Applications to Boundary Value Problems and Homotopy Theory via Tripled Fixed Point Techniques in Partially Metric Spaces

Abstract

In this manuscript, some tripled fixed point results were derived under $(\phi ,\rho ,\ell )$-contraction in the framework of ordered partially metric spaces. Moreover, we furnish an example which supports our theorem. Furthermore, some results about a homotopy results are obtained. Finally, theoretical results are involved in some applications, such as finding the unique solution to the boundary value problems and homotopy theory.

1. Introduction

Fixed point theory is one of the important and indispensable branches of non-linear analysis due to the proliferation of its applications in many disciplines such as engineering, computer science, physics, economics, biology, chemistry, etc. In mathematics, this technique is credited with clarifying and studying the behavior of dynamical systems, statistical methods, game theory models, differential equations, and many others. Specifically, this technique studies the existence and uniqueness of the solution to many integral and fractional equations, which facilitates the way to find numerical solutions to such problems, see [1,2,3,4,5,6,7,8,9,10,11].

Homotopy theory is a fundamental branch of algebraic topology where topological objects are studied up to homotopy equivalence. In the last century, strong links have emerged between this theory and many branches of mathematics. For example, this trend plays a prominent role in strengthening ties between homotopy theory and category theory (higher-dimensional), which have received considerable attention in recent years, see [12,13,14,15]. Furthermore, it is useful in quantum mechanics for dealing with Hamiltonian manifolds.

The idea of a partially metric space (PMS) was presented by Matthews [16] as a part of the study of denotational semantics of data flow networks. In fact, it is widely shared that PMSs play a prominent role in model building in computational and field theory in computer science, see [17,18,19,20,21,22].

Fixed point results for a single mapping in partially ordered metric space (POMSs) were introduced by Matthews [16,23], Oltra and Valero [24] and Altun et al. [25]. For more papers in common fixed point consequences in abstract spaces, we mention [26,27,28,29,30,31,32].

In [33], the notion of a coupled fixed point was presented and some pivotal results from it in the setting of PMSs are studied. Later, coupled fixed and coupled common fixed point theorems are proved by [34,35,36,37].

In 2011, Berinde and Borcut [38] generalized the idea of a coupled fixed point to a tripled fixed point (TFP) in the setting of POMSs. Under this space, Borcut [39,40], Karapnar et al. [41], Radenović [42] and Aydi et al. [43] introduced some circular theorems concerning TFP theorems. Moreover, more applications in this line are presented by Mustafa et al. [44] and Hammad and De la Sen [45,46].

The purpose of this manuscript is to prove some TFP consequences via $(\phi ,\rho ,\ell )$-contraction in the setting of ordered PMSs. In addition, to support our theoretical results, we give an example. Moreover, as applications, the existence and uniqueness of the solution to an initial value problem (IVP) and a homotopy theory are discussed.

2. Preliminaries

This part is devoted to recall the standard definition of a homotopy and some elementary properties for PMSs.

A homotopy between two functions is defined as follows: consider two continuous functions from a topological space to another; the two functions are considered homotopic if one can be continuously deformed into the other. This deformity is called a homotopy between the two functions. It can be formulated as follows:

Definition 1

([12]). Assume that $U,V:D\to E$ are continuous mappings defined on topological spaces D and E. Then the continuous function $H:D\times [0,1]\to E$ so that $H(\vartheta ,0)=U\vartheta$ and $H(\vartheta ,1)=V\vartheta$ is called a homotopy from U to $V.$ Furthermore, U and V are called homotopic mappings.

Definition 2

([16]). Let $\daleth \ne \varnothing .$ The function $\mathsf{\Lambda }:\daleth \times \daleth \to {\mathbb{R}}^{+}$ is called a partial metric on ℸ if for all ${\vartheta }_1,{\vartheta }_2,{\vartheta }_3\in \daleth ,$ the hypotheses below hold:

$\left({\mathsf{\Lambda }}_1\right)$

${\vartheta }_1={\vartheta }_2$ iff $\mathsf{\Lambda }({\vartheta }_1,{\vartheta }_1)=\mathsf{\Lambda }({\vartheta }_1,{\vartheta }_2)=\mathsf{\Lambda }({\vartheta }_2,{\vartheta }_2);$

$\left({\mathsf{\Lambda }}_2\right)$

$\mathsf{\Lambda }({\vartheta }_1,{\vartheta }_1)\le \mathsf{\Lambda }({\vartheta }_1,{\vartheta }_2)$ and $\mathsf{\Lambda }({\vartheta }_2,{\vartheta }_2)\le \mathsf{\Lambda }({\vartheta }_1,{\vartheta }_2);$

$\left({\mathsf{\Lambda }}_3\right)$

$\mathsf{\Lambda }({\vartheta }_1,{\vartheta }_2)=\mathsf{\Lambda }({\vartheta }_2,{\vartheta }_1);$

$\left({\mathsf{\Lambda }}_4\right)$

$\mathsf{\Lambda }({\vartheta }_1,{\vartheta }_2)\le \mathsf{\Lambda }({\vartheta }_1,{\vartheta }_3)+\mathsf{\Lambda }({\vartheta }_3,{\vartheta }_2)-\mathsf{\Lambda }({\vartheta }_3,{\vartheta }_3).$

We say that $\left(\daleth ,\mathsf{\Lambda }\right)$ is a PMS.

It should be noted that if $\mathsf{\Lambda }$ is partial metric on $\daleth ,$ then the function ${\mathsf{\Omega }}_{\mathsf{\Lambda }}:\daleth \times \daleth \to {\mathbb{R}}^{+},$ described by

$${\mathsf{\Omega }}_{\mathsf{\Lambda }}({\vartheta }_1,{\vartheta }_2)\le 2\mathsf{\Lambda }({\vartheta }_1,{\vartheta }_2)-\mathsf{\Lambda }({\vartheta }_1,{\vartheta }_1)-\mathsf{\Lambda }({\vartheta }_2,{\vartheta }_2),$$

is a metric on $\daleth .$

Example 1

([23]). Let $\daleth =[0,\infty ).$ The pair $\left(\daleth ,\mathsf{\Lambda }\right)$ is a PMS under the distance $\mathsf{\Lambda }({\vartheta }_1,{\vartheta }_2)=max\{{\vartheta }_1,{\vartheta }_2\}.$ It is obvious that Λ is not a usual metric and in this case ${\mathsf{\Omega }}_{\mathsf{\Lambda }}({\vartheta }_1,{\vartheta }_2)=\left|{\vartheta }_1-{\vartheta }_2\right|.$

Example 2.

Assume $\daleth =\{[{\vartheta }_1,{\vartheta }_2]:{\vartheta }_1,{\vartheta }_2\in \mathbb{R},{\vartheta }_1\le {\vartheta }_2\}$ and define

$$\mathsf{\Lambda }(\left[{\vartheta }_1,{\vartheta }_3\right],\left[{\vartheta }_2,{\vartheta }_4\right])=max\{{\vartheta }_3,{\vartheta }_4\}-min\{{\vartheta }_1,{\vartheta }_2\}.$$

Then $\left(\daleth ,\mathsf{\Lambda }\right)$ is a PMS.

Every partial metric $\mathsf{\Lambda }$ on $\daleth$ produces a ${\mathsf{{\rm Y}}}_0$ topology ${\chi }_{\mathsf{\Lambda }}$ on $\daleth .$ The base of a topology ${\chi }_{\mathsf{\Lambda }}$ is a family of open $\mathsf{\Lambda }$-balls $\{O_{\mathsf{\Lambda }}(z,ϵ),z\in \daleth ,ϵ>0\}$, where $O_{\mathsf{\Lambda }}(z,ϵ)=\{\vartheta \in \daleth :\mathsf{\Lambda }(z,\vartheta )<\mathsf{\Lambda }(z,z)+ϵ\},$ $\forall z\in \daleth$ and $ϵ>0.$

Now, we state some topological properties on PMSs.

Definition 3

([16]). A sequence $\left\{{\vartheta }^{\omega }\right\}$ in a PMS $\left(\daleth ,\mathsf{\Lambda }\right)$ is called:

(1) 

converges to the limit ϑ iff $\mathsf{\Lambda }(\vartheta ,\vartheta )={lim}_{\omega \to \infty }\mathsf{\Lambda }(\vartheta ,{\vartheta }^{\omega }).$

(2) 

a Cauchy sequence if ${lim}_{\nu ,\omega \to \infty }\mathsf{\Lambda }({\vartheta }^{\nu },{\vartheta }^{\omega })$ exists and is finite.

Definition 4

([16]). (i) A PMS $\left(\daleth ,\mathsf{\Lambda }\right)$ is called complete if every Cauchy sequence $\left\{{\vartheta }^{\omega }\right\}$ in ℸ converges (with respect to ${\chi }_{\mathsf{\Lambda }})$ to a point $\vartheta \in \daleth$ so that $\mathsf{\Lambda }(\vartheta ,\vartheta )={lim}_{\nu ,\omega \to \infty }\mathsf{\Lambda }({\vartheta }^{\nu },{\vartheta }^{\omega }).$

(ii) 

A mapping $\xi :\daleth \to \daleth$ is said to be continuous at ${\vartheta }^0\in \daleth ,$ if $\forall ϵ>0,$ $\exists \delta >0$ so that $\xi \left(O_{\mathsf{\Lambda }}({\vartheta }^0,\delta )\right)\subseteq O_{\mathsf{\Lambda }}(\xi {\vartheta }^0,ϵ).$

The following lemmas are very important in the following.

Lemma 1

([16]). (i) We say that $\left\{{\vartheta }^{\omega }\right\}$ is a Cauchy sequence in the PMS $\left(\daleth ,\mathsf{\Lambda }\right)$ if it is a Cauchy sequence in the metric space $\left(\daleth ,{\mathsf{\Omega }}_{\mathsf{\Lambda }}\right).$

(ii) 

If the metric space $\left(\daleth ,{\mathsf{\Omega }}_{\mathsf{\Lambda }}\right)$ is complete, the PMS $\left(\daleth ,\mathsf{\Lambda }\right)$ is too. Furthermore,

$$\underset{\omega \to \infty }{lim}{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left(\vartheta ,{\vartheta }^{\omega }\right)=0\iff \mathsf{\Lambda }(\vartheta ,\vartheta )=\underset{\omega \to \infty }{lim}{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left(\vartheta ,{\vartheta }^{\omega }\right)=\underset{\omega ,\upsilon \to \infty }{lim}{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\vartheta }^{\upsilon },{\vartheta }^{\omega }\right).$$

Lemma 2

([47]). Suppose that $\left(\daleth ,\mathsf{\Lambda }\right)$ is a PMS. If $\left\{{\vartheta }^{\omega }\right\}\in \daleth$ so that ${lim}_{\omega \to \infty }{\vartheta }^{\omega }=r$ and $\mathsf{\Lambda }(r,r)=0.$ Then ${lim}_{\omega \to \infty }\mathsf{\Lambda }({\vartheta }^{\omega },s)={lim}_{\omega \to \infty }\mathsf{\Lambda }(r,s)$ for every $r,s\in \daleth .$

Lemma 3

([47]). Assume that $\left(\daleth ,\mathsf{\Lambda }\right)$ is a PMS. Then

(a) 

If $\mathsf{\Lambda }({\vartheta }_1,{\vartheta }_2)=0,$ then ${\vartheta }_1={\vartheta }_2$;

(b) 

If ${\vartheta }_1\ne {\vartheta }_2,$ then $\mathsf{\Lambda }({\vartheta }_1,{\vartheta }_2)>0.$

Remark 1

([47]). If ${\vartheta }_1={\vartheta }_2,$ then $\mathsf{\Lambda }({\vartheta }_1,{\vartheta }_2)$ may not be $0.$

Definition 5

([46]). A mapping $\xi :{\daleth }^3\to \daleth$ (where $\daleth \times \daleth \times \daleth ={\daleth }^3)$ on a partially ordered set (POS) $\left(\daleth ,⪯\right),$ has a mixed-monotone property, if for any $\vartheta ,\theta ,\omega \in \daleth ,$

$$\begin{array}{ccc}\hfill {\vartheta }_1,{\vartheta }_2& \in & \daleth ,{\vartheta }_1⪯{\vartheta }_2\Rightarrow \xi ({\vartheta }_1,\theta ,\omega )⪯\xi ({\vartheta }_2,\theta ,\omega ),\hfill \\ \hfill {\theta }_1,{\theta }_2& \in & \daleth ,{\theta }_1⪯{\theta }_2\Rightarrow \xi (\vartheta ,{\theta }_1,\omega )⪰\xi (\vartheta ,{\theta }_2,\omega ),\hfill \\ \hfill {\omega }_1,{\omega }_2& \in & \daleth ,{\omega }_1⪯{\omega }_2\Rightarrow \xi (\vartheta ,\theta ,{\omega }_1)⪯\xi (\vartheta ,\theta ,{\omega }_2).\hfill \end{array}$$

Definition 6

([43]). A mapping $\xi :{\daleth }^3\to \daleth$ on a POS $\left(\daleth ,⪯\right)$ has a mixed ${\xi }^{*}-$ monotone property where ${\xi }^{*}:\daleth \to \daleth ,$ if for any $\vartheta ,\theta ,\omega \in \daleth ,$

$$\begin{array}{ccc}\hfill {\vartheta }_1,{\vartheta }_2& \in & \daleth ,{\xi }^{*}{\vartheta }_1⪯{\xi }^{*}{\vartheta }_2\Rightarrow \xi ({\vartheta }_1,\theta ,\omega )⪯\xi ({\vartheta }_2,\theta ,\omega ),\hfill \\ \hfill {\theta }_1,{\theta }_2& \in & \daleth ,{\xi }^{*}{\theta }_1⪯{\xi }^{*}{\theta }_2\Rightarrow \xi (\vartheta ,{\theta }_1,\omega )⪰\xi (\vartheta ,{\theta }_2,\omega ),\hfill \\ \hfill {\omega }_1,{\omega }_2& \in & \daleth ,{\xi }^{*}{\omega }_1⪯{\xi }^{*}{\omega }_2\Rightarrow \xi (\vartheta ,\theta ,{\omega }_1)⪯\xi (\vartheta ,\theta ,{\omega }_2).\hfill \end{array}$$

Definition 7

([39]). We say that a trio $(\vartheta ,\theta ,\omega )\in {\daleth }^3$ is a TFP of the self-mapping $\xi :{\daleth }^3\to \daleth$ if $\vartheta =\xi (\vartheta ,\theta ,\omega ),$ $\theta =\xi (\theta ,\omega ,\vartheta )$ and $\omega =\xi (\omega ,\vartheta ,\theta ).$

Definition 8

([40]). A trio $(\vartheta ,\theta ,\omega )\in {\daleth }^3$ is called a tripled coincidence point (TCP) of the two self-mappings $\xi :{\daleth }^3\to \daleth$ and ${\xi }^{*}:\daleth \to \daleth$ if ${\xi }^{*}\vartheta =\xi (\vartheta ,\theta ,\omega ),$ ${\xi }^{*}\theta =\xi (\theta ,\omega ,\vartheta )$ and ${\xi }^{*}\omega =\xi (\omega ,\vartheta ,\theta ).$

Definition 9

([40]). Assume that $\daleth \ne \varnothing$ is a set, a trio $(\vartheta ,\theta ,\omega )\in {\daleth }^3$ is called a common TFP of $\xi :{\daleth }^3\to \daleth$ and ${\xi }^{*}:\daleth \to \daleth ,$ if $\vartheta ={\xi }^{*}\vartheta =\xi (\vartheta ,\theta ,\omega ),$ $\theta ={\xi }^{*}\theta =\xi (\theta ,\omega ,\vartheta )$ and $\omega ={\xi }^{*}\omega =\xi (\omega ,\vartheta ,\theta ).$

Definition 10

([48]). The mappings $\xi :{\daleth }^3\to \daleth$ and ${\xi }^{*}:\daleth \to \daleth$ are called $w-$compatible if ${\xi }^{*}\left(\xi \left(\vartheta ,\theta ,\omega \right)\right)=\xi \left({\xi }^{*}\vartheta ,{\xi }^{*}\theta ,{\xi }^{*}\omega \right)$, ${\xi }^{*}\left(\xi \left(\theta ,\omega ,\vartheta \right)\right)=\xi \left({\xi }^{*}\theta ,{\xi }^{*}\omega ,{\xi }^{*}\vartheta \right)$ and ${\xi }^{*}\left(\xi \left(\omega ,\vartheta ,\theta \right)\right)=\xi \left({\xi }^{*}\omega ,{\xi }^{*}\vartheta ,{\xi }^{*}\theta \right),$ whenever ${\xi }^{*}\vartheta =\xi \left(\vartheta ,\theta ,\omega \right),$ ${\xi }^{*}\theta =\xi \left(\theta ,\omega ,\vartheta \right)$ and ${\xi }^{*}\omega =\xi \left(\omega ,\vartheta ,\theta \right).$

3. Theorems and Discussion

We begin this part with the following definition:

Definition 11.

Assume that $\left(\daleth ,\mathsf{\Lambda }\right)$ is a PMS, $\xi :{\daleth }^3\to \daleth$ and ${\xi }^{*}:\daleth \to \daleth$ are mappings. We say that ξ verifies a $\left(\phi ,\rho ,\ell \right)-$contraction w.r.t ${\xi }^{*}$ if there are $\phi ,\rho ,\ell :[0,\infty )\to [0,\infty )$ verifying the assertions below:

$\left(a_1\right)$

φ is continuous and monotonically non-decreasing, ρ is continuous and ℓ is lower semi continuous (LSC);

$\left(a_2\right)$

$\phi \left(\nu \right)=0$ iff $\nu =0,$ $\rho \left(0\right)=\ell \left(0\right)=0;$

$\left(a_3\right)$

$\phi \left(\nu \right)-\rho \left(\nu \right)+\ell \left(\nu \right)>0;$

$\left(a_4\right)$

for all ${\theta }_1,{\theta }_2,{\theta }_3,{\vartheta }_1,{\vartheta }_2,{\vartheta }_3\in \daleth$ with ${\xi }^{*}{\theta }_1⪯{\xi }^{*}{\vartheta }_1,$ ${\xi }^{*}{\theta }_2⪰{\xi }^{*}{\vartheta }_2$ and ${\xi }^{*}{\theta }_3⪯{\xi }^{*}{\vartheta }_3,$ we have

$$\phi \left(\mathsf{\Lambda }\left(\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right),\xi \left({\vartheta }_1,{\vartheta }_2,{\vartheta }_3\right)\right)\right)\le \rho \left(\beth \left({\theta }_1,{\theta }_2,{\theta }_3,{\vartheta }_1,{\vartheta }_2,{\vartheta }_3\right)\right)-\ell \left(\beth \left({\theta }_1,{\theta }_2,{\theta }_3,{\vartheta }_1,{\vartheta }_2,{\vartheta }_3\right)\right),$$

where

$$\begin{array}{ccc}& & \beth \left({\theta }_1,{\theta }_2,{\theta }_3,{\vartheta }_1,{\vartheta }_2,{\vartheta }_3\right)\hfill \\ & =& max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1,{\xi }^{*}{\vartheta }_1\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2,{\xi }^{*}{\vartheta }_2\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3,{\xi }^{*}{\vartheta }_3\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\theta }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2,\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3,\xi \left({\theta }_3,{\theta }_1,{\theta }_2\right)\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\vartheta }_1,\xi \left({\vartheta }_1,{\vartheta }_2,{\vartheta }_3\right)\right),\mathsf{\Lambda }\left({\xi }^{*}{\vartheta }_2,\xi \left({\vartheta }_2,{\vartheta }_3,{\vartheta }_1\right)\right),\mathsf{\Lambda }\left({\xi }^{*}{\vartheta }_3,\xi \left({\vartheta }_3,{\vartheta }_1,{\vartheta }_2\right)\right),\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2,\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1,{\xi }^{*}{\vartheta }_1\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2,{\xi }^{*}{\vartheta }_2\right)+\mathsf{\Lambda }\left(\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right),\xi \left({\vartheta }_1,{\vartheta }_2,{\vartheta }_3\right)\right)},\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3,\xi \left({\theta }_3,{\theta }_1,{\theta }_2\right)\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1,{\xi }^{*}{\vartheta }_1\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3,{\xi }^{*}{\vartheta }_3\right)+\mathsf{\Lambda }\left(\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right),\xi \left({\vartheta }_1,{\vartheta }_2,{\vartheta }_3\right)\right)},\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\vartheta }_1,\xi \left({\vartheta }_1,{\vartheta }_2,{\vartheta }_3\right)\right)\mathsf{\Lambda }\left({\xi }^{*}{\vartheta }_2,\xi \left({\vartheta }_2,{\vartheta }_3,{\vartheta }_1\right)\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1,{\xi }^{*}{\vartheta }_1\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2,{\xi }^{*}{\vartheta }_2\right)+\mathsf{\Lambda }\left(\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right),\xi \left({\vartheta }_1,{\vartheta }_2,{\vartheta }_3\right)\right)},\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\vartheta }_1,\xi \left({\vartheta }_1,{\vartheta }_2,{\vartheta }_3\right)\right)\mathsf{\Lambda }\left({\xi }^{*}{\vartheta }_3,\xi \left({\vartheta }_3,{\vartheta }_1,{\vartheta }_2\right)\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1,{\xi }^{*}{\vartheta }_1\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2,{\xi }^{*}{\vartheta }_2\right)+\mathsf{\Lambda }\left(\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right),\xi \left({\vartheta }_1,{\vartheta }_2,{\vartheta }_3\right)\right)}\end{array}\right\}.\hfill \end{array}$$

Theorem 1.

Let $\left(\daleth ,⪯\right)$ be a POS and $\mathsf{\Lambda }$ be a partial metric so that $(\daleth ,\mathsf{\Lambda })$ is a PMS. Assume that $\xi :{\daleth }^3\to \daleth$ and ${\xi }^{*}:\daleth \to \daleth$ are mappings which satisfy

(${⊳}_i$)

ξ verifies a $\left(\phi ,\rho ,\ell \right)-$contraction with respect to ${\xi }^{*};$

(${⊳}_{ii}$)

$\xi \left({\daleth }^3\right)\subseteq {\xi }^{*}\left(\daleth \right)$ and ${\xi }^{*}\left(\daleth \right)$ is a complete subspace of $\daleth ;$

(${⊳}_{iii}$)

ξ has a mixed ${\xi }^{*}-$monotone property;

(${⊳}_{iv}$)

(1) if non-decreasing sequences $\left\{{\theta }_1^n\right\}\to {\theta }_1$ and $\left\{{\theta }_3^n\right\}\to {\theta }_3,$ then ${\theta }_1^n⪯{\theta }_1$ and ${\theta }_3^n⪯{\theta }_3$ for all n;

(2) if a non-increasing sequence $\left\{{\theta }_2^n\right\}\to {\theta }_2,$ then ${\theta }_2⪯{\theta }_2^n,$ for all $n.$

If there are ${\theta }_1^0,{\theta }_2^0,{\theta }_3^0\in \daleth$ so that ${\xi }^{*}{\theta }_1^0⪯\xi \left({\theta }_1^0,{\theta }_2^0,{\theta }_3^0\right),$ ${\xi }^{*}{\theta }_2^0⪰\xi \left({\theta }_2^0,{\theta }_3^0,{\theta }_1^0\right)$ and ${\xi }^{*}{\theta }_3^0⪯\xi \left({\theta }_3^0,{\theta }_1^0,{\theta }_2^0\right),$ then ξ and ${\xi }^{*}$ have a TCP in ${\daleth }^3.$

Proof. 

Assume that ${\theta }_1^0,{\theta }_2^0,{\theta }_3^0\in {\daleth }^3$ such that ${\xi }^{*}{\theta }_1^0⪯\xi \left({\theta }_1^0,{\theta }_2^0,{\theta }_3^0\right),$ ${\xi }^{*}{\theta }_2^0⪰\xi \left({\theta }_2^0,{\theta }_3^0,{\theta }_1^0\right)$ and ${\xi }^{*}{\theta }_3^0⪯\xi \left({\theta }_3^0,{\theta }_1^0,{\theta }_2^0\right).$ Because $\xi \left({\daleth }^3\right)\subseteq {\xi }^{*}\left(\daleth \right),$ we select ${\theta }_1^1,{\theta }_2^1,{\theta }_3^1\in \daleth$ so that

$${\xi }^{*}{\theta }_1^0⪯\xi \left({\theta }_1^0,{\theta }_2^0,{\theta }_3^0\right)={\xi }^{*}{\theta }_1^1,{\xi }^{*}{\theta }_2^0⪰\xi \left({\theta }_2^0,{\theta }_3^0,{\theta }_1^0\right)={\xi }^{*}{\theta }_2^1\mathrm{and}{\xi }^{*}{\theta }_3^0⪯\xi \left({\theta }_3^0,{\theta }_1^0,{\theta }_2^0\right)={\xi }^{*}{\theta }_3^1,$$

and select ${\theta }_1^2,{\theta }_2^2,{\theta }_3^2\in \daleth$ so that

$$\xi \left({\theta }_1^1,{\theta }_2^1,{\theta }_3^1\right)={\xi }^{*}{\theta }_1^2,\xi \left({\theta }_2^1,{\theta }_3^1,{\theta }_1^1\right)={\xi }^{*}{\theta }_2^2\mathrm{and}\xi \left({\theta }_3^1,{\theta }_1^1,{\theta }_2^1\right)={\xi }^{*}{\theta }_3^2.$$

Because $\xi$ has the mixed ${\xi }^{*}$-monotone property, we obtain

$${\xi }^{*}{\theta }_1^0⪯{\xi }^{*}{\theta }_1^1⪯{\xi }^{*}{\theta }_1^2,{\xi }^{*}{\theta }_2^0⪰{\xi }^{*}{\theta }_2^1⪰{\xi }^{*}{\theta }_2^2\mathrm{and}{\xi }^{*}{\theta }_3^0⪯{\xi }^{*}{\theta }_3^1⪯{\xi }^{*}{\theta }_3^3.$$

Continuing with the same scenario, we build the sequences $\left\{{\theta }_1^{\omega }\right\},$ $\left\{{\theta }_2^{\omega }\right\}$ and $\left\{{\theta }_3^{\omega }\right\}$ in ℸ so that

$${\xi }^{*}{\theta }_1^{\omega +1}=\xi \left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega }\right),{\xi }^{*}{\theta }_2^{\omega +1}=\xi \left({\theta }_2^{\omega },{\theta }_3^{\omega },{\theta }_1^{\omega }\right)\mathrm{and}{\xi }^{*}{\theta }_3^{\omega +1}=\xi \left({\theta }_3^{\omega },{\theta }_1^{\omega },{\theta }_2^{\omega }\right),\omega =0,1,2,\dots$$

with

$$\begin{array}{ccc}\hfill {\xi }^{*}{\theta }_1^0& ⪯& {\xi }^{*}{\theta }_1^1⪯{\xi }^{*}{\theta }_1^2⪯\dots ,\hfill \\ \hfill {\xi }^{*}{\theta }_2^0& ⪰& {\xi }^{*}{\theta }_2^1⪰{\xi }^{*}{\theta }_2^2⪰\dots ,\hfill \\ \hfill \mathrm{and}{\xi }^{*}{\theta }_3^0& ⪯& {\xi }^{*}{\theta }_3^1⪯{\xi }^{*}{\theta }_3^3⪯\dots .\hfill \end{array}$$

Now, if ${\xi }^{*}{\theta }_1^{\omega }={\xi }^{*}{\theta }_1^{\omega +1},$ ${\xi }^{*}{\theta }_2^{\omega }={\xi }^{*}{\theta }_2^{\omega +1}$ and ${\xi }^{*}{\theta }_3^{\omega }={\xi }^{*}{\theta }_3^{\omega +1},$ for some $\omega ,$ then a trio $({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega })$ is a TCP in ${\daleth }^3$ and nothing proof. So, we assume that ${\xi }^{*}{\theta }_1^{\omega }={\xi }^{*}{\theta }_1^{\omega +1}$ or ${\xi }^{*}{\theta }_2^{\omega }={\xi }^{*}{\theta }_2^{\omega +1}$ or ${\xi }^{*}{\theta }_3^{\omega }={\xi }^{*}{\theta }_3^{\omega +1},$ for all $\omega .$ Because ${\xi }^{*}{\theta }_1^{\omega }⪯{\xi }^{*}{\theta }_1^{\omega +1},$ ${\xi }^{*}{\theta }_2^{\omega }⪰{\xi }^{*}{\theta }_2^{\omega +1}$ and ${\xi }^{*}{\theta }_3^{\omega }⪯{\xi }^{*}{\theta }_3^{\omega +1},$ from assertion $\left(a_4\right)$, we can write

$$\begin{array}{ccc}\hfill \phi \left(\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right)\right)& =& \phi \left(\mathsf{\Lambda }\left(\xi \left({\theta }_1^{\omega -1},{\theta }_2^{\omega -1},{\theta }_3^{\omega -1}\right),\xi \left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega }\right)\right)\right)\hfill \\ & \le & \rho \left(\beth \left({\theta }_1^{\omega -1},{\theta }_2^{\omega -1},{\theta }_3^{\omega -1},{\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega }\right)\right)-\ell \left(\beth \left({\theta }_1^{\omega -1},{\theta }_2^{\omega -1},{\theta }_3^{\omega -1},{\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega }\right)\right),\hfill \end{array}$$

where

$$\begin{array}{ccc}& & \beth \left({\theta }_1^{\omega -1},{\theta }_2^{\omega -1},{\theta }_3^{\omega -1},{\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega }\right)\hfill \\ & =& max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},{\xi }^{*}{\theta }_3^{\omega }\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},\xi \left({\theta }_1^{\omega -1},{\theta }_2^{\omega -1},{\theta }_3^{\omega -1}\right)\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},\xi \left({\theta }_2^{\omega -1},{\theta }_3^{\omega -1},{\theta }_1^{\omega -1}\right)\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},\xi \left({\theta }_3^{\omega -1},{\theta }_1^{\omega -1},{\theta }_2^{\omega -1}\right)\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },\xi \left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega }\right)\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },\xi \left({\theta }_2^{\omega },{\theta }_3^{\omega },{\theta }_1^{\omega }\right)\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },\xi \left({\theta }_3^{\omega },{\theta }_1^{\omega },{\theta }_2^{\omega }\right)\right),\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},\xi \left({\theta }_1^{\omega -1},{\theta }_2^{\omega -1},{\theta }_3^{\omega -1}\right)\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},\xi \left({\theta }_2^{\omega -1},{\theta }_3^{\omega -1},{\theta }_1^{\omega -1}\right)\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right)+\mathsf{\Lambda }\left(\xi \left({\theta }_1^{\omega -1},{\theta }_2^{\omega -1},{\theta }_3^{\omega -1}\right),\xi \left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega }\right)\right)},\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},\xi \left({\theta }_1^{\omega -1},{\theta }_2^{\omega -1},{\theta }_3^{\omega -1}\right)\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},\xi \left({\theta }_3^{\omega -1},{\theta }_1^{\omega -1},{\theta }_2^{\omega -1}\right)\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},{\xi }^{*}{\theta }_3^{\omega }\right)+\mathsf{\Lambda }\left(\xi \left({\theta }_1^{\omega -1},{\theta }_2^{\omega -1},{\theta }_3^{\omega -1}\right),\xi \left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega }\right)\right)},\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },\xi \left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega }\right)\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },\xi \left({\theta }_2^{\omega },{\theta }_3^{\omega },{\theta }_1^{\omega }\right)\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right)+\mathsf{\Lambda }\left(\xi \left({\theta }_1^{\omega -1},{\theta }_2^{\omega -1},{\theta }_3^{\omega -1}\right),\xi \left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega }\right)\right)},\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },\xi \left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega }\right)\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },\xi \left({\theta }_3^{\omega },{\theta }_1^{\omega },{\theta }_2^{\omega }\right)\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right)+\mathsf{\Lambda }\left(\xi \left({\theta }_1^{\omega -1},{\theta }_2^{\omega -1},{\theta }_3^{\omega -1}\right),\xi \left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega }\right)\right)}\end{array}\right\}\hfill \\ & =& max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},{\xi }^{*}{\theta }_3^{\omega }\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},{\xi }^{*}{\theta }_3^{\omega }\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right),\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right)},\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},{\xi }^{*}{\theta }_3^{\omega }\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},{\xi }^{*}{\theta }_3^{\omega }\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right)},\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right)},\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},{\xi }^{*}{\theta }_3^{\omega }\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right)}\end{array}\right\}.\hfill \end{array}$$

However,

$$\begin{array}{ccc}\hfill \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right)}& \le & max\{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right)\},\hfill \\ \hfill \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},{\xi }^{*}{\theta }_3^{\omega }\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},{\xi }^{*}{\theta }_3^{\omega }\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right)}& \le & max\{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},{\xi }^{*}{\theta }_3^{\omega }\right)\},\hfill \\ \hfill \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right)}& \le & \mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right),\hfill \end{array}$$

and

$$\frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},{\xi }^{*}{\theta }_3^{\omega }\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right)}\le \mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right).$$

Then

$$\beth \left({\theta }_1^{\omega -1},{\theta }_2^{\omega -1},{\theta }_3^{\omega -1},{\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega }\right)=max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},{\xi }^{*}{\theta }_3^{\omega }\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right)\end{array}\right\}.$$

Hence

$$\begin{array}{ccc}\hfill \phi \left(\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right)\right)& \le & \rho \left(max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},{\xi }^{*}{\theta }_3^{\omega }\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right)\end{array}\right\}\right)\hfill \\ & & -\ell \left(max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},{\xi }^{*}{\theta }_3^{\omega }\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right)\end{array}\right\}\right).\hfill \end{array}$$

Analogously, for the second and third components, we can write

$$\begin{array}{ccc}\hfill \phi \left(\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right)\right)& \le & \rho \left(max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},{\xi }^{*}{\theta }_3^{\omega }\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right)\end{array}\right\}\right)\hfill \\ & & -\ell \left(max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},{\xi }^{*}{\theta }_3^{\omega }\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right)\end{array}\right\}\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \phi \left(\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right)\right)& \le & \rho \left(max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},{\xi }^{*}{\theta }_3^{\omega }\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right)\end{array}\right\}\right)\hfill \\ & & -\ell \left(max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},{\xi }^{*}{\theta }_3^{\omega }\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right)\end{array}\right\}\right).\hfill \end{array}$$

Set ${\nabla }_{\omega }=max\{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right)\}.$ Let us consider for all $\omega \ge 1,$ ${\nabla }_{\omega }\ne 0.$ Moreover, let, if possible for some $\omega ,$ ${\nabla }_{\omega -1}<{\nabla }_{\omega }.$ Now

$$\begin{array}{ccc}\hfill \phi \left({\nabla }_{\omega }\right)& =& \phi \left(max\{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right)\}\right)\hfill \\ & =& max\{\phi \left(\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right)\right),\phi \left(\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right)\right),\phi \left(\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right)\right)\}\hfill \\ & \le & \rho \left(max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},{\xi }^{*}{\theta }_3^{\omega }\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right)\end{array}\right\}\right)\hfill \\ & & -\ell \left(max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega -1},{\xi }^{*}{\theta }_1^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega -1},{\xi }^{*}{\theta }_2^{\omega }\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega -1},{\xi }^{*}{\theta }_3^{\omega }\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right)\end{array}\right\}\right)\hfill \\ & =& \rho \left(max\{{\nabla }_{\omega -1},{\nabla }_{\omega }\}\right)-\ell \left(max\{{\nabla }_{\omega -1},{\nabla }_{\omega }\}\right)\hfill \\ & =& \rho \left({\nabla }_{\omega }\right)-\ell \left({\nabla }_{\omega }\right).\hfill \end{array}$$

It follows from $\left(a_2\right)$ and $\left(a_3\right)$ that ${\nabla }_{\omega }=0,$ a contradiction. Hence ${\nabla }_{\omega -1}\le {\nabla }_{\omega }.$ This proves that the sequence $\left\{{\nabla }_{\omega }\right\}$ is a non-increasing and must converge to a real number $\varpi$(say)$\ge 0.$ Furthermore,

$$\phi \left({\nabla }_{\omega }\right)\le \rho \left({\nabla }_{\omega -1}\right)-\ell \left({\nabla }_{\omega -1}\right).$$

Passing $\omega \to \infty ,$ we have

$$\phi \left(\varpi \right)\le \rho \left(\varpi \right)-\ell \left(\varpi \right).$$

Based on assertions $\left(a_2\right)$ and $\left(a_3\right),$ we have $\varpi =0$. Thus

$$\underset{\omega \to \infty }{lim}max\{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right)\}=0,$$

this leads to

$$\underset{\omega \to \infty }{lim}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right)=0,\underset{\omega \to \infty }{lim}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right)=0\mathrm{and}\underset{\omega \to \infty }{lim}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right)=0.$$

(1)

By $\left({\mathsf{\Lambda }}_2\right),$ one can write

$$\underset{\omega \to \infty }{lim}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega }\right)=0,\underset{\omega \to \infty }{lim}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega }\right)=0\mathrm{and}\underset{\omega \to \infty }{lim}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega }\right)=0.$$

(2)

It follows from (1), (2) and the definition of ${\mathsf{\Omega }}_{\mathsf{\Lambda }}$ that

$$\underset{\omega \to \infty }{lim}{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right)=0,\underset{\omega \to \infty }{lim}{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right)=0\mathrm{and}\underset{\omega \to \infty }{lim}{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right)=0.$$

(3)

Now, using the contradiction method, we’re going to prove that $\left\{{\xi }^{*}{\theta }_1^{\omega }\right\},$ $\left\{{\xi }^{*}{\theta }_2^{\omega }\right\}$ and $\left\{{\xi }^{*}{\theta }_3^{\omega }\right\}$ are Cauchy sequences. Assume that $\left\{{\xi }^{*}{\theta }_1^{\omega }\right\}$ or $\left\{{\xi }^{*}{\theta }_2^{\omega }\right\}$ or $\left\{{\xi }^{*}{\theta }_3^{\omega }\right\}$ is not Cauchy. This means ${\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{\upsilon },{\xi }^{*}{\theta }_1^{\omega }\right)\nrightarrow 0$ or ${\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{\upsilon },{\xi }^{*}{\theta }_2^{\omega }\right)\nrightarrow 0$ or ${\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{\upsilon },{\xi }^{*}{\theta }_3^{\omega }\right)\nrightarrow 0$ as $\omega ,\upsilon \to \infty .$ Consequently,

$$max\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{\upsilon },{\xi }^{*}{\theta }_1^{\omega }\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{\upsilon },{\xi }^{*}{\theta }_2^{\omega }\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{\upsilon },{\xi }^{*}{\theta }_3^{\omega }\right)\}\nrightarrow 0,\mathrm{as}\omega ,\upsilon \to \infty .$$

Then there is an $ϵ>0$ and monotone increasing sequences $\left\{{\upsilon }_j\right\}$ and $\left\{{\omega }_j\right\}$ so that ${\omega }_j>{\upsilon }_j>j,$

$$max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)\right\}\ge ϵ,$$

(4)

and

$$max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j-1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\omega }_j-1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\omega }_j-1}\right)\right\}<ϵ.$$

(5)

From (4) and (5), we obtain that

$$\begin{array}{ccc}\hfill ϵ& \le & max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)\right\}\hfill \\ & \le & max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j-1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\omega }_j-1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\omega }_j-1}\right)\right\}\hfill \\ & & +max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\omega }_j-1},{\xi }^{*}{\theta }_1^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\omega }_j-1},{\xi }^{*}{\theta }_2^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\omega }_j-1},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)\right\}\hfill \\ & <& ϵ+max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\omega }_j-1},{\xi }^{*}{\theta }_1^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\omega }_j-1},{\xi }^{*}{\theta }_2^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\omega }_j-1},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)\right\}.\hfill \end{array}$$

As $j\to \infty$ and applying (3), one sees that

$$\underset{j\to \infty }{lim}max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)\right\}=ϵ.$$

(6)

According to the definition of ${\mathsf{\Omega }}_{\mathsf{\Lambda }}$ and (2), one can obtain

$$\underset{j\to \infty }{lim}max\left\{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\omega }_j}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)\right\}=\frac{ϵ}{2}.$$

(7)

From (4), we find that

$$\begin{array}{ccc}\hfill ϵ& \le & max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)\right\}\hfill \\ & \le & max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\upsilon }_j-1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\upsilon }_j-1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\upsilon }_j-1}\right)\right\}\hfill \\ & & +max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\upsilon }_j-1},{\xi }^{*}{\theta }_2^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\upsilon }_j-1},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)\right\}\hfill \\ & \le & 2max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\upsilon }_j-1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\upsilon }_j-1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\upsilon }_j-1}\right)\right\}\hfill \\ & & +max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\upsilon }_j-1},{\xi }^{*}{\theta }_2^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\upsilon }_j-1},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)\right\}.\hfill \end{array}$$

(8)

Taking $j\to \infty$ and applying (3), (6) and (8), we have

$$\underset{j\to \infty }{lim}max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\upsilon }_j-1},{\xi }^{*}{\theta }_2^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\upsilon }_j-1},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)\right\}=ϵ.$$

(9)

Hence, we obtain

$$\underset{j\to \infty }{lim}max\left\{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\omega }_j}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\upsilon }_j-1},{\xi }^{*}{\theta }_2^{{\omega }_j}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\upsilon }_j-1},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)\right\}=\frac{ϵ}{2}.$$

(10)

From (5), we obtain that

$$\begin{array}{ccc}\hfill ϵ& \le & max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)\right\}\hfill \\ & \le & max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\upsilon }_j-1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\upsilon }_j-1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\upsilon }_j-1}\right)\right\}\hfill \\ & & +max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\upsilon }_j-1},{\xi }^{*}{\theta }_2^{{\omega }_j+1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\upsilon }_j-1},{\xi }^{*}{\theta }_3^{{\omega }_j+1}\right)\right\}\hfill \\ & & +max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\omega }_j+1},{\xi }^{*}{\theta }_1^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\omega }_j+1},{\xi }^{*}{\theta }_2^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\omega }_j+1},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)\right\}\hfill \\ & \le & 2max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\upsilon }_j-1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\upsilon }_j-1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\upsilon }_j-1}\right)\right\}\hfill \\ & & +max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)\right\}\hfill \\ & & +2max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\omega }_j+1},{\xi }^{*}{\theta }_1^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\omega }_j+1},{\xi }^{*}{\theta }_2^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\omega }_j+1},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)\right\}.\hfill \end{array}$$

(11)

Letting $j\to \infty$ in (11), using (3) and (9), we have

$$\underset{j\to \infty }{lim}max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\upsilon }_j-1},{\xi }^{*}{\theta }_2^{{\omega }_j+1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\upsilon }_j-1},{\xi }^{*}{\theta }_3^{{\omega }_j+1}\right)\right\}=ϵ.$$

Hence, we obtain

$$\underset{j\to \infty }{lim}max\left\{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\upsilon }_j-1},{\xi }^{*}{\theta }_2^{{\omega }_j+1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\upsilon }_j-1},{\xi }^{*}{\theta }_3^{{\omega }_j+1}\right)\right\}=\frac{ϵ}{2}.$$

Again, from (4), one can obtain

$$\begin{array}{ccc}\hfill ϵ& \le & max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)\right\}\hfill \\ & \le & max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\omega }_j+1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\omega }_j+1}\right)\right\}\hfill \\ & & +max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\omega }_j+1},{\xi }^{*}{\theta }_1^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\omega }_j+1},{\xi }^{*}{\theta }_2^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\omega }_j+1},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)\right\}\hfill \end{array}$$

Setting $j\to \infty$, from(2) and (3), one sees that

$$\begin{array}{ccc}\hfill ϵ& \le & {\displaystyle \underset{j\to \infty }{lim}}max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\omega }_j+1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\omega }_j+1}\right)\right\}+0\hfill \\ & \le & {\displaystyle \underset{j\to \infty }{lim}}max\left\{\begin{array}{c}2\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right)-\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\upsilon }_j}\right)-\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\omega }_j+1},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right),\\ 2\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\omega }_j+1}\right)-\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\upsilon }_j}\right)-\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\omega }_j+1},{\xi }^{*}{\theta }_2^{{\omega }_j+1}\right),\\ 2\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\omega }_j+1}\right)-\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\upsilon }_j}\right)-\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\omega }_j+1},{\xi }^{*}{\theta }_3^{{\omega }_j+1}\right)\end{array}\right\}\hfill \\ & =& 2{\displaystyle \underset{j\to \infty }{lim}}max\left\{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\omega }_j+1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\omega }_j+1}\right)\right\}.\hfill \end{array}$$

Thus

$$\frac{ϵ}{2}\le \underset{j\to \infty }{lim}max\left\{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\omega }_j+1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\omega }_j+1}\right)\right\}.$$

By the definition of $\phi ,$

$$\begin{array}{ccc}\hfill \phi \left(\frac{ϵ}{2}\right)& \le & {\displaystyle \underset{j\to \infty }{lim}}\phi \left(max\left\{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\omega }_j+1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\omega }_j+1}\right)\right\}\right)\hfill \\ & =& {\displaystyle \underset{j\to \infty }{lim}}max\left\{\phi \left(\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right)\right),\phi \left(\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\omega }_j+1}\right)\right),\phi \left(\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\omega }_j+1}\right)\right)\right\}.\hfill \end{array}$$

(12)

Consider

$$\begin{array}{ccc}\hfill \phi \left(\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right)\right)& =& \phi \left(\mathsf{\Lambda }\left(\xi \left({\theta }_1^{{\upsilon }_j-1},{\theta }_2^{{\upsilon }_j-1},{\theta }_3^{{\upsilon }_j-1}\right),\xi \left({\theta }_1^{{\omega }_j},{\theta }_2^{{\omega }_j},{\theta }_3^{{\omega }_j}\right)\right)\right)\hfill \\ & \le & \rho \left(\beth \left({\theta }_1^{{\upsilon }_j-1},{\theta }_2^{{\upsilon }_j-1},{\theta }_3^{{\upsilon }_j-1},{\theta }_1^{{\omega }_j},{\theta }_2^{{\omega }_j},{\theta }_3^{{\omega }_j}\right)\right)-\ell \left(\beth \left({\theta }_1^{{\upsilon }_j-1},{\theta }_2^{{\upsilon }_j-1},{\theta }_3^{{\upsilon }_j-1},{\theta }_1^{{\omega }_j},{\theta }_2^{{\omega }_j},{\theta }_3^{{\omega }_j}\right)\right)\hfill \\ & =& \rho \left(max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\omega }_j}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\upsilon }_j-1},{\xi }^{*}{\theta }_2^{{\omega }_j}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\upsilon }_j-1},{\xi }^{*}{\theta }_3^{{\omega }_j}\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\upsilon }_j}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\upsilon }_j-1},{\xi }^{*}{\theta }_2^{{\upsilon }_j}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\upsilon }_j-1},{\xi }^{*}{\theta }_3^{{\upsilon }_j}\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\omega }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\omega }_j},{\xi }^{*}{\theta }_2^{{\omega }_j+1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\omega }_j},{\xi }^{*}{\theta }_3^{{\omega }_j+1}\right),\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\upsilon }_j}\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\upsilon }_j-1},{\xi }^{*}{\theta }_2^{{\upsilon }_j}\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\omega }_j}\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\upsilon }_j-1},{\xi }^{*}{\theta }_2^{{\omega }_j}\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right)},\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\upsilon }_j}\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\upsilon }_j-1},{\xi }^{*}{\theta }_3^{{\upsilon }_j}\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\omega }_j}\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\upsilon }_j-1},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right)},\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\omega }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\omega }_j},{\xi }^{*}{\theta }_2^{{\omega }_j+1}\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\omega }_j}\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\upsilon }_j-1},{\xi }^{*}{\theta }_2^{{\omega }_j}\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right)},\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\omega }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\omega }_j},{\xi }^{*}{\theta }_3^{{\omega }_j+1}\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\omega }_j}\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\upsilon }_j-1},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right)}\end{array}\right\}\right)\hfill \\ & & -\ell \left(max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\omega }_j}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\upsilon }_j-1},{\xi }^{*}{\theta }_2^{{\omega }_j}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\upsilon }_j-1},{\xi }^{*}{\theta }_3^{{\omega }_j}\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\upsilon }_j}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\upsilon }_j-1},{\xi }^{*}{\theta }_2^{{\upsilon }_j}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\upsilon }_j-1},{\xi }^{*}{\theta }_3^{{\upsilon }_j}\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\omega }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\omega }_j},{\xi }^{*}{\theta }_2^{{\omega }_j+1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\omega }_j},{\xi }^{*}{\theta }_3^{{\omega }_j+1}\right),\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\upsilon }_j}\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\upsilon }_j-1},{\xi }^{*}{\theta }_2^{{\upsilon }_j}\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\omega }_j}\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\upsilon }_j-1},{\xi }^{*}{\theta }_2^{{\omega }_j}\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right)},\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\upsilon }_j}\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\upsilon }_j-1},{\xi }^{*}{\theta }_3^{{\upsilon }_j}\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\omega }_j}\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\upsilon }_j-1},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right)},\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\omega }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\omega }_j},{\xi }^{*}{\theta }_2^{{\omega }_j+1}\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\omega }_j}\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\upsilon }_j-1},{\xi }^{*}{\theta }_2^{{\omega }_j}\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right)},\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\omega }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\omega }_j},{\xi }^{*}{\theta }_3^{{\omega }_j+1}\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j-1},{\xi }^{*}{\theta }_1^{{\omega }_j}\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\upsilon }_j-1},{\xi }^{*}{\theta }_3^{{\omega }_j}\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right)}\end{array}\right\}\right).\hfill \end{array}$$

Passing $j\to \infty ,$ we can write

$$\underset{j\to \infty }{lim}\phi \left(\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{{\upsilon }_j},{\xi }^{*}{\theta }_1^{{\omega }_j+1}\right)\right)\le \rho \left(\frac{ϵ}{2}\right)-\ell \left(\frac{ϵ}{2}\right).$$

Analogously, we obtain

$$\underset{j\to \infty }{lim}\phi \left(\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{{\upsilon }_j},{\xi }^{*}{\theta }_2^{{\omega }_j+1}\right)\right)\le \rho \left(\frac{ϵ}{2}\right)-\ell \left(\frac{ϵ}{2}\right),$$

and

$$\underset{j\to \infty }{lim}\phi \left(\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{{\upsilon }_j},{\xi }^{*}{\theta }_3^{{\omega }_j+1}\right)\right)\le \rho \left(\frac{ϵ}{2}\right)-\ell \left(\frac{ϵ}{2}\right).$$

So, from (12), one can write

$$\rho \left(\frac{ϵ}{2}\right)\le \rho \left(\frac{ϵ}{2}\right)-\ell \left(\frac{ϵ}{2}\right).$$

It follows from hypotheses $\left(a_2\right)$ and $\left(a_3\right)$ that $\frac{ϵ}{2}=0,$ a contradiction. Hence, $\left\{{\xi }^{*}{\theta }_1^{\omega }\right\},$ $\left\{{\xi }^{*}{\theta }_2^{\omega }\right\}$ and $\left\{{\xi }^{*}{\theta }_3^{\omega }\right\}$ are Cauchy sequences in the metric space $\left(\daleth ,{\mathsf{\Omega }}_{\mathsf{\Lambda }}\right)$. Therefore, ${\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{\upsilon },{\xi }^{*}{\theta }_1^{\omega }\right)\to 0$, ${\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{\upsilon },{\xi }^{*}{\theta }_2^{\omega }\right)\to 0$ and ${\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{\upsilon },{\xi }^{*}{\theta }_3^{\omega }\right)\to 0$ as $\upsilon ,\omega \to \infty .$ Thus, by (2) and the definition of ${\mathsf{\Omega }}_{\mathsf{\Lambda }},$ we have

$$\underset{\upsilon ,\omega \to \infty }{lim}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\upsilon },{\xi }^{*}{\theta }_1^{\omega }\right)=0,\underset{\upsilon ,\omega \to \infty }{lim}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\upsilon },{\xi }^{*}{\theta }_2^{\omega }\right)=0\mathrm{and}\underset{\upsilon ,\omega \to \infty }{lim}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\upsilon },{\xi }^{*}{\theta }_3^{\omega }\right)=0.$$

(13)

Because ${\xi }^{*}\left(\daleth \right)$ is a complete subspace of ℸ and $\left\{{\xi }^{*}{\theta }_1^{\omega }\right\},$ $\left\{{\xi }^{*}{\theta }_2^{\omega }\right\}$ and $\left\{{\xi }^{*}{\theta }_3^{\omega }\right\}$ are Cauchy sequences in a complete metric space $\left({\xi }^{*}\left(\daleth \right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\right),$ then $\left\{{\xi }^{*}{\theta }_1^{\omega }\right\},$ $\left\{{\xi }^{*}{\theta }_2^{\omega }\right\}$ and $\left\{{\xi }^{*}{\theta }_3^{\omega }\right\}$ converges to some ${\varrho }_1,$ ${\varrho }_2$ and ${\varrho }_3$ in ${\xi }^{*}\left(\daleth \right)$, respectively. Thus,

$$\underset{\omega \to \infty }{lim}{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_1^{\omega },{\varrho }_1\right)=0,\underset{\omega \to \infty }{lim}{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_2^{\omega },{\varrho }_2\right)=0\mathrm{and}\underset{\omega \to \infty }{lim}{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\xi }^{*}{\theta }_3^{\omega },{\varrho }_3\right)=0.$$

for some ${\varrho }_1,{\varrho }_2,{\varrho }_3\in {\xi }^{*}\left(\daleth \right).$ Since ${\varrho }_1,{\varrho }_2,{\varrho }_3\in {\xi }^{*}\left(\daleth \right),$ there are ${\theta }_1,{\theta }_2,{\theta }_3\in \daleth$ so that ${\varrho }_1={\xi }^{*}{\theta }_1,$ ${\varrho }_2={\xi }^{*}{\theta }_2$ and ${\varrho }_3={\xi }^{*}{\theta }_3.$ Because $\left\{{\xi }^{*}{\theta }_1^{\omega }\right\},$ $\left\{{\xi }^{*}{\theta }_2^{\omega }\right\}$ and $\left\{{\xi }^{*}{\theta }_3^{\omega }\right\}$ are Cauchy sequences, then $\left\{{\xi }^{*}{\theta }_1^{\omega }\right\}\to {\varrho }_1,$ $\left\{{\xi }^{*}{\theta }_2^{\omega }\right\}\to {\varrho }_2,$ $\left\{{\xi }^{*}{\theta }_3^{\omega }\right\}\to {\varrho }_3,$ $\left\{{\xi }^{*}{\theta }_1^{\omega +1}\right\}\to {\varrho }_1,$ $\left\{{\xi }^{*}{\theta }_2^{\omega +1}\right\}\to {\varrho }_2$ and $\left\{{\xi }^{*}{\theta }_3^{\omega +1}\right\}\to {\varrho }_3.$ Applying Lemma 1 (ii) and (13), we obtain

$$\begin{array}{ccc}\hfill \mathsf{\Lambda }({\varrho }_1,{\varrho }_1)& =& {\displaystyle \underset{\omega \to \infty }{lim}}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\varrho }_1\right)=\mathsf{\Lambda }({\varrho }_2,{\varrho }_2)\hfill \\ & =& {\displaystyle \underset{\omega \to \infty }{lim}}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\varrho }_2\right)=\mathsf{\Lambda }({\varrho }_3,{\varrho }_3)={\displaystyle \underset{\omega \to \infty }{lim}}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\varrho }_3\right)=0.\hfill \end{array}$$

(14)

Next, we want to show that

$$\underset{\omega \to \infty }{lim}\mathsf{\Lambda }\left(\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right),{\xi }^{*}{\theta }_1^{\omega }\right)=\mathsf{\Lambda }\left(\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right),{\varrho }_1\right).$$

Based on definition of ${\mathsf{\Omega }}_{\mathsf{\Lambda }},$ we obtain

$$\begin{array}{ccc}\hfill {\mathsf{\Omega }}_{\mathsf{\Lambda }}\left(\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right),{\xi }^{*}{\theta }_1^{\omega }\right)& =& 2\mathsf{\Lambda }\left(\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right),{\xi }^{*}{\theta }_1^{\omega }\right)\hfill \\ & & -\mathsf{\Lambda }\left(\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right),\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right)-\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega }\right),\hfill \end{array}$$

Passing $\omega \to \infty$ and using (2),

$${\mathsf{\Omega }}_{\mathsf{\Lambda }}\left(\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right),{\varrho }_1\right)=2\underset{\omega \to \infty }{lim}\mathsf{\Lambda }\left(\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right),{\xi }^{*}{\theta }_1^{\omega }\right)-\mathsf{\Lambda }\left(\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right),\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right)-0$$

According to definition of ${\mathsf{\Omega }}_{\mathsf{\Lambda }}$ and (13), one can write

$$\underset{\omega \to \infty }{lim}\mathsf{\Lambda }\left(\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right),{\xi }^{*}{\theta }_1^{\omega }\right)=\mathsf{\Lambda }\left(\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right),{\varrho }_1\right),$$

Similarly

$$\underset{\omega \to \infty }{lim}\mathsf{\Lambda }\left(\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right),{\xi }^{*}{\theta }_2^{\omega }\right)=\mathsf{\Lambda }\left(\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right),{\varrho }_2\right),$$

and

$$\underset{\omega \to \infty }{lim}\mathsf{\Lambda }\left(\xi \left({\theta }_3,{\theta }_1,{\theta }_2\right),{\xi }^{*}{\theta }_3^{\omega }\right)=\mathsf{\Lambda }\left(\xi \left({\theta }_3,{\theta }_1,{\theta }_2\right),{\varrho }_3\right).$$

From $\left({\mathsf{\Lambda }}_4\right)$, we obtain

$$\mathsf{\Lambda }\left({\varrho }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right)\le \mathsf{\Lambda }\left({\varrho }_1,{\xi }^{*}{\theta }_1^{\omega +1}\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega +1},\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right).$$

Setting $\omega \to \infty ,$ we have

$$\mathsf{\Lambda }\left({\varrho }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right)\le 0+\underset{\omega \to \infty }{lim}\mathsf{\Lambda }\left(\xi \left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega }\right),\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right).$$

(15)

From $\left(a_1\right),$ $\left(a_2\right)$ and (15), one can obtain

$$\begin{array}{ccc}\hfill \phi \left(\mathsf{\Lambda }\left({\varrho }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right)\right)& \le & {\displaystyle \underset{\omega \to \infty }{lim}}\phi \left(\mathsf{\Lambda }\left(\xi \left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega }\right),\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right)\right)\hfill \\ & \le & {\displaystyle \underset{\omega \to \infty }{lim}}\left[\rho \left(\beth \left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega },{\theta }_1,{\theta }_2,{\theta }_3\right)\right)-\ell \left(\beth \left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega },{\theta }_1,{\theta }_2,{\theta }_3\right)\right)\right],\hfill \end{array}$$

where

$$\begin{array}{ccc}& & \beth \left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega },{\theta }_1,{\theta }_2,{\theta }_3\right)\hfill \\ & =& max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\varrho }_1\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\varrho }_2\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\varrho }_3\right),\\ \mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right),\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right),\\ \mathsf{\Lambda }\left({\varrho }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right),\mathsf{\Lambda }\left({\varrho }_2,\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)\right),\mathsf{\Lambda }\left({\varrho }_3,\xi \left({\theta }_3,{\theta }_1,{\theta }_2\right)\right),\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\xi }^{*}{\theta }_2^{\omega +1}\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\varrho }_1\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\varrho }_2\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega +1},\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right)},\\ \frac{\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\xi }^{*}{\theta }_1^{\omega +1}\right)\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\xi }^{*}{\theta }_3^{\omega +1}\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\varrho }_1\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\varrho }_3\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega +1},\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right)},\\ \frac{\mathsf{\Lambda }\left({\varrho }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right)\mathsf{\Lambda }\left({\varrho }_2,\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\varrho }_1\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_2^{\omega },{\varrho }_2\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega +1},\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right)},\\ \frac{\mathsf{\Lambda }\left({\varrho }_2,\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)\right)\mathsf{\Lambda }\left({\varrho }_3,\xi \left({\theta }_3,{\theta }_1,{\theta }_2\right)\right)}{1+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega },{\varrho }_1\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_3^{\omega },{\varrho }_3\right)+\mathsf{\Lambda }\left({\xi }^{*}{\theta }_1^{\omega +1},\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right)}\end{array}\right\}\hfill \\ & \to & max\{\mathsf{\Lambda }\left({\varrho }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right),\mathsf{\Lambda }\left({\varrho }_2,\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)\right),\mathsf{\Lambda }\left({\varrho }_3,\xi \left({\theta }_3,{\theta }_1,{\theta }_2\right)\right)\}\mathrm{as}\omega \to \infty .\hfill \end{array}$$

Hence

$$\begin{array}{ccc}\hfill \phi \left(\mathsf{\Lambda }\left({\varrho }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right)\right)& \le & \rho \left(max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\varrho }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right),\\ \mathsf{\Lambda }\left({\varrho }_2,\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)\right),\\ \mathsf{\Lambda }\left({\varrho }_3,\xi \left({\theta }_3,{\theta }_1,{\theta }_2\right)\right)\end{array}\right\}\right)\hfill \\ & & -\ell \left(max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\varrho }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right),\\ \mathsf{\Lambda }\left({\varrho }_2,\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)\right),\\ \mathsf{\Lambda }\left({\varrho }_3,\xi \left({\theta }_3,{\theta }_1,{\theta }_2\right)\right)\end{array}\right\}\right),\hfill \end{array}$$

Analogously,

$$\begin{array}{ccc}\hfill \phi \left(\mathsf{\Lambda }\left({\varrho }_2,\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)\right)\right)& \le & \rho \left(max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\varrho }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right),\\ \mathsf{\Lambda }\left({\varrho }_2,\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)\right),\\ \mathsf{\Lambda }\left({\varrho }_3,\xi \left({\theta }_3,{\theta }_1,{\theta }_2\right)\right)\end{array}\right\}\right)\hfill \\ & & -\ell \left(max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\varrho }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right),\\ \mathsf{\Lambda }\left({\varrho }_2,\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)\right),\\ \mathsf{\Lambda }\left({\varrho }_3,\xi \left({\theta }_3,{\theta }_1,{\theta }_2\right)\right)\end{array}\right\}\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \phi \left(\mathsf{\Lambda }\left({\varrho }_3,\xi \left({\theta }_3,{\theta }_2,{\theta }_1\right)\right)\right)& \le & \rho \left(max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\varrho }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right),\\ \mathsf{\Lambda }\left({\varrho }_2,\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)\right),\\ \mathsf{\Lambda }\left({\varrho }_3,\xi \left({\theta }_3,{\theta }_1,{\theta }_2\right)\right)\end{array}\right\}\right)\hfill \\ & & -\ell \left(max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\varrho }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right),\\ \mathsf{\Lambda }\left({\varrho }_2,\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)\right),\\ \mathsf{\Lambda }\left({\varrho }_3,\xi \left({\theta }_3,{\theta }_1,{\theta }_2\right)\right)\end{array}\right\}\right).\hfill \end{array}$$

Therefore

$$\begin{array}{ccc}\hfill \phi \left(max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\varrho }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right),\\ \mathsf{\Lambda }\left({\varrho }_2,\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)\right),\\ \mathsf{\Lambda }\left({\varrho }_3,\xi \left({\theta }_3,{\theta }_2,{\theta }_1\right)\right)\end{array}\right\}\right)& =& max\left\{\begin{array}{c}\phi \left(\mathsf{\Lambda }\left({\varrho }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right)\right),\\ \phi \left(\mathsf{\Lambda }\left({\varrho }_2,\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)\right)\right),\\ \phi \left(\mathsf{\Lambda }\left({\varrho }_3,\xi \left({\theta }_3,{\theta }_2,{\theta }_1\right)\right)\right)\end{array}\right\}\hfill \\ & \le & \rho \left(max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\varrho }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right),\\ \mathsf{\Lambda }\left({\varrho }_2,\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)\right),\\ \mathsf{\Lambda }\left({\varrho }_3,\xi \left({\theta }_3,{\theta }_1,{\theta }_2\right)\right)\end{array}\right\}\right)\hfill \\ & & -\ell \left(max\left\{\begin{array}{c}\mathsf{\Lambda }\left({\varrho }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right),\\ \mathsf{\Lambda }\left({\varrho }_2,\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)\right),\\ \mathsf{\Lambda }\left({\varrho }_3,\xi \left({\theta }_3,{\theta }_1,{\theta }_2\right)\right)\end{array}\right\}\right).\hfill \end{array}$$

This implies that

$$max\left\{\mathsf{\Lambda }\left({\varrho }_1,\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\right),\mathsf{\Lambda }\left({\varrho }_2,\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)\right),\mathsf{\Lambda }\left({\varrho }_3,\xi \left({\theta }_3,{\theta }_1,{\theta }_2\right)\right)\right\}=0.$$

So ${\varrho }_1=\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right),$ ${\varrho }_2=\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)$ and ${\varrho }_3=\xi \left({\theta }_3,{\theta }_1,{\theta }_2\right).$ This leads to $\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)={\xi }^{*}{\theta }_1={\varrho }_1,$ $\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)={\xi }^{*}{\theta }_2={\varrho }_2$ and $\xi \left({\theta }_3,{\theta }_1,{\theta }_2\right)={\xi }^{*}{\theta }_3={\varrho }_3.$ Therefore, $\xi$ and ${\xi }^{*}$ have a coincidence point in ${\daleth }^3.$ □

The following theorem gives the uniqueness of a TFP:

Theorem 2.

Adding to hypotheses of Theorem 1 the following hypothesis:

Let for each $\left({\theta }_1,{\theta }_2,{\theta }_3\right),\left({\vartheta }_1,{\vartheta }_2,{\vartheta }_3\right)\in {\daleth }^3$ there is a trio $\left({\varrho }_1,{\varrho }_2,{\varrho }_3\right)\in {\daleth }^3$ so that a trio ($\xi \left({\varrho }_1,{\varrho }_2,{\varrho }_3\right),\xi \left({\varrho }_2,{\varrho }_3,{\varrho }_1\right)$, $\xi \left({\varrho }_3,{\varrho }_1,{\varrho }_2\right)$) is comparable to ($\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right),\xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)$, $\xi \left({\theta }_3,{\theta }_1,{\theta }_2\right)$) and $\left(\xi \left({\vartheta }_1,{\vartheta }_2,{\vartheta }_3\right),\xi \left({\vartheta }_2,{\vartheta }_3,{\vartheta }_1\right),\xi \left({\vartheta }_3,{\vartheta }_1,{\vartheta }_2\right)\right).$ If $\left({\theta }_1,{\theta }_2,{\theta }_3\right)$ and $\left({\vartheta }_1,{\vartheta }_2,{\vartheta }_3\right)$ are TCPs of ξ and ${\xi }^{*}$ then

$$\begin{array}{ccc}\hfill \xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)& =& {\xi }^{*}{\theta }_1={\xi }^{*}{\vartheta }_1=\xi \left({\vartheta }_1,{\vartheta }_2,{\vartheta }_3\right),\hfill \\ \hfill \xi \left({\theta }_2,{\theta }_3,{\theta }_1\right)& =& {\xi }^{*}{\theta }_2={\xi }^{*}{\vartheta }_2=\xi \left({\vartheta }_2,{\vartheta }_3,{\vartheta }_1\right),\hfill \\ \hfill and\xi \left({\theta }_3,{\theta }_1,{\theta }_2\right)& =& {\xi }^{*}{\theta }_3={\xi }^{*}{\vartheta }_3=\xi \left({\vartheta }_3,{\vartheta }_1,{\vartheta }_2\right).\hfill \end{array}$$

Furthermore, if ξ and ${\xi }^{*}$ are $w-$compatible, then there is a unique common TFP of ξ and ${\xi }^{*}$ in ${\daleth }^3.$

Proof. 

The proof follows immediately from Theorem 1 and the concept of comparability. □

The result below follows from Theorem 1 and it is important in the next section.

Corollary 1.

Let $\left(\daleth ,⪯\right)$ be a POS and Λ be a partial metric so that $(\daleth ,\mathsf{\Lambda })$ is a PMS. Assume that $\xi :{\daleth }^3\to \daleth$ is a mapping so that

$$\begin{array}{ccc}\hfill \phi \left(\mathsf{\Lambda }\left(\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right),\xi \left({\vartheta }_1,{\vartheta }_2,{\vartheta }_3\right)\right)\right)& \le & \rho \left(max\left\{\mathsf{\Lambda }\left({\theta }_1,{\vartheta }_1\right),\mathsf{\Lambda }\left({\theta }_2,{\vartheta }_2\right),\mathsf{\Lambda }\left({\theta }_3,{\vartheta }_3\right)\right\}\right)\hfill \\ & & -\ell \left(max\left\{\mathsf{\Lambda }\left({\theta }_1,{\vartheta }_1\right),\mathsf{\Lambda }\left({\theta }_2,{\vartheta }_2\right),\mathsf{\Lambda }\left({\theta }_3,{\vartheta }_3\right)\right\}\right),\hfill \end{array}$$

(16)

for all ${\theta }_1,{\theta }_2,{\theta }_3,{\vartheta }_1,{\vartheta }_2,{\vartheta }_3\in \daleth$ with ${\theta }_1⪯{\vartheta }_1,$ ${\theta }_2⪰{\vartheta }_2$ and ${\theta }_3⪯{\vartheta }_3,$ where $\phi ,\rho$ and ℓ are described in Definition 11 and

(i) If non-decreasing sequences $\left\{{\theta }_1^n\right\}\to {\theta }_1$ and $\left\{{\theta }_3^n\right\}\to {\theta }_3,$ then ${\theta }_1^n⪯{\theta }_1$ and ${\theta }_3^n⪯{\theta }_3$ for all n;

(ii) If a non-increasing sequence $\left\{{\theta }_2^n\right\}\to {\theta }_2,$ then ${\theta }_2⪯{\theta }_2^n,$ for all $n.$

If there are ${\theta }_1^0,{\theta }_2^0,{\theta }_3^0\in \daleth$ so that ${\theta }_1^0⪯\xi \left({\theta }_1^0,{\theta }_2^0,{\theta }_3^0\right),$ ${\theta }_2^0⪰\xi \left({\theta }_2^0,{\theta }_3^0,{\theta }_1^0\right)$ and ${\theta }_3^0⪯\xi \left({\theta }_3^0,{\theta }_1^0,{\theta }_2^0\right),$ then ξ has a TCP in ${\daleth }^3.$

Example 3.

Suppose that $\daleth =[0,1]$. Describe a partially ordered ⪯ on ℸ as

$${\theta }_1⪯{\theta }_2⟺{\theta }_1\le {\theta }_2.$$

Define the mapping $\xi :{\daleth }^3\to \daleth$ by $\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)=\frac{{\theta }_1^2+{\theta }_2^2+{\theta }_3^2}{8({\theta }_1+{\theta }_2+{\theta }_3+1)}$ and $\mathsf{\Lambda }:\daleth \times \daleth \to [0,\infty )$ by $\mathsf{\Lambda }\left({\theta }_1,{\theta }_2\right)=max\{{\theta }_1,{\theta }_2\}.$ It is clear that $(\daleth ,\mathsf{\Lambda })$ is a PMS. Define $\phi ,\rho ,\ell :[0,\infty )\to [0,\infty )$ by $\phi \left(\nu \right)=\nu ,$ $\rho \left(\nu \right)=\frac{5\nu }{8}$ and $\ell \left(\nu \right)=\frac{\nu }{4}.$

Consider

$$\begin{array}{ccc}& & \mathsf{\Lambda }\left(\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right),\xi \left({\vartheta }_1,{\vartheta }_2,{\vartheta }_3\right)\right)\hfill \\ & =& max\left\{\frac{{\theta }_1^2+{\theta }_2^2+{\theta }_3^2}{8({\theta }_1+{\theta }_2+{\theta }_3+1)},\frac{{\vartheta }_1^2+{\vartheta }_2^2+{\vartheta }_3^2}{8({\vartheta }_1+{\vartheta }_2+{\vartheta }_3+1)}\right\}\hfill \\ & =& \frac{1}{8}\left(max\left\{\frac{{\theta }_1^2}{{\theta }_1+{\theta }_2+{\theta }_3+1},\frac{{\vartheta }_1^2}{{\vartheta }_1+{\vartheta }_2+{\vartheta }_3+1}\right\}\right.\hfill \\ & & +max\left\{\frac{{\theta }_2^2}{{\theta }_1+{\theta }_2+{\theta }_3+1},\frac{{\vartheta }_2^2}{{\vartheta }_1+{\vartheta }_2+{\vartheta }_3+1}\right\}\hfill \\ & & \left.+max\left\{\frac{{\theta }_3^2}{{\theta }_1+{\theta }_2+{\theta }_3+1},\frac{{\vartheta }_3^2}{{\vartheta }_1+{\vartheta }_2+{\vartheta }_3+1}\right\}\right)\hfill \\ & \le & \frac{1}{8}\left(max\left\{\frac{{\theta }_1^2}{{\theta }_1+1},\frac{{\vartheta }_1^2}{{\vartheta }_1+1}\right\}+max\left\{\frac{{\theta }_2^2}{{\theta }_2+1},\frac{{\vartheta }_2^2}{{\vartheta }_2+1}\right\}+max\left\{\frac{{\theta }_3^2}{{\theta }_3+1},\frac{{\vartheta }_3^2}{{\vartheta }_3+1}\right\}\right)\hfill \\ & \le & \frac{1}{8}\left(max\left\{\frac{{\theta }_1}{{\theta }_1+1},\frac{{\vartheta }_1}{{\vartheta }_1+1}\right\}+max\left\{\frac{{\theta }_2}{{\theta }_2+1},\frac{{\vartheta }_2}{{\vartheta }_2+1}\right\}+max\left\{\frac{{\theta }_3}{{\theta }_3+1},\frac{{\vartheta }_3}{{\vartheta }_3+1}\right\}\right)\hfill \\ & \le & \frac{1}{8}\left(max\left\{{\theta }_1,{\vartheta }_1\right\}+max\left\{{\theta }_2,{\vartheta }_2\right\}+max\left\{{\theta }_3,{\vartheta }_3\right\}\right)\hfill \\ & =& \frac{1}{8}\left(\mathsf{\Lambda }\left({\theta }_1,{\vartheta }_1\right)+\mathsf{\Lambda }\left({\theta }_2,{\vartheta }_2\right)+\mathsf{\Lambda }\left({\theta }_3,{\vartheta }_3\right)\right)\hfill \\ & \le & \frac{3}{8}max\left\{\mathsf{\Lambda }\left({\theta }_1,{\vartheta }_1\right),\mathsf{\Lambda }\left({\theta }_2,{\vartheta }_2\right),\mathsf{\Lambda }\left({\theta }_3,{\vartheta }_3\right)\right\}\hfill \\ & =& \frac{5}{8}max\left\{\mathsf{\Lambda }\left({\theta }_1,{\vartheta }_1\right),\mathsf{\Lambda }\left({\theta }_2,{\vartheta }_2\right),\mathsf{\Lambda }\left({\theta }_3,{\vartheta }_3\right)\right\}-\frac{1}{4}max\left\{\mathsf{\Lambda }\left({\theta }_1,{\vartheta }_1\right),\mathsf{\Lambda }\left({\theta }_2,{\vartheta }_2\right),\mathsf{\Lambda }\left({\theta }_3,{\vartheta }_3\right)\right\}\hfill \\ & =& \rho \left(max\left\{\mathsf{\Lambda }\left({\theta }_1,{\vartheta }_1\right),\mathsf{\Lambda }\left({\theta }_2,{\vartheta }_2\right),\mathsf{\Lambda }\left({\theta }_3,{\vartheta }_3\right)\right\}\right)-\ell \left(max\left\{\mathsf{\Lambda }\left({\theta }_1,{\vartheta }_1\right),\mathsf{\Lambda }\left({\theta }_2,{\vartheta }_2\right),\mathsf{\Lambda }\left({\theta }_3,{\vartheta }_3\right)\right\}\right).\hfill \end{array}$$

Therefore, all assertions of Corollary 1 are fulfilled and $(0,0,0)$ is a unique TFP of ξ on ${\daleth }^3.$

4. Application to IVPs

In the setting of PMSs, this part is devoted to discussing the existence of a uniqueness solution to the IVP below:

$$\left\{\begin{array}{c}{\theta }_1^{\prime }\left(\nu \right)=\mathsf{\Xi }\left(\nu ,{\theta }_1\left(\nu \right),{\theta }_1\left(\nu \right),{\theta }_1\left(\nu \right)\right),\nu \in \chi =[0,1],\\ {\theta }_1\left(0\right)={\theta }_1^0,\end{array}\right.$$

(17)

where $\mathsf{\Xi }:\chi \times [\frac{{\theta }_1^0}{5},\infty )\times [\frac{{\theta }_1^0}{5},\infty )\times [\frac{{\theta }_1^0}{5},\infty )\to [\frac{{\theta }_1^0}{5},\infty )$ is a continuous function for ${\theta }_1^0\in \mathbb{R}.$

Now, we state and prove our main theorem in this part.

Theorem 3.

Assume that the IVP (17) with $\mathsf{\Xi }\in C\left(\chi \times [\frac{{\theta }_1^0}{5},\infty )\times [\frac{{\theta }_1^0}{5},\infty )\times [\frac{{\theta }_1^0}{5},\infty )\right)$ and

$$\underset{0}{\overset{\nu }{\int }}\mathsf{\Xi }\left(\gamma ,{\theta }_1\left(\gamma \right),{\theta }_2\left(\gamma \right),{\theta }_3\left(\gamma \right)\right)d\gamma \le max\left\{\begin{array}{c}\frac{1}{5}\underset{0}{\overset{\nu }{\int }}\mathsf{\Xi }\left(\gamma ,{\theta }_1\left(\gamma \right),{\theta }_1\left(\gamma \right),{\theta }_1\left(\gamma \right)\right)d\gamma -\frac{16{\theta }_1^0}{25},\\ \frac{1}{5}\underset{0}{\overset{\nu }{\int }}\mathsf{\Xi }\left(\gamma ,{\theta }_2\left(\gamma \right),{\theta }_2\left(\gamma \right),{\theta }_2\left(\gamma \right)\right)d\gamma -\frac{16{\theta }_1^0}{25},\\ \frac{1}{5}\underset{0}{\overset{\nu }{\int }}\mathsf{\Xi }\left(\gamma ,{\theta }_3\left(\gamma \right),{\theta }_3\left(\gamma \right),{\theta }_3\left(\gamma \right)\right)d\gamma -\frac{16{\theta }_1^0}{25}\end{array}\right\}.$$

Then the IVP (17) has a unique solution in $C\left(\chi ,[\frac{{\theta }_1^0}{4},\infty )\right).$

Proof. 

The IVP (17) is equivalent to the following integral equation:

$${\theta }_1\left(\nu \right)={\theta }_1^0+\underset{0}{\overset{\nu }{\int }}\mathsf{\Xi }\left(\gamma ,{\theta }_1\left(\gamma \right),{\theta }_2\left(\gamma \right),{\theta }_3\left(\gamma \right)\right)d\gamma .$$

(18)

Assume that $\daleth =C\left(\chi ,[\frac{{\theta }_1^0}{5},\infty )\right)$ and $\mathsf{\Lambda }({\theta }_1,{\theta }_2)=max\left\{{\theta }_1-\frac{{\theta }_1^0}{5},{\theta }_2-\frac{{\theta }_1^0}{5},{\theta }_3-\frac{{\theta }_1^0}{5}\right\}$ for ${\theta }_1,{\theta }_2\in \daleth .$ Define $\phi ,\rho ,\ell :[0,\infty )\to [0,\infty )$ by $\phi \left(\nu \right)=\nu ,$ $\rho \left(\nu \right)=\frac{4\nu }{5}$ and $\ell \left(\nu \right)=\frac{3\nu }{5}.$ Describe the mapping $\xi :{\daleth }^3\to \daleth$ as

$$\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)\left(\nu \right)={\theta }_1^0+\underset{0}{\overset{\nu }{\int }}\mathsf{\Xi }\left(\gamma ,{\theta }_1\left(\gamma \right),{\theta }_2\left(\gamma \right),{\theta }_3\left(\gamma \right)\right)d\gamma .$$

Now

$$\begin{array}{ccc}& & \mathsf{\Lambda }\left(\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right),\xi \left({\vartheta }_1,{\vartheta }_2,{\vartheta }_3\right)\right)\hfill \\ & =& max\left\{\xi \left({\theta }_1,{\theta }_2,{\theta }_3\right)-\frac{{\theta }_1^0}{5},\xi \left({\vartheta }_1,{\vartheta }_2,{\vartheta }_3\right)-\frac{{\theta }_1^0}{5}\right\}\hfill \\ & =& max\left\{\frac{4{\theta }_1^0}{5}+\underset{0}{\overset{\nu }{\int }}\mathsf{\Xi }\left(\gamma ,{\theta }_1\left(\gamma \right),{\theta }_2\left(\gamma \right),{\theta }_3\left(\gamma \right)\right)d\gamma ,\frac{4{\theta }_1^0}{5}+\underset{0}{\overset{\nu }{\int }}\mathsf{\Xi }\left(\gamma ,{\vartheta }_1\left(\gamma \right),{\vartheta }_2\left(\gamma \right),{\vartheta }_3\left(\gamma \right)\right)d\gamma \right\}\hfill \\ & \le & max\left\{\begin{array}{c}\frac{4{\theta }_1^0}{5}+max\left\{\begin{array}{c}\frac{1}{5}\underset{0}{\overset{\nu }{\int }}\mathsf{\Xi }\left(\gamma ,{\theta }_1\left(\gamma \right),{\theta }_1\left(\gamma \right),{\theta }_1\left(\gamma \right)\right)d\gamma -\frac{16{\theta }_1^0}{25},\\ \frac{1}{5}\underset{0}{\overset{\nu }{\int }}\mathsf{\Xi }\left(\gamma ,{\theta }_2\left(\gamma \right),{\theta }_2\left(\gamma \right),{\theta }_2\left(\gamma \right)\right)d\gamma -\frac{16{\theta }_1^0}{25},\\ \frac{1}{5}\underset{0}{\overset{\nu }{\int }}\mathsf{\Xi }\left(\gamma ,{\theta }_3\left(\gamma \right),{\theta }_3\left(\gamma \right),{\theta }_3\left(\gamma \right)\right)d\gamma -\frac{16{\theta }_1^0}{25}\end{array}\right\},\\ \frac{4{\theta }_1^0}{5}+max\left\{\begin{array}{c}\frac{1}{5}\underset{0}{\overset{\nu }{\int }}\mathsf{\Xi }\left(\gamma ,{\vartheta }_1\left(\gamma \right),{\vartheta }_1\left(\gamma \right),{\vartheta }_1\left(\gamma \right)\right)d\gamma -\frac{16{\theta }_1^0}{25},\\ \frac{1}{5}\underset{0}{\overset{\nu }{\int }}\mathsf{\Xi }\left(\gamma ,{\vartheta }_2\left(\gamma \right),{\vartheta }_2\left(\gamma \right),{\vartheta }_2\left(\gamma \right)\right)d\gamma -\frac{16{\theta }_1^0}{25},\\ \frac{1}{5}\underset{0}{\overset{\nu }{\int }}\mathsf{\Xi }\left(\gamma ,{\vartheta }_3\left(\gamma \right),{\vartheta }_3\left(\gamma \right),{\vartheta }_3\left(\gamma \right)\right)d\gamma -\frac{16{\theta }_1^0}{25}\end{array}\right\}\end{array}\right\}\hfill \\ & =& max\left\{\begin{array}{c}max\left\{\frac{{\theta }_1\left(\nu \right)}{5}-\frac{{\theta }_1^0}{25},\frac{{\vartheta }_1\left(\nu \right)}{5}-\frac{{\theta }_1^0}{25}\right\},\\ max\left\{\frac{{\theta }_2\left(\nu \right)}{5}-\frac{{\theta }_1^0}{25},\frac{{\vartheta }_2\left(\nu \right)}{5}-\frac{{\theta }_1^0}{25}\right\},\\ max\left\{\frac{{\theta }_3\left(\nu \right)}{5}-\frac{{\theta }_1^0}{25},\frac{{\vartheta }_3\left(\nu \right)}{5}-\frac{{\theta }_1^0}{25}\right\}\end{array}\right\}\hfill \\ & =& \frac{1}{5}max\left\{\begin{array}{c}max\left\{{\theta }_1\left(\nu \right)-\frac{{\theta }_1^0}{5},{\vartheta }_1\left(\nu \right)-\frac{{\theta }_1^0}{5}\right\},\\ max\left\{{\theta }_2\left(\nu \right)-\frac{{\theta }_1^0}{5},{\vartheta }_2\left(\nu \right)-\frac{{\theta }_1^0}{5}\right\},\\ max\left\{{\theta }_3\left(\nu \right)-\frac{{\theta }_1^0}{5},{\vartheta }_3\left(\nu \right)-\frac{{\theta }_1^0}{5}\right\}\end{array}\right\}\hfill \\ & =& \frac{1}{5}max\left\{\mathsf{\Lambda }\left({\theta }_1,{\vartheta }_1\right),\mathsf{\Lambda }\left({\theta }_2,{\vartheta }_2\right),\mathsf{\Lambda }\left({\theta }_3,{\vartheta }_3\right)\right\}\hfill \\ & =& \frac{4}{5}max\left\{\mathsf{\Lambda }\left({\theta }_1,{\vartheta }_1\right),\mathsf{\Lambda }\left({\theta }_2,{\vartheta }_2\right),\mathsf{\Lambda }\left({\theta }_3,{\vartheta }_3\right)\right\}-\frac{3}{5}max\left\{\mathsf{\Lambda }\left({\theta }_1,{\vartheta }_1\right),\mathsf{\Lambda }\left({\theta }_2,{\vartheta }_2\right),\mathsf{\Lambda }\left({\theta }_3,{\vartheta }_3\right)\right\}\hfill \\ & =& \rho \left(max\left\{\mathsf{\Lambda }\left({\theta }_1,{\vartheta }_1\right),\mathsf{\Lambda }\left({\theta }_2,{\vartheta }_2\right),\mathsf{\Lambda }\left({\theta }_3,{\vartheta }_3\right)\right\}\right)-\ell \left(max\left\{\mathsf{\Lambda }\left({\theta }_1,{\vartheta }_1\right),\mathsf{\Lambda }\left({\theta }_2,{\vartheta }_2\right),\mathsf{\Lambda }\left({\theta }_3,{\vartheta }_3\right)\right\}\right).\hfill \end{array}$$

Therefore, $\xi$ verifies the stipulation (16) of Corollary 1. Thus, $\xi$ has a unique TFP $\left({\theta }_1,{\theta }_2,{\theta }_3\right)$ with ${\theta }_1={\theta }_2={\theta }_3$, which is a unique solution of the IVP (17). □

5. Application to a Homotopy

Here, we discuss a unique solution to homotopy theory.

Theorem 4.

Suppose that $\left(\daleth ,\mathsf{\Lambda }\right)$ is a complete PMS, A is an open subset of ℸ and $\overline{A}$ is a closed subset of ℸ so that $A\subseteq \overline{A}.$ Assume that $H:\overline{A}\times \overline{A}\times \overline{A}\times [0,1]\to \daleth$ is an operator satisfies the hypotheses below:

$\left({\aleph }_1\right)$

${\theta }_1\ne H\left({\theta }_1,{\theta }_2,{\theta }_3,\sigma \right),$${\theta }_2\ne H\left({\theta }_2,{\theta }_3,{\theta }_1,\sigma \right)$ and ${\theta }_3\ne H\left({\theta }_3,{\theta }_1,{\theta }_2,\sigma \right),$ for each ${\theta }_1,{\theta }_2,{\theta }_3\in \partial A$ (here $\partial A$ refer to the boundary of A in ℸ) and $\sigma \in [0,1];$

$\left({\aleph }_2\right)$

 

$$\begin{array}{ccc}\hfill \phi \left(\mathsf{\Lambda }\left(H\left({\theta }_1,{\theta }_2,{\theta }_3,\sigma \right),H\left({\vartheta }_1,{\vartheta }_2,{\vartheta }_3,\sigma \right)\right)\right)& \le & \rho \left(max\left\{\mathsf{\Lambda }\left({\theta }_1,{\vartheta }_1\right),\mathsf{\Lambda }\left({\theta }_2,{\vartheta }_2\right),\mathsf{\Lambda }\left({\theta }_3,{\vartheta }_3\right)\right\}\right)\hfill \\ & & -\ell \left(max\left\{\mathsf{\Lambda }\left({\theta }_1,{\vartheta }_1\right),\mathsf{\Lambda }\left({\theta }_2,{\vartheta }_2\right),\mathsf{\Lambda }\left({\theta }_3,{\vartheta }_3\right)\right\}\right),\hfill \end{array}$$

for all ${\theta }_1,{\theta }_2,{\theta }_3,{\vartheta }_1,{\vartheta }_2,{\vartheta }_3\in \overline{A}$ and $\sigma \in [0,1],$ where $\phi ,\rho :[0,\infty )\to [0,\infty )$ are continuous and non-decreasing and $\ell :[0,\infty )\to [0,\infty )$ is an LSC with $\phi \left(\nu \right)-\rho \left(\nu \right)+\ell \left(\nu \right)>0,$ for $\nu >0;$

$\left({\aleph }_3\right)$

There exists $C\ge 0$ so that

$$\mathsf{\Lambda }\left(H\left({\theta }_1,{\theta }_2,{\theta }_3,\sigma \right),H\left({\theta }_1,{\theta }_2,{\theta }_3,{\sigma }^{*}\right)\right)\le C\left|\sigma -{\sigma }^{*}\right|,$$

for each ${\theta }_1,{\theta }_2,{\theta }_3\in \overline{A}$ and $\sigma ,{\sigma }^{*}\in [0,1].$

Then $H(.,0)$ has a TFP, whenever $H(.,1)$ has a TFP.

Proof. 

Define the set

$$\mho =\{\sigma \in [0,1]:\left({\theta }_1,{\theta }_2,{\theta }_3\right)=H\left({\theta }_1,{\theta }_2,{\theta }_3,\sigma \right)\mathrm{for}\mathrm{some}{\theta }_1,{\theta }_2,{\theta }_3\in A\}.$$

Because $H(.,0)$ has a TFP in $\nabla ,$ we obtain $0\in \mho ,$ this proves that $\mho \ne \varnothing .$

We claim that ℧ is open and closed in $[0,1]$ so by the connectedness, we obtain $\mho =[0,1].$ Consequently, $H(.,1)$ has a TFP in $\nabla .$ Initially, we shall show that ℧ is open and closed in $[0,1].$ To do this, assume ${\left\{{\sigma }^{\omega }\right\}}_{\omega =1}^{\infty }\subseteq \mho$ with ${\sigma }^{\omega }\to \sigma \in [0,1]$ as $\omega \to \infty .$ It must be shown that $\sigma \in \mho .$

Because ${\sigma }^{\omega }\in \mho ,$ for $\omega \ge 1,$ there are ${\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega }\in A$ with ${\varrho }^{\omega }=\left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega }\right)=H\left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega },{\sigma }^{\omega }\right).$ Consider

$$\begin{array}{ccc}\hfill \mathsf{\Lambda }\left({\theta }_1^{\omega },{\theta }_1^{\omega +1}\right)& =& \mathsf{\Lambda }\left(H\left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega },{\sigma }^{\omega }\right),H\left({\theta }_1^{\omega +1},{\theta }_2^{\omega +1},{\theta }_3^{\omega +1},{\sigma }^{\omega +1}\right)\right)\hfill \\ & \le & \mathsf{\Lambda }\left(H\left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega },{\sigma }^{\omega }\right),H\left({\theta }_1^{\omega +1},{\theta }_2^{\omega +1},{\theta }_3^{\omega +1},{\sigma }^{\omega }\right)\right)\hfill \\ & & +\mathsf{\Lambda }\left(H\left({\theta }_1^{\omega +1},{\theta }_2^{\omega +1},{\theta }_3^{\omega +1},{\sigma }^{\omega }\right),H\left({\theta }_1^{\omega +1},{\theta }_2^{\omega +1},{\theta }_3^{\omega +1},{\sigma }^{\omega +1}\right)\right)\hfill \\ & & -\mathsf{\Lambda }\left(H\left({\theta }_1^{\omega +1},{\theta }_2^{\omega +1},{\theta }_3^{\omega +1},{\sigma }^{\omega }\right),H\left({\theta }_1^{\omega +1},{\theta }_2^{\omega +1},{\theta }_3^{\omega +1},{\sigma }^{\omega }\right)\right)\hfill \\ & \le & \mathsf{\Lambda }\left(H\left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega },{\sigma }^{\omega }\right),H\left({\theta }_1^{\omega +1},{\theta }_2^{\omega +1},{\theta }_3^{\omega +1},{\sigma }^{\omega }\right)\right)+C\left|{\sigma }^{\omega }-{\sigma }^{\omega +1}\right|.\hfill \end{array}$$

As $\omega \to \infty$ in the above inequality, we have

$$\underset{\omega \to \infty }{lim}\mathsf{\Lambda }\left({\theta }_1^{\omega },{\theta }_1^{\omega +1}\right)\le \underset{\omega \to \infty }{lim}\mathsf{\Lambda }\left(H\left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega },{\sigma }^{\omega }\right),H\left({\theta }_1^{\omega +1},{\theta }_2^{\omega +1},{\theta }_3^{\omega +1},{\sigma }^{\omega }\right)\right)+0,$$

Since $\phi$ is non-decreasing and continuous, we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \underset{\omega \to \infty }{lim}}\phi \left(\mathsf{\Lambda }\left({\theta }_1^{\omega },{\theta }_1^{\omega +1}\right)\right)& \le & {\displaystyle \underset{\omega \to \infty }{lim}}\phi \left[\mathsf{\Lambda }\left(H\left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega },{\sigma }^{\omega }\right),H\left({\theta }_1^{\omega +1},{\theta }_2^{\omega +1},{\theta }_3^{\omega +1},{\sigma }^{\omega }\right)\right)\right]\hfill \\ & \le & {\displaystyle \underset{\omega \to \infty }{lim}}\left[\begin{array}{c}\rho \left(max\left\{\mathsf{\Lambda }\left({\theta }_1^{\omega },{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\theta }_2^{\omega },{\theta }_2^{\omega +1}\right),\mathsf{\Lambda }\left({\theta }_3^{\omega },{\theta }_3^{\omega +1}\right)\right\}\right)\\ -\ell \left(max\left\{\mathsf{\Lambda }\left({\theta }_1^{\omega },{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\theta }_2^{\omega },{\theta }_2^{\omega +1}\right),\mathsf{\Lambda }\left({\theta }_3^{\omega },{\theta }_3^{\omega +1}\right)\right\}\right)\end{array}\right].\hfill \end{array}$$

Analogously,

$$\underset{\omega \to \infty }{lim}\phi \left(\mathsf{\Lambda }\left({\theta }_2^{\omega },{\theta }_2^{\omega +1}\right)\right)\le \underset{\omega \to \infty }{lim}\left[\begin{array}{c}\rho \left(max\left\{\mathsf{\Lambda }\left({\theta }_1^{\omega },{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\theta }_2^{\omega },{\theta }_2^{\omega +1}\right),\mathsf{\Lambda }\left({\theta }_3^{\omega },{\theta }_3^{\omega +1}\right)\right\}\right)\\ -\ell \left(max\left\{\mathsf{\Lambda }\left({\theta }_1^{\omega },{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\theta }_2^{\omega },{\theta }_2^{\omega +1}\right),\mathsf{\Lambda }\left({\theta }_3^{\omega },{\theta }_3^{\omega +1}\right)\right\}\right)\end{array}\right],$$

and

$$\underset{\omega \to \infty }{lim}\phi \left(\mathsf{\Lambda }\left({\theta }_3^{\omega },{\theta }_3^{\omega +1}\right)\right)\le \underset{\omega \to \infty }{lim}\left[\begin{array}{c}\rho \left(max\left\{\mathsf{\Lambda }\left({\theta }_1^{\omega },{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\theta }_2^{\omega },{\theta }_2^{\omega +1}\right),\mathsf{\Lambda }\left({\theta }_3^{\omega },{\theta }_3^{\omega +1}\right)\right\}\right)\\ -\ell \left(max\left\{\mathsf{\Lambda }\left({\theta }_1^{\omega },{\theta }_1^{\omega +1}\right),\mathsf{\Lambda }\left({\theta }_2^{\omega },{\theta }_2^{\omega +1}\right),\mathsf{\Lambda }\left({\theta }_3^{\omega },{\theta }_3^{\omega +1}\right)\right\}\right)\end{array}\right].$$

This implies that

$$\underset{\omega \to \infty }{lim}\mathsf{\Lambda }\left({\theta }_1^{\omega },{\theta }_1^{\omega +1}\right)=0,\underset{\omega \to \infty }{lim}\mathsf{\Lambda }\left({\theta }_2^{\omega },{\theta }_2^{\omega +1}\right)=0\mathrm{and}\underset{\omega \to \infty }{lim}\mathsf{\Lambda }\left({\theta }_3^{\omega },{\theta }_3^{\omega +1}\right)=0.$$

(19)

It follows from $\left({\mathsf{\Lambda }}_2\right)$ that

$$\underset{\omega \to \infty }{lim}\mathsf{\Lambda }\left({\theta }_1^{\omega },{\theta }_1^{\omega }\right)=0,\underset{\omega \to \infty }{lim}\mathsf{\Lambda }\left({\theta }_2^{\omega },{\theta }_2^{\omega }\right)=0\mathrm{and}\underset{\omega \to \infty }{lim}\mathsf{\Lambda }\left({\theta }_3^{\omega },{\theta }_3^{\omega }\right)=0.$$

(20)

From the definition of ${\mathsf{\Omega }}_{\mathsf{\Lambda }},$ we can write

$$\underset{\omega \to \infty }{lim}{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_1^{\omega },{\theta }_1^{\omega +1}\right)=0,\underset{\omega \to \infty }{lim}{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_2^{\omega },{\theta }_2^{\omega +1}\right)=0\mathrm{and}\underset{\omega \to \infty }{lim}{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_3^{\omega },{\theta }_3^{\omega +1}\right)=0.$$

(21)

In order to prove that $\left\{{\theta }_1^{\omega }\right\},$ $\left\{{\theta }_2^{\omega }\right\}$ and $\left\{{\theta }_3^{\omega }\right\}$ are Cauchy sequences, assume that $\left\{{\theta }_1^{\omega }\right\}$ or $\left\{{\theta }_2^{\omega }\right\}$ or $\left\{{\theta }_3^{\omega }\right\}$ is not a Cauchy. Then there is an $ϵ>0$ and monotone increasing sequences $\left\{{\upsilon }_j\right\}$ and $\left\{{\omega }_j\right\}$ so that ${\omega }_j>{\upsilon }_j>j,$

$$max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_1^{{\upsilon }_j},{\theta }_1^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_2^{{\upsilon }_j},{\theta }_2^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_3^{{\upsilon }_j},{\theta }_3^{{\omega }_j}\right)\right\}\ge ϵ,$$

(22)

and

$$max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_1^{{\upsilon }_j},{\theta }_1^{{\omega }_j-1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_2^{{\upsilon }_j},{\theta }_2^{{\omega }_j-1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_3^{{\upsilon }_j},{\theta }_3^{{\omega }_j-1}\right)\right\}<ϵ.$$

(23)

Using (22) and (23), we obtain

$$\begin{array}{ccc}\hfill ϵ& \le & max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_1^{{\upsilon }_j},{\theta }_1^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_2^{{\upsilon }_j},{\theta }_2^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_3^{{\upsilon }_j},{\theta }_3^{{\omega }_j}\right)\right\}\hfill \\ & \le & max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_1^{{\upsilon }_j},{\theta }_1^{{\omega }_j-1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_2^{{\upsilon }_j},{\theta }_2^{{\omega }_j-1}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_3^{{\upsilon }_j},{\theta }_3^{{\omega }_j-1}\right)\right\}\hfill \\ & & +max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_1^{{\omega }_j-1},{\theta }_1^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_2^{{\omega }_j-1},{\theta }_2^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_3^{{\omega }_j-1},{\theta }_3^{{\omega }_j}\right)\right\}\hfill \\ & <& ϵ+max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_1^{{\omega }_j-1},{\theta }_1^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_2^{{\omega }_j-1},{\theta }_2^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_3^{{\omega }_j-1},{\theta }_3^{{\omega }_j}\right)\right\}.\hfill \end{array}$$

Letting $j\to \infty$ and using (21), we have

$$\underset{j\to \infty }{lim}max\left\{{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_1^{{\upsilon }_j},{\theta }_1^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_2^{{\upsilon }_j},{\theta }_2^{{\omega }_j}\right),{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_3^{{\upsilon }_j},{\theta }_3^{{\omega }_j}\right)\right\}=ϵ.$$

(24)

Based on the definition of ${\mathsf{\Omega }}_{\mathsf{\Lambda }}$ and by (2), one can obtain

$$\underset{j\to \infty }{lim}max\left\{\mathsf{\Lambda }\left({\theta }_1^{{\upsilon }_j},{\theta }_1^{{\omega }_j}\right),\mathsf{\Lambda }\left({\theta }_2^{{\upsilon }_j},{\theta }_2^{{\omega }_j}\right),\mathsf{\Lambda }\left({\theta }_3^{{\upsilon }_j},{\theta }_3^{{\omega }_j}\right)\right\}=\frac{ϵ}{2}.$$

(25)

Taking $j\to \infty$ and applying (24) and (21) in

$$\left|{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_1^{{\upsilon }_j},{\theta }_1^{{\omega }_j+1}\right)-{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_1^{{\upsilon }_j},{\theta }_1^{{\omega }_j}\right)\right|\le {\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_1^{{\omega }_j},{\theta }_1^{{\omega }_j+1}\right),$$

we have

$$\underset{j\to \infty }{lim}{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_1^{{\upsilon }_j},{\theta }_1^{{\omega }_j+1}\right)=ϵ,$$

hence, we obtain

$$\underset{j\to \infty }{lim}\mathsf{\Lambda }\left({\theta }_1^{{\upsilon }_j},{\theta }_1^{{\omega }_j+1}\right)=\frac{ϵ}{2}.$$

By the same manner, one can obtain

$$\underset{j\to \infty }{lim}\mathsf{\Lambda }\left({\theta }_2^{{\upsilon }_j},{\theta }_2^{{\omega }_j+1}\right)=\frac{ϵ}{2},$$

and

$$\underset{j\to \infty }{lim}\mathsf{\Lambda }\left({\theta }_3^{{\upsilon }_j},{\theta }_3^{{\omega }_j+1}\right)=\frac{ϵ}{2}.$$

Let

$$\begin{array}{ccc}\hfill {\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_1^{{\upsilon }_j},{\theta }_1^{{\omega }_j+1}\right)& =& \mathsf{\Lambda }\left(H\left({\theta }_1^{{\upsilon }_j},{\theta }_2^{{\upsilon }_j},{\theta }_{13}^{{\upsilon }_j},{\sigma }^{{\upsilon }_j}\right),H\left({\theta }_1^{{\omega }_j+1},{\theta }_2^{{\omega }_j+1},{\theta }_3^{{\omega }_j+1},{\sigma }^{{\omega }_j+1}\right)\right)\hfill \\ & \le & \mathsf{\Lambda }\left(H\left({\theta }_1^{{\upsilon }_j},{\theta }_2^{{\upsilon }_j},{\theta }_3^{{\upsilon }_j},{\sigma }^{{\upsilon }_j}\right),H\left({\theta }_1^{{\upsilon }_j},{\theta }_2^{{\upsilon }_j},{\theta }_3^{{\upsilon }_j},{\sigma }^{{\omega }_j+1}\right)\right)\hfill \\ & & +\mathsf{\Lambda }\left(H\left({\theta }_1^{{\upsilon }_j},{\theta }_2^{{\upsilon }_j},{\theta }_3^{{\upsilon }_j},{\sigma }^{{\omega }_j+1}\right),H\left({\theta }_1^{{\omega }_j+1},{\theta }_2^{{\omega }_j+1},{\theta }_3^{{\omega }_j+1},{\sigma }^{{\omega }_j+1}\right)\right)\hfill \\ & & -\mathsf{\Lambda }\left(H\left({\theta }_1^{{\upsilon }_j},{\theta }_2^{{\upsilon }_j},{\theta }_3^{{\upsilon }_j},{\sigma }^{{\omega }_j+1}\right),H\left({\theta }_1^{{\upsilon }_j},{\theta }_2^{{\upsilon }_j},{\theta }_3^{{\upsilon }_j},{\sigma }^{{\omega }_j+1}\right)\right)\hfill \\ & \le & C\left|{\sigma }^{{\upsilon }_j}-{\sigma }^{{\omega }_j+1}\right|+\mathsf{\Lambda }\left(H\left({\theta }_1^{{\upsilon }_j},{\theta }_2^{{\upsilon }_j},{\theta }_3^{{\upsilon }_j},{\sigma }^{{\omega }_j+1}\right),H\left({\theta }_1^{{\omega }_j+1},{\theta }_2^{{\omega }_j+1},{\theta }_3^{{\omega }_j+1},{\sigma }^{{\omega }_j+1}\right)\right).\hfill \end{array}$$

Letting $j\to \infty$ in the above and since $\left\{{\sigma }^{{\omega }_j}\right\}$ is Cauchy, we obtain

$$\frac{ϵ}{2}\le \underset{j\to \infty }{lim}\mathsf{\Lambda }\left(H\left({\theta }_1^{{\upsilon }_j},{\theta }_2^{{\upsilon }_j},{\theta }_3^{{\upsilon }_j},{\sigma }^{{\omega }_j+1}\right),H\left({\theta }_1^{{\omega }_j+1},{\theta }_2^{{\omega }_j+1},{\theta }_3^{{\omega }_j+1},{\sigma }^{{\omega }_j+1}\right)\right)$$

Because $\phi$ is non-decreasing and continuous, we have

$$\begin{array}{ccc}\hfill \phi \left(\frac{ϵ}{2}\right)& \le & {\displaystyle \underset{j\to \infty }{lim}}\phi \left[\mathsf{\Lambda }\left(H\left({\theta }_1^{{\upsilon }_j},{\theta }_2^{{\upsilon }_j},{\theta }_3^{{\upsilon }_j},{\sigma }^{{\omega }_j+1}\right),H\left({\theta }_1^{{\omega }_j+1},{\theta }_2^{{\omega }_j+1},{\theta }_3^{{\omega }_j+1},{\sigma }^{{\omega }_j+1}\right)\right)\right]\hfill \\ & \le & {\displaystyle \underset{\omega \to \infty }{lim}}\left[\begin{array}{c}\rho \left(max\left\{\mathsf{\Lambda }\left({\theta }_1^{{\upsilon }_j},{\theta }_1^{{\omega }_j+1}\right),\mathsf{\Lambda }\left({\theta }_2^{{\upsilon }_j},{\theta }_2^{{\omega }_j+1}\right),\mathsf{\Lambda }\left({\theta }_3^{{\upsilon }_j},{\theta }_3^{{\omega }_j+1}\right)\right\}\right)\\ -\ell \left(\mathsf{\Lambda }\left({\theta }_1^{{\upsilon }_j},{\theta }_1^{{\omega }_j+1}\right),\mathsf{\Lambda }\left({\theta }_2^{{\upsilon }_j},{\theta }_2^{{\omega }_j+1}\right),\mathsf{\Lambda }\left({\theta }_3^{{\upsilon }_j},{\theta }_3^{{\omega }_j+1}\right)\right)\end{array}\right]\hfill \\ & \le & \rho \left(\frac{ϵ}{2}\right)-\ell \left(\frac{ϵ}{2}\right),\hfill \end{array}$$

this implies that $ϵ\le 0,$ which is a contradiction. Hence, $\left\{{\theta }_1^{\omega }\right\}$ is Cauchy sequence. Similarly, $\left\{{\theta }_2^{\omega }\right\}$ and $\left\{{\theta }_3^{\omega }\right\}$ are too in $\left(\daleth ,{\mathsf{\Omega }}_{\mathsf{\Lambda }}\right)$ and ${\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_1^{\upsilon },{\theta }_1^{\omega }\right)\to 0$, ${\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_2^{\upsilon },{\theta }_2^{\omega }\right)\to 0$ and ${\mathsf{\Omega }}_{\mathsf{\Lambda }}\left({\theta }_3^{\upsilon },{\theta }_3^{\omega }\right)\to 0$ as $\upsilon ,\omega \to \infty .$ Thus, by (20) and the definition of ${\mathsf{\Omega }}_{\mathsf{\Lambda }},$ we have

$$\underset{\upsilon ,\omega \to \infty }{lim}\mathsf{\Lambda }\left({\theta }_1^{\upsilon },{\theta }_1^{\omega }\right)=0,\underset{\upsilon ,\omega \to \infty }{lim}\mathsf{\Lambda }\left({\theta }_2^{\upsilon },{\theta }_2^{\omega }\right)=0\mathrm{and}\underset{\upsilon ,\omega \to \infty }{lim}\mathsf{\Lambda }\left({\theta }_3^{\upsilon },{\theta }_3^{\omega }\right)=0.$$

It follows from Lemma 1 (i) that $\left\{{\theta }_1^{\omega }\right\},$ $\left\{{\theta }_2^{\omega }\right\}$ and $\left\{{\theta }_3^{\omega }\right\}$ are Cauchy sequences in $\left(\daleth ,\mathsf{\Lambda }\right).$

Because $\left(\daleth ,\mathsf{\Lambda }\right)$ is a complete, from Lemma 1 (ii), there exist ${\varrho }_1,{\varrho }_2,{\varrho }_3\in \daleth$ with

$$\begin{array}{ccc}\hfill \mathsf{\Lambda }\left({\varrho }_1,{\varrho }_1\right)& =& {\displaystyle \underset{\omega \to \infty }{lim}}\mathsf{\Lambda }\left({\theta }_1^{\omega },{\varrho }_1\right)={\displaystyle \underset{\omega \to \infty }{lim}}\mathsf{\Lambda }\left({\theta }_1^{\omega +1},{\varrho }_1\right)={\displaystyle \underset{\omega ,\upsilon \to \infty }{lim}}\mathsf{\Lambda }\left({\theta }_1^{\omega },{\theta }_1^{\upsilon }\right),\hfill \\ \hfill \mathsf{\Lambda }\left({\varrho }_2,{\varrho }_2\right)& =& {\displaystyle \underset{\omega \to \infty }{lim}}\mathsf{\Lambda }\left({\theta }_2^{\omega },{\varrho }_2\right)={\displaystyle \underset{\omega \to \infty }{lim}}\mathsf{\Lambda }\left({\theta }_2^{\omega +1},{\varrho }_2\right)={\displaystyle \underset{\omega ,\upsilon \to \infty }{lim}}\mathsf{\Lambda }\left({\theta }_2^{\omega },{\theta }_2^{\upsilon }\right),\hfill \\ \hfill \mathsf{\Lambda }\left({\varrho }_3,{\varrho }_3\right)& =& {\displaystyle \underset{\omega \to \infty }{lim}}\mathsf{\Lambda }\left({\theta }_3^{\omega },{\varrho }_3\right)={\displaystyle \underset{\omega \to \infty }{lim}}\mathsf{\Lambda }\left({\theta }_3^{\omega +1},{\varrho }_3\right)={\displaystyle \underset{\omega ,\upsilon \to \infty }{lim}}\mathsf{\Lambda }\left({\theta }_3^{\omega },{\theta }_3^{\upsilon }\right),\hfill \end{array}$$

Using Lemma 2, we have

$${\displaystyle \underset{\omega \to \infty }{lim}}\mathsf{\Lambda }\left({\theta }_1^{\omega },H\left({\varrho }_1,{\varrho }_2,{\varrho }_3,\sigma \right)\right)=\mathsf{\Lambda }\left({\varrho }_1,H\left({\varrho }_1,{\varrho }_2,{\varrho }_3,\sigma \right)\right).$$

Now

$$\begin{array}{ccc}\hfill \mathsf{\Lambda }\left({\theta }_1^{\omega },H\left({\varrho }_1,{\varrho }_2,{\varrho }_3,\sigma \right)\right)& =& \mathsf{\Lambda }\left(H\left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega },{\sigma }^{\omega }\right),H\left({\varrho }_1,{\varrho }_2,{\varrho }_3,\sigma \right)\right)\hfill \\ & \le & \mathsf{\Lambda }\left(H\left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega },{\sigma }^{\omega }\right),H\left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega },\sigma \right)\right)\hfill \\ & & +\mathsf{\Lambda }\left(H\left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega },\sigma \right),H\left({\varrho }_1,{\varrho }_2,{\varrho }_3,\sigma \right)\right)\hfill \\ & & -\mathsf{\Lambda }\left(H\left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega },\sigma \right),H\left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega },\sigma \right)\right)\hfill \\ & =& C\left|{\sigma }^{\omega }-\sigma \right|+\mathsf{\Lambda }\left(H\left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega },\sigma \right),H\left({\varrho }_1,{\varrho }_2,{\varrho }_3,\sigma \right)\right).\hfill \end{array}$$

Passing $\omega \to \infty ,$ we obtain

$$\mathsf{\Lambda }\left({\varrho }_1,H\left({\varrho }_1,{\varrho }_2,{\varrho }_3,\sigma \right)\right)\le {\displaystyle \underset{\omega \to \infty }{lim}}\mathsf{\Lambda }\left(H\left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega },\sigma \right),H\left({\varrho }_1,{\varrho }_2,{\varrho }_3,\sigma \right)\right).$$

Because $\phi$ is non-decreasing and continuous, we obtain

$$\begin{array}{ccc}\hfill \phi \left(\mathsf{\Lambda }\left({\varrho }_1,H\left({\varrho }_1,{\varrho }_2,{\varrho }_3,\sigma \right)\right)\right)& \le & {\displaystyle \underset{\omega \to \infty }{lim}}\phi \left[\mathsf{\Lambda }\left(H\left({\theta }_1^{\omega },{\theta }_2^{\omega },{\theta }_3^{\omega },\sigma \right),H\left({\varrho }_1,{\varrho }_2,{\varrho }_3,\sigma \right)\right)\right]\hfill \\ & \le & {\displaystyle \underset{\omega \to \infty }{lim}}\left[\begin{array}{c}\rho \left(\mathsf{\Lambda }\left({\theta }_1^{\omega },{\varrho }_1\right),\mathsf{\Lambda }\left({\theta }_2^{\omega },{\varrho }_2\right),\mathsf{\Lambda }\left({\theta }_3^{\omega },{\varrho }_3\right)\right)\\ -\ell \left(\mathsf{\Lambda }\left({\theta }_1^{\omega },{\varrho }_1\right),\mathsf{\Lambda }\left({\theta }_2^{\omega },{\varrho }_2\right),\mathsf{\Lambda }\left({\theta }_3^{\omega },{\varrho }_3\right)\right)\end{array}\right]\hfill \\ & =& 0.\hfill \end{array}$$

This implies that $\mathsf{\Lambda }\left({\varrho }_1,H\left({\varrho }_1,{\varrho }_2,{\varrho }_3,\sigma \right)\right)=0.$ Thus, ${\varrho }_1=H\left({\varrho }_1,{\varrho }_2,{\varrho }_3,\sigma \right).$ Analogously, ${\varrho }_2=H\left({\varrho }_2,{\varrho }_3,{\varrho }_1,\sigma \right)$ and ${\varrho }_3=H\left({\varrho }_3,{\varrho }_1,{\varrho }_2,\sigma \right).$ Hence, $\sigma \in \mho ,$ this proves that ℧ is closed in $[0,1].$

Assume that ${\sigma }^0\in \mho .$ Then there are ${\theta }_1^0,{\theta }_2^0,{\theta }_3^0\in A$ with ${\theta }_1^0=H({\theta }_1^0,{\theta }_2^0,{\theta }_3^0,{\sigma }^0).$ Because ℧ is open, then there exists $z>0$ so that $O_{\mathsf{\Lambda }}({\theta }_1^0,z)\subseteq \mho .$ Select $\sigma \in ({\sigma }^0-ϵ,{\sigma }^0+ϵ)$ so that $\left|\sigma -{\sigma }^0\right|\le \frac{1}{C^{\omega }}<ϵ.$

Then ${\theta }_1\in \overline{O_{\mathsf{\Lambda }}({\theta }_1^0,z)}=\{{\theta }_1\in \daleth /\mathsf{\Lambda }({\theta }_1,{\theta }_1^0)\le z+\mathsf{\Lambda }({\theta }_1^0,{\theta }_1^0).$ We obtain

$$\begin{array}{ccc}\hfill \mathsf{\Lambda }(H\left({\theta }_1,{\theta }_2,{\theta }_3,\sigma \right),{\theta }_1^0)& =& \mathsf{\Lambda }(H\left({\theta }_1,{\theta }_2,{\theta }_3,\sigma \right),H({\theta }_1^0,{\theta }_2^0,{\theta }_3^0,{\sigma }^0))\hfill \\ & \le & \mathsf{\Lambda }(H\left({\theta }_1,{\theta }_2,{\theta }_3,\sigma \right),H({\theta }_1,{\theta }_2,{\theta }_3,{\sigma }^0))\hfill \\ & & +\mathsf{\Lambda }(H\left({\theta }_1,{\theta }_2,{\theta }_3,{\sigma }^0\right),H({\theta }_1^0,{\theta }_2^0,{\theta }_3^0,{\sigma }^0))\hfill \\ & & -\mathsf{\Lambda }(H\left({\theta }_1,{\theta }_2,{\theta }_3,{\sigma }^0\right),H({\theta }_1,{\theta }_2,{\theta }_3,{\sigma }^0))\hfill \\ & \le & C\left|\sigma -{\sigma }^0\right|+\mathsf{\Lambda }(H\left({\theta }_1,{\theta }_2,{\theta }_3,{\sigma }^0\right),H({\theta }_1^0,{\theta }_2^0,{\theta }_3^0,{\sigma }^0))\hfill \\ & \le & \frac{1}{C^{\omega -1}}+\mathsf{\Lambda }(H\left({\theta }_1,{\theta }_2,{\theta }_3,{\sigma }^0\right),H({\theta }_1^0,{\theta }_2^0,{\theta }_3^0,{\sigma }^0)).\hfill \end{array}$$

Letting $\omega \to \infty ,$ we have

$$\mathsf{\Lambda }(H\left({\theta }_1,{\theta }_2,{\theta }_3,\sigma \right),{\theta }_1^0)\le \mathsf{\Lambda }(H\left({\theta }_1,{\theta }_2,{\theta }_3,{\sigma }^0\right),H({\theta }_1^0,{\theta }_2^0,{\theta }_3^0,{\sigma }^0)).$$

Because $\phi$ is non-decreasing and continuous, we obtain

$$\begin{array}{ccc}\hfill \phi \left(\mathsf{\Lambda }(H\left({\theta }_1,{\theta }_2,{\theta }_3,\sigma \right),{\theta }_1^0)\right)& \le & \phi \left[\mathsf{\Lambda }\left(H\left({\theta }_1,{\theta }_2,{\theta }_3,{\sigma }^0\right),H({\theta }_1^0,{\theta }_2^0,{\theta }_3^0,{\sigma }^0)\right)\right]\hfill \\ & \le & \rho \left(\mathsf{\Lambda }\left({\theta }_1,{\theta }_1^0\right),\mathsf{\Lambda }\left({\theta }_2,{\theta }_2^0\right),\mathsf{\Lambda }\left({\theta }_3,{\theta }_3^0\right)\right)-\ell \left(\mathsf{\Lambda }\left({\theta }_1,{\theta }_1^0\right),\mathsf{\Lambda }\left({\theta }_2,{\theta }_2^0\right),\mathsf{\Lambda }\left({\theta }_3,{\theta }_3^0\right)\right).\hfill \end{array}$$

By the same scenario, one can write

$$\phi \left(\mathsf{\Lambda }(H\left({\theta }_2,{\theta }_3{\theta }_1,\sigma \right),{\theta }_2^0)\right)\le \rho \left(\mathsf{\Lambda }\left({\theta }_1,{\theta }_1^0\right),\mathsf{\Lambda }\left({\theta }_2,{\theta }_2^0\right),\mathsf{\Lambda }\left({\theta }_3,{\theta }_3^0\right)\right)-\ell \left(\mathsf{\Lambda }\left({\theta }_1,{\theta }_1^0\right),\mathsf{\Lambda }\left({\theta }_2,{\theta }_2^0\right),\mathsf{\Lambda }\left({\theta }_3,{\theta }_3^0\right)\right),$$

and

$$\phi \left(\mathsf{\Lambda }(H\left({\theta }_3,{\theta }_1,{\theta }_2,\sigma \right),{\theta }_3^0)\right)\le \rho \left(\mathsf{\Lambda }\left({\theta }_1,{\theta }_1^0\right),\mathsf{\Lambda }\left({\theta }_2,{\theta }_2^0\right),\mathsf{\Lambda }\left({\theta }_3,{\theta }_3^0\right)\right)-\ell \left(\mathsf{\Lambda }\left({\theta }_1,{\theta }_1^0\right),\mathsf{\Lambda }\left({\theta }_2,{\theta }_2^0\right),\mathsf{\Lambda }\left({\theta }_3,{\theta }_3^0\right)\right).$$

Hence

$$\begin{array}{ccc}& & \phi \left(max\left\{\mathsf{\Lambda }(H\left({\theta }_1,{\theta }_2,{\theta }_3,\sigma \right),{\theta }_1^0),\mathsf{\Lambda }(H\left({\theta }_2,{\theta }_3{\theta }_1,\sigma \right),{\theta }_2^0),\mathsf{\Lambda }(H\left({\theta }_3,{\theta }_1,{\theta }_2,\sigma \right),{\theta }_3^0)\right\}\right)\hfill \\ & \le & \rho \left(max\left\{\mathsf{\Lambda }\left({\theta }_1,{\theta }_1^0\right),\mathsf{\Lambda }\left({\theta }_2,{\theta }_2^0\right),\mathsf{\Lambda }\left({\theta }_3,{\theta }_3^0\right)\right\}\right)-\ell \left(max\left\{\mathsf{\Lambda }\left({\theta }_1,{\theta }_1^0\right),\mathsf{\Lambda }\left({\theta }_2,{\theta }_2^0\right),\mathsf{\Lambda }\left({\theta }_3,{\theta }_3^0\right)\right\}\right)\hfill \\ & \le & \phi \left(max\left\{\mathsf{\Lambda }\left({\theta }_1,{\theta }_1^0\right),\mathsf{\Lambda }\left({\theta }_2,{\theta }_2^0\right),\mathsf{\Lambda }\left({\theta }_3,{\theta }_3^0\right)\right\}\right).\hfill \end{array}$$

Since $\phi$ is non-decreasing, we obtain

$$\begin{array}{ccc}& & max\left\{\mathsf{\Lambda }(H\left({\theta }_1,{\theta }_2,{\theta }_3,\sigma \right),{\theta }_1^0),\mathsf{\Lambda }(H\left({\theta }_2,{\theta }_3{\theta }_1,\sigma \right),{\theta }_2^0),\mathsf{\Lambda }(H\left({\theta }_3,{\theta }_1,{\theta }_2,\sigma \right),{\theta }_3^0)\right\}\hfill \\ & \le & max\left\{\mathsf{\Lambda }\left({\theta }_1,{\theta }_1^0\right),\mathsf{\Lambda }\left({\theta }_2,{\theta }_2^0\right),\mathsf{\Lambda }\left({\theta }_3,{\theta }_3^0\right)\right\}\hfill \\ & \le & max\left\{z+\mathsf{\Lambda }\left({\theta }_1^0,{\theta }_1^0\right),z+\mathsf{\Lambda }\left({\theta }_2^0,{\theta }_2^0\right),z+\mathsf{\Lambda }\left({\theta }_3^0,{\theta }_3^0\right)\right\}.\hfill \end{array}$$

Therefore, for each $\sigma \in ({\sigma }^0-ϵ,{\sigma }^0+ϵ),$ we have $H:\overline{O_{\mathsf{\Lambda }}({\theta }_1^0,z)}\to \overline{O_{\mathsf{\Lambda }}({\theta }_1^0,z)}.$

Moreover, because assertion $\left({\aleph }_2\right)$ holds and $\phi ,\rho :[0,\infty )\to [0,\infty )$ are continuous and non-decreasing and $\ell :[0,\infty )\to [0,\infty )$ is LSC with $\phi \left(\nu \right)-\rho \left(\nu \right)+\ell \left(\nu \right)>0,$ for $\nu >0.$ Then, all hypotheses of Corollary 1 are fulfilled. Hence, we conclude that $H(.,\sigma )$ has a TFP in $\overline{A}.$ Since this TFP must be contained in A since $\left({\aleph }_1\right)$ is satisfied. Thus, $\sigma \in \mho$ for any $\sigma \in ({\sigma }^0-ϵ,{\sigma }^0+ϵ).$ Hence, $({\sigma }^0-ϵ,{\sigma }^0+ϵ)\subseteq \mho$ and therefore ℧ is open in $[0,1].$ For the reverse implication, we use the same strategy. This finishes the proof. □

Corollary 2.

Suppose that $\left(\daleth ,\mathsf{\Lambda }\right)$ is a complete PMS, A is an open subset of ℸ and $H:\overline{A}\times \overline{A}\times \overline{A}\times [0,1]\to \daleth$ with hypotheses below:

(i)

${\theta }_1\ne H\left({\theta }_1,{\theta }_2,{\theta }_3,\sigma \right),$${\theta }_2\ne H\left({\theta }_2,{\theta }_3,{\theta }_1,\sigma \right)$ and ${\theta }_3\ne H\left({\theta }_3,{\theta }_1,{\theta }_2,\sigma \right),$ for each ${\theta }_1,{\theta }_2,{\theta }_3\in \partial A$ (here $\partial A$ refer to the boundary of A in ℸ) and $\sigma \in [0,1];$

(ii)

There are ${\theta }_1,{\theta }_2,{\theta }_3,{\vartheta }_1,{\vartheta }_2,{\vartheta }_3\in \overline{A}$ and $\sigma \in [0,1],$ $K\in [0,1)$ so that

$$\mathsf{\Lambda }\left(H\left({\theta }_1,{\theta }_2,{\theta }_3,\sigma \right),H\left({\vartheta }_1,{\vartheta }_2,{\vartheta }_3,\sigma \right)\right)\le Kmax\left\{\mathsf{\Lambda }\left({\theta }_1,{\vartheta }_1\right),\mathsf{\Lambda }\left({\theta }_2,{\vartheta }_2\right),\mathsf{\Lambda }\left({\theta }_3,{\vartheta }_3\right)\right\},$$

(iii)

There exist $C\ge 0$ so that

$$\mathsf{\Lambda }\left(H\left({\theta }_1,{\theta }_2,{\theta }_3,\sigma \right),H\left({\theta }_1,{\theta }_2,{\theta }_3,{\sigma }^{*}\right)\right)\le C\left|\sigma -{\sigma }^{*}\right|,$$

for each ${\theta }_1,{\theta }_2,{\theta }_3\in \overline{A}$ and $\sigma ,{\sigma }^{*}\in [0,1].$

Then $H(.,0)$ has a TFP, whenever $H(.,1)$ has a TFP.

Proof. 

The proof follows immediately from Theorem 4 by putting $\phi \left(\nu \right)=\nu ,$ $\rho \left(\nu \right)=K\nu -\nu$ with $K\in [0,1)$ and $\ell \left(\nu \right)=\nu ,$ for $\nu >0.$ □