On p-Hardy–Rogers and p-Zamfirescu Contractions in Complete Metric Spaces: Existence and Uniqueness Results

Abstract

In this paper, we introduce and investigate two generalized forms of classical contraction mappings, namely the p-Hardy–Rogers and p-Zamfirescu contractions. By incorporating the integer parameter $p\ge 1$, these new definitions extend the traditional Hardy–Rogers and Zamfirescu conditions to iterated mappings ${\mathit{ħ}}^p$. We establish fixed-point theorems, ensuring both existence and uniqueness of fixed points for continuous self-maps on complete metric spaces that satisfy these p-contractive conditions. The proofs are constructed via geometric estimates on the iterates and by transferring the fixed point from the p-th iterate ${\mathit{ħ}}^p$ to the original mapping ħ. Our results unify and broaden several well-known fixed-point theorems reported in previous studies, including those of Banach, Hardy–Rogers, and Zamfirescu as special cases.

1. Introduction

The theory of fixed points has become one of the most powerful and elegant tools in modern mathematical analysis, with extensive applications in nonlinear functional analysis, optimization, differential and integral equations, and dynamic systems. The cornerstone of this theory is the celebrated Banach contraction principle, which guarantees the existence and uniqueness of a fixed point for a contractive self-map on a complete metric space. Over the years, many authors have developed numerous generalizations of this principle, among which the Hardy–Rogers and Zamfirescu contractions occupy a central role.

The classical Hardy–Rogers contraction [1] refines the Banach contraction by incorporating mixed terms involving the distances between $\zeta ,\mathit{ħ}\zeta ,\eta$, and $\mathit{ħ}\eta$. On the other hand, Zamfirescu [2] introduced a broader condition that unifies several contraction types, including those of Banach, Kannan, and Chatterjea. Both concepts have inspired an impressive body of literature devoted to extending and combining these mappings across distinct abstract settings, notably in b-metric spaces, partial metric spaces, and ordered structures. In this paper, we propose two new generalizations of these classical contractions by introducing the integer parameter $p\ge 1$ into the framework. Specifically, we define and study the notions of p-Hardy–Rogers and p-Zamfirescu contractions. The idea is to apply the contractive condition not to the mapping ħ itself, but to its p-th iterate ${\mathit{ħ}}^p$. This approach, inspired by the work of Bekri and Fabiano [3], captures a wider class of iterative processes and provides a natural unification between the classical and iterated forms of contraction mappings. We establish fixed-point theorems, ensuring the existence and uniqueness of fixed points for such p-contractive mappings on complete metric spaces. The proofs are constructed by analyzing the sequence of iterates ${\left\{{\mathit{ħ}}^{np}\left({\zeta }_0\right)\right\}}_{n\ge 0}$, showing its Cauchy property, and transferring the obtained fixed point from ${\mathit{ħ}}^p$ to ħ. Furthermore, we demonstrate that our results properly generalize the theorems of Banach, Hardy–Rogers, and Zamfirescu as special cases when $p=1$.

The structure of this paper is as follows. In Section 2, we recall the basic definitions. Section 3 presents the core results concerning p-Hardy–Rogers and p-Zamfirescu contractions together with their proofs. In Section 4, we highlight practical examples inspired by basic theories. In Section 5, an application of the results to a nonlinear boundary-value problem is discussed, demonstrating the utility of the theory in the study of integral equations. Finally, Section 6 provides concluding remarks and possible directions for future research.

2. Preliminaries

In the preliminary part of this work, we introduce the notions of p-Zamfirescu contraction and p-Hardy–Rogers contraction, both formulated within the framework of the generalized Singh [4,5] concept applied to the Kannan [6,7] setting. Furthermore, we provide rigorous proofs of two fundamental theorems.

Suppose that $(\mathrm{\Xi },d)$ denotes a metric space, and $\mathit{ħ}:\mathrm{\Xi }\to \mathrm{\Xi }$ represents a self-mapping. We denote by $I_p$ the p-th iterate of ħ, defined by

$${\mathit{ħ}}^p\left(\zeta \right)=\underset{p\mathrm{times}}{\underbrace{\mathit{ħ}\left(\mathit{ħ}\right(\cdots \mathit{ħ}\left(\zeta \right)\cdots \left)\right)}}.$$

Definition 1

(Hardy–Rogers contraction [1,8]). A mapping $\mathit{ħ}:\mathrm{\Xi }\to \mathrm{\Xi }$ is called a Hardy–Rogers contraction if there exist nonnegative constants ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5$, satisfying

$${\lambda }_1+{\lambda }_2+{\lambda }_3+2({\lambda }_4+{\lambda }_5)<1,$$

such that for all $\zeta ,\eta \in \mathrm{\Xi }$,

$$d(\mathit{ħ}\zeta ,\mathit{ħ}\eta )\le {\lambda }_{1}d(\zeta ,\eta )+{\lambda }_{2}d(\zeta ,\mathit{ħ}\zeta )+{\lambda }_{3}d(\eta ,\mathit{ħ}\eta )+{\lambda }_{4}d(\zeta ,\mathit{ħ}\eta )+{\lambda }_{5}d(\eta ,\mathit{ħ}\zeta ).$$

Definition 2

(p-Hardy–Rogers contraction). Let $(\mathrm{\Xi },d)$ be a metric space and let $p\ge 1$ be an integer. A mapping $\mathit{ħ}:\mathrm{\Xi }\to \mathrm{\Xi }$ is said to satisfy the p-Hardy–Rogers contraction condition if there exist nonnegative constants ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5$, such that

$${\lambda }_1+{\lambda }_2+{\lambda }_3+2({\lambda }_4+{\lambda }_5)<1,$$

and for all $\zeta ,\eta \in \mathrm{\Xi }$,

$$d({\mathit{ħ}}^p\zeta ,{\mathit{ħ}}^p\eta )\le {\lambda }_{1}d(\zeta ,\eta )+{\lambda }_{2}d(\zeta ,{\mathit{ħ}}^p\zeta )+{\lambda }_{3}d(\eta ,{\mathit{ħ}}^p\eta )+{\lambda }_{4}d(\zeta ,{\mathit{ħ}}^p\eta )+{\lambda }_{5}d(\eta ,{\mathit{ħ}}^p\zeta ).$$

Definition 3

(Zamfirescu contraction [2,8]). A self-map $\mathit{ħ}:\mathrm{\Xi }\to \mathrm{\Xi }$ is called a Zamfirescu contraction if there exist constants $\lambda ,\beta ,\gamma$, such that

$$0<\lambda <1,0<\beta ,\gamma <\frac{1}{2},$$

and for all $\zeta ,\eta \in \mathrm{\Xi }$, at least one of the following inequalities holds:

(i)   

$d(\mathit{ħ}\zeta ,\mathit{ħ}\eta )\le \lambda d(\zeta ,\eta );$

(ii)  

$d(\mathit{ħ}\zeta ,\mathit{ħ}\eta )\le \beta \left(d(\zeta ,\mathit{ħ}\zeta )+d(\eta ,\mathit{ħ}\eta )\right);$

(iii) 

$d(\mathit{ħ}\zeta ,\mathit{ħ}\eta )\le \gamma \left(d(\zeta ,\mathit{ħ}\eta )+d(\eta ,\mathit{ħ}\zeta )\right).$

Definition 4

(p-Zamfirescu contraction). Let $(\mathrm{\Xi },d)$ be a metric space and let $p\ge 1$. A self-map $\mathit{ħ}:\mathrm{\Xi }\to \mathrm{\Xi }$ satisfies the p-Zamfirescu condition if there exist real numbers $\lambda ,\beta ,\gamma$ satisfying

$$0<\lambda <1,0<\beta ,\gamma <\frac{1}{2},$$

such that for all $\zeta ,\eta \in \mathrm{\Xi }$, at least one of the following holds:

(i)   

$d({\mathit{ħ}}^p\zeta ,{\mathit{ħ}}^p\eta )\le \lambda d(\zeta ,\eta );$

(ii)  

$d({\mathit{ħ}}^p\zeta ,{\mathit{ħ}}^p\eta )\le \beta \left(d(\zeta ,{\mathit{ħ}}^p\zeta )+d(\eta ,{\mathit{ħ}}^p\eta )\right);$

(iii) 

$d({\mathit{ħ}}^p\zeta ,{\mathit{ħ}}^p\eta )\le \gamma \left(d(\zeta ,{\mathit{ħ}}^p\eta )+d(\eta ,{\mathit{ħ}}^p\zeta )\right).$

3. Core Results

Within this section, we establish the core theorems in fixed-point theory for the p-Hardy–Rogers and p-Zamfirescu contractions introduced above.

Theorem 1.

Suppose $(\mathrm{\Xi },d)$ is a complete metric space, and let $\mathit{ħ}:\mathrm{\Xi }\to \mathrm{\Xi }$ be a continuous mapping. Fix an integer $p\ge 1$. Assume that ħ satisfies the p-Hardy–Rogers contraction condition; that is, there exist nonnegative constants ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5$, such that

$${\lambda }_1+{\lambda }_2+{\lambda }_3+2({\lambda }_4+{\lambda }_5)<1,$$

and for all $\zeta ,\eta \in \mathrm{\Xi }$,

$$d({\mathit{ħ}}^p\zeta ,{\mathit{ħ}}^p\eta )\le {\lambda }_{1}d(\zeta ,\eta )+{\lambda }_{2}d(\zeta ,{\mathit{ħ}}^p\zeta )+{\lambda }_{3}d(\eta ,{\mathit{ħ}}^p\eta )+{\lambda }_{4}d(\zeta ,{\mathit{ħ}}^p\eta )+{\lambda }_{5}d(\eta ,{\mathit{ħ}}^p\zeta ).$$

Then ħ admits a unique fixed point $z\in \mathrm{\Xi }$. Furthermore, for any initial point ${\zeta }_0\in \mathrm{\Xi }$, the Picard iterates ${\zeta }_{n+1}=\mathit{ħ}\left({\zeta }_n\right)$ converge to z as $n\to \infty$.

Proof. 

The proof proceeds by applying the Hardy–Rogers inequality to the iterates of ${\mathit{ħ}}^p$ and then transferring the fixed point from ${\mathit{ħ}}^p$ to ħ.

Step 1.

Reduction to the sequence of p-iterates.

Fix ${\zeta }_0\in \mathrm{\Xi }$ and define

$${\eta }_n:={\left({\mathit{ħ}}^p\right)}^n\left({\zeta }_0\right)={\mathit{ħ}}^{np}\left({\zeta }_0\right),n=0,1,2,\dots$$

Set

$${\Delta }_n:=d({\eta }_n,{\eta }_{n+1})=d({\mathit{ħ}}^{np}\left({\zeta }_0\right),{\mathit{ħ}}^{(n+1)p}\left({\zeta }_0\right)).$$

Apply the p-Hardy–Rogers inequality to the pair $({\eta }_n,{\eta }_{n+1})$. Since ${\mathit{ħ}}^p\left({\eta }_n\right)={\eta }_{n+1}$ and ${\mathit{ħ}}^p\left({\eta }_{n+1}\right)={\eta }_{n+2}$, we obtain

$$\begin{gathered} {\Delta }_{n+1}=d({\mathit{ħ}}^p\left({\eta }_n\right),{\mathit{ħ}}^p\left({\eta }_{n+1}\right))\le {\lambda }_{1}d({\eta }_n,{\eta }_{n+1})+{\lambda }_{2}d({\eta }_n,{\eta }_{n+1})+{\lambda }_{3}d({\eta }_{n+1},{\eta }_{n+2})+ \\ {\lambda }_{4}d({\eta }_n,{\eta }_{n+2})+{\lambda }_{5}d({\eta }_{n+1},{\eta }_{n+1}). \end{gathered}$$

Because $d({\eta }_{n+1},{\eta }_{n+1})=0$ and by the triangle inequality $d({\eta }_n,{\eta }_{n+2})\le {\Delta }_n+{\Delta }_{n+1}$, we get

$${\Delta }_{n+1}\le ({\lambda }_1+{\lambda }_2+{\lambda }_4){\Delta }_n+({\lambda }_3+{\lambda }_4){\Delta }_{n+1}.$$

Rearranging gives

$$(1-{\lambda }_3-{\lambda }_4){\Delta }_{n+1}\le ({\lambda }_1+{\lambda }_2+{\lambda }_4){\Delta }_n,$$

and therefore,

$${\Delta }_{n+1}\le q{\Delta }_n,\mathrm{where}q={\displaystyle \frac{{\lambda }_1+{\lambda }_2+{\lambda }_4}{1-{\lambda }_3-{\lambda }_4}}.$$

We must check $0\le q<1$. From the hypothesis,

$${\lambda }_1+{\lambda }_2+{\lambda }_3+2({\lambda }_4+{\lambda }_5)<1,$$

and ${\lambda }_5\ge 0$, we deduce

$${\lambda }_1+{\lambda }_2+{\lambda }_3+2{\lambda }_4<1\Rightarrow {\lambda }_1+{\lambda }_2+{\lambda }_4<1-{\lambda }_3-{\lambda }_4.$$

Thus, the denominator $1-{\lambda }_3-{\lambda }_4$ is positive and $q<1$. Also $q\ge 0$ because all ${\lambda }_i\ge 0$. Hence, a strict geometric contraction factor $q\in [0,1)$ exists. Consequently, by induction,

$${\Delta }_n<q^n{\Delta }_0\mathrm{for}\mathrm{all}n\ge 0,$$

so, in particular, ${\Delta }_n\to 0$ as $n\to \infty$.

Step 2.

The sequence $\left\{{\eta }_n\right\}$ is Cauchy.

For $m>n$, we have by telescoping and the geometric bound

$$d({\eta }_n,{\eta }_m)\le \sum _{k=n}^{m-1}{\Delta }_k\le {\Delta }_n\sum _{j=0}^{\infty }{q}^j={\displaystyle \frac{{\Delta }_n}{1-q}}.$$

Since ${\Delta }_n\to 0$, the right-hand side can be made arbitrarily small for large n, so $\left\{{\eta }_n\right\}$ is Cauchy. By completeness of $\mathrm{\Xi }$, there exists $z\in \mathrm{\Xi }$ with ${\eta }_n\to z$. Passing to the limit in ${\eta }_{n+1}={\mathit{ħ}}^p\left({\eta }_n\right)$ and using continuity of ${\mathit{ħ}}^p$ (which follows from continuity of ħ), we obtain

$$z=\underset{n\to \infty }{lim}{\eta }_{n+1}=\underset{n\to \infty }{lim}{\mathit{ħ}}^p\left({\eta }_n\right)={\mathit{ħ}}^p\left(z\right),$$

so z is a fixed point for ${\mathit{ħ}}^p$.

Step 3.

Uniqueness property of the fixed point of ${\mathit{ħ}}^p$.

Suppose $\theta \ne \vartheta$ are two distinct fixed points of ${\mathit{ħ}}^p$. Apply the contraction to $(\theta ,\vartheta )$.

$$d(\theta ,\vartheta )=d({\mathit{ħ}}^p\left(\theta \right),{\mathit{ħ}}^p\left(\vartheta \right))\le {\lambda }_{1}d(\theta ,\vartheta )+{\lambda }_2·0+{\lambda }_3·0+{\lambda }_{4}d(\theta ,\vartheta )+{\lambda }_{5}d(\theta ,\vartheta ).$$

Hence,

$$d(\theta ,\vartheta )\le ({\lambda }_1+{\lambda }_4+{\lambda }_5)d(\theta ,\vartheta )<d(\theta ,\vartheta ),$$

which is impossible because ${\lambda }_1+{\lambda }_4+{\lambda }_5<1$ (since ${\lambda }_1+{\lambda }_2+{\lambda }_3+2({\lambda }_4+{\lambda }_5)<1$). Therefore, $\theta =\vartheta$, and the fixed point of ${\mathit{ħ}}^p$ is unique; thus, the limit z is the single fixed point of ${\mathit{ħ}}^p$.

Step 4.

Transfer of the fixed point from ${\mathit{ħ}}^p$ to $\mathit{ħ}$.

We show that z is also a fixed point of ħ. From ${\eta }_n\to z$, we have for every $r\in \{0,1,\dots ,p-1\}$ the subsequence

$${\zeta }_{np+r}={\mathit{ħ}}^r\left({\eta }_n\right)$$

satisfies ${\zeta }_{np+r}\to {\mathit{ħ}}^r\left(z\right)$. On the other hand, those subsequences are exactly the subsequences of the Picard sequence ${\zeta }_m={\mathit{ħ}}^m\left({\zeta }_0\right)$. Because the subsequence with $r=0$ converges to z and that with $r=1$ converges to $\mathit{ħ}\left(z\right)$, but ${\zeta }_{np+1}=\mathit{ħ}\left({\eta }_n\right)\to z$ (since ${\eta }_{n+1}\to z$), it follows that $\mathit{ħ}\left(z\right)=z$. Thus, z is a fixed point for ħ.

Step 5.

Uniqueness of the fixed point of $\mathit{ħ}$ and convergence of Picard iterates.

Assume the points $\theta$ and $\vartheta$ are fixed under ħ; then they are fixed for ${\mathit{ħ}}^p$ as well. Hence, by uniqueness in Step 3, we have $\theta =\vartheta$. Thus, the fixed point of ħ is unique. Finally, every subsequence modulo p of the Picard sequence ${\zeta }_n={\mathit{ħ}}^n\left({\zeta }_0\right)$ converges to z (since ${\zeta }_{np+r}={\mathit{ħ}}^r\left({\eta }_n\right)\to {\mathit{ħ}}^r\left(z\right)=z$). Therefore, the whole sequence ${\zeta }_n$ converges to z. With this, the proof is completed. □

Theorem 2.

Take $(\mathrm{\Xi },d)$ to be a complete metric space along with a continuous mapping $\mathit{ħ}:\mathrm{\Xi }\to \mathrm{\Xi }$. Fix an integer $p\ge 1$. Assume that ħ satisfies the p-Zamfirescu condition; that is, there exist constants $0<\lambda <1$ and $0<\beta ,\gamma <\frac{1}{2}$, such that for every $\zeta ,\eta \in \mathrm{\Xi }$, any one of the following inequalities holds

$$\begin{array}{cc}\hfill \mathit{\left(}i\mathit{\right)}& d({\mathit{ħ}}^p\zeta ,{\mathit{ħ}}^p\eta )\le \lambda d(\zeta ,\eta ),\hfill \\ \hfill \mathit{\left(}ii\mathit{\right)}& d({\mathit{ħ}}^p\zeta ,{\mathit{ħ}}^p\eta )\le \beta \left(d(\zeta ,{\mathit{ħ}}^p\zeta )+d(\eta ,{\mathit{ħ}}^p\eta )\right),\hfill \\ \hfill \mathit{\left(}iii\mathit{\right)}& d({\mathit{ħ}}^p\zeta ,{\mathit{ħ}}^p\eta )\le \gamma \left(d(\zeta ,{\mathit{ħ}}^p\eta )+d(\eta ,{\mathit{ħ}}^p\zeta )\right).\hfill \end{array}$$

Then ħ admits a single fixed point $z\in \mathrm{\Xi }$. Furthermore, for any initial point ${\zeta }_0\in \mathrm{\Xi }$, the Picard iterates ${\zeta }_{n+1}={\mathit{ħ}}^p\left({\zeta }_n\right)$ converge to z.

Proof. 

The proof proceeds by applying the Zamfirescu inequality to the iterates of ${\mathit{ħ}}^p$ and then transferring the fixed point from ${\mathit{ħ}}^p$ to ħ.

Step 1.

On the existence and uniqueness of a fixed point of ${\mathit{ħ}}^p$.

Fix ${\zeta }_0\in \mathrm{\Xi }$ and define the sequence

$${\eta }_n={\left({\mathit{ħ}}^p\right)}^n\left({\zeta }_0\right)={\mathit{ħ}}^{np}\left({\zeta }_0\right),n=0,1,2,\dots$$

For brevity, denote

$${\Delta }_n:=d({\eta }_n,{\eta }_{n+1})=d({\mathit{ħ}}^{np}\left({\zeta }_0\right),{\mathit{ħ}}^{(n+1)p}\left({\zeta }_0\right)).$$

By the p-Zamfirescu condition, for each n, at least one of the three inequalities holds with $(\zeta ,\eta )=({\eta }_n,{\eta }_{n+1})$.

$$\begin{array}{cc}\hfill {\Delta }_{n+1}& \le \lambda d({\eta }_n,{\eta }_{n+1})=\lambda {\Delta }_n,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill {\Delta }_{n+1}& \le \beta \left(d({\eta }_n,{\mathit{ħ}}^p\left({\eta }_n\right))+d({\eta }_{n+1},{\mathit{ħ}}^p\left({\eta }_{n+1}\right))\right)=\beta ({\Delta }_n+{\Delta }_{n+1}),\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\Delta }_{n+1}& \le \gamma \left(d({\eta }_n,{\mathit{ħ}}^p\left({\eta }_{n+1}\right))+d({\eta }_{n+1},{\mathit{ħ}}^p\left({\eta }_n\right))\right)=\gamma (d({\eta }_n,{\eta }_{n+2})+0).\hfill \end{array}$$

(3)

We analyze the long-term behavior of $\left({\Delta }_n\right)$.

Claim A.

${\Delta }_n\to 0$ as $n\to \infty$.

Proof of Claim A.

Assume by contradiction that ${lim\; sup}_{n\to \infty }{\Delta }_n=\delta >0$, and pick a subsequence $\left({\Delta }_{n_k}\right)$ with ${\Delta }_{n_k}\to \delta$. For each index $n_k$, at least one of (1)–(3) holds.

• If infinitely many of these indices satisfy (1), then along that sub-subsequence,

  $${\Delta }_{n_k+1}\le \lambda {\Delta }_{n_k}.$$

  Passing to the limit gives $lim\; sup{\Delta }_{n_k+1}\le \lambda \delta <\delta$, contradicting that $\delta$ is a limsup for the whole sequence.
• Thus, only finitely many of the $n_k$ satisfy (1). For infinitely many large k, either (2) or (3) must hold.

  –

  If (2) holds for infinitely many such indices, then

  $${\Delta }_{n_k+1}\le \beta ({\Delta }_{n_k}+{\Delta }_{n_k+1})\Rightarrow (1-\beta ){\Delta }_{n_k+1}<\beta {\Delta }_{n_k},$$

  hence,

  $${\Delta }_{n_k+1}\le {\displaystyle \frac{\beta }{1-\beta }}{\Delta }_{n_k}.$$

  Since $0<\beta <\frac{1}{2}$, we have ${\displaystyle \frac{\beta }{1-\beta }}<1$, and passing to the limit yields $lim\; sup{\Delta }_{n_k+1}<\delta$, again a contradiction.

  –

  If (3) holds for infinitely many such indices then, using $d({\eta }_{n_k},{\eta }_{n_k+2})\le {\Delta }_{n_k}+{\Delta }_{n_k+1}$,

  $${\Delta }_{n_k+1}\le \gamma ({\Delta }_{n_k}+{\Delta }_{n_k+1}),$$

  so

  $$(1-\gamma ){\Delta }_{n_k+1}\le \gamma {\Delta }_{n_k},$$

  and therefore,

  $${\Delta }_{n_k+1}\le {\displaystyle \frac{\gamma }{1-\gamma }}{\Delta }_{n_k}.$$

  Again, $0<\gamma <\frac{1}{2}$ implies ${\displaystyle \frac{\gamma }{1-\gamma }}<1$, contradicting $lim\; sup=\delta$.

All possibilities lead to contradiction, so $\delta >0$ is impossible; hence, ${\Delta }_n\to 0$. □

Now we show that $\left\{{\eta }_n\right\}$ is Cauchy. Since ${\Delta }_n\to 0$, there exists N and a constant $q\in (0,1)$, such that for all $n\ge N$,

$${\Delta }_{n+1}\le q{\Delta }_n.$$

Therefore, for $m>n\ge N$,

$$d({\eta }_n,{\eta }_m)\le \sum _{k=n}^{m-1}{\Delta }_k\le {\Delta }_n\sum _{j=0}^{\infty }{q}^j={\displaystyle \frac{{\Delta }_n}{1-q}},$$

and because ${\Delta }_n\to 0$, the right-hand side can be made arbitrarily small; thus, $\left\{{\eta }_n\right\}$ is Cauchy. By completeness of $\mathrm{\Xi }$, there exists $z\in \mathrm{\Xi }$ with ${\eta }_n\to z$. Passing to the limit in ${\eta }_{n+1}={\mathit{ħ}}^p\left({\eta }_n\right)$ and using continuity of ${\mathit{ħ}}^p$, we obtain $z={\mathit{ħ}}^p\left(z\right)$, i.e., z is a fixed point of ${\mathit{ħ}}^p$.

Uniqueness. 

Suppose $\theta \ne \vartheta$ are two distinct fixed points of ${\mathit{ħ}}^p$. Applying the p-Zamfirescu condition to $(\theta ,\vartheta )$ yields one of

$$d(\theta ,\vartheta )\le \lambda d(\theta ,\vartheta ),d(\theta ,\vartheta )\le \beta (0+0),d(\theta ,\vartheta )\le 2\gamma d(\theta ,\vartheta ),$$

all impossible since the right-hand sides are strictly less than $d(\theta ,\vartheta )$. Hence, $\theta =\vartheta$ and the fixed point of ${\mathit{ħ}}^p$ is unique.

Step 2.

The unique fixed point $z$ of ${\mathit{ħ}}^p$ also serves as a fixed point of $\mathit{ħ}$.

Since ${\eta }_n={\mathit{ħ}}^{np}\left({\zeta }_0\right)\to z$, for each $r\in \{0,1,\dots ,p-1\}$, the subsequence ${\mathit{ħ}}^r\left({\eta }_n\right)={\mathit{ħ}}^{np+r}\left({\zeta }_0\right)$ converges to ${\mathit{ħ}}^r\left(z\right)$ by continuity. But these subsequences partition the Picard sequence ${\zeta }_m={\mathit{ħ}}^m\left({\zeta }_0\right)$. As ${\eta }_{n+1}=\mathit{ħ}\left({\eta }_n\right)\to z$, we obtain $\mathit{ħ}\left(z\right)=z$. Thus z is a fixed point of ħ.

Step 3.

Uniqueness property of the fixed point of $\mathit{ħ}$.

If $\theta$ and $\vartheta$ serve as two fixed points of ħ, then they are fixed points of ${\mathit{ħ}}^p$ as well. Owing to the uniqueness of the fixed point of ${\mathit{ħ}}^p$ proved above (in Step 1 in (Uniqueness)), we get $\theta =\vartheta$. Thus, the fixed point of ħ is unique.

Step 4.

Convergence of the Picard iterates.

We have already shown that the subsequence ${\eta }_n={\mathit{ħ}}^{np}\left({\zeta }_0\right)$ converges to z. For each fixed $r\in \{0,1,\dots ,p-1\}$, the subsequence ${\zeta }_{np+r}={\mathit{ħ}}^r\left({\eta }_n\right)$ converges to ${\mathit{ħ}}^r\left(z\right)=z$ because ${\mathit{ħ}}^r$ is continuous and ${\mathit{ħ}}^r\left(z\right)=z$ (as $\mathit{ħ}\left(z\right)=z$). Hence, every subsequence modulo p converges to z; therefore, the whole Picard sequence ${\zeta }_n={\mathit{ħ}}^n\left({\zeta }_0\right)$ converges to z.

• This completes the proof of Theorem. □

Corollary 1.

Let $(\mathrm{\Xi },d)$ be a complete metric space and $\mathit{ħ}:\mathrm{\Xi }\to \mathrm{\Xi }$ a continuous mapping.

1. 

If ħ satisfies the p-Hardy–Rogers condition with $p=1$, then Theorem 1 reduces to the classical Hardy–Rogers fixed-point theorem [1].

2. 

If ħ satisfies the p-Zamfirescu condition with $p=1$, then Theorem 2 reduces to the classical Zamfirescu fixed-point theorem [2].

3. 

In particular, if in the p-Hardy–Rogers condition, we take

$${\lambda }_2={\lambda }_3={\lambda }_4={\lambda }_5=0,0\le {\lambda }_1<1,$$

then Theorem 1 reduces to the Banach contraction principle [9].

4. 

If in the p-Hardy–Rogers condition, we set

$${\lambda }_1={\lambda }_3={\lambda }_4={\lambda }_5=0,0\le {\lambda }_2<1,$$

then, we obtain Kannan’s fixed-point theorem [6].

5. 

If in the p-Hardy–Rogers condition, we set

$${\lambda }_1={\lambda }_2={\lambda }_3={\lambda }_5=0,0\le {\lambda }_4<{\displaystyle \frac{1}{2}},$$

then we obtain Chatterjea’s fixed-point theorem [10].

6. 

The p-Zamfirescu condition with $p=1$ already unifies the conditions of Banach, Kannan, and Chatterjea, as originally shown by Zamfirescu [2].

Thus, Theorems 1 and 2 provide a comprehensive framework that extends these classical results to iterated mappings ${\mathit{ħ}}^p$.

4. Illustrative Examples

In this section, we present two examples that demonstrate the applicability of our main theorems.

Example 1

(p-Hardy–Rogers contraction). Assume $\mathrm{\Xi }=[0,1]$ with the usual metric $d(\zeta ,\eta )=|\zeta -\eta |$, and define the mapping

$$\mathit{ħ}\left(\zeta \right)=\frac{1}{4}(\zeta +{\zeta }^2).$$

We compute

$${\mathit{ħ}}^2\left(\zeta \right)=\mathit{ħ}\left(\mathit{ħ}\left(\zeta \right)\right)=\frac{1}{4}\left(\frac{1}{4}(\zeta +{\zeta }^2)+{\left(\frac{1}{4}(\zeta +{\zeta }^2)\right)}^2\right)=\frac{1}{16}(\zeta +{\zeta }^2)+\frac{1}{64}{(\zeta +{\zeta }^2)}^2.$$

For all $\zeta ,\eta \in [0,1]$, one can estimate

$$|{\mathit{ħ}}^2\left(\zeta \right)-{\mathit{ħ}}^2\left(\eta \right)|\le \frac{3}{8}|\zeta -\eta |.$$

Hence, the inequality

$${\lambda }_1+{\lambda }_2+{\lambda }_3+2({\lambda }_4+{\lambda }_5)=\frac{3}{8}<1$$

is satisfied (take the constants so that the left-hand side equals $3/8$), so ħ is a p-Hardy–Rogers contraction with $p=2$. By Theorem 1, ħ has a unique fixed point in $[0,1]$. Solving $\mathit{ħ}\left(\zeta \right)=\zeta$ gives $\zeta =0$ as the unique fixed point, and the Picard iteration converges to $0$ for any starting value in $[0,1]$.

Example 2

(p-Zamfirescu contraction). Suppose $\mathrm{\Xi }=[0,1]$ with the standard metric $d(\zeta ,\eta )=|\zeta -\eta |$, and define

$$\mathit{ħ}\left(\zeta \right)=\sqrt{{\displaystyle \frac{\zeta +1}{10}}},\zeta \in [0,1].$$

It fulfills the condition

$$\mathit{ħ}\left([0,1]\right)=[\sqrt{0.1},\sqrt{0.2}]\approx [0.316,0.447]\subset [0,1].$$

Then

$${\mathit{ħ}}^2\left(\zeta \right)=\mathit{ħ}\left(\mathit{ħ}\left(\zeta \right)\right)=\sqrt{{\displaystyle \frac{\sqrt{\frac{\zeta +1}{10}}+1}{10}}}.$$

For any $\zeta ,\eta \in [0,1]$, one can obtain an estimate of the form

$$|{\mathit{ħ}}^2\left(\zeta \right)-{\mathit{ħ}}^2\left(\eta \right)|\le \frac{1}{4}|\zeta -\eta |.$$

Thus, the mapping ħ satisfies the p-Zamfirescu condition with $\lambda =\frac{1}{4}$, $\beta =\gamma =\frac{1}{4}$ and $p=2$. By Theorem 2, ħ has a unique fixed point in $[0,1]$. Solving $\mathit{ħ}\left(\zeta \right)=\zeta$ yields the fixed point (the equation can be solved numerically or symbolically); the Picard sequence ${\zeta }_{n+1}=\mathit{ħ}\left({\zeta }_n\right)$ converges to that unique fixed point for any starting ${\zeta }_0\in [0,1]$.

5. Application to a Nonlinear Boundary Value Problem

To illustrate the applicability of our main results, we consider a nonlinear second-order boundary value problem (BVP).

$$\left\{\begin{array}{c}-{\theta }^{\prime \prime }\left(\zeta \right)=\mathit{ħ}(\zeta ,\theta \left(\zeta \right)),\zeta \in [0,1],\hfill \\ \theta \left(0\right)=\theta \left(1\right)=0,\hfill \end{array}\right.$$

(4)

where $\mathit{ħ}:[0,1]\times \mathbb{R}\to \mathbb{R}$ is a continuous function satisfying certain growth conditions.

It is well-known that a function $\theta \in C^2[0,1]$ is a solution of (4) if and only if it satisfies the Hammerstein integral equation

$$\theta \left(\zeta \right)={\int }_0^{1}G(\zeta ,\eta )\mathit{ħ}(\eta ,\theta \left(\eta \right))d\eta =:\left(\mathcal{T}\theta \right)\left(\zeta \right),$$

(5)

where G is the Green’s function associated with the differential operator $-{\displaystyle \frac{d^2}{d{t}^2}}$ and the given boundary conditions

$$G(t,s)=\left\{\begin{array}{cc}\zeta (1-\eta ),\hfill & 0\le \zeta \le \eta \le 1,\hfill \\ \eta (1-\zeta ),\hfill & 0\le \eta <\zeta \le 1.\hfill \end{array}\right.$$

Define the complete metric space $(\mathrm{\Xi },d)$ by $\mathrm{\Xi }=C[0,1]$ equipped with the supremum norm

$${\parallel \theta \parallel }_{\infty }=\underset{\zeta \in [0,1]}{max}\left|\theta \left(\zeta \right)\right|,d(\theta ,\vartheta )={\parallel \theta -\vartheta \parallel }_{\infty }.$$

Assume that there exist constants $\alpha ,L\ge 0$ with $\alpha +2L<{\displaystyle \frac{1}{\parallel G\parallel }}$, where

$$\parallel G\parallel =\underset{\zeta \in [0,1]}{max}{\int }_0^1\left|G(\zeta ,\eta )\right|ds={\displaystyle \frac{1}{8}},$$

where $\mathcal{T}\theta$ is evaluated pointwise via (5).

Then, for a sufficiently large integer $p\ge 1$, the integral operator $\mathcal{T}$ defined in (5) satisfies the p-Zamfirescu condition on $C[0,1]$. Consequently, by Theorem 2, the BVP (4) admits a unique solution ${\theta }^{*}\in C[0,1]$, and the Picard iteration ${\theta }_{n+1}=\mathcal{T}{\theta }_n$ converges uniformly to ${\theta }^{*}$ for any initial guess ${\theta }_0\in C[0,1]$.

6. Conclusions

In this paper, we have introduced two new generalized classes of contractive mappings, namely the p-Hardy–Rogers and p-Zamfirescu contractions, by extending the classical Hardy–Rogers and Zamfirescu frameworks through the introduction of the integer parameter $p\ge 1$. This approach allowed us to investigate the fixed-point properties of iterated mappings ${\mathit{ħ}}^p$ and to transfer the obtained results to the original mapping ħ. We proved that under suitable contractive conditions, each mapping ħ admits a unique fixed point in a complete metric space and that the Picard iteration converges to this point.

The results presented here unify and extend several classical theorems in fixed-point theory, including those of Banach, Hardy–Rogers, and Zamfirescu, as particular cases corresponding to $p=1$. The p-iterative framework also provides a natural setting for analyzing the stability and convergence behavior of nonlinear iterative processes, which are of growing interest in various fields such as nonlinear functional analysis, integral and differential equations, and optimization.

Future research may explore several directions. One possible extension involves studying the p-Hardy–Rogers and p-Zamfirescu contractions in more general spaces including b-metric spaces, modular, or partial metric spaces. Another direction could focus on relaxing the continuity assumption or incorporating control functions and altering distance conditions. Such developments would further enhance the applicability of the proposed framework to broader classes of nonlinear problems.