Fixed Point Results for Large Contraction Mappings in b-Metric Spaces with Applications to Fractional Differential Equations

Abstract

In this paper, we establish new fixed point results for Burton-type large contractions in complete b-metric spaces and introduce a Rakotch-type generalization in this setting. We establish existence and uniqueness results for fixed points, with an example to illustrate the applicability. Furthermore, an application to a fractional differential equation is presented. Our results generalize classical fixed point theorems and contribute to the theory of nonuniform contractions in generalized metric spaces.

1. Introduction

Fixed point theory plays a fundamental role in nonlinear analysis and has wide applications in differential and integral equations. One of the most celebrated results in this area is the Banach contraction principle, which guarantees the existence and uniqueness of fixed points for contractive mappings in complete metric spaces [1]. This classical result has become a cornerstone of modern analysis and has inspired numerous generalizations.

Over the years, many extensions of the Banach contraction principle have been developed. In particular, several authors have introduced more flexible contractive conditions such as altering distance functions and generalized Lipschitz-type mappings [2,3,4,5,6]. These approaches have significantly broadened the scope of fixed point theory and its applications.

To extend this framework, Czerwik introduced the concept of b-metric spaces, which generalize classical metric spaces by relaxing the triangle inequality [7]. This generalization has opened the door to studying fixed point theory in more flexible and nonstandard settings. Since then, considerable research has been devoted to the development of fixed point results in b-metric spaces. For instance, Koleva and Zlatanov [8] established fixed point results for Chatterjea-type mappings, showing that classical contraction principles can be extended to this generalized framework. Berinde and Păcurar [9] provided a comprehensive survey of early developments in fixed point theory on b-metric spaces, highlighting the evolution and significance of this area. Moreover, Aydi et al. [10] extended fixed point theory to set-valued quasi-contractions, demonstrating the richness of the structure of b-metric spaces and their applicability to more general mappings. George et al. [11] introduced and studied rectangular b-metric spaces, further generalizing the concept and establishing corresponding contraction principles. In addition, Miculescu and Mihail [12] developed new fixed point results for set-valued contractions, while Kir and Kiziltunc [13] revisited several classical fixed point theorems and showed that they remain valid under the weaker assumptions of b-metric spaces. Also, in [14], some fixed points in complete extended b-metric spaces for single-valued and multi-valued mappings were studied.

These contributions collectively confirm that many classical fixed point results can be successfully extended to b-metric spaces, thereby enriching the theory and expanding its range of applications. In parallel, Burton introduced the notion of large contraction mappings as a nonuniform generalization of classical contraction mappings [15]. Unlike Banach contractions, where the contractive behavior is governed by a fixed constant, large contractions allow the contraction rate to depend on the distance between points, making the framework more flexible and suitable for nonlinear analysis. This concept has since attracted considerable attention. In particular, necessary and sufficient conditions for large contractions were further investigated in later works, providing a deeper understanding of their structural properties [16]. Moreover, several extensions and applications have been developed in different directions. For instance, Dehici et al. [17] studied large Kannan contraction mappings and their applications to fixed point problems. Mesmouli et al. [18] studied large Chatterjea contraction mappings and their applications to delay fractional differential equations, highlighting the relevance of large contractions in applied analysis. More recently, further generalizations have been proposed, including large triangle perimeter contractions in metric spaces [19], coupled large Kannan contractions with applications to integral equations [20], and cyclic large contractions under perturbation frameworks [21]. These developments demonstrate that the theory of large contractions is rapidly evolving and has become an active area of research with significant applications in nonlinear analysis and differential equations [22].

However, the study of large contraction mappings in the context of b-metric spaces remains limited. Motivated by this gap, we extend Burton’s concept of large contractions to complete b-metric spaces and establish new fixed point theorems for this class of mappings. Unlike the classical metric setting, the proof in b-metric spaces requires the additional control of the weighted geometric terms generated by the coefficient s in the generalized triangle inequality. This creates technical difficulties that do not appear in the standard metric setting and shows that the extension is not merely formal.

The main contributions of this paper are threefold. First, we prove a Burton-type fixed point theorem for large contractions in complete b-metric spaces. Second, we introduce a Rakotch-type version of large contractions adapted to the coefficient s of the b-metric space. Third, we provide a nontrivial example and apply the obtained results to a Caputo-type fractional boundary value problem.

Compared with existing results on generalized contractions in b-metric spaces, the present work focuses on nonuniform contractive conditions depending on distance thresholds rather than global Lipschitz constants. Therefore, the obtained results extend and complement several classical Banach-type, Burton-type, and Rakotch-type fixed point theorems.

The remainder of this paper is organized as follows. In Section 2, we recall some preliminaries on b-metric spaces and large contractions. Section 3 contains the main fixed point results for large contraction mappings in b-metric spaces and related generalizations. In Section 4, we provide an application to fractional differential equations. Finally, the Section 5 concludes this paper with some remarks and possible future directions.

2. Preliminaries

In this section, we recall some basic definitions and fundamental results that will be used throughout this paper. These notions play a central role in the study of fixed point theory in b-metric spaces and contractive mappings.

Definition 1.

Let ℵ be a nonempty set and $s\ge 1$. A function $\wp :\aleph \times \aleph \to [0,\infty )$ is called a b-metric if for all $\rho ,\sigma ,z\in \aleph$, the following conditions hold:

1.

$\wp (\rho ,\sigma )=0$ if and only if $\rho =\sigma$.

2.

$\wp (\rho ,\sigma )=\wp (\sigma ,\rho )$.

3.

$\wp (\rho ,z)\le s(\wp (\rho ,\sigma )+\wp (\sigma ,z\left)\right)$.

Then $(\aleph ,\wp )$ is called a b-metric space.

Remark 1.

It is worth noting that every metric space is a b-metric space with $s=1$; however the converse is not necessarily true. The relaxation introduced by the coefficient $s\ge 1$ allows for a wider class of spaces, which is particularly useful in applications where the classical triangle inequality is too restrictive. This generalization has led to significant developments in fixed point theory.

Example 1.

Let $\aleph =\mathbb{R}$, and define

$$\wp (\rho ,\sigma )={|\rho -\sigma |}^p,p>1.$$

Then $(\aleph ,\wp )$ is a b-metric space with constant $s=2^{p-1}$.

• Indeed, for all $\rho ,\sigma ,z\in \mathbb{R}$, using the inequality

$${|\rho -z|}^p\le {\left(\right|\rho -\sigma |+|\sigma -z\left|\right)}^p\le 2^{p-1}{\left(\right|\rho -\sigma |}^p{+|\sigma -z|}^p),$$

we obtain

$$\wp (\rho ,z)\le s(\wp (\rho ,\sigma )+\wp (\sigma ,z\left)\right).$$

Thus all conditions of a b-metric are satisfied.

We now recall a fundamental result that extends the classical Banach contraction principle to the setting of b-metric spaces, see [7,13].

Theorem 1

([7,13]). Let $(\aleph ,\wp )$ be a complete b-metric space, and let $\mathrm{{\rm Y}}:\aleph \to \aleph$ satisfy

$$\wp (\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\sigma )\le k\wp (\rho ,\sigma ),0<k<1.$$

Then $\mathrm{{\rm Y}}$ has a unique fixed point in ℵ.

In order to further relax the contractive condition, several authors have introduced more general notions of contractions. One such concept is the notion of a large contraction, which allows the contraction behavior to depend on the distance scale between points.

Definition 2

(Large contraction). Let $(\aleph ,\wp )$ be a b-metric space. A mapping $\mathrm{{\rm Y}}:\aleph \to \aleph$ is called a large contraction if $\wp (\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\sigma )<\wp (\rho ,\sigma )$ for all $\rho \ne \sigma$ and if for every $\epsilon >0$, there exists $\delta \left(\epsilon \right)\in (0,1)$ such that

$$\wp (\rho ,\sigma )\ge \epsilon \Rightarrow \wp (\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\sigma )\le \delta \left(\epsilon \right)\wp (\rho ,\sigma ).$$

The following theorem is given by Burton in [15] for a large contraction mapping in a metric space.

Theorem 2

([15]). Let $\left(\aleph ,\wp \right)$ be a complete metric space and $\mathrm{{\rm Y}}:\aleph \to \aleph$ be a large contraction mapping. Suppose there exist ${\rho }_0\in \aleph$ and $L>0$ such that

$$\wp ({\rho }_0,{\mathrm{{\rm Y}}}^n{\rho }_0)\le L,\forall n\ge 1.$$

Then $\mathrm{{\rm Y}}$ has a unique fixed point in ℵ.

Example 2.

Let $\aleph =\mathbb{R}$ equipped with the usual metric $\wp (\rho ,\sigma )=|\rho -\sigma |$, and define

$$\hslash \left(\rho \right)=\rho -{\rho }^3.$$

Then ℏ exhibits a large contraction.

The following remarks have been noted in [17].

Remark 2.

Observe that if $\left(\aleph ,\wp \right)$ is a compact metric space, then the assumption that there exist ${\rho }_0\in \aleph$ and $L>0$ such that

$$\wp \left({\rho }_0,{\mathrm{{\rm Y}}}^n{\rho }_0\right)\le L\mathit{for}\mathit{all}n\ge 1$$

can be omitted. Indeed, in this case, the existence and uniqueness of a fixed point follow directly from Edelstein’s theorem.

Remark 3.

If $\left(\aleph ,\wp \right)$ is a bounded complete metric space, then for any ${\rho }_0\in \aleph$ and for all integers $n\ge 1$, we have

$$\wp \left({\rho }_0,{\mathrm{{\rm Y}}}^n{\rho }_0\right)\le diam(\aleph ),$$

where $diam(\aleph )$ denotes the diameter of ℵ. Hence, the boundedness condition on the orbit is automatically satisfied in this setting.

3. Main Results

In this section, we will investigate fixed point results for large contractions in the setting of b-metric spaces and establish new extensions of classical fixed point theorems.

Theorem 3.

Let $(\aleph ,\wp )$ be a complete b-metric space with constant $s\ge 1$, and let $\mathrm{{\rm Y}}:\aleph \to \aleph$ be a large contraction mapping. Suppose there exist ${\rho }_0\in \aleph$ and $L>0$ such that

$$\wp ({\rho }_0,{\mathrm{{\rm Y}}}^n{\rho }_0)\le L,\forall n\ge 1.$$

Then $\mathrm{{\rm Y}}$ has a unique fixed point in ℵ.

Proof. 

Let ${\rho }_0\in \aleph$, and define ${\rho }_n={\mathrm{{\rm Y}}}^n{\rho }_0$.

• If there exists $n_0\ge 1$ such that ${\rho }_{n_0}={\rho }_{n_0+1}$, then ${\rho }_{n_0}$ is a fixed point of $\mathrm{{\rm Y}}$.
• Assume now that ${\rho }_n\ne {\rho }_{n+1}$ for all $n\ge 0$.
• Step 1. Since $\mathrm{{\rm Y}}$ is a large contraction, we have

  $$\wp ({\rho }_{n+1},{\rho }_n)=\wp (\mathrm{{\rm Y}}{\rho }_n,\mathrm{{\rm Y}}{\rho }_{n-1})<\wp ({\rho }_n,{\rho }_{n-1}).$$

Hence the sequence

$${\zeta }_n=\wp ({\rho }_{n+1},{\rho }_n)$$

is strictly decreasing and converges to some $\gamma \ge 0$.

Suppose that $\gamma >0$. Then ${\zeta }_n\ge \gamma$ for all n. By the large contraction condition, there exists $\delta \left(\gamma \right)\in (0,1)$ such that

$${\zeta }_n=\wp ({\rho }_{n+1},{\rho }_n)\le \delta \left(\gamma \right)\wp ({\rho }_n,{\rho }_{n-1})=\delta \left(\gamma \right){\zeta }_{n-1}.$$

Iterating this inequality gives

$${\zeta }_n\le \delta {\left(\gamma \right)}^n{\zeta }_0.$$

Letting $n\to \infty$, we obtain ${\zeta }_n\to 0$, which contradicts $\gamma >0$. Therefore,

$$\underset{n\to \infty }{lim}\wp ({\rho }_{n+1},{\rho }_n)=0.$$

• Step 2. We prove that $\left({\rho }_n\right)$ is a Cauchy sequence.

Assume, to the contrary, that $\left({\rho }_n\right)$ is not Cauchy. Then there exist $\epsilon >0$ and subsequences $\left({\rho }_{m_j}\right)$ and $\left({\rho }_{n_j}\right)$ such that

$$m_j>n_j\to \infty$$

and

$$\wp ({\rho }_{m_j},{\rho }_{n_j})\ge \epsilon \mathrm{for}\mathrm{all}j.$$

Since $\mathrm{{\rm Y}}$ is a large contraction, there exists $\delta =\delta \left(\epsilon \right)\in (0,1)$ such that

$$\wp (x,y)\ge \epsilon ⟹\wp (\mathrm{{\rm Y}}x,\mathrm{{\rm Y}}y)\le \delta \wp (x,y).$$

Moreover, from the strict contractive property,

$$\wp ({\rho }_{m_j},{\rho }_{n_j})<\wp ({\rho }_{m_j-1},{\rho }_{n_j-1})<\cdots <\wp ({\rho }_{m_j-q},{\rho }_{n_j-q})$$

for every fixed positive integer q and all sufficiently large j. Thus, each of the distances appearing above is at least $\epsilon$.

• Therefore, applying the large contraction condition q times, we obtain

  $$\wp ({\rho }_{m_j},{\rho }_{n_j})\le {\delta }^q\wp ({\rho }_{m_j-q},{\rho }_{n_j-q}).$$

Now, using the b-metric inequality and the boundedness assumption on the orbit, we get

$$\wp ({\rho }_{m_j-q},{\rho }_{n_j-q})\le s\left(\wp ({\rho }_{m_j-q},{\rho }_0)+\wp ({\rho }_0,{\rho }_{n_j-q})\right)\le 2sL.$$

Hence

$$\wp ({\rho }_{m_j},{\rho }_{n_j})\le 2sL{\delta }^q.$$

Since $\delta \in (0,1)$, we may choose q sufficiently large such that

$$2sL{\delta }^q<\epsilon .$$

Thus,

$$\wp ({\rho }_{m_j},{\rho }_{n_j})<\epsilon ,$$

which contradicts

$$\wp ({\rho }_{m_j},{\rho }_{n_j})\ge \epsilon .$$

Therefore, $\left({\rho }_n\right)$ is a Cauchy sequence.

Since $(\aleph ,\wp )$ is complete, there exists ${\rho }^{*}\in \aleph$ such that

$${\rho }_n\to {\rho }^{*}.$$

• Step 3. We show that ${\rho }^{*}$ is a fixed point of $\mathrm{{\rm Y}}$.

Using the b-metric inequality, we have

$$\wp (\mathrm{{\rm Y}}{\rho }^{*},{\rho }^{*})\le s\left(\wp (\mathrm{{\rm Y}}{\rho }^{*},\mathrm{{\rm Y}}{\rho }_n)+\wp (\mathrm{{\rm Y}}{\rho }_n,{\rho }^{*})\right).$$

Since $\mathrm{{\rm Y}}$ is a large contraction,

$$\wp (\mathrm{{\rm Y}}{\rho }^{*},\mathrm{{\rm Y}}{\rho }_n)<\wp ({\rho }^{*},{\rho }_n)\to 0,$$

and

$$\wp (\mathrm{{\rm Y}}{\rho }_n,{\rho }^{*})=\wp ({\rho }_{n+1},{\rho }^{*})\to 0.$$

Therefore,

$$\wp (\mathrm{{\rm Y}}{\rho }^{*},{\rho }^{*})=0,$$

and hence

$$\mathrm{{\rm Y}}{\rho }^{*}={\rho }^{*}.$$

Uniqueness. Let ${\sigma }^{*}$ be another fixed point of $\mathrm{{\rm Y}}$. If ${\rho }^{*}\ne {\sigma }^{*}$, then

$$\wp ({\rho }^{*},{\sigma }^{*})=\wp (\mathrm{{\rm Y}}{\rho }^{*},\mathrm{{\rm Y}}{\sigma }^{*})<\wp ({\rho }^{*},{\sigma }^{*}),$$

which is impossible. Hence ${\rho }^{*}={\sigma }^{*}$. □

Remark 4.

The boundedness assumption on the orbit

$$\wp ({\rho }_0,{\mathrm{{\rm Y}}}^n{\rho }_0)\le L,n\ge 1,$$

plays an essential role in the proof of Theorem 3, particularly in the derivation of the Cauchy property of the Picard iteration sequence. Indeed, in the setting of b-metric spaces, the generalized triangle inequality generates weighted estimates involving the coefficient s, and the boundedness condition allows us to control these terms.

It is worth mentioning that, under stronger contractive assumptions or additional geometric conditions on the space, this boundedness assumption may possibly be weakened or removed. See the previous Remarks 2 and 3.

Example 3.

Let $\aleph =\mathbb{R}$, and define

$$\wp (\rho ,\sigma )={|\rho -\sigma |}^2.$$

Then $(\aleph ,\wp )$ is a b-metric space with constant $s=2$, since

$${|\rho -z|}^2\le {\left(\right|\rho -\sigma |+|\sigma -z\left|\right)}^2\le {2\left(\right|\rho -\sigma |}^2{+|\sigma -z|}^2).$$

Define $\mathrm{{\rm Y}}:\aleph \to \aleph$ by

$$\mathrm{{\rm Y}}\left(\rho \right)={\displaystyle \frac{\rho }{1+\left|\rho \right|}}.$$

Then $\mathrm{{\rm Y}}$ has a unique fixed point in ℵ.

Proof. 

First, $(\aleph ,\wp )$ is a b-metric space with constant $s=2$.

We show that $\mathrm{{\rm Y}}$ is not a Banach contraction. Indeed, for $\rho >0$ and $\sigma =0$, we have

$${\displaystyle \frac{\left|\mathrm{{\rm Y}}\right(\rho )-\mathrm{{\rm Y}}(0\left)\right|}{|\rho -0|}}={\displaystyle \frac{1}{1+\rho }}\to 1\mathrm{as}\rho \to 0^{+}.$$

Hence there is no constant $k\in (0,1)$ such that

$$\wp (\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\sigma )\le k\wp (\rho ,\sigma ),\forall \rho ,\sigma \in \mathbb{R}.$$

Thus the Banach contraction principle is not applicable.

• Now we prove that $\mathrm{{\rm Y}}$ is a large contraction. Let $\rho ,\sigma \in \mathbb{R}$, $\rho \ne \sigma$.
• If $\rho ,\sigma \ge 0$, then

  $$|\mathrm{{\rm Y}}\left(\rho \right)-\mathrm{{\rm Y}}\left(\sigma \right)|={\displaystyle \frac{|\rho -\sigma |}{(1+\rho )(1+\sigma )}}<|\rho -\sigma |.$$

Similarly, if $\rho ,\sigma \le 0$, the same estimate holds.

If $\rho \ge 0$ and $\sigma \le 0$, write $\rho =a$ and $\sigma =-b$, where $a,b\ge 0$. Then

$$|\rho -\sigma |=a+b$$

and

$$|\mathrm{{\rm Y}}\left(\rho \right)-\mathrm{{\rm Y}}\left(\sigma \right)|={\displaystyle \frac{a}{1+a}}+{\displaystyle \frac{b}{1+b}}.$$

Since the function $g\left(t\right)={\displaystyle \frac{t}{1+t}}$ is concave on $[0,\infty )$, we have

$${\displaystyle \frac{a}{1+a}}+{\displaystyle \frac{b}{1+b}}\le 2g\left({\displaystyle \frac{a+b}{2}}\right)={\displaystyle \frac{a+b}{1+\frac{a+b}{2}}}.$$

Thus

$$|\mathrm{{\rm Y}}\left(\rho \right)-\mathrm{{\rm Y}}\left(\sigma \right)|\le {\displaystyle \frac{1}{1+\frac{|\rho -\sigma |}{2}}}|\rho -\sigma |<|\rho -\sigma |.$$

Therefore,

$$\wp (\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\sigma )<\wp (\rho ,\sigma ),$$

for all $\rho \ne \sigma$.

Now let $\epsilon >0$, and suppose that

$$\wp (\rho ,\sigma )\ge \epsilon .$$

Then

$$|\rho -\sigma |\ge \sqrt{\epsilon }.$$

From the previous estimates, we obtain

$$|\mathrm{{\rm Y}}\left(\rho \right)-\mathrm{{\rm Y}}\left(\sigma \right)|\le {\displaystyle \frac{1}{1+\frac{\sqrt{\epsilon }}{2}}}|\rho -\sigma |.$$

Consequently,

$$\wp (\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\sigma )\le {\left({\displaystyle \frac{1}{1+\frac{\sqrt{\epsilon }}{2}}}\right)}^2\wp (\rho ,\sigma ).$$

Set

$$\delta \left(\epsilon \right)={\left({\displaystyle \frac{1}{1+\frac{\sqrt{\epsilon }}{2}}}\right)}^2.$$

Then $\delta \left(\epsilon \right)\in (0,1)$, and hence $\mathrm{{\rm Y}}$ is a large contraction.

Finally, the fixed point equation is

$$\rho ={\displaystyle \frac{\rho }{1+\left|\rho \right|}}.$$

If $\rho \ne 0$, then

$$\rho (1+|\rho \left|\right)=\rho ,$$

which gives

$$\rho \left|\rho \right|=0,$$

a contradiction. Hence $\rho =0$. Therefore, the unique fixed point is

$${\rho }^{*}=0.$$

□

Corollary 1.

Let $(\aleph ,\wp )$ be a complete b-metric space and $\mathrm{{\rm Y}}:\aleph \to \aleph$ such that ${\mathrm{{\rm Y}}}^{m_0}$ is a large contraction for some $m_0\ge 1$. Then $\mathrm{{\rm Y}}$ has a unique fixed point.

Proof. 

By Theorem 3, there exists $z_0\in \aleph$ such that ${\mathrm{{\rm Y}}}^{m_0}{z}_0=z_0$.

Then

$${\mathrm{{\rm Y}}}^{m_0}\left(\mathrm{{\rm Y}}{z}_0\right)={\mathrm{{\rm Y}}}^{m_0+1}{z}_0=\mathrm{{\rm Y}}{z}_0,$$

so $\mathrm{{\rm Y}}{z}_0$ is also a fixed point of ${\mathrm{{\rm Y}}}^{m_0}$.

• By uniqueness, $\mathrm{{\rm Y}}{z}_0=z_0$.
• If $z_1$ is another fixed point of $\mathrm{{\rm Y}}$, then it is also a fixed point of ${\mathrm{{\rm Y}}}^{m_0}$; hence $z_1=z_0$. □

To further generalize this setting, we combine the approach of large contractions with the functional control of Rakotch-type mappings [23]. This leads to a more flexible class of contractions in which the contractive behavior is governed by a family of control functions depending on distance thresholds.

Unlike the classical metric case, the generalized triangle inequality in b-metric spaces generates additional weighted terms involving the coefficient s. Therefore, stronger control conditions are required in order to compensate for the influence of this coefficient.

Let ${\Omega }_s$ denote the class of real-valued control functions satisfying

$${\Omega }_s=\left\{\hslash :(0,\infty )\to \left[0,{\displaystyle \frac{1}{s}}\right):\forall \epsilon >0,\underset{t\ge \epsilon }{sup}\hslash \left(t\right)<{\displaystyle \frac{1}{s}}\right\}.$$

The class ${\Omega }_s$ can be regarded as a b-metric analog of the control functions appearing in Rakotch-type contractions. The restriction involving $1/s$ is imposed to control the weighted estimates generated by the b-metric inequality. In particular, when $s=1$, the condition reduces to the usual Rakotch-type control in metric spaces.

Although the proof of the following theorem follows the general Picard iteration strategy used in Theorem 3, the essential novelty lies in replacing the constant threshold contraction factor by a family of Rakotch-type control functions adapted to the b-metric coefficient s.

Theorem 4.

Let $(\aleph ,\wp )$ be a complete b-metric space with constant $s\ge 1$, and let $\mathrm{{\rm Y}}:\aleph \to \aleph$ be a mapping such that

$$\wp (\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\sigma )<\wp (\rho ,\sigma )$$

for all $\rho \ne \sigma$. Assume that for every $\epsilon >0$, there exists a function ${\hslash }_{\epsilon }\in {\Omega }_s$ such that

$$\wp (\rho ,\sigma )\ge \epsilon \Rightarrow \wp (\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\sigma )\le {\hslash }_{\epsilon }(\wp (\rho ,\sigma ))\wp (\rho ,\sigma ).$$

Then $\mathrm{{\rm Y}}$ has a unique fixed point $z_0\in \aleph$. Moreover, for any ${\rho }_0\in \aleph$, the sequence

$${\rho }_n={\mathrm{{\rm Y}}}^n{\rho }_0$$

converges to $z_0$.

Proof. 

Let ${\rho }_0\in \aleph$, and define

$${\rho }_n={\mathrm{{\rm Y}}}^n{\rho }_0,n\ge 0.$$

If there exists $n_0\ge 0$ such that ${\rho }_{n_0}={\rho }_{n_0+1}$, then ${\rho }_{n_0}$ is a fixed point of $\mathrm{{\rm Y}}$.

• Assume now that ${\rho }_n\ne {\rho }_{n+1}$ for all $n\ge 0$.
• Step 1. Let

  $${\zeta }_n=\wp ({\rho }_n,{\rho }_{n+1}).$$

Since

$${\zeta }_{n+1}=\wp (\mathrm{{\rm Y}}{\rho }_n,\mathrm{{\rm Y}}{\rho }_{n+1})<\wp ({\rho }_n,{\rho }_{n+1})={\zeta }_n,$$

the sequence $\left({\zeta }_n\right)$ is decreasing. Hence there exists $\gamma \ge 0$ such that

$${\zeta }_n\to \gamma .$$

Suppose that $\gamma >0$. Then, for all sufficiently large n,

$${\zeta }_n\ge {\displaystyle \frac{\gamma }{2}}.$$

Putting

$$\epsilon ={\displaystyle \frac{\gamma }{2}}.$$

By the assumption, there exists ${\hslash }_{\epsilon }\in {\Omega }_s$ such that

$${\zeta }_{n+1}\le {\hslash }_{\epsilon }\left({\zeta }_n\right){\zeta }_n.$$

Let

$$k_{\epsilon }=\underset{t\ge \epsilon }{sup}{\hslash }_{\epsilon }\left(t\right).$$

Since ${\hslash }_{\epsilon }\in {\Omega }_s$, we have

$$k_{\epsilon }<{\displaystyle \frac{1}{s}}\le 1.$$

Therefore, for all sufficiently large n,

$${\zeta }_{n+1}\le k_{\epsilon }{\zeta }_n.$$

By iteration, we obtain ${\zeta }_n\to 0$, which contradicts $\gamma >0$. Hence

$$\underset{n\to \infty }{lim}\wp ({\rho }_n,{\rho }_{n+1})=0.$$

Step 2. We show that the orbit $\left\{{\rho }_n\right\}$ is bounded.

Set

$$a=\wp ({\rho }_0,{\rho }_1).$$

If $a=0$, then ${\rho }_0$ is a fixed point, and there is nothing to prove. Assume $a>0$.

Fix $\eta >0$, and let

$$k_{\eta }=\underset{t\ge \eta }{sup}{\hslash }_{\eta }\left(t\right)<{\displaystyle \frac{1}{s}}.$$

Put

$$\lambda =s{k}_{\eta }.$$

Then

$$0<\lambda <1.$$

Let

$$D_n=\wp ({\rho }_0,{\rho }_n).$$

Using the b-metric inequality, we obtain

$$D_{n+1}=\wp ({\rho }_0,{\rho }_{n+1})\le s\left(\wp ({\rho }_0,{\rho }_1)+\wp ({\rho }_1,{\rho }_{n+1})\right).$$

Since

$$\wp ({\rho }_1,{\rho }_{n+1})=\wp (\mathrm{{\rm Y}}{\rho }_0,\mathrm{{\rm Y}}{\rho }_n),$$

we consider two cases.

• If $D_n\ge \eta$, then

  $$\wp ({\rho }_1,{\rho }_{n+1})\le k_{\eta }{D}_n.$$

Hence

$$D_{n+1}\le sa+s{k}_{\eta }{D}_n=sa+\lambda D_n.$$

If $D_n<\eta$, then by the strict contractive condition,

$$\wp ({\rho }_1,{\rho }_{n+1})=\wp (\mathrm{{\rm Y}}{\rho }_0,\mathrm{{\rm Y}}{\rho }_n)<\wp ({\rho }_0,{\rho }_n)=D_n<\eta .$$

Thus

$$D_{n+1}\le s(a+\eta ).$$

Choose

$$B=max\left\{D_0,D_1,s(a+\eta ),{\displaystyle \frac{sa}{1-\lambda }}\right\}.$$

It follows by induction that

$$D_n\le B,\forall n\ge 0.$$

Therefore,

$$\wp ({\rho }_0,{\rho }_n)\le B,\forall n\ge 0,$$

and the orbit of ${\rho }_0$ is bounded.

• Step 3. We now show that $\mathrm{{\rm Y}}$ satisfies the large contraction condition in the sense of Burton.

Indeed, for every $\epsilon >0$, since ${\hslash }_{\epsilon }\in {\Omega }_s$, we have

$$k_{\epsilon }=\underset{t\ge \epsilon }{sup}{\hslash }_{\epsilon }\left(t\right)<{\displaystyle \frac{1}{s}}<1.$$

Hence, whenever $\wp (\rho ,\sigma )\ge \epsilon$,

$$\wp (\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\sigma )\le {\hslash }_{\epsilon }(\wp (\rho ,\sigma ))\wp (\rho ,\sigma )\le k_{\epsilon }\wp (\rho ,\sigma ).$$

Thus $\mathrm{{\rm Y}}$ is a large contraction.

Since the orbit of ${\rho }_0$ is bounded, all assumptions of Theorem 3 are satisfied. Therefore, $\mathrm{{\rm Y}}$ has a unique fixed point $z_0\in \aleph$, and the Picard sequence

$${\rho }_n={\mathrm{{\rm Y}}}^n{\rho }_0$$

converges to $z_0$. □

Remark 5.

Theorem 4 extends Burton’s large contraction principle and Rakotch-type contractions to the setting of complete b-metric spaces. In contrast to the classical metric case, the convergence analysis in the present setting requires additional control related to the coefficient s of the generalized triangle inequality.

The following result shows that our main theorem reduces to a new class of Rakotch-type large contractions in the classical metric setting, which has not been explicitly studied in this form in the literature.

Corollary 2.

Let $(\aleph ,\wp )$ be a complete metric space and $\mathrm{{\rm Y}}:\aleph \to \aleph$ be a mapping such that $\wp (\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\sigma )<\wp (\rho ,\sigma )$ for all $\rho \ne \sigma$ and for every $\epsilon >0$, there exists a function ${\hslash }_{\epsilon }\in \Omega$ such that

$$\wp (\rho ,\sigma )\ge \epsilon \Rightarrow \wp (\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\sigma )\le {\hslash }_{\epsilon }(\wp (\rho ,\sigma ))\wp (\rho ,\sigma ).$$

Then $\mathrm{{\rm Y}}$ has a unique fixed point in ℵ.

Proof. 

The result follows directly from Theorem 4 by taking $s=1$, which reduces the b-metric space to a classical metric space. □

Remark 6.

The obtained results extend classical Banach-type and Rakotch-type contraction principles in several directions. In the Banach contraction principle, the contractive condition depends on a uniform Lipschitz constant $k\in (0,1)$, while in the present work, the contractive behavior depends on the distance scale through large contraction conditions.

Moreover, Rakotch contractions involve variable contractive factors depending on the distance between points, but the current approach combines this idea with Burton-type large contractions in the setting of complete b-metric spaces. This provides greater flexibility in the study of nonlinear operators, especially in situations where global uniform contractions are not available.

In particular, the obtained results remain applicable to certain nonlinear problems where classical Banach contractions fail, as illustrated in Example 3 and in the application to fractional differential equations.

4. Application to Fractional Differential Equation

In this section, we apply the obtained fixed point results for large contractions in b-metric spaces to establish the existence and uniqueness of solutions for a fractional differential equation.

Fractional differential equations have attracted significant attention due to their ability to model memory and hereditary properties in various physical and engineering systems. In particular, Caputo-type fractional derivatives are widely used because they allow for classical boundary conditions.

Let $\aleph =C\left(\right[0,1],\mathbb{R})$, and define

$$\wp (\rho ,\sigma )={\parallel \rho -\sigma \parallel }_{\infty }^2.$$

Then $(\aleph ,\wp )$ is a complete b-metric space with coefficient $s=2$.

Consider the fractional boundary value problem

$${}^{C}D^{\alpha }\rho \left(t\right)+\hslash (t,\rho \left(t\right))=0,t\in [0,1],1<\alpha \le 2,$$

with boundary conditions

$$\rho \left(0\right)=0,\rho \left(1\right)=0,$$

where ${}^{C}D^{\alpha }$ denotes the Caputo fractional derivative.

It is well known (see, for example, [24,25]) that the above fractional boundary value problem is equivalent to the integral equation

$$\rho \left(t\right)={\int }_0^{1}G(t,s)\hslash (s,\rho \left(s\right))ds,$$

where the associated Green function is given by

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

Moreover, $G(t,s)$ is continuous and nonnegative on $[0,1]\times [0,1]$.

Assume that:

• $\hslash :[0,1]\times \mathbb{R}\to \mathbb{R}$ is continuous;
• There exists a nondecreasing function $\varphi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ such that

  $$|\hslash (t,u)-\hslash (t,v\left)\right|\le \varphi \left(\right|u-v\left|\right),\forall t\in [0,1],u,v\in \mathbb{R};$$
• $\varphi$ satisfies the large contraction condition: for every $\epsilon >0$, there exists $\delta \left(\epsilon \right)\in (0,1)$ such that

  $$r\ge \epsilon \Rightarrow \varphi \left(r\right)\le \delta \left(\epsilon \right)r;$$
• If

  $$M=\underset{t\in [0,1]}{sup}{\int }_0^{1}G(t,s)ds,$$

  then

  $$M\delta \left(\epsilon \right)<1$$

  for every $\epsilon >0$.

Accordingly, solutions of the fractional boundary value problem correspond to fixed points of the operator $\mathrm{{\rm Y}}:\aleph \to \aleph$ defined by

$$\left(\mathrm{{\rm Y}}\rho \right)\left(t\right)={\int }_0^{1}G(t,s)\hslash (s,\rho \left(s\right))ds.$$

Proof. 

Let $\rho ,\sigma \in \aleph$. Then

$$|\left(\mathrm{{\rm Y}}\rho \right)\left(t\right)-\left(\mathrm{{\rm Y}}\sigma \right)\left(t\right)|\le {\int }_0^{1}G(t,s)|\hslash (s,\rho \left(s\right))-\hslash (s,\sigma \left(s\right))|ds.$$

Using the assumption on ℏ and the monotonicity of $\varphi$, we get

$$|\hslash (s,\rho \left(s\right))-\hslash (s,\sigma \left(s\right))|\le \varphi \left(\right|\rho \left(s\right)-\sigma \left(s\right)\left|\right)\le {\varphi (\parallel \rho -\sigma \parallel }_{\infty }).$$

Hence

$$|\left(\mathrm{{\rm Y}}\rho \right)\left(t\right)-\left(\mathrm{{\rm Y}}\sigma \right)\left(t\right)|\le {\varphi (\parallel \rho -\sigma \parallel }_{\infty }){\int }_0^{1}G(t,s)ds.$$

Taking the supremum over $t\in [0,1]$, we obtain

$${\parallel \mathrm{{\rm Y}}\rho -\mathrm{{\rm Y}}\sigma \parallel }_{\infty }\le {M\varphi (\parallel \rho -\sigma \parallel }_{\infty }).$$

Let $\rho \ne \sigma$, and set

$$r={\parallel \rho -\sigma \parallel }_{\infty }>0.$$

By the large contraction condition applied with $\epsilon =r$, we have

$$\varphi \left(r\right)\le \delta \left(r\right)r.$$

Therefore,

$${\parallel \mathrm{{\rm Y}}\rho -\mathrm{{\rm Y}}\sigma \parallel }_{\infty }\le M\delta \left(r\right){\parallel \rho -\sigma \parallel }_{\infty }.$$

Since $M\delta \left(r\right)<1$, it follows that

$${\parallel \mathrm{{\rm Y}}\rho -\mathrm{{\rm Y}}\sigma \parallel }_{\infty }<{\parallel \rho -\sigma \parallel }_{\infty }.$$

Consequently,

$$\wp (\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\sigma )<\wp (\rho ,\sigma ).$$

Now let $\epsilon >0$, and suppose that

$$\wp (\rho ,\sigma )\ge \epsilon .$$

Then

$${\parallel \rho -\sigma \parallel }_{\infty }\ge \sqrt{\epsilon }.$$

Using the large contraction condition for $\varphi$, we get

$${\varphi (\parallel \rho -\sigma \parallel }_{\infty })\le \delta \left(\sqrt{\epsilon }\right){\parallel \rho -\sigma \parallel }_{\infty }.$$

Thus,

$${\parallel \mathrm{{\rm Y}}\rho -\mathrm{{\rm Y}}\sigma \parallel }_{\infty }\le M\delta \left(\sqrt{\epsilon }\right){\parallel \rho -\sigma \parallel }_{\infty }.$$

Set

$${\delta }_1\left(\epsilon \right)={\left(M\delta \left(\sqrt{\epsilon }\right)\right)}^2.$$

By assumption, ${\delta }_1\left(\epsilon \right)\in (0,1)$. Therefore,

$$\wp (\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\sigma )\le {\delta }_1\left(\epsilon \right)\wp (\rho ,\sigma ),\mathrm{whenever}\wp (\rho ,\sigma )\ge \epsilon .$$

Hence $\mathrm{{\rm Y}}$ is a large contraction on the complete b-metric space $(\aleph ,\wp )$.

By Theorem 3, $\mathrm{{\rm Y}}$ has a unique fixed point in ℵ. Therefore, the fractional boundary value problem admits a unique solution on $[0,1]$. □

5. Conclusions

In this paper, we studied fixed point results for a class of mappings known as large contractions in b-metric spaces. This class extends the classical Banach contraction principle by allowing the contractive condition to depend on the distance scale rather than requiring a uniform Lipschitz constant. We also derived theorems and corollaries using the functional control of Rakotch, showing that our results generalize classical fixed point theorems.

As an application, we studied a Caputo-type fractional boundary value problem by transforming it into an equivalent integral equation. We showed that the associated operator satisfies the large contraction condition, which guarantees the existence and uniqueness of solutions.

Finally, future work may focus on extending these results to more general spaces, such as partial b-metric spaces or generalized b-metric spaces, and to more complex fractional systems with delay or impulsive effects.