Abstraction of Interpolative Reich-Rus-Ćirić-Type Contractions and Simplest Proof Technique

Abstract

The concept of symmetry is a very vast topic that is involved in the studies of several phenomena. This concept enables us to discuss the phenomenon in some systematic pattern depending upon the type of phenomenon. Each phenomenon has its own type of symmetry. The phenomenon that is used in the discussion of this article is a symmetric distance-measuring function. This article presents the notions of abstract interpolative Reich-Rus-Ćirić-type contractions with a shrink map and examines the existence of $\varphi$-fixed points for such maps in complete metric space. These notions are defined through special types of simulation functions. The proof technique of the results presented in this article is easy to understand compared with the existing literature on interpolative Reich-Rus-Ćirić-type contractions.

1. Introduction and Preliminaries

Metric fixed point theory has a significant contribution to nonlinear analysis with its applications. This branch of fixed point theory is based on the work of the famous mathematician Banach. He proved that [1], on a complete metric space, every contraction map possesses a unique fixed point. Later on, Kannan [2] and Chatterjea [3] modified the contraction inequality to study the existence of fixed points of discontinuous self-maps on a complete metric space. Afterward, this field has flourished with several interesting results. A few results have been obtained for the following aspects:

(1)

Modifying contraction inequality,

(2)

Modifying distance measuring function.

Recently, Karapınar [4] derived the interpolative Kannan contraction, which can be considered a modified form of the Kannan contraction. Inspiration from this work led several researchers to extend the existing contraction type inequalities in the pattern of interpolative Kannan contraction.

A few generalizations of contraction inequality have been obtained using some special types of simulation functions, for example [5,6].

Symmetry is a very vast topic that is involved in the studies of several phenomena. Each phenomenon has its own definition of symmetry, which helps to discuss the phenomenon in a systematic pattern. Metric space is a symmetric distance measuring function, which is used in the discussion of this article. In the literature related to interpolative Kannan contractions, we have seen several results based on the symmetric distance measuring function, for example, [7,8], and the asymmetric distance measuring function, for example, [9,10].

In this article, we use special types of simulation functions to extend interpolative Reich-Rus-Ćirić-type contraction inequalities. The proof technique of the fixed point results involving interpolative contraction type inequalities is more complicated than the proof technique of the fixed point results involving contraction type inequalities. With the help of a simulation function, we have tried minimizing these complications of the proof technique, and now the presented proofs are easier to understand.

Before moving on to the next section, we will recall some basic concepts such as interpolative Kannan contraction, a few generalizations of the interpolative Kannan contraction, well-known simulation functions and some other notions that are required for the next section.

Let $(V,d_V)$ be a metric space and let $Q:V\to V$ be a self map. Then, we have the following notions.

• A map $Q:V\to V$ is said to be an interpolative Kannan contraction [4], if

  $$d_V(Qk,Ql)\le \eta d_V{(k,Qk)}^{{\omega }_1}{d}_V{(l,Ql)}^{1-{\omega }_1}$$

  for all $k,l\in V$ with $k\ne Qk$, where $\eta \in [0,1)$ and ${\omega }_1\in (0,1)$.
  Later on, it was observed by Karapinar et al. [11] that the above inequality does not ensure the existence of a unique fixed point of a map in complete metric space. Hence, to discuss the uniqueness of a fixed point, the above inequality was redefined in the following way.
• A map $Q:V\to V$ is said to be an improved interpolative Kannan contraction [11], if

  $$d_V(Qk,Ql)\le \eta d_V{(k,Qk)}^{{\omega }_1}{d}_V{(l,Ql)}^{1-{\omega }_1}$$

  for all $k,l\in V\backslash Fix\left(Q\right)$, where $\eta \in [0,1)$, ${\omega }_1\in (0,1)$ and $Fix\left(Q\right)=\{k\in V:Qk=k\}$.
• A map $Q:V\to V$ is said to be an interpolative Reich-Rus-Ćirić-type contraction [12], if

  $$\begin{array}{ccc}\hfill d_V(Qk,Ql)& \le & \eta d_V{(k,l)}^{{\omega }_1}{d}_V{(k,Qk)}^{{\omega }_2}{d}_V{(l,Ql)}^{1-{\omega }_1-{\omega }_2}\hfill \end{array}$$

  for each $k,l\in V\backslash Fix\left(Q\right)$, where $\eta \in [0,1)$ and ${\omega }_1,{\omega }_2\in (0,1)$ with ${\omega }_1+{\omega }_2<1$.

In the literature, $CB\left(V\right)$ represents the collection of all nonvoid closed and bounded subsets of V and the Pompeiu–Hausdorff distance is a map $H_V:CB\left(V\right)\times CB\left(V\right)\to [0,\infty )$ defined by

$$H_V(E,F)=max\{\underset{e\in E}{sup}{d}_V(e,F),\underset{f\in F}{sup}{d}_V(f,E)\}$$

where $d_V(f,E)=inf\{d_V(f,e):e\in E\}$.

A set-valued generalization of interpolative Reich-Rus-Ćirić-type contraction is defined in the way: A map $Q:V\to CB\left(V\right)$ is said to be a set-valued interpolative Reich-Rus-Ćirić-type contraction [13], if

$$\begin{array}{ccc}\hfill H_V(Qk,Ql)& \le & \eta d_V{(k,l)}^{{\omega }_1}{d}_V{(k,Qk)}^{{\omega }_2}{d}_V{(l,Ql)}^{1-{\omega }_1-{\omega }_2}\hfill \end{array}$$

for each $k,l\in V\backslash Fix\left(Q\right)$, where $\eta \in [0,1)$ and ${\omega }_1,{\omega }_2\in (0,1)$ with ${\omega }_1+{\omega }_2<1$.

In the literature, we have seen many auxiliary type functions from $[0,\infty )\times [0,\infty )$ into $\mathbb{R}$, for example, simulation functions, R-functions and C-class functions. Recently, Karapinar [14] used the simulation function $\zeta :[0,\infty )\times [0,\infty )\to \mathbb{R}$ given by Khojasteh et al. [15] to define the following notion.

A map $Q:V\to V$ is said to be an interpolative Hardy–Rogers type Z-contraction, if

$$\begin{array}{c}\hfill \zeta (d_V(Qk,Ql),C(k,l))\ge 0,\end{array}$$

for each $k,l\in V\backslash Fix\left(Q\right)$, where ${\omega }_1,{\omega }_2,{\omega }_3\in (0,1)$ with ${\omega }_1+{\omega }_2+{\omega }_3<1$, and

$$C(k,l)=d_V{(k,l)}^{{\omega }_1}{d}_V{(k,Qk)}^{{\omega }_2}{d}_V{(l,Ql)}^{{\omega }_3}{\left[\frac{d_V(k,Ql)+d_V(l,Qk)}{2}\right]}^{1-{\omega }_1-{\omega }_2-{\omega }_3}.$$

A few more studies related to interpolative type contractions are available in [16,17,18].

In the next section, we use the following family of functions defined in [19]:

${\Theta }_F$ is the collection of functions ${\theta }_f:{[0,\infty )}^4\to [0,\infty )$ with the given properties

θ1:

${\theta }_f(d,b,c,0)=0\forall d,b,c\in [0,\infty )$;

θ2:

continuous and nondecreasing.

It is well-known that for a self-map $Q:V\to V$, a point $v\in V$ with $v=Qv$ is called a fixed point of Q. If v is a fixed point of Q with $\varphi \left(v\right)=0$ for a map $\varphi :V\to [0,\infty )$, then v is called a $\varphi$-fixed point of Q. This notion is presented in [20].

2. Results

In this section, we denote ${\Xi }_F$ by the collection of functions ${\xi }_f:{[0,\infty )}^3\to [0,\infty )$ such that

(f1)

${\xi }_f$ is nondecreasing in each coordinate;

(f2)

${\xi }_f(g^{{\omega }_1},g^{{\omega }_2},g^{{\omega }_3})\le g$ for each $g\in (0,\infty )$ and for each ${\omega }_1,{\omega }_2,{\omega }_3\in [0,1]$ with ${\omega }_1+{\omega }_2+{\omega }_3=1$.

Example 1.

The following functions belong to ${\Xi }_F$.

(E1)

${\xi }_f(a,b,c)=abc$;

(E2)

${\xi }_f(a,b,c)=\left(\frac{ac}{1+b}\right)\left(\frac{ab}{1+c}\right)\left(\frac{bc}{1+a}\right)$.

Throughout this article, ${\xi }_f$ belongs to ${\Xi }_f$, ${\theta }_f$ belongs to ${\Theta }_F$, $\varphi$ represents a map from V into $[0,\infty )$, and $(V,d_V)$ is a metric space.

The following definition is the first form of abstract interpolative Reich-Rus-Ćirić type contraction with a shrink map.

Definition 1.

A self-map $Q:V\to V$ is called an abstract interpolative Reich-Rus-Ćirić type-I contraction with ϕ shrink, if the below-stated inequalities hold:

$$\begin{array}{ccc}\hfill d_V(Qk,Ql)& \le & \eta {\xi }_f\left(d_V{(k,l)}^{{\omega }_1},d_V{(k,Qk)}^{{\omega }_2},d_V{(l,Ql)}^{{\omega }_3}\right)\hfill \\ & & +L{\theta }_f\left(d_V{(k,l)}^{{\omega }_1},d_V{(k,Qk)}^{{\omega }_2},d_V{(l,Ql)}^{{\omega }_3},d_V{(l,Qk)}^{{\omega }_4}\right)\hfill \end{array}$$

(1)

for each $k,l\in V\backslash Fix\left(Q\right)$ with $l\ne k$, where ${\omega }_1,{\omega }_2,{\omega }_3\in [0,1]$ with ${\omega }_1+{\omega }_2+{\omega }_3=1$, ${\omega }_4>0$, and $L\ge 0$;

for every $l\in V$, we have

$$\begin{array}{c}\hfill \varphi \left(Ql\right)\le \eta \varphi \left(l\right),\end{array}$$

(2)

where $\eta \in [0,1)$ and $Fix\left(Q\right)=\{v\in V:v=Qv\}$.

The following theorem ensures the existence of $\varphi$-fixed points of the map Q satisfying the above definition.

Theorem 1.

Let $Q:V\to V$ be an abstract interpolative Reich-Rus-Ćirić type-I contraction with ϕ shrink on a complete metric space $(V,d_V)$. Then at least one ϕ-fixed point of Q exists in V.

Proof. 

Take an arbitrary point $l_0\in V$, and define an iterative sequence $l_n=Q{l}_{n-1}\forall n\in \mathbb{N}$. If $l_{n_0}=l_{n_0+1}$ for some $n_0$, then $l_{n_0}$ is a fixed point of Q. Moreover, by (2) we get $\varphi \left(l_{n_0}\right)=\varphi \left(Q{l}_{n_0}\right)\le \lambda \varphi \left(l_{n_0}\right)$. This gives $\varphi \left(l_{n_0}\right)=0$. Hence, $l_{n_0}$ is a $\varphi$-fixed point of Q. Now, consider $l_{n-1}\ne l_n\forall n\in \mathbb{N}$. By (1), for each $n\in \mathbb{N}$, we get

$$\begin{array}{ccc}\hfill d_V(Q{l}_{n-1},Q{l}_n)& \le & \eta {\xi }_f\left(d_V{(l_{n-1},l_n)}^{{\omega }_1},d_V{(l_{n-1},Q{l}_{n-1})}^{{\omega }_2},d_V{(l_n,Q{l}_n)}^{{\omega }_3}\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_{n-1},l_n)}^{{\omega }_1},d_V{(l_{n-1},Q{l}_{n-1})}^{{\omega }_2},d_V{(l_n,Q{l}_n)}^{{\omega }_3},d_V{(l_n,Q{l}_{n-1})}^{{\omega }_4}\right).\hfill \end{array}$$

(3)

That is,

$$\begin{array}{ccc}\hfill d_V(l_n,l_{n+1})& \le & \eta {\xi }_f\left(d_V{(l_{n-1},l_n)}^{{\omega }_1},d_V{(l_{n-1},l_n)}^{{\omega }_2},d_V{(l_n,l_{n+1})}^{{\omega }_3}\right)\forall n\in \mathbb{N}.\hfill \end{array}$$

(4)

Now, claim that $d_V(l_n,l_{n+1})<d_V(l_{n-1},l_n)\forall n\in \mathbb{N}$. If it is wrong, then we have $m_0\in N$ with $d_V(l_{m_0},l_{m_0+1})\ge d_V(l_{m_0-1},l_{m_0})$. By (4) we get

$$\begin{array}{ccc}\hfill d_V(l_{m_0},l_{m_0+1})& \le & \eta {\xi }_f\left(d_V{(l_{m_0-1},l_{m_0})}^{{\omega }_1},d_V{(l_{m_0-1},l_{m_0})}^{{\omega }_2},d_V{(l_{m_0},l_{m_0+1})}^{{\omega }_3}\right)\hfill \\ & \le & \eta {\xi }_f\left(d_V{(l_{m_0},l_{m_0+1})}^{{\omega }_1},d_V{(l_{m_0},l_{m_0+1})}^{{\omega }_2},d_V{(l_{m_0},l_{m_0+1})}^{{\omega }_3}\right)\hfill \\ & \le & \eta d_V(l_{m_0},l_{m_0+1})\hfill \end{array}$$

which is only possible when $d_V(l_{m_0},l_{m_0+1})=0$, and it contradicts our assumption. Thus, the claim is true. Since $d_V(l_n,l_{n+1})<d_V(l_{n-1},l_n)\forall n\in \mathbb{N}$, then (4) we get

$$\begin{array}{ccc}\hfill d_V(l_n,l_{n+1})& \le & \eta {\xi }_f\left(d_V{(l_{n-1},l_n)}^{{\omega }_1},d_V{(l_{n-1},l_n)}^{{\omega }_2},d_V{(l_n,l_{n+1})}^{{\omega }_3}\right)\hfill \\ & \le & \eta {\xi }_f\left(d_V{(l_{n-1},l_n)}^{{\omega }_1},d_V{(l_{n-1},l_n)}^{{\omega }_2},d_V{(l_{n-1},l_n)}^{{\omega }_3}\right)\hfill \\ & \le & \eta d_V(l_{n-1},l_n)\forall n\in \mathbb{N}.\hfill \end{array}$$

(5)

The above inequality implies that

$$\begin{array}{c}\hfill d_V(l_n,l_{n+1})\le {\eta }^n{d}_V(l_0,l_1)\forall n\in \mathbb{N}.\end{array}$$

(6)

To verify that the sequence $\left\{l_n\right\}$ is Cauchy. Consider $m,n\in \mathbb{N}$ with $n>m$. By triangle inequality and (6) we obtain

$$\begin{array}{ccc}\hfill d_V(l_m,l_n)& \le & \sum _{j=m}^{n-1}{d}_V(l_j,l_{j+1})\le \sum _{j=m}^{n-1}{\eta }^j{d}_V(l_0,l_1).\hfill \end{array}$$

Since ${\sum }_{j=1}^{\infty }{\eta }^j$ is a convergent series, thus, by the above inequality, we get ${lim}_{n,m\to \infty }$$d_V(l_m,l_n)=0$. As $(V,d_V)$ is complete and $\left\{l_n\right\}$ is Cauchy in V, then there exists an element $l^{*}\in V$ with $l_n\to l^{*}$. Now, claim that $l^{*}=Q{l}^{*}$. If it is wrong, then $d_V(l^{*},Q{l}^{*})>0$. Since $\left\{l_n\right\}$ is an iterative sequence with $l_n\to l^{*}$, thus, we get

$$max\{d_V(l_n,l^{*}),d_V(l_n,l_{n+1}),d_V(l^{*},Q{l}^{*})\}=d_V(l^{*},Q{l}^{*})\forall n\ge N_0$$

(7)

for some $N_0\in \mathbb{N}$. By (1), for each $n\in \mathbb{N}$, we obtain

$$\begin{array}{ccc}\hfill d_V(Q{l}_n,Q{l}^{*})& \le & \eta {\xi }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3}\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3},d_V{(l^{*},Q{l}_n)}^{{\omega }_4}\right).\hfill \end{array}$$

(8)

From (7) and (8), for each $n\ge N_0$, we get

$$\begin{array}{ccc}\hfill d_V(l_{n+1},Q{l}^{*})& \le & \eta {\xi }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,l_{n+1})}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3}\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3},d_V{(l^{*},l_{n+1})}^{{\omega }_4}\right)\hfill \\ & \le & \eta {\xi }_f\left(d_V{(l^{*},Q{l}^{*})}^{{\omega }_1},d_V{(l^{*},Q{l}^{*})}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3}\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3},d_V{(l^{*},l_{n+1})}^{{\omega }_4}\right)\hfill \\ & \le & \eta d_V(l^{*},Q{l}^{*})\hfill \\ & & +L{\theta }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3},d_V{(l^{*},l_{n+1})}^{{\omega }_4}\right).\hfill \end{array}$$

(9)

By applying the limit $n\to \infty$ in (9), we get

$$d_V(l^{*},Q{l}^{*})\le \eta d_V(l^{*},Q{l}^{*}).$$

As $\eta <1$, thus, the above inequality, only exists when $d_V(l^{*},Q{l}^{*})=0$. Hence, the claim is correct. Since $l^{*}=Q{l}^{*}$, then by (2) we get

$$\varphi \left(l^{*}\right)=\varphi \left(Q{l}^{*}\right)\le \lambda \varphi \left(l^{*}\right).$$

This implies that $\varphi \left(l^{*}\right)=0$. Hence, $l^{*}$ is $\varphi$-fixed point of Q. □

By letting ${\xi }_f(a,b,c)=abc$ and ${\theta }_f(a,b,c,d)=abcd$ in Theorem 1, we get the following result.

Corollary 1.

Let $(V,d_V)$ be a complete metric space. Let $Q:V\to V$ and $\varphi :V\to [0,\infty )$ be two maps such that

$$\begin{array}{ccc}\hfill d_V(Qk,Ql)& \le & \eta d_V{(k,l)}^{{\omega }_1}{d}_V{(k,Qk)}^{{\omega }_2}{d}_V{(l,Ql)}^{{\omega }_3}\hfill \\ & & +L{d}_V{(k,l)}^{{\omega }_1}{d}_V{(k,Qk)}^{{\omega }_2}{d}_V{(l,Ql)}^{{\omega }_3}{d}_V{(l,Qk)}^{{\omega }_4}\hfill \end{array}$$

for each $k,l\in V\backslash Fix\left(Q\right)$ with $l\ne k$, where ${\omega }_1,{\omega }_2,{\omega }_3\in [0,1]$ with ${\omega }_1+{\omega }_2+{\omega }_3=1$ and ${\omega }_4>0$; further, for every $l\in V$, we have

$$\begin{array}{c}\hfill \varphi \left(Ql\right)\le \eta \varphi \left(l\right),\end{array}$$

where $\eta \in [0,1)$ and $L\ge 0$. Then at least one ϕ-fixed point of Q exists in V.

By taking ${\omega }_1={\omega }_4=1$ and ${\omega }_2={\omega }_3=0$ in the above mentioned corollary, we obtain the following result.

Corollary 2.

Let $(V,d_V)$ be a complete metric space. Let $Q:V\to V$ and $\varphi :V\to [0,\infty )$ be two maps such that

$$\begin{array}{ccc}\hfill d_V(Qk,Ql)& \le & \eta d_V(k,l)+L{d}_V(k,l)d_V(l,Qk)\hfill \end{array}$$

for each $k,l\in V\backslash Fix\left(Q\right)$ with $l\ne k$; further, for every $l\in V$, we have

$$\begin{array}{c}\hfill \varphi \left(Ql\right)\le \eta \varphi \left(l\right),\end{array}$$

where $\eta \in [0,1)$ and $L\ge 0$. Then at least one ϕ-fixed point of Q exists in V.

Corollary 3.

Let $(V,d_V)$ be a complete metric space. Let $Q:V\to V$ be a map such that

$$\begin{array}{ccc}\hfill d_V(Qk,Ql)& \le & \eta d_V{(k,l)}^{{\omega }_1}{d}_V{(k,Qk)}^{{\omega }_2}{d}_V{(l,Ql)}^{{\omega }_3}\hfill \end{array}$$

(10)

for each $k,l\in V\backslash Fix\left(Q\right)$ with $l\ne k$, where ${\omega }_1,{\omega }_2,{\omega }_3\in [0,1]$ with ${\omega }_1+{\omega }_2+{\omega }_3=1$, and $\eta \in [0,1)$. Then a fixed point of Q exists in V.

The conclusion of the above result can be concluded from Corollary 1 by considering $L=0$ and $\varphi \left(k\right)=0$ $\forall k\in V$.

The following corollary follows from Corollary 3 by defining ${\omega }_1={\tau }_1$, ${\omega }_2={\tau }_2$ and ${\omega }_3=1-{\tau }_1-{\tau }_2$.

Corollary 4.

Let $(V,d_V)$ be a complete metric space. Let $Q:V\to V$ be a map such that

$$\begin{array}{ccc}\hfill d_V(Qk,Ql)& \le & \eta d_V{(k,l)}^{{\tau }_1}{d}_V{(k,Qk)}^{{\tau }_2}{d}_V{(l,Ql)}^{1-{\tau }_1-{\tau }_2}\hfill \end{array}$$

(11)

for each $k,l\in V\backslash Fix\left(Q\right)$ with $l\ne k$, where ${\tau }_1,{\tau }_2\in (0,1)$ with ${\tau }_1+{\tau }_2<1$, and $\eta \in [0,1)$. Then fixed point of Q exists in V.

Inequality (12) can be considered as a rational type interpolative contraction inequality obtained through (1) by taking ${\xi }_f(a,b,c)=\left(\frac{ac}{1+b}\right)\left(\frac{ab}{1+c}\right)\left(\frac{bc}{1+a}\right)$ and $L=0$. Some interesting results related to rational type contraction conditions are given in [21].

Corollary 5.

Let $(V,d_V)$ be a complete metric space. Let $Q:V\to V$ and $\varphi :V\to [0,\infty )$ be two maps such that

$$\begin{array}{ccc}\hfill & & d_V(Qk,Ql)\le \eta \left(\frac{d_V{(k,l)}^{{\omega }_1}{d}_V{(l,Ql)}^{{\omega }_3}}{1+d_V{(k,Qk)}^{{\omega }_2}}\right)\left(\frac{d_V{(k,l)}^{{\omega }_1}{d}_V{(k,Qk)}^{{\omega }_2}}{1+d_V{(l,Ql)}^{{\omega }_3}}\right)\left(\frac{d_V{(k,Qk)}^{{\omega }_2}{d}_V{(l,Ql)}^{{\omega }_3}}{1+d_V{(k,l)}^{{\omega }_1}}\right)\hfill \end{array}$$

(12)

for each $k,l\in V\backslash Fix\left(Q\right)$ with $k\ne l$, where ${\omega }_1,{\omega }_2,{\omega }_3\in [0,1]$ with ${\omega }_1+{\omega }_2+{\omega }_3=1$; further, for every $l\in V$, we have

$$\begin{array}{c}\hfill \varphi \left(Ql\right)\le \eta \varphi \left(l\right)\end{array}$$

where $\eta \in [0,1)$. Then at least one ϕ-fixed point of Q exists in V.

Consider a simulation function ${\beta }_{\psi }:{[0,\infty )}^2\to \mathbb{R}$ with the properties:

(b1)

${\beta }_{\psi }(0,0)=0$;

(b2)

${\beta }_{\psi }(t,s)\le \psi \left(s\right)-t$;

where $\psi :[0,\infty )\to [0,\infty )$ is a nondecreasing function that fulfills that ${\sum }_{j=1}^{\infty }{\psi }^j\left(s\right)$ is convergent for each $s>0$, moreover, $\psi \left(0\right)=0$ and $\psi \left(s\right)<s$ if $s>0$.

Example 2.

A function ${\beta }_{\psi }:[0,\infty )\times [0,\infty )\to \mathbb{R}$ defined by ${\beta }_{\psi }(k,l)=\alpha l-k$ for each $k,l\in [0,\infty )$, where $\psi \left(l\right)=\alpha l$ and $\alpha \in (0,1)$, is the simplest example of the above-defined simulation function.

Throughout the article, ${\beta }_{\psi }$ represents the above simulation function. Now, we define an abstract interpolative Reich-Rus-Ćirić type-II contraction with $\varphi$ shrink by using the simulation function ${\beta }_{\psi }$.

Definition 2.

A self-map $Q:V\to V$ is called an abstract interpolative Reich-Rus-Ćirić type-II contraction with ϕ shrink, if the below-stated inequalities hold:

$$\begin{array}{ccc}\hfill & & {\beta }_{\psi }\left(d_V(Qk,Ql),{\xi }_f\left(d_V{(k,l)}^{{\omega }_1},d_V{(k,Qk)}^{{\omega }_2},d_V{(l,Ql)}^{{\omega }_3}\right)\right)\hfill \\ & & +L{\theta }_f\left(d_V{(k,l)}^{{\omega }_1},d_V{(k,Qk)}^{{\omega }_2},d_V{(l,Ql)}^{{\omega }_3},d_V{(l,Qk)}^{{\omega }_4}\right)\ge 0\hfill \end{array}$$

(13)

for each $k,l\in V\backslash Fix\left(Q\right)$ with $l\ne k$, where ${\omega }_1,{\omega }_2,{\omega }_3\in [0,1]$ with ${\omega }_1+{\omega }_2+{\omega }_3=1$, ${\omega }_4>0$, and $L\ge 0$;

for every $l\in V$, we have

$$\begin{array}{c}\hfill {\beta }_{\psi }\left(\varphi \left(Ql\right),\varphi \left(l\right)\right)\ge 0.\end{array}$$

(14)

Now, we discuss the following $\varphi$-fixed point result for self-maps satisfying the above definition.

Theorem 2.

Let $Q:V\to V$ be an abstract interpolative Reich-Rus-Ćirić type-II contraction with ϕ shrink on a complete metric space $(V,d_V)$. Then at least one ϕ-fixed point of Q exists in V.

Proof. 

Define an iterative sequence $\left\{l_n\right\}$, that is $l_n=Q{l}_{n-1}\forall n\in \mathbb{N}$, for an arbitrary point $l_0\in V$. If $l_{n_0}=l_{n_0+1}$ for some $n_0$, then $l_{n_0}$ is a fixed point of Q. Moreover, from (14) we obtain $0\le {\beta }_{\psi }\left(\varphi \left(Q{l}_{n_0}\right),\varphi \left(l_{n_0}\right)\right)\le \psi \left(\varphi \left(l_{n_0}\right)\right)-\varphi \left(Q{l}_{n_0}\right)$; that is $\varphi \left(l_{n_0}\right)=\varphi \left(Q{l}_{n_0}\right)\le \psi \left(\varphi \left(l_{n_0}\right)\right)$. This gives $\varphi \left(l_{n_0}\right)=0$. Hence, $l_{n_0}$ is a $\varphi$-fixed point of Q. To work with the proof, we consider $l_{n-1}\ne l_n\forall n\in \mathbb{N}$. By (13), for each $n\in \mathbb{N}$, we get

$$\begin{array}{ccc}\hfill & & {\beta }_{\psi }\left(d_V(Q{l}_{n-1},Q{l}_n),{\xi }_f\left(d_V{(l_{n-1},l_n)}^{{\omega }_1},d_V{(l_{n-1},Q{l}_{n-1})}^{{\omega }_2},d_V{(l_n,Q{l}_n)}^{{\omega }_3}\right)\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_{n-1},l_n)}^{{\omega }_1},d_V{(l_{n-1},Q{l}_{n-1})}^{{\omega }_2},d_V{(l_n,Q{l}_n)}^{{\omega }_3},d_V{(l_n,Q{l}_{n-1})}^{{\omega }_4}\right)\ge 0.\hfill \end{array}$$

(15)

Using (b2) and (15), we get

$$\begin{array}{ccc}& & \psi \left({\xi }_f\left(d_V{(l_{n-1},l_n)}^{{\omega }_1},d_V{(l_{n-1},Q{l}_{n-1})}^{{\omega }_2},d_V{(l_n,Q{l}_n)}^{{\omega }_3}\right)\right)-d_V(Q{l}_{n-1},Q{l}_n)\hfill \\ & & +L{\theta }_f\left(d_V{(l_{n-1},l_n)}^{{\omega }_1},d_V{(l_{n-1},Q{l}_{n-1})}^{{\omega }_2},d_V{(l_n,Q{l}_n)}^{{\omega }_3},d_V{(l_n,Q{l}_{n-1})}^{{\omega }_4}\right)\hfill \\ & & \ge {\beta }_{\psi }\left(d_V(Q{l}_{n-1},Q{l}_n),{\xi }_f\left(d_V{(l_{n-1},l_n)}^{{\omega }_1},d_V{(l_{n-1},Q{l}_{n-1})}^{{\omega }_2},d_V{(l_n,Q{l}_n)}^{{\omega }_3}\right)\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_{n-1},l_n)}^{{\omega }_1},d_V{(l_{n-1},Q{l}_{n-1})}^{{\omega }_2},d_V{(l_n,Q{l}_n)}^{{\omega }_3},d_V{(l_n,Q{l}_{n-1})}^{{\omega }_4}\right)\ge 0.\hfill \end{array}$$

This implies

$$\begin{array}{ccc}\hfill d_V(Q{l}_{n-1},Q{l}_n)& \le & \psi \left({\xi }_f\left(d_V{(l_{n-1},l_n)}^{{\omega }_1},d_V{(l_{n-1},Q{l}_{n-1})}^{{\omega }_2},d_V{(l_n,Q{l}_n)}^{{\omega }_3}\right)\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_{n-1},l_n)}^{{\omega }_1},d_V{(l_{n-1},Q{l}_{n-1})}^{{\omega }_2},d_V{(l_n,Q{l}_n)}^{{\omega }_3},d_V{(l_n,Q{l}_{n-1})}^{{\omega }_4}\right).\hfill \end{array}$$

(16)

That is,

$$\begin{array}{ccc}\hfill d_V(l_n,l_{n+1})& \le & \psi \left({\xi }_f\left(d_V{(l_{n-1},l_n)}^{{\omega }_1},d_V{(l_{n-1},Q{l}_{n-1})}^{{\omega }_2},d_V{(l_n,Q{l}_n)}^{{\omega }_3}\right)\right)\forall n\in \mathbb{N}.\hfill \end{array}$$

(17)

Now, let us claim that $d_V(l_n,l_{n+1})<d_V(l_{n-1},l_n)\forall n\in \mathbb{N}$. Assume that the claim is wrong, then we have $m_0\in N$ with $d_V(l_{m_0},l_{m_0+1})\ge d_V(l_{m_0-1},l_{m_0})$. By (17) we get

$$\begin{array}{ccc}\hfill d_V(l_{m_0},l_{m_0+1})& \le & \psi \left({\xi }_f\left(d_V{(l_{m_0-1},l_{m_0})}^{{\omega }_1},d_V{(l_{m_0-1},l_{m_0})}^{{\omega }_2},d_V{(l_{m_0},l_{m_0+1})}^{{\omega }_3}\right)\right)\hfill \\ & \le & \psi \left({\xi }_f\left(d_V{(l_{m_0},l_{m_0+1})}^{{\omega }_1},d_V{(l_{m_0},l_{m_0+1})}^{{\omega }_2},d_V{(l_{m_0},l_{m_0+1})}^{{\omega }_3}\right)\right)\hfill \\ & \le & \psi \left(d_V(l_{m_0},l_{m_0+1})\right)\hfill \end{array}$$

which is impossible, since $l_{m_0}\ne l_{m_0+1}$. Hence, the claim holds. As $d_V(l_n,l_{n+1})<d_V(l_{n-1},l_n)\forall n\in \mathbb{N}$, then (17) we get

$$\begin{array}{ccc}\hfill d_V(l_n,l_{n+1})& \le & \psi \left({\xi }_f\left(d_V{(l_{n-1},l_n)}^{{\omega }_1},d_V{(l_{n-1},l_n)}^{{\omega }_2},d_V{(l_n,l_{n+1})}^{{\omega }_3}\right)\right)\hfill \\ & \le & \psi \left({\xi }_f\left(d_V{(l_{n-1},l_n)}^{{\omega }_1},d_V{(l_{n-1},l_n)}^{{\omega }_2},d_V{(l_{n-1},l_n)}^{{\omega }_3}\right)\right)\hfill \\ & \le & \psi \left(d_V(l_{n-1},l_n)\right)\forall n\in \mathbb{N}.\hfill \end{array}$$

(18)

This yields

$$\begin{array}{c}\hfill d_V(l_n,l_{n+1})\le {\psi }^n\left(d_V(l_0,l_1)\right)\forall n\in \mathbb{N}.\end{array}$$

(19)

Consider $m,n\in \mathbb{N}$ with $n>m$. By triangle inequality and (19) we obtain

$$\begin{array}{ccc}\hfill d_V(l_m,l_n)& \le & \sum _{j=m}^{n-1}{d}_V(l_j,l_{j+1})\le \sum _{j=m}^{n-1}{\psi }^j\left(d_V(l_0,l_1)\right).\hfill \end{array}$$

Since ${\sum }_{j=1}^{\infty }{\psi }^j\left(s\right)$ is a convergent series for each $s>0$, hence, by the above inequality we get ${lim}_{n,m\to \infty }{d}_V(l_m,l_n)=0$. The completeness of $(V,d_V)$ confirms the existence of an element $l^{*}\in V$ with $l_n\to l^{*}$. Now, let us claim that $l^{*}=Q{l}^{*}$. Let us suppose that the claim is wrong, then $d_V(l^{*},Q{l}^{*})>0$. Since $\left\{l_n\right\}$ is an iterative sequence with $l_n\to l^{*}$, thus, we get

$$max\{d_V(l_n,l^{*}),d_V(l_n,l_{n+1}),d_V(l^{*},Q{l}^{*})\}=d_V(l^{*},Q{l}^{*})\forall n\ge N_0$$

(20)

for some $N_0\in \mathbb{N}$. By (13), for each $n\in \mathbb{N}$, we obtain

$$\begin{array}{ccc}\hfill & & {\beta }_{\psi }\left(d_V(Q{l}_n,Q{l}^{*}),{\xi }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3}\right)\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3},d_V{(l^{*},Q{l}_n)}^{{\omega }_4}\right)\ge 0.\hfill \end{array}$$

(21)

This gives

$$\begin{array}{ccc}\hfill d_V(Q{l}_n,Q{l}^{*})& \le & \psi \left({\xi }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3}\right)\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3},d_V{(l^{*},Q{l}_n)}^{{\omega }_4}\right).\hfill \end{array}$$

(22)

By (20) and (22), for each $n\ge N_0$, we get

$$\begin{array}{ccc}\hfill d_V(l_{n+1},Q{l}^{*})& \le & \psi \left({\xi }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,l_{n+1})}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3}\right)\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3},d_V{(l^{*},l_{n+1})}^{{\omega }_4}\right)\hfill \\ & \le & \psi \left({\xi }_f\left(d_V{(l^{*},Q{l}^{*})}^{{\omega }_1},d_V{(l^{*},Q{l}^{*})}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3}\right)\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3},d_V{(l^{*},l_{n+1})}^{{\omega }_4}\right)\hfill \\ & \le & \psi \left(d_V(l^{*},Q{l}^{*})\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3},d_V{(l^{*},l_{n+1})}^{{\omega }_4}\right).\hfill \end{array}$$

(23)

Letting $n\to \infty$ in (23), we get

$$d_V(l^{*},Q{l}^{*})\le \psi \left(d_V(l^{*},Q{l}^{*})\right).$$

The above inequality, only holds when $d_V(l^{*},Q{l}^{*})=0$. Hence, the claim is correct, $l^{*}=Q{l}^{*}$. By (14) we get $0\le {\beta }_{\psi }\left(\varphi \left(Q{l}^{*}\right),\varphi \left(l^{*}\right)\right)\le \psi \left(\varphi \left(l^{*}\right)\right)-\varphi \left(Q{l}^{*}\right)$; that is $\varphi \left(l^{*}\right)=\varphi \left(Q{l}^{*}\right)\le \psi \left(\varphi \left(l^{*}\right)\right)$. This implies that $\varphi \left(l^{*}\right)=0$. Hence, $l^{*}$ is a $\varphi$-fixed point of Q. □

We will extend the above results by considering Q as a set-valued map. In the following, $CB\left(V\right)$ represents the collection of all nonvoid closed and bounded subsets of V and $CL\left(V\right)$ represents the collection of all nonvoid closed subsets of V.

Definition 3.

A set-valued map $Q:V\to CB\left(V\right)$ is called an abstract interpolative Reich-Rus-Ćirić type-I set-valued contraction with ϕ shrink, if the below-stated inequalities hold:

$$\begin{array}{ccc}\hfill H_V(Qk,Ql)& \le & \eta {\xi }_f\left(d_V{(k,l)}^{{\omega }_1},d_V{(k,Qk)}^{{\omega }_2},d_V{(l,Ql)}^{{\omega }_3}\right)\hfill \\ & & +L{\theta }_f\left(d_V{(k,l)}^{{\omega }_1},d_V{(k,Qk)}^{{\omega }_2},d_V{(l,Ql)}^{{\omega }_3},d_V{(l,Qk)}^{{\omega }_4}\right)\hfill \end{array}$$

(24)

for each $k,l\in V\backslash Fix\left(Q\right)$ with $l\ne k$, where ${\omega }_1,{\omega }_2,{\omega }_3\in [0,1]$ with ${\omega }_1+{\omega }_2+{\omega }_3=1$, ${\omega }_4>0$, and $L\ge 0$;

for every $k\in V$, we have

$$\begin{array}{c}\hfill \underset{l\in Qk}{sup}\varphi \left(l\right)\le \eta \varphi \left(k\right),\end{array}$$

(25)

where $\eta \in (0,1)$ and $Fix\left(Q\right)=\{v\in V:v\in Qv\}$.

The following theorem can be used to validate the existence of $\varphi$-fixed points for a map satisfying the above definition.

Theorem 3.

Let $Q:V\to CB\left(V\right)$ be an abstract interpolative Reich-Rus-Ćirić type-I set-valued contraction with ϕ shrink on a complete metric space $(V,d_V)$. Then at least one ϕ-fixed point of Q exists in V; that is, there exists a point $v^{*}$ in V with $v^{*}\in Q{v}^{*}$ and $\varphi \left(v^{*}\right)=0$.

Proof. 

For an arbitrary point $l_0\in V$, we get some $l_1\in Q{l}_0$. If $l_0=l_1$, then $l_0$ is a fixed point of Q. Moreover, by (25) we get $\varphi \left(l_0\right)\le {sup}_{l\in Q{l}_0}\varphi \left(l\right)\le \eta \varphi \left(l_0\right)$; that is $\varphi \left(l_0\right)=0$. Hence, $l_0$ is a $\varphi$-fixed point of Q. Suppose that neither $l_0$ nor $l_1$ is a fixed point of Q, then by (24) we get

$$\begin{array}{ccc}\hfill d_V(l_1,Q{l}_1)& \le & H_V(Q{l}_0,Q{l}_1)\hfill \\ & \le & \eta {\xi }_f\left(d_V{(l_0,l_1)}^{{\omega }_1},d_V{(l_0,Q{l}_0)}^{{\omega }_2},d_V{(l_1,Q{l}_1)}^{{\omega }_3}\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_0,l_1)}^{{\omega }_1},d_V{(l_0,Q{l}_0)}^{{\omega }_2},d_V{(l_1,Q{l}_1)}^{{\omega }_3},d_V{(l_1,Q{l}_0)}^{{\omega }_4}\right).\hfill \end{array}$$

(26)

That is,

$$\begin{array}{ccc}\hfill d_V(l_1,Q{l}_1)& \le & \eta {\xi }_f\left(d_V{(l_0,l_1)}^{{\omega }_1},d_V{(l_0,Q{l}_0)}^{{\omega }_2},d_V{(l_1,Q{l}_1)}^{{\omega }_3}\right).\hfill \end{array}$$

(27)

Since $\eta \in (0,1)$, thus, for $\frac{1}{\sqrt{\eta }}>1$ we have $l_2\in Q{l}_1$ satisfying the given inequality

$$\begin{array}{c}\hfill d_V(l_1,l_2)\le \frac{1}{\sqrt{\eta }}{d}_V(l_1,Q{l}_1).\end{array}$$

(28)

To proceed with the proof, we assume that $l_1\ne l_2$, otherwise $l_2$ is a $\varphi$-fixed point. From (27) and (28), we get

$$\begin{array}{ccc}\hfill d_V(l_1,l_2)& \le & \sqrt{\eta }{\xi }_f\left(d_V{(l_0,l_1)}^{{\omega }_1},d_V{(l_0,Q{l}_0)}^{{\omega }_2},d_V{(l_1,Q{l}_1)}^{{\omega }_3}\right).\hfill \end{array}$$

(29)

From the facts that $l_1\in Q{l}_0$, $l_2\in Q{l}_1$, and nondecreasing property of ${\xi }_f$, by (29), we get

$$\begin{array}{ccc}\hfill d_V(l_1,l_2)& \le & \sqrt{\eta }{\xi }_f\left(d_V{(l_0,l_1)}^{{\omega }_1},d_V{(l_0,l_1)}^{{\omega }_2},d_V{(l_1,l_2)}^{{\omega }_3}\right).\hfill \end{array}$$

(30)

If $d_V(l_0,l_1)\le d_V(l_1,l_2)$, then from the above inequality we get $d_V(l_1,l_2)=0$, which is impossible. Thus, $d_V(l_1,l_2)<d_V(l_0,l_1)$. Now, by (30), we get

$$\begin{array}{ccc}\hfill d_V(l_1,l_2)& \le & \sqrt{\eta }{\xi }_f\left(d_V{(l_0,l_1)}^{{\omega }_1},d_V{(l_0,l_1)}^{{\omega }_2},d_V{(l_1,l_2)}^{{\omega }_3}\right)\hfill \\ & \le & \sqrt{\eta }{\xi }_f\left(d_V{(l_0,l_1)}^{{\omega }_1},d_V{(l_0,l_1)}^{{\omega }_2},d_V{(l_0,l_1)}^{{\omega }_3}\right)\hfill \\ & \le & \sqrt{\eta }{d}_V(l_0,l_1).\hfill \end{array}$$

(31)

Continuing the proof on the above lines we can obtain a sequence $\left\{l_n\right\}$ with $l_n\in Q{l}_{n-1}\forall n\in \mathbb{N}$, $l_{n-1}\ne l_n\forall n\in \mathbb{N}$, and

$$d_V(l_n,l_{n+1})\le {\left(\sqrt{\eta }\right)}^n{d}_V(l_0,l_1)\forall n\in \mathbb{N}.$$

Moreover, it is trivial to conclude that $\left\{l_n\right\}$ is a Cauchy sequence in a complete metric space $(V,d_V)$, thus, there is a point $l^{*}\in V$ with $l_n\to l^{*}$. Now, we claim that $l^{*}\in Q{l}^{*}$. If it is wrong, then $d_V(l^{*},Q{l}^{*})>0$. Thus, we can obtain $N_0\in \mathbb{N}$ such that

$$max\{d_V(l_n,l^{*}),d_V(l_n,l_{n+1}),d_V(l^{*},Q{l}^{*})\}=d_V(l^{*},Q{l}^{*})\forall n\ge N_0.$$

(32)

By (24), for $k=l_n$ and $l=l^{*}$, we obtain

$$\begin{array}{ccc}\hfill d_V(l_{n+1},Q{l}^{*})& \le & H_V(Q{l}_n,Q{l}^{*})\hfill \\ & \le & \eta {\xi }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3}\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3},d_V{(l^{*},Q{l}_n)}^{{\omega }_4}\right)\forall n\in \mathbb{N}.\hfill \end{array}$$

(33)

From (32) and (33), for each $n\ge N_0$, we get

$$\begin{array}{ccc}\hfill d_V(l_{n+1},Q{l}^{*})& \le & \eta {\xi }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,l_{n+1})}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3}\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3},d_V{(l^{*},l_{n+1})}^{{\omega }_4}\right)\hfill \\ & \le & \eta {\xi }_f\left(d_V{(l^{*},Q{l}^{*})}^{{\omega }_1},d_V{(l^{*},Q{l}^{*})}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3}\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3},d_V{(l^{*},l_{n+1})}^{{\omega }_4}\right)\hfill \\ & \le & \eta d_V(l^{*},Q{l}^{*})\hfill \\ & & +L{\theta }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3},d_V{(l^{*},l_{n+1})}^{{\omega }_4}\right).\hfill \end{array}$$

(34)

By applying the limit $n\to \infty$ in (34), we get

$$d_V(l^{*},Q{l}^{*})\le \eta d_V(l^{*},Q{l}^{*}).$$

The existence of the above inequality is impossible when $d_V(l^{*},Q{l}^{*})>0$. Hence, the claim is correct, $l^{*}\in Q{l}^{*}$. By (25) we get

$$\varphi \left(l^{*}\right)\le \underset{l\in Q{l}^{*}}{sup}\varphi \left(l\right)\le \lambda \varphi \left(l^{*}\right).$$

This implies that $\varphi \left(l^{*}\right)=0$. Hence, $l^{*}$ is a $\varphi$-fixed point of Q. □

The following result examines the existence of $\varphi$-fixed points for a set-valued map $Q:V\to CL\left(V\right)$.

Theorem 4.

Let $(V,d_V)$ be a complete metric space and let $Q:V\to CL\left(V\right)$ be a set-valued map and $\varphi :V\to [0,\infty )$ be another map fulfilling the following inequalities:

$$\begin{array}{ccc}\hfill d_V(l,Ql)& \le & \eta {\xi }_f\left(d_V{(k,l)}^{{\omega }_1},d_V{(k,Qk)}^{{\omega }_2},d_V{(l,Ql)}^{{\omega }_3}\right)\hfill \end{array}$$

(35)

for each $k,l\in V\backslash Fix\left(Q\right)$ with $l\in Qk$, where ${\omega }_1,{\omega }_2,{\omega }_3\in [0,1]$ with ${\omega }_1+{\omega }_2+{\omega }_3=1$, and ${\omega }_3\ne 1$; further, for every $k\in V$, we have

$$\begin{array}{c}\hfill \underset{l\in Qk}{sup}\varphi \left(l\right)\le \eta \varphi \left(k\right),\end{array}$$

(36)

where $\eta \in (0,1)$. Moreover, assume that $Graph\left(Q\right)=\left\{\right(k,l):k\in V,l\in Qk\}$ is closed. Then at least one ϕ-fixed point of Q exists in V.

Proof. 

Following the proof of Theorem 3, here, one can easily obtain a Cauchy sequence $\left\{l_n\right\}$ in a complete metric space $(V,d_V)$ with $l_n\in Q{l}_{n-1}\forall n\in \mathbb{N}$, $l_{n-1}\ne l_n\forall n\in \mathbb{N}$, and

$$d_V(l_n,l_{n+1})\le {\left(\sqrt{\eta }\right)}^n{d}_V(l_0,l_1)\forall n\in \mathbb{N}.$$

Furthermore, there exists a point $l^{*}\in V$ with $l_n\to l^{*}$. Since $l_n\in Q{l}_{n-1}\forall n\in \mathbb{N}$, thus, $(l_{n-1},l_n)\in Graph\left(Q\right)\forall n\in \mathbb{N}$. As given that $Graph\left(Q\right)$ is closed, thus, $(l^{*},l^{*})\in Graph\left(Q\right)$, that is $l^{*}\in Q{l}^{*}$. Hence, $l^{*}$ is a fixed point of Q. By considering (36), we conclude that $l^{*}$ is a $\varphi$-fixed point of Q. □

Now we present the definition of the abstract interpolative Reich-Rus-Ćirić type-II set-valued contraction with $\varphi$ shrink.

Definition 4.

A set-valued map $Q:V\to CB\left(V\right)$ is called an abstract interpolative Reich-Rus-Ćirić type-II set-valued contraction with ϕ shrink, if the below-stated inequalities are fulfilled:

$$\begin{array}{ccc}\hfill & & {\beta }_{\psi }\left(H_V(Qk,Ql),{\xi }_f\left(d_V{(k,l)}^{{\omega }_1},d_V{(k,Qk)}^{{\omega }_2},d_V{(l,Ql)}^{{\omega }_3}\right)\right)\hfill \\ & & +L{\theta }_f\left(d_V{(k,l)}^{{\omega }_1},d_V{(k,Qk)}^{{\omega }_2},d_V{(l,Ql)}^{{\omega }_3},d_V{(l,Qk)}^{{\omega }_4}\right)\ge 0\hfill \end{array}$$

(37)

for each $k,l\in V\backslash Fix\left(Q\right)$ with $l\ne k$, where ${\omega }_1,{\omega }_2,{\omega }_3\in [0,1]$ with ${\omega }_1+{\omega }_2+{\omega }_3=1$, ${\omega }_3\ne 0$, ${\omega }_4>0$, and $L\ge 0$;

for every $k\in V$, we have

$$\begin{array}{c}\hfill {\beta }_{\psi }(\underset{l\in Qk}{sup}\varphi \left(l\right),\varphi \left(k\right))\ge 0.\end{array}$$

(38)

In the following theorems, we assume that ${\xi }_f$ and $\psi$ are strictly increasing instead of nondecreasing.

Theorem 5.

Let $Q:V\to CB\left(V\right)$ be an abstract interpolative Reich-Rus-Ćirić type-II set-valued contraction with ϕ shrink on a complete metric space $(V,d_V)$. Then at least one ϕ-fixed point of Q exists in V.

Proof. 

For an arbitrary point $l_0\in V$, we get a point $l_1\in Q{l}_0$. If $l_0=l_1$, then $l_0$ is a fixed point of Q. Moreover, by (38), we get $0\le {\beta }_{\psi }\left({sup}_{l\in Q{l}_0}\varphi \left(l\right),\varphi \left(l_0\right)\right)\le \psi \left(\varphi \left(l_0\right)\right)-{sup}_{l\in Q{l}_0}\varphi \left(l\right)$, this implies $\varphi \left(l_0\right)\le \psi \left(\varphi \left(l_0\right)\right)$, hence, $l_0$ is a $\varphi$-fixed point of Q. Suppose that neither $l_0$ nor $l_1$ is a fixed point of Q, then by (37) we get

$$\begin{array}{ccc}\hfill & & {\beta }_{\psi }(H_V(Q{l}_0,Q{l}_1),{\xi }_f\left(d_V{(l_0,l_1)}^{{\omega }_1},d_V{(l_0,Q{l}_0)}^{{\omega }_2},d_V{(l_1,Q{l}_1)}^{{\omega }_3}\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_0,l_1)}^{{\omega }_1},d_V{(l_0,Q{l}_0)}^{{\omega }_2},d_V{(l_1,Q{l}_1)}^{{\omega }_3},d_V{(l_1,Q{l}_0)}^{{\omega }_4}\right)\ge 0.\hfill \end{array}$$

(39)

This implies that

$$\begin{array}{ccc}\hfill H_V(Q{l}_0,Q{l}_1)& \le & \psi \left({\xi }_f\left(d_V{(l_0,l_1)}^{{\omega }_1},d_V{(l_0,Q{l}_0)}^{{\omega }_2},d_V{(l_1,Q{l}_1)}^{{\omega }_3}\right)\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_0,l_1)}^{{\omega }_1},d_V{(l_0,Q{l}_0)}^{{\omega }_2},d_V{(l_1,Q{l}_1)}^{{\omega }_3},d_V{(l_1,Q{l}_0)}^{{\omega }_4}\right).\hfill \end{array}$$

(40)

Since $l_1\in Q{l}_0$, thus, by the above inequality we get

$$\begin{array}{ccc}\hfill d_V(l_1,Q{l}_1)& \le & \psi \left({\xi }_f\left(d_V{(l_0,l_1)}^{{\omega }_1},d_V{(l_0,l_1)}^{{\omega }_2},d_V{(l_1,Q{l}_1)}^{{\omega }_3}\right)\right).\hfill \end{array}$$

(41)

If $d_V(l_0,l_1)\le d_V(l_1,Q{l}_1)$, then by (41) we get $d_V(l_1,Q{l}_1)\le \psi \left(d_V(l_1,Q{l}_1)\right)<d_V(l_1,Q{l}_1)$, which is impossible. Thus, we conclude $d_V(l_0,l_1)>d_V(l_1,Q{l}_1)$. By considering strictly increasing behavior of $\psi$, ${\xi }_f$, and using (41) we get

$$\begin{array}{ccc}\hfill d_V(l_1,Q{l}_1)& \le & \psi \left({\xi }_f\left(d_V{(l_0,l_1)}^{{\omega }_1},d_V{(l_0,l_1)}^{{\omega }_2},d_V{(l_1,Q{l}_1)}^{{\omega }_3}\right)\right)\hfill \\ & <& \psi \left({\xi }_f\left(d_V{(l_0,l_1)}^{{\omega }_1},d_V{(l_0,l_1)}^{{\omega }_2},d_V{(l_0,l_1)}^{{\omega }_3}\right)\right)\hfill \\ & \le & \psi \left(d_V(l_0,l_1)\right).\hfill \end{array}$$

(42)

As $d_V(l_1,Q{l}_1)<\psi (d_V(l_0,l_1))$, there exists some real number ${ϵ}_1>0$ such that $d_V(l_1,Q{l}_1)+{ϵ}_1=\psi (d_V(l_0,l_1))$. Thus, we get $l_2\in Q{l}_1$ such that $d_V(l_1,l_2)\le d_V(l_1,Q{l}_1)+{ϵ}_1$. Hence, we conclude that

$$\begin{array}{c}\hfill d_V(l_1,l_2)\le \psi \left(d_V(l_0,l_1)\right).\end{array}$$

(43)

Continuing the proof on the above lines we can obtain a sequence $\left\{l_n\right\}$ with $l_n\in Q{l}_{n-1}\forall n\in \mathbb{N}$, $l_{n-1}\ne l_n\forall n\in \mathbb{N}$, and

$$d_V(l_n,l_{n+1})\le {\psi }^n\left(d_V(l_0,l_1)\right)\forall n\in \mathbb{N}.$$

Further, it can be seen that $\left\{l_n\right\}$ is a Cauchy sequence in a complete metric space $(V,d_V)$ and there exists $l^{*}\in V$ with $l_n\to l^{*}$. Now, we claim that $l^{*}\in Q{l}^{*}$. If it is wrong then $d_V(l^{*},Q{l}^{*})>0$. Thus, we can obtain $N_0\in \mathbb{N}$ such that

$$max\{d_V(l_n,l^{*}),d_V(l_n,l_{n+1}),d_V(l^{*},Q{l}^{*})\}=d_V(l^{*},Q{l}^{*})\forall n\ge N_0.$$

(44)

By (37), for $k=l_n$ and $l=l^{*}$, we get

$$\begin{array}{ccc}\hfill & & {\beta }_{\psi }\left(H_V(Q{l}_n,Q{l}^{*}),{\xi }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3}\right)\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3},d_V{(l^{*},Q{l}_n)}^{{\omega }_4}\right)\forall n\in \mathbb{N}.\hfill \end{array}$$

(45)

From the above inequality, we obtain

$$\begin{array}{ccc}\hfill d_V(l_{n+1},Q{l}^{*})& \le & H_V(Q{l}_n,Q{l}^{*})\hfill \\ & \le & \psi \left({\xi }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3}\right)\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3},d_V{(l^{*},Q{l}_n)}^{{\omega }_4}\right)\forall n\in \mathbb{N}.\hfill \end{array}$$

(46)

From (44) and (46), for each $n\ge N_0$, we get

$$\begin{array}{ccc}\hfill d_V(l_{n+1},Q{l}^{*})& \le & \psi \left({\xi }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,l_{n+1})}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3}\right)\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3},d_V{(l^{*},l_{n+1})}^{{\omega }_4}\right)\hfill \\ & \le & \psi \left({\xi }_f\left(d_V{(l^{*},Q{l}^{*})}^{{\omega }_1},d_V{(l^{*},Q{l}^{*})}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3}\right)\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3},d_V{(l^{*},l_{n+1})}^{{\omega }_4}\right)\hfill \\ & \le & \psi \left(d_V(l^{*},Q{l}^{*})\right)\hfill \\ & & +L{\theta }_f\left(d_V{(l_n,l^{*})}^{{\omega }_1},d_V{(l_n,Q{l}_n)}^{{\omega }_2},d_V{(l^{*},Q{l}^{*})}^{{\omega }_3},d_V{(l^{*},l_{n+1})}^{{\omega }_4}\right).\hfill \end{array}$$

(47)

By letting $n\to \infty$ in (47), we get

$$d_V(l^{*},Q{l}^{*})\le \psi \left(d_V(l^{*},Q{l}^{*})\right)$$

which is impossible for $d_V(l^{*},Q{l}^{*})>0$. Hence, the claim is correct, $l^{*}\in Q{l}^{*}$. Moreover, by (38) we get $0\le {\beta }_{\psi }\left({sup}_{l\in Q{l}^{*}}\varphi \left(l\right),\varphi \left(l^{*}\right)\right)\le \psi \left(\varphi \left(l^{*}\right)\right)-{sup}_{l\in Q{l}^{*}}\varphi \left(l\right)$. As $l^{*}\in Q{l}^{*}$, thus, $\varphi \left(l^{*}\right)\le {sup}_{l\in Q{l}^{*}}\varphi \left(l\right)\le \psi \left(\varphi \left(l^{*}\right)\right)$. This implies that $\varphi \left(l^{*}\right)=0$. Hence, $l^{*}$ is a $\varphi$-fixed point of Q. □

The following theorem can examine $\varphi$-fixed points of set-valued map $Q:V\to CL\left(V\right)$.

Theorem 6.

Let $(V,d_V)$ be a complete metric space and let $Q:V\to CL\left(V\right)$ be a set-valued map and $\varphi :V\to [0,\infty )$ be another map fulfilling the following inequalities:

$$\begin{array}{c}\hfill {\beta }_{\psi }\left(d_V(l,Ql),{\xi }_f\left(d_V{(k,l)}^{{\omega }_1},d_V{(k,Qk)}^{{\omega }_2},d_V{(l,Ql)}^{{\omega }_3}\right)\right)\ge 0\end{array}$$

(48)

for each $k,l\in V\backslash Fix\left(Q\right)$ with $l\in Qk$, where ${\omega }_1,{\omega }_2\in [0,1]$ and ${\omega }_3\in (0,1)$ with ${\omega }_1+{\omega }_2+{\omega }_3=1$; further, for every $k\in V$, we have

$$\begin{array}{c}\hfill {\beta }_{\psi }(\underset{l\in Qk}{sup}\varphi \left(l\right),\varphi \left(k\right))\ge 0.\end{array}$$

(49)

Furthermore, assume that $Graph\left(Q\right)=\left\{\right(k,l):k\in V,l\in Qk\}$ is closed. Then at least one ϕ-fixed point of Q exists in V.

3. Application

A suitable application of the work can be seen as an existence theorem for the following type of fractional-order integral equation:

$$k\left(t\right)=q\left(t\right)+\frac{\mu }{{[\Gamma \left(\alpha \right)]}^2}{\int }_0^{p\left(t\right)}{(p\left(t\right)-s)}^{\alpha -1}w(s,k\left(s\right))ds,\alpha \in (0,1),t\in J=[a,b]$$

(50)

where $q:J\to \mathbb{R}$, $p:J\to {\mathbb{R}}^{+}=[0,\infty )$, and $w:J\times \mathbb{R}\to \mathbb{R}$ are continuous functions, $\mu$ is constant real number, and $\Gamma$ is the Euler gamma function; that is $\Gamma \left(\alpha \right)={\int }_0^{\infty }{t}^{\alpha -1}{e}^{-t}dt$.

Consider $V=\left(C\right[a,b],\mathbb{R})$ is the space of all continuous and bounded real-valued functions defined on $J=[a,b]$. Define a metric on V by

$$d_V(k,l)=\parallel k-l\parallel =\underset{t\in J}{max}|k\left(t\right)-l\left(t\right)|\forall k,l\in V.$$

Clearly, $(V,d_V)$ is a complete metric space.

Now, we move towards the existence theorem of (50).

Theorem 7.

Consider $V=\left(C\right[a,b],\mathbb{R})$ and consider the operator

$$Q:V\to V,Qk\left(t\right)=q\left(t\right)+\frac{\mu }{{[\Gamma \left(\alpha \right)]}^2}{\int }_0^{p\left(t\right)}{(p\left(t\right)-s)}^{\alpha -1}w(s,k\left(s\right))ds,\alpha \in (0,1),t\in J$$

where $q:J\to \mathbb{R}$, $p:J\to {\mathbb{R}}^{+}=[0,\infty )$, and $w:J\times \mathbb{R}\to \mathbb{R}$ are continuous functions, μ is constant, and Γ is the Euler gamma function; that is $\Gamma \left(\alpha \right)={\int }_0^{\infty }{t}^{\alpha -1}{e}^{-t}dt$. Moreover, consider that there are ${\omega }_1,{\omega }_2,{\omega }_3\in [0,1]$ with ${\omega }_1+{\omega }_2+{\omega }_3=1$ satisfying

$$\frac{\left|w\right(s,k\left(s\right))-w(s,l\left(s\right)\left)\right|}{{\parallel k-Qk\parallel }^{{\omega }_2}{\parallel l-Ql\parallel }^{{\omega }_3}}\le {[\Gamma (\alpha +1)]}^2{|k\left(s\right)-l\left(s\right)|}^{{\omega }_1}$$

(51)

for all $s\in J$ and for each $k,l\in V$ with $min\{\parallel k-l\parallel ,\parallel k-Qk\parallel ,\parallel l-Ql\parallel \}>0$, moreover,

$$\underset{t\in J}{sup}\left|\mu {\left(p\left(t\right)\right)}^{\alpha }\right|\le 1.$$

Then, (50) possesses at least one solution.

Proof. 

For each $k,l\in V$ with $min\{\parallel k-l\parallel ,\parallel k-Qk\parallel ,\parallel l-Ql\parallel \}>0$, we obtain

$$\begin{array}{ccc}\hfill \left|Qk\right(t)-Ql(t\left)\right|& =& \left|\frac{\mu }{{[\Gamma \left(\alpha \right)]}^2}{\int }_0^{p\left(t\right)}{(p\left(t\right)-s)}^{\alpha -1}[w(s,k\left(s\right))-w(s,l\left(s\right))]ds\right|\hfill \\ & \le & \left|\frac{\mu }{{[\Gamma \left(\alpha \right)]}^2}{\int }_0^{p\left(t\right)}{(p\left(t\right)-s)}^{\alpha -1}ds\right|{[\Gamma (\alpha +1)]}^2{\parallel k-l\parallel }^{{\omega }_1}{\parallel k-Qk\parallel }^{{\omega }_2}{\parallel l-Ql\parallel }^{{\omega }_3}\hfill \\ & =& \left|\frac{\mu }{{[\Gamma \left(\alpha \right)]}^2}\frac{{\left(p\left(t\right)\right)}^{\alpha }}{\alpha }\right|{[\alpha \Gamma \left(\alpha \right)]}^2{\parallel k-l\parallel }^{{\omega }_1}{\parallel k-Qk\parallel }^{{\omega }_2}{\parallel l-Ql\parallel }^{{\omega }_3}\hfill \\ & =& \alpha |\mu {\left(p\left(t\right)\right)}^{\alpha }{|\parallel k-l\parallel }^{{\omega }_1}{\parallel k-Qk\parallel }^{{\omega }_2}{\parallel l-Ql\parallel }^{{\omega }_3}\forall t\in J.\hfill \end{array}$$

Thus, we get

$$\begin{array}{c}\hfill \parallel Qk-Ql\parallel \le {\alpha \parallel k-l\parallel }^{{\omega }_1}{\parallel k-Qk\parallel }^{{\omega }_2}{\parallel l-Ql\parallel }^{{\omega }_3}\end{array}$$

for each $k,l\in V\backslash Fix\left(Q\right)$ with $k\ne l$. Thus, by Corollary 3, a fixed point of Q occurs; that is, the integral Equation (50) possesses at least one solution. □

Example 3.

Consider $V=\{0,1,2\cdots ,20\}$ and define

$$d_V(k,l)=\left\{\begin{array}{c}0,k=l\hfill \\ \max \{k,l\},k\ne l.\hfill \end{array}\right.$$

Define $Q:V\to V$ and $\varphi :V\to [0,\infty )$ by

$$Q\left(k\right)=\left\{\begin{array}{c}0,k=0\hfill \\ k-1,otherwise\hfill \end{array}\right.$$

and

$$\varphi \left(k\right)=\frac{k}{2}.$$

Then, it is easy to verify that the axioms of Theorem 1 are valid, by taking ${\xi }_f(a,b,c)=abc$, ${\omega }_1=$ 0.99, ${\omega }_2=$ 0.005, ${\omega }_3=$ 0.005, $L=0$ and $\eta =\frac{99}{100}$. Thus, there is an element $k\in V$ with $Qk=k$ and $\varphi \left(k\right)=0$.

Example 4.

Consider $V=\mathbb{W}$ the set of all whole numbers and define

$$d_V(k,l)=\left\{\begin{array}{c}0,k=l\hfill \\ \max \{k,l\},k\ne l.\hfill \end{array}\right.$$

Define $Q:V\to CB\left(V\right)$ and $\varphi :V\to [0,\infty )$ by

$$Q\left(k\right)=\left\{\begin{array}{c}\left\{0\right\},k\in \{0,1\}\hfill \\ \{0,k-1\},k\in \{2,3,\cdots ,10\}\hfill \\ \{0,k\},otherwise\hfill \end{array}\right.$$

and

$$\varphi \left(k\right)=\left\{\begin{array}{c}k/2,k\in \{1,2,\cdots ,10\}\hfill \\ 0,otherwise.\hfill \end{array}\right.$$

Then, it is easy to check that the axioms of Theorem 6 are valid, by taking ${\xi }_f(a,b,c)=abc$, ${\beta }_{\psi }(k,l)=(49/50)l-k$, ${\omega }_1=$ 0.99, ${\omega }_2=$ 0.005, and ${\omega }_3=$ 0.005. Since

$${(k-1)}^{0.995}\le (49/50)k^{0.995}\mathrm{for}\mathrm{each}k\in \{1,2,\cdots ,10\}.$$

Hence, there is an element $k\in V$ with $k\in Qk$ and $\varphi \left(k\right)=0$.

4. Conclusions

In this article, we have studied the existence of $\varphi$-fixed points for the mappings satisfying abstract interpolative Reich-Rus-Ćirić-type contractions with a shrink map on a complete metric space. Abstract interpolative Reich-Rus-Ćirić-type contraction with a shrink map has the following characteristics:

• It is an extended form of interpolative Reich-Rus-Ćirić-type contraction.
• It provides an easier proof of the results, ensuring $\varphi$-fixed points.

Finally, we have studied the existence of a solution for a fractional-order integral equation using our results.