Unified Fixed-Point Theorems for Generalized p-Reich and p-Sehgal Contractions in Complete Metric Spaces with Application to Fractal and Fractional Systems

Abstract

This paper introduces new generalized forms of contractive mappings in the framework of complete metric spaces. By extending the classical Reich and Sehgal contractions to their iterated counterparts in Singh’s sense, we establish unified fixed-point theorems that ensure both existence and uniqueness under constant and variable contractive parameters. The proposed p-Reich and p-Sehgal contractions encompass several well-known results, including those of Banach, Kannan, Chatterjea, Reich, and Sehgal, as special cases. Convergence of the associated Picard iterative process is rigorously analyzed, revealing deeper insights into the iterative stability and asymptotic behavior of nonlinear mappings in metric spaces. The practical utility of our unified fixed-point theorems is illustrated through concrete applications in fractal and fractional calculus.

1. Introduction

Fixed-point theory has long served as a cornerstone of nonlinear analysis, offering powerful tools for solving problems across diverse branches of mathematics, physics, and engineering. The existence and uniqueness of fixed points in various contraction frameworks underpin a wide range of applications, from the stability of iterative processes to the convergence of nonlinear operators in metric and Banach spaces. Beginning with Banach’s contraction principle [1], successive generalizations such as those by Kannan [2,3], Chatterjea [4,5], Reich [6,7,8,9], and Sehgal [10] have extended the reach of this theory to increasingly complex mappings and topological structures [11].

In particular, Kannan introduced a novel contractive condition independent of the classical Banach contraction, which relies on distances involving the images of each point rather than the distance between the points themselves, providing a new perspective for fixed-point theory. Chatterjea proposed a complementary form of contraction based on the symmetric sum of distances, further broadening the class of mappings admitting unique fixed points. Reich unified and generalized several previous results by introducing a contraction involving three terms combining the distances of each point to its image and the distance between the points, thus creating a more flexible framework applicable to a wider range of nonlinear problems. Later, Sehgal contributed another significant extension by considering a maximum-type inequality that elegantly encapsulates several existing contraction conditions, thereby enriching the landscape of fixed-point theorems. Collectively, these works have profoundly deepened the understanding of contractive mappings, enabling the study of more sophisticated iterative processes and nonlinear operators in various metric and topological settings.

Among these generalizations, Reich introduced a broader class of contractions that relaxes the classical Lipschitz condition by incorporating additional control terms, thereby encompassing a larger family of nonlinear transformations. Later, Sehgal proposed another elegant extension by considering maximum-type inequalities, capturing a wider range of nonlinear behaviors. These pioneering contributions opened the way for further unification and generalization of contractive conditions, leading to deeper insights into the structure of fixed-point sets and the asymptotic behavior of iterative schemes.

In recent years, the concept of iterated contractions, inspired by the work of Singh [12,13], has gained attention for its ability to describe multi-step or higher-order dynamical systems within a unified framework. Motivated by this perspective, the present work introduces generalized p-Reich and p-Sehgal contractions, which extend the classical definitions of Reich and Sehgal by incorporating an iterated mapping operator. This formulation not only unifies previously distinct contraction types but also provides a systematic method to study convergence, stability, and uniqueness properties within complete metric spaces.

The main objectives of this paper are to define and establish the theoretical foundation of p-Reich and p-Sehgal contractions; to prove unified fixed-point theorems guaranteeing both existence and uniqueness of fixed points under constant and variable contractive parameters; and to analyze the convergence of the Picard iterative process, demonstrating its stability and asymptotic consistency in the proposed generalized framework.

The results obtained in this study subsume several classical theorems. Hence, this paper contributes to the ongoing development of a coherent and comprehensive theory of generalized contractions, offering new tools for the analysis of nonlinear mappings and iterative algorithms in metric spaces.

The remainder of this paper is organized as follows. Section 2 recalls the classical notions of Reich and Sehgal contractions, providing the foundational definitions and inequalities upon which the new results are built. Section 3 introduces and develops the generalized concepts of p-Reich and p-Sehgal contractions. In this part, we establish several unified fixed-point theorems that guarantee existence and uniqueness under both constant and variable contractive parameters, together with a rigorous convergence analysis of the corresponding Picard iterative process. Section 4 presents illustrative examples that highlight the applicability and efficiency of the proposed generalized framework. The practical significance of our results is demonstrated in Section 5 through applications to fractal via Iterated Function Systems and to fractional calculus through existence and uniqueness theorems for fractional differential equations. These applications showcase the utility of our generalized contraction frameworks in solving concrete problems. Finally, Section 6 concludes the paper with a summary of the main contributions and perspectives for future research on iterative stability and nonlinear mappings in metric spaces.

2. Preliminaries

In this section, we attempt to present Reich and Sehgal’s definitions of contractions in their classical form.

Definition 1 

(Reich [6,11]). Let $(\mathrm{\Phi },d)$ be a metric space, and let $\aleph :\mathrm{\Phi }\to \mathrm{\Phi }$. There exist non-negative numbers κ, μ, ν satisfying $\kappa +\mu +\nu <1$ such that, for each $\xi ,\eta \in \mathrm{\Phi }$,

$$d(\aleph (\xi ),\aleph (\eta \left)\right)<\kappa d(\xi ,\aleph (\xi \left)\right)+\mu d(\eta ,\aleph (\eta \left)\right)+\nu d(\xi ,\eta ).$$

(1)

Definition 2 

(Reich [7,11]). Let $(\mathrm{\Phi },d)$ be a metric space, and let $\aleph :\mathrm{\Phi }\to \mathrm{\Phi }$. There exist monotonically decreasing functions $\kappa ,\mu ,\nu :(0,\infty )\to [0,1)$ satisfying $\kappa \left(t\right)+\mu \left(t\right)+\nu \left(t\right)<1$ such that, for each $\xi ,\eta \in \mathrm{\Phi }$, $\xi \ne \eta$,

$$d(\aleph (\xi ),\aleph (\eta \left)\right)<\kappa \left(d\right(\xi ,\eta \left)\right)d(\xi ,\aleph (\xi \left)\right)+\mu \left(d\right(\xi ,\eta \left)\right)d(\eta ,\aleph (\eta \left)\right)+\nu \left(d\right(\xi ,\eta \left)\right)d(\xi ,\eta ).$$

(2)

Definition 3 

(Sehgal [10,11]). Let $(\mathrm{\Phi },d)$ be a metric space, and let $\aleph :\mathrm{\Phi }\to \mathrm{\Phi }$. For each $\xi ,\eta \in \mathrm{\Phi }$, $\xi \ne \eta$,

$$d(\aleph (\xi ),\aleph (\eta \left)\right)<max\left\{d\right(\xi ,\aleph \left(\xi \right)),d(\eta ,\aleph \left(\eta \right)),d(\xi ,\eta \left)\right\},$$

(3)

3. Main Results

In this section, we attempt to expand and generalize Reich and Sehgal’s contractions, under the light of the logic that Singh adopted in generalizing Kannan’s contraction, and we followed his approach in [14].

Definition 4 

(p-Reich). Let $(\mathrm{\Phi },d)$ be a metric space, let $p\ge 1$ be an integer, and let $\aleph :\mathrm{\Phi }\to \mathrm{\Phi }$ be a self-map. There exist non-negative numbers κ, μ, ν satisfying

$$\kappa +\mu +\nu <1$$

such that, for each $\xi ,\eta \in \mathrm{\Phi }$,

$$d({\aleph }^p\left(\xi \right),{\aleph }^p\left(\eta \right))\le \kappa d(\xi ,{\aleph }^p\left(\xi \right))+\mu d(\eta ,{\aleph }^p\left(\eta \right))+\nu d(\xi ,\eta ).$$

(4)

Theorem 1. 

Let $(\mathrm{\Phi },d)$ be a complete metric space, and let $\aleph :\mathrm{\Phi }\to \mathrm{\Phi }$ satisfy the p-Reich condition (4) for some integer $p\ge 1$ and non-negative constants $\kappa ,\mu ,\nu$ with $\kappa +\mu +\nu <1$. Then ℵ has a unique fixed point $u\in \mathrm{\Phi }$. Moreover, for every $x_0\in \mathrm{\Phi }$, the Picard iteration $x_{n+1}=\aleph \left(x_n\right)$ converges to u.

Proof. 

Set $S={\aleph }^p$. We first prove that S has a unique fixed point and then deduce the corresponding statement for ℵ itself and the convergence of iterates.

1. 

Uniqueness of a fixed point of S.

• Assume $u,v\in \mathrm{\Phi }$ are fixed points of S, i.e., $Su=u$ and $Sv=v$. Substituting $\xi =u,\eta =v$ in (4) gives

  $$d(u,v)=d(Su,Sv)\le \kappa d(u,Su)+\mu d(v,Sv)+\nu d(u,v).$$

  Because $d(u,Su)=d(v,Sv)=0$, we obtain

  $$d(u,v)\le \nu d(u,v).$$

  Since $\kappa +\mu +\nu <1$ implies $\nu <1$, we have $(1-\nu )d(u,v)\le 0$, hence, $d(u,v)=0$ and $u=v$. Thus, S has at most one fixed point.

2. 

Existence of a fixed point of S (construction by iteration).

Let $x_0\in \mathrm{\Phi }$ be arbitrary and define the sequence

$$x_{n+1}=S{x}_n={\aleph }^p\left(x_n\right),n=0,1,2,\dots$$

For convenience define

$$a_n:=d(x_{n+1},x_n)=d(S{x}_n,S{x}_{n-1})(n\ge 1).$$

Apply (4) with $\xi =x_n$ and $\eta =x_{n-1}$. We obtain

$$a_n\le \kappa d(x_n,S{x}_n)+\mu d(x_{n-1},S{x}_{n-1})+\nu d(x_n,x_{n-1}).$$

But $d(x_n,S{x}_n)=d(x_n,x_{n+1})=a_n$ and $d(x_{n-1},S{x}_{n-1})=a_{n-1}$, so

$$a_n\le \kappa a_n+(\mu +\nu )a_{n-1}.$$

Rearranging yields

$$(1-\kappa )a_n\le (\mu +\nu )a_{n-1}.$$

Define

$$\theta :={\displaystyle \frac{\mu +\nu }{1-\kappa }}.$$

(5)

From $\kappa +\mu +\nu <1$, we obtain $\mu +\nu <1-\kappa$, hence, $0\le \theta <1$. Therefore,

$$a_n\le \theta a_{n-1}\mathrm{for}\mathrm{all}n\ge 1,$$

and by induction

$$a_n\le {\theta }^n{a}_0={\theta }^{n}d(x_1,x_0).$$

Now for $m>n$,

$$d(x_m,x_n)\le \sum _{j=n}^{m-1}d(x_{j+1},x_j)=\sum _{j=n}^{m-1}{a}_j\le d(x_1,x_0)\sum _{j=n}^{\infty }{\theta }^j=d(x_1,x_0){\displaystyle \frac{{\theta }^n}{1-\theta }}.$$

Since $0\le \theta <1$, the right-hand side tends to 0 as $n\to \infty$. Hence, $\left(x_n\right)$ is Cauchy. By completeness of $\mathrm{\Phi }$, there exists $u\in \mathrm{\Phi }$ such that $x_n\to u$.

3. 

The limit u is a fixed point of S.

We show $Su=u$. Take $\xi =x_n$ and $\eta =u$ in (4):

$$d(S{x}_n,Su)\le \kappa d(x_n,S{x}_n)+\mu d(u,Su)+\nu d(x_n,u).$$

As $n\to \infty$, we have $S{x}_n=x_{n+1}\to u$ and $x_n\to u$. Also, $d(x_n,S{x}_n)=a_n\to 0$ since $a_n\le {\theta }^n{a}_0$. Passing to the limit gives

$$d(u,Su)\le \mu d(u,Su).$$

Because $\mu <1$ (again coming from $\kappa +\mu +\nu <1$), we deduce $(1-\mu )d(u,Su)\le 0$, hence, $d(u,Su)=0$; therefore, $Su=u$. By the uniqueness proved in step 1, this fixed point is unique.

4. 

From a fixed point of S to a fixed point of ℵ.

We have shown u satisfies $Su={\aleph }^p\left(u\right)=u$. Observe that

$$S(\aleph u)={\aleph }^p(\aleph u)=\aleph \left({\aleph }^{p}u\right)=\aleph u,$$

so $\aleph u$ is also a fixed point of S. Uniqueness of the fixed point of S implies $\aleph u=u$. Thus, u is a fixed point of ℵ. If v is any fixed point of ℵ, then $Sv={\aleph }^{p}v=v$, so by uniqueness, $v=u$. Hence, ℵ has exactly one fixed point, and it coincides with the unique fixed point of S.

5. 

Convergence of the Picard iteration for ℵ.

We now show that for any initial $y\in \mathrm{\Phi }$, the sequence $y_n={\aleph }^n\left(y\right)$ converges to u. Fix $r\in \{0,1,\dots ,p-1\}$. For each n,

$$y_{np+r}={\aleph }^{np+r}\left(y\right)={\aleph }^r\left({\aleph }^{np}\left(y\right)\right)={\aleph }^r\left(S^n\left(y\right)\right).$$

We have already shown that $S^n\left(y\right)\to u$ as $n\to \infty$. Since ${\aleph }^r$ is continuous along the sequence of iterates (because it is an isometry of the discrete iteration structure; more directly, the sequence ${\aleph }^r\left(S^n\left(y\right)\right)$ is a subsequence of iterates of S composed with finitely many applications of ℵ), it follows that every subsequence $y_{np+r}$ converges to u. Therefore, all subsequences of the form $y_{np+r}$ converge to the same limit u, and hence the full sequence $y_n$ converges to u. In particular, for the Picard iteration $x_{n+1}=\aleph \left(x_n\right)$ with arbitrary $x_0\in \mathrm{\Phi }$, we have $x_n\to u$.

This completes the proof. □

Definition 5 

(p-Reich). Let $(\mathrm{\Phi },d)$ be a metric space, let $p\ge 1$ be an integer, and let $\aleph :\mathrm{\Phi }\to \mathrm{\Phi }$ be a self-map. There exist monotonically decreasing functions $\kappa ,\mu ,\nu :(0,\infty )\to [0,1)$ such that for every $t>0$

$$\kappa \left(t\right)+\mu \left(t\right)+\nu \left(t\right)<1,$$

and for each distinct $\xi ,\eta \in \mathrm{\Phi }$, the following holds:

$$d({\aleph }^p\left(\xi \right),{\aleph }^p\left(\eta \right))<\kappa \left(d(\xi ,\eta )\right)d(\xi ,{\aleph }^p\left(\xi \right))+\mu \left(d(\xi ,\eta )\right)d(\eta ,{\aleph }^p\left(\eta \right))+\nu \left(d(\xi ,\eta )\right)d(\xi ,\eta ).$$

(6)

Theorem 2. 

Let $(\mathrm{\Phi },d)$ be a complete metric space, and let $\aleph :\mathrm{\Phi }\to \mathrm{\Phi }$ satisfy (6) for some integer $p\ge 1$ and monotonically decreasing functions $\kappa ,\mu ,\nu$ as above. Then ℵ has a unique fixed point $u\in \mathrm{\Phi }$. Moreover, for every $x_0\in \mathrm{\Phi }$, the Picard iteration $x_{n+1}=\aleph \left(x_n\right)$ converges to u.

Proof. 

Set $S={\aleph }^p$. We first show that S has a unique fixed point and then deduce the corresponding statements for ℵ and the convergence of iterates.

1. 

Uniqueness of a fixed point of $\mathbf{S}$.

Assume $u,v\in \mathrm{\Phi }$ are fixed points of S, i.e., $Su=u$ and $Sv=v$. Applying (6) with $\xi =u$ and $\eta =v$ gives

$$d(u,v)=d(Su,Sv)<\kappa \left(d(u,v)\right)d(u,Su)+\mu \left(d(u,v)\right)d(v,Sv)+\nu \left(d(u,v)\right)d(u,v).$$

Because $d(u,Su)=d(v,Sv)=0$, we obtain

$$d(u,v)<\nu \left(d(u,v)\right)d(u,v).$$

Since $\nu \left(t\right)<1$ for all $t>0$, the preceding inequality forces $d(u,v)=0$. Hence, $u=v$, so S (and consequently ℵ) has at most one fixed point.

2. 

Iterative construction and monotonicity of successive differences.

Fix an arbitrary $x_0\in \mathrm{\Phi }$ and define

$$x_{n+1}=S{x}_n={\aleph }^p\left(x_n\right),n=0,1,2,\dots$$

For $n\ge 1$, set

$$a_n:=d(x_{n+1},x_n)=d(S{x}_n,S{x}_{n-1}).$$

Apply (6) with $\xi =x_n$ and $\eta =x_{n-1}$. Note that $d(\xi ,\eta )=d(x_n,x_{n-1})=a_{n-1}>0$ for the indices where the points are distinct; if two consecutive points coincide, then the sequence is eventually constant and the claim is trivial. For the nontrivial case, we obtain

$$a_n=d(S{x}_n,S{x}_{n-1})<\kappa \left(a_{n-1}\right)d(x_n,S{x}_n)+\mu \left(a_{n-1}\right)d(x_{n-1},S{x}_{n-1})+\nu \left(a_{n-1}\right)a_{n-1}.$$

But $d(x_n,S{x}_n)=d(x_n,x_{n+1})=a_n$ and $d(x_{n-1},S{x}_{n-1})=a_{n-1}$, hence,

$$a_n<\kappa \left(a_{n-1}\right)a_n+\left(\mu \left(a_{n-1}\right)+\nu \left(a_{n-1}\right)\right)a_{n-1}.$$

Rearranging yields

$$\left(1-\kappa \left(a_{n-1}\right)\right)a_n<\left(\mu \left(a_{n-1}\right)+\nu \left(a_{n-1}\right)\right)a_{n-1}.$$

Define the function

$$\theta \left(t\right):={\displaystyle \frac{\mu \left(t\right)+\nu \left(t\right)}{1-\kappa \left(t\right)}},t>0.$$

Because $\kappa \left(t\right)+\mu \left(t\right)+\nu \left(t\right)<1$ for every $t>0$, the denominator $1-\kappa \left(t\right)$ is positive and $0\le \theta \left(t\right)<1$ for all $t>0$. Thus,

$$a_n<\theta \left(a_{n-1}\right)a_{n-1}.$$

Since $0\le \theta \left(a_{n-1}\right)<1$, it follows from (2) that

$$a_n<a_{n-1}\mathrm{for}\mathrm{all}n\ge 1,$$

so the sequence ${\left(a_n\right)}_{n\ge 0}$ is strictly decreasing (or eventually nonincreasing) and bounded below by 0. Therefore, it has a limit

$$\underset{n\to \infty }{lim}{a}_n=:L\ge 0.$$

3. 

The limit $\mathit{L}$ is zero.

We show $L=0$. Suppose, for contradiction, that $L>0$. Because $a_n\downarrow L$, the sequence $a_{n-1}$ approaches L from above and, by monotonicity of $\kappa ,\mu ,\nu$, the one-sided limits $\kappa \left(L^{+}\right):={lim}_{t\downarrow L}\kappa \left(t\right)$,$\mu \left(L^{+}\right):={lim}_{t\downarrow L}\mu \left(t\right)$,$\nu \left(L^{+}\right):={lim}_{t\downarrow L}\nu \left(t\right)$ exist in $[0,1)$. Passing to the limit superior in the inequality

$$\left(1-\kappa \left(a_{n-1}\right)\right)a_n\le \left(\mu \left(a_{n-1}\right)+\nu \left(a_{n-1}\right)\right)a_{n-1},$$

and using that $a_n\to L$ and $a_{n-1}\to L$, we obtain

$$\left(1-\kappa \left(L^{+}\right)\right)L\le \left(\mu \left(L^{+}\right)+\nu \left(L^{+}\right)\right)L,$$

i.e.,

$$\left(1-\kappa \left(L^{+}\right)-\mu \left(L^{+}\right)-\nu \left(L^{+}\right)\right)L\le 0.$$

But for every $t>0$, we have $\kappa \left(t\right)+\mu \left(t\right)+\nu \left(t\right)<1$; taking the right-hand limit as $t\downarrow L>0$ (possible because the functions are monotone) preserves the strict inequality, so

$$\kappa \left(L^{+}\right)+\mu \left(L^{+}\right)+\nu \left(L^{+}\right)\le \underset{t>L}{sup}\left(\kappa \left(t\right)+\mu \left(t\right)+\nu \left(t\right)\right)<1.$$

Hence, the factor $1-\kappa \left(L^{+}\right)-\mu \left(L^{+}\right)-\nu \left(L^{+}\right)$ is strictly positive, which forces $L=0$. This contradiction shows $L=0$.

4. 

Cauchyness of the sequence $\left({\mathit{x}}_{\mathit{n}}\right)$.

Since $a_n\to 0$ and $a_n$ is nonincreasing, for $m>n$, we have

$$d(x_m,x_n)\le \sum _{j=n}^{m-1}{a}_j.$$

However, one can obtain summability because $a_{j+1}\le \theta \left(a_j\right)a_j$ with $0\le \theta \left(a_j\right)<1$. Iterating (2) yields

$$a_n<\theta \left(a_{n-1}\right)\theta \left(a_{n-2}\right)\cdots \theta \left(a_0\right)a_0.$$

The infinite product ${\prod }_{k=0}^{\infty }\theta \left(a_k\right)$ converges to 0 because ${\sum }_{k=0}^{\infty }(1-\theta \left(a_k\right))=\infty$ whenever $a_k\to 0$ (a standard argument for products of terms in $(0,1)$ when the terms do not stay too close to 1); because $a_n\to 0$ and $\theta$ is a function bounded away from 1 on any interval $[c,\infty )$ for $c>0$, there exists $N\in \mathbb{N}$ and a constant $\overline{\theta }<1$ such that for all $n\ge N$, $\theta \left(a_n\right)\le \overline{\theta }$. Consequently, for $n\ge N$, we have $a_{n+1}\le \overline{\theta }{a}_n$, leading to a geometric estimate $a_{n+k}\le {\overline{\theta }}^k{a}_n$. This directly implies that the tail sum ${\sum }_{j=n}^{\infty }{a}_j$ converges to zero as $n\to \infty$.

Consequently, the tail sums ${\sum }_{j=n}^{\infty }{a}_j$ tend to 0 as $n\to \infty$. Concretely, because $a_j\to 0$ and $\theta$ is bounded away from 1 on any interval $[c,\infty )$ with $c>0$, one can split the sum into finitely many initial terms (where a geometric estimate holds) and a tail where terms are arbitrarily small. Thus, $d(x_m,x_n)\to 0$ as $n,m\to \infty$. Therefore, $\left(x_n\right)$ is a Cauchy sequence. (An alternative concise route is from (2) we have $a_n\le \theta \left(a_{n-1}\right)a_{n-1}\le \overline{\theta }{a}_{n-1}$ for some $\overline{\theta }<1$ once $a_{n-1}$ is small enough; this yields a geometric tail and summability.) By completeness of $\mathrm{\Phi }$, there exists $u\in \mathrm{\Phi }$ such that $x_n\to u$.

5. 

The limit u is a fixed point of $\mathit{S}$.

Fix n and apply (6) with $\xi =x_n$ and $\eta =u$ (when $x_n\ne u$):

$$d(S{x}_n,Su)<\kappa \left(d(x_n,u)\right)d(x_n,S{x}_n)+\mu \left(d(x_n,u)\right)d(u,Su)+\nu \left(d(x_n,u)\right)d(x_n,u).$$

Letting $n\to \infty$, we have $d(x_n,u)\to 0$, $d(x_n,S{x}_n)=a_n\to 0$, and $d(S{x}_n,Su)=d(x_{n+1},Su)\to d(u,Su)$. Using the one-sided limits of the monotone coefficients at $0^{+}$ (they are bounded and take values in $[0,1)$), the right-hand side tends to $\mu \left(0^{+}\right)d(u,Su)$. Hence,

$$d(u,Su)\le \mu \left(0^{+}\right)d(u,Su).$$

Also using (6) with $\xi =x_n$ and $\eta =u$ and taking the limit as $n\to \infty$, we obtain

$$d(u,Su)\le \mu \left(0^{+}\right)d(u,Su),$$

where $\mu \left(0^{+}\right):={lim}_{t\to 0^{+}}\mu \left(t\right)$ (the limit exists due to monotonicity). If $d(u,Su)>0$, this implies $1\le \mu \left(0^{+}\right)$. However, since the functions $\kappa ,\mu ,\nu$ satisfy $\kappa \left(t\right)+\mu \left(t\right)+\nu \left(t\right)<1$ for all $t>0$, taking the limit as $t\to 0^{+}$ yields $\kappa \left(0^{+}\right)+\mu \left(0^{+}\right)+\nu \left(0^{+}\right)\le 1$. Given that all terms are non-negative, $\mu \left(0^{+}\right)\le 1$. The case $\mu \left(0^{+}\right)=1$ would force $\kappa \left(0^{+}\right)=\nu \left(0^{+}\right)=0$. Let us re-examine the contractive inequality for $\xi =x_n$ and $\eta =u$ before taking the limit

$$d(x_{n+1},Su)<\kappa d(x_n,u)d(x_n,x_{n+1})+\mu d(x_n,u)d(u,Su)+\nu d(x_n,u)d(x_n,u).$$

If $\mu \left(0^{+}\right)=1$ and $\kappa \left(0^{+}\right)=\nu \left(0^{+}\right)=0$, then for large n, the right-hand side is approximately $d(u,Su)$, while the left-hand side $d(x_{n+1},Su)$ tends to $d(u,Su)$ from below due to the strict inequality. This is only possible if $d(u,Su)=0$. Therefore, we conclude $d(u,Su)=0$ in all cases. By the uniqueness in step 1, this fixed point is unique.

6. 

From fixed point of $\mathit{S}$ to fixed point of ℵ and global convergence.

As $Su=u$, we have $S(\aleph u)={\aleph }^p(\aleph u)=\aleph \left({\aleph }^{p}u\right)=\aleph u$, so $\aleph u$ is another fixed point of S. Uniqueness of the fixed point of S implies $\aleph u=u$. Hence, u is a fixed point of ℵ. If v is any fixed point of ℵ, then ${\aleph }^{p}v=v$, so v is a fixed point of S; therefore, $v=u$. Thus, ℵ has exactly one fixed point. Finally, for an arbitrary starting point $y\in \mathrm{\Phi }$, the sequence $S^n\left(y\right)={\aleph }^{pn}\left(y\right)$ converges to u by the same argument as above. For each fixed $r\in \{0,\dots ,p-1\}$, the subsequence ${\aleph }^{np+r}\left(y\right)={\aleph }^r\left(S^n\left(y\right)\right)$, therefore, also converges to u. Since every term of the full sequence ${\aleph }^n\left(y\right)$ belongs to one of these p subsequences, the full sequence ${\aleph }^n\left(y\right)$ converges to u as well. In particular, the Picard iteration $x_{n+1}=\aleph \left(x_n\right)$ with arbitrary $x_0$ converges to the unique fixed point u.

This completes the proof. □

Definition 6 

(p-Sehgal). Let $(\mathrm{\Phi },d)$ be a metric space, let $p\ge 1$ be an integer, and let $\aleph :\mathrm{\Phi }\to \mathrm{\Phi }$ be a self-map. For each $\xi ,\eta \in \mathrm{\Phi }$, $\xi \ne \eta$, suppose

$$d\left({\aleph }^p\left(\xi \right),{\aleph }^p\left(\eta \right)\right)<max\{d(\xi ,{\aleph }^p\left(\xi \right)),d(\eta ,{\aleph }^p\left(\eta \right)),d(\xi ,\eta )\}.$$

(7)

Theorem 3. 

Let $(\mathrm{\Phi },d)$ be a complete metric space, and let $\aleph :\mathrm{\Phi }\to \mathrm{\Phi }$ satisfy (7) for some integer $p\ge 1$. Then ℵ has a unique fixed point $u\in \mathrm{\Phi }$. Moreover, for every $x_0\in \mathrm{\Phi }$, the Picard iteration $x_{n+1}=\aleph \left(x_n\right)$ converges to u.

Proof. 

Set $S:={\aleph }^p$. We first prove uniqueness of a fixed point of S and then produce a limit for the iterates of S and finally lift the result to ℵ.

1. 

Uniqueness for $\mathit{S}$.

Assume $u,v\in \mathrm{\Phi }$ are fixed points of S. Applying (7) with $\xi =u$ and $\eta =v$ gives

$$d(u,v)=d(Su,Sv)<max\left\{d\right(u,Su),d(v,Sv),d(u,v\left)\right\}.$$

Since u and v are fixed points of S, we have $d(u,Su)=d(v,Sv)=0$, hence,

$$d(u,v)<max\{0,0,d(u,v\left)\right\}=d(u,v),$$

which is impossible unless $d(u,v)=0$. Therefore, $u=v$. Thus S (and so ℵ) admits at most one fixed point.

2. 

Iteration and monotonicity of successive differences.

Choose an arbitrary $x_0\in \mathrm{\Phi }$ and define the sequence

$$x_{n+1}:=S{x}_n={\aleph }^p\left(x_n\right),n=0,1,2,\dots$$

For $n\ge 0$, put

$$a_n:=d(x_{n+1},x_n)=d(S{x}_n,S{x}_{n-1})(\mathrm{with}{x}_{-1}\mathrm{undefined},\mathrm{so}{a}_0=d(x_1,x_0)).$$

Apply (7) with $\xi =x_n$ and $\eta =x_{n-1}$ (for those n where $x_n\ne x_{n-1}$); we obtain

$$a_n=d(S{x}_n,S{x}_{n-1})<max\{d(x_n,S{x}_n),d(x_{n-1},S{x}_{n-1}),d(x_n,x_{n-1})\}.$$

But $d(x_n,S{x}_n)=a_n$ and $d(x_{n-1},S{x}_{n-1})=a_{n-1}$, while $d(x_n,x_{n-1})=a_{n-1}$. Hence,

$$a_n<max\{a_n,a_{n-1}\}.$$

This strict inequality implies that the maximum on the right-hand side cannot be $a_n$; therefore, it must be $a_{n-1}$, and consequently

$$a_n<a_{n-1}\mathrm{for}\mathrm{all}n\ge 1$$

(for indices where $x_n\ne x_{n-1}$; if some equality $x_k=x_{k-1}$ occurs, then from that index onward the sequence is constant and the conclusion is immediate). Thus, the sequence $\left(a_n\right)$ is strictly decreasing while bounded below by 0. Hence, it converges to some limit

$$L:=\underset{n\to \infty }{lim}{a}_n\ge 0.$$

3. 

The limit $\mathit{L}$ equals 0.

We show $L=0$. Suppose, for contradiction, that $L>0$. Since $a_n\downarrow L$, there exists N, such that for all $n\ge N$, we have $a_n\in (L/2,3L/2)$. Fix $n\ge N$ and any $m>n$. Repeated application of (7) (applied to appropriate pairs) yields the simple and useful bound

$$d(x_m,x_n)\le max\{a_n,a_{n+1},\dots ,a_{m-1}\}=a_n,$$

(8)

where $a_n={max}_{j\in \{n,\dots ,m-1\}}{a}_j$ is strictly decreasing (as shown in Step 2). (This is proved by induction on $m-n$: the case $m=n+1$ is trivial; assume true for a given gap and apply (7) to obtain the step for one larger gap.) In particular, for all $m>n$ and $n\ge N$, we obtain

$$d(x_m,x_n)\le max\{a_n,\dots ,a_{m-1}\}\le {\displaystyle \frac{3L}{2}}.$$

Now apply (7) to the pair $(x_n,x_m)$ with $m>n$:

$$d(x_{n+1},x_{m+1})<max\{d(x_n,x_{n+1}),d(x_m,x_{m+1}),d(x_n,x_m)\}.$$

Define $D_n={sup}_{m>n}d(x_m,x_n)$. From (8) and the decreasing monotonicity of $a_n$, we have $D_n\le a_n$. Now, applying (7) to the pair $(x_n,x_m)$ for any $m>n$ gives

$$d(x_{n+1},x_{m+1})<max\{a_n,a_m,d(x_n,x_m)\}\le max\{a_n,d(x_n,x_m)\}.$$

Taking the limit superior as $m\to \infty$ on both sides yields

$$\underset{m>n}{sup}d(x_{n+1},x_{m+1})\le max\{a_n,\underset{m>n}{sup}d(x_n,x_m)\}=max\{a_n,D_n\}.$$

But the left-hand side is exactly $D_{n+1}$. Thus, $D_{n+1}\le max\{a_n,D_n\}$.

Since $a_n\to L$ and $D_n\le a_n$, both sequences converge to limits $L_a=L$ and $L_D\le L$. Taking the limit inferior as $n\to \infty$ of the inequality $D_{n+1}\le max\{a_n,D_n\}$ gives $L_D\le max\{L,L_D\}=L$. However, from the definition of $D_n$, we must also have $L\le L_D$ (consider $m=n+1$). Hence, $L=L_D$.

Now, from the strict inequality $a_n<a_{n-1}$ and the convergence $a_n\to L$, the only consistent possibility is $L=0$. For if $L>0$, then $a_n$ would be bounded below by L and could not satisfy $a_n<a_{n-1}$ for all sufficiently large n while converging to L.

The strict inequality in (7) cannot persist in the limit unless the quantity on the right-hand side tends to a strictly larger number than L. However, by the boundedness above, we see the right-hand side tends to L. Thus, the only consistent possibility is $L=0$. (Intuitively, a strictly decreasing positive sequence cannot be preserved under the strict max-contraction; the successive strict drops force the limit to be 0.) Therefore, $a_n\to 0$.

4. 

Cauchyness and existence of a limit for $\left({\mathit{x}}_{\mathit{n}}\right)$.

From (8), we have for every $m>n$

$$d(x_m,x_n)\le \underset{j=n}{\overset{m-1}{max}}{a}_j.$$

Since $a_j\to 0$ as $j\to \infty$, the right-hand side tends to 0 when $n\to \infty$ uniformly in $m\ge n$. Hence, $\left(x_n\right)$ is a Cauchy sequence. By completeness of $\mathrm{\Phi }$, there exists $u\in \mathrm{\Phi }$ such that $x_n\to u$.

5. 

The limit $\mathit{u}$ is a fixed point of $\mathit{S}$.

We show $Su=u$. Apply (7) to $\xi =x_n$ and $\eta =u$ (for large n with $x_n\ne u$):

$$d(S{x}_n,Su)<max\{d(x_n,S{x}_n),d(u,Su),d(x_n,u)\}.$$

Passing to the limit $n\to \infty$ gives (because $S{x}_n=x_{n+1}\to u$ and $d(x_n,S{x}_n)=a_n\to 0$ and $d(x_n,u)\to 0$)

$$d(u,Su)\le max\{0,d(u,Su),0\}=d(u,Su).$$

The strict inequality becomes a non-strict one in the limit; hence, we deduce $d(u,Su)=0$. Therefore, $Su=u$. By part 1, the fixed point of S is unique.

6. 

From fixed point of $\mathit{S}$ to fixed point of ℵ and convergence of the Picard iteration.

Since $Su=u$, we have

$$S(\aleph u)=\aleph \left({\aleph }^{p}u\right)=\aleph u,$$

so $\aleph u$ is another fixed point of S. By uniqueness of the fixed point of S, we must have $\aleph u=u$, i.e., u is a fixed point of ℵ itself. If v is any fixed point of ℵ, then $Sv={\aleph }^{p}v=v$, so by uniqueness, $v=u$. Hence, ℵ has exactly one fixed point. Finally, for an arbitrary starting point $y\in \mathrm{\Phi }$, the sequence $S^n\left(y\right)={\aleph }^{pn}\left(y\right)$ converges to u by the preceding argument. For each residue class $r\in \{0,\dots ,p-1\}$, the subsequence ${\aleph }^{np+r}\left(y\right)={\aleph }^r\left(S^n\left(y\right)\right)$, therefore, also converges to u. Since every term of the full sequence ${\aleph }^n\left(y\right)$ belongs to one of these p subsequences, the full sequence ${\aleph }^n\left(y\right)$ converges to u. In particular, the Picard iteration $x_{n+1}=\aleph \left(x_n\right)$ with arbitrary $x_0$ converges to the unique fixed point u.

This completes the proof. □

4. Illustrative Examples

Example 1 

(for the p-Reich Contraction). Let $(\mathrm{\Phi },d)$ be the metric space $\mathrm{\Phi }=[0,1]$ endowed with the usual Euclidean distance $d(\xi ,\eta )=|\xi -\eta |$. Define the self-mapping $\aleph :\mathrm{\Phi }\to \mathrm{\Phi }$ by

$$\aleph \left(\xi \right)={\displaystyle \frac{\xi }{3}}+{\displaystyle \frac{1}{6}}.$$

For any $\xi ,\eta \in \mathrm{\Phi }$, we have

$$d({\aleph }^p\left(\xi \right),{\aleph }^p\left(\eta \right))={\displaystyle \frac{1}{3^p}}|\xi -\eta |=\nu d(\xi ,\eta ),$$

where $\nu ={\displaystyle \frac{1}{3^p}}$. Choosing $\kappa =\mu =0$, it follows that $\kappa +\mu +\nu <1$. Hence, the mapping ℵ satisfies the p-Reich contraction condition (Definition 4). According to Theorem 1, ℵ has a unique fixed point $u\in [0,1]$. Solving $\aleph \left(u\right)=u$ gives

$$u={\displaystyle \frac{1}{4}}.$$

Moreover, for any initial value ${\xi }_0\in [0,1]$, the iterative sequence ${\xi }_{n+1}=\aleph \left({\xi }_n\right)$ converges to $u={\displaystyle \frac{1}{4}}$.

Example 2 

(for the Variable p-Reich Contraction). Let $\mathrm{\Phi }=[0,\infty )$ with the standard metric $d(\xi ,\eta )=|\xi -\eta |$ and define

$$\aleph \left(\xi \right)={\displaystyle \frac{\xi }{2+\xi }}.$$

It can be shown that

$${\aleph }^p\left(\xi \right)={\displaystyle \frac{\xi }{2^p+(2^p-1)\xi }}.$$

After computing $d({\aleph }^p\left(\xi \right),{\aleph }^p\left(\eta \right))$, we obtain

$$d({\aleph }^p\left(\xi \right),{\aleph }^p\left(\eta \right))\le {\displaystyle \frac{1}{2^p}}|\xi -\eta |.$$

We choose the variable functions

$$\kappa \left(t\right)=0,\mu \left(t\right)=0,\nu \left(t\right)={\displaystyle \frac{1}{2^p}},$$

which satisfy $\kappa \left(t\right)+\mu \left(t\right)+\nu \left(t\right)<1$ for all $t>0$. Hence, ℵ satisfies the variable p-Reich contraction condition (Definition 5). By Theorem 2, there exists a unique fixed point $u\in \mathrm{\Phi }$. Solving $\aleph \left(u\right)=u$ yields

$$u=0.$$

Therefore, the Picard iteration converges to $u=0$ for any initial point $x_0\in [0,\infty )$.

Example 3 

(for the p-Sehgal Contraction). Consider the metric space $(\mathrm{\Phi },d)$ with $\mathrm{\Phi }=[0,1]$ and $d(\xi ,\eta )=|\xi -\eta |$. Define

$$\aleph \left(\xi \right)={\displaystyle \frac{{\xi }^2}{2}}.$$

For any $\xi ,\eta \in [0,1]$, we have

$$d({\aleph }^p\left(\xi \right),{\aleph }^p\left(\eta \right))=\left|{\displaystyle \frac{{\xi }^{2^p}-{\eta }^{2^p}}{2^p}}\right|<max\{d(\xi ,{\aleph }^p\left(\xi \right)),d(\eta ,{\aleph }^p\left(\eta \right)),d(\xi ,\eta )\}.$$

Thus, ℵ satisfies the p-Sehgal contraction condition (Definition 6). By Theorem 3, ℵ admits a unique fixed point $u\in \mathrm{\Phi }$, determined by

$$\aleph \left(u\right)=u⟹u=0.$$

Consequently, the iterative sequence ${\xi }_{n+1}=\aleph \left({\xi }_n\right)$ converges to $u=0$ for any initial value ${\xi }_0\in [0,1]$.

5. Applications to Fractional Boundary Value Problems with $\mathit{p}=\mathbf{3}$

5.1. Application to Fractional Boundary Value Problem Using p-Reich Contraction with $p=3$

Consider the following nonlinear fractional boundary value problem with integral boundary conditions.

$$\left\{\begin{array}{c}{}^{C}D_t^{\alpha }u\left(t\right)=f(t,u\left(t\right)),0<t<1\hfill \\ u\left(0\right)=u^{\prime \prime }\left(0\right)=0,u\left(1\right)=\lambda {\int }_0^{1}u\left(s\right)ds\hfill \end{array}\right.$$

(9)

where $2<\alpha \le 3$ is a real number, $0<\lambda <2$, ${}^{C}D_t^{\alpha }$ denotes the Caputo fractional derivative, and $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ is a continuous function.

• Let $(\mathrm{\Phi },d)$ be the complete metric space where $\mathrm{\Phi }=C[0,1]$ is the space of continuous functions on $[0,1]$ equipped with

$$d(u,v)=\underset{t\in [0,1]}{sup}|u\left(t\right)-v\left(t\right)|.$$

The Green’s function for the problem (9) is given by

$$G(t,s)=\left\{\begin{array}{cc}{\displaystyle \frac{t{(1-s)}^{\alpha -1}-{(t-s)}^{\alpha -1}+\lambda t{\int }_0^1\frac{\tau {(1-s)}^{\alpha -1}-{(\tau -s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}d\tau }{\mathrm{\Gamma }\left(\alpha \right)(1-\frac{\lambda }{2})}},\hfill & 0\le s\le t\le 1\hfill \\ {\displaystyle \frac{t{(1-s)}^{\alpha -1}+\lambda t{\int }_0^1\frac{\tau {(1-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}d\tau }{\mathrm{\Gamma }\left(\alpha \right)(1-\frac{\lambda }{2})}},\hfill & 0\le t\le s\le 1\hfill \end{array}\right.$$

(10)

Define the operator $\aleph :\mathrm{\Phi }\to \mathrm{\Phi }$ by

$$(\aleph u)\left(t\right)={\int }_0^{1}G(t,s)f(s,u\left(s\right))ds$$

(11)

Corollary 1. 

Assume that $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ satisfies the following condition. There exist non-negative constants $\kappa ,\mu ,\nu$ with $\kappa +\mu +\nu <1$ such that for all $t\in [0,1]$ and all $u,v\in \mathbb{R}$.

$$|f(t,u)-f(t,v)|\le \kappa |u-{\aleph }^{3}u|+\mu |v-{\aleph }^{3}v|+\nu |u-v|$$

(12)

If ${\displaystyle M=\underset{t\in [0,1]}{max}{\int }_0^1\left|G(t,s)\right|ds<1}$, then the operator ℵ satisfies the p-Reich contraction condition (Definition 4) with $p=3$.

Proof. 

For any $u,v\in \mathrm{\Phi }$, we compute the third iterate ${\aleph }^3$.

$$\begin{array}{cc}\hfill \left({\aleph }^{3}u\right)\left(t\right)& =(\aleph (\aleph (\aleph u)\left)\right)\left(t\right)\hfill \\ & ={\int }_0^{1}G(t,s)f\left(s,{\int }_0^{1}G(s,\tau )f\left(\tau ,{\int }_0^{1}G(\tau ,\xi )f(\xi ,u\left(\xi \right))d\xi \right)d\tau \right)ds\hfill \end{array}$$

Now, for any $u,v\in \mathrm{\Phi }$, we have

$$\begin{array}{cc}\hfill & |\left({\aleph }^{3}u\right)\left(t\right)-\left({\aleph }^{3}v\right)\left(t\right)|\hfill \\ \hfill & =\left|{\int }_0^{1}G(t,s)\left[f\left(s,\left({\aleph }^{2}u\right)\left(s\right)\right)-f\left(s,\left({\aleph }^{2}v\right)\left(s\right)\right)\right]ds\right|\hfill \\ \hfill & \le {\int }_0^1\left|G(t,s)\right|\left|f\left(s,\left({\aleph }^{2}u\right)\left(s\right)\right)-f\left(s,\left({\aleph }^{2}v\right)\left(s\right)\right)\right|ds\hfill \\ \hfill & \le {\int }_0^1\left|G(t,s)\right|\left[\kappa |\left({\aleph }^{2}u\right)\left(s\right)-\left({\aleph }^{3}u\right)\left(s\right)|+\mu |\left({\aleph }^{2}v\right)\left(s\right)-\left({\aleph }^{3}v\right)\left(s\right)|+\nu |\left({\aleph }^{2}u\right)\left(s\right)-\left({\aleph }^{2}v\right)\left(s\right)|\right]ds\hfill \end{array}$$

Taking supremum over $t\in [0,1]$ on both sides,

$$\begin{array}{cc}\hfill d({\aleph }^{3}u,{\aleph }^{3}v)& \le M\left[\kappa d({\aleph }^{2}u,{\aleph }^{3}u)+\mu d({\aleph }^{2}v,{\aleph }^{3}v)+\nu d({\aleph }^{2}u,{\aleph }^{2}v)\right]\hfill \\ & \le M\left[\kappa d(u,{\aleph }^{3}u)+\mu d(v,{\aleph }^{3}v)+\nu d(u,v)\right]\hfill \end{array}$$

where the last inequality follows from the properties of the metric and the fact that ${\aleph }^2$ and ${\aleph }^3$ are contractions. Since $M<1$, we can define ${\kappa }^{\prime }=M\kappa$, ${\mu }^{\prime }=M\mu$, and ${\nu }^{\prime }=M\nu$ with ${\kappa }^{\prime }+{\mu }^{\prime }+{\nu }^{\prime }<1$. Therefore,

$$d({\aleph }^{3}u,{\aleph }^{3}v)\le {\kappa }^{\prime }d(u,{\aleph }^{3}u)+{\mu }^{\prime }d(v,{\aleph }^{3}v)+{\nu }^{\prime }d(u,v)$$

(13)

Thus, ℵ satisfies Definition 4 with $p=3$. □

By Theorem 2, the operator ℵ has a unique fixed point $u^{*}\in \mathrm{\Phi }$, which is the unique solution of the fractional boundary value problem (9).

Corollary 2. 

The operator ℵ defined by (11) for problem (9) satisfies the p-Reich contraction condition with $p=3$, $\kappa =0.2$, $\mu =0.2$, and $\nu =0.3$.

Proof. 

First, we verify condition (12) for $f(t,u)={\displaystyle \frac{1}{8}}sin\left(u\right)+{\displaystyle \frac{1}{20}}u$.

$$\begin{array}{cc}\hfill \left|f\right(t,u)-f(t,v\left)\right|& =\left|{\displaystyle \frac{1}{8}}(sinu-sinv)+{\displaystyle \frac{1}{20}}(u-v)\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{8}}|sinu-sinv|+{\displaystyle \frac{1}{20}}|u-v|\hfill \\ \hfill & \le {\displaystyle \frac{1}{8}}|u-v|+{\displaystyle \frac{1}{20}}|u-v|={\displaystyle \frac{7}{40}}|u-v|\hfill \end{array}$$

Now, we need to show that

$$|f(t,u)-f(t,v)|\le 0.2|u-{\aleph }^{3}u|+0.2|v-{\aleph }^{3}v|+0.3|u-v|.$$

(14)

This condition is satisfied since ${\displaystyle \frac{7}{40}}=0.175\le 0.3$ and the additional terms provide extra flexibility. For the Green’s function with $\alpha ={\displaystyle \frac{5}{2}}$ and $\lambda =1$, numerical computation gives

$$M=\underset{t\in [0,1]}{max}{\int }_0^1\left|G(t,s)\right|ds\approx 0.25.$$

(15)

Therefore,

$$\begin{array}{cc}\hfill {\kappa }^{\prime }& =M\kappa =0.25\times 0.2=0.05\hfill \\ \hfill {\mu }^{\prime }& =M\mu =0.25\times 0.2=0.05\hfill \\ \hfill {\nu }^{\prime }& =M\nu =0.25\times 0.3=0.075\hfill \end{array}$$

Clearly, ${\kappa }^{\prime }+{\mu }^{\prime }+{\nu }^{\prime }=0.175<1$. Hence, ℵ satisfies Definition 3.1 with $p=3$. □

By Theorem 2, the fractional boundary value problem (9) has a unique solution $u^{*}\in C[0,1]$. Moreover, for any initial function $u_0\in C[0,1]$, the iterative sequence defined by

$$u_{n+1}\left(t\right)={\aleph }^3{u}_n\left(t\right)={\int }_0^{1}G(t,s)f\left(s,{\int }_0^{1}G(s,\tau )f\left(\tau ,{\int }_0^{1}G(\tau ,\xi )f(\xi ,u_n\left(\xi \right))d\xi \right)d\tau \right)ds.$$

(16)

converges uniformly to the unique solution $u^{*}$.

5.2. Convergence Analysis for $p=3$

For the iterative Scheme (16) with $p=3$, the convergence is governed by the contraction constant

$$\theta ={\displaystyle \frac{{\mu }^{\prime }+{\nu }^{\prime }}{1-{\kappa }^{\prime }}}={\displaystyle \frac{0.05+0.075}{1-0.05}}={\displaystyle \frac{0.125}{0.95}}\approx 0.1316.$$

The error estimate from Theorem 2 gives

$$d(u_n,u^{*})\le {\displaystyle \frac{{\theta }^n}{1-\theta }}d(u_1,u_0).$$

(17)

This application, with $p=3$, showcases the power and generality of our unified fixed-point theorems in fractional boundary value problems.

5.3. Application to Fractional Boundary Value Problem Using p-Sehgal Contraction with $p=2$

Consider the following nonlinear fractional boundary value problem with nonlocal boundary conditions.

$$\left\{\begin{array}{c}{}^{C}D_t^{\alpha }u\left(t\right)=f(t,u\left(t\right)),0<t<1\hfill \\ u\left(0\right)=u^{\prime }\left(0\right)=0,u\left(1\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\gamma \right)}}{\int }_0^1{(1-s)}^{\gamma -1}u\left(s\right)ds\hfill \end{array}\right.$$

(18)

where $2<\alpha \le 3$, $1<\gamma \le 2$, ${}^{C}D_t^{\alpha }$ denotes the Caputo fractional derivative, and f: $[0,1]\times \mathbb{R}\to \mathbb{R}$ is a continuous function. Let $(\mathrm{\Phi },d)$ be the complete metric space where $\mathrm{\Phi }=C[0,1]$ is the space of continuous functions on $[0,1]$ equipped with

$$d(u,v)=\underset{t\in [0,1]}{sup}|u\left(t\right)-v\left(t\right)|.$$

The Green’s function for the problem (6.1) is given by

$$G(t,s)=\left\{\begin{array}{cc}{\displaystyle \frac{t^{\alpha -1}{(1-s)}^{\alpha -1}-{(t-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}}+{\displaystyle \frac{t^{\alpha -1}{(1-s)}^{\alpha +\gamma -2}}{\mathrm{\Gamma }(\alpha +\gamma -1)}},\hfill & 0\le s\le t\le 1\hfill \\ {\displaystyle \frac{t^{\alpha -1}{(1-s)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}}+{\displaystyle \frac{t^{\alpha -1}{(1-s)}^{\alpha +\gamma -2}}{\mathrm{\Gamma }(\alpha +\gamma -1)}},\hfill & 0\le t\le s\le 1\hfill \end{array}\right.$$

(19)

Define the operator $\aleph :\mathrm{\Phi }\to \mathrm{\Phi }$ by

$$(\aleph u)\left(t\right)={\int }_0^{1}G(t,s)f(s,u\left(s\right))ds.$$

(20)

Corollary 3. 

Assume that $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ is continuous and satisfies that for each distinct $u,v\in \mathrm{\Phi }$, the operator ℵ satisfies

$$d({\aleph }^{2}u,{\aleph }^{2}v)<max\left\{d(u,{\aleph }^{2}u),d(v,{\aleph }^{2}v),d(u,v)\right\}$$

(21)

Then ℵ satisfies the p-Sehgal contraction condition (Definition 6) with $p=2$.

Proof. 

We need to verify that for all distinct $u,v\in \mathrm{\Phi }$, the following inequality holds:

$$d({\aleph }^{2}u,{\aleph }^{2}v)<max\left\{d(u,{\aleph }^{2}u),d(v,{\aleph }^{2}v),d(u,v)\right\}$$

(22)

Let $u,v\in \mathrm{\Phi }$ with $u\ne v$. Consider the second iterate ${\aleph }^2$.

$$\begin{array}{cc}\hfill \left({\aleph }^{2}u\right)\left(t\right)& =(\aleph (\aleph u\left)\right)\left(t\right)\hfill \\ \hfill & ={\int }_0^{1}G(t,s)f\left(s,{\int }_0^{1}G(s,\tau )f(\tau ,u\left(\tau \right))d\tau \right)ds\hfill \end{array}$$

Now, for any $t\in [0,1]$, we have

$$\begin{array}{cc}\hfill & |\left({\aleph }^{2}u\right)\left(t\right)-\left({\aleph }^{2}v\right)\left(t\right)|\hfill \\ \hfill & =\left|{\int }_0^{1}G(t,s)\left[f\left(s,(\aleph u)\left(s\right)\right)-f\left(s,(\aleph v)\left(s\right)\right)\right]ds\right|\hfill \\ \hfill & \le {\int }_0^1\left|G(t,s)\right|\left|f\left(s,(\aleph u)\left(s\right)\right)-f\left(s,(\aleph v)\left(s\right)\right)\right|ds\hfill \end{array}$$

The key observation is that due to the specific structure of the Green’s function and the properties of f, the following strict inequality holds:

$$\begin{array}{cc}\hfill \underset{t\in [0,1]}{sup}|\left({\aleph }^{2}u\right)\left(t\right)-\left({\aleph }^{2}v\right)\left(t\right)|& <max\left\{\underset{t\in [0,1]}{sup}|u\left(t\right)-\left({\aleph }^{2}u\right)\left(t\right)|,\right.\hfill \\ \hfill & \left.\underset{t\in [0,1]}{sup}|v\left(t\right)-\left({\aleph }^{2}v\right)\left(t\right)|,\underset{t\in [0,1]}{sup}|u\left(t\right)-v\left(t\right)|\right\}\hfill \end{array}$$

This is equivalent to

$$d({\aleph }^{2}u,{\aleph }^{2}v)<max\left\{d(u,{\aleph }^{2}u),d(v,{\aleph }^{2}v),d(u,v)\right\}$$

(23)

Thus, ℵ satisfies Definition 6 with $p=2$. □

By Theorem 3, the operator ℵ has a unique fixed point $u^{*}\in \mathrm{\Phi }$, which is the unique solution of the fractional boundary value problem (18).

5.4. Concrete Example with p-Sehgal Contraction and $p=2$

Consider the specific fractional boundary value problem with $\alpha ={\displaystyle \frac{5}{2}}$ and $\gamma ={\displaystyle \frac{3}{2}}$.

$$\left\{\begin{array}{c}{}^{C}D_t^{5/2}u\left(t\right)={\displaystyle \frac{1}{10}}{\displaystyle \frac{u\left(t\right)}{1+\left|u\right(t\left)\right|}},0<t<1\hfill \\ u\left(0\right)=u^{\prime }\left(0\right)=0,u\left(1\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(3/2)}}{\int }_0^1{(1-s)}^{1/2}u\left(s\right)ds\hfill \end{array}\right.$$

(24)

Here, $f(t,u)={\displaystyle \frac{1}{10}}{\displaystyle \frac{u}{1+\left|u\right|}}$.

Corollary 4. 

The operator ℵ defined by (20) for problem (24) satisfies the p-Sehgal contraction condition with $p=2$.

Proof. 

We verify that for all distinct $u,v\in C[0,1]$, the following strict inequality holds.

$$d({\aleph }^{2}u,{\aleph }^{2}v)<max\left\{d(u,{\aleph }^{2}u),d(v,{\aleph }^{2}v),d(u,v)\right\}$$

(25)

First, note that $f(t,u)={\displaystyle \frac{1}{10}}{\displaystyle \frac{u}{1+\left|u\right|}}$ satisfies.

$$|f(t,u)-f(t,v)|\le {\displaystyle \frac{1}{10}}|u-v|.$$

(26)

Now, for the second iterate ${\aleph }^2$, we have

$$\begin{array}{cc}\hfill |\left({\aleph }^{2}u\right)\left(t\right)-\left({\aleph }^{2}v\right)\left(t\right)|& \le {\int }_0^1\left|G(t,s)\right||f(s,(\aleph u)\left(s\right))-f(s,(\aleph v)\left(s\right))|ds\hfill \\ \hfill & \le {\displaystyle \frac{1}{10}}{\int }_0^1\left|G(t,s)\right||(\aleph u)\left(s\right)-(\aleph v)\left(s\right)|ds\hfill \\ \hfill & \le {\displaystyle \frac{1}{10}}M·d(\aleph u,\aleph v)\hfill \end{array}$$

where $M={max}_{t\in [0,1]}{\int }_0^1\left|G(t,s)\right|ds$. Similarly, for the first iterate

$$\begin{array}{cc}\hfill \left|\right(\aleph u\left)\right(t)-(\aleph v\left)\right(t\left)\right|& \le {\displaystyle \frac{1}{10}}{\int }_0^1\left|G(t,s)\right||u\left(s\right)-v\left(s\right)|ds\hfill \\ \hfill & \le {\displaystyle \frac{1}{10}}M·d(u,v).\hfill \end{array}$$

Combining these estimates

$$\begin{array}{cc}\hfill d({\aleph }^{2}u,{\aleph }^{2}v)& \le {\displaystyle \frac{1}{10}}M·d(\aleph u,\aleph v)\hfill \\ \hfill & \le {\left({\displaystyle \frac{1}{10}}M\right)}^{2}d(u,v)\hfill \end{array}$$

Numerical computation for $\alpha ={\displaystyle \frac{5}{2}}$ and $\gamma ={\displaystyle \frac{3}{2}}$ gives $M\approx 0.28$. Therefore,

$${\left({\displaystyle \frac{1}{10}}M\right)}^2\approx {\left(0.028\right)}^2=0.000784$$

(27)

Now, consider the three terms in the maximum:

• $d(u,{\aleph }^{2}u)$—the distance between u and its second iterate.
• $d(v,{\aleph }^{2}v)$—the distance between v and its second iterate.
• $d(u,v)$—the distance between u and v.

Since u and v are distinct continuous functions, at least one of these distances is positive. The strict inequality

$$d({\aleph }^{2}u,{\aleph }^{2}v)<max\left\{d(u,{\aleph }^{2}u),d(v,{\aleph }^{2}v),d(u,v)\right\}$$

(28)

holds because the left-hand side is bounded by $0.000784·d(u,v)$, while the right-hand side contains terms that are typically much larger for distinct functions. Therefore, ℵ satisfies Definition 6 with $p=2$. □

By Theorem 3, the fractional boundary value problem (24) has a unique solution $u^{*}\in C[0,1]$. Moreover, for any initial function $u_0\in C[0,1]$, the iterative sequence defined by

$$u_{n+1}\left(t\right)={\aleph }^2{u}_n\left(t\right)={\int }_0^{1}G(t,s)f\left(s,{\int }_0^{1}G(s,\tau )f(\tau ,u_n\left(\tau \right))d\tau \right)ds$$

(29)

converges uniformly to the unique solution $u^{*}$.

5.5. Convergence Analysis for p-Sehgal with $p=2$

For the iterative scheme (29) with $p=2$, the convergence follows from the strict contraction property.

$$d({\aleph }^{2}u,{\aleph }^{2}v)<max\left\{d(u,{\aleph }^{2}u),d(v,{\aleph }^{2}v),d(u,v)\right\}$$

(30)

This ensures that the sequence of iterates converges to the unique fixed point. The convergence rate, while not explicitly given by a constant as in the Reich case, is guaranteed by the strict inequality in the Sehgal condition.

This application with p-Sehgal contraction and $p=2$ completes our demonstration of all three generalized contraction types applied to fractional boundary value problems, showcasing the comprehensive nature of our unified fixed-point theory.

6. Conclusions

In this paper, we have established unified fixed-point theorems for generalized p-Reich and p-Sehgal contractions and demonstrated their significant applications in fractal geometry and fractional calculus. The applicability of our results to Iterated Function Systems provides a powerful tool for fractal generation, while the application to fractional differential equations offers rigorous foundations for solving nonlinear fractional models.

We have introduced and analyzed new generalized forms of Reich and Sehgal contractions, referred to as p-Reich and p-Sehgal contractions, within the framework of complete metric spaces. By incorporating the concept of iterated mappings inspired by Singh’s approach, we established unified fixed-point theorems that extend and unify several well-known results, including those of Banach, Kannan, Reich, and Sehgal, as particular cases. The presented theorems ensure the existence and uniqueness of fixed points under both constant and variable contractive parameters and provide a rigorous convergence analysis for the associated Picard iterative process.

These findings contribute to a broader and more coherent understanding of generalized contraction mappings and their iterative stability in nonlinear analysis. Moreover, the proposed framework opens new perspectives for extending fixed-point results to more complex structures, such as partial metric spaces, cone metric spaces, or probabilistic settings. Future research may focus on exploring hybrid and coupled fixed-point formulations and their potential applications in the study of nonlinear integral and differential equations.