 Research
 Open Access
 Published:
Revising the Hardy–Rogers–Suzukitype Zcontractions
Advances in Difference Equations volume 2021, Article number: 413 (2021)
Abstract
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.
1 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., [3–41] 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., [19–29]), admissible mapping (see, e.g., [9–18]), and Suzukitype contraction (see [4, 5]).
In 2014, Popescu [21] suggested an interesting notion, the socalled ωorbital admissible mappings, which is a smart expansion of the notion of αadmissible mappings, see Samet et al. [19]. In this work, Popescu [21] showed that each admissible mapping is an ωorbital admissible mapping, but the converse is not true.
Definition 1
([21])
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 ωorbital admissible if for all u in Y, we have
One of the interesting uses of ωadmissible mapping is that it is ωregular in the setting of metric spaces. This was a condition that helps refine the continuity condition on the selfmapping accompanied with some additional conditions; see, e.g., [19].
Definition 2
A metric space \((Y,d)\) is called ωregular 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:
 \((\zeta _{1})\):

\(\zeta (t,s)< st\) 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
$$ \limsup_{n\rightarrow \infty }\zeta (t_{n},s_{n})< 0. $$(1.1)
We demonstrate here some examples of simulation functions from [10–18].
Example 4
For \(i=1,2\), we define the mappings \(\zeta _{i}:[0,\infty )\times {}[ 0,\infty )\rightarrow \mathbb{R}\), as follows:

(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 \(\zeta _{B}=\lambda st\) for all \(s,t\in {}[ 0,\infty )\).

(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 Suzukitype contraction mappings.
Definition 5
([5])
A selfmapping H on a metric space \(( Y,d ) \) is called a Suzukitype contraction if for all \(x,y\in Y\) with \(x\neq y\), we have
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, 30–41]. Recently, interpolative Hardy–Rogerstype contractions have been investigated by many authors (see [6–8]). In particular, in [38], Karapınar used simulation functions to introduce the notion of interpolative Hardy–Rogerstype \(\mathcal{Z}\)contraction mappings and prove some related fixed point results. The aim of our work is to combine the latter contractions with those of the Suzukitype 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–Rogerstype \(\mathcal{Z}\)contraction mappings is given as follows.
Definition 6
([38])
Let H be a selfmapping 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
for all \(x,y \in Y\backslash \operatorname{Fix}(H)\), where \(\operatorname{Fix}(H)\) is the set of all fixed point of H, and
then we say that H is an interpolative Hardy–Rogerstype \(\mathcal{Z}\)contraction with respect to ζ.
2 Main results
We introduce now our new contraction type mapping in the following definition.
Definition 7
Let H be a selfmapping on a metric space \(( Y,d ) \). We say that H is an interpolative HardyRogers–Suzukitype \(\mathcal{Z}\)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
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 selfmapping on Y. Assume that

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

(ii)
H is ωorbital admissible;

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

(iv)
Y is ωregular.
Then H has a fixed point.
Proof
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 ωorbital admissible, \(\omega ( u_{n},u_{n+1} ) \geq 1\) for all \(n\in \mathbb{N}\). And as H is an interpolative Hardy–Rogers–Suzukitype \(\mathcal{Z}\)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
which turns into
As \(\omega ( u_{n},u_{n+1} ) \geq 1\) for all \(n\in \mathbb{N}\), we have
which implies that
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
So from (2.3) we have
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
which implies that
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
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 ωregular 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
or
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
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 ωregularity of Y we have
Using \(\zeta _{2}\), we have
As the limit of the righthand 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. □
2.1 Consequences
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 selfmapping on Y. Assume that

(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
$$\begin{aligned} &\frac{1}{2}d ( x,Hx ) \leq d ( x,y ) \\ &\quad \Longrightarrow \quad \omega ( x,y ) d ( Hx,Hy ) \leq \lambda C ( x,y ); \end{aligned}$$(2.7) 
(ii)
H is ωorbital admissible;

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

(iv)
Y is ωregular.
Then H has a fixed point.
Sketch of the proof
It is sufficient to replace \(\zeta =\lambda st \) 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 selfmapping on Y. Assume that

(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 semicontinuous mapping \(\eta :[0,\infty )\rightarrow {}[ 0,\infty )\) with \(\eta (t)< t\) for all \(t>0\) and \(\eta (0)=0\) such that
$$\begin{aligned} &\frac{1}{2}d ( x,Hx ) \leq d ( x,y ) \\ &\quad \Longrightarrow \quad \omega ( x,y ) d ( Hx,Hy ) \leq \eta \bigl( C ( x,y ) \bigr); \end{aligned}$$(2.8) 
(ii)
H is ωorbital admissible;

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

(iv)
Y is ωregular.
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 selfmapping 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
for all x, \(y\notin \operatorname{Fix}(H)\) where

(ii)
H is ωorbital admissible;

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

(iv)
Y is ωregular.
Then H has a fixed point.
Proof
By analogue of the proof of Theorem 8. □
3 Conclusion
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.
References
Banach, S.: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, 133–181 (1922)
Caccioppoli, R.: Una teorema generale sull’esistenza di elementi uniti in una transformazione funzionale. Rend. Accad. Naz. Lincei 11, 794–799 (1930)
Rus, I.A.: Generalized Contractions and Applications. Cluj University Press, ClujNapoca (2001)
Suzuki, T.: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc. 136(5), 1861–1869 (2008)
Suzuki, T.: A new type of fixed point theorem in metric spaces. Nonlinear Anal. 71, 5313–5317 (2009)
Karapınar, E., Aydi, H., Fulga, A.: On phybrid Wardowski contractions. J. Math. 2020, Article ID 1632526 (2020)
Khan, M.S., Singh, Y.M., Karapınar, E.: On the interpolative \((\phi , \psi )\)type Zcontraction. UPB Sci. Bull., Ser. A 83(2) (2021)
Yesilkaya, S.S.: On interpolative Hardy–Rogers contractive of Suzuki type mappings. Topol. Algebra Appl. 9(1), 13–19 (2021)
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)
Alsulami, H.H., Karapınar, E., Khojasteh, F., Roldan Lopez de Hierro, A.F.: A proposal to the study of contractions in quasimetric spaces. Discrete Dyn. Nat. Soc. 2014, Article ID 269286 (2014)
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)
Alsubaie, R., Alqahtani, B., Karapınar, E., Roldan Lopez de Hierro, A.F.: Extended simulation function via rational expressions. Mathematics 8(5), 710 (2020)
Alqahtani, O., Karapınar, E.: A bilateral contraction via simulation function. Filomat 33(15), 4837–4843 (2019)
Alghamdi, M., GulyazOzyurt, S., Karapınar, E.: A note on extended Zcontraction. Mathematics 8(2), 195 (2020)
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)
Aydi, H., Karapınar, E., Rakocevic, V.: Nonunique fixed point theorems on bmetric spaces via simulation functions. Jordan J. Math. Stat. 12(3), 265–288 (2019)
Karapınar, E., Khojasteh, F.: An approach to best proximity points results via simulation functions. J. Fixed Point Theory Appl. 19(3), 1983–1995 (2017)
Karapınar, E.: Fixed points results via simulation functions. Filomat 30(8), 2343–2350 (2016)
Samet, B., Vetro, C., Vetro, P.: Fixed point theorems for α–ψcontractive type mappings. Nonlinear Anal., Theory Methods Appl. 75(4), 2154–2165 (2012)
Karapınar, E., Samet, B.: Generalized α–ψ contractive type mappings and related fixed point theorems with applications. Abstr. Appl. Anal. 2012, Article ID 793486 (2012)
Popescu, O.: Some new fixed point theorems for αGeraghty contraction type maps in metric spaces. Fixed Point Theory Appl. 2014(1), 190 (2014)
Aksoy, U., Karapınar, E., Erhan, I.M.: Fixed points of generalized alphaadmissible contractions on bmetric spaces with an application to boundary value problems. J. Nonlinear Convex Anal. 17(6), 1095–1108 (2016)
Alsulami, H.H., Gulyaz, S., Karapınar, E., Erhan, I.M.: Fixed point theorems for a class of αadmissible contractions and applications to boundary value problem. Abstr. Appl. Anal. 2014, Article ID 187031 (2014)
AlMezel, S.A., Chen, C.M., Karapınar, E., Rakocevic, V.: Fixed point results for various αadmissible contractive mappings on metriclike spaces. Abstr. Appl. Anal. 2014, Article ID 379358 (2014)
Chen, C.M., Abkar, A., Ghods, S., Karapınar, E.: Fixed point theory for the αadmissible Meir–Keeler type set contractions having KKM* property on almost convex sets. Appl. Math. Inf. Sci. 11(1), 171–176 (2017)
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)
Hammache, K., Karapınar, E., OuldHammouda, A.: On admissible weak contractions in bmetriclike space. J. Math. Anal. 8(3), 167–180 (2017)
Aydi, H., Karapınar, E., Yazidi, H.: Modified Fcontractions via αadmissible mappings and application to integral equations. Filomat 31(5), 1141–1148 (2017)
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)
Chifu, I.C., Karapınar, E.: Admissible hybrid Zcontractions in bmetric spaces. Axioms 9(1), 2 (2020)
Fulga, A., Yesilkaya, S.S.: On some interpolative contractions of Suzuki type mappings. J. Funct. Spaces 2021, Article ID 6596096 (2021). https://doi.org/10.1155/2021/6596096
Gaba, Y.U., Karapınar, E.: A new approach to the interpolative contractions. Axioms 8, 110 (2019)
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)
Aydi, H., Chen, C.M., Karapınar, E.: Interpolative Ćirić–Reich–Rus type contractions via the Branciari distance. Mathematics 7(1), 84 (2019)
Aydi, H., Karapınar, E., Roldan Lopez de Hierro, A.F.: ωInterpolative Ćirić–Reich–Rustype contractions. Mathematics 7, 57 (2019)
Karapınar, E., Alqahtani, O., Aydi, H.: On interpolative Hardy–Rogers type contractions. Symmetry 11(1), 8 (2019). https://doi.org/10.3390/sym11010008
Karapınar, E., Agarwal, R., Aydi, H.: Interpolative Reich–Rus–Ćirić type contractions on partial metric spaces. Mathematics 6, 256 (2018). https://doi.org/10.3390/math6110256
Karapınar, E.: Revisiting simulation functions via interpolative contractions. Appl. Anal. Discrete Math. 13, 859–870 (2019)
Karapınar, E.: Revisiting the Kannan type contractions via interpolation. Adv. Theory Nonlinear Anal. Appl. 2, 85–87 (2018)
Karapınar, E., Fulga, A.: New hybrid contractions on bmetric spaces. Mathematics 7(7), 578 (2019)
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)
Acknowledgements
The author thanks the reviewers for reviewing this paper.
Funding
The research was not funded.
Author information
Authors and Affiliations
Contributions
The author read and approved the final manuscript.
Corresponding author
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 http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Noorwali, M. Revising the Hardy–Rogers–Suzukitype Zcontractions. Adv Differ Equ 2021, 413 (2021). https://doi.org/10.1186/s13662021035668
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662021035668
MSC
 47H10
 54H25
Keywords
 Fixed point
 Admissible contractions
 Interpolative
 Suzukitype contractions
 wregular