White background with black patterned and
green-shaded borders. Green highlighting, circles, and text accents. (13)
Text:  What Issues Matter to You? #WhatMattersin2020. Visuals:  Dark gray
image of protesters in background with poster that says ”Demilitarize the
Police.” ”Matter” is in yellow cursive and text is in white and yellow. (14)
Text:  Court Support Needed!! BLMLA Organizer, Melina Abdullah is being
prosecuted for her activism. When they come for one of us, they come for all
of us. Join us!! Friday 9/28 8:30 AM 210 W. Temple St. (Criminal Courts
Building) Dept. 48. **Wear your #BlackLivesMatter gear, if you have it.**
Visuals:  Background is a black and white image of Abdullah with a megaphone
along with yellow, black, and white patterns and shapes. (15) Text:  End
Prison Slavery! Nationwide prison strike Aug 21-Sept 9, 2018. Visuals:  The
head and shoulders of two men is pictured in front of a pink, orange, and
yellow gradient. In front, a smaller individual holds a flag reading ”Strike”
atop a layer of orange and pink roses. (16) Text:  #SayHerName. A week of
action to end violence against Black women and girls. June 11th - 17th.
bit.ly/sayhername2018. Visuals:  Dark gray background with yellow outlined
circle enclosing the text. Word accents in green and red. A protester with a
megaphone is pictured at the bottom. (17) Text:  APTP/BLM Sacramento and
Justice Network Call For A National Day of Action. 50th Anniversary of MLK’s
Assasination. DA’s Office, 901 G ST, Wed Apr 4, 3-5:30. We have nothing to
lose but our chains!. We love you Stephon. The time is always right to do what
is right. Visuals: Purple background, text in yellow and white. Graphical
yellow and white images of Assata Shakur, Stephon, and MLK with the quotes
listed respectively. (18) Text:  Flint. Flint will have clean and free water.
There will be a day when watter warriors will happily retire. Visuals:  An
individual with white face paint and a turquoise shirt stands in front of a
wall of the ”Flint will have clean and free water” quote repeated in white
text, with ”Flint” listed large and in white at the top. (19) Text:  Beyond
the Moment. Day of Political Education. National Prep Call. Tues. March 28th
8PM EST / 5PM PST. Join a broad mutli-racial, cross-sector coalition of
organizers from around the country as we share plans for a mass day of
poltiical education that will fall ont he 50th anniversary of Dr. King’s
”Beyond Vietnam” speec on april 4th. Find out how to plug in and organize your
own. Visuals:  The ”Beyond the Moment” header is slanted and a black and white
image of protesters with the American flag is at the top. Background is gray
and turquoise. (20) Text:  $40K in 4 Days. For #4YearsofBLM. Please donate at
celebrateBLM.com. ”When Black people get free, we all get free.” Visuals:
Large black and gray text atop a black and white image of a protest.

###  A.2. Figure 2

Left to right: (1) Text:  Allyship is a constant process of actions, not
beliefs. Visuals:  Black background with pink bold text. (2) Text:  Use Your
Platform. However large or small, you have an opportunity to reach people. Use
it. Share news stories, and updates—especially if your timeline is not very
diverse. Work to diversify your timeline. Raise awareness amongst those who
may otherwise not see. Understand that your voices can travel to places that
ours can’t. But, remember—posting is not taking action. Emails, phone calls,
letters, donations, conversations are actions. Visuals:  Pink background with
white and black bold text. (3) Text:  What to Say When People Deny the Reality
of What’s Happening Right Now Part 2. Another quick guide of possible
responses to common remarks made in regard to recent6 protests sparked by
George Floyd’s murder, most of which are made by, you guessed it, white
people. Visuals:  Beige background with black text. (4) Text:  7 Tips to
Maintain Momentum & Stay Involved. Get Involved and Wield Your Dollar. Protest
& Aid Protestors. Help Vulnerable Population and the Homeless. Stay Connected
to the Movement, Organizations, and Activists. Support Black Businesses,
Friends, Family, and Colleagues. Use What You’re Good At. Hold Racists and
Racist Institutions Accountable. Visuals:  Overlayed dark gray, green, and
maroon squares. White text.

###  A.3. Figure 3

Left to right: (1) Text:  You can spell pretty without exotic. Visuals:  Peach
background with an eye in the center of the text. Text in maroon, brown, red,
teal, and yellow. (2) So… Sanrio’s your new aesthetic? Visuals:  Pink
background. Image of Hello Kitty picking cherries from a tree. Text is in red
with yellow highlight. (3) Text:  Stop fetishizing Asians. Visuals:  Peach and
beige waves in the background. A brown silhouette of a person accompanies the
text in brown. (4) Text:  Voguish. Problematic. Visuals:  Peach background.
Two eyes framed by two cupped hands. Yellow flowers at the top. ”Problematic”
is enclosed in brackets and red and ”Voguish” is in mild yellow at the top.
(5) Text:  They can’t burn us all. Visuals:  Three individuals with fists
raised frame the text, which is black with yellow highlight. Red background.
(6) Text:  Trump is banning WeChat. Visuals:  Mild yellow background with a
person holding a phone. The phone has an icon of Trump and an icon of the
WeChat logo. Text is in dark red.

###  A.4. Figure 5

Four images are on the left and four are on the right. For the 2014 images on
the left, from left to right: (1) A crowd is seated in a church. (2) Two
individuals hold a poster with a person’s face and the text ”My Life Matters.”
(3) An individual is being arrested on a street, surrounded by police. (4) An
individual is painting art on concrete. For the 2020 images on the right, from
left to right: (1) Black Voters Showed Up! This time in Georgia! Thank you for
your work: Fair Fight and New Georgia Project! (2) Text:  Black Women & Femmes
Are Divine: A Reclamation in the Name of Breonna Taylor. Saturday, 10/3 12PM
PST. Visuals:  A pink-tinted image of four women/femmes in white T-shirts. (3)
Text:  BREATHE Day. IG Take Over. From the Streets to Congress: A BREATHE Day
conversation with Jamaal Bowman. Monifa Bandele, Jamaal Bowman
(jamaalbowmanny). September 29th 11:00AM ET. The BREATHE Act. M4BL. Visuals:
Orange, cyan, and purple patterns in the background. Images of Bowman and
Bandele in the center. (4) Text:  Seven Years of Growth: BLM’s Founder and
incoming Executive Director reflects on the movement. Visuals:  The background
is a purple, red, and yellow tinted collage of posters and protesting
individuals, including a Black Lives Matter Sign and a BLM sign with a yellow
fist.

###  A.5. Figure 6

Left to right: (1) Text:  Free Angela! Free All of Us! Angela Davis Finally
Acquitted of False Charges. Vote for Survival: Bobby Seale for Mayor! Elaine
Brown for City Council! Visuals:  An image of Davis’s face on the right and
Davis standing on the left. Bright pink background with bold black text (Pra,
[n.d.]). (2) Text:  Murdered. Sam Napier. One word is a thousand words to a
thousand ears. Sam Napier spread the people’s word. Even now we hear it
resound ten thousand times. ”Circulate to Educate.” Sam Napier, Black Panther
Party, Intercommunal News Service, Circulation Manager, Murdered by Fascists.
Visuals:  ”Murdered” is in bold text at the top. Napiers face is pictured
large in the center with rays extending from it. Light blue background (Lib,
2012). (3) Text:  Caustion: Surviving is Criminal: Events in Two Black Men’s
Lives Dramatize Why. Prison Camps U.S.A. A-000000000. The Unknown Slaves.
Visuals:  Two heads appear with shackles about the neck. Title is in blue text
and Prison Camp text appears as a label at the bottom (Sum, 2017). (4) Text:
Our Fight is Not in Vietnam. Free the GI’s. A crying individual appears with a
helmet and weaponry. In their helmet is an image of war and destruction. The
text is on a protest sign (Mckinley and Russonello, 2016). (5) Text:  Afro-
American solidarity with the oppressed People of the world. Visuals:  An
individual with a spear and gun in a pink and teal jacket stares forward. The
background is comprised of pink and gray alternating rays (Jones, 2020). (6)
Text: Free the NY 21. And All Political Prisoners. Visuals:  Orange
background. A picture of protesters with fists raised lines the bottom of the
image (Jones, 2020). (7) Text:  Political Prisoners of USA Fascism. Visuals:
Two individuals in leather jackets, and one with a gun stand (Jones, 2020).
The text is in bold blue. (8) Text:  Free Angela. Visuals:  A headshot of
Davis speaking. Text is in green (Jones, 2020).

###  A.6. Figure 8

Left to right: (1) Farmworkers stand distraught behind barbed wire in front of
farmhouses (Noriega, [n.d.]). (2) Large text reads ”We are NOT a minority!!”
An individual to the right of the text points to the viewer. Background is a
blue sky over a horizon with greenery at the bottom (Acosta, [n.d.]).


# Doing Things Twice: Strategies to Identify Studies for Targeted Validation

Gopal P. Sarma
School of Medicine, Emory University
Email<EMAIL_ADDRESS>
## I. Introduction

In recent years, significant attention has been given to problems with
reproducibility in many areas of science. Some of these analyses have been
theoretical in nature [1] while others have been focused efforts aimed at
replicating large numbers of studies in a specific field [2, 3] .

The issue has become sufficiently high profile that it has been dubbed the
“reproducibility crisis,” and is now a major topic of and debate in both the
scientific [4, 5, 6, 7, 8] and popular press [9, 10, 11, 12, 13].

One thing is clear—we simply do not know what the “reproducibility
distribution” looks like for the entirety of science. Taking this position as
a starting point, how then do we identify and prioritize published results to
investigate in greater detail? Should reproducibility initiatives be strictly
local and originate from individual scientists themselves, or should there be
more global, distributed efforts as well? In this brief note, I examine this
and other questions and propose several strategies for identifying key results
to be the focus of validation efforts.

## II. Uncovering “Linchipin” Results

It goes without saying that all scientific results, even the true ones, are
not created equally. In order to use resources efficiently, we would ideally
identify “linchipin” results, that is, those studies which would carry the
highest impact if we had greater certainty in their outcome. For instance, in
cases where poorly conducted or fraudulent studies form the basis for
guidelines or procedures in medicine, lengthy retractions can have
significantly deleterious consequences for the public [14]. To use an
ecological metaphor, these studies might be described as “keystone species” of
the scientific ecosystem. How can we uncover such results?

### Reproduction Versus Validation

Although the phrase “reproducibility crisis” has taken root in contemporary
discussions, simply re-doing an experiment may not always be the most
appropriate course of action. For instance, there might be linchipin
theoretical results which simply need greater scrutiny or investigation with
alternative methods. The same may be true of certain experimental results,
where the highest value would be gained from re-thinking a given experimental
design using alternative techniques.

Therefore, to broaden the scope of the discussion to include all scientific
results, not just experimental ones, as well as approaches other than simply
repeating the original study under question, I will use the phrase “validation
effort” rather than “reproducibility effort.”

### Have Individual Scientists Initiate Validation Efforts

A purely “local” approach would be for validation efforts to be initiated by
investigators themselves. For instance, Schooler and several colleagues at
UCLA arrived at an agreement whereby each researcher’s lab would attempt to
replicate the results of the others prior to publication [15].

However, not all scientists are in a position to create such arrangements.
Therefore, formal mechanisms for arranging validation efforts should be
encouraged. As an example, this is the principle behind ScienceExchange’s
Reproducibility Project111http://validation.scienceexchange.com/, a
marketplace for scientists to identify researchers from a network of
laboratories to validate their research.

### Polling Scientists or Crowdsourcing

This approach would be more globally oriented and could be initiated by
funding agencies, individual laboratories, or by “open science” projects. The
strategy would be to distribute polls to scientists in different disciplines
asking them what they believe to be high-value results. Like any poll, a
number of practical issues will arise in arriving at reliable data. Questions
will likely have to be written by scientists with sufficient experience in a
given field to elicit reliable answers and to follow-up on ambiguities in
responses. The questions will have to be framed appropriately. The
administrators of such polls will have to be mindful of the fact that
individuals could use a call for reproducibility as a political tactic for
attacking a competitor’s research.

Nonetheless, polling scientists is likely a very straightforward strategy to
uncover high-value results. The results of such polls could be used by funding
agencies, independent foundations, or individual scientists themselves in
deciding how to allocate resources for validation efforts.

### Let Larger Research Agendas be the Focal Points

This strategy would be to focus validation efforts around those results that
form the foundation of major research agendas. For example, while soliciting
proposals for new programs, funding agencies could ask scientists to submit a
bibliography containing results upon which their proposed research relies. Or
more directly, researchers could be asked to submit a separate document
alongside their grant proposals suggesting experiments that merit additional
investigation and which would advance their own research. Scientists could
also publish such documents independent of grant applications, perhaps along
the lines of a review article.

### Scientific Data Science

Scientific data science refers to the use of data analytic techniques to treat
the scientific corpus itself as a massive data set for analysis. Although the
growth of data science has largely been driven by commercial applications in
social media and business intelligence, we are now beginning to see the
applications of data science to the scientific literature as well.

At the most basic level, article recommendation by Google Scholar or the many
reference managers used by researchers is an example of data science applied
to scientific papers. Other examples of contemporary research in scientific
data science include fraud detection (applying natural language processing to
uncover linguistic signatures of fraudulent research) [16], characterizing the
emergence of global scientific trends (using $n$-grams and patent citation
networks to model the flow of ideas and technological development) [17], and
resource allocation in the biomedical sciences (developing metrics which
incorporate disease burden, research literature coverage, and clinical trial
coverage to uncover underfunded areas of research) [18].

One can easily imagine using techniques from machine learning and natural
language processing to identify linchpin results, perhaps by examining
citations networks, or using entity extraction to model the emergence of new
terminology and concepts. The process would not need to be fully automated. We
could employ a hybrid approach whereby data analytic techniques allow us to
narrow down a corpus of tens of thousands of research papers to a few dozen or
a hundred. Subsequently, with the guidance of experts and manual curation, we
would arrive at a list of candidate results or studies to be the focus of
targeted validation efforts.

## III. Conclusion

Modern science is witnessing many growing pains, one of which is an increasing
concern about the quality of research, as measured by reproducibility, in a
number of distinct areas of inquiry. Although this concern has quickly grown
to the point of being labeled a “crisis,” the reality is that we simply do not
know what the distribution of reproducibility rates looks like for the
entirety of science.

Nonetheless, this very uncertainty should be sufficient motivation to put into
place procedures and incentives to increase the reliability of published
results. There are many approaches to take in addressing this problem, a few
of which have been outlined above.

Most of these ideas have been discussed or attempted in some form or another
in recent years. The proposal that has received the least attention in the
context of the reproducibility crisis is scientific data science. One of the
primary roadblocks to open-ended, exploratory data analysis with a large
corpus of scientific papers is restrictions on their availability—in other
words, closed access publishing models. Therefore, researchers should consider
the applications of scientific data science to uncovering linchpin results to
be a key motivating factor in encouraging a complete transition to an open
access publishing model.

### Acknowledgments

I would like to thank Adam Safron, P. Ravi Sarma, and Daniel Weissman for
insightful discussions and feedback on the manuscript. A special thanks to
A.S. for suggesting the phrase “keystone species.”

## References

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