Skip to main content

Revising the Hardy–Rogers–Suzuki-type Z-contractions


The aim of this study is to introduce a new interpolative contractive mapping combining the Hardy–Rogers contractive mapping of Suzuki type and \(\mathcal{Z}\)-contraction. We investigate the existence of a fixed point of this type of mappings and prove some corollaries. The new results of the paper generalize a number of existing results which were published in the last two decades.

Introduction and preliminaries

A century ago, the notion of fixed point theory appeared in the papers that were written to solve certain differential equations. The first independent fixed point result was given by Banach [1] in the setting of a complete normed space. The analog of this result in the framework of the complete metric space was reported by Caccioppoli [2] in 1930. After then, metric fixed point theory has advanced in many directions in the setting of several abstract spaces. Regarding the appearance of the notion, fixed point theory is one of the useful and crucial tools in several disciplines. Most of the daily life problems can be restated in the context of fixed point theory, see, e.g., the book of Rus [3] for interesting examples.

In the last fourth decades, an enormous number of publications were reported on the advances of metric fixed point theory regarding very distinct aspects in various settings, see, e.g., [341] and related reference therein. As a natural consequence of this fact, some authors proposed new notions to combine and unify this tremendous number of publications in the literature. Here, we mention and use three interesting notions that were proposed for this purpose, namely simulation function (see, e.g., [1929]), admissible mapping (see, e.g., [918]), and Suzuki-type contraction (see [4, 5]).

In 2014, Popescu [21] suggested an interesting notion, the so-called ω-orbital admissible mappings, which is a smart expansion of the notion of α-admissible mappings, see Samet et al. [19]. In this work, Popescu [21] showed that each admissible mapping is an ω-orbital admissible mapping, but the converse is not true.

Definition 1


Let \(\omega :Y\times Y\rightarrow {}[ 0,\infty )\) be a function where Y is a any nonempty set. A self mapping H on Y is called ω-orbital admissible if for all u in Y, we have

$$ \omega ( u,Hu ) \geq 1\quad \Longrightarrow \quad\omega \bigl( Hu,H^{2}u \bigr) \geq 1. $$

One of the interesting uses of ω-admissible mapping is that it is ω-regular in the setting of metric spaces. This was a condition that helps refine the continuity condition on the self-mapping accompanied with some additional conditions; see, e.g., [19].

Definition 2

A metric space \((Y,d)\) is called ω-regular if for every sequence \(\{ {u_{n}} \} \) in Y, which converges to some \(z\in Y\) and satisfies \(\omega ( u_{n},u_{n+1} ) \geq 1\) for each \(n\in \mathbb{N}\), we have \(\omega ( u_{n},z ) \geq 1\).

Later in 2015, the concept of a simulation function had been introduced by Khojasteh et al. [9]. These functions cover many types of the existing contractions. We give now the definition of simulation function as it was redefined by Argoubi [11].

Definition 3

A simulation function is a mapping \(\zeta :[0,\infty )\times {}[ 0,\infty )\rightarrow \mathbb{R}\) satisfying the following conditions:

\((\zeta _{1})\):

\(\zeta (t,s)< s-t\) for all \(t,s>0\);

\((\zeta _{2})\):

if \(\{t_{n}\}\), \(\{s_{n}\}\) are sequences in \((0,\infty )\) such that \(\lim_{n\rightarrow \infty }t_{n}=\lim_{n \rightarrow \infty }s_{n}>0\), then

$$ \limsup_{n\rightarrow \infty }\zeta (t_{n},s_{n})< 0. $$

We demonstrate here some examples of simulation functions from [1018].

Example 4

For \(i=1,2\), we define the mappings \(\zeta _{i}:[0,\infty )\times {}[ 0,\infty )\rightarrow \mathbb{R}\), as follows:

  1. (i)

    \(\zeta _{1}(t,s)=\phi _{1}(s)-\phi _{2}(t)\) for all \(t, s\in {}[ 0,\infty )\), where \(\phi _{1},\phi _{2}:[0,\infty )\rightarrow {}[ 0,\infty )\) are two continuous functions such that \(\phi _{1}(t)=\phi _{2}(t)=0\) if and only if \(t=0\) and \(\phi _{1}(t)< t\leq \phi _{2}(t)\) for all \(t>0\).

    If we take \(\phi _{2}(t)=t\), \(\phi _{1}(t)=\lambda t\) where \(\lambda \in {}[ 0,1)\), we get the special case \(\zeta _{B}=\lambda s-t\) for all \(s,t\in {}[ 0,\infty )\).

  2. (ii)

    \(\zeta _{2}(t,s)=\eta (s)-t\) for all \(s,t\in {}[ 0,\infty )\), where \(\eta :[0,\infty )\rightarrow {}[ 0,\infty )\) is an upper semicontinuous mapping such that \(\eta (t)< t\) for all \(t>0\) and \(\eta (0)=0\).

It is clear that each function \(\zeta _{i}\) (\(i=1,2\)) forms a simulation function.

The next definition presents the Suzuki-type contraction mappings.

Definition 5


A self-mapping H on a metric space \(( Y,d ) \) is called a Suzuki-type contraction if for all \(x,y\in Y\) with \(x\neq y\), we have

$$ \frac{1}{2}d ( x,Hx ) \leq d ( x,y )\quad \Longrightarrow\quad d ( Hx,Hy ) \leq d ( x,y ).$$

One of the interesting results in metric fixed point theory was given by Karapınar [39], which involves interpolation. After these initial results, interpolative contraction has been investigated by several authors, e.g., [15, 3041]. Recently, interpolative Hardy–Rogers-type contractions have been investigated by many authors (see [68]). In particular, in [38], Karapınar used simulation functions to introduce the notion of interpolative Hardy–Rogers-type \(\mathcal{Z}\)-contraction mappings and prove some related fixed point results. The aim of our work is to combine the latter contractions with those of the Suzuki-type and investigate the existence of fixed points of this new type of mappings under some conditions.

Karapınar’s definition that introduced the notion of interpolative Hardy–Rogers-type \(\mathcal{Z}\)-contraction mappings is given as follows.

Definition 6


Let H be a self-mapping defined on a metric space \((Y,d)\). If there exist \(\alpha ,\beta ,\gamma \in (0,1)\) with \(\alpha +\beta +\gamma <1\), and \(\zeta \in \mathcal{Z}\) such that

$$ \zeta \bigl(d(Hx,Hy),C(x,y)\bigr)\geq 0 , $$

for all \(x,y \in Y\backslash \operatorname{Fix}(H)\), where \(\operatorname{Fix}(H)\) is the set of all fixed point of H, and

$$ C(x,y):= \bigl[ d ( x,y ) \bigr] ^{\beta } \cdot \bigl[ d ( x,Hx ) \bigr] ^{\alpha } \cdot \bigl[ d ( y,Hy ) \bigr] ^{\gamma } \cdot \biggl[ \frac{1}{2}\bigl(d ( x,Hy ) + d ( y,Hx ) \bigr) \biggr] ^{1-\alpha -\beta -\gamma },$$

then we say that H is an interpolative Hardy–Rogers-type \(\mathcal{Z}\)-contraction with respect to ζ.

Main results

We introduce now our new contraction type mapping in the following definition.

Definition 7

Let H be a self-mapping on a metric space \(( Y,d ) \). We say that H is an interpolative Hardy-Rogers–Suzuki-type \(\mathcal{Z}\)-contraction with respect to some \(\zeta \in \mathcal{Z}\) if there exists \(\alpha ,\beta ,\gamma \in ( 0,1 ) \) with \(\alpha +\beta +\gamma <1\), \(\zeta \in \mathcal{Z}\) and a function \(\omega :Y\times Y\rightarrow {}[ 0,\infty )\) such that

$$\begin{aligned} & \frac{1}{2}d ( x,Hx ) \leq d ( x,y ) \\ &\quad \Longrightarrow\quad \zeta \bigl( \omega ( x,y ) d ( Hx,Hy ) ,C ( x,y ) \bigr) \geq 0 \end{aligned}$$

for all x, y \(\notin \operatorname{Fix}(H)\) where \(C(x,y)\) is given by (1.2).

Our main result is the following theorem:

Theorem 8

Let \(( Y,d ) \) be a complete metric space and let H be a self-mapping on Y. Assume that

  1. (i)

    H is an interpolative Hardy–Rogers–Suzuki-type \(\mathcal{Z}\)-contraction with respect to some \(\zeta \in \mathcal{Z}\);

  2. (ii)

    H is ω-orbital admissible;

  3. (iii)

    there exists \(u_{0}\in Y\) such that \(\omega ( u_{0},Hu_{0} ) \geq 1\);

  4. (iv)

    Y is ω-regular.

Then H has a fixed point.


Define the sequence \({u_{n}}\) by \(u_{n}=H^{n}u_{0}\). If there exists \(k\in \mathbb{N}\) such that \(u_{k}=u_{k+1}\), then \(u_{k}\) is a fixed point of H. Assume that \(u_{n}\neq u_{n+1}\) for all \(n\in \mathbb{N}\). Now as \(\omega ( u_{0},Hu_{0} ) \geq 1\) and H is ω-orbital admissible, \(\omega ( u_{n},u_{n+1} ) \geq 1\) for all \(n\in \mathbb{N}\). And as H is an interpolative Hardy–Rogers–Suzuki-type \(\mathcal{Z}\)-contraction with respect to some \(\zeta \in \mathcal{Z}\) with \(\frac{1}{2}d ( u_{n},Hu_{n} ) =\frac{1}{2}d ( u_{n},u_{n+1} ) \leq d ( u_{n},u_{n+1} ) \), we have

$$ \zeta \bigl( \omega ( u_{n},u_{n+1} ) d ( u_{n+1},u_{n+2} ) ,C ( u_{n},u_{n+1} ) \bigr) \geq 0 $$

which turns into

$$\begin{aligned}& \begin{aligned} 0&\leq \zeta \bigl( \omega ( u_{n},u_{n+1} ) d ( u_{n+1},u_{n+2} ) ,C ( u_{n},u_{n+1} ) \bigr) \\ &< C ( u_{n},u_{n+1} ) -\omega ( u_{n},u_{n+1} ) d ( u_{n+1},u_{n+2} ) \end{aligned} \\& \quad \Longrightarrow\quad \omega ( u_{n},u_{n+1} ) d ( u_{n+1},u_{n+2} ) < C ( u_{n},u_{n+1} ). \end{aligned}$$

As \(\omega ( u_{n},u_{n+1} ) \geq 1\) for all \(n\in \mathbb{N}\), we have

$$ d ( u_{n+1},u_{n+2} ) \leq \omega ( u_{n},u_{n+1} ) d ( u_{n+1},u_{n+2} ) < C ( u_{n},u_{n+1} ), $$

which implies that

$$\begin{aligned} d ( u_{n+1},u_{n+2} ) & < d ( u_{n},u_{n+1} ) ^{ \alpha }d ( u_{n},u_{n+1} ) ^{\beta }d ( u_{n+1},u_{n+2} ) ^{\gamma } \\ &\quad {} \times \biggl[ \frac{1}{2} \bigl( d ( u_{n},u_{n+2} ) +d ( u_{n+1},u_{n+1} ) \bigr) \biggr] ^{1- \alpha -\beta -\gamma }, \end{aligned}$$

and, using the triangular inequality with the fact that the function \(f ( x ) =x^{1-\alpha -\beta -\gamma }\) is increasing for \(x>0\), we obtain

$$ \biggl[ \frac{1}{2}d ( u_{n},u_{n+2} ) \biggr] ^{1- \alpha -\beta -\gamma }\leq \biggl[ \frac{1}{2} \bigl( d ( u_{n},u_{n+1} ) +d ( u_{n+1},u_{n+2} ) \bigr) \biggr] ^{1- \alpha -\beta -\gamma }. $$

So from (2.3) we have

$$\begin{aligned} d ( u_{n+1},u_{n+2} ) & < d ( u_{n},u_{n+1} ) ^{ \alpha +\beta } d ( u_{n+1},u_{n+2} ) ^{\gamma } \\ &\quad {} \times \biggl[ \frac{1}{2} \bigl( d ( u_{n},u_{n+1} ) +d ( u_{n+1},u_{n+2} ) \bigr) \biggr] ^{1- \alpha -\beta -\gamma }. \end{aligned}$$

If we suppose that \(d ( u_{n},u_{n+1} ) < d ( u_{n+1},u_{n+2} ) \) for all \(n\in \mathbb{N}\), then (2.4) yields

$$ d ( u_{n+1},u_{n+2} ) ^{1-\gamma }< d ( u_{n},u_{n+1} ) ^{\alpha +\beta }+d ( u_{n+1},u_{n+2} ) ^{1- \alpha -\beta -\gamma }, $$

which implies that

$$ d ( u_{n+1},u_{n+2} ) ^{\alpha +\beta }< d ( u_{n},u_{n+1} ) ^{\alpha +\beta }, $$

a contradiction. Hence \(d ( u_{n+1},u_{n+2} ) \leq d ( u_{n},u_{n+1} )\) for all \(n\in \mathbb{N}\). So, we deduce that the sequence \(\{ d ( u_{n},u_{n+1} ) \} \) is nonincreasing, and as \(d ( u_{n},u_{n+1} ) \geq 0\) for all \(n\in \mathbb{N}\), \(\{ d ( u_{n},u_{n+1} ) \} \) is a bounded monotone sequence of real numbers, which implies that there exists \(t\geq 0\) such that \(\lim_{n\rightarrow \infty }d ( u_{n},u_{n+1} ) =t\). We have to prove that \(t=0\). It is easy to see that \(\lim_{n\rightarrow \infty }C ( u_{n},u_{n+1} ) =t\). So from (2.2) we have \(\lim_{n\rightarrow \infty } \omega ( u_{n},u_{n+1} ) d ( u_{n+1},u_{n+2} ) =t \) by the squeeze theorem. Accordingly, if we suppose that \(t>0\), we can apply \(\zeta _{2}\) to get

$$ 0\leq \zeta \bigl( \omega ( u_{n},u_{n+1} ) d ( u_{n+1},u_{n+2} ) ,C ( u_{n},u_{n+1} ) \bigr) < 0, $$

which is a contradiction. Hence \(t=0\), which implies that \(\{ u_{n} \} \) is a Cauchy sequence. By completeness of Y, there exists \(v\in Y\) such that \(\lim_{n\rightarrow \infty }u_{n}=v\). We will prove that v is a fixed point of H. Note that as Y is ω-regular and \(\omega ( u_{n},u_{n+1} ) \geq 1\) for all \(n\in \mathbb{N}\), so \(\omega ( u_{n},v ) \geq 1\) for all \(n\in \mathbb{N}\). Now either

$$ \frac{1}{2}d ( u_{n},Hu_{n} ) \leq d ( u_{n},v ) $$


$$ \frac{1}{2}d \bigl( Hu_{n},H^{2}u_{n} \bigr) \leq d ( Hu_{n},v ), $$

for if we suppose that \(\frac{1}{2}d ( u_{n},Hu_{n} ) >d ( u_{n},v ) \) and \(\frac{1}{2}d ( Hu_{n},H^{2}u_{n} ) >d ( Hu_{n},v ) \) then, using the triangular inequality together with the fact that \(\{ d ( u_{n},u_{n+1} ) \} \) is a nonincreasing sequence, we will get

$$\begin{aligned} d ( u_{n},u_{n+1} ) &=d ( u_{n},Hu_{n} ) \leq d ( u_{n},v ) +d ( v,Hu_{n} ) \\ &< \frac{1}{2}d ( u_{n},Hu_{n} ) + \frac{1}{2}d \bigl( Hu_{n},H^{2}u_{n} \bigr) \\ &=\frac{1}{2}d ( u_{n},u_{n+1} ) + \frac{1}{2}d ( u_{n+1},u_{n+2} ) \\ &\leq \frac{1}{2}d ( u_{n},u_{n+1} ) + \frac{1}{2}d ( u_{n},u_{n+1} ) \\ &=d ( u_{n},u_{n+1} ), \end{aligned}$$

which is a contradiction. So either (2.5) or (2.6) holds. If we assume that (2.5) holds and v is not a fixed point of H, then by ω-regularity of Y we have

$$ 0\leq \zeta \bigl( \omega ( u_{n},v ) d ( Hu_{n},Hv ) ,C ( u_{n},v ) \bigr). $$

Using \(\zeta _{2}\), we have

$$\begin{aligned}& 0\leq C ( u_{n},v ) -\omega ( u_{n},v ) d ( u_{n},Hv ) \\& \begin{aligned} \quad \Longrightarrow\quad d ( u_{n},Hv ) &\leq \omega ( u_{n},v ) d ( u_{n},Hv ) \\ &\leq C ( u_{n},v ) \\ &= \bigl[ d ( u_{n},u_{n+1} ) \bigr] ^{\alpha } \bigl[ d ( u_{n},v ) \bigr] ^{\beta } \bigl[ d ( v,Hv ) \bigr] ^{\gamma } \\ &\quad {} \times \biggl[ \frac{1}{2} \bigl( d ( u_{n},Hv ) +d ( v,u_{n+1} ) \bigr) \biggr] ^{1-\alpha -\beta - \gamma }. \end{aligned} \end{aligned}$$

As the limit of the right-hand side of the previous inequality as \(n\to \infty \) is zero, by the squeeze theorem, \(\lim_{n\rightarrow \infty }d ( u_{n},Hv ) =0\). Hence, by the uniqueness of the limit, we have \(v=Hv\). Similarly, if (2.6) holds, we can prove that v is a fixed point of H, as wanted. □


We get the following corollaries by using different examples of the function ζ.

Corollary 9

Let \(( Y,d ) \) be a complete metric space and let H be a self-mapping on Y. Assume that

  1. (i)

    there exists \(\alpha ,\beta ,\gamma \in ( 0,1 ) \), \(\lambda \in {}[ 0,1)\) with \(\alpha +\beta +\gamma <1\), and a function \(\omega :Y\times Y\rightarrow {}[ 0,\infty )\) such that

    $$\begin{aligned} &\frac{1}{2}d ( x,Hx ) \leq d ( x,y ) \\ &\quad \Longrightarrow \quad \omega ( x,y ) d ( Hx,Hy ) \leq \lambda C ( x,y ); \end{aligned}$$
  2. (ii)

    H is ω-orbital admissible;

  3. (iii)

    there exists \(u_{0}\in Y\) such that \(\omega ( u_{0}Hu_{0} ) \geq 1\);

  4. (iv)

    Y is ω-regular.

Then H has a fixed point.

Sketch of the proof

It is sufficient to replace \(\zeta =\lambda s-t \) in Theorem 8 where \(\lambda \in {}[ 0,1)\) for all \(s,t\in {}[ 0,\infty )\).

Corollary 10

Let \(( Y,d ) \) be a complete metric space and let H be a self-mapping on Y. Assume that

  1. (i)

    there exists \(\alpha ,\beta ,\gamma \in ( 0,1 ) \), with \(\alpha +\beta +\gamma <1\), a function \(\omega :Y\times Y\rightarrow {}[ 0,\infty )\) and an upper semi-continuous mapping \(\eta :[0,\infty )\rightarrow {}[ 0,\infty )\) with \(\eta (t)< t\) for all \(t>0\) and \(\eta (0)=0\) such that

    $$\begin{aligned} &\frac{1}{2}d ( x,Hx ) \leq d ( x,y ) \\ &\quad \Longrightarrow \quad \omega ( x,y ) d ( Hx,Hy ) \leq \eta \bigl( C ( x,y ) \bigr); \end{aligned}$$
  2. (ii)

    H is ω-orbital admissible;

  3. (iii)

    there exists \(u_{0}\in Y\) such that \(\omega ( u_{0}Hu_{0} ) \geq 1\);

  4. (iv)

    Y is ω-regular.

Then H has a fixed point.

Sketch of the proof

It is sufficient to replace \(\zeta (t,s)=\eta (s)-t\) for all \(s,t\in {}[ 0,\infty )\) in Theorem 8.

We can obtain more results by reducing the terms in Theorem 8 as follows:

Theorem 11

Let \(( Y,d ) \) be a complete metric space and let H be a self-mapping on Y. Assume that there exists \(\alpha ,\beta ,\in ( 0,1 ) \) with \(\alpha +\beta <1\), \(\zeta \in \mathcal{Z}\) and a function \(\omega :Y\times Y\rightarrow {}[ 0,\infty )\) such that

$$\begin{aligned} (\mathrm{i})&\quad \frac{1}{2}d ( x,Hx ) \leq d ( x,y ) \\ \hphantom{(\mathrm{i})}&\qquad \Longrightarrow\quad \zeta \bigl( \omega ( x,y ) d ( Hx,Hy ) ,D ( x,y ) \bigr) \geq 0 \end{aligned}$$

for all x, \(y\notin \operatorname{Fix}(H)\) where

$$ D(x,y):= \bigl[ d ( x,y ) \bigr] ^{\beta } \bigl[ d ( x,Hx ) \bigr] ^{\alpha } \bigl[ d ( y,Hy ) \bigr] ^{1-\alpha -\beta };$$
  1. (ii)

    H is ω-orbital admissible;

  2. (iii)

    there exists \(u_{0}\in Y\) such that \(\omega ( u_{0},Hu_{0} ) \geq 1\);

  3. (iv)

    Y is ω-regular.

Then H has a fixed point.


By analogue of the proof of Theorem 8. □


In conclusion, we can use the results of the paper to generate more results by using different examples of the simulation function. Moreover, we can follow the same argument of the proof of the main result to prove more results with less terms; this will enrich the fixed point theory.

Availability of data and materials

Not applicable.


  1. 1.

    Banach, S.: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, 133–181 (1922)

    Article  Google Scholar 

  2. 2.

    Caccioppoli, R.: Una teorema generale sull’esistenza di elementi uniti in una transformazione funzionale. Rend. Accad. Naz. Lincei 11, 794–799 (1930)

    MATH  Google Scholar 

  3. 3.

    Rus, I.A.: Generalized Contractions and Applications. Cluj University Press, Cluj-Napoca (2001)

    MATH  Google Scholar 

  4. 4.

    Suzuki, T.: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc. 136(5), 1861–1869 (2008)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Suzuki, T.: A new type of fixed point theorem in metric spaces. Nonlinear Anal. 71, 5313–5317 (2009)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Karapınar, E., Aydi, H., Fulga, A.: On p-hybrid Wardowski contractions. J. Math. 2020, Article ID 1632526 (2020)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Khan, M.S., Singh, Y.M., Karapınar, E.: On the interpolative \((\phi , \psi )\)-type Z-contraction. UPB Sci. Bull., Ser. A 83(2) (2021)

  8. 8.

    Yesilkaya, S.S.: On interpolative Hardy–Rogers contractive of Suzuki type mappings. Topol. Algebra Appl. 9(1), 13–19 (2021)

    MathSciNet  Google Scholar 

  9. 9.

    Khojasteh, F., Shukla, S., Radenovic, S.: A new approach to the study of fixed point theory for simulation functions. Filomat 29(6), 1189–1194 (2015)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Alsulami, H.H., Karapınar, E., Khojasteh, F., Roldan Lopez de Hierro, A.F.: A proposal to the study of contractions in quasi-metric spaces. Discrete Dyn. Nat. Soc. 2014, Article ID 269286 (2014)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Argoubi, H., Samet, B., Vetro, C.: Nonlinear contractions involving simulation functions in a metric space with a partial order. J. Nonlinear Sci. Appl. 8(6), 1082–1094 (2015)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Alsubaie, R., Alqahtani, B., Karapınar, E., Roldan Lopez de Hierro, A.F.: Extended simulation function via rational expressions. Mathematics 8(5), 710 (2020)

    Article  Google Scholar 

  13. 13.

    Alqahtani, O., Karapınar, E.: A bilateral contraction via simulation function. Filomat 33(15), 4837–4843 (2019)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Alghamdi, M., Gulyaz-Ozyurt, S., Karapınar, E.: A note on extended Z-contraction. Mathematics 8(2), 195 (2020)

    Article  Google Scholar 

  15. 15.

    Karapınar, E., Agarwal, R.P.: Interpolative Rus–Reich–Ćirić type contractions via simulation functions. An. Ştiinţ. Univ. ‘Ovidius’ Constanţa, Ser. Mat. 27, 137–152 (2019)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Aydi, H., Karapınar, E., Rakocevic, V.: Nonunique fixed point theorems on b-metric spaces via simulation functions. Jordan J. Math. Stat. 12(3), 265–288 (2019)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Karapınar, E., Khojasteh, F.: An approach to best proximity points results via simulation functions. J. Fixed Point Theory Appl. 19(3), 1983–1995 (2017)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Karapınar, E.: Fixed points results via simulation functions. Filomat 30(8), 2343–2350 (2016)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Samet, B., Vetro, C., Vetro, P.: Fixed point theorems for αψ-contractive type mappings. Nonlinear Anal., Theory Methods Appl. 75(4), 2154–2165 (2012)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Karapınar, E., Samet, B.: Generalized αψ contractive type mappings and related fixed point theorems with applications. Abstr. Appl. Anal. 2012, Article ID 793486 (2012)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Popescu, O.: Some new fixed point theorems for α-Geraghty contraction type maps in metric spaces. Fixed Point Theory Appl. 2014(1), 190 (2014)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Aksoy, U., Karapınar, E., Erhan, I.M.: Fixed points of generalized alpha-admissible contractions on b-metric spaces with an application to boundary value problems. J. Nonlinear Convex Anal. 17(6), 1095–1108 (2016)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Alsulami, H.H., Gulyaz, S., Karapınar, E., Erhan, I.M.: Fixed point theorems for a class of α-admissible contractions and applications to boundary value problem. Abstr. Appl. Anal. 2014, Article ID 187031 (2014)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Al-Mezel, S.A., Chen, C.M., Karapınar, E., Rakocevic, V.: Fixed point results for various α-admissible contractive mappings on metric-like spaces. Abstr. Appl. Anal. 2014, Article ID 379358 (2014)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Chen, C.M., Abkar, A., Ghods, S., Karapınar, E.: Fixed point theory for the α-admissible Meir–Keeler type set contractions having KKM* property on almost convex sets. Appl. Math. Inf. Sci. 11(1), 171–176 (2017)

    Article  Google Scholar 

  26. 26.

    Alharbi, A.S., Alsulami, H.H., Karapınar, E.: On the power of simulation and admissible functions in metric fixed point theory. J. Funct. Spaces 2017, Article ID 2068163 (2017)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Hammache, K., Karapınar, E., Ould-Hammouda, A.: On admissible weak contractions in b-metric-like space. J. Math. Anal. 8(3), 167–180 (2017)

    MathSciNet  Google Scholar 

  28. 28.

    Aydi, H., Karapınar, E., Yazidi, H.: Modified F-contractions via α-admissible mappings and application to integral equations. Filomat 31(5), 1141–1148 (2017)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Aydi, H., Karapınar, E., Zhang, D.: A note on generalized admissible Meir–Keeler contractions in the context of generalized metric spaces. Results Math. 71(1), 73–92 (2017)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Chifu, I.C., Karapınar, E.: Admissible hybrid Z-contractions in b-metric spaces. Axioms 9(1), 2 (2020)

    Article  Google Scholar 

  31. 31.

    Fulga, A., Yesilkaya, S.S.: On some interpolative contractions of Suzuki type mappings. J. Funct. Spaces 2021, Article ID 6596096 (2021).

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Gaba, Y.U., Karapınar, E.: A new approach to the interpolative contractions. Axioms 8, 110 (2019)

    Article  Google Scholar 

  33. 33.

    Karapınar, E., Aydi, H., Mitrovic, Z.D.: On interpolative Boyd–Wong and Matkowski type contractions. TWMS J. Pure Appl. Math. 11(2), 204–212 (2020)

    MathSciNet  Google Scholar 

  34. 34.

    Aydi, H., Chen, C.M., Karapınar, E.: Interpolative Ćirić–Reich–Rus type contractions via the Branciari distance. Mathematics 7(1), 84 (2019)

    Article  Google Scholar 

  35. 35.

    Aydi, H., Karapınar, E., Roldan Lopez de Hierro, A.F.: ω-Interpolative Ćirić–Reich–Rus-type contractions. Mathematics 7, 57 (2019)

    Article  Google Scholar 

  36. 36.

    Karapınar, E., Alqahtani, O., Aydi, H.: On interpolative Hardy–Rogers type contractions. Symmetry 11(1), 8 (2019).

    Article  MATH  Google Scholar 

  37. 37.

    Karapınar, E., Agarwal, R., Aydi, H.: Interpolative Reich–Rus–Ćirić type contractions on partial metric spaces. Mathematics 6, 256 (2018).

    Article  MATH  Google Scholar 

  38. 38.

    Karapınar, E.: Revisiting simulation functions via interpolative contractions. Appl. Anal. Discrete Math. 13, 859–870 (2019)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Karapınar, E.: Revisiting the Kannan type contractions via interpolation. Adv. Theory Nonlinear Anal. Appl. 2, 85–87 (2018)

    MATH  Google Scholar 

  40. 40.

    Karapınar, E., Fulga, A.: New hybrid contractions on b-metric spaces. Mathematics 7(7), 578 (2019)

    Article  Google Scholar 

  41. 41.

    Karapınar, E., Fulga, A., Roldan Lopez de Hierro, A.F.: Fixed point theory in the setting of \((\alpha ,\beta ,\psi ,\phi ) \)-interpolative contractions. Adv. Differ. Equ. 2021(1), 339 (2021)

    MathSciNet  Article  Google Scholar 

Download references


The author thanks the reviewers for reviewing this paper.


The research was not funded.

Author information




The author read and approved the final manuscript.

Corresponding author

Correspondence to Maha Noorwali.

Ethics declarations

Competing interests

The author declares that she has no competing interests.

Consent for publication

Not applicable.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Noorwali, M. Revising the Hardy–Rogers–Suzuki-type Z-contractions. Adv Differ Equ 2021, 413 (2021).

Download citation


  • 47H10
  • 54H25


  • Fixed point
  • Admissible contractions
  • Interpolative
  • Suzuki-type contractions
  • w-regular