ABSTRACT

In the present paper, we define Dislocated Soft Metric Space and discuss about the existence and uniqueness of soft fixed point of a cyclic mapping in soft dislocated metric space. We also prove the unique soft fixed point theorems of a cyclic mapping in the context of dislocated soft metric space. Examples are given for support of the results.

Keywords:

Soft Set, Soft Fixed Point, Soft Point of Soft Mapping, Soft Metric Space, Dislocated Soft Metric Space

1. Introduction

The soft set theory is one of the branches of mathematics, which aims to describe phenomena and concepts of an ambiguous, undefined vague and imprecise meaning, which was initiated by Molodtsov [1] . This theory is applicable where there are no clearly defined mathematical models. Recently many papers concerning soft sets have been published (see [2] - [8] ). In many aspects of Mathematics, fixed point theory has wonderful applications. Shabir and Naz [9] presented soft topological spaces and they investigated some properties of soft topological spaces. Later many researchers were studied about soft topological spaces. In these studies, concepts of soft fixed point are expressed by different approaches. Das and Samanta [10] [11] introduced a different notion of soft metric space by using different concept of soft point and investigated some basic properties of these. In 2000, Hitzler and Seda [12] introduced the notion of dislocated metric space, in which self distance of a point need not be equal to zero. Aage and Salunke [13] established some important fixed point theorem in single and pair of mappings in dislocated metric space. Later Karapnar and Salimi [14] discussed the existence and uniqueness of fixed point of a cyclic mapping in the context of metric like space. The study of common fixed point of mapping in dislocated metric space satisfying certain contractive condition has been at the Centre of vigorous research activity (see [15] [16] [17] [18] ). Dislocated metric space plays very important role in topology, logical programming and in electronic engineering. Recently Wadkar et al. [19] , Mishra et al. [20] [21] [22] [23] , Deepmala and pathak [24] , Wadkar et al. [25] , [26] discussed and proved fixed point theorems by employing different concepts.

In the present paper, we discuss about the investigations concerning the existence and uniqueness of soft fixed point of a cyclic mapping in soft dislocated metric space. We also prove the unique soft fixed point theorems of a cyclic mapping in the context of dislocated metric space. To check the validity of the result we give the examples. Before starting to prove main result, some basic definitions are required.

Definition 1.1: Let X and E are respectively an initial inverse set and a parameter set. A soft set over X is pair denoted by (Y, E) if and only if Y is a mapping from E into the set of all subsets of the set X, i.e. Y: E → P(x), where P(x) is the power set of X.

Example of Soft Set: Let $\left(\stackrel{⌣}{C},\stackrel{⌣}{D}\right)$ be soft set, which describes the Nature of workers at Industry. Suppose that $U=\left\{u_1,u_2,u_3,u_4\right\}$ , i.e. universe of four workers. Let a set of decision parameters be $\stackrel{⌣}{D}=\left\{s_1,s_2,s_3,s_4,s_5\right\}$ . Now $s_i\left(i=1,2,3,4,5\right)$ stand for the parameters: high working speed, slow working speed, average working speed, work delay working speed and no working speed properly respectively.

Now consider $\tilde{C}\left(s_1\right)=\left\{u_1,u_3\right\}$ , $\tilde{C}\left(s_2\right)=\left\{u_1,u_3,u_4\right\}$ , $\tilde{C}\left(s_3\right)=U$ , $\tilde{C}\left(s_4\right)=\left\{u_2,u_4\right\}$ and $\tilde{C}\left(s_5\right)=\text{empty}$ . By consisting of the following collection of approximations, the soft set $\left(\stackrel{⌣}{C},\stackrel{⌣}{D}\right)$ can be viewed.

$\begin{array}{l}\left(\tilde{C},\tilde{D}\right)=\{\left(\text{highworkingspeed,}\left\{u_1,u_3\right\}\right)\text{,}\left(\text{slowworkingspeed,}\left\{u_1,u_3,u_4\right\}\right)\text{,}\\ \left(\text{averageworkingspeed,}U\right)\text{,}\left(\text{workdelayworkingspeed,}\left\{u_2,u_4\right\}\right)\text{,}\\ \left(\text{noworkingspeed,empty}\right)\}\text{.}\end{array}$

Definition 1.2: The intersection of two soft sets $\left(Y,A\right)$ and $\left(Z,B\right)$ over X is a soft set over X denoted by $\left(I,C\right)$ and is given by $\left(Y,A\right)\tilde{\cap }\left(Z,B\right)=\left(I,C\right)$ , where $C=A\cap B$ and $\forall \epsilon \in C$ , $I\left(\epsilon \right)=Y\left(\epsilon \right)\cap Z\left(\epsilon \right).$

Definition 1.3: The union of two soft sets $\left(Y,A\right)$ and $\left(Z,B\right)$ over X is the soft set $\left(I,C\right)$ , where $C=A\cup B$ and for all k in C,

$I\left(k\right)=\{\begin{array}{l}Y\left(k\right),\text{if}k\text{isanelementof}A-B,\hfill \\ Z\left(k\right),\text{if}k\text{isanelementof}B-A,\hfill \\ Y\left(k\right)\cup Z\left(k\right),\text{if}k\text{isanelementof}A\cap B.\hfill \end{array}$

This relationship is denoted by $\left(Y,A\right)\tilde{\cup }\left(Z,B\right)=\left(I,C\right).$

Definition 1.4: A soft set (Y, A) over X is said to be a null soft set if $Y\left(k\right)=\text{empty}$ , for all k in A and is denoted by $\Phi$ .

Definition 1.5: For all $k\in A$ , if $Y\left(k\right)=X$ then $\left(Y,A\right)$ is called an absolute soft set over X.

Definition 1.6: The difference of two soft sets $\left(F,E\right)$ and $\left(G,E\right)$ over X is a soft set $\left(H,E\right)$ over X, denoted by $\left(F,E\right)\backslash \left(G,E\right)$ and is defined as $H\left(x\right)=F\left(x\right)\backslash G\left(x\right)$ , $\forall x\in E$ .

Definition 1.7: The complement of soft set $\left(Y,A\right)$ is denoted by ${\left(Y,A\right)}^c$ and is defined as ${\left(Y,A\right)}^c=\left(Y^c,A\right)$ , where $Y^c:A\to P\left(X\right)$ is a mapping given by $Y^c\left(\beta \right)=X-Y\left(\beta \right)$ , for all $\beta$ .

Definition 1.8: Let B(R) be the collection of all non-empty bounded subsets of R and E taken as a set of parameters. Then the mapping $Y:E\to B\left(R\right)$ is called a soft real set. It is denoted by (Y, E).

Definition 1.9: For two soft real numbers $\tilde{m}$ and $\tilde{n}$ the following conditions holds:

(i) $\tilde{m}\le \tilde{n}$ if $\tilde{m}\left(s\right)\le \tilde{n}\left(s\right)$ , for all $s\in E$ ;

(ii) $\tilde{m}\ge \tilde{n}$ if $\tilde{m}\left(s\right)\ge \tilde{n}\left(s\right)$ , for all $s\in E$ ;

(iii) $\tilde{m}<\tilde{n}$ if $\tilde{m}\left(s\right)<\tilde{n}\left(s\right)$ , for all $s\in E$ ;

(iv) $\tilde{m}>\tilde{n}$ if $\tilde{m}\left(s\right)>\tilde{n}\left(s\right)$ , for all $s\in E$ .

Definition 1.10: A soft set (P, E) over X is said to have a soft point if there is exactly one $s\in E$ such that $P\left(s\right)=\left\{x\right\}$ , for some $x\in X$ , also $P\left(s^{\prime }\right)=\phi$ , $\forall s^{\prime }\in E/\left\{s\right\}$ . It will be denoted by ${\tilde{x}}_s$ .

Definition 1.11: Two soft points ${\tilde{x}}_i$ , ${\tilde{y}}_j$ are said to be equal if $i=j$ and $P\left(i\right)=P\left(j\right)$ i.e. x = y. Hence ${\tilde{x}}_i\ne {\tilde{y}}_j\iff x\ne y$ or $i\ne j$ .

Definition 1.12: A mapping $\tilde{\rho }:SP\left(\tilde{X}\right)\times SP\left(\tilde{X}\right)\to R\left(E\right)*$ is soft metric on soft set $\tilde{X}$ with following properties.

SM1. for all ${\tilde{x}}_{s_1},{\tilde{y}}_{s_2}\in \tilde{X},$ $\tilde{\rho }\left({\tilde{x}}_{s_1},{\tilde{y}}_{s_2}\right)\tilde{\ge }\bar{0};$

SM2. $\tilde{\rho }\left({\tilde{x}}_{s_1},{\tilde{y}}_{s_2}\right)=\bar{0}$ , if and only if ${\tilde{x}}_{s_1}={\tilde{y}}_{s_2};$

SM3. for all ${\tilde{x}}_{s_1},{\tilde{y}}_{s_2}\in \tilde{X},$ $\tilde{\rho }\left({\tilde{x}}_{s_1},{\tilde{y}}_{s_2}\right)=\tilde{\rho }\left({\tilde{y}}_{s_2},{\tilde{x}}_{s_1}\right);$

SM4. for all ${\tilde{x}}_{s_1},{\tilde{y}}_{s_2},{\tilde{z}}_{s_3}\in \tilde{X},$ $\tilde{\rho }\left({\tilde{x}}_{s_1},{\tilde{z}}_{s_3}\right)\le \tilde{\rho }\left({\tilde{x}}_{s_1},{\tilde{y}}_{s_2}\right)+\tilde{\rho }\left({\tilde{y}}_{s_2},{\tilde{z}}_{s_3}\right).$

The soft set $\tilde{X}$ with a soft metric $\tilde{\rho }$ defined on $\tilde{X}$ is called a soft metric space and denoted by $\left(\tilde{X},\tilde{\rho },E\right)$ .

Definition 1.13: Let us consider a soft metric $\left(\tilde{X},\tilde{\rho },E\right)$ and $\tilde{\alpha }$ be a non-negative soft real number. The soft open ball with center at ${\tilde{x}}_s$ and radius $\tilde{\alpha }$ is given by

$B\left({\tilde{x}}_s,\tilde{\alpha }\right)=\left\{{\tilde{y}}_{s\text{'}}\in \tilde{X}:\tilde{\rho }\left({\tilde{x}}_s,{\tilde{y}}_{s^{\prime }}\right)\le \tilde{\alpha }\right\}\subset SP\left(\tilde{X}\right),$

and the soft closed ball with center at ${\tilde{x}}_{s^{\prime }}$ and radius $\tilde{\alpha }$ is given by

$B\left({\tilde{x}}_s,\tilde{\alpha }\right)=\left\{{\tilde{x}}_{s^{\prime }}\in \tilde{X}:\tilde{\rho }\left({\tilde{x}}_{s^{\prime }},{\tilde{y}}_{s^{\prime }}\right)\le \tilde{\alpha }\right\}\subset SP\left(\tilde{X}\right).$

Definition 1.14: A sequence $\left\{{\tilde{x}}_{{\lambda }_n}^n\right\}$ of soft points in soft metric space $\left(\tilde{X},\tilde{\rho },E\right)$ is said to be convergent in $\left(\tilde{X},\tilde{\rho },E\right)$ if there is a soft point ${\tilde{y}}_{\mu }\tilde{\in }\tilde{X}$ such that $\tilde{\rho }\left({\tilde{x}}_{{\lambda }_n}^n,{\tilde{y}}_{\mu }\right)\to \tilde{0}$ as $n\to \infty ,$ that is for every $\tilde{\epsilon }>\bar{0},$ there is a natural number $N=N\left(\tilde{\epsilon }\right)$ such that $\bar{0}\le \tilde{\rho }\left({\tilde{x}}_{{\lambda }_n}^n,{\tilde{y}}_{\mu }\right)\widetilde{<}\tilde{\epsilon }$ , whenever $n>N.$

Definition 1.15: Let $\left(\tilde{X},\tilde{\rho },E\right)$ be a soft metric space, then the sequence $\left\{{\tilde{x}}_{{\lambda }_n}^n\right\}$ of soft points in $\left(\tilde{X},\tilde{\rho },E\right)$ is said to be a Cauchy sequence in $\tilde{X},$ if corresponding to every $\tilde{\epsilon }\tilde{\ge }\bar{0},$ there exist $m\in N$ such that $\tilde{\rho }\left({\tilde{x}}_{{\lambda }_i}^i,{\tilde{y}}_{{\lambda }_j}^j\right)\tilde{\le }\tilde{\epsilon },\forall i,j\ge m,$ i.e. $\tilde{\rho }\left({\tilde{x}}_{{\lambda }_i}^i,{\tilde{y}}_{{\lambda }_j}^j\right)\to \bar{0}$ as $i,j\to \infty .$

Definition 1.16: The soft metric space $\left(\tilde{X},\tilde{\rho },E\right)$ is called complete, if every Cauchy sequence in $\tilde{X}$ converges to some point of $\tilde{X}$ .

Definition 1.17: Let $\left(\tilde{X},\tilde{\rho },E\right)$ be a soft metric space. A function $\left(f,\phi \right):\left(\tilde{X},\tilde{\rho },E\right)\to \left(\tilde{X},\tilde{\rho },E\right)$ is called a soft contraction mapping if there is a soft real number $\alpha \in R,\bar{0}\le \alpha <\bar{1}$ such that for every point ${\tilde{x}}_{\lambda },{\tilde{y}}_{\mu }\in SP\left(X\right),$ we have

$\tilde{\rho }\left(\left(f,\phi \right)\left({\tilde{x}}_{\lambda }\right),\left(f,\phi \right)\left({\tilde{y}}_{\mu }\right)\right)\le \alpha \tilde{\rho }\left({\tilde{x}}_{\lambda },{\tilde{y}}_{\mu }\right)$ .

Definition 1.18: A mapping $\tilde{\rho }:SP\left(\tilde{X}\right)\times SP\left(\tilde{X}\right)\to R\left(E\right)*$ is said to be dislocated soft metric on the soft set $\tilde{X}$ if $\tilde{\rho }$ satisfies the following conditions:

(d1) $\tilde{\rho }\left({\tilde{x}}_{s_1},{\tilde{y}}_{s_2}\right)=\bar{0}$ then ${\tilde{x}}_{s_1}={\tilde{y}}_{s_2},$

(d2) $\tilde{\rho }\left({\tilde{x}}_{s_1},{\tilde{y}}_{s_2}\right)=\tilde{\rho }\left({\tilde{y}}_{s_2},{\tilde{x}}_{s_1}\right)$ , for all ${\tilde{x}}_{s_1},{\tilde{y}}_{s_2}\in \tilde{X},$

(d3) $\tilde{\rho }\left({\tilde{x}}_{s_1},{\tilde{z}}_{s_3}\right)\le \tilde{\rho }\left({\tilde{x}}_{s_1},{\tilde{y}}_{s_2}\right)+\tilde{\rho }\left({\tilde{y}}_{s_2},{\tilde{z}}_{s_3}\right)$ , for all ${\tilde{x}}_{s_1},{\tilde{y}}_{s_2},{\tilde{z}}_{s_3}\in \tilde{X}.$

The soft set $\tilde{X}$ with soft dislocated metric $\tilde{\rho }$ defined on $\tilde{X}$ is called a dislocated soft metric space and denoted by $\left(\tilde{X},\tilde{\rho },E\right)$ .

2. Main Results

Theorem 2.1: Let A and B be two non-empty closed subsets of a complete dislocated soft metric space $\left(\tilde{X},\tilde{\rho },E\right)$ . Suppose $\left(f,\phi \right):A\cup B\to A\cup B$ is cyclic and satisfy the following:

(c) There exist a constant $k\in \left(0,1\right)$ such that $\tilde{\rho }\left(\left(f,\phi \right)x_{\lambda },\left(f,\phi \right)y_{\mu }\right)\le k\tilde{\rho }\left(x_{\lambda },y_{\mu }\right)$ for all $x_{\lambda }\in A$ , $y_{\mu }\in B.$

Then $\left(f,\phi \right)$ has a unique soft fixed point that belongs to $A\cap B$ .

Definition 2.2: Let $\left(\tilde{X},\tilde{\rho },E\right)$ be dislocated soft metric space and U be a subset of $\tilde{X}$ . We say that U is $\tilde{\rho }$ -open subset of X if $\forall x\in \tilde{X},$ there exists r > 0 such that $B_{\rho }\left(x,r\right)\subseteq U.$ Also, $V\subseteq \tilde{X}$ is a $\tilde{\rho }$ -closed subset of $\tilde{X}$ if $\left(\tilde{X}/V\right)$ is $\tilde{\rho }$ -open subset of $\tilde{X}$ .

Lemma 2.3: Let $\left(\tilde{X},\tilde{\rho },E\right)$ be dislocated soft metric space and V be a $\tilde{\rho }$ - closed subset of X. Let $\left\{x_{{\lambda }_n}^n\right\}$ be a sequence in V. If $x_{{\lambda }_n}^n\to x_{\lambda }$ as $n\to \infty ,$ then $x_{\lambda }\in V.$

Proof: Let $x_{\lambda }\notin V,$ by definition 2.2, $X\backslash V$ is a $\tilde{\rho }$ -open set then there exist r > 0 such that $B_{\tilde{\rho }}\left(x_{\lambda },r\right)\subseteq X\backslash V.$ On the other hand, we have $\underset{n\to \infty }{\lim }\left|\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{\lambda }\right)-\tilde{\rho }\left(x_{\lambda },x_{\lambda }\right)\right|=0$ , Since $x_{{\lambda }_n}^n\to x_{\lambda }$ as $n\to \infty .$ So there exist $n_0\in N$ such that

$\left|\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{\lambda }\right)-\tilde{\rho }\left(x_{\lambda },x_{\lambda }\right)\right|<r,$

for all $n>n_0$ . Hence we conclude that $\left\{x_{{\lambda }_n}^n\right\}\subseteq B_{\rho }\left(x_{\lambda },r\right)\subseteq X\backslash V,$ for all $n>n_0$ . This is contradiction since $\left\{x_{{\lambda }_n}^n\right\}\subseteq V,$ for all n in N.

Lemma 2.4: Let $\left(\tilde{X},\tilde{\rho },E\right)$ be dislocated soft metric space and $\left\{x_{{\lambda }_n}^n\right\}$ be a sequence in $\left(\tilde{X},\tilde{\rho },E\right)$ such that $\left\{x_{{\lambda }_n}^n\right\}\to x_{\lambda }$ as $n\to \infty$ and $\tilde{\rho }\left(x_{\lambda },x_{\lambda }\right)=0.$ Then

$\underset{n\to \infty }{\lim }\tilde{\rho }\left(x_{{\lambda }_n}^n,y_{\mu }\right)=\tilde{\rho }\left(x_{\lambda },y_{\mu }\right)$ , for all $y_{\mu }\in \left(\tilde{X},\tilde{\rho },E\right)$ .

Lemma 2.5: If $\left(\tilde{X},\tilde{\rho },E\right)$ be soft dislocated metric space, then the following conditions holds

A. If $\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_n}^n\right)=0,$ then $\tilde{\rho }\left(x_{\lambda }{x}_{\lambda }\right)=\tilde{\rho }\left(y_{\mu },y_{\mu }\right)=0;$

B. If $\left\{x_{{\lambda }_n}^n\right\}$ be a sequence such that $\underset{n\to \infty }{\lim }\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_{n+1}}^{n+1}\right)=0,$ then we have $\underset{n\to \infty }{\lim }\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_n}^n\right)=\underset{n\to \infty }{\lim }\tilde{\rho }\left(x_{{\lambda }_{n+1}}^{n+1},x_{{\lambda }_{n+1}}^{n+1}\right)=0;$

C. If $x_{\lambda }\ne y_{\mu },$ then $\tilde{\rho }\left(x_{\lambda },y_{\mu }\right)>0;$

$\tilde{\rho }\left(x_{\lambda }{x}_{\lambda }\right)=\frac{2}{n}{\displaystyle \underset{i=1}{\overset{n}{\sum }}\tilde{\rho }\left(x_{\lambda },x_{{\lambda }_i}^i\right)},$ holds for all $x_{\lambda },x_{{\lambda }_i}^i\in \tilde{X},$ where $1\le i\le n$ .

D. At first we define the class of Ф and Ψ by the following ways:

$\Psi =\left\{\psi :\left[0,\infty \right)\to \left[0,\infty \right)\text{suchthat}\psi \text{isnondecreasingandcontinuous}\right\}$ and $\Phi =\left\{\varphi :\left[0,\infty \right)\to \left[0,\infty \right)\text{suchthat}\varphi \text{islowersemicontinuous}\right\}$ .

Definition 2.6: Let $\left(\tilde{X},\tilde{\rho },E\right)$ be a soft dislocated metric space, $m\in N$ , let $A_1,A_2,\cdots ,A_m$ be $\tilde{\rho }$ -closed non empty subsets of $\tilde{X}$ and let $\tilde{Y}={{\displaystyle \cup }}_{i=1}^{i=m}{A}_i$ . We say that $\left(f,\phi \right)$ is a cyclic generalized $\varphi -\psi$ contractive mapping if

i. $\tilde{Y}={{\displaystyle \cup }}_{i=1}^{i=m}{A}_i$ is a cyclic representation of $\tilde{Y}$ with respect to $(f,\phi ),$

ii. $\psi \left(t\right)-\psi \left(s\right)+\varphi \left(s\right)>0$ for all $t>0$ and $s=t$ or $s=0$ and

$\psi \left(\tilde{\rho }\left(\left(f,\phi \right)x_{\lambda },\left(f,\phi \right)y_{\mu }\right)\right)\prec \psi \left(M_{\tilde{\rho }}\left(x_{\lambda },y_{\mu }\right)\right)-\varphi \left(M_{\tilde{\rho }}\left(x_{\lambda },y_{\mu }\right)\right),$ (1)

for any $x_{\lambda }\in A_i\text{,}{y}_{\mu }\in A_{i+1},$ $i=1,2,3,\cdots ,m$ , where $A_{i+1}=A_1,\varphi \in \Phi ,\psi \in \Psi$ and

$\begin{array}{l}{M}_{\tilde{\rho }}\left(x_{\lambda },y_{\mu }\right)=\text{max}\{\tilde{\rho }\left(x_{\lambda },y_{\mu }\right),\tilde{\rho }\left(x_{\lambda },\left(f,\phi \right)x_{\lambda }\right),\tilde{\rho }\left(y_{\mu },\left(f,\phi \right)y_{\mu }\right),\begin{array}{c}\underset{}{}\\ \end{array}\\ \frac{\tilde{\rho }\left(y_{\mu },\left(f,\phi \right)x_{\lambda }\right)+\tilde{\rho }\left(y_{\mu },\left(f,\phi \right)\text{}{y}_{\mu }\right)+\tilde{\rho }\left(x_{\lambda },\left(f,\phi \right)x_{\lambda }\right)}{6}\}.\end{array}$

Let $\tilde{X}$ be a non-empty set and $\left(f,\phi \right):\left(\tilde{X},\tilde{\rho },E\right)\to \left(\tilde{X},\tilde{\rho },E\right)$ be given map. The set of all soft fixed points of $\left(f,\phi \right)$ will be denoted by $Fix\left(\left(f,\phi \right)\right)$ i.e. $Fix\left(\left(f,\phi \right)\right)=\left\{x_{\lambda }\in \tilde{X}:x_{\lambda }=\left(f,\phi \right)x_{\lambda }\right\}$ .

Theorem 2.7: Let $\left(\tilde{X},\tilde{\rho },E\right)$ be a complete dislocated soft metric space, $m\in N$ , let $A_1,A_2,\cdots ,A_m$ be non-empty $\tilde{\rho }$ -closed subsets of $\left(\tilde{X},\tilde{\rho },E\right)$ and

let $\tilde{Y}={{\displaystyle \cup }}_{i=1}^{i=m}{A}_i$ . Suppose that $\left(f,\phi \right):\tilde{Y}\to \tilde{Y}$ is a cyclic generalized $\varphi -\psi$ con-

tractive mapping. Then $\left(f,\phi \right)$ has fixed point in ${{\displaystyle \cap }}_{i=1}^n{A}_i$ . Moreover if $\tilde{\rho }\left(x_{\lambda },y_{\mu }\right)\ge \tilde{\rho }\left(x_{\lambda },x_{\lambda }\right)$ for all $x,y\in Fix\left(f,\phi \right)$ , then $\left(f,\phi \right)$ has a unique fixed point in ${{\displaystyle \cap }}_{i=1}^n{A}_i.$

Proof: Let $x_{{\lambda }_0}^0$ be an arbitrary point of $\tilde{Y},$ so there exists some $i_0$ such

that $x_{{\lambda }_0}^0\in A_{i_0}.$ We know that $\left(f,\phi \right)\left(A_{i_0}\right)\subseteq A_{i_0+1},$ we conclude that $\left(f,\phi \right)\left(x_{{\lambda }_0}^0\right)\in A_{i_0+1}.$ Thus there exist $x_{{\lambda }_1}^1$ in $A_{i_0+1}$ such that $\left(f,\phi \right)x_{{\lambda }_0}^0=x_{{\lambda }_1}^1.$ Recursive $\left(f,\phi \right)x_{{\lambda }_n}^n=x_{{\lambda }_{n+1}}^{n+1},$ where $x_{{\lambda }_n}^n\in A_{i_n}.$ Hence for $n\ge 0$ there exist $i_n\in \left\{1,2,3,\cdots ,m\right\}$ such that $x_{{\lambda }_n}^n\in A_{i_n}$ . In case $x_{{\lambda }_{n_0}}^{n_0}=x_{{\lambda }_{n_0+1}}^{n_0+1}$ for some $n_0=0,1,2,\cdots$ , then it is clear that $x_{{\lambda }_{n_0}}^{n_0}$ is a soft fixed point of $\left(f,\phi \right)$ . Now assume that $x_{{\lambda }_n}^n\ne x_{{\lambda }_{n+1}}^{n+1}$ for all n. Hence by lemma 2.5(c) we have $\tilde{\rho }\left(x_{{\lambda }_{n-1}}^{n-1},x_{{\lambda }_n}^n\right)>0$ for all n. We shall show that the sequence $\left\{d_n\right\}$ is non-in- creasing, where $d_n=\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_{n+1}}^{n+1}\right)$ . Assume that there exists some $n_0\in N$ such that

$\tilde{\rho }\left(x_{{\lambda }_{n_0-1}}^{{}_{n_0-1}},x_{n_0}^{n_0}\right)\le \tilde{\rho }\left(x_{{\lambda }_{n_0}}^{{}_{n_0}},x_{n_0+1}^{n_0+1}\right).$

Hence we get

$\psi \left(\tilde{\rho }\left(x_{n_0-1}^{n_0-1},x_{n_0}^{n_0}\right)\right)\le \psi \left(\tilde{\rho }\left(x_{n_0}^{n_0},x_{n_0+1}^{n_0+1}\right)\right).$ (2)

Set ${\tilde{x}}_{{\lambda }_1}^1=\left(f,\phi \right)\left({\tilde{x}}_{\lambda }^0\right)={\left(f\left({\tilde{x}}_{\lambda }^0\right)\right)}_{\phi \left(\lambda \right)},{\tilde{x}}_{{\lambda }_2}^2=\left(f,\phi \right)\left({\tilde{x}}_{{\lambda }_1}^1\right)={\left(f^2\left({\tilde{x}}_{\lambda }^0\right)\right)}_{{\phi }^2\left(\lambda \right)},\cdots ,$

${\tilde{x}}_{{\lambda }_{n+1}}^{n+1}=\left(f,\phi \right)\left({\tilde{x}}_{{\lambda }_n}^n\right)={\left(f^{n+1}\left({\tilde{x}}_{\lambda }^0\right)\right)}_{{\phi }^{n+1}\left(\lambda \right)},\cdots .$

Using conditions (1) together with (2), we get

(3)

On the other hand, from lemma 2.5(D) we have $\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_n}^n\right)\le \tilde{\rho }\left(x_{{\lambda }_{n-1}}^{n-1},x_{{\lambda }_n}^n\right)+\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_{n+1}}^{n+1}\right)$ and $\frac{\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_n}^n\right)+\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_{n+1}}^{n+1}\right)+\tilde{\rho }\left(x_{{\lambda }_{n-1}}^{n-1},x_{{\lambda }_n}^n\right)}{6}\le \frac{\tilde{\rho }\left(x_{{\lambda }_{n-1}}^{n-1},x_{{\lambda }_n}^n\right)+\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_{n+1}}^{n+1}\right)}{3}.$

That is $\begin{array}{l}\max \left\{\tilde{\rho }\left(x_{{\lambda }_{n-1}}^{n-1},x_{{\lambda }_n}^n\right),\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_{n+1}}^{n+1}\right),\frac{\tilde{\rho }\left(x_{{\lambda }_{n-1}}^{n-1},x_{{\lambda }_n}^n\right)+\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_{n+1}}^{n+1}\right)}{3}\right\}\\ \le \max \left\{\tilde{\rho }\left(x_{{\lambda }_{n-1}}^{n-1},x_{{\lambda }_n}^n\right),\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_{n+1}}^{n+1}\right)\right\}.\end{array}$

Therefore from (3) we get

$\begin{array}{c}\psi \left(\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_{n+1}}^{n+1}\right)\right)\le \psi \left(\max \left\{\tilde{\rho }\left(x_{{\lambda }_{n-1}}^{n-1},x_{{\lambda }_n}^n\right),\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_{n+1}}^{n+1}\right)\right\}\right)\\ -\varphi \left(\max \left\{\tilde{\rho }\left(x_{{\lambda }_{n-1}}^{n-1},x_{{\lambda }_n}^n\right),\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_{n+1}}^{n+1}\right)\right\}\right).\end{array}$

Now, if $\max \left\{\tilde{\rho }\left(x_{{\lambda }_{n-1}}^{n-1},x_{{\lambda }_n}^n\right),\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_{n+1}}^{n+1}\right)\right\}=\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_{n+1}}^{n+1}\right),$ then

$\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_{n+1}}^{n+1}\right)\le \alpha \tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_{n+1}}^{n+1}\right)-\beta \tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_{n+1}}^{n+1}\right)$ .

This is contradiction. Hence we have

$\psi \tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_{n+1}}^{n+1}\right)\le \psi \left\{\tilde{\rho }\left(x_{{\lambda }_{n-1}}^{n-1},x_{{\lambda }_n}^n\right)\right\}-\varphi \left\{\tilde{\rho }\left(x_{{\lambda }_{n-1}}^{n-1},x_{{\lambda }_n}^n\right)\right\},$ (4)

for all $n\in N$ . By taking $x_{\lambda }=x_{{\lambda }_{n_0-1}}^{n_0-1}$ and $y_{\mu }=x_{{\lambda }_{n_0}}^{n_0}$ in (4) and keeping (2) in

mind, we deduce that

$\psi \tilde{\rho }\left(x_{{\lambda }_n{}_{{}_0-1}}^{n_0-1},x_{{\lambda }_{n_0}}^{n_0}\right)\le \psi \left\{\tilde{\rho }\left(x_{{\lambda }_{n_0-1}}^{n_0-1},x_{{\lambda }_{n_0}}^{n_0}\right)\right\}-\varphi \left\{\tilde{\rho }\left(x_{{\lambda }_{n_0-1}}^{n_0-1},x_{{\lambda }_{n_0}}^{n_0}\right)\right\}$ .

This is a contradiction. Hence we conclude that $d_n<d_{n-1}$ i.e.

$\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_{n+1}}^{n+1}\right)\le \tilde{\rho }\left(x_{{\lambda }_{n-1}}^{n-1},x_{{\lambda }_n}^n\right)$ hold for all $n\in N$ . Thus there exist $r\ge 0$ such that $\underset{n\to \infty }{\lim }{d}_n=\underset{n\to \infty }{\lim }\tilde{\rho }\left(x_{{\lambda }_n}^n,x_{{\lambda }_{n+1}}^{n+1}\right)=r$ . We shall show that $r=0$ by the method of

reductio ad absurdum. For this purpose, we assume that $r>0$ . By (4) together with the property of $\varphi$ and $\psi$ we have

$\psi \left(r\right)=\underset{n\to \infty }{\lim }\sup \psi \left(d_n\right)\le \underset{n\to \infty }{\lim }\sup \left[\psi \left(d_{n-1}\right)-\varphi \left(d_{n-1}\right)\right]\le \psi \left(r\right)-\varphi \left(r\right)$ .

This yields that $\phi \left(r\right)\le 0$ . This is contradiction. Hence we obtain that

$\underset{n\to \infty }{\lim }{d}_n\le \underset{n\to \infty }{\lim }\left(x_{{\lambda }_n}^n,x_{{\lambda }_{n+1}}^{n+1}\right)=0$ . (5)

We shall show that $\left\{x_{{\lambda }_n}^n\right\}$ is a $\tilde{\rho }$ -Cauchy sequence. To reach this goal, first we prove the followings claim:

(k) For every $ϵ>0,$ there exists $n\in N$ such that if $r,q\ge n$ with

$r-q\equiv 1\left(m\right),$ then $\tilde{\rho }\left(x_{{\lambda }_{r_n}}^{r_n},x_{{\lambda }_{q_n}}^{q_n}\right)<ϵ$ .

Suppose, on the contrary that there exist $ϵ>0$ such that for any $n\in N$ , we can find $r_n>q_n\ge n$ with $r_n-q_n\equiv 1\left(m\right)$ satisfying

$\tilde{\rho }\left(x_{{\lambda }_{q_n}}^{q_n},x_{{\lambda }_{r_n}}^{r_n}\right)\ge ϵ$ . (6)

Now we consider $n>2m.$ Then, corresponding to $q_n\ge n$ , we can choose $r_n$ in such a way that it is the smallest integer with $r_n>q_n$ satisfying

$r_n-q_n\equiv 1\left(m\right)$ and $\tilde{\rho }\left(x_{{\lambda }_{q_n}}^{q_n},x_{{\lambda }_{r_n}}^{r_n}\right)\ge ϵ.$ Therefore $\tilde{\rho }\left(x_{{\lambda }_{q_n}}^{q_n},x_{{\lambda }_{r_n-m}}^{r_n-m}\right)\le ϵ$ . By using

triangular inequality, we obtain

$\begin{array}{c}ϵ\le \tilde{\rho }\left(x_{{\lambda }_{q_n}}^{q_n},x_{{\lambda }_{r_n}}^{r_n}\right)\\ \le \tilde{\rho }\left(x_{{\lambda }_{q_n}}^{q_n},x_{{\lambda }_{q_n-m}}^{q_n-m}\right)+{\displaystyle \underset{i=1}{\overset{m}{\sum }}\tilde{\rho }\left(x_{{\lambda }_{r_n-i}}^{r_n-i},x_{{\lambda }_{r_n-i+1}}^{r_n-i+1}\right)}\\ \le ϵ+{\displaystyle \underset{i=1}{\overset{m}{\sum }}\tilde{\rho }\left(x_{{\lambda }_{r_n-i}}^{r_n-i},x_{{\lambda }_{r_n-i+1}}^{r_n-i+1}\right)}\end{array}$ .

Passing to the limit as $n\to \infty$ in the last inequality and taking (5) into account, we obtain that

$\underset{n\to \infty }{\lim }\tilde{\rho }\left(x_{{\lambda }_{q_n}}^{q_n},x_{{\lambda }_{r_n}}^{r_n}\right)=ϵ$ . (7)

Again by (d₃) we derive that

$\begin{array}{c}ϵ\le \tilde{\rho }\left(x_{{\lambda }_{q_n}}^{q_n},x_{{\lambda }_{r_n}}^{r_n}\right)\\ \le \tilde{\rho }\left(x_{{\lambda }_{q_n}}^{q_n},x_{{\lambda }_{q_{n+1}}}^{q_{n+1}}\right)+\tilde{\rho }\left(x_{{\lambda }_{q_{n+1}}}^{q_{n+1}},x_{{\lambda }_{r_n+1}}^{r_n+1}\right)+\tilde{\rho }\left(x_{{\lambda }_{r_n+1}}^{r_n+1},x_{{\lambda }_{r_n}}^{r_n}\right)\\ \le \tilde{\rho }\left(x_{{\lambda }_{q_n}}^{q_n},x_{{\lambda }_{q_{n+1}}}^{q_{n+1}}\right)+\tilde{\rho }\left(x_{{\lambda }_{q_{n+1}}}^{q_{n+1}},x_{{\lambda }_{q_n}}^{q_n}\right)+\tilde{\rho }\left(x_{{\lambda }_{q_n}}^{q_n},x_{{\lambda }_{r_n}}^{r_n}\right)\\ +\tilde{\rho }\left(x_{{\lambda }_{r_n}}^{r_n},x_{{\lambda }_{r_n}+1}^{r_n+1}\right)+\tilde{\rho }\left(x_{{\lambda }_{r_n}+1}^{r_n+1},x_{{\lambda }_{r_n}}^{r_n}\right)\\ \le 2\tilde{\rho }\left(x_{{\lambda }_{q_n}}^{q_n},x_{{\lambda }_{q_{n+1}}}^{q_{n+1}}\right)+\tilde{\rho }\left(x_{{\lambda }_{q_n}}^{q_n},x_{{\lambda }_{r_n}}^{r_n}\right)+2\tilde{\rho }\left(x_{{\lambda }_{r_n}+1}^{r_n+1},x_{{\lambda }_{r_n}}^{r_n}\right).\end{array}$

Taking (5) and (7) in account we get

$\underset{n\to \infty }{\lim }\tilde{\rho }\left(x_{{\lambda }_{q_{n+1}}}^{q_{n+1}},x_{{\lambda }_{r_n+1}}^{r_n+1}\right)=ϵ$ . (8)

By ( $d_3$ ) we have the following inequality

$\tilde{\rho }\left(x_{{\lambda }_{q_n}}^{q_n},x_{{\lambda }_{r_n+1}}^{r_n+1}\right)\le \tilde{\rho }\left(x_{{\lambda }_{q_n}}^{q_n},x_{{\lambda }_{r_n}}^{r_n}\right)+\tilde{\rho }\left(x_{{\lambda }_{r_n}}^{r_n},x_{{\lambda }_{r_n+1}}^{r_n+1}\right)$ (9)

and $\tilde{\rho }\left(x_{{\lambda }_{q_n}}^{q_n},x_{{\lambda }_{r_n}}^{r_n}\right)\le \tilde{\rho }\left(x_{{\lambda }_{q_n}}^{q_n},x_{{\lambda }_{r_n+1}}^{r_n+1}\right)+\tilde{\rho }\left(x_{{\lambda }_{r_n}}^{r_n},x_{{\lambda }_{r_n+1}}^{r_n+1}\right)$ . (10)

Letting $\lim n\to \infty$ in (9) and (10), we have

$\underset{n\to \infty }{\lim }\tilde{\rho }\left(x_{{\lambda }_{q_n}}^{q_n},x_{{\lambda }_{r_n+1}}^{r_n+1}\right)=ϵ$ . (11)

Again by ( $d_3$ ) we have

$\tilde{\rho }\left(x_{{\lambda }_{r_n}}^{r_n},x_{{\lambda }_{q_n+1}}^{q_n+1}\right)\le \tilde{\rho }\left(x_{{\lambda }_{r_n}}^{r_n},x_{{\lambda }_{r_n+1}}^{r_n+1}\right)+\tilde{\rho }\left(x_{{\lambda }_{r_n+1}}^{r_n+1},x_{{\lambda }_{q_n+1}}^{q_n+1}\right)$ (12)

and $\tilde{\rho }\left(x_{{\lambda }_{r_n+1}}^{r_n+1},x_{{\lambda }_{q_n+1}}^{q_n+1}\right)\le \tilde{\rho }\left(x_{{\lambda }_{r_n+1}}^{r_n+1},x_{{\lambda }_{r_n}}^{r_n}\right)+\tilde{\rho }\left(x_{{\lambda }_{r_n}}^{r_n},x_{{\lambda }_{q_n+1}}^{q_n+1}\right)$ . (13)

Letting $n\to \infty$ in (12) and (13), we derive that

$\underset{n\to \infty }{\lim }\tilde{\rho }\left(x_{{\lambda }_{r_n}}^{r_n},x_{{\lambda }_{q_n+1}}^{q_n+1}\right)=ϵ$ . (14)

Since $x_{{\lambda }_{q_n}}^{q_n}$ and $x_{{\lambda }_{r_n}}^{r_n}$ lie in different adjacently labeled sets and for certain $1\le i\le m$ . By using (5), (7), (8), (11) and (14) together with the fact that $\left(f,\phi \right)$ is a generalized cyclic $\varphi -\psi$ contractive mappings, we find that