P-Type Contractive Mappings in b-Metric Spaces and an Application to a (p,q)-Difference Langevin Problem

Abstract

This work investigates fixed point results for mappings satisfying generalized P-type contractive conditions in the framework of b-metric spaces. Several existence and uniqueness theorems are established by employing appropriate iterative techniques adapted to the b-metric setting. Illustrative examples are provided to clarify the relationship between P-contractions and classical contractions. In addition, an application to a boundary value problem involving a second-order $(p,q)$-difference Langevin equation is presented to demonstrate the effectiveness of the theoretical results.

1. Introduction

Distance-based structures constitute a fundamental tool in mathematical analysis, providing a natural framework for measuring proximity between elements. The classical concept of a metric has been extended in various directions by relaxing or modifying some of its defining axioms, leading to several generalized metric-type spaces.

One of the most prominent generalizations is the notion of a b-metric space, which is obtained by modifying the triangle inequality with a control constant. This structure preserves many useful features of metric spaces while offering greater flexibility for applications. The theory of b-metric spaces has undergone substantial development in recent years, supported by survey and foundational contributions such as Berinde and Păcurar [1], where an up-to-date overview of early advances and structural aspects is given. Various extensions of the classical framework include probabilistic b-metric environments [2,3], graphical b-metric structures [4,5], and multivalued versions [6,7,8], each broadening the applicability of fixed point techniques.

Fixed point theory plays a central role in the study of nonlinear mappings defined on such spaces. The existence of invariant points for self-maps has significant implications in differential equations, integral equations, and iterative algorithms. Numerous fixed point results have been established in this setting, including generalized cyclic p-contractions [9], rational-type contractions in extended b-metric spaces [10], and several extensions of Geraghty- and Boyd–Wong-type contractions [11,12,13,14,15,16].

Several generalizations of the Banach contraction principle have also been proposed to capture a wider class of nonlinear mappings. Within this framework, we investigate P-type contractive mappings defined on b-metric spaces. The results obtained in this study provide a unified approach that extends a number of existing fixed point theorems in the literature. In particular, P-contractive and generalized P-contractive mappings have attracted considerable attention due to their unifying nature; see Altun et al. [17,18], Şahin and Demir [19], and recent developments involving extended P-type contractions [20]. Best proximity versions and coupled fixed point counterparts have also been explored in [4,21].

As an application of the theoretical findings, we examine the solvability of a boundary value problem associated with a second-order $(p,q)$-difference Langevin equation under suitable conditions. Moreover, recent works have demonstrated that b-metric structures provide an effective framework for analyzing nonlinear differential and difference equations [22,23], further motivating the study of generalized contractive conditions such as those considered in this paper.

2. Preliminaries

The concept of a b-metric space was first introduced by Czerwik in [24].

Definition 1

([24]). Let X be a nonempty set and let $s\ge 1$. A function $\rho :X\times X\to [0,\infty )$ is a b-metric if it satisfies symmetry, identity of indiscernibles, and the relaxed triangle inequality

$$\rho (\xi ,\zeta )\le s\left(\rho (\xi ,\eta )+\rho (\eta ,\zeta )\right),$$

for all $\xi ,\eta ,\zeta \in X$. The triple $(X,\rho ,s)$ is called a b-metric space.

It is worth emphasizing that the class of b-metric spaces strictly contains the class of metric spaces, since a b-metric reduces to a metric when $s=1$.

Numerous examples of b-metric spaces can be found in the literature. For completeness, we recall one illustrative example.

Example 1

([25]). Let $(X,\sigma )$ be a metric space and let $p>1$ be a real number. Define $\rho (\xi ,\eta )={\left(\sigma (\xi ,\eta )\right)}^p$. Then $(X,\rho ,2^{p-1})$ is a b-metric space. Indeed, conditions (b1) and (b2) are immediate. Moreover, since the function $f\left(t\right)=t^p$ for $t\ge 0$ is convex, we have

$${\left({\displaystyle \frac{a+b}{2}}\right)}^p\le {\displaystyle \frac{1}{2}}(a^p+b^p),$$

which implies

$${(a+b)}^p\le 2^{p-1}(a^p+b^p),a,b\ge 0.$$

Consequently, for all $\xi ,\eta ,\zeta \in X$,

$$\rho (\xi ,\eta )={\left(\sigma (\xi ,\eta )\right)}^p\le 2^{p-1}[\rho (\xi ,\zeta )+\rho (\zeta ,\eta )],$$

and thus condition (b3) holds.

Remark 1

([26]). Every b-metric ρ on a nonempty set X induces a topology ${\tau }_{\rho }$ generated by the family of open balls

$$B_{\rho }({\xi }_0,r)=\{\eta \in X:\rho ({\xi }_0,\eta )<r\},$$

where ${\xi }_0\in X$ and $r>0$. A subset $U\subset X$ is open in $(X,\rho )$ if and only if for every $\xi \in U$ there exists $r_{\xi }>0$ such that $B_{\rho }(\xi ,r_{\xi })\subset U$. The collection of all such open sets forms the topology ${\tau }_{\rho }$ on X.

Definition 2

([6]). Let $(X,\rho ,s)$ be a b-metric space and let $\left\{{\xi }_n\right\}$ be a sequence in X.

(a) 

The sequence is said to be convergent if there exists $\xi \in X$ such that $\rho ({\xi }_n,\xi )\to 0$ as $n\to \infty$.

(b) 

The sequence is said to be Cauchy if $\rho ({\xi }_n,{\xi }_m)\to 0$ as $n,m\to \infty$.

The space $(X,\rho ,s)$ is called complete if every Cauchy sequence in X is convergent.

Proposition 1

([6]). Let $(X,\rho ,s)$ be a b-metric space. The following assertions hold:

1. 

A convergent sequence in X admits a unique limit.

2. 

Convergence of a sequence in X implies that it is Cauchy.

3. 

Unlike the metric case, the function ρ is not necessarily continuous on $X\times X$.

Although a b-metric is not necessarily continuous, it satisfies the following useful property.

Lemma 1

([25]). Let $(X,\rho ,s)$ be a b-metric space and let $\left\{{\xi }_n\right\}$, $\left\{{\eta }_n\right\}$ be sequences converging to ξ and η, respectively. Then

$${\displaystyle \frac{1}{s^2}}\rho (\xi ,\eta )\le \underset{n\to \infty }{lim\; inf}\rho ({\xi }_n,{\eta }_n)\le \underset{n\to \infty }{lim\; sup}\rho ({\xi }_n,{\eta }_n)\le s^2\rho (\xi ,\eta ).$$

In particular, if $\xi =\eta$, then $\rho ({\xi }_n,{\eta }_n)\to 0$. Moreover, for any $u\in X$,

$${\displaystyle \frac{1}{s}}\rho (\xi ,u)\le \underset{n\to \infty }{lim\; inf}\rho ({\xi }_n,u)\le \underset{n\to \infty }{lim\; sup}\rho ({\xi }_n,u)\le s\rho (\xi ,u).$$

Definition 3

([22]). Let $(X,{\rho }_1,s_1)$ and $(Y,{\rho }_2,s_2)$ be two b-metric spaces and $F:X\to Y$ be a mapping. The map F is said to be continuous at ${\xi }_0\in X$ if for every $\epsilon >0$ there exists $\delta >0$ such that

$${\rho }_1(\xi ,{\xi }_0)<\delta \Rightarrow {\rho }_2(F\xi ,F{\xi }_0)<\epsilon .$$

If F is continuous at every point of X, then F is said to be continuous on X.

The notion of continuity in b-metric spaces is closely related to sequential continuity, as stated below.

Proposition 2

([22]). Let $(X,{\rho }_1,s_1)$ and $(Y,{\rho }_2,s_2)$ be b-metric spaces and let $F:X\to Y$ be a mapping. For a point $\xi \in X$, the mapping F is continuous at ξ if and only if for every sequence $\left\{{\xi }_n\right\}\subset X$ converging to ξ with respect to ${\rho }_1$, the sequence $\left\{F{\xi }_n\right\}$ converges to $F\xi$ with respect to ${\rho }_2$.

The main objective of this paper is to establish new fixed point results in b-metric spaces inspired by classical metric fixed point theory. It is well known that proofs of fixed point theorems in metric spaces generally proceed through three fundamental steps: First, an iterative sequence, typically a Picard iteration, is constructed. Second, it is shown that this sequence is Cauchy, which constitutes the main difficulty of the proof. Finally, completeness of the space guarantees the convergence of the sequence, and the limit point is shown to be a fixed point of the underlying mapping.

In a metric space $(X,\rho )$, if there exists $\lambda \in [0,1)$ such that

$$\rho ({\xi }_n,{\xi }_{n+1})\le \lambda \rho ({\xi }_{n-1},{\xi }_n),$$

(1)

for all $n\in \mathbb{N}$, then the sequence $\left\{{\xi }_n\right\}$ is Cauchy. Indeed, using inequality (1) together with the triangle inequality, we obtain for $m>n$,

$$\rho ({\xi }_n,{\xi }_m)\le \sum _{k=n}^{m-1}\rho ({\xi }_k,{\xi }_{k+1})\le {\displaystyle \frac{{\lambda }^n}{1-\lambda }}\rho ({\xi }_0,{\xi }_1),$$

which shows that $\left\{{\xi }_n\right\}$ is a Cauchy sequence. Moreover, the convergence of the series

$$\sum _{n=1}^{\infty }\rho ({\xi }_n,{\xi }_{n+1})$$

(2)

also guarantees that $\left\{{\xi }_n\right\}$ is Cauchy.

However, these arguments do not directly extend to the framework of b-metric spaces. In particular, Suzuki [27] provided an example showing that, in a b-metric space, the convergence of the series (2) does not necessarily imply that the corresponding sequence is Cauchy. This observation highlights the need for additional tools when dealing with fixed point theory in b-metric spaces.

The following results play a crucial role in overcoming these difficulties.

Lemma 2

([13]). Let $(X,\rho ,s)$ be a b-metric space and $\left\{{\xi }_n\right\}$ a sequence in X. If there exists $\gamma >{log}_{2}s$ such that the series

$$\sum _{n=1}^{\infty }{n}^{\gamma }\rho ({\xi }_n,{\xi }_{n+1})$$

is convergent, then $\left\{{\xi }_n\right\}$ is a Cauchy sequence.

Remark 2

([15]). Lemma 2 does not hold when $\gamma ={log}_{2}s$. Indeed, let $X=\mathbb{R}$, $\rho (\xi ,\eta )={(\xi -\eta )}^2$ and

$${\xi }_n=\sum _{k=2}^n{\displaystyle \frac{1}{klnk}}.$$

Then $(X,\rho ,2)$ is a b-metric space and the series

$$\sum _{n=2}^{\infty }n\rho ({\xi }_n,{\xi }_{n+1})$$

is convergent, while the sequence $\left\{{\xi }_n\right\}$ is not Cauchy.

Lemma 3

([13]). Let $(X,\rho ,s)$ be a b-metric space and $\left\{{\xi }_n\right\}$ a sequence in X. If there exists $\alpha >1$ such that the series

$$\sum _{n=1}^{\infty }{\alpha }^n\rho ({\xi }_n,{\xi }_{n+1})$$

is convergent, then $\left\{{\xi }_n\right\}$ is a Cauchy sequence.

Another important observation is that inequality (1) remains sufficient to guarantee the Cauchy property in a b-metric space, even when $\lambda <1$.

Lemma 4

([27,28]). Let $(X,\rho ,s)$ be a b-metric space and $\left\{{\xi }_n\right\}$ a sequence in X. If there exists $\alpha \in [0,1)$ such that

$$\rho ({\xi }_n,{\xi }_{n+1})\le \alpha \rho ({\xi }_{n-1},{\xi }_n)$$

for all $n\in \mathbb{N}$, then $\left\{{\xi }_n\right\}$ is a Cauchy sequence.

Motivated by these results, numerous fixed point theorems have been established in b-metric spaces. For instance, Mitrovic [14] extended the Banach contraction principle to b-metric spaces, while further generalizations involving rational and max-type contractions were obtained in [29].

More recently, the theory of contraction mappings has been enriched by the introduction of weaker contractive conditions. Fulga [11] proposed the concept of ${\phi }_E$-Geraghty contractions, and Altun [17] introduced P-contractive mappings in metric spaces, showing that every contraction is P-contractive, whereas the converse does not hold. Subsequently, Suzuki-type P-contractive mappings were investigated in [18], and fixed point results for P-contractive mappings in M-metric spaces were obtained in [30].

In this paper, we extend the concepts of P-contraction and P-contractive mappings to the setting of b-metric spaces. We provide illustrative examples to clarify the relationship between these notions and classical contractions. Furthermore, we establish several fixed point theorems that generalize and unify the existing results in the literature. Finally, in order to demonstrate the applicability of the obtained theoretical results, we investigate the existence and uniqueness of solutions for a second-order $(p,q)$-difference Langevin equation. For further developments on enriched P-contractions, we refer the reader to [19].

3. Main Results

Definition 4.

Let $(X,\rho ,s)$ be a b-metric space and let $F:X\to X$ be a self-mapping. The mapping F is said to satisfy a P-contractive condition if there exists a constant $L\in [0,1)$ such that

$$\rho (F\xi ,F\eta )\le L\left(\rho (\xi ,\eta )+\left|\rho (\xi ,F\xi )-\rho (\eta ,F\eta )\right|\right),$$

(3)

for all $\xi ,\eta \in X$. In this case, F is referred to as a P-contraction. Moreover, if

$$\rho (F\xi ,F\eta )<\rho (\xi ,\eta )+\left|\rho (\xi ,F\xi )-\rho (\eta ,F\eta )\right|$$

holds for all distinct $\xi ,\eta \in X$, then F is calledP-contractive.

It is immediate that every contraction mapping on a b-metric space is also a P-contraction. However, the converse implication does not hold in general, as demonstrated by the following example:

Example 2.

Let $X=\mathbb{N}\cup \left\{0\right\}$ and define

$$\rho (\xi ,\eta )={(\xi -\eta )}^2.$$

Then $(X,\rho ,2)$ is a b-metric space. Define a mapping $F:X\to X$ by

$$F\xi =\left\{\begin{array}{cc}1,\hfill & \xi \in \{0,1\},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

The mapping F is not a contraction, since

$$\rho \left(F\right(1),F(2\left)\right)=\rho (1,0)=1=\rho (1,2).$$

However, F satisfies the P-contractive condition with $L={\displaystyle \frac{1}{5}}$. To verify this, let $\xi ,\eta \in X$ be arbitrary. If $F\xi =F\eta$, then $\rho (F\xi ,F\eta )=0$ and the inequality holds trivially. Otherwise, $F\xi \ne F\eta$, which implies that exactly one of ξ and η belongs to $\{0,1\}$. Since the P-contractive inequality is symmetric in the variables ξ and η, it is enough to consider the cases where one of them belongs to $\{0,1\}$ while the other satisfies $\eta \ge 2$. Without loss of generality, assume that $\xi \in \{0,1\}$ and $\eta \ge 2$. Then

$$\begin{array}{cc}\hfill \rho (\xi ,\eta )+\left|\rho \right(\xi ,F\xi )-\rho (\eta ,F\eta \left)\right|& ={(\xi -\eta )}^2+|{(\xi -1)}^2-{\eta }^2|\hfill \\ \hfill & ={(\xi -\eta )}^2+{\eta }^2-{(\xi -1)}^2\hfill \\ \hfill & \ge {\eta }^2+{(\eta -1)}^2-1\hfill \\ \hfill & \ge 5.\hfill \end{array}$$

Therefore,

$$\rho (F\xi ,F\eta )=1\le {\displaystyle \frac{1}{5}}\left[\rho (\xi ,\eta )+\left|\rho \right(\xi ,F\xi )-\rho (\eta ,F\eta \left)\right|\right],$$

which shows that F is a P-contraction.

The next example illustrates that a P-contractive mapping need not be a P-contraction in a b-metric space.

Example 3.

Let $X=C[0,1]$ be the space of all real-valued continuous functions on $[0,1]$, and define

$$\rho (u,v)=\underset{t\in [0,1]}{sup}{|u\left(t\right)-v\left(t\right)|}^2,$$

for all $u,v\in X$. Then $(X,\rho ,2)$ is a b-metric space. Consider the subset

$$X^{*}=\{u\in X:0=u\left(0\right)\le u\left(t\right)\le u\left(1\right)=1\},$$

and define a mapping $F:X^{*}\to X^{*}$ by $Fu\left(t\right)=\sqrt{t}u\left(t\right)$ for $t\in [0,1]$.

Define sequences $u_n\left(t\right)=t^n$ and $v_n\left(t\right)=t^{n/2}$ for $t\in [0,1]$. A direct computation shows that

$$\rho (u_n,v_n)=\underset{t\in [0,1]}{sup}{|t^n-t^{n/2}|}^2\to {\displaystyle \frac{1}{16}},$$

and

$$\rho (F{u}_n,F{v}_n)=\underset{t\in [0,1]}{sup}{|t^{n+\frac{1}{2}}-t^{\frac{n+1}{2}}|}^2\to {\displaystyle \frac{1}{16}}.$$

Moreover,

$$|\rho (u_n,F{u}_n)-\rho (v_n,F{v}_n)|\to 0.$$

Consequently,

$${\displaystyle \frac{\rho (F{u}_n,F{v}_n)}{\rho (u_n,v_n)+|\rho (u_n,F{u}_n)-\rho (v_n,F{v}_n)|}}\to 1,$$

which shows that F is not P-contraction. On the other hand, for all $u,v\in X^{*}$,

$$\begin{array}{ccc}\hfill \rho (Fu,Fv)& =& \underset{t\in [0,1]}{sup}{|\sqrt{t}u\left(t\right)-\sqrt{t}v\left(t\right)|}^2\hfill \\ & <& \underset{t\in [0,1]}{sup}{|u\left(t\right)-v\left(t\right)|}^2\hfill \\ & =& \rho (u,v)\hfill \\ & \le & \rho (u,v)+\left|\rho \right(u,Fu)-\rho (v,Fv\left)\right|.\hfill \end{array}$$

Therefore, F is P-contractive but not a P-contraction.

Now we are ready to prove our main theoretical result.

Theorem 1.

Let $(X,\rho ,s)$ be a complete b-metric space and let $F:X\to X$ be a P-contraction. Then F admits a unique fixed point, provided that F is continuous or $sL<1$.

Proof. 

Choose an initial point ${\xi }_0\in X$ and construct a sequence $\left\{{\xi }_n\right\}$ recursively by ${\xi }_n=F{\xi }_{n-1}$ for all $n\ge 1$. If there exists an index $n_0\in \mathbb{N}$ such that $\rho ({\xi }_{n_0+1},{\xi }_{n_0})=0$, then it follows immediately that ${\xi }_{n_0}$ is a fixed point of F. Consequently, in what follows we restrict our attention to the case where $\rho ({\xi }_{n+1},{\xi }_n)>0$ for every $n\in \mathbb{N}$.

Using the P-contraction condition, we obtain

$$\begin{array}{ccc}\hfill \rho ({\xi }_{n+1},{\xi }_n)& =& \rho (F{\xi }_n,F{\xi }_{n-1})\hfill \\ & \le & L\left[\rho ({\xi }_n,{\xi }_{n-1})+|\rho ({\xi }_n,F{\xi }_n)-\rho ({\xi }_{n-1},F{\xi }_{n-1})|\right]\hfill \\ & =& L\left[\rho ({\xi }_n,{\xi }_{n-1})+|\rho ({\xi }_n,{\xi }_{n+1})-\rho ({\xi }_{n-1},{\xi }_n)|\right].\hfill \end{array}$$

(4)

Suppose that there exists $k\in \mathbb{N}$ such that $\rho ({\xi }_k,{\xi }_{k+1})\ge \rho ({\xi }_{k-1},{\xi }_k)$. Then, by (4), we obtain

$$\rho ({\xi }_k,{\xi }_{k+1})\le L\rho ({\xi }_k,{\xi }_{k+1}),$$

which is impossible since $L<1$ and $\rho ({\xi }_k,{\xi }_{k+1})>0$. Hence,

$$\rho ({\xi }_n,{\xi }_{n+1})<\rho ({\xi }_{n-1},{\xi }_n)\mathrm{for}\mathrm{all}n\in \mathbb{N}.$$

Using this fact in (4), we conclude that

$$\rho ({\xi }_n,{\xi }_{n+1})\le \gamma \rho ({\xi }_{n-1},{\xi }_n),$$

for all $n\in \mathbb{N}$, where

$$\gamma ={\displaystyle \frac{2L}{1+L}}\in [0,1).$$

By Lemma 4, the sequence $\left\{{\xi }_n\right\}$ is Cauchy in $(X,\rho )$. Since the space is complete, there exists $u\in X$ such that

$$\underset{n\to \infty }{lim}\rho ({\xi }_n,u)=0.$$

Assume first that F is continuous. Then

$$\underset{n\to \infty }{lim}\rho (F{\xi }_n,Fu)=\underset{n\to \infty }{lim}\rho ({\xi }_{n+1},Fu)=0,$$

which implies $Fu=u$ by the uniqueness of limits in a b-metric space.

Now assume that $sL<1$. Then

$$\begin{array}{ccc}\hfill \rho (u,Fu)& \le & s\left[\rho (u,{\xi }_n)+\rho ({\xi }_n,Fu)\right]\hfill \\ & =& s\left[\rho ({\xi }_n,u)+\rho (F{\xi }_{n-1},Fu)\right]\hfill \\ & \le & s\left[\rho ({\xi }_n,u)+L\left(\rho ({\xi }_{n-1},u)+|\rho ({\xi }_{n-1},{\xi }_n)-\rho (u,Fu)|\right)\right].\hfill \end{array}$$

Letting $n\to \infty$ yields

$$\rho (u,Fu)\le sL\rho (u,Fu),$$

which implies $\rho (u,Fu)=0$ since $sL<1$. Thus, u is a fixed point of F.

The uniqueness of the fixed point follows directly from (3). □

We can also obtain the following results for P-contractive mappings:

Theorem 2.

Let $(X,\rho ,s)$ be a b-metric space. Suppose that F: $X\to X$ is a P-contractive self-mapping and define f: $X\to \mathbb{R}$ by $f\left(\xi \right)=\rho (\xi ,F\xi )$. If there exists ${\xi }_0\in X$ for which $f\left({\xi }_0\right)\le f\left(F{\xi }_0\right)$ holds, then F possesses exactly one fixed point.

Proof. 

Let ${\xi }_0\in X$ be as assumed. If $\rho ({\xi }_0,F{\xi }_0)>0$, then by the P-contractivity of F, we obtain

$$\begin{array}{ccc}\hfill f\left(F{\xi }_0\right)& =& \rho (F{\xi }_0,FF{\xi }_0)\hfill \\ & <& \rho ({\xi }_0,F{\xi }_0)+|\rho ({\xi }_0,F{\xi }_0)-\rho (F{\xi }_0,FF{\xi }_0)|\hfill \\ & =& f\left({\xi }_0\right)+|f\left({\xi }_0\right)-f\left(F{\xi }_0\right)|\hfill \\ & =& f\left(F{\xi }_0\right),\hfill \end{array}$$

which is a contradiction. Hence, $\rho ({\xi }_0,F{\xi }_0)=0$, and thus ${\xi }_0$ is a fixed point of F. The uniqueness follows immediately from the P-contractivity of F. □

Remark 3.

If the function $f\left(\xi \right)=\rho (\xi ,F\xi )$ attains its minimum on X, then there exists ${\xi }_0\in X$ such that $f\left({\xi }_0\right)=inff\left(X\right)$. In this case, $f\left({\xi }_0\right)\le f\left(F{\xi }_0\right)$, and therefore ${\xi }_0$ is the unique fixed point of F by Theorem 2.

Before presenting the next result, we note that if the assumptions in Theorem 2 guarantee the existence of a minimum point of the functional $f\left(\xi \right)=\rho (\xi ,F\xi )$, then, together with Remark 3, these conditions ensure the existence of a fixed point of F. Consequently, the following theorem can be obtained directly from Theorem 2.

Theorem 3.

Suppose that $(X,\rho ,s)$ is a compact b-metric space and that F: $X\to X$ satisfies a P-contractive condition. If the mapping $f\left(\xi \right)=\rho (\xi ,F\xi )$ is lower semicontinuous on X, then there exists a unique point ${\xi }^{*}\in X$ such that $F{\xi }^{*}={\xi }^{*}$.

Proof. 

Since X is compact and f: $X\to \mathbb{R}$ is lower semicontinuous, the function f attains its minimum on X. The conclusion then follows directly from Theorem 2 and Remark 3. □

4. Application

This section is devoted to an application of Theorem 1 to a boundary value problem involving a second-order $(p,q)$-difference Langevin equation. More precisely, we study the existence and uniqueness of solutions of the problem

$$\left\{\begin{array}{c}{D}_{p,q}\left(D_{p,q}+\gamma \right)\xi \left(t\right)=f(t,\xi \left(t\right)),t\in (0,1],\hfill \\ \xi \left(0\right)=\alpha ,D_{p,q}\xi \left(0\right)=\beta ,\hfill \end{array}\right.$$

(5)

where the function f: $[0,1]\times \mathbb{R}\to \mathbb{R}$ is continuous; $0<q<p\le 1$; and $\gamma ,\alpha ,\beta$ are given real constants.

We recall some basic notions of $(p,q)$-calculus (see [23]). The $(p,q)$-derivative of a function g is defined by

$$D_{p,q}g\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{g\left(pt\right)-g\left(qt\right)}{(p-q)t}},\hfill & t\ne 0,\hfill \\ \underset{t\to 0}{lim}{D}_{p,q}g\left(t\right),\hfill & t=0,\hfill \end{array}\right.$$

and the $(p,q)$-integral is given by

$${\int }_0^{t}g\left(s\right)d_{p,q}s=(p-q)t\sum _{n=0}^{\infty }{\displaystyle \frac{q^n}{p^{n+1}}}g\left({\displaystyle \frac{q^n}{p^{n+1}}}t\right),$$

provided that the series converges.

The following relations hold:

$$\begin{array}{cc}\hfill {\int }_a^{b}g\left(pt\right)D_{p,q}h\left(t\right)d_{p,q}t& =g\left(t\right)h\left(t\right){|}_a^b-{\int }_a^{b}h\left(qt\right)D_{p,q}g\left(t\right)d_{p,q}t,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill D_{p,q}\left({\int }_0^{t}g\left(s\right)d_{p,q}s\right)& =g\left(t\right),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\int }_0^t{D}_{p,q}g\left(s\right)d_{p,q}s& =g\left(t\right)-g\left(0\right).\hfill \end{array}$$

(8)

Assume that $f(t,\xi )=0$ for all $t\in [p,1]$. Then problem (5) is equivalent to the integral equation

$$\xi \left(t\right)=\alpha +(\beta +\gamma \alpha )t-\gamma {\int }_0^t\xi \left(s\right)d_{p,q}s+{\int }_0^{t/p}(t-pqs)f(s,\xi \left(s\right))d_{p,q}s.$$

(9)

Let $X=C[0,1]$ be the space of real-valued continuous functions on $[0,1]$, endowed with the b-metric

$$\rho (u,v)=\underset{t\in [0,1]}{sup}{|u\left(t\right)-v\left(t\right)|}^2.$$

Then $(X,\rho ,2)$ is a complete b-metric space. Define the operator F: $X\to X$ by

$$\left(Fu\right)\left(t\right)=\alpha +(\beta +\gamma \alpha )t-\gamma {\int }_0^{t}u\left(s\right)d_{p,q}s+{\int }_0^{t/p}(t-pqs)f(s,u\left(s\right))d_{p,q}s.$$

Clearly, fixed points of F correspond to solutions of (5).

We impose the following assumptions:

(A1)

f: $[0,1]\times [0,\infty )\to [0,\infty )$ is continuous and $f(t,\xi )=0$ for all $t\in [p,1]$.

(A2)

$\alpha (1+\gamma )+\beta =0$.

(A3)

there exists $L\ge 0$ such that $\left|f\right(t,\xi )-f(t,\eta \left)\right|\le L|\xi -\eta |$ for all $\xi ,\eta \in [0,\infty )$.

Lemma 5.

Assume that conditions (A1)–(A3) hold. Then the operator F: $C[0,1]\to C[0,1]$, defined by

$$\left(Fu\right)\left(t\right)=\alpha +(\beta +\gamma \alpha )t-\gamma {\int }_0^{t}u\left(s\right)d_{p,q}s+{\int }_0^{{\displaystyle \frac{t}{p}}}(t-pqs)f(s,u\left(s\right))d_{p,q}s,$$

is well-defined and continuous on b-metric space $\left(C\right[0,1],\rho ,2)$.

Proof. 

Let $u\in C[0,1]$ be arbitrary. Since u is continuous on $[0,1]$ and f is continuous by (A1), the function $f(·,u(·\left)\right)$ is continuous on $[0,1]$. Moreover, by the properties of the $(p,q)$-integral, the mappings

$$t\mapsto {\int }_0^{t}u\left(s\right)d_{p,q}s\mathrm{and}t\mapsto {\int }_0^{{\displaystyle \frac{t}{p}}}(t-pqs)f(s,u\left(s\right))d_{p,q}s$$

are continuous on $[0,1]$.

Hence, $\left(Fu\right)\left(t\right)$ is a sum of continuous functions, which implies that $Fu\in C[0,1]$. Therefore, the operator F is well-defined from $C[0,1]$ into itself.

Next, let $\left\{u_n\right\}\subset C[0,1]$ be a sequence such that $u_n\to u$ in $C[0,1]$, that is,

$$\rho (u_n,u)\to 0\mathrm{as}n\to \infty .$$

Using assumption (A3), there exists $L>0$ such that

$$|f(t,u_n\left(t\right))-f(t,u\left(t\right))|\le L|u_n\left(t\right)-u\left(t\right)|,t\in [0,1].$$

Then, for each $t\in [0,1]$, we have

$$\begin{array}{ccc}\hfill |\left(F{u}_n\right)\left(t\right)-\left(Fu\right)\left(t\right)|& \le & \left|\gamma \right|{\int }_0^t|u_n\left(s\right)-u\left(s\right)|d_{p,q}s\hfill \\ & & +{\int }_0^{{\displaystyle \frac{t}{p}}}(t-pqs)|f(s,u_n\left(s\right))-f(s,u\left(s\right))|d_{p,q}s\hfill \\ & \le & \left|\gamma \right|\sqrt{\rho (u_n,u)}{\int }_0^t{d}_{p,q}s\hfill \\ & & +L\sqrt{\rho (u_n,u)}{\int }_0^{{\displaystyle \frac{t}{p}}}(t-pqs)d_{p,q}s.\hfill \end{array}$$

Since

$${\int }_0^t{d}_{p,q}s\le 1\mathrm{and}{\int }_0^{\frac{t}{p}}(t-pqs)d_{p,q}s\le {\displaystyle \frac{p+q-q{p}^2}{p^2(p+q)}},$$

it follows from the last inequality and the above bounds that

$$|\left(F{u}_n\right)\left(t\right)-\left(Fu\right)\left(t\right)|\le \left(\left|\gamma \right|+L{\displaystyle \frac{p+q-q{p}^2}{p^2(p+q)}}\right)\sqrt{\rho (u_n,u)}.$$

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

$$\rho (F{u}_n,Fu)\le {\left(\left|\gamma \right|+L{\displaystyle \frac{p+q-q{p}^2}{p^2(p+q)}}\right)}^2\rho (u_n,u).$$

Thus, $F{u}_n\to Fu$ as $n\to \infty$, which shows that the operator F is continuous on $\left(C\right[0,1],\rho ,2)$. □

Theorem 4.

Assume that $\left(A1\right)$–$\left(A3\right)$ hold. If

$$\left|\gamma \right|+L{\displaystyle \frac{p+q-q{p}^2}{p^2(p+q)}}<1,$$

then the $(p,q)$-difference Langevin Equation (5) admits a unique solution in $C[0,1]$.

Proof. 

Let $u,v\in X$. Using $\left(A3\right)$ and properties of the $(p,q)$-integral, we obtain

$$\begin{array}{cc}\hfill \left|Fu\right(t)-Fv(t\left)\right|& \le \left|\gamma \right|{\int }_0^t|u\left(s\right)-v\left(s\right)|d_{p,q}s\hfill \\ \hfill & +L{\int }_0^{t/p}(t-pqs)|u\left(s\right)-v\left(s\right)|d_{p,q}s.\hfill \end{array}$$

Taking supremum and squaring yields

$$\rho (Fu,Fv)\le {\left(\left|\gamma \right|+L{\displaystyle \frac{p+q-q{p}^2}{p^2(p+q)}}\right)}^2\rho (u,v).$$

Let

$$\lambda :=\left|\gamma \right|+L{\displaystyle \frac{p+q-q{p}^2}{p^2(p+q)}}<1.$$

Then

$$\rho (Fu,Fv)\le \lambda \left[\rho (u,v)+\left|\rho \right(u,Fu)-\rho (v,Fv\left)\right|\right],$$

which shows that F is a P-contraction on $(X,\rho ,2)$. By Lemma 5 and Theorem 1, F has a unique fixed point in X. Hence, problem (5) has a unique solution. □

We illustrate the applicability of the obtained results with a concrete example.

Example 4.

Let

$$p={\displaystyle \frac{3}{4}},q={\displaystyle \frac{1}{2}},\gamma ={\displaystyle \frac{1}{6}},\alpha =0,\beta =0,$$

and define

$$f(t,\xi )=t^2\xi ,(t,\xi )\in [0,1]\times [0,\infty ).$$

Clearly, f is continuous and satisfies assumption $\left(A1\right)$. Moreover,

$$|f(t,\xi )-f(t,\eta )|=t^2|\xi -\eta |\le |\xi -\eta |,$$

so that $L=1$. Now, we compute

$$\left|\gamma \right|+L{\displaystyle \frac{p+q-q{p}^2}{p^2(p+q)}}={\displaystyle \frac{1}{6}}+{\displaystyle \frac{\frac{3}{4}+\frac{1}{2}-\frac{1}{2}·\frac{9}{16}}{\frac{9}{16}·\frac{5}{4}}}={\displaystyle \frac{1}{6}}+{\displaystyle \frac{31}{45}}<1.$$

Hence, all the assumptions of the main theorem in the Application Section are satisfied. Therefore, the $(p,q)$-difference Langevin Equation (5) admits a unique solution in $C[0,1]$.

Corollary 1.

Let all the assumptions of the main application theorem hold. If $p\to 1$, then the $(p,q)$-difference Langevin Equation (5) reduces to the q-difference Langevin equation

$$D_q(D_q+\gamma )\xi \left(t\right)=f(t,\xi \left(t\right)),$$

with the same boundary conditions. Moreover, if

$$\left|\gamma \right|+{\displaystyle \frac{L}{1+q}}<1,$$

then the q-difference Langevin equation admits a unique solution in $C[0,1]$.

Proof. 

Letting $p\to 1$ in the $(p,q)$-integral and $(p,q)$-derivative, we recover the classical q-calculus operators. The contraction estimate obtained in the Application section passes to the limit, and the conclusion follows directly from Theorem 1. □

5. Conclusions

In this paper, we introduced a new class of contractive mappings, called P-contractions, in the setting of b-metric spaces. This class properly extends the classical Banach contraction principle and allows the treatment of nonlinear mappings that are not contractions in the usual sense. Several examples were provided to illustrate the differences between contractions, P-contractions, and P-contractive mappings in b-metric spaces.

We established existence and uniqueness results for fixed points of P-contraction mappings under mild conditions. In particular, it was shown that a P-contraction admits a unique fixed point in a complete b-metric space whenever the mapping is continuous or the condition $sL<1$ is satisfied. Furthermore, for P-contractive mappings, fixed point results were obtained via minimization of the functional $f\left(\xi \right)=\rho (\xi ,F\xi )$ without requiring completeness of the space. These results generalize and unify several existing fixed point theorems in the literature.

As an application, the developed theory was applied to a second-order $(p,q)$-difference Langevin equation with boundary conditions. By transforming the problem into an equivalent integral equation, the existence and uniqueness of solutions were obtained using the introduced P-contraction framework. A numerical example was provided to demonstrate the effectiveness of the theoretical results, and the limiting case of the q-difference Langevin equation was also discussed as a corollary.

The results presented in this work indicate that the concept of P-contractions is a powerful and flexible tool for studying nonlinear problems arising in discrete fractional calculus and difference equations. Future research directions may include the investigation of multivalued P-contractions, extensions to partial b-metric spaces, and applications to more general classes of fractional and stochastic differential equations.