# Non-stationary $\phi$-contractions and associated fractals

Amit<EMAIL_ADDRESS>Vineeta Basotia<EMAIL_ADDRESS>Ajay Prajapati
<EMAIL_ADDRESS>Department of Mathematics, JJT University, Jhunjhunu,
Rajasthan, India, 333001  Department of Computer Science, Banaras Hindu
University, Varansi, India, 221005

###### Abstract

In this study we provide several significant generalisations of Banach
contraction principle where the Lipschitz constant is substituted by real-
valued control function that is a comparison function. We study non-stationary
variants of fixed-point. In particular, this article looks into “trajectories
of maps defined by function systems” which are regarded as generalizations of
traditional iterated function system. The importance of forward and backward
trajectories of general sequences of mappings is analyzed. The convergence
characteristics of these trajectories determined a non-stationary variant of
the traditional fixed-point theory. Unlike the normal fractals which have
self-similarity at various scales, the attractors of these trajectories of
maps which defined by function systems that may have various structures at
various scales. In this literature we also study the sequence of countable IFS
having some generalized contractions on a complete metric space

###### keywords:

Forward Trajectories , Backward Trajectories , Comparison Function , Iterated
Function Systems , Fractal interpolation

##  1 Introduction

The most renowned contribution in the study of fixed points theory is the
Banach contraction principle. In [21] we see some valuable generalizations of
Banach contraction principle where the Lipschitz’s constant is shifted by some
control function with real-values. In complete metric space, Boyd and Wong
[20] revealed the existence of fixed points of mappings and also noted the
Banach contraction principle that have some of generalized contractions
condition. Dyn et al. [9, 11] studied the non-stationary fractals. It should
also be noted that Graf [13] had already studied this problem under the title
of statistically self-similar sets.

The purpose of this article is to getting the theorems of fixed point by
generalizing the contractions or also introducing the trajectories which
induced with sequence of many countable transformations or also establish its
elementary properties (see [11]). The concept of fractal interpolation
function(FIF) was introduced by Barnsley[4] through iterated function system
(IFS). An IFS have contractions mapping that belongs to finite family over
some complete metric space $X\to X$. An IFS induces an operator [24] on
$K(X)$(collection of all nonempty compact subsets of $X$) into itself which is
a contraction on this complete metric space hence, according to Banach’s
Contraction Principle, has a unique set “fixed-point” ( also called attractor
of the respective IFS) which is, generally, a fractal [25]. Generalized
countable IFS (GCIFS) are expansion of countable IFS rather to define
contraction on complete metric space $X$, contraction define on $X\times X\to
X$ (see [26]).[27] examined that there exist a subset of plane which is an
attractor of generalized IFS, however its not an attractor of IFS also find
out an examples of Cantors set that is not an attractor of generalized IFS.

We first explain some fundamental results before establishing the fixed point
theorems of any generalised contraction. Fractal dimension is an integral part
of fractal geometry. Many recent works on fractal dimensions can be seen in
the literature. Liang [17] confirmed that any function which is continuous and
of bounded variation on $[0,1]$ having one box dimension. He also confirmed
that the Riemann Liouville fractional integral of any continuous function
which is defined on bounded variation on $[0,1]$ have one box dimension. The
effect of the Riemann-Liouville fractional integral operator on unbounded
variation points of a continuous function can be seen [29]. When we move from
unit interval to rectangular domain, we have different ideas of bounded
variations such as Arzelá, Toneli, Hahn and Peirpont, for instance, see [10].
By using the concept of bounded variation with respect to Arzelá, Verma and
Viswanathan gave results on the fractal dimension of the graph of the mixed
Riemann-Liouville fractional integrals over rectangular domain in [31]. In
this order, Chandra and Abbas [7] estimated the fractal dimension of the graph
of the mixed Riemann-Liouville fractional integrals of different choice of
continuous functions over rectangular domain. They also estimated the box
dimension of the graph of the mixed Katugampola fractional integrals of a two
dimensional continuous functions in [8]. The fractal dimensional results of
the mixed Weyl-Marchaud fractional derivative can be seen in [6]. Chandra and
Abbas[5] examined the partial integrals as well as partial derivatives about
the bivariate fractal interpolation functions (FIFs) and also study the
integral transforms and fractional order of integral transforms of bivariate
FIFs. Gowrisankar and Uthayakumar [12] studied about the presence of fractal
interpolation function (FIF) for the data sequence given as
$\\{(a_{n},b_{n}):n\geq 2\\}$ having countable iterated function system where
the sequence $a_{n}$ is bounded as well as monotone and sequence $b_{n}$ is
bounded, also the order integral of FIF for sequence data is evaluate if the
values at initial endpoint and final endpoint of the integral is known.
Jha and Verma[16] provided a hard study of fractal dimension of
$\alpha-$fractal functions on various function spaces. Jha [15] shown the
method proposed generalize the existing stationary interpolant in terms of IFS
and also worked on the proposed interpolant’s basic properties and figured out
its box dimension as well as Hausdorff dimension. Sahu and Priyadrshi
estimated the box dimension of Graph of harmonic function on Sierpiński gasket
in [23]. Verma and Sahu [30] introduced the notion of bounded variation on
Sierpiński gasket. Also, they have estimated the fractal dimension of a
continuous function which is of bounded vartion on Sierpiński gasket. In this
order, Agrawal and Som [3] have presented the concept of dimension preserving
approximation for the bivariate continuous function on a rectangular domain.
Agrawal and Som [1] have estimated fractal dimensions of $\alpha$-fractal
functions on the Sierpiński Gasket. Further, the same authors [2] have
investigated the approximation of functions by fractal functions corresponding
to the $L^{p}$-norm on the Sierpiński gasket. New idea for construction of FIF
by using Rakotch and Matkowski fixed point theorems is given in [22]. Verma et
al. [33] initiated theory on fractal dimension of vector valued function.

Mihail [18] proved the existence of an analogue of Hutchinson measure
associated with a GIFS with probabilities and present some of its properties.
Pandey et al. [19] developed some theory about the dimension of vector valued
function graph also figured out the fractal dimension of the graphs of this
Katugampola fractional integral about vector-valued continuous functions
having bounded variation which defined on closed and bounded interval of
$\mathbb{R}$. Verma [28] investigated approximation issues in relation to the
fractal dimensions of function as well as their derivative.

The organization of this paper is as follows. In Section 2, we discuss about
basics definition and examples relates to our result that we generalize in
this paper. In Section 3, we discuss about some contractive condition and its
example and properties related to contraction i.e convergence of forward as
well as backward trajectories. Further, Section 4 is about the trajectories of
sequence of function systems (SFSs) asymptotic similarity of SFS trajectories
and convergence of forward and backward SFS trajectories and some result based
on $\phi-$ contraction. In Section 5, we discuss about the countable iterated
function system consisting of $\phi-$ contractions.

##  2 Preliminary facts

We will revisit several well-known principles of fixed point theory in this
part, that will be included in the follow.

######  Definition 2.1.

(Contraction mapping) Assume $X$ is a metric space with metric $d$. A function
$f:X\to X$ is Known as Lipschitz continuous if $\exists~{}\lambda\geq 0$ in
the sense that

$d(f(a),f(b))\leq\lambda d(a,b),~{}\forall~{}a,b\in X.$

The Lipschitz constant of $f$ is the lowest $\lambda$ for which the preceding
inequality holds. If $\lambda<1$ is true, then $f$ is a contraction mapping.

######  Definition 2.2.

(Banach contraction principle) Consider the contraction mapping $f$ on
complete metric space $X$ then there exist unique fixed point $\bar{x}\in X$
of $f$.

######  Example 2.3.

Let $(\mathbb{R},d)$ be a metric space, where $d$ be a Euclidean distance
metric that is, $d(a,b)=|a-b|$. The mapping $f:\mathbb{R}\to\mathbb{R}$
defined as $f(x)=\frac{x}{2}+k$, clearly it is a contraction mapping. Hence,
definition 2.2 yields $f$ has a unique fixed point and one can easily see that
$2k$ is the unique fixed point of the defined function $f$.

######  Definition 2.4.

(Iterated function system IFS) Let $f=\\{f_{1},f_{2},\dots,f_{n}\\}$ is a
collection of $n$ contraction mappings defined on a closed and bounded subset
$D\subset\mathbb{R}^{n}$ that is, for each $k\in\\{1,2,\dots
n\\},~{}f_{k}:D\to D$. Thus, using the definition (2.1) there exists a
constant $0\leq\lambda_{k}<1$ such that
$d(f_{k}(a),f_{k}(b))\leq\lambda_{k}d(a,b)~{}~{}\forall~{}~{}a,b\in D$. Hence,
we get an IFS $\\{D;f_{i},i=1,\ldots,k\\}$. Let
$F:\mathcal{C}(D)\to\mathcal{C}(D)$ be a map on the collection of all the
compact subsets of $D$, which is denoted by $\mathcal{C}(D)$. For any subset
$A\subset D$, $F$ is defined by

$F(A)={\bigcup_{i=1}^{n}f_{i}(A)}.$

The map $F$ follows the definition 2.2 if it is endowed with Hausdorff metric,
therefore $F$ has a unique set, say $A^{*}\subset D$ which is the “fixed
point” of $F$ i.e

$F(A^{*})=A^{*}.$

Thus, this $A^{*}$ is an attractor of IFS.
For example consider following two maps on $[0,1]$: $f_{1}(x)=\frac{x}{3}$ ,
$f_{2}(x)=\frac{x}{3}+\frac{2}{3}$ clearly we can see Cantor set
$C\subset[0,1]$ is the attractor/fixed point i.e $C=f_{1}(C)\cup f_{2}(C)$

######  Definition 2.5.

Consider $(X,d)$ be a metric space and $P(X)$ denotes the class of all
nonempty subsets of $X$ , $K(X)$ denotes the collection of all nonempty
compact subsets of $X$. The function $h:P(X)\times
P(X)\to\mathbb{R_{+}}\cup\\{\infty\\}$ ,

$h(A,B)=\max\\{d(A,B),d(B,A)\\}=\max\left\\{\max_{x\in A}\min_{y\in
B}d(x,y),\max_{y\in B}\min_{x\in A}d(x,y)\right\\},$

$\forall A,B\in P(X)$ is called Hausdorff semi–metric. When we use $K(X)$
instead of $P(X)$, the previously described mapping $h$ is called Hausdorff
metric.

######  Lemma 2.6.

If $(X,d)$ is complete and compact, then the metric space $(K(X),h)$ is
complete and compact respectively.

######  Example 2.7.

Let the metric space $\mathbb{R}$ having usual metric $d$ which induced the
absolute value, that is $d(a,b)=|a-b|,~{}a,b\in\mathbb{R}$ . Let $X=[0,20]$
and $Y=[22,31]$. It’s worth noting that for each $x\in X$, the closest point
in $Y$ that gives the smallest distance will always be $y=22$. Therefore, we
find that $d(X,Y)=\sup\\{d(x,22):x\in X\\}$. The point $x=0$ in $X$ maximizes
this distance. Therefore

$d(X,y)=d(0,22)=|22-0|=22.$

Similarly, we find that $d(Y,X)=\sup\\{d(y,20):y\in Y\\}$, since the point
$x=20$ will give the smallest distance to any point in $Y$. The point $y=31$
in $Y$ maximizes this distance, so we have

$d(Y,X)=d(20,31)=|20-31|=11.$

It follows that

$h(X,Y)=\max\\{d_{2}(X,Y),d_{2}(Y,X)\\}=22.$

Note that $d(Y,X)$ and $d(X,Y)$ are not always equal.

######  Example 2.8.

Let the metric space $\mathbb{R}$ having usual metric $d$ which induced the
absolute value, that is $d(a,b)=|a-b|,~{}a,b\in\mathbb{R}$ . Consider
$A=(0,1]~{}~{}{\mbox{and}}~{}~{}B=[-1,0)$. Similarly as in previous example we
get

$\displaystyle h(A,B)=1.$

##  3 Sequences of transformations and trajectories

For basic terminologies and definitions used in this section, we refer the
reader to [9, 14, 15]. Consider a complete metric space $(X,d)$. Let a
sequence of continuous transformations $\\{T_{i}:X\to X\mid
i\in\mathbb{N}\\}$.

######  Definition 3.1.

(Forward and backward procedures) Consider $\\{T_{i}:X\to X\mid
i\in\mathbb{N}\\}$ be the sequence of maps, forward and backward procedures
are defined as

  * 1.

$\Phi=T_{k}\circ T_{k-1}\circ\dots\circ T_{1},$

  * 2.

$\Psi=T_{1}\circ T_{2}\circ\dots\circ T_{k}.$

######  Definition 3.2.

(Forward and backward trajectories) We define forward and backward
trajectories in $X$ as a result of the forward and backward processes,
beginning with $x\in X,\\{\Phi_{k}(x)\\}$ and $\\{\Psi_{k}(x)\\}$ are given as

$\Phi_{k}(x)=T_{k}\circ T_{k-1}\circ\dots\circ
T_{1}(x)=T_{k}(x)\circ\Phi_{k-1}(x),k\in\mathbb{N},$ $\Psi_{k}(x)=T_{1}\circ
T_{2}\circ\dots\circ T_{k}(x)=T_{1}(x)\circ\Psi_{k-1}(x),k\in\mathbb{N}.$

We will look at the convergence of both forward and backward trajectories in
this section. Let us begin by defining the following term before stating our
next argument.

######  Definition 3.3.

(Asymptotic similarity) Consider $(X,d)$ be a metric space then two sequences
$\\{x_{i}\\}_{i\in N}$, $\\{y_{i}\\}_{i\in N}$ are claimed to be
asymptotically similar if

$d(x_{i},y_{i})\to 0~{}~{}as~{}~{}i\to\infty.$

This relation is denoted by

$\\{x_{i}\\}\sim\\{y_{i}\\}.$

###  3.1 Some generalized contractions

Let $(X,d)$ be a complete metric space, now consider $f:X\to X$. We review
some contractive-type conditions here. We say that
$\phi:[0,\infty)\to[0,\infty)$ is a comparison function if it is increasing
and $\phi^{p}(t)\to 0$ as $p\to\infty$, for every $t\geq 0$, where
$\phi^{p}=\phi^{p}\circ\phi^{p-1}$, which denotes p-times compositions of
function $\phi$ . One can deduce that, if $\phi$ is a comparison function,
then $\phi(t)<t$ for all t $>$ 0, $\phi$ is right continuous at 0 and
$\phi(0)=0.$

$d(f(x_{1}),f(x_{2}))\leq\phi(d(x_{1},x_{2})),~{}~{}\forall x_{1},x_{2}\in X.$

This mapping $f$ is called $\phi$ contraction mapping. Then mapping $f$ has
unique fixed point in $X$ i.e $\bar{x}\in X$. Furthermore, for any $x_{0}\in
X$ then the sequence $f_{n}(x_{0})$ always converges to $\bar{x}$.

Obviously, it is a specific case of this outcome, for $\phi(t)=rt,$ where
$r\in(0,1)$ of Banach contraction principle, $\phi(t)=\frac{t}{t+3}$ and
$\phi(t)=\ln(t+2)$ are examples of comparison functions. Throughout this paper
the notation $\phi$ is reserved to denote a comparison function.

######  Definition 3.4.

If we have a self-map $f$ over a metric space $(X,d)$ for some function
$\phi:[0,\infty)\to[0,\infty)$ and we have ,

$d(f(x_{1}),f(x_{2}))\leq\phi(d(x_{1},x_{2})),~{}~{}\forall x_{1},x_{2}\in X.$

then we say that $f$ is a $\phi$-contraction and if $\phi$ is non-decreasing
mapping such that $\phi^{p}(t)\to 0$ as $p\to\infty$, for all $t>0$, then $f$
is called a Matkowski contraction and if $\alpha(t)=\frac{\phi(t)}{t}<1$ for
any $t>0$, the function $\alpha$ is non-increasing then, at that point we
consider such a mapping a Rakotch contraction .

######  Example 3.5.

Consider $X=[0,\infty)$ and $d(x,y)=|x-y|$ . Let $f_{1}:X\to[0,1]$ and
$f_{2}:X\to[0,1]$, where $[0,1]\subset X$ and defined $f_{1}(x)=\frac{1}{1+x}$
and $f_{2}(x)=\frac{x}{1+x}$. Then

$d(f_{1}(x),f_{1}(y))=\frac{|x-y|}{(x+1)(y+1)}\leq\frac{|x-y|}{(|x-y|+1)}=\alpha(d(x,y))d(x,y).$
$d(f_{1}(x),f_{1}(y))\leq\alpha(d(x,y))d(x,y).$

Similarly

$d(f_{2}(x),f_{2}(y))\leq\alpha(d(x,y))d(x,y),$

where $\alpha(t)=\frac{1}{1+t}$ where $t=d(x,y)\geq 0$. Obviously $f_{1}$ and
$f_{2}$ are Rakotch contraction mappings on $X\subset\mathbb{R}$ over the
usual metric but these are not Banach contraction maps.

######  Proposition 3.6.

(Asymptotic similarity of trajectories) Consider sequence of transformations
$\\{T_{i}\\}_{i\in\mathbb{N}}$ on metric space $X$, where every $\\{T_{i}\\}$
is a $\phi$ contraction mapping along with each comparison mapping $\phi_{i}$.
If

$\lim_{k\to\infty}\phi_{1}\circ\phi_{2}\circ\phi_{3}\dots\circ\phi_{k}=0.$

then for any $x,y\in X,$

$\\{\Phi_{k}(x)\\}\sim\\{\Phi_{k}(y)\\},$

$\\{\Psi_{k}(x)\\}\sim\\{\Psi_{k}(y)\\}.$

###### Proof.

Consider $x,y\in X$ and let the trajectories $\\{\Psi_{k}(x)\\}$ and
$\\{\Psi_{k}(y)\\}$. Utilizing the way that $\\{T_{i}\\}$ is a $\phi$
contraction mapping with each has comparison mapping $\phi_{i}$.

$\displaystyle d(\Psi_{k}(x),\Psi_{k}(y))$
$\displaystyle\leq\phi_{1}(d({T_{2}\circ T_{3}\circ\dots\circ
T_{k}(x)},{T_{2}\circ T_{3}\circ\dots\circ T_{k}(y)})$
$\displaystyle\leq\phi_{1}\circ\phi_{2}(d({T_{3}\circ T_{4}\circ\dots\circ
T_{k}(x)},{T_{3}\circ T_{4}\circ\dots\circ T_{k}(y)})$
$\displaystyle\leq\phi_{1}\circ\phi_{2}\circ\phi_{3}(d({T_{4}\circ\dots\circ
T_{k}(x)},{T_{4}\circ\dots\circ T_{k}(y)})$
$\displaystyle\leq\phi_{1}\circ\phi_{2}\circ\phi_{3}\dots\circ\phi_{k}(d((x,y))\to
0~{}~{}~{}~{}\text{as}~{}k\to\infty.$

By using property of $\phi$ mapping that is,
$\lim_{k\to\infty}\phi_{1}\circ\phi_{2}\circ\phi_{3}\dots\circ\phi_{k}=0.$ ∎

######  Remark 3.7.

The proof is alike for the backward and the forward trajectories.

######  Remark 3.8.

The condition
$\lim_{k\to\infty}\phi_{1}\circ\phi_{2}\circ\phi_{3}\dots\circ\phi_{k}=0$
condition expressed in proposition 3.6 doesn’t guarantee convergence of the
trajectories $\\{\Phi_{k}(x)\\}$.

If $T_{i}=T~{}\forall~{}i\in\mathbb{N}$, and $T$ is a $\phi$ contraction
mapping with comparison mapping $\phi$ such that $\phi(t)<$ t for all t $>$ 0
then, at that point both kind of trajectories are only the fixed-point
iteration trajectories ${T^{k}(x)}$, where $T^{k}$ is the k-times composition
of $T$ itself and $T^{k}$ converge to unique limit for taking any starting
point $x\in X$. Which is also known from the principle of Banach contraction
(see [20]) that ${T^{k}(x)}$ converges to unique limit for any $x\in X$. The
query now emerges with regard to the convergence of general trajectories that
is, which conditions ensure the convergence of the backward and the forward
trajectories.

######  Definition 3.9.

(Invariant domain) Any subset of $X$ i.e consider $C\subseteq X$ is called an
invariant domain of sequence of transformations $\\{T_{i}\\}_{i\in N}$ if
$\text{take any}~{}~{}x\in C,\text{then}~{}T_{i}(x)\in C$,
$\forall~{}i\in\mathbb{N}$.

######  Proposition 3.10.

(Convergence of forward trajectories) Consider $\\{T_{i}\\}_{i\in\mathbb{N}}$
be a sequence of transformations defined on $X$, also having compact invariant
domain $C$, and consider $\\{T_{i}\\}_{i\in\mathbb{N}}$ converges uniformly to
a map $T$ on $C$, where $T$ is a $\phi$ contraction mapping with comparison
mapping $\phi$. Then, for taking any $x\in C$, the trajectory
$\\{\Phi_{i}(x)\\}_{i\in\mathbb{N}}$ will converges to fixed-point $p$ of $T$
i.e,

$\lim_{k\to\infty}d(\Phi_{k}(x),p)=0.$

###### Proof.

Denoting $~{}\epsilon_{i}=\sup_{x\in C}d(T_{i}(x),T(x)),i\in\mathbb{N}$, it
follows that

$\lim_{k\to\infty}\epsilon_{i}=0.$

As $T$ is $\phi$ contraction mapping with comparison mapping $\phi$ such that
$\phi(t)<t$ for all $t>0$, the fixed-point iterations $T^{k}(x)$ converge to
$p\in X$ which is unique fixed-point for taking any initial element $x$. It
additionally obey that $C$ is an invariant domain of $T$. Beginning with
taking any point $x\in C$, which gives that $\Phi_{k}(x)\subseteq C$. Utilize
the triangle inequality of metric space $(X,d)$ and $\phi$ contraction
property of $T$, we get

$\displaystyle d(\Phi_{k+m}(x),T^{m}\Phi_{k}(x))\leq(d({T_{k+m}\circ
T_{k+m-1}\circ\dots\circ T_{k+1}\circ\Phi_{k}(x)},T^{m}\Phi_{k}(x))$
$\displaystyle\leq(d({T_{k+m}\circ T_{k+m-1}\circ\dots\circ
T_{k+1}\circ\Phi_{k}(x)},T\circ T_{k+m-1}\circ\dots\circ
T_{k+1}\circ\Phi_{k}(x))$ $\displaystyle+(d(T\circ T_{k+m-1}\circ\dots\circ
T_{k+1}\circ\Phi_{k}(x),T^{2}\circ T_{k+m-2}\circ\dots\circ
T_{k+1}\circ\Phi_{k}(x))$ $\displaystyle+(d(T^{2}\circ
T_{k+m-2}\circ\dots\circ T_{k+1}\circ\Phi_{k}(x),T^{3}\circ
T_{k+m-3}\circ\dots\circ T_{k+1}\circ\Phi_{k}(x))$
$\displaystyle+\dots+(d(T^{m-1}\circ
T_{k+1}\circ\Phi_{k}(x),T^{m}\Phi_{k}(x))$
$\displaystyle\leq\epsilon_{k+m}+\phi\epsilon_{k+m-1}+\phi^{2}\epsilon_{k+m-2}+\dots+\phi^{m-1}\epsilon_{k+1}$
$\displaystyle\leq\max_{1\leq i\leq
m}\\{\epsilon_{k+i}\\}\times\frac{1}{1-\phi}.$

Now utilize the relation

$d(\Phi_{k+m}(x),p)\leq
d(\Phi_{k+m}(x),T^{m}\Phi_{k}(x))+d(T^{m}\Phi_{k}(x),p),$

the outcome follows by seeing that for $k$ large enough $\max_{1\leq i\leq
m}\\{\epsilon_{k+i}\\}$ can constructed as small as we need and for such $k$,
for its sufficient large value , $d(T^{m}\Phi_{k}(x),p)$ is as small as
required. ∎

######  Proposition 3.11.

(Convergence of backward trajectories) Consider $\\{T_{i}\\}_{i\in N}$ be a
sequence of transformations defined on $X$, also $d(T(x_{0}),x_{0})$ is
bounded for some $x_{0}\in X$, and assume $T_{i}$ is a $\phi$\- contraction
mapping with comparison mapping $\phi$ such that $\phi(t)<t$ for all $t>0$,

$\sum_{k=1}^{\infty}\phi_{1}\circ\phi_{2}\circ\phi_{3}\dots\circ\phi_{k}(x)<\infty,$

then the backward trajectories $\Psi_{k}(x)=T_{1}\circ T_{2}\circ\dots\circ
T_{k}(x),k\in N$, converge for taking any initial element $x\in X$ to a unique
limit point in $X$.

###### Proof.

We have $d(T(x_{0}),x_{0})$ is bounded for some $x_{0}\in X$, that is

$\sup_{1\leq i<\infty}d(T_{i}(x_{0}),x_{0}))\leq M<\infty,$

$d(T_{i}(x_{0}),x_{0}))\leq M$ then
$\phi_{i}(d(T_{i}(x_{0}),x_{0}))\leq\phi_{i}(M)$ because $\phi_{i}$ is
increasing function. We know $\\{\Psi_{k}(x)\\}\sim\\{\Psi_{k}(y)\\}$ for any
$x,y\in X$ that is we need to show $\Psi_{k}(x_{0})$ is convergent then
$\Psi_{k}(x)$ it is convergent for any $x\in X$ by asymptotic similarity.

$\displaystyle d(\Psi_{k+1}(x_{0}),\Psi_{k}(x_{0}))$ $\displaystyle\leq
d(\Psi_{k}(T_{k+1}(x_{0}),\Psi_{k}(x_{0}))$
$\displaystyle\leq\phi_{1}\circ\phi_{2}\circ\phi_{3}\dots\circ\phi_{k}(d(T_{k+1}(x_{0}),x_{0}))\to
0~{}as~{}k\to\infty.$

For taking $m,k\in\mathbb{N},m>k$, we get

$\displaystyle d(\Psi_{m}(x_{0}),\Psi_{k}(x_{0}))$ $\displaystyle\leq
d(\Psi_{m}(x_{0}),\Psi_{m-1}(x_{0}))+\dots+d(\Psi_{k+2}(x_{0}),\Psi_{k+1}(x_{0}))+d(\Psi_{k+1}(x_{0}),\Psi_{k}(x_{0}))$
$\displaystyle\leq\phi_{1}\circ\dots\circ\phi_{m-1}d(T_{m}(x_{0}),x_{0})+\dots+\phi_{1}\circ\dots\circ\phi_{k+1}(d(T_{k+2}(x_{0}),x_{0}))$
$\displaystyle+\phi_{1}\circ\dots\circ\phi_{k}(d(T_{k+1}(x_{0}),x_{0})$
$\displaystyle\leq\phi_{1}\circ\dots\circ\phi_{m-1}(M)+\phi_{1}\circ\dots\circ\phi_{k+1}(M)+\phi_{1}\circ\dots\circ\phi_{k}(M)$
$\displaystyle=S_{m-1}(M)-S_{k}(M),$

where
$S_{k}(t)=\sum_{k=1}^{\infty}\phi_{1}\circ\phi_{2}\circ\phi_{3}\dots\circ\phi_{k}(t)$
for any $t>0$. Since
$\sum_{k=1}^{\infty}\phi_{1}\circ\phi_{2}\circ\phi_{3}\dots\circ\phi_{k}(t)<\infty$
for any $t>0$, it follows that $S_{m-1}(M)-S_{k}(M)\to 0$. From this, we get

$d(\Psi_{m}(x_{0}),\Psi_{k}(x_{0}))\to 0~{}\text{as}~{}k\to\infty.$

That is, $\\{\Psi_{k}(x_{0})\\}_{k\in\mathbb{N}}$ becomes a Cauchy sequence
and because of completeness property of $(X,d)$, this is convergent for some
$x_{0}\in X$. The uniqueness of the limit point is attain by the equivalence
property of all trajectories. ∎

######  Remark 3.12.

(Differences between forward and backward trajectories) 1) We observe that if
$T_{i}\to T$ and $T$ is a $\phi$ contraction mapping with comparison mapping
$\phi$ such that $\phi(t)<t$ for all $t>0$,

$\sum_{k=1}^{\infty}\phi_{1}\circ\phi_{2}\circ\phi_{3}\dots\circ\phi_{k}<\infty$

and both trajectories, the backward and the forward trajectories converge.

2) The sufficient condition for the asymptotic similarity result for the both
backward and forward trajectories is

$\lim_{k\to\infty}\phi_{1}\circ\phi_{2}\circ\phi_{3}\dots\circ\phi_{k}=0.$

Under the more severe conditions

$\sum_{k=1}^{\infty}\phi_{1}\circ\phi_{2}\circ\phi_{3}\dots\circ\phi_{k}<\infty$

and the existence of some $x_{0}\in X$ such that $d(T(x_{0}),x_{0})$ is
bounded then for the backward trajectories, we achieve convergence.

3) In numerous cases the forward trajectories don’t converge while the
backward trajectories converge. In order to explain this, consider the metric
space $R$ with distance function define as $d(x,y)=|x-y|$ and let us consider
the simple sequence of contractive transformations $\phi_{2i-1}(x)$ =$x/2$,
$\phi_{2i}=x/2+c$, $i\geq 1$ . The backward trajectories converge to the fixed
point of $\Phi_{1}=\phi_{1}\circ\phi_{2}$, which is $2c/3$. The forward
trajectories have two accumulation points, which are the fixed point of
$\Phi_{1}$ i.e., $2c/3$, and the fixed point of
$\Psi_{2}=\phi_{2}\circ\phi_{1}$ which is $4c/3$.

##  4 Trajectories of sequences of function systems

Classical generalisation of IFS, let us consider a sequence of function
systems (SFS), as well as its trajectories. Assume that $(X,d)$ be a complete
metric space. Let an SFS $\\{F_{i}\\}_{i\in N}$ characterised by
$F_{i}=\\{X;f_{1,i},f_{2,i},\dots,f_{ni,i}\\},$

where $f_{r,i}:X\to X$ all are continuous maps. The set-valued maps that go
with it are given as $F_{i}:H(X)\to H(X)$ and

$F_{i}(A)={\bigcup_{r=1}^{n_{i}}f_{r,i}(A)},$

where denoting $f_{r,i}~{}\phi$-contraction mapping with comparison mapping
$\phi_{r,i}$ such that $\phi_{r,i}(t)<t$ for all $t>0$ and for
$r=1,2,\dots,n_{i}$. We recall that , the comparison mapping of $F_{i}$ in
$(H(X),h)$ is $max_{r=1,2,\dots,n_{i}}\phi_{r,i}=\phi_{i}$. The attractor is
addressed in classical IFS theory apparently, the set that is a map $F_{i}$’s
fixed-point. In this section, we assume the trajectories of sequence of
function system (SFS) maps $\\{F_{i}\\}_{i\in N}$, we call it forward and
backward SFS trajectories

$\Phi_{k}(A)={F_{k}\circ F_{k-1}\circ\dots\circ F_{1}}(A),k\in N,$
$\Psi_{k}(A)={F_{1}\circ F_{2}\circ\dots\circ F_{k}}(A),k\in N$

respectively. $H(X)$, possessing the Hausdorff metric $h$ and will be complete
metric space if given that $(X,d)$ is complete metric space.

######  Corollary 4.1.

(Asymptotic similarity of SFS trajectories) Let an SFS characterised by
$F_{i}=\\{X;f_{1,i},f_{2,i},\dots,f_{ni,i}\\},i\in N$, here $f_{r,i}:X\to X$
are ${\phi}$-contraction mapping having comparison mapping $\phi_{r,i}$.
Further, assume the set valued maps $F_{i}$ on $(H(X),h)$ with the
corresponding comparison mapping $\phi_{i}$ satisfy for any $t>0,$

$\lim_{k\to\infty}\phi_{1}\circ\phi_{2}\circ\phi_{3}\dots\circ\phi_{k}(t)=0.$

As a result, all of $F_{i}$’s forward trajectories are asymptotically similar
and all of $F_{i}$’s backward trajectories are asymptotically similar that is
$\forall A,B\in H(X)$

$\\{\Phi_{k}(A)\\}\sim\\{\Phi_{k}(B)\\},$
$\\{\Psi_{k}(A)\\}\sim\\{\Psi_{k}(B)\\}.$

######  Remark 4.2.

For both backward and forward trajectories, the proof is the same.

######  Corollary 4.3.

(Convergence of forward SFS trajectories)
Consider $\\{F_{i}\\}_{i\in N}$ as in the corollary 4.1, there are an equal
number of mappings, $n_{i}=n$ and consider
$F_{i}=\\{X;f_{1},f_{2},\dots,f_{n}\\},~{}i\in N$. Suppose $\exists$
$C\subseteq X$, which is compact invariant domain for ${f_{r,i}}$ as well as
for each $r=1,2,\dots,n$ and the $\\{f_{r,i}\\}_{i\in N}$ sequence converges
to $f_{r}$ uniformly over $C$ as i $\to$ $\infty$. Also suppose that $F$ is
${\phi}$-contraction mapping then the forward trajectories ${\Phi_{k}(A)}$
converge for taking any initial subset of $c$ i.e $P\subseteq C$ to a unique
attractor of $F$.

######  Corollary 4.4.

(Convergence of backward SFS trajectories)
Let an SFS $F_{i}=\\{X;f_{1},f_{2},\dots,f_{n}\\},$ $i\in N$ defined as in
corollary 4.1 and $d(F_{i}(A_{1}),A_{1})$ is bounded for some $A_{1}\subseteq
H(X)$ assume that for any $t>0$ and

$\sum_{k=1}^{\infty}\phi_{1}\circ\phi_{2}\circ\phi_{3}\dots\circ\phi_{k}(t)<\infty,$

then the backward trajectories $\Psi_{k}(A)=F_{1}\circ F_{2}\circ\dots\circ
F_{k}(A)~{},k\in N$ converge for any $P\subseteq C$ is unique set (attractor)
in $C$.

######  Lemma 4.5.

Consider $(X,d)$ be a metric space, $\phi:[0,+\infty)\to[0,+\infty)$ be non-
decreasing function and $f$ be a $\phi$-contractive. Then
$h_{d}(F(A),F(B))\leq\phi(h_{d}(A,B))$ for all $A,B\in H(X)$. That is,
$F:H(X)\to H(X)$ is also a $\phi$-contractive (where $\phi$ is comparison
function), where $\forall D~{}\in H(X),~{}~{}~{}F(D)=f(D)$.

###### Proof.

The distance function $d(x,y)$ for fixed $x\in X$ is continuous in $y\in X$.
Let $A,B\in H(X)$ and set any $x_{0}\in A$. Using the compactness of $A$,
there is $y_{x_{0}}\in B$ such that

$\min_{y\in B}d(x_{0},y)=d(x_{0},y_{x_{0}}),$

then we obtain

$\inf_{y\in B}\phi(d(x_{0},y))\leq\phi(d(x_{0},y_{x_{0}}))=\phi(\min_{y\in
B}d(x_{0},y)),$

because $\phi:[0,+\infty)\to[0,+\infty)$ is non-decreasing, it follows that

$\phi(\min_{y\in B}d(x_{0},y))\leq\phi(\max_{x\in A}\min_{y\in
B}d(x,y))\leq\phi(h_{d}(A,B)).$

Since $x_{0}$ was arbitrary,

$\sup_{x\in A}\phi(\min_{y\in B}d(x,y))\leq\phi(h_{d}(A,B)).$

Hence,

$\sup_{x\in A}\inf_{y\in B}\phi(d(x,y))\leq\sup_{x\in A}\phi(\min_{y\in
B}d(x,y))\leq\phi(h_{d}(A,B)).$

Similarly,

$\sup_{y\in B}\inf_{x\in A}\phi(d(x,y))\leq\phi(h_{d}(A,B)).$

So

$\displaystyle d(F(A),F(B))$ $\displaystyle=\max_{f(x)\in F(A)}\min_{f(y)\in
F(B)}d(f(x),f(y))$ $\displaystyle=\max_{x\in A}\min_{y\in B}d(f(x),f(y))$
$\displaystyle\leq\sup_{x\in A}\inf_{y\in B}\phi(d(x,y))$
$\displaystyle\leq\phi(h_{d}(A,B)).$

Similarly,

$d(F(A),F(B))\leq\sup_{x\in A}\inf_{y\in B}\phi(d(x,y))\leq\phi(h_{d}(A,B)).$

Hence, we have

$h_{d}(F(A),F(B))=\max\\{d(F(A),F(B)),d(F(B),F(A))\\}\leq\phi(h_{d}(A,B)),$

this completing the proof. ∎

In the next theorem, we show an application of the non-stationary fixed point
results established previously to fractal interpolation theory.

######  Theorem 4.6.

Let $\\{(x_{i},y_{i}):i=0,1,\dots,N\\}$ be a data set such that
$x_{0}<x_{1}<\dots<x_{N}$ and $I:=[x_{0},x_{N}]$. We consider
$l_{i}:I\to[x_{i-1},x_{i}].$ Assume that maps $F_{i,k}$ are Matkowski
contractions (with the same function $\phi$) with respect to the second
variable, that is for some non-decreasing function
$\phi:\mathbb{R_{+}}\to\mathbb{R_{+}}$ with $\phi^{n}(t)\to 0$ for $t>0$ each
map $F_{i}$ satisfies

$|F_{i,k}(x,y)-F_{i,k}(x,y^{\prime})|\leq\phi(|y-y^{\prime}|)~{}~{}~{}\forall
x\in I,~{}~{}\forall y,y^{\prime}\in[a,b]$

Then the operator $T_{k}:C^{*}(I)\to C^{*}(I))$ defined by

$(T_{k}f)(x)=F_{i,k}(l_{i}^{-1}(x),f(l_{i}^{-1}(x)),$

is a Matkowski contraction. Then there is a unique continuous function
$f^{*}:I\to[a,b]$ which is the limit of the backword trajectories
$\Psi_{k}(g)=T_{1}\circ T_{2}\circ\dots\circ T_{k}(g)$ for any $g\in C^{*}(I)$
satisfying $f^{*}(x_{i})=y_{i}$ for $i=0,1,\dots,N$.

###### Proof.

For all $g,h\in C^{*}(I),$ we have

$\displaystyle d_{C(I)}(T_{k}g,T_{k}h)$
$\displaystyle=\max_{x\in[x_{0},x_{N}]}|T_{k}g(x)-T_{k}h(x)|$
$\displaystyle=\max_{i=1,2,...,N}\max_{x\in[x_{i-1},x_{i}]}|T_{k}g(x)-T_{k}h(x)|$
$\displaystyle=\max_{i=1,2,...,N}\max_{x\in[x_{i-1},x_{i}]}|F_{i,k}(l_{i}^{-1}(x),g(l_{i}^{-1})(x))-F_{i,k}(l_{i}^{-1}(x),h(l_{i}^{-1})(x))|$
$\displaystyle\leq\max_{i=1,2,...,N}\sup_{x\in[x_{i-1},x_{i}]}\phi(|g({l_{i}}^{-1}(x)-h({l_{i}}^{-1}(x))|).$

Since $\phi:(0,+\infty)\to(0,+\infty)$ is non-decreasing function and
${l_{i}}^{-1}:[x_{i-1},x_{i}]\to[x_{0},x_{N}]$ for all $i=1,2,...,N$, we
obtain that for $i_{0}\in{1,2,...,N}$ and $x_{*}\in[x_{i_{0}-1},x_{i_{0}}]$,

$\displaystyle\phi(|g({l_{i_{0}}}^{-1}(x_{*})-h({l_{i_{0}}}^{-1}(x_{*}))|)$
$\displaystyle\leq\phi(\max_{x\in[x_{i_{0}-1},x_{i_{0}}]}|g({l_{i_{0}}}^{-1}(x)-h({l_{i_{0}}}^{-1}(x))|)$
$\displaystyle\leq\phi(\max_{x\in[x_{0},x_{N}]}|g(x)-h(x)|$
$\displaystyle=\phi(d_{C(I)}(g,h)).$

Since $x_{*}$ was arbitrary, therefore

$\sup_{x\in[x_{i_{0}-1},x_{i_{0}}]}\phi(|g({l_{i_{0}}}^{-1}(x))-h({l_{i_{0}}}^{-1}(x))|)\leq\phi(d_{C(I)}(g,h)),$

and since $i_{0}$ was arbitrary

$\max_{i=1,2,...,N}\sup_{x\in[x_{i-1},x_{i}]}\phi(|g({l_{i}}^{-1}(x)-h({l_{i}}^{-1}(x))|)\leq\phi(d_{C(I)}(g,h)).$

Hence, we obtain

$\displaystyle d_{C(I)}(T_{k}g,T_{k}h)$
$\displaystyle\leq\max_{i=1,2,...,N}\sup_{x\in[x_{i-1},x_{i}]}\phi(|g({l_{i}}^{-1}(x)-h({l_{i}}^{-1}(x))|)$
$\displaystyle\leq\phi(d_{C(I)}(g,h)).$

So we conclude that $T_{k}$ is a Matkowski $\phi$-contractive map on the
complete metric space $(C^{*}(I),d_{C(I)})$. It is easy to check that all
hypotheses of Proposition 3.11 are satisfied. Now, by using Proposition 3.11,
we have a unique function $f^{*}:I\to[a,b]$ satisfying $f^{*}(x_{i})=y_{i}$
for $i=0,1,...,N$, which is the limit of the backward trajectories
$\Psi_{k}(g)=T_{1}\circ T_{2}\circ\dots\circ T_{k}(g)$ for any $g\in
C^{*}(I).$ This completes the proof. ∎

######  Remark 4.7.

The above result generalizes several results available in the fractal
interpolation theory, see, for instance, [4, 21, 22, 32].

##  5  Countable Iterated Function System

A sequence of continuous maps $(f_{n})_{n\geq 1}$ on a topological space $X$
into itself is called a Countable Iterated Function System (CIFS) on X. The
associated set function $F:P(X)\to P(X)$ is defined by

$F(A)=\overline{{\bigcup_{i=1}^{\infty}f_{i}(A)}},~{}~{}~{}\forall A\in H(X).$

A set $A\in P(X)$ for which $F(A)=A$, is named set fixed point (s.f.p.) of the
CIFS. The set $A$ is called attractor of the CIFS whenever $A\in K(X)$ ( where
$K(X)$ is the class of all compact sets of $P(X)$) and it is the unique s.f.p.
for the respective CIFS.

######  Theorem 5.1.

Let $(X,d)$ be a metric space, $\phi:[0,+\infty)\to[0,+\infty)$ be non-
decreasing function and $f$ be a $\phi$-contractive. Then
$h_{d}(F(A),F(B))\leq\phi(h_{d}(A,B))$ for all $A,B\in H(X)$. That is,
$F:H(X)\to H(X)$ is also a $\phi$-contractive (where $\phi$ is comparison
function), where

$F(A)=\overline{{\bigcup_{i=1}^{\infty}f_{i}(A)}},~{}~{}~{}~{}~{}\forall A\in
H(X).$

###### Proof.

The distance function $d(x,y)$ for fixed $x\in X$ is continuous in $y\in X$.
Let $A,B\in H(X)$ and set any $x_{0}\in A$. Using the compactness of $A$,
there is $y_{x_{0}}\in B$ such that

$\min_{y\in B}d(x_{0},y)=d(x_{0},y_{x_{0}}),$

then we obtain

$\inf_{y\in B}\phi(d(x_{0},y))\leq\phi(d(x_{0},y_{x_{0}}))=\phi(\min_{y\in
B}d(x_{0},y)),$

because $\phi:[0,+\infty)\to[0,+\infty)$ is non-decreasing, it follows that

$\phi(\min_{y\in B}d(x_{0},y))\leq\phi(\max_{x\in A}\min_{y\in
B}d(x,y))\leq\phi(h_{d}(A,B)).$

Since $x_{0}$ was arbitrary,

$\sup_{x\in A}\phi(\min_{y\in B}d(x,y))\leq\phi(h_{d}(A,B)).$

Hence,

$\sup_{x\in A}\inf_{y\in B}\phi(d(x,y))\leq\sup_{x\in A}\phi(\min_{y\in
B}d(x,y))\leq\phi(h_{d}(A,B)).$

Similarly,

$\sup_{y\in B}\inf_{x\in A}\phi(d(x,y))\leq\phi(h_{d}(A,B)).$

So

$\displaystyle d(F(A),F(B))$
$\displaystyle=d\Big{(}\overline{\bigcup_{i=1}^{\infty}f_{i}(A)},\overline{\bigcup_{i=1}^{\infty}f_{i}(B)}\Big{)}$
$\displaystyle\leq\sup_{i\in N}d(f_{i}(A),f_{i}(B))$ $\displaystyle=\sup_{i\in
N}\\{\max_{f_{i}(x)\in f_{i}(A)}\min_{f_{i}(y)\in
f_{i}(B)}d(f_{i}(x),f_{i}(y))\\}$ $\displaystyle=\sup_{i\in N}\\{\max_{x\in
A}\min_{y\in B}d(f_{i}(x),f_{i}(y))\\}$ $\displaystyle\leq{\sup_{x\in
A}\inf_{y\in B}\phi(d(x,y))}$ $\displaystyle\leq\phi(h_{d}(A,B)).$

Similarly,

$d(F(A),F(B))\leq\sup_{x\in A}\inf_{y\in B}\phi(d(x,y))\leq\phi(h_{d}(A,B)).$

Hence, we get

$h_{d}(F(A),F(B))=\max\\{d(F(A),F(B)),d(F(B),F(A))\\}\leq\phi(h_{d}(A,B)),$

establishing the claim. ∎

##  6 Acknowledgements

The work of first author is financially supported by the CSIR, India with
grant no: 09/1028(0019)/2020-EMR-I.

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