PPF dependent fixed point theorems for ${\alpha }_c$-admissible rational type contractive mappings in Banach spaces

Abstract

In this paper, we prove some PPF dependent fixed point theorems in the Razumikhin class for some rational type contractive mappings involving ${\alpha }_c$-admissible mappings where the domain and range of the mappings are not the same. As applications of these results, we derive some PPF dependent fixed point theorems for these nonself-contractions whenever the range space is endowed with a graph. Our results extend and generalize some results in the literature.

MSC:46N40, 47H10, 54H25, 46T99.

1 Introduction

The fixed point theory in Banach spaces plays an important role and is useful in mathematics. In fact, fixed point theory can be applied for solving equilibrium problems, variational inequalities and optimization problems. In particular, a very powerful tool is the Banach fixed point theorem, which was generalized and extended in various directions (see [1–37]). In 1977, Bernfeld et al. [2] introduced the concept of PPF dependent fixed point or the fixed point with PPF dependence which is a fixed point for mappings that have different domains and ranges. They also proved the existence of PPF dependent fixed point theorems in the Razumikhin class for Banach type contraction mappings. Very recently, some authors established the existence and uniqueness of PPF dependent fixed point for different types of contractive mappings and generalized some results of Bernfeld et al. [2] (see [1, 4, 12, 15, 20], and [33]).

In order to generalize the Banach contraction principle, Geraghty [9] proved the following theorem.

Theorem 1 (Geraghty [9])

Let $(X,d)$ be a complete metric space and $T:X\to X$ be an operator. Suppose that there exists $\beta :[0,+\mathrm{\infty })\to [0,1)$ satisfying the condition

$$\beta (t_n)\to 1\mathit{\text{implies}}{t}_n\to 0,\mathit{\text{as }}n\to +\mathrm{\infty }.$$

If T satisfies the following inequality:

$$d(Tx,Ty)\le \beta (d(x,y))d(x,y)\mathit{\text{for all }}x,y\in X,$$

(1.1)

then T has a unique fixed point.

Throughout this paper, let $(E,{\parallel \cdot \parallel }_E)$ be a Banach space, I denotes a closed interval $[a,b]$ in ℝ and $E_0=C(I,E)$ denotes the set of all continuous E-valued functions on I equipped with the supremum norm ${\parallel \cdot \parallel }_{E_0}$ defined by

$${\parallel \varphi \parallel }_{E_0}=\underset{t\in I}{sup}{\parallel \varphi (t)\parallel }_E.$$

For a fixed element $c\in I$, the Razumikhin or minimal class of functions in $E_0$ is defined by

$${\mathcal{R}}_c=\{\varphi \in E_0:{\parallel \varphi \parallel }_{E_0}={\parallel \varphi (c)\parallel }_E\}.$$

Clearly, every constant function from I to E is a member of ${\mathcal{R}}_c$. It is easy to see that the class ${\mathcal{R}}_c$ is algebraically closed with respect to difference, i.e., $\varphi -\xi \in {\mathcal{R}}_c$ when $\varphi ,\xi \in {\mathcal{R}}_c$. Also the class ${\mathcal{R}}_c$ is topologically closed if it is closed with respect to the topology on $E_0$ generated by the norm ${\parallel \cdot \parallel }_{E_0}$.

Definition 1 ([2])

A mapping $\varphi \in E_0$ is said to be a PPF dependent fixed point or a fixed point with PPF dependence of mapping $T:E_0\to E$ if $T\varphi =\varphi (c)$ for some $c\in I$.

Definition 2 ([2])

The mapping $T:E_0\to E$ is called a Banach type contraction if there exists $k\in [0,1)$ such that

$${\parallel T\varphi -T\xi \parallel }_E\le k{\parallel \varphi -\xi \parallel }_{E_0}$$

for all $\varphi ,\xi \in E_0$.

Samet in 2012 introduced the concepts of α-ψ-contractive and α-admissible mappings. Karapınar and Samet generalized these notions to obtain other fixed point results. Many authors generalized these notions to obtain fixed point results (see [18, 19, 21–23], and [32]).

Samet et al. [31], defined the notion of α-admissible mappings as follows:

Definition 3 ([31])

Let T be a self-mapping on X and $\alpha :X\times X\to [0,\mathrm{\infty })$ be a function. We say that T is an α-admissible mapping if

$$x,y\in X,\alpha (x,y)\ge 1⟹\alpha (Tx,Ty)\ge 1.$$

Definition 4 ([17])

Let $f:X\to X$ and $\alpha :X\times X\to [0,+\mathrm{\infty })$. We say that f is a triangular α-admissible mapping if

(T1) $\alpha (x,y)\ge 1$ implies $\alpha (fx,fy)\ge 1$, $x,y\in X$,

(T2) $\{\begin{array}{l}\alpha (x,z)\ge 1,\\ \alpha (z,y)\ge 1\end{array}$ implies $\alpha (x,y)\ge 1$, $x,y,z\in X$.

The concept of ${\alpha }_c$-admissible mapping was introduced by Agarwal et al. in 2013 (see [1]).

Definition 5 ([1])

Let $c\in I$, $T:E_0\to E$, and $\alpha :E\times E\to [0,\mathrm{\infty })$. We say T is an ${\alpha }_c$-admissible mapping if for $\varphi ,\xi \in E_0$

$$\alpha (\varphi (c),\xi (c))\ge 1⟹\alpha (T\varphi ,T\xi )\ge 1.$$

(1.2)

Definition 6 ([4])

Let $c\in I$, $T:E_0\to E$, and $\alpha :E\times E\to [0,\mathrm{\infty })$. We say T is a triangular ${\alpha }_c$-admissible mapping if

(T1) $\alpha (\varphi (c),\xi (c))\ge 1$ implies $\alpha (T\varphi ,T\xi )\ge 1$,

(T2) $\alpha (\varphi (c),\mu (c))\ge 1$ and $\alpha (\mu (c),\xi (c))\ge 1$ implies $\alpha (\varphi (c),\xi (c))\ge 1$

for $\varphi ,\xi ,\mu \in E_0$.

Lemma 1 ([4])

Let $T:E_0\to E$ be a triangular ${\alpha }_c$-admissible mapping. Define the sequence $\{{\varphi }_n\}$ in the following way:

$$T{\varphi }_{n-1}={\varphi }_n(c)$$

for all $n\in \mathbb{N}$, where ${\varphi }_0\in {\mathcal{R}}_c$ is such that $\alpha ({\varphi }_0(c),T{\varphi }_0)\ge 1$. Then

$$\alpha ({\varphi }_n(c),{\varphi }_m(c))\ge 1\mathit{\text{for all }}m,n\in \mathbb{N}\mathit{\text{ with }}m<n.$$

2 Main results

Let ℱ denotes the class of all functions $\beta :[0,+\mathrm{\infty })\to [0,1)$ satisfying the following condition:

$$\beta (t_n)\to 1\text{implies}{t}_n\to 0,\text{as }n\to +\mathrm{\infty }.$$

(2.1)

Definition 7 Let $T:E_0\to E$ be a nonself-mapping and $\alpha :E\times E\to [0,\mathrm{\infty })$ be a function. We say T is a rational Geraghty contraction of type I if there exist $\beta \in \mathcal{F}$ and $c\in I$ such that

$$\alpha (\varphi (c),T\varphi )\alpha (\xi (c),T\xi ){\parallel T\varphi -T\xi \parallel }_E\le \beta \left(M(\varphi (c),\xi (c))\right)M(\varphi (c),\xi (c))$$

for all $\varphi ,\xi \in E_0$, where

$$\begin{array}{rcl}M(\varphi (c),\xi (c))& =& max\{{\parallel \varphi -\xi \parallel }_{E_0},\frac{{\parallel \varphi (c)-T\varphi \parallel }_E{\parallel \xi (c)-T\xi \parallel }_E}{1+{\parallel \varphi -\xi \parallel }_{E_0}},\\ \frac{{\parallel \varphi (c)-T\varphi \parallel }_E{\parallel \xi (c)-T\xi \parallel }_E}{1+{\parallel T\varphi -T\xi \parallel }_E}\}.\end{array}$$

Theorem 2 Let $T:E_0\to E$ and $\alpha :E\times E\to [0,\mathrm{\infty })$ be two mappings satisfying the following assertions:

• (a) there exists $c\in I$ such that ${\mathcal{R}}_c$ is topologically closed and algebraically closed with respect to difference,

• (b) T is an ${\alpha }_c$-admissible,

• (c) T is a rational Geraghty contractive mapping of type I,

• (d) if $\{{\varphi }_n\}$ is a sequence in $E_0$ such that ${\varphi }_n\to \varphi$ as $n\to \mathrm{\infty }$ and $\alpha ({\varphi }_n(c),T{\varphi }_n)\ge 1$, then $\alpha (\varphi (c),T\varphi )\ge 1$ for all $n\in \mathbb{N}$,

• (e) there exists ${\varphi }_0\in {\mathcal{R}}_c$ such that $\alpha ({\varphi }_0(c),T{\varphi }_0)\ge 1$.

Then T has a unique PPF dependent fixed point ${\varphi }^{\ast }\in {\mathcal{R}}_c$. Moreover, for a fixed ${\varphi }_0\in {\mathcal{R}}_c$, if the sequence $\{{\varphi }_n\}$ of iterates of T is defined by $T{\varphi }_{n-1}={\varphi }_n(c)$ for all $n\in \mathbb{N}$, then $\{{\varphi }_n\}$ converges to ${\varphi }^{\ast }\in {\mathcal{R}}_c$.

Proof Let ${\varphi }_0$ is a point in ${\mathcal{R}}_c\subset E_0$ such that $\alpha ({\varphi }_0(c),T{\varphi }_0)\ge 1$. Since $T{\varphi }_0\in E$, there exists $x_1\in E$ such that $T{\varphi }_0=x_1$. Choose ${\varphi }_1\in {\mathcal{R}}_c$ such that $x_1={\varphi }_1(c)$. Since ${\varphi }_1\in {\mathcal{R}}_c\subset E_0$ and, by hypothesis, we get $T{\varphi }_1\in E$. This implies that there exists $x_2\in E$ such that $T{\varphi }_1=x_2$. Thus, we can choose ${\varphi }_2\in {\mathcal{R}}_c$ such that $x_2={\varphi }_2(c)$. Continuing this process, by induction, we can build the sequence $\{{\varphi }_n\}$ in ${\mathcal{R}}_c\subset E_0$ such that $T{\varphi }_{n-1}={\varphi }_n(c)$ for all $n\in \mathbb{N}$. It follows from the fact that ${\mathcal{R}}_c$ is algebraically closed with respect to difference

$${\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}={\parallel {\varphi }_{n-1}(c)-{\varphi }_n(c)\parallel }_E\text{for all }n\in \mathbb{N}.$$

Since T is ${\alpha }_c$-admissible and $\alpha ({\varphi }_0(c),{\varphi }_1(c))=\alpha ({\varphi }_0(c),T{\varphi }_0)\ge 1$, we deduce that

$$\alpha ({\varphi }_1(c),T{\varphi }_1)=\alpha (T{\varphi }_0,T{\varphi }_1)\ge 1.$$

By continuing this process, we get $\alpha ({\varphi }_{n-1}(c),T{\varphi }_{n-1})\ge 1$ for all $n\in \mathbb{N}$. Since T is a rational Geraghty contraction of type I, we have

$$\begin{array}{rcl}{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}& =& {\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E={\parallel T{\varphi }_{n-1}-T{\varphi }_n\parallel }_E\\ \le & \alpha ({\varphi }_{n-1}(c),T{\varphi }_{n-1})\alpha ({\varphi }_n(c),T{\varphi }_n){\parallel T{\varphi }_{n-1}-T{\varphi }_n\parallel }_E\\ \le & \beta \left(M({\varphi }_{n-1}(c),{\varphi }_n(c))\right)M({\varphi }_{n-1}(c),{\varphi }_n(c)).\end{array}$$

(2.2)

On the other hand,

$$\begin{array}{rcl}M({\varphi }_{n-1}(c),{\varphi }_n(c))& =& max\{{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0},\\ \frac{{\parallel {\varphi }_{n-1}(c)-T{\varphi }_{n-1}\parallel }_E{\parallel {\varphi }_n(c)-T{\varphi }_n\parallel }_E}{1+{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}},\\ \frac{{\parallel {\varphi }_{n-1}(c)-T{\varphi }_{n-1}\parallel }_E{\parallel {\varphi }_n(c)-T{\varphi }_n\parallel }_E}{1+{\parallel T{\varphi }_{n-1}-T{\varphi }_n\parallel }_E}\}\\ =& max\{{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0},\\ \frac{{\parallel {\varphi }_{n-1}(c)-{\varphi }_n(c)\parallel }_E{\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E}{1+{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}},\\ \frac{{\parallel {\varphi }_{n-1}(c)-{\varphi }_n\parallel }_E{\parallel {\varphi }_n(c)-{\varphi }_{n+1}\parallel }_E}{1+{\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E}\}\\ \le & max\{{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0},{\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E\}\\ =& max\{{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0},{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\}.\end{array}$$

If

$$max\{{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0},{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\}={\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0},$$

from (2.2) we have

$${\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\le \beta \left({\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\right){\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}<{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0},$$

(2.3)

which is a contradiction. So,

$$max\{{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0},{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\}={\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}.$$

By (2.2) we conclude

$${\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\le \beta \left({\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}\right){\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}<{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}$$

(2.4)

for all $n\in \mathbb{N}$. This implies that the sequence $\{{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\}$ is decreasing in ${\mathbb{R}}_{+}$. So, it is convergent. Suppose that there exists $r\ge 0$ such that ${lim}_{n\to +\mathrm{\infty }}{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}=r$. Assume that $r>0$. Taking the limit as $n\to +\mathrm{\infty }$ from (2.4) we conclude

$$r\le \underset{n\to +\mathrm{\infty }}{lim}\beta \left({\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}\right)r,$$

which implies $1\le {lim}_{n\to +\mathrm{\infty }}\beta ({\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0})$. So,

$$\underset{n\to +\mathrm{\infty }}{lim}\beta \left({\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}\right)=1,$$

and since $\beta \in \mathcal{F}$, ${lim}_{n\to +\mathrm{\infty }}{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}=0$, which is a contradiction. Hence, $r=0$. This means

$$\underset{n\to +\mathrm{\infty }}{lim}{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}=0.$$

(2.5)

We prove that the sequence $\{{\varphi }_n\}$ is a Cauchy sequence in ${\mathcal{R}}_c$. Assume that $\{{\varphi }_n\}$ is not a Cauchy sequence, then

$$\underset{m,n\to +\mathrm{\infty }}{lim}{\parallel {\varphi }_m-{\varphi }_n\parallel }_{E_0}>0.$$

(2.6)

Since T is a rational Geraghty contraction of type I, we have

$$\begin{array}{rcl}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}& \le & {\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}+{\parallel {\varphi }_{n+1}-{\varphi }_{m+1}\parallel }_{E_0}+{\parallel {\varphi }_{m+1}-{\varphi }_m\parallel }_{E_0}\\ \le & {\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}+\alpha ({\varphi }_n(c),T{\varphi }_n)\alpha ({\varphi }_m(c),T{\varphi }_m){\parallel T{\varphi }_n-T{\varphi }_m\parallel }_{E_0}\\ +{\parallel {\varphi }_{m+1}-{\varphi }_m\parallel }_{E_0}\\ \le & {\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}+\beta \left(M({\varphi }_n(c),{\varphi }_m(c))\right)M({\varphi }_n(c),{\varphi }_m(c))\\ +{\parallel {\varphi }_{m+1}-{\varphi }_m\parallel }_{E_0}.\end{array}$$

Taking the limit when $m,n\to \mathrm{\infty }$ in the above inequality and applying (2.5) we deduce

$$\underset{m,n\to \mathrm{\infty }}{lim}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}\le \underset{m,n\to \mathrm{\infty }}{lim}\beta \left(M({\varphi }_n(c),{\varphi }_m(c))\right)\underset{m,n\to \mathrm{\infty }}{lim}M({\varphi }_n(c),{\varphi }_m(c)),$$

(2.7)

where

$$\begin{array}{rcl}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}& \le & M({\varphi }_n(c),{\varphi }_m(c))\\ =& max\{{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0},\frac{{\parallel {\varphi }_n(c)-T{\varphi }_n\parallel }_E{\parallel {\varphi }_m(c)-T{\varphi }_m\parallel }_E}{1+{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}},\\ \frac{{\parallel {\varphi }_n(c)-T{\varphi }_n\parallel }_E{\parallel {\varphi }_m(c)-T{\varphi }_m\parallel }_E}{1+{\parallel T{\varphi }_n-T{\varphi }_m\parallel }_E}\}\\ =& max\{{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0},\frac{{\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E{\parallel {\varphi }_m(c)-{\varphi }_{m+1}(c)\parallel }_E}{1+{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}},\\ \frac{{\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E{\parallel {\varphi }_m(c)-{\varphi }_{m+1}(c)\parallel }_E}{1+{\parallel {\varphi }_{n+1}(c)-{\varphi }_{m+1}(c)\parallel }_E}\}\\ =& max\{{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0},\frac{{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}{\parallel {\varphi }_m-{\varphi }_{m+1}\parallel }_{E_0}}{1+{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}},\\ \frac{{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}{\parallel {\varphi }_m-{\varphi }_{m+1}\parallel }_{E_0}}{1+{\parallel {\varphi }_{n+1}-{\varphi }_{m+1}\parallel }_{E_0}}\}.\end{array}$$

(2.8)

Letting $m,n\to \mathrm{\infty }$ in the above inequality and applying (2.5), we get

$$\underset{m,n\to +\mathrm{\infty }}{lim}M({\varphi }_n(c),{\varphi }_m(c))=\underset{m,n\to +\mathrm{\infty }}{lim}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}.$$

(2.9)

So, by (2.7) and (2.9), we have

$$\underset{m,n\to +\mathrm{\infty }}{lim\hspace{0.17em}sup}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}\le \underset{m,n\to +\mathrm{\infty }}{lim\hspace{0.17em}sup}\beta \left({\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}\right)\underset{m,n\to +\mathrm{\infty }}{lim\hspace{0.17em}sup}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}$$

and hence from (2.6) we get $1\le {lim\hspace{0.17em}sup}_{m,n\to +\mathrm{\infty }}\beta ({\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0})$. This means

$$\underset{m,n\to +\mathrm{\infty }}{lim}\beta \left({\parallel {\varphi }_m-{\varphi }_n\parallel }_{E_0}\right)=1$$

and since $\beta \in \mathcal{F}$, we conclude

$$\underset{m,n\to +\mathrm{\infty }}{lim}{\parallel {\varphi }_m-{\varphi }_n\parallel }_{E_0}=0,$$

which is a contradiction. Consequently,

$$\underset{m,n\to +\mathrm{\infty }}{lim}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}=0$$

and hence $\{{\varphi }_n\}$ is a Cauchy sequence in ${\mathcal{R}}_c\subseteq E_0$. By Completeness of $E_0$, we find that $\{{\varphi }_n\}$ converges to a point ${\varphi }^{\ast }\in E_0$, this means ${\varphi }_n\to {\varphi }^{\ast }$, as $n\to +\mathrm{\infty }$. Since ${\mathcal{R}}_c$ is topologically closed, we deduce, ${\varphi }^{\ast }\in {\mathcal{R}}_c$. By condition b, we have $\alpha ({\varphi }^{\ast }(c),T{\varphi }^{\ast })\ge 1$. Now, since T is a rational Geraghty contraction of type I, we have

$$\begin{array}{c}{\parallel T{\varphi }^{\ast }-{\varphi }^{\ast }(c)\parallel }_E\hfill \\ \le {\parallel T{\varphi }^{\ast }-{\varphi }_n(c)\parallel }_E+{\parallel {\varphi }_n(c)-{\varphi }^{\ast }(c)\parallel }_E\hfill \\ ={\parallel T{\varphi }^{\ast }-T{\varphi }_{n-1}\parallel }_E+{\parallel {\varphi }_n-{\varphi }^{\ast }\parallel }_{E_0}\hfill \\ \le \alpha ({\varphi }^{\ast }(c),T{\varphi }^{\ast })\alpha ({\varphi }_{n-1}(c),T{\varphi }_{n-1}){\parallel T{\varphi }^{\ast }-T{\varphi }_{n-1}\parallel }_E+{\parallel {\varphi }_n-{\varphi }^{\ast }\parallel }_{E_0}\hfill \\ \le \beta \left(M({\varphi }^{\ast }(c),{\varphi }_{n-1}(c))\right)M({\varphi }^{\ast }(c),{\varphi }_{n-1}(c)).\hfill \end{array}$$

Taking the limit as $n\to \mathrm{\infty }$ in the above inequality, we get

$${\parallel T{\varphi }^{\ast }-{\varphi }^{\ast }(c)\parallel }_E\le \underset{n\to \mathrm{\infty }}{lim}\beta \left(M({\varphi }^{\ast }(c),{\varphi }_{n-1}(c))\right)\underset{n\to \mathrm{\infty }}{lim}M({\varphi }^{\ast }(c),{\varphi }_{n-1}(c)).$$

(2.10)

But

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim}M({\varphi }^{\ast }(c),{\varphi }_{n-1}(c))& =& \underset{n\to \mathrm{\infty }}{lim}max\{{\parallel {\varphi }^{\ast }-{\varphi }_{n-1}\parallel }_{E_0},\\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E{\parallel {\varphi }_{n-1}(c)-T{\varphi }_{n-1}\parallel }_E}{1+{\parallel {\varphi }^{\ast }-{\varphi }_{n-1}\parallel }_{E_0}},\\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E{\parallel {\varphi }_{n-1}(c)-T{\varphi }_{n-1}\parallel }_E}{1+{\parallel T{\varphi }^{\ast }-T{\varphi }_{n-1}\parallel }_E}\}\\ =& \underset{n\to \mathrm{\infty }}{lim}max\{{\parallel {\varphi }^{\ast }-{\varphi }_{n-1}\parallel }_{E_0},\\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E{\parallel {\varphi }_{n-1}(c)-{\varphi }_n(c)\parallel }_E}{1+{\parallel {\varphi }^{\ast }-{\varphi }_{n-1}\parallel }_{E_0}},\\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E{\parallel {\varphi }_{n-1}(c)-{\varphi }_n(c)\parallel }_E}{1+{\parallel T{\varphi }^{\ast }-{\varphi }_n(c)\parallel }_E}\}\\ =& \underset{n\to \mathrm{\infty }}{lim}max\{{\parallel {\varphi }^{\ast }-{\varphi }_{n-1}\parallel }_{E_0},\\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}}{1+{\parallel {\varphi }^{\ast }-{\varphi }_{n-1}\parallel }_{E_0}}\\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}}{1+{\parallel T{\varphi }^{\ast }-{\varphi }_n(c)\parallel }_E}\}=0.\end{array}$$

(2.11)

Therefore, from (2.10) and (2.11), we deduce

$${\parallel T{\varphi }^{\ast }-{\varphi }^{\ast }(c)\parallel }_E=0,$$

that is,

$$T{\varphi }^{\ast }={\varphi }^{\ast }(c),$$

which implies that ${\varphi }^{\ast }$ is a PPF dependent fixed point of T in ${\mathcal{R}}_c$. Now, we show that T has a unique PPF dependent fixed point in ${\mathcal{R}}_c$. Suppose on the contrary that ${\varphi }^{\ast }$ and ${\phi }^{\ast }$ are two PPF dependent fixed points of T in ${\mathcal{R}}_c$ such that ${\varphi }^{\ast }\ne {\phi }^{\ast }$. Then

$$\begin{array}{rcl}{\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0}& =& {\parallel {\varphi }^{\ast }(c)-{\phi }^{\ast }(c)\parallel }_E={\parallel T{\varphi }^{\ast }-T{\phi }^{\ast }\parallel }_E\\ \le & \alpha ({\varphi }^{\ast }(c),T{\varphi }^{\ast })\alpha ({\phi }^{\ast }(c),T{\phi }^{\ast }){\parallel T{\varphi }^{\ast }-T{\phi }^{\ast }\parallel }_E\\ \le & \beta \left(M({\varphi }^{\ast }(c),{\phi }^{\ast }(c))\right)M({\varphi }^{\ast }(c),{\phi }^{\ast }(c)),\end{array}$$

where

$$\begin{array}{rcl}M({\varphi }^{\ast }(c),{\phi }^{\ast }(c))& =& max\{{\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0},\frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E{\parallel {\phi }^{\ast }(c)-T{\phi }^{\ast }\parallel }_E}{1+{\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0}},\\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E{\parallel {\phi }^{\ast }(c)-T{\phi }^{\ast }\parallel }_E}{1+{\parallel T{\varphi }^{\ast }-T{\phi }^{\ast }\parallel }_E}\}={\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0}.\end{array}$$

Therefore,

$${\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0}\le \beta \left({\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0}\right){\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0}<{\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0},$$

which is a contradiction. Hence, ${\varphi }^{\ast }={\phi }^{\ast }$. Then T has a unique PPF dependent fixed point in ${\mathcal{R}}_c$. □

Definition 8 Let $\alpha :E\times E\to [0,\mathrm{\infty })$ and $T:E_0\to E$. We say that T is a rational Geraghty contraction of type II if there exist $\beta \in \mathcal{F}$ and $c\in I$ such that

$$\alpha (\varphi (c),T\varphi )\alpha (\xi (c),T\xi ){\parallel T\varphi -T\xi \parallel }_E\le \beta \left(M(\varphi (c),\xi (c))\right)M(\varphi (c),\xi (c))$$

for all $\varphi ,\xi \in E_0$, where

$$\begin{array}{rcl}M(\varphi (c),\xi (c))& =& max\{{\parallel \varphi -\xi \parallel }_{E_0},\\ \frac{{\parallel \varphi (c)-T\varphi \parallel }_E{\parallel \varphi (c)-T\xi \parallel }_E+{\parallel \xi (c)-T\xi \parallel }_E{\parallel \xi (c)-T\varphi \parallel }_E}{1+{\parallel \varphi (c)-T\varphi \parallel }_E+{\parallel \xi (c)-T\xi \parallel }_E},\\ \frac{{\parallel \varphi (c)-T\varphi \parallel }_E{\parallel \varphi (c)-T\xi \parallel }_E+{\parallel \xi (c)-T\xi \parallel }_E{\parallel \xi (c)-T\varphi \parallel }_E}{1+{\parallel \varphi (c)-T\xi \parallel }_E+{\parallel \xi (c)-T\varphi \parallel }_E}\}.\end{array}$$

Theorem 3 Let $T:E_0\to E$ and $\alpha :E\times E\to [0,\mathrm{\infty })$ be two mappings satisfying the following assertions:

• (a) there exists $c\in I$ such that ${\mathcal{R}}_c$ is topologically closed and algebraically closed with respect to difference,

• (b) T is an ${\alpha }_c$-admissible,

• (c) T is a rational Geraghty contractive mapping of type II,

• (d) if $\{{\varphi }_n\}$ is a sequence in $E_0$ such that ${\varphi }_n\to \varphi$ as $n\to \mathrm{\infty }$ and $\alpha ({\varphi }_n(c),T{\varphi }_n)\ge 1$, then $\alpha (\varphi (c),T\varphi )\ge 1$ for all $n\in \mathbb{N}$,

• (e) there exists ${\varphi }_0\in {\mathcal{R}}_c$ such that $\alpha ({\varphi }_0(c),T{\varphi }_0)\ge 1$.

Then T has a unique PPF dependent fixed point ${\varphi }^{\ast }\in {\mathcal{R}}_c$. Moreover, for a fixed ${\varphi }_0\in {\mathcal{R}}_c$, if the sequence $\{{\varphi }_n\}$ of iterates of T is defined by $T{\varphi }_{n-1}={\varphi }_n(c)$ for all $n\in \mathbb{N}$, then $\{{\varphi }_n\}$ converges to ${\varphi }^{\ast }\in {\mathcal{R}}_c$.

Proof Suppose that ${\varphi }_0$ is a point in ${\mathcal{R}}_c\subset E_0$ such that $\alpha ({\varphi }_0(c),T{\varphi }_0)\ge 1$. Since $T{\varphi }_0\in E$, there exists $x_1\in E$ such that $T{\varphi }_0=x_1$. Choose ${\varphi }_1\in {\mathcal{R}}_c$ such that $x_1={\varphi }_1(c)$. Since ${\varphi }_1\in {\mathcal{R}}_c\subset E_0$ and, by hypothesis, we get $T{\varphi }_1\in E$. This implies that there exists $x_2\in E$ such that $T{\varphi }_1=x_2$. Thus, we can choose ${\varphi }_2\in {\mathcal{R}}_c$ such that $x_2={\varphi }_2(c)$. Continuing this process, by induction, we can build the sequence $\{{\varphi }_n\}$ in ${\mathcal{R}}_c\subset E_0$ such that $T{\varphi }_{n-1}={\varphi }_n(c)$ for all $n\in \mathbb{N}$. It follows from the fact that ${\mathcal{R}}_c$ is algebraically closed with respect to difference

$${\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}={\parallel {\varphi }_{n-1}(c)-{\varphi }_n(c)\parallel }_E\text{for all }n\in \mathbb{N}.$$

Since T is ${\alpha }_c$-admissible and $\alpha ({\varphi }_0(c),{\varphi }_1(c))=\alpha ({\varphi }_0(c),T{\varphi }_0)\ge 1$, we deduce that

$$\alpha ({\varphi }_1(c),T{\varphi }_1)=\alpha (T{\varphi }_0,T{\varphi }_1)\ge 1.$$

Continuing this process, we get $\alpha ({\varphi }_{n-1}(c),T{\varphi }_{n-1})\ge 1$ for all $n\in \mathbb{N}$. Since T is a rational Geraghty contraction of type II, we have

$$\begin{array}{rcl}{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}& =& {\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E={\parallel T{\varphi }_{n-1}-T{\varphi }_n\parallel }_E\\ \le & \alpha ({\varphi }_{n-1}(c),T{\varphi }_{n-1})\alpha ({\varphi }_n(c),T{\varphi }_n){\parallel T{\varphi }_{n-1}-T{\varphi }_n\parallel }_E\\ \le & \beta \left(M({\varphi }_{n-1}(c),{\varphi }_n(c))\right)M({\varphi }_{n-1}(c),{\varphi }_n(c)).\end{array}$$

(2.12)

On the other hand,

$$\begin{array}{c}M({\varphi }_{n-1}(c),{\varphi }_n(c))\hfill \\ =max\{{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0},\hfill \\ \frac{{\parallel {\varphi }_{n-1}(c)-T{\varphi }_{n-1}\parallel }_E{\parallel {\varphi }_{n-1}(c)-T{\varphi }_n\parallel }_E+{\parallel {\varphi }_n(c)-T{\varphi }_n\parallel }_E{\parallel {\varphi }_n(c)-T{\varphi }_{n-1}\parallel }_E}{1+{\parallel {\varphi }_{n-1}(c)-T{\varphi }_{n-1}\parallel }_E+{\parallel {\varphi }_n(c)-T{\varphi }_n\parallel }_E},\hfill \\ \frac{{\parallel {\varphi }_{n-1}(c)-T{\varphi }_{n-1}\parallel }_E{\parallel {\varphi }_{n-1}(c)-T{\varphi }_n\parallel }_E+{\parallel {\varphi }_n(c)-T{\varphi }_n\parallel }_E{\parallel {\varphi }_n(c)-T{\varphi }_{n-1}\parallel }_E}{1+{\parallel {\varphi }_{n-1}(c)-T{\varphi }_n\parallel }_E+{\parallel {\varphi }_n(c)-T{\varphi }_{n-1}\parallel }_E}\}\hfill \\ =max\{{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0},\hfill \\ \frac{{\parallel {\varphi }_{n-1}(c)-{\varphi }_n(c)\parallel }_E{\parallel {\varphi }_{n-1}(c)-{\varphi }_{n+1}(c)\parallel }_E+{\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E{\parallel {\varphi }_n(c)-{\varphi }_n(c)\parallel }_E}{1+{\parallel {\varphi }_{n-1}(c)-{\varphi }_n(c)\parallel }_E+{\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E},\hfill \\ \frac{{\parallel {\varphi }_{n-1}(c)-{\varphi }_n(c)\parallel }_E{\parallel {\varphi }_{n-1}(c)-{\varphi }_{n+1}(c)\parallel }_E+{\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E{\parallel {\varphi }_n(c)-{\varphi }_n(c)\parallel }_E}{1+{\parallel {\varphi }_{n-1}(c)-{\varphi }_{n+1}(c)\parallel }_E+{\parallel {\varphi }_n(c)-{\varphi }_n(c)\parallel }_E}\}\hfill \\ =max\{{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0},\hfill \\ \frac{{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}{\parallel {\varphi }_{n-1}-{\varphi }_{n+1}\parallel }_{E_0}+{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}{\parallel {\varphi }_n-{\varphi }_n\parallel }_{E_0}}{1+{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}+{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}},\hfill \\ \frac{{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}{\parallel {\varphi }_{n-1}-{\varphi }_{n+1}\parallel }_{E_0}+{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}{\parallel {\varphi }_n-{\varphi }_n\parallel }_{E_0}}{1+{\parallel {\varphi }_{n-1}-{\varphi }_{n+1}\parallel }_{E_0}+{\parallel {\varphi }_n-{\varphi }_n\parallel }_{E_0}}\}\hfill \\ ={\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}.\hfill \end{array}$$

From (2.12) we conclude

$${\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\le \beta \left({\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}\right){\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}<{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}$$

(2.13)

for all $n\in \mathbb{N}$. So, the sequence $\{{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\}$ is decreasing in ${\mathbb{R}}_{+}$ and there exists $r\ge 0$ such that ${lim}_{n\to +\mathrm{\infty }}{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}=r$. Reviewing the proof of Theorem 2, we can show that $r=0$, i.e.,

$$\underset{n\to +\mathrm{\infty }}{lim}{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}=0.$$

(2.14)

Now, we prove that the sequence $\{{\varphi }_n\}$ is Cauchy in ${\mathcal{R}}_c$. If not, then

$$\underset{m,n\to +\mathrm{\infty }}{lim}{\parallel {\varphi }_m-{\varphi }_n\parallel }_{E_0}>0.$$

(2.15)

From the fact that T is a rational Geraghty contraction of type II, we have

$$\begin{array}{rcl}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}& \le & {\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}+{\parallel {\varphi }_{n+1}-{\varphi }_{m+1}\parallel }_{E_0}+{\parallel {\varphi }_{m+1}-{\varphi }_m\parallel }_{E_0}\\ \le & {\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}+\alpha ({\varphi }_n(c),T{\varphi }_n)\alpha ({\varphi }_m(c),T{\varphi }_m){\parallel T{\varphi }_n-T{\varphi }_m\parallel }_E\\ +{\parallel {\varphi }_{m+1}-{\varphi }_m\parallel }_{E_0}\\ \le & {\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}+\beta \left(M({\varphi }_n(c),{\varphi }_m(c))\right)M({\varphi }_n(c),{\varphi }_m(c))\\ +{\parallel {\varphi }_{m+1}-{\varphi }_m\parallel }_{E_0}.\end{array}$$

Letting $m,n\to \mathrm{\infty }$ in the above inequality and applying (2.14) we deduce

$$\underset{m,n\to \mathrm{\infty }}{lim}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}\le \underset{m,n\to \mathrm{\infty }}{lim}\beta \left(M({\varphi }_n(c),{\varphi }_m(c))\right)\underset{m,n\to \mathrm{\infty }}{lim}M({\varphi }_n(c),{\varphi }_m(c)),$$

(2.16)

where

$$\begin{array}{c}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}\hfill \\ \le M({\varphi }_n(c),{\varphi }_m(c))\hfill \\ =max\{{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0},\hfill \\ \frac{{\parallel {\varphi }_n(c)-T{\varphi }_n\parallel }_E{\parallel {\varphi }_n(c)-T{\varphi }_m\parallel }_E+{\parallel {\varphi }_m(c)-T{\varphi }_m\parallel }_E{\parallel {\varphi }_m(c)-T{\varphi }_n\parallel }_E}{1+{\parallel {\varphi }_n(c)-T{\varphi }_n\parallel }_E+{\parallel {\varphi }_m(c)-T{\varphi }_m\parallel }_E},\hfill \\ \frac{{\parallel {\varphi }_n(c)-T{\varphi }_n\parallel }_E{\parallel {\varphi }_n(c)-T{\varphi }_m\parallel }_E+{\parallel {\varphi }_m(c)-T{\varphi }_m\parallel }_E{\parallel {\varphi }_m(c)-T{\varphi }_n\parallel }_E}{1+{\parallel {\varphi }_n(c)-T{\varphi }_m\parallel }_E+{\parallel {\varphi }_m(c)-T{\varphi }_n\parallel }_E}\}\hfill \\ =max\{{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0},\hfill \\ \frac{{\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E{\parallel {\varphi }_n(c)-{\varphi }_{m+1}(c)\parallel }_E+{\parallel {\varphi }_m(c)-{\varphi }_{m+1}\parallel }_E{\parallel {\varphi }_m(c)-{\varphi }_{n+1}(c)\parallel }_E}{1+{\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E+{\parallel {\varphi }_m(c)-{\varphi }_{m+1}(c)\parallel }_E},\hfill \\ \frac{{\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E{\parallel {\varphi }_n(c)-{\varphi }_{m+1}(c)\parallel }_E+{\parallel {\varphi }_m(c)-{\varphi }_{m+1}(c)\parallel }_E{\parallel {\varphi }_m(c)-{\varphi }_{n+1}(c)\parallel }_E}{1+{\parallel {\varphi }_n(c)-{\varphi }_{m+1}(c)\parallel }_E+{\parallel {\varphi }_m(c)-{\varphi }_{n+1}(c)\parallel }_E}\}\hfill \\ =max\{{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0},\hfill \\ \frac{{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}{\parallel {\varphi }_n-{\varphi }_{m+1}\parallel }_{E_0}+{\parallel {\varphi }_m-{\varphi }_{m+1}\parallel }_{E_0}{\parallel {\varphi }_m-{\varphi }_{n+1}\parallel }_{E_0}}{1+{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}+{\parallel {\varphi }_m-{\varphi }_{m+1}\parallel }_{E_0}},\hfill \\ \frac{{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}{\parallel {\varphi }_n-{\varphi }_{m+1}\parallel }_{E_0}+{\parallel {\varphi }_m-{\varphi }_{m+1}\parallel }_{E_0}{\parallel {\varphi }_m-{\varphi }_{n+1}\parallel }_{E_0}}{1+{\parallel {\varphi }_n-{\varphi }_{m+1}\parallel }_{E_0}+{\parallel {\varphi }_m-{\varphi }_{n+1}\parallel }_{E_0}}\}.\hfill \end{array}$$

(2.17)

Letting $m,n\to \mathrm{\infty }$ in the above inequality and applying (2.14), we get

$$\underset{m,n\to +\mathrm{\infty }}{lim}M({\varphi }_n(c),{\varphi }_m(c))=\underset{m,n\to +\mathrm{\infty }}{lim}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}.$$

(2.18)

So, from (2.16) and (2.18), we obtain

$$\underset{m,n\to +\mathrm{\infty }}{lim\hspace{0.17em}sup}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}\le \underset{m,n\to +\mathrm{\infty }}{lim\hspace{0.17em}sup}\beta \left({\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}\right)\underset{m,n\to +\mathrm{\infty }}{lim\hspace{0.17em}sup}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}$$

and so by (2.15) we get, $1\le {lim\hspace{0.17em}sup}_{m,n\to +\mathrm{\infty }}\beta ({\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0})$. That is,

$$\underset{m,n\to +\mathrm{\infty }}{lim}\beta \left({\parallel {\varphi }_m-{\varphi }_n\parallel }_{E_0}\right)=1$$

and since $\beta \in \mathcal{F}$, we deduce

$$\underset{m,n\to +\mathrm{\infty }}{lim}{\parallel {\varphi }_m-{\varphi }_n\parallel }_{E_0}=0,$$

which is a contradiction. Consequently,

$$\underset{m,n\to +\mathrm{\infty }}{lim}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}=0$$

and hence $\{{\varphi }_n\}$ is a Cauchy sequence in ${\mathcal{R}}_c\subseteq E_0$. By completeness of $E_0$, we find that $\{{\varphi }_n\}$ converges to a point ${\varphi }^{\ast }\in E_0$, this means that ${\varphi }_n\to {\varphi }^{\ast }$, as $n\to +\mathrm{\infty }$. Since ${\mathcal{R}}_c$ is topologically closed, we deduce that ${\varphi }^{\ast }\in {\mathcal{R}}_c$. Now, since T is a rational Geraghty contraction of type II, we have

$$\begin{array}{c}{\parallel T{\varphi }^{\ast }-{\varphi }^{\ast }(c)\parallel }_E\hfill \\ \le {\parallel T{\varphi }^{\ast }-{\varphi }_n(c)\parallel }_E+{\parallel {\varphi }_n(c)-{\varphi }^{\ast }(c)\parallel }_E\hfill \\ ={\parallel T{\varphi }^{\ast }-T{\varphi }_{n-1}\parallel }_E+{\parallel {\varphi }_n-{\varphi }^{\ast }\parallel }_{E_0}\hfill \\ \le \alpha ({\varphi }^{\ast }(c),T{\varphi }^{\ast })\alpha ({\varphi }_{n-1}(c),T{\varphi }_{n-1}){\parallel T{\varphi }^{\ast }-T{\varphi }_{n-1}\parallel }_E+{\parallel {\varphi }_n-{\varphi }^{\ast }\parallel }_{E_0}\hfill \\ \le \beta \left(M({\varphi }^{\ast }(c),{\varphi }_{n-1}(c))\right)M({\varphi }^{\ast }(c),{\varphi }_{n-1}(c)).\hfill \end{array}$$

Taking the limit as $n\to \mathrm{\infty }$ in the above inequality, we get

$${\parallel T{\varphi }^{\ast }-{\varphi }^{\ast }(c)\parallel }_E\le \underset{n\to \mathrm{\infty }}{lim}\beta \left(M({\varphi }^{\ast }(c),{\varphi }_{n-1}(c))\right)\underset{n\to \mathrm{\infty }}{lim}M({\varphi }^{\ast }(c),{\varphi }_{n-1}(c)).$$

(2.19)

But

$$\begin{array}{c}M({\varphi }^{\ast }(c),{\varphi }_{n-1}(c))\hfill \\ =max\{{\parallel {\varphi }^{\ast }-{\varphi }_{n-1}\parallel }_{E_0},\hfill \\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E{\parallel {\varphi }^{\ast }(c)-T{\varphi }_{n-1}\parallel }_E+{\parallel {\varphi }_{n-1}(c)-T{\varphi }_{n-1}\parallel }_E{\parallel {\varphi }_{n-1}(c)-T{\varphi }^{\ast }\parallel }_E}{1+{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E+{\parallel {\varphi }_{n-1}(c)-T{\varphi }_{n-1}\parallel }_E},\hfill \\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E{\parallel {\varphi }^{\ast }(c)-T{\varphi }_{n-1}\parallel }_E+{\parallel {\varphi }_{n-1}(c)-T{\varphi }_{n-1}\parallel }_E{\parallel {\varphi }_{n-1}(c)-T{\varphi }^{\ast }\parallel }_E}{1+{\parallel {\varphi }^{\ast }(c)-T{\varphi }_{n-1}\parallel }_E+{\parallel {\varphi }_{n-1}(c)-T{\varphi }^{\ast }\parallel }_E}\}\hfill \\ =max\{{\parallel {\varphi }^{\ast }-{\varphi }_{n-1}\parallel }_{E_0},\hfill \\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E{\parallel {\varphi }^{\ast }(c)-{\varphi }_n(c)\parallel }_E+{\parallel {\varphi }_{n-1}(c)-{\varphi }_n(c)\parallel }_E{\parallel {\varphi }_{n-1}(c)-T{\varphi }^{\ast }\parallel }_E}{1+{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E+{\parallel {\varphi }_{n-1}(c)-{\varphi }_n(c)\parallel }_E},\hfill \\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E{\parallel {\varphi }^{\ast }(c)-{\varphi }_n(c)\parallel }_E+{\parallel {\varphi }_{n-1}(c)-{\varphi }_n(c)\parallel }_E{\parallel {\varphi }_{n-1}(c)-T{\varphi }^{\ast }\parallel }_E}{1+{\parallel {\varphi }^{\ast }(c)-{\varphi }_n(c)\parallel }_E+{\parallel {\varphi }_{n-1}(c)-T{\varphi }^{\ast }\parallel }_E}\}\hfill \\ =max\{{\parallel {\varphi }^{\ast }-{\varphi }_{n-1}\parallel }_{E_0},\hfill \\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E{\parallel {\varphi }^{\ast }-{\varphi }_n\parallel }_{E_0}+{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}{\parallel {\varphi }_{n-1}(c)-T{\varphi }^{\ast }\parallel }_E}{1+{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E+{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}},\hfill \\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E{\parallel {\varphi }^{\ast }-{\varphi }_n\parallel }_{E_0}+{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}{\parallel {\varphi }_{n-1}(c)-T{\varphi }^{\ast }\parallel }_E}{1+{\parallel {\varphi }^{\ast }-{\varphi }_n\parallel }_{E_0}+{\parallel {\varphi }_{n-1}-T{\varphi }^{\ast }\parallel }_{E_0}}\}.\hfill \end{array}$$

(2.20)

So,

$$\underset{n\to \mathrm{\infty }}{lim}M({\varphi }^{\ast }(c),{\varphi }_{n-1}(c))=0,$$

and by (2.19) and (2.20), we conclude

$${\parallel T{\varphi }^{\ast }-{\varphi }^{\ast }(c)\parallel }_E=0,$$

that is,

$$T{\varphi }^{\ast }={\varphi }^{\ast }(c),$$

which implies that ${\varphi }^{\ast }$ is a PPF dependent fixed point of T in ${\mathcal{R}}_c$. Finally, we prove the uniqueness of the PPF dependent fixed point of T in ${\mathcal{R}}_c$. Let ${\varphi }^{\ast }$ and ${\phi }^{\ast }$ be two PPF dependent fixed points of T in ${\mathcal{R}}_c$ such that ${\varphi }^{\ast }\ne {\phi }^{\ast }$. So, we obtain

$$\begin{array}{rcl}{\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0}& =& {\parallel {\varphi }^{\ast }(c)-{\phi }^{\ast }(c)\parallel }_E\\ =& {\parallel T{\varphi }^{\ast }-T{\phi }^{\ast }\parallel }_E\\ \le & \alpha ({\varphi }^{\ast }(c),T{\varphi }^{\ast })\alpha ({\phi }^{\ast }(c),T{\phi }^{\ast }){\parallel T{\varphi }^{\ast }-T{\phi }^{\ast }\parallel }_E\\ \le & \beta \left(M({\varphi }^{\ast }(c),{\phi }^{\ast }(c))\right)M({\varphi }^{\ast }(c),{\phi }^{\ast }(c)),\end{array}$$

where

$$\begin{array}{rcl}M({\varphi }^{\ast }(c),{\phi }^{\ast }(c))& =& max\{{\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0},\\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E{\parallel {\varphi }^{\ast }(c)-T{\phi }^{\ast }\parallel }_E+{\parallel {\phi }^{\ast }(c)-T{\phi }^{\ast }\parallel }_E{\parallel {\phi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E}{1+{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E+{\parallel {\phi }^{\ast }(c)-T{\phi }^{\ast }\parallel }_E},\\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E{\parallel {\varphi }^{\ast }(c)-T{\phi }^{\ast }\parallel }_E+{\parallel {\phi }^{\ast }(c)-T{\phi }^{\ast }\parallel }_E{\parallel {\phi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E}{1+{\parallel {\varphi }^{\ast }(c)-T{\phi }^{\ast }\parallel }_E+{\parallel {\phi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E}\}\\ =& {\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0}.\end{array}$$

Therefore,

$${\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0}\le \beta \left({\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0}\right){\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0}<{\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0},$$

which is a contradiction. Hence, ${\varphi }^{\ast }={\phi }^{\ast }$. Therefore, T has a unique PPF dependent fixed point in ${\mathcal{R}}_c$. This completes the proof. □

Definition 9 Let $\alpha :E\times E\to [0,\mathrm{\infty })$ and $T:E_0\to E$. We say that T is a rational Geraghty contraction of type III if there exist $\beta \in \mathcal{F}$ and $c\in I$ such that

$$\alpha (\varphi (c),T\varphi )\alpha (\xi (c),T\xi ){\parallel T\varphi -T\xi \parallel }_E\le \beta \left(M(\varphi (c),\xi (c))\right)M(\varphi (c),\xi (c))$$

for all $\varphi ,\xi \in E_0$, where

$$\begin{array}{rcl}M(\varphi (c),\xi (c))& =& max\{{\parallel \varphi -\xi \parallel }_{E_0},\frac{{\parallel \varphi (c)-T\varphi \parallel }_E{\parallel \xi (c)-T\xi \parallel }_E}{1+{\parallel \varphi -\xi \parallel }_{E_0}+{\parallel \varphi (c)-T\xi \parallel }_E+{\parallel \xi (c)-T\varphi \parallel }_E},\\ \frac{{\parallel \varphi (c)-T\xi \parallel }_E{\parallel \varphi (c)-\xi (c)\parallel }_E}{1+{\parallel \varphi (c)-T\varphi \parallel }_E+{\parallel \xi (c)-T\varphi \parallel }_E+{\parallel \xi (c)-T\xi \parallel }_E}\}.\end{array}$$

Theorem 4 Let $T:E_0\to E$ and $\alpha :E\times E\to [0,\mathrm{\infty })$ be two mappings satisfying the following assertions:

• (a) there exists $c\in I$ such that ${\mathcal{R}}_c$ is topologically closed and algebraically closed with respect to difference,

• (b) T is an ${\alpha }_c$-admissible,

• (c) T is a rational Geraghty contractive mapping of type III,

• (d) if $\{{\varphi }_n\}$ is a sequence in $E_0$ such that ${\varphi }_n\to \varphi$ as $n\to \mathrm{\infty }$ and $\alpha ({\varphi }_n(c),T{\varphi }_n)\ge 1$, then $\alpha (\varphi (c),T\varphi )\ge 1$ for all $n\in \mathbb{N}$,

• (e) there exists ${\varphi }_0\in {\mathcal{R}}_c$ such that $\alpha ({\varphi }_0(c),T{\varphi }_0)\ge 1$.

Then T has a unique PPF dependent fixed point ${\varphi }^{\ast }\in {\mathcal{R}}_c$. Moreover, for a fixed ${\varphi }_0\in {\mathcal{R}}_c$, if the sequence $\{{\varphi }_n\}$ of iterates of T defined by $T{\varphi }_{n-1}={\varphi }_n(c)$ for all $n\in \mathbb{N}$, then $\{{\varphi }_n\}$ converges to the PPF dependent fixed point of T in ${\mathcal{R}}_c$.

Proof Suppose that ${\varphi }_0$ be a point in ${\mathcal{R}}_c\subset E_0$ such that $\alpha ({\varphi }_0(c),T{\varphi }_0)\ge 1$. Since $T{\varphi }_0\in E$, there exists $x_1\in E$ such that $T{\varphi }_0=x_1$. Choose ${\varphi }_1\in {\mathcal{R}}_c$ such that $x_1={\varphi }_1(c)$. Since ${\varphi }_1\in {\mathcal{R}}_c\subset E_0$ and, by hypothesis, we get $T{\varphi }_1\in E$. This implies that there exists $x_2\in E$ such that $T{\varphi }_1=x_2$. Thus, we can choose ${\varphi }_2\in {\mathcal{R}}_c$ such that $x_2={\varphi }_2(c)$. Repeating this process, by induction, we can construct the sequence $\{{\varphi }_n\}$ in ${\mathcal{R}}_c\subset E_0$ such that $T{\varphi }_{n-1}={\varphi }_n(c)$ for all $n\in \mathbb{N}$. From the fact that ${\mathcal{R}}_c$ is algebraically closed with respect to difference it follows that

$${\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}={\parallel {\varphi }_{n-1}(c)-{\varphi }_n(c)\parallel }_E\text{for all }n\in \mathbb{N}.$$

Since T is ${\alpha }_c$-admissible and $\alpha ({\varphi }_0(c),{\varphi }_1(c))=\alpha ({\varphi }_0(c),T{\varphi }_0)\ge 1$, we deduce

$$\alpha ({\varphi }_1(c),T{\varphi }_1)=\alpha (T{\varphi }_0,T{\varphi }_1)\ge 1.$$

Continuing this process, we get $\alpha ({\varphi }_{n-1}(c),T{\varphi }_{n-1})\ge 1$ for all $n\in \mathbb{N}$. By the fact that T is a rational Geraghty contraction of type III, we have

$$\begin{array}{rl}{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}& ={\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E={\parallel T{\varphi }_{n-1}-T{\varphi }_n\parallel }_E\\ \le \alpha ({\varphi }_{n-1}(c),T{\varphi }_{n-1})\alpha ({\varphi }_n(c),T{\varphi }_n){\parallel T{\varphi }_{n-1}-T{\varphi }_n\parallel }_E\\ \le \beta \left(M({\varphi }_{n-1}(c),{\varphi }_n(c))\right)M({\varphi }_{n-1}(c),{\varphi }_n(c)).\end{array}$$

(2.21)

On the other hand,

$$\begin{array}{rcl}M({\varphi }_{n-1}(c),{\varphi }_n(c))& =& max\{{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0},\\ \frac{{\parallel {\varphi }_{n-1}(c)-T{\varphi }_{n-1}\parallel }_E{\parallel {\varphi }_n(c)-T{\varphi }_n\parallel }_E}{1+{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}+{\parallel {\varphi }_{n-1}(c)-T{\varphi }_n\parallel }_E+{\parallel {\varphi }_n(c)-T{\varphi }_{n-1}\parallel }_E},\\ \frac{{\parallel {\varphi }_{n-1}(c)-T{\varphi }_n\parallel }_E{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}}{1+{\parallel {\varphi }_{n-1}(c)-T{\varphi }_{n-1}\parallel }_E+{\parallel {\varphi }_n(c)-T{\varphi }_{n-1}\parallel }_E+{\parallel {\varphi }_n(c)-T{\varphi }_n\parallel }_E}\}\\ =& max\{{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0},\\ \frac{{\parallel {\varphi }_{n-1}(c)-{\varphi }_n(c)\parallel }_E{\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E}{1+{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}+{\parallel {\varphi }_{n-1}(c)-{\varphi }_{n+1}(c)\parallel }_E+{\parallel {\varphi }_n(c)-{\varphi }_n(c)\parallel }_E},\\ \frac{{\parallel {\varphi }_{n-1}(c)-{\varphi }_{n+1}(c)\parallel }_E{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}}{1+{\parallel {\varphi }_{n-1}(c)-{\varphi }_n(c)\parallel }_E+{\parallel {\varphi }_n(c)-{\varphi }_n(c)\parallel }_E+{\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E}\}\\ \le & max\{{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0},\\ \frac{{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}({\parallel {\varphi }_n-{\varphi }_{n-1}\parallel }_{E_0}+{\parallel {\varphi }_{n-1}-{\varphi }_{n+1}\parallel }_{E_0})}{1+{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}+{\parallel {\varphi }_{n-1}-{\varphi }_{n+1}\parallel }_{E_0}+{\parallel {\varphi }_n-{\varphi }_n\parallel }_{E_0}},\\ \frac{({\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}+{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}){\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}}{1+{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}+{\parallel {\varphi }_n-{\varphi }_n\parallel }_{E_0}+{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}}\}\\ =& {\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}.\end{array}$$

From (2.21) we conclude

$${\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\le \beta \left({\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}\right){\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}<{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}$$

(2.22)

for all $n\in \mathbb{N}$. This implies that the sequence $\{{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\}$ is decreasing in ${\mathbb{R}}_{+}$. Then there exists $r\ge 0$ such that ${lim}_{n\to +\mathrm{\infty }}{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}=r$. Repeating the proof of Theorem 2, we conclude that $r=0$. That is,

$$\underset{n\to +\mathrm{\infty }}{lim}{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}=0.$$

(2.23)

Now, we prove that the sequence $\{{\varphi }_n\}$ is Cauchy in ${\mathcal{R}}_c$. If not, then

$$\underset{m,n\to +\mathrm{\infty }}{lim}{\parallel {\varphi }_m-{\varphi }_n\parallel }_{E_0}>0.$$

(2.24)

Since T is a rational Geraghty contraction of type III, we have

$$\begin{array}{rcl}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}& \le & {\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}+{\parallel {\varphi }_{n+1}-{\varphi }_{m+1}\parallel }_{E_0}+{\parallel {\varphi }_{m+1}-{\varphi }_m\parallel }_{E_0}\\ \le & {\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}+\alpha ({\varphi }_n(c),T{\varphi }_n)\alpha ({\varphi }_m(c),T{\varphi }_m){\parallel T{\varphi }_n-T{\varphi }_m\parallel }_{E_0}\\ +{\parallel {\varphi }_{m+1}-{\varphi }_m\parallel }_{E_0}\\ \le & \beta \left(M({\varphi }_n(c),{\varphi }_m(c))\right)M({\varphi }_n(c),{\varphi }_m(c))\\ +{\parallel {\varphi }_{m+1}-{\varphi }_m\parallel }_{E_0}.\end{array}$$

Making $m,n\to \mathrm{\infty }$ in the above inequality and applying (2.23) we have

$$\underset{m,n\to \mathrm{\infty }}{lim}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}\le \underset{m,n\to \mathrm{\infty }}{lim}\beta \left(M({\varphi }_n(c),{\varphi }_m(c))\right)\underset{m,n\to \mathrm{\infty }}{lim}M({\varphi }_n(c),{\varphi }_m(c)).$$

(2.25)

Also,

$$\begin{array}{rcl}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}& \le & M({\varphi }_n(c),{\varphi }_m(c))\\ =& max\{{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0},\\ \frac{{\parallel {\varphi }_n(c)-T{\varphi }_n\parallel }_E{\parallel {\varphi }_m(c)-T{\varphi }_m\parallel }_E}{1+{\parallel {\varphi }_n(c)-{\varphi }_m(c)\parallel }_{E_0}+{\parallel {\varphi }_n(c)-T{\varphi }_m\parallel }_E+{\parallel {\varphi }_m(c)-T{\varphi }_n\parallel }_E},\\ \frac{{\parallel {\varphi }_n(c)-T{\varphi }_m\parallel }_E{\parallel {\varphi }_n(c)-{\varphi }_m(c)\parallel }_E}{1+{\parallel {\varphi }_n(c)-T{\varphi }_n\parallel }_E+{\parallel {\varphi }_m(c)-T{\varphi }_n\parallel }_E+{\parallel {\varphi }_m(c)-T{\varphi }_m\parallel }_E}\}\\ =& max\{{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0},\\ \frac{{\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E{\parallel {\varphi }_m(c)-{\varphi }_{m+1}(c)\parallel }_E}{1+{\parallel {\varphi }_n(c)-{\varphi }_m(c)\parallel }_{E_0}+{\parallel {\varphi }_n(c)-{\varphi }_{m+1}(c)\parallel }_E+{\parallel {\varphi }_m(c)-{\varphi }_{n+1}(c)\parallel }_E},\\ \frac{{\parallel {\varphi }_n(c)-{\varphi }_{m+1}(c)\parallel }_E{\parallel {\varphi }_n(c)-{\varphi }_m(c)\parallel }_E}{1+{\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E+{\parallel {\varphi }_m(c)-{\varphi }_{n+1}(c)\parallel }_E+{\parallel {\varphi }_m(c)-{\varphi }_{m+1}(c)\parallel }_E}\}\\ \le & max\{{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0},\frac{{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}{\parallel {\varphi }_m-{\varphi }_{m+1}\parallel }_{E_0}}{1+{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}+{\parallel {\varphi }_n-{\varphi }_{m+1}\parallel }_{E_0}+{\parallel {\varphi }_m-{\varphi }_{n+1}\parallel }_{E_0}},\\ \frac{({\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}+{\parallel {\varphi }_m-{\varphi }_{n+1}\parallel }_{E_0}+{\parallel {\varphi }_m-{\varphi }_{m+1}\parallel }_{E_0}){\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}}{1+{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}+{\parallel {\varphi }_m-{\varphi }_{n+1}\parallel }_{E_0}+{\parallel {\varphi }_m-{\varphi }_{m+1}\parallel }_{E_0}}\}.\end{array}$$

Letting $m,n\to \mathrm{\infty }$ in the above inequality and applying (2.23), we get

$$\underset{m,n\to +\mathrm{\infty }}{lim}M({\varphi }_n,{\varphi }_m)=\underset{m,n\to +\mathrm{\infty }}{lim}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}.$$

(2.26)

Hence, from (2.25) and (2.26), we obtain

$$\underset{m,n\to +\mathrm{\infty }}{lim\hspace{0.17em}sup}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}\le \underset{m,n\to +\mathrm{\infty }}{lim\hspace{0.17em}sup}\beta \left({\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}\right)\underset{m,n\to +\mathrm{\infty }}{lim\hspace{0.17em}sup}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}$$

and so by (2.24) we get $1\le {lim\hspace{0.17em}sup}_{m,n\to +\mathrm{\infty }}\beta ({\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0})$. That is,

$$\underset{m,n\to +\mathrm{\infty }}{lim}\beta \left({\parallel {\varphi }_m-{\varphi }_n\parallel }_{E_0}\right)=1$$

and since $\beta \in \mathcal{F}$, we deduce

$$\underset{m,n\to +\mathrm{\infty }}{lim}{\parallel {\varphi }_m-{\varphi }_n\parallel }_{E_0}=0,$$

which is a contradiction. Consequently,

$$\underset{m,n\to +\mathrm{\infty }}{lim}{\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}=0$$

and hence $\{{\varphi }_n\}$ is a Cauchy sequence in ${\mathcal{R}}_c\subseteq E_0$. Completeness of $E_0$ shows that $\{{\varphi }_n\}$ converges to a point ${\varphi }^{\ast }\in E_0$, this means that ${\varphi }_n\to {\varphi }^{\ast }$, as $n\to +\mathrm{\infty }$. Since ${\mathcal{R}}_c$ is topologically closed, we deduce that ${\varphi }^{\ast }\in {\mathcal{R}}_c$. Now, since T is a rational Geraghty contraction of type III, we have

$$\begin{array}{c}{\parallel T{\varphi }^{\ast }-{\varphi }^{\ast }(c)\parallel }_E\hfill \\ \le {\parallel T{\varphi }^{\ast }-{\varphi }_n(c)\parallel }_E+{\parallel {\varphi }_n(c)-{\varphi }^{\ast }(c)\parallel }_E\hfill \\ ={\parallel T{\varphi }^{\ast }-T{\varphi }_{n-1}\parallel }_E+{\parallel {\varphi }_n-{\varphi }^{\ast }\parallel }_{E_0}\hfill \\ \le \alpha ({\varphi }^{\ast }(c),T{\varphi }^{\ast })\alpha ({\varphi }_{n-1}(c),T{\varphi }_{n-1}){\parallel T{\varphi }^{\ast }-T{\varphi }_{n-1}\parallel }_E+{\parallel {\varphi }_n-{\varphi }^{\ast }\parallel }_{E_0}\hfill \\ \le \beta \left(M({\varphi }^{\ast }(c),{\varphi }_{n-1}(c))\right)M({\varphi }^{\ast }(c),{\varphi }_{n-1}(c)).\hfill \end{array}$$

Taking the limit as $n\to \mathrm{\infty }$ in the above inequality, we get

$${\parallel T{\varphi }^{\ast }-{\varphi }^{\ast }(c)\parallel }_E\le \underset{n\to \mathrm{\infty }}{lim}\beta \left(M({\varphi }^{\ast }(c),{\varphi }_{n-1}(c))\right)\underset{n\to \mathrm{\infty }}{lim}M({\varphi }^{\ast }(c),{\varphi }_{n-1}(c)).$$

(2.27)

But

$$\begin{array}{rcl}M({\varphi }^{\ast }(c),{\varphi }_{n-1}(c))& =& max\{{\parallel {\varphi }^{\ast }-{\varphi }_{n-1}\parallel }_{E_0},\\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E{\parallel {\varphi }_{n-1}(c)-T{\varphi }_{n-1}\parallel }_E}{1+{\parallel {\varphi }^{\ast }-{\varphi }_{n-1}\parallel }_{E_0}+{\parallel {\varphi }^{\ast }(c)-T{\varphi }_{n-1}\parallel }_E+{\parallel {\varphi }_{n-1}(c)-T{\varphi }^{\ast }\parallel }_E},\\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }_{n-1}\parallel }_E{\parallel {\varphi }^{\ast }-{\varphi }_{n-1}\parallel }_{E_0}}{1+{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E+{\parallel {\varphi }_{n-1}(c)-T{\varphi }^{\ast }\parallel }_E+{\parallel {\varphi }_{n-1}(c)-T{\varphi }_{n-1}\parallel }_E}\}\\ =& max\{{\parallel {\varphi }^{\ast }-{\varphi }_{n-1}\parallel }_{E_0},\\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E{\parallel {\varphi }_{n-1}(c)-{\varphi }_n(c)\parallel }_E}{1+{\parallel {\varphi }^{\ast }-{\varphi }_{n-1}\parallel }_{E_0}+{\parallel {\varphi }^{\ast }(c)-{\varphi }_n(c)\parallel }_E+{\parallel {\varphi }_{n-1}(c)-T{\varphi }^{\ast }\parallel }_E},\\ \frac{{\parallel {\varphi }^{\ast }(c)-{\varphi }_n(c)\parallel }_E{\parallel {\varphi }^{\ast }-{\varphi }_{n-1}\parallel }_{E_0}}{1+{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E+{\parallel {\varphi }_{n-1}(c)-T{\varphi }^{\ast }\parallel }_E+{\parallel {\varphi }_{n-1}(c)-{\varphi }_n(c)\parallel }_E}\}\\ =& max\{{\parallel {\varphi }^{\ast }-{\varphi }_{n-1}\parallel }_{E_0},\\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}}{1+{\parallel {\varphi }^{\ast }-{\varphi }_{n-1}\parallel }_{E_0}+{\parallel {\varphi }^{\ast }-{\varphi }_n\parallel }_{E_0}+{\parallel {\varphi }_{n-1}(c)-T{\varphi }^{\ast }\parallel }_E},\\ \frac{{\parallel {\varphi }^{\ast }-{\varphi }_n\parallel }_{E_0}{\parallel {\varphi }^{\ast }-{\varphi }_{n-1}\parallel }_{E_0}}{1+{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E+{\parallel {\varphi }_{n-1}(c)-T{\varphi }^{\ast }\parallel }_E+{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}}\}.\end{array}$$

(2.28)

Therefore, from (2.27) and (2.28), we deduce that

$${\parallel T{\varphi }^{\ast }-{\varphi }^{\ast }(c)\parallel }_E=0,$$

that is,

$$T{\varphi }^{\ast }={\varphi }^{\ast }(c),$$

which implies that ${\varphi }^{\ast }$ is a PPF dependent fixed point of T in ${\mathcal{R}}_c$. Suppose that ${\varphi }^{\ast }$ and ${\phi }^{\ast }$ are two PPF dependent fixed points of T in ${\mathcal{R}}_c$ such that ${\varphi }^{\ast }\ne {\phi }^{\ast }$. So,

$$\begin{array}{rcl}{\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0}& =& {\parallel {\varphi }^{\ast }(c)-{\phi }^{\ast }(c)\parallel }_E={\parallel T{\varphi }^{\ast }-T{\phi }^{\ast }\parallel }_E\\ \le & \alpha ({\varphi }^{\ast }(c),T{\varphi }^{\ast })\alpha ({\phi }^{\ast }(c),T{\phi }^{\ast }){\parallel T{\varphi }^{\ast }-T{\phi }^{\ast }\parallel }_E\\ \le & \beta \left(M({\varphi }^{\ast }(c),{\phi }^{\ast }(c))\right)M({\varphi }^{\ast }(c),{\phi }^{\ast }(c)),\end{array}$$

where

$$\begin{array}{rcl}M({\varphi }^{\ast }(c),{\phi }^{\ast }(c))& =& max\{{\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0},\\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E{\parallel {\phi }^{\ast }(c)-T{\phi }^{\ast }\parallel }_E}{1+{\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0}+{\parallel {\varphi }^{\ast }(c)-T{\phi }^{\ast }\parallel }_E+{\parallel {\phi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E},\\ \frac{{\parallel {\varphi }^{\ast }(c)-T{\phi }^{\ast }\parallel }_E{\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0}}{1+{\parallel {\varphi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E+{\parallel {\phi }^{\ast }(c)-T{\varphi }^{\ast }\parallel }_E+{\parallel {\phi }^{\ast }(c)-T{\phi }^{\ast }\parallel }_E}\}\\ =& {\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0}.\end{array}$$

Therefore,

$${\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0}\le \beta \left({\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0}\right){\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0}<{\parallel {\varphi }^{\ast }-{\phi }^{\ast }\parallel }_{E_0},$$

which is a contradiction. Hence, ${\varphi }^{\ast }={\phi }^{\ast }$. Then T has a unique PPF dependent fixed point in ${\mathcal{R}}_c$. □

Corollary 1 Let $T:E_0\to E$ and $\alpha :E\times E\to [0,\mathrm{\infty })$ be two mappings satisfying the following assertions:

• (a) there exists $c\in I$ such that ${\mathcal{R}}_c$ is topologically closed and algebraically closed with respect to difference,

• (b) T is an ${\alpha }_c$-admissible,

• (c) assume that

  $$\alpha (\varphi (c),T\varphi )\alpha (\xi (c),T\xi ){\parallel T\varphi -T\xi \parallel }_E\le rM(\varphi (c),\xi (c))$$

for all $\varphi ,\xi \in E_0$, where $0\le r<1$ and

$$\begin{array}{rcl}M(\varphi (c),\xi (c))& =& max\{{\parallel \varphi -\xi \parallel }_{E_0},\frac{{\parallel \varphi (c)-T\varphi \parallel }_E{\parallel \xi (c)-T\xi \parallel }_E}{1+{\parallel \varphi -\xi \parallel }_{E_0}},\\ \frac{{\parallel \varphi (c)-T\varphi \parallel }_E{\parallel \xi (c)-T\xi \parallel }_E}{1+{\parallel T\varphi -T\xi \parallel }_E}\}\end{array}$$

or

$$\begin{array}{rcl}M(\varphi (c),\xi (c))& =& max\{{\parallel \varphi -\xi \parallel }_{E_0},\\ \frac{{\parallel \varphi (c)-T\varphi \parallel }_E{\parallel \varphi (c)-T\xi \parallel }_E+{\parallel \xi (c)-T\xi \parallel }_E{\parallel \xi (c)-T\varphi \parallel }_E}{1+{\parallel \varphi -T\varphi \parallel }_E+{\parallel \xi -T\xi \parallel }_E},\\ \frac{{\parallel \varphi (}_{}}{}\end{array}$$