On Ćirić-Type Fixed Point Results on Interpolative b-Metric Spaces with Application to Volterra Integral Equations

Abstract

This paper introduces a new class of generalized metric structures, called interpolative b-metric spaces, which unify and extend both b-metric spaces and interpolative metric spaces in a non-trivial way. By incorporating a nonlinear correction term alongside a multiplicative scaling parameter into the triangle inequality, this framework enables broader contractive conditions and refined control of convergence behavior. We develop the foundational theory of interpolative b-metric spaces and establish a generalized Ćirić-type fixed point theorem, along with Banach, Kannan, and Bianchini-type results as corollaries. To highlight the originality and applicability of our approach, we apply the main theorem to a nonlinear Volterra-type integral equation, demonstrating that interpolative b-metrics effectively accommodate nonlinear solution structures beyond the scope of traditional metric models. This work offers a unified platform for fixed point analysis and opens new directions in nonlinear and functional analysis.

1. Introduction

The study of fixed point theory in generalized metric spaces has been an active area of research since the pioneering work of Banach, Caristi, and Edelstein. Over time, various extensions of metric spaces have emerged to broaden the scope of fixed point theorems beyond classical settings. Among these, b-metric spaces introduced by Bakhtin and Czerwik, partial metric spaces by Matthews, G-metric spaces by Mustafa and Sims, and cone metric spaces by Huang and Zhang have offered fertile ground for theoretical advancements [1,2,3,4,5,6].

In their famous book, Petrusel and Rus [7] presented some outstanding developments of the metric geometry in the literature. Among the several metric spaces, the concepts of b-metric spaces [2] and interpolative metric spaces [8,9] have attracted substantial attention. We also refer to the interesting works of Younis and $\mathrm{Ö}\mathrm{z}\mathrm{t}\ddot{\mathrm{r}}\mathrm{k}$ [10], Zainul Abidin et al. [11,12] and Debnath [13,14].

Karapınar introduced the concept of $(\alpha ,c)$-interpolative metric spaces, offering a flexible modification of the triangle inequality via a nonlinear correction term [15]. These spaces generalize standard metric and cone metric spaces by incorporating nonlinear interpolative bounds, thus unifying and extending a wide class of fixed point results.

However, a limitation of interpolative metric spaces lies in their restriction to the standard triangle inequality. In contrast, b-metric spaces allow a multiplicative distortion factor $s\ge 1$, accommodating a broader range of distance functions, particularly those arising in functional analysis, fractal geometry, and computer science applications [16,17,18].

To bridge these frameworks, we introduce the notion of interpolative b-metric spaces, which combine the multiplicative distortion of b-metrics with the nonlinear interpolation of Karapınar-type metrics. Specifically, an $(\alpha ,c,s)$-interpolative b-metric space permits both a scaling parameter $s\ge 1$ and an interpolative term $c{\left[\mathcal{M}(x,z)\mathcal{M}(z,y)\right]}^{\alpha }$, generalizing both the interpolative metric and b-metric spaces. This structure allows for more refined analysis of convergence and contraction behavior.

In this paper, we achieve the following:

• Define the structure of interpolative b-metric spaces and provide illustrative examples showing their proper generalization over existing frameworks.
• Prove a Ćirić-type fixed point theorem in this new setting, extending the result of Karapınar [15].
• Derive analogues of classical fixed point results such as those of Bianchini [19], Kannan [20], and Banach.
• Present examples demonstrating that the new space is strictly more general than interpolative metric spaces and b-metric spaces.

2. Preliminaries

In this section, we provide key definitions and foundational results that form the basis for our study of interpolative b-metric spaces and their fixed point properties.

We begin with the classical concept of a metric space, the foundation of distance-based analysis in fixed point theory.

Definition 1 

(Metric Space). A function $\mathcal{M}:\mathcal{E}\times \mathcal{E}\to [0,\infty )$ is called a metric on a nonempty set $\mathcal{E}$ if for all $\mathfrak{p},\mathfrak{q},z\in \mathcal{E}$, the following conditions hold:

1. 

$\mathcal{M}(\mathfrak{p},\mathfrak{q})=0\iff \mathfrak{p}=\mathfrak{q}$,

2. 

$\mathcal{M}(\mathfrak{p},\mathfrak{q})=\mathcal{M}(\mathfrak{q},\mathfrak{p})$,

3. 

$\mathcal{M}(\mathfrak{p},\mathfrak{q})\le \mathcal{M}(\mathfrak{p},\mathfrak{r})+\mathcal{M}(\mathfrak{r},\mathfrak{q})$ (triangle inequality).

The following structure generalizes metrics by allowing a relaxation of the triangle inequality via a multiplicative constant.

Definition 2 

(b-Metric Space [2]). Let $s\ge 1$. A function $\mathcal{M}:\mathcal{E}\times \mathcal{E}\to [0,\infty )$ is called a b-metric on $\mathcal{E}$ if the following are met:

1. 

$\mathcal{M}(\mathfrak{p},\mathfrak{q})=0\iff \mathfrak{p}=\mathfrak{q}$,

2. 

$\mathcal{M}(\mathfrak{p},\mathfrak{q})=\mathcal{M}(\mathfrak{q},\mathfrak{p})$,

3. 

$\mathcal{M}(\mathfrak{p},\mathfrak{q})\le s\left[\mathcal{M}\right(\mathfrak{p},\mathfrak{r})+\mathcal{M}(\mathfrak{r},\mathfrak{q}\left)\right]\mathit{for}\mathit{all}\mathfrak{p},\mathfrak{q},\mathfrak{r}\in \mathcal{E}$.

The pair $(\mathcal{E},\mathcal{M})$ is then called a b-metric space.

Example 1. 

Let $\mathcal{E}=\mathbb{R}$ and define $\mathcal{M}:\mathcal{E}\times \mathcal{E}\to [0,\infty )$ by the following:

$$\mathcal{M}(\mathfrak{p},\mathfrak{q})={|\mathfrak{p}-\mathfrak{q}|}^2,\mathit{for}\mathit{all}\mathfrak{p},\mathfrak{q}\in E.$$

Then $(\mathcal{E},\mathcal{M})$ is a b-metric space with $s=2$. Indeed, $\mathcal{M}$ satisfies the identity of indiscernibles and symmetry, and for all $\mathfrak{p},\mathfrak{q},\mathfrak{r}\in \mathcal{E}$:

$$\mathcal{M}(\mathfrak{p},\mathfrak{q})={|\mathfrak{p}-\mathfrak{q}|}^2\le 2\left({|\mathfrak{p}-\mathfrak{r}|}^2+{|\mathfrak{r}-\mathfrak{q}|}^2\right)=s[\mathcal{M}(\mathfrak{p},\mathfrak{r})+\mathcal{M}(\mathfrak{r},\mathfrak{q})].$$

Thus, all conditions of Definition 2 are satisfied. Note that $\mathcal{M}$ is not a metric, since it fails the standard triangle inequality when $s=1$.

To enrich the classical metric structure, Karapınar introduced interpolative metrics that incorporate a nonlinear correction term into the triangle inequality.

Definition 3 

(($\alpha ,c$)-Interpolative Metric Space [15]). Let $\mathcal{E}$ be a nonempty set. A function $\mathcal{M}:\mathcal{E}\times \mathcal{E}\to [0,\infty )$ is said to be an $(\alpha ,c)$-interpolative metric if the following are met:

1. 

$\mathcal{M}(\mathfrak{p},\mathfrak{q})=0\iff \mathfrak{p}=\mathfrak{q}$,

2. 

$\mathcal{M}(\mathfrak{p},\mathfrak{q})=\mathcal{M}(\mathfrak{q},\mathfrak{p})$,

3. 

$\mathcal{M}(\mathfrak{p},\mathfrak{q})\le \mathcal{M}(\mathfrak{p},\mathfrak{r})+\mathcal{M}(\mathfrak{r},\mathfrak{q})+c·[\mathcal{M}{(\mathfrak{p},\mathfrak{r})}^{\alpha }·\mathcal{M}{(\mathfrak{r},\mathfrak{q})}^{1-\alpha }]$,

For all $\mathfrak{p},\mathfrak{q},\mathfrak{r}\in \mathcal{E}$, with fixed constants $\alpha \in (0,1)$ and $c\ge 0$.

Remark 1. 

If $c=0$, then the interpolative inequality reduces to the classical triangle inequality, hence every metric space is trivially an interpolative metric space.

Example 2. 

Let $\mathcal{E}=\mathbb{R}$ and define $\mathcal{M}:\mathcal{E}\times \mathcal{E}\to [0,\infty )$ by the following:

$$\mathcal{M}(\mathfrak{p},\mathfrak{q})={|\mathfrak{p}-\mathfrak{q}|+|\mathfrak{p}-\mathfrak{q}|}^2,\mathit{for}\mathit{all}\mathfrak{p},\mathfrak{q}\in \mathcal{E}.$$

Take $\alpha ={\displaystyle \frac{1}{2}}$ and $c=1$. It is immediate that $\mathcal{M}$ satisfies (1) and (2) of Definition 3. For condition (3), let

$$a=|\mathfrak{p}-\mathfrak{r}|,b=|\mathfrak{r}-\mathfrak{q}|,$$

so that

$$\mathcal{M}(\mathfrak{p},\mathfrak{q})=(a+b)+{(a+b)}^2=a+b+a^2+2ab+b^2.$$

On the other hand, the following is calculated:

$$\mathcal{M}(\mathfrak{p},\mathfrak{r})+\mathcal{M}(\mathfrak{r},\mathfrak{q})=a+a^2+b+b^2,$$

and the extra term $2ab$ can be bounded by

$$\mathcal{M}{(\mathfrak{p},\mathfrak{r})}^{\alpha }\mathcal{M}{(\mathfrak{r},\mathfrak{q})}^{1-\alpha }={\left(a+a^2\right)}^{1/2}{\left(b+b^2\right)}^{1/2}.$$

Hence, we obtain the following:

$$\mathcal{M}(\mathfrak{p},\mathfrak{q})\le \mathcal{M}(\mathfrak{p},\mathfrak{r})+\mathcal{M}(\mathfrak{r},\mathfrak{q})+c\mathcal{M}{(\mathfrak{p},\mathfrak{r})}^{\alpha }\mathcal{M}{(\mathfrak{r},\mathfrak{q})}^{1-\alpha },$$

showing that $(\mathcal{E},\mathcal{M})$ is indeed an $(\alpha ,c)$-interpolative metric space.

The following definitions clarify the notions of convergence and completeness in these generalized metric settings.

Definition 4 

(Convergence and Completeness [15]). Let $(\mathcal{E},\mathcal{M})$ be an $(\alpha ,c)$-interpolative metric space.

• A sequence $\left\{{\mathfrak{p}}_n\right\}\subset \mathcal{E}$ is said to converge to $\mathfrak{p}\in \mathcal{E}$ if $\mathcal{M}({\mathfrak{p}}_n,\mathfrak{p})\to 0$ as $n\to \infty$.
• $\left\{{\mathfrak{p}}_n\right\}$ is called a Cauchy sequence if ${lim}_{n\to \infty }{sup}_{m>n}\mathcal{M}({\mathfrak{p}}_n,{\mathfrak{p}}_m)=0$.
• The space $(\mathcal{E},\mathcal{M})$ is called complete if every Cauchy sequence converges in $\mathcal{E}$.

3. Interpolative b-Metric Spaces

Here we introduce and define an Interpolative b-Metric Space. This definition allows the simultaneous presence of multiplicative and nonlinear perturbations in the triangle inequality. As a result, interpolative b-metric spaces can model a broader range of analytical and geometric behaviors and serve as a fertile ground for establishing generalized fixed point theorems.

Definition 5. 

Let $\mathcal{E}$ be a non-empty set and let $\mathcal{M}:\mathcal{E}\times \mathcal{E}\to [0,\infty )$ be a function. For given constants $\alpha \in \left(\frac{1}{2},1\right)$, $c>0$, and $s\ge 1$, the function $\mathcal{M}$ is said to be an $(\alpha ,c,s)$-interpolative b-metric on $\mathcal{E}$ if the following conditions hold for all $\mathfrak{p},\mathfrak{q},\mathfrak{r}\in \mathcal{E}$:

(IB1) 

$\mathcal{M}(\mathfrak{p},\mathfrak{q})=0\iff \mathfrak{p}=\mathfrak{q}$,

(IB2) 

$\mathcal{M}(\mathfrak{p},\mathfrak{q})=\mathcal{M}(\mathfrak{q},\mathfrak{p})$,

(IB3) 

$\mathcal{M}(\mathfrak{p},\mathfrak{q})\le s\left[\mathcal{M}(\mathfrak{p},\mathfrak{r})+\mathcal{M}(\mathfrak{r},\mathfrak{q})\right]+c{\left[\mathcal{M}(\mathfrak{p},\mathfrak{r})·\mathcal{M}(\mathfrak{r},\mathfrak{q})\right]}^{\alpha }.$

The pair $(\mathcal{E},\mathcal{M})$ is then called an $(\alpha ,c,s)$-interpolative b-metric space.

Note that this definition generalizes various known spaces:

• If $c=0$, we recover a standard b-metric space as defined by Czerwik [2].
• If $s=1,\alpha ={\displaystyle \frac{1}{2}}$, the space becomes an $(\alpha ,c)$-interpolative metric space as introduced in [8,9].
• If $c=0$ and $s=1$, it reduces to a classical metric space.

In [9], Karapınar established a version of the Ćirić-type fixed point theorem in the framework of $(\alpha ,c)$-interpolative metric spaces, where the standard triangle inequality is modified via a nonlinear interpolative term. While this approach effectively generalizes several fixed point results in classical metric settings, it still assumes the standard linearity of metric addition.

Remark 2. 

The restriction $\alpha \in \left(\frac{1}{2},1\right)$ is essential for ensuring that interpolative b-metrics define a genuinely broader class than standard b-metrics, while also retaining mathematical relevance in fixed point analysis.

• If $\alpha \le \frac{1}{2}$, then for all $a,b\ge 0$ one has ${\left(ab\right)}^{\alpha }\le a+b$. Consequently, the additional interpolative term ${\left[\mathcal{M}(\mathfrak{p},\mathfrak{r})\mathcal{M}(\mathfrak{r},\mathfrak{q})\right]}^{\alpha }$ is always absorbed into the linear part $\mathcal{M}(\mathfrak{p},\mathfrak{r})+\mathcal{M}(\mathfrak{r},\mathfrak{q})$, reducing the inequality to that of an ordinary b-metric with a larger constant. Hence, no genuine generalization occurs.
• If $\alpha >1$, then for small distances the term ${\left[\mathcal{M}(\mathfrak{p},\mathfrak{r})\mathcal{M}(\mathfrak{r},\mathfrak{q})\right]}^{\alpha }$ becomes negligible compared to the linear terms. This makes the interpolative component irrelevant in convergence and Cauchy sequence arguments, weakening the applicability to fixed point theory.
• For $\alpha \in (\frac{1}{2},1)$, the nonlinear term can dominate the linear part for large values, yet remains non-negligible near fixed points. This guarantees that the inequality does not collapse into the classical b-metric form and that the interpolative term plays a meaningful role in both structure and analysis.

Thus, the choice $\alpha \in (\frac{1}{2},1)$ ensures that interpolative b-metrics are both non-trivial and theoretically significant.

3.1. Example of an Interpolative b-Metric Space

Example 3. 

Let $\mathcal{E}=\mathbb{R}$ and define the following:

$$\mathcal{M}(p,q)={|p-q|}^2+{|p-q|}^4,\forall p,q\in \mathbb{R}.$$

We claim that $(\mathbb{R},\mathcal{M})$ is an $(\alpha ,c,s)$-interpolative b-metric space for some choice of parameters with $\alpha \in (\frac{1}{2},1)$.

• Verification:

(IB1) 

Clearly, $\mathcal{M}(p,q)=0\iff p=q$.

(IB2) 

Symmetry follows since $\mathcal{M}(p,q)=\mathcal{M}(q,p)$.

(IB3) 

Let $a=|p-r|$ and $b=|r-q|$. Then we obtain the following:

$$\mathcal{M}(p,q)={(a+b)}^2+{(a+b)}^4.$$

Using the following elementary inequalities:

$${(a+b)}^2\le 2(a^2+b^2),{(a+b)}^4\le 8(a^4+b^4),$$

we obtain

$$\mathcal{M}(p,q)\le 2(a^2+b^2)+8(a^4+b^4).$$

Since we have the following:

$$\mathcal{M}(p,r)=a^2+a^4,\mathcal{M}(r,q)=b^2+b^4,$$

it follows that

$$\mathcal{M}(p,q)\le s\left(\mathcal{M}(p,r)+\mathcal{M}(r,q)\right)+c{\left(ab\right)}^{\alpha },$$

with $s=8$, $c=1$, and $\alpha \in (\frac{1}{2},1)$.

Non-reducibility to a b-metric:   To see that the nonlinear term cannot always be absorbed into the linear part, consider $a=\mathcal{M}(p,r)$ and $b=\mathcal{M}(r,q)$. If we take $a=5$, $b=10$, and $\alpha =0.75$, then the following is calculated:

$${\left(ab\right)}^{\alpha }={\left(50\right)}^{0.75}\approx 18.80,a+b=15.$$

Thus, the interpolative contribution exceeds the linear sum and cannot be dominated by any constant multiple of $a+b$. Hence, $\mathcal{M}$ cannot be reduced to a standard b-metric.

Therefore, $(\mathbb{R},\mathcal{M})$ provides a genuine example of an $(\alpha ,c,s)$-interpolative b-metric space with $\alpha \in (\frac{1}{2},1)$, illustrating that this class of spaces strictly extends the family of b-metric spaces.

Remark 3. 

The above function $\mathcal{M}(p,q)={|p-q|}^2+{|p-q|}^4$ demonstrates that interpolative b-metric spaces form a strictly larger class than the previously known frameworks.

• Not a metric: The quadratic and quartic terms prevent $\mathcal{M}$ from satisfying the standard triangle inequality of a metric.
• Not a b-metric:  In a b-metric, one must have the following:

  $$\mathcal{M}(p,q)\le s\left(\mathcal{M}(p,r)+\mathcal{M}(r,q)\right).$$
  However, as shown numerically (e.g., for $a=5,b=10,\alpha =0.75$), the nonlinear term ${\left(ab\right)}^{\alpha }$ can exceed $a+b$. Thus no constant s can absorb the nonlinear contribution into the linear part, and hence $\mathcal{M}$ is not a b-metric.
• Not an interpolative metric:  An interpolative metric requires $s=1$. In this case, the quartic growth of ${(a+b)}^4$ cannot be controlled by a linear sum of $(a^2+a^4)$ and $(b^2+b^4)$, so no choice of constants yields $s=1$. Therefore, $(\mathcal{E},\mathcal{M})$ does not reduce to an interpolative metric space.

This confirms that the class of interpolative b-metric spaces properly contains the classes of metric spaces, b-metric spaces, and interpolative metric spaces.

Remark 4. 

In Example 3 we chose the specific form $\mathcal{M}(p,q)={|p-q|}^2+{|p-q|}^4$ for concreteness. However, the construction is not limited to this choice. Indeed, one may replace the exponents 2 and 4 with a general power $\delta >1$, defining the following:

$${\mathcal{M}}_{\delta }(p,q)={|p-q|}^{\delta },p,q\in \mathbb{R}.$$

For suitable constants $s\ge 1$, $c>0$, and $\alpha \in \left(\frac{1}{2},1\right)$, the interpolative inequality (IB3) remains valid, since the linear part controls small values while the superlinear interpolative term compensates for cross-growth at large values. Thus, Example 3 illustrates only one representative case, and the framework naturally accommodates broader families of exponents $\delta >1$.

3.2. Topological Properties of Interpolative b-Metric Spaces

The concept of an $(\alpha ,c,s)$-interpolative b-metric naturally induces a topology analogous to that of b-metric spaces. For any $\mathfrak{p}\in \mathcal{E}$ and $\epsilon >0$, define the open ball as follows:

$$B_{\mathcal{M}}(\mathfrak{p},\epsilon )=\{\mathfrak{q}\in \mathcal{E}:\mathcal{M}(\mathfrak{p},\mathfrak{q})<\epsilon \}.$$

The collection of such sets forms a base for a topology ${\tau }_{\mathcal{M}}$ on $\mathcal{E}$. The following properties are direct consequences of the structure of $\mathcal{M}$.

Proposition 1. 

Let $(\mathcal{E},\mathcal{M})$ be an $(\alpha ,c,s)$-interpolative b-metric space. Then we interpret the following:

(i) 

$(\mathcal{E},{\tau }_{\mathcal{M}})$ is a Hausdorff topological space;

(ii) 

$(\mathcal{E},{\tau }_{\mathcal{M}})$ is first countable, hence sequential;

(iii) 

A sequence $\left\{{\mathfrak{p}}_n\right\}\subset \mathcal{E}$ converges to $\mathfrak{p}$ if and only if $\mathcal{M}({\mathfrak{p}}_n,\mathfrak{p})\to 0$;

(iv) 

Limits are unique: if $\mathcal{M}({\mathfrak{p}}_n,\mathfrak{p})\to 0$ and $\mathcal{M}({\mathfrak{p}}_n,\mathfrak{q})\to 0$, then $\mathfrak{p}=\mathfrak{q}$.

Proof. 

(i) If $\mathfrak{p}\ne \mathfrak{q}$, then $\mathcal{M}(\mathfrak{p},\mathfrak{q})>0$. Set $\epsilon ={\displaystyle \frac{1}{3s}}\mathcal{M}(\mathfrak{p},\mathfrak{q})$; the corresponding open balls are disjoint by (IB3), showing Hausdorffness.

(ii) The family ${\left\{B_{\mathcal{M}}(\mathfrak{p},1/n)\right\}}_{n\in \mathbb{N}}$ forms a countable neighborhood base, proving first countability.

(iii)–(iv) follow from (IB1) and the definition of convergence via $\mathcal{M}$. The arguments mirror those for b-metric spaces (cf. Czerwik, 1993) and require no modification. □

Remark 5. 

Thus, every $(\alpha ,c,s)$-interpolative b-metric space is a Hausdorff, first-countable topological space in which Cauchy and convergence concepts coincide with the standard generalization of metric-space behavior. This justifies the use of sequence-based convergence and completeness arguments in Theorem 1.

4. Main Results

We now formulate and prove an analogue of the Čirić-type theorem within the more general framework of $(\alpha ,c,s)$-interpolative b-metric spaces. This setting extends the classical b-metric and interpolative metric spaces by incorporating the following:

• The multiplicative constant $s\ge 1$ from the theory of b-metrics, which allows additional flexibility in distance scaling;
• The nonlinear interpolative correction term $c·{\left[\mathcal{M}(\mathfrak{p},\mathfrak{r})·\mathcal{M}(\mathfrak{r},\mathfrak{q})\right]}^{\alpha }$, with $\alpha \in (\frac{1}{2},1)$, which guarantees that the interpolative component contributes meaningfully without collapsing into the linear part or becoming negligible.

The resulting structure accommodates both types of perturbations simultaneously, providing a richer framework for the analysis of convergence and contractive mappings. The fixed point theorem established in this setting extends the applicability of Ćirić-type contractions to a wider class of nonlinear metric-like spaces, while preserving the essential guarantees of existence and uniqueness of fixed points under appropriate completeness and contractiveness assumptions.

Theorem 1 

(Ćirić-type Fixed Point Theorem). Let $(\mathcal{E},\mathcal{M})$ be a complete $(\alpha ,c,s)$-interpolative b-metric space with parameters $\alpha \in (\frac{1}{2},1)$, $c>0$, and $s\ge 1$. Suppose that the mapping $\mathrm{\Theta }:\mathcal{E}\to \mathcal{E}$ satisfies the following contractive condition:

$$\mathcal{M}(\mathrm{\Theta }\mathfrak{p},\mathrm{\Theta }\mathfrak{q})\le q·\mathcal{N}(\mathfrak{p},\mathfrak{q}),\forall \mathfrak{p},\mathfrak{q}\in \mathcal{E},$$

for some constant $q\in (0,1)$, where

$$\mathcal{N}(\mathfrak{p},\mathfrak{q})=max\left\{\mathcal{M}(\mathfrak{p},\mathfrak{q}),\mathcal{M}(\mathfrak{p},\mathrm{\Theta }\mathfrak{p}),\mathcal{M}(\mathfrak{q},\mathrm{\Theta }\mathfrak{q})\right\}.$$

Then Θ has a unique fixed point in $\mathcal{E}$.

Proof. 

Let ${\mathfrak{p}}_0\in \mathcal{E}$ be arbitrary, and define a sequence $\left\{{\mathfrak{p}}_n\right\}$ by ${\mathfrak{p}}_{n+1}=\mathrm{\Theta }{\mathfrak{p}}_n$ for all $n\in {\mathbb{N}}_0$.

If ${\mathfrak{p}}_n={\mathfrak{p}}_{n+1}$ for some n, then a fixed point is already obtained. Otherwise, by the contractive condition, the following is calculated:

$$\mathcal{M}({\mathfrak{p}}_{n+1},{\mathfrak{p}}_{n+2})=\mathcal{M}(\mathrm{\Theta }{\mathfrak{p}}_n,\mathrm{\Theta }{\mathfrak{p}}_{n+1})\le q·\mathcal{N}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1}),$$

where

$$\mathcal{N}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1})=max\left\{\mathcal{M}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1}),\mathcal{M}({\mathfrak{p}}_n,\mathrm{\Theta }{\mathfrak{p}}_n),\mathcal{M}({\mathfrak{p}}_{n+1},\mathrm{\Theta }{\mathfrak{p}}_{n+1})\right\}.$$

Since ${\mathfrak{p}}_{n+1}=\mathrm{\Theta }{\mathfrak{p}}_n$, this reduces to the following:

$$\mathcal{N}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1})=max\{\mathcal{M}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1}),\mathcal{M}({\mathfrak{p}}_{n+1},{\mathfrak{p}}_{n+2})\}.$$

We define the following:

$$A_n=max\{\mathcal{M}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1}),\mathcal{M}({\mathfrak{p}}_{n+1},{\mathfrak{p}}_{n+2})\}.$$

Then, we define the following:

$$\mathcal{M}({\mathfrak{p}}_{n+1},{\mathfrak{p}}_{n+2})\le q{A}_n,$$

which further implies

$$A_{n+1}=max\{\mathcal{M}({\mathfrak{p}}_{n+1},{\mathfrak{p}}_{n+2}),\mathcal{M}({\mathfrak{p}}_{n+2},{\mathfrak{p}}_{n+3})\}\le q{A}_n.$$

By induction, we define the following:

$$A_n\le q^n{A}_0,\mathrm{with}{A}_0=max\{\mathcal{M}({\mathfrak{p}}_0,{\mathfrak{p}}_1),\mathcal{M}({\mathfrak{p}}_1,{\mathfrak{p}}_2)\}.$$

Since $q\in (0,1)$, we conclude the following:

$$\underset{n\to \infty }{lim}{A}_n=0,\mathrm{and}\mathrm{hence}\mathcal{M}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1})\to 0.$$

Cauchy property. Using the interpolative b-metric inequality,

$$\mathcal{M}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+2})\le s\left(\mathcal{M}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1})+\mathcal{M}({\mathfrak{p}}_{n+1},{\mathfrak{p}}_{n+2})\right)+c{\left(\mathcal{M}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1})\mathcal{M}({\mathfrak{p}}_{n+1},{\mathfrak{p}}_{n+2})\right)}^{\alpha }.$$

Substituting the bounds from $A_n\le q^n{A}_0$, we obtain the following:

$$\mathcal{M}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+2})\le s(q^n+q^{n+1})A_0+c{A}_0^{2\alpha }{q}^{\alpha (2n+1)}.$$

Since $q\in (0,1)$ and $\alpha >\frac{1}{2}$, both terms vanish as $n\to \infty$.

More generally, by repeated application one shows that for every $k\in \mathbb{N}$, the following is obtained:

$$\mathcal{M}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+k})\le C_k\sum _{i=0}^{k-1}\mathcal{M}({\mathfrak{p}}_{n+i},{\mathfrak{p}}_{n+i+1})+(\mathrm{interpolative}\mathrm{terms}).$$

The first sum is bounded by the following:

$$A_0{q}^n{\displaystyle \frac{1-q^k}{1-q}}\to 0\mathrm{as}n\to \infty ,$$

while the interpolative terms involve factors of the form

$${\left(\mathcal{M}({\mathfrak{p}}_{n+i},{\mathfrak{p}}_{n+i+1})\mathcal{M}({\mathfrak{p}}_{n+i+1},{\mathfrak{p}}_{n+i+2})\right)}^{\alpha }\le {\left(q^{2n+2i+1}{A}_0^2\right)}^{\alpha },$$

which also vanish since $\alpha \in (\frac{1}{2},1)$. Thus, we obtain the following:

$$\underset{n\to \infty }{lim}\mathcal{M}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+k})=0\mathrm{for}\mathrm{all}k\in \mathbb{N}.$$

Hence $\left\{{\mathfrak{p}}_n\right\}$ is Cauchy in $(\mathcal{E},\mathcal{M})$.

Existence of a fixed point. Since the space is complete, there exists ${\mathfrak{p}}^{*}\in \mathcal{E}$ with ${\mathfrak{p}}_n\to {\mathfrak{p}}^{*}$. Applying the interpolative b-metric inequality to $\mathcal{M}({\mathfrak{p}}^{*},\mathrm{\Theta }{\mathfrak{p}}^{*})$ and passing to the limit, we obtain the following:

$$\mathcal{M}({\mathfrak{p}}^{*},\mathrm{\Theta }{\mathfrak{p}}^{*})=0,$$

hence ${\mathfrak{p}}^{*}=\mathrm{\Theta }{\mathfrak{p}}^{*}$.

Uniqueness. If ${\mathfrak{q}}^{*}$ is another fixed point, then we obtain the following:

$$\mathcal{M}({\mathfrak{p}}^{*},{\mathfrak{q}}^{*})=\mathcal{M}(\mathrm{\Theta }{\mathfrak{p}}^{*},\mathrm{\Theta }{\mathfrak{q}}^{*})\le q·\mathcal{M}({\mathfrak{p}}^{*},{\mathfrak{q}}^{*}).$$

Thus $(1-q)\mathcal{M}({\mathfrak{p}}^{*},{\mathfrak{q}}^{*})\le 0$, which forces $\mathcal{M}({\mathfrak{p}}^{*},{\mathfrak{q}}^{*})=0$, so ${\mathfrak{p}}^{*}={\mathfrak{q}}^{*}$.

Hence $\mathrm{\Theta }$ has a unique fixed point in $\mathcal{E}$. □

Next, we provide an example to support our main result.

Example 4. 

Let $\mathcal{E}=[0,\infty )$, and define the following function:

$$\mathcal{M}:\mathcal{E}\times \mathcal{E}\to [0,\infty ),\mathcal{M}(p,q)={|p-q|}^2.$$

We show that $(\mathcal{E},\mathcal{M})$ is an $(\alpha ,c,s)$-interpolative b-metric space, and that the self-map, written as follows:

$$\mathrm{\Theta }:\mathcal{E}\to \mathcal{E},\mathrm{\Theta }\left(p\right)=\frac{p}{2},$$

satisfies the Ćirić-type contractive condition.

• Verifying interpolative b-metric properties.

(IB1) 

$\mathcal{M}(p,q)=0\iff p=q$.

(IB2) 

Symmetry holds since $\mathcal{M}(p,q)=\mathcal{M}(q,p)$.

(IB3) 

Let $a=|p-r|$, $b=|r-q|$. Then, we obtain the following:

$$\mathcal{M}(p,q)={|p-q|}^2\le {(a+b)}^2=a^2+b^2+2ab.$$

Using AM–GM, $2ab\le a^2+b^2$, and since ${\left(ab\right)}^{2\alpha }\ge 0$ for $\alpha \in (\frac{1}{2},1)$, we obtain the following:

$$\mathcal{M}(p,q)\le 2(a^2+b^2)+{\left(ab\right)}^{2\alpha }.$$

That is, the following:

$$\mathcal{M}(p,q)\le s\left(\mathcal{M}(p,r)+\mathcal{M}(r,q)\right)+c{\left(\mathcal{M}(p,r)\mathcal{M}(r,q)\right)}^{\alpha },$$

with $s=2$, $c=1$, and $\alpha \in (\frac{1}{2},1)$. Hence $(\mathcal{E},\mathcal{M})$ is an interpolative b-metric space.

• Checking the contractive condition.

For $p,q\in \mathcal{E}$, we obtain the following:

$$\mathcal{M}(\mathrm{\Theta }p,\mathrm{\Theta }q)={\left|\frac{p}{2}-\frac{q}{2}\right|}^2=\frac{1}{4}{|p-q|}^2=\frac{1}{4}\mathcal{M}(p,q).$$

Thus, we obtain the following:

$$\mathcal{M}(\mathrm{\Theta }p,\mathrm{\Theta }q)\le q{M}_{\mathcal{M}}(p,q),$$

with $q=\frac{1}{4}$, where

$$M_{\mathcal{M}}(p,q)=max\{\mathcal{M}(p,q),\mathcal{M}(p,\mathrm{\Theta }p),\mathcal{M}(q,\mathrm{\Theta }q)\}.$$

• Fixed point and convergence.

Starting with any $p_0\in \mathcal{E}$ and iterating, we obtain the following:

$$p_{n+1}=\mathrm{\Theta }\left(p_n\right)=\frac{p_n}{2},\mathrm{so}{p}_n=\frac{p_0}{2^n}.$$

Then $p_n\to 0$ as $n\to \infty$. Moreover, we obtain the following:

$$\mathcal{M}(p_n,0)=|\frac{p_0}{2^n}-0{|}^2=\frac{p_0^2}{4^n}\to 0=\mathcal{M}(0,0).$$

Thus, the sequence $\left\{p_n\right\}$ converges to the unique fixed point $p^{*}=0$.

Conclusion: The function $\mathrm{\Theta }\left(p\right)=p/2$ satisfies the assumptions of the Ćirić-type fixed point theorem in the interpolative b-metric space $(\mathcal{E},\mathcal{M})$, and the unique fixed point is $p^{*}=0$.

Now, we provide one more example to validate the last result.

Example 5. 

Let $\mathcal{E}=[0,\infty )$ and define the following:

$$\mathcal{M}(p,q)={|p-q|}^2+{|p-q|}^4,p,q\in \mathcal{E}.$$

We show that $(\mathcal{E},\mathcal{M})$ is an $(\alpha ,c,s)$–interpolative b–metric space with parameters $\alpha \in (\frac{1}{2},1)$, $c=1$, and $s=8$, and that a nontrivial self-mapping on this space satisfies the Čirić-type contraction.

• Verification of (IB1)–(IB3).

(IB1) 

$\mathcal{M}(p,q)=0$ if and only if $p=q$.

(IB2) 

Symmetry is obvious.

(IB3) 

For any $p,q,r\in \mathcal{E}$, set $a=|p-r|$, $b=|r-q|$. Then we obtain the following:

$$\mathcal{M}(p,q)={(a+b)}^2+{(a+b)}^4\le 2(a^2+b^2)+8(a^4+b^4)\le s[\mathcal{M}(p,r)+\mathcal{M}(r,q)]+c{\left[\mathcal{M}(p,r)\mathcal{M}(r,q)\right]}^{\alpha },$$

for $s=8$, $c=1$, $\alpha \in (\frac{1}{2},1)$.

Hence $(\mathcal{E},\mathcal{M})$ is an $(\alpha ,c,s)$–interpolative b–metric space.

Why it is not a b–metric or an interpolative metric. The quartic term ${|p-q|}^4$ prevents reduction to a b–metric since no fixed s can absorb ${\left(ab\right)}^{\alpha }$ into $a^2+a^4+b^2+b^4$. Similarly, setting $s=1$ fails to control ${(a+b)}^4$ by a linear sum, hence it is not an interpolative metric either.

Nontrivial mapping. Define $\mathrm{\Theta }:\mathcal{E}\to \mathcal{E}$ by the following:

$$\left(\mathrm{\Theta }p\right)={\displaystyle \frac{p}{1+p}},p\ge 0.$$

Then for any $p,q\in \mathcal{E}$, the following is calculated:

$$|\mathrm{\Theta }p-\mathrm{\Theta }q|={\displaystyle \frac{|p-q|}{(1+p)(1+q)}}\le |p-q|,$$

so that

$$\mathcal{M}(\mathrm{\Theta }p,\mathrm{\Theta }q)={|\mathrm{\Theta }p-\mathrm{\Theta }q|}^2+{|\mathrm{\Theta }p-\mathrm{\Theta }q|}^4\le L^2\left({|p-q|}^2+{|p-q|}^4\right)=L^2\mathcal{M}(p,q),$$

with $L={\displaystyle \frac{1}{{(1+min\{p,q\})}^2}}<1$. Thus, for each bounded subset of $\mathcal{E}$, Θ is a Čirić-type contraction with constant $q=L^2<1$.

This example therefore provides a genuine case of an $(\alpha ,c,s)$–interpolative b-metric space that is neither a b-metric nor an interpolative metric, and in which Theorem 1 (the Čirić-type fixed point theorem) applies nontrivially.

Corollary 1 

(Bianchini-Type Fixed Point Theorem). Let $(\mathcal{E},d)$ be a complete $(\alpha ,c,s)$-interpolative b-metric space, and let $\mathrm{\Theta }:\mathcal{E}\to \mathcal{E}$ be a mapping. Suppose there exists a constant $q\in (0,1)$ such that the following is calculated:

$$\mathcal{M}(\mathrm{\Theta }\mathfrak{p},\mathrm{\Theta }\mathfrak{q})\le q·N_{\mathcal{M}}(\mathfrak{p},\mathfrak{q}),\mathit{for}\mathit{all}\mathfrak{p},\mathfrak{q}\in \mathcal{E},$$

where

$$N_{\mathcal{M}}(\mathfrak{p},\mathfrak{q})=max\left\{\mathcal{M}(\mathfrak{p},\mathrm{\Theta }\mathfrak{p}),\mathcal{M}(\mathfrak{q},\mathrm{\Theta }\mathfrak{q})\right\}.$$

Then Θ has a unique fixed point in $\mathcal{E}$.

Proof. 

The proof follows the same structure as the theorem. Construct the sequence ${\mathfrak{p}}_{n+1}=\mathrm{\Theta }{\mathfrak{p}}_n$. The recursive inequality becomes the following:

$$\mathcal{M}({\mathfrak{p}}_{n+1},{\mathfrak{p}}_{n+2})=\mathcal{M}(\mathrm{\Theta }{\mathfrak{p}}_n,\mathrm{\Theta }{\mathfrak{p}}_{n+1})\le q·max\{\mathcal{M}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1}),\mathcal{M}({\mathfrak{p}}_{n+1},{\mathfrak{p}}_{n+2})\}.$$

As in the main theorem, this leads to a geometric decay and shows $\left\{{\mathfrak{p}}_n\right\}$ is Cauchy. The remainder follows by completeness and uniqueness from the contraction inequality. □

The following corollary provides an important application, and as such, we provide a more detailed exposition.

Corollary 2 

(Kannan-Type Fixed Point Theorem). Let $(\mathcal{E},\mathcal{M})$ be a complete $(\alpha ,c,s)$-interpolative b-metric space with $\alpha \in \left(\frac{1}{2},1\right)$, and let $\mathrm{\Theta }:\mathcal{E}\to \mathcal{E}$ be a mapping. Suppose there exists a constant $q\in (0,1)$ such that the following is written:

$$\mathcal{M}(\mathrm{\Theta }\mathfrak{p},\mathrm{\Theta }\mathfrak{q})\le q{K}_{\mathcal{M}}(\mathfrak{p},\mathfrak{q})\mathit{for}\mathit{all}\mathfrak{p},\mathfrak{q}\in \mathcal{E},$$

where

$$K_{\mathcal{M}}(\mathfrak{p},\mathfrak{q})=\frac{1}{2}\left[\mathcal{M}(\mathfrak{p},\mathrm{\Theta }\mathfrak{p})+\mathcal{M}(\mathfrak{q},\mathrm{\Theta }\mathfrak{q})\right].$$

Then Θ admits a unique fixed point in $\mathcal{E}$.

Proof. 

Choose an arbitrary ${\mathfrak{p}}_0\in \mathcal{E}$ and define the iterative sequence as follows:

$${\mathfrak{p}}_{n+1}=\mathrm{\Theta }{\mathfrak{p}}_n,n\ge 0.$$

Applying the Kannan-type condition to $({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1})$ yields the following:

$$\mathcal{M}({\mathfrak{p}}_{n+1},{\mathfrak{p}}_{n+2})\le q\frac{\mathcal{M}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1})+\mathcal{M}({\mathfrak{p}}_{n+1},{\mathfrak{p}}_{n+2})}{2}.$$

Rearranging, results in the following:

$$\left(1-\frac{q}{2}\right)\mathcal{M}({\mathfrak{p}}_{n+1},{\mathfrak{p}}_{n+2})\le \frac{q}{2}\mathcal{M}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1}),$$

so that

$$\mathcal{M}({\mathfrak{p}}_{n+1},{\mathfrak{p}}_{n+2})\le \lambda \mathcal{M}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1}),\lambda =\frac{q}{2-q}\in (0,1).$$

By induction, the following is obtained:

$$\mathcal{M}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1})\le {\lambda }^n\mathcal{M}({\mathfrak{p}}_0,{\mathfrak{p}}_1),n\ge 0,$$

implying ${lim}_{n\to \infty }\mathcal{M}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1})=0$.

Cauchy property. For $m>n$, repeated use of the interpolative b-metric inequality (IB3) gives the following:

$$\mathcal{M}({\mathfrak{p}}_n,{\mathfrak{p}}_m)\le s\sum _{i=n}^{m-1}\mathcal{M}({\mathfrak{p}}_i,{\mathfrak{p}}_{i+1})+c\sum _{i=n}^{m-2}{\left[\mathcal{M}({\mathfrak{p}}_i,{\mathfrak{p}}_{i+1})\mathcal{M}({\mathfrak{p}}_{i+1},{\mathfrak{p}}_{i+2})\right]}^{\alpha }.$$

Since $\mathcal{M}({\mathfrak{p}}_i,{\mathfrak{p}}_{i+1})\le {\lambda }^i\mathcal{M}({\mathfrak{p}}_0,{\mathfrak{p}}_1)$ and $\alpha \in (\frac{1}{2},1)$, both sums vanish as $n\to \infty$. Thus, $\left\{{\mathfrak{p}}_n\right\}$ is a Cauchy sequence.

Existence and uniqueness. By completeness, there exists ${\mathfrak{p}}^{*}\in \mathcal{E}$ such that the following is calculated: ${\mathfrak{p}}_n\to {\mathfrak{p}}^{*}$. Passing to the limit in ${\mathfrak{p}}_{n+1}=\mathrm{\Theta }{\mathfrak{p}}_n$ yields $\mathrm{\Theta }{\mathfrak{p}}^{*}={\mathfrak{p}}^{*}$.

If ${\mathfrak{q}}^{*}$ is another fixed point, then the following is calculated:

$$\mathcal{M}({\mathfrak{p}}^{*},{\mathfrak{q}}^{*})=\mathcal{M}(\mathrm{\Theta }{\mathfrak{p}}^{*},\mathrm{\Theta }{\mathfrak{q}}^{*})\le q{K}_{\mathcal{M}}({\mathfrak{p}}^{*},{\mathfrak{q}}^{*})=0,$$

so ${\mathfrak{p}}^{*}={\mathfrak{q}}^{*}$.

Hence $\mathrm{\Theta }$ has a unique fixed point in $\mathcal{E}$. □

Remark 6. 

This corollary extends the classical Kannan fixed point theorem from standard metric spaces to the broader class of $(\alpha ,c,s)$-interpolative b-metric spaces.

• If $c=0$ and $s=1$, the defining inequality (IB3) reduces to the usual metric triangle inequality, and the contractive condition becomes precisely Kannan’s condition in a metric space.
• If $c=0$ but $s>1$, the result recovers the Kannan-type theorem in b-metric spaces.
• If $s=1$ but $c>0$ and $\alpha \in (\frac{1}{2},1)$, the result specializes to interpolative metric spaces in the sense of Karapınar, where the nonlinear term contributes in a nontrivial way.

Thus, the present theorem unifies and extends Kannan-type fixed point results from metric spaces, b-metric spaces, and interpolative metric spaces into a single framework of interpolative b-metric spaces.

Corollary 3 

(Banach-Type Fixed Point Theorem). Let $(\mathcal{E},\mathcal{M})$ be a complete $(\alpha ,c,s)$-interpolative b-metric space, and let $\mathrm{\Theta }:\mathcal{E}\to \mathcal{E}$ be a mapping. Suppose that there exists a constant $q\in (0,1)$ such that the following is calculated:

$$\mathcal{M}(\mathrm{\Theta }\mathfrak{p},\mathrm{\Theta }\mathfrak{q})\le q·\mathcal{M}(\mathfrak{p},\mathfrak{q}),\mathit{for}\mathit{all}\mathfrak{p},\mathfrak{q}\in \mathcal{E}.$$

Then Θ has a unique fixed point in $\mathcal{E}$.

Proof. 

This is a direct consequence of Theorem 1, since the following assumption:

$$\mathcal{M}(\mathrm{\Theta }\mathfrak{p},\mathrm{\Theta }\mathfrak{q})\le q·\mathcal{M}(\mathfrak{p},\mathfrak{q})$$

implies the Ćirić-type condition with

$$\mathcal{M}(\mathrm{\Theta }\mathfrak{p},\mathrm{\Theta }\mathfrak{q})\le q·max\{\mathcal{M}(\mathfrak{p},\mathfrak{q}),\mathcal{M}(\mathfrak{p},\mathrm{\Theta }\mathfrak{p}),\mathcal{M}(\mathfrak{q},\mathrm{\Theta }\mathfrak{q})\}=q·M_{\mathcal{M}}(\mathfrak{p},\mathfrak{q}).$$

Thus, the hypotheses of Theorem 1 are satisfied. The sequence $\left\{{\mathfrak{p}}_n\right\}$ defined by ${\mathfrak{p}}_{n+1}=\mathrm{\Theta }{\mathfrak{p}}_n$ satisfies the following:

$$\mathcal{M}({\mathfrak{p}}_{n+1},{\mathfrak{p}}_{n+2})\le q·\mathcal{M}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1}),$$

leading to $\mathcal{M}({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1})\le q^n\mathcal{M}({\mathfrak{p}}_0,{\mathfrak{p}}_1)$ and hence Cauchy convergence in $(\mathcal{E},\mathcal{M})$. The rest of the argument follows identically from the proof of Theorem 1. □

Remark 7. 

Corollary 3 recovers the classical Banach contraction principle as a special case.

Indeed, if we take $s=1$, $c=0$, and α arbitrary, then the definition of an $(\alpha ,c,s)$-interpolative b-metric space reduces to that of a standard metric space, and the contractive condition in Corollary 3 coincides exactly with Banach’s contraction condition. Hence, the present framework properly extends the Banach contraction theorem.

5. Application to Nonlinear Volterra-Type Integral Equation in an Interpolative b-Metric Space

Consider the following nonlinear integral equation:

$$\mathfrak{p}\left(t\right)={\int }_0^{t}K(t,s,\mathfrak{p}\left(s\right))ds,t\in [0,T],$$

(1)

where $K:[0,T]\times [0,T]\times \mathbb{R}\to \mathbb{R}$ is continuous. We show that the operator associated with (1) admits a unique fixed point in a suitable $(\alpha ,c,s)$-interpolative b-metric space, demonstrating that the new framework extends beyond the scope of the classical Banach contraction theorem.

5.1. Interpolative b-Metric Structure

Let the following be valid:

$$\mathcal{E}={C\left([0,T]\right),\parallel \mathfrak{p}\parallel }_{\infty }=\underset{t\in [0,T]}{sup}\left|\mathfrak{p}\left(t\right)\right|.$$

We define the following:

$$\mathcal{M}(\mathfrak{p},\mathfrak{q})={\parallel \mathfrak{p}-\mathfrak{q}\parallel }_{\infty }^2+{\left({\parallel \mathfrak{p}-\mathfrak{q}\parallel }_{\infty }^2\right)}^{\alpha },\mathfrak{p},\mathfrak{q}\in \mathcal{E},$$

with parameters $\alpha \in (\frac{1}{2},1)$, $c=1$, and $s=3$.

Proposition 2. 

The pair $(\mathcal{E},\mathcal{M})$ is a complete $(\alpha ,c,s)$-interpolative b-metric space.

Proof. 

(IB1) holds trivially since $\mathcal{M}(\mathfrak{p},\mathfrak{q})=0\iff \mathfrak{p}=\mathfrak{q}$.

(IB2) follows from the symmetry of the norm.

For (IB3), let $A={\parallel \mathfrak{p}-\mathfrak{r}\parallel }_{\infty }$ and $B={\parallel \mathfrak{r}-\mathfrak{q}\parallel }_{\infty }$. By the triangle inequality, ${\parallel \mathfrak{p}-\mathfrak{q}\parallel }_{\infty }\le A+B$, hence we obtain the following:

$${\parallel \mathfrak{p}-\mathfrak{q}\parallel }_{\infty }^2\le 2(A^2+B^2){,(\parallel \mathfrak{p}-\mathfrak{q}\parallel }_{\infty }^2{)}^{\alpha }\le 2^{\alpha }(A^{2\alpha }+B^{2\alpha }).$$

Therefore, we obtain the following:

$$\mathcal{M}(\mathfrak{p},\mathfrak{q})\le s[\mathcal{M}(\mathfrak{p},\mathfrak{r})+\mathcal{M}(\mathfrak{r},\mathfrak{q})]+c{\left(\mathcal{M}(\mathfrak{p},\mathfrak{r})\mathcal{M}(\mathfrak{r},\mathfrak{q})\right)}^{\alpha }$$

with $s=3$, $c=1$, and $\alpha \in (\frac{1}{2},1)$, verifying (IB3). Completeness follows because the uniform norm space $(C\left([0,T]\right),\parallel ·{\parallel }_{\infty })$ is Banach and $\mathcal{M}$-Cauchy sequences are norm-Cauchy. □

5.2. Operator and Contractive Condition

Define the integral operator $\mathrm{\Theta }:\mathcal{E}\to \mathcal{E}$ by the following:

$$\left(\mathrm{\Theta }\mathfrak{p}\right)\left(t\right)={\int }_0^{t}K(t,s,\mathfrak{p}\left(s\right))ds.$$

Assume K satisfies a Lipschitz condition: there exists $L>0$ such that the following is calculated:

$$\left|K\right(t,s,\mathfrak{p})-K(t,s,\mathfrak{q}\left)\right|\le L|\mathfrak{p}-\mathfrak{q}|,t,s\in [0,T],\mathfrak{p},\mathfrak{q}\in \mathbb{R}.$$

Then, for $\mathfrak{p},\mathfrak{q}\in \mathcal{E}$ and $t\in [0,T]$, we obtain the following:

$$|\left(\mathrm{\Theta }\mathfrak{p}\right)\left(t\right)-\left(\mathrm{\Theta }\mathfrak{q}\right)\left(t\right)|\le L{\int }_0^t|\mathfrak{p}\left(s\right)-\mathfrak{q}\left(s\right)|ds\le {LT\parallel \mathfrak{p}-\mathfrak{q}\parallel }_{\infty }.$$

Hence, we obtain the following:

$${\parallel \mathrm{\Theta }\mathfrak{p}-\mathrm{\Theta }\mathfrak{q}\parallel }_{\infty }^2\le {\left(LT\right)}^2{\parallel \mathfrak{p}-\mathfrak{q}\parallel }_{\infty }^2.$$

Substituting into the definition of $\mathcal{M}$ gives the following:

$$\mathcal{M}(\mathrm{\Theta }\mathfrak{p},\mathrm{\Theta }\mathfrak{q})\le {\left(LT\right)}^2\mathcal{M}(\mathfrak{p},\mathfrak{q})+{\left[{\left(LT\right)}^2\mathcal{M}(\mathfrak{p},\mathfrak{q})\right]}^{\alpha }.$$

That is, the following is calculated:

$$\mathcal{M}(\mathrm{\Theta }\mathfrak{p},\mathrm{\Theta }\mathfrak{q})\le q_1\mathcal{M}(\mathfrak{p},\mathfrak{q})+q_2{\left[\mathcal{M}(\mathfrak{p},\mathfrak{q})\right]}^{\alpha },$$

(2)

where $q_1={\left(LT\right)}^2$, $q_2={\left(LT\right)}^{2\alpha }$. Since $LT<1$, both $q_1,q_2\in (0,1)$, so (2) defines an interpolative contraction of the form required by Theorem 1.

5.3. Existence and Uniqueness

By Theorem 1, there exists a unique ${\mathfrak{p}}^{*}\in \mathcal{E}$ such that the following is written:

$$\mathrm{\Theta }{\mathfrak{p}}^{*}={\mathfrak{p}}^{*},$$

i.e.,

$${\mathfrak{p}}^{*}\left(t\right)={\int }_0^{t}K(t,s,{\mathfrak{p}}^{*}\left(s\right))ds,t\in [0,T].$$

Therefore, under the condition $LT<1$, the integral Equation (1) admits a unique continuous solution in the $(\alpha ,c,s)$-interpolative b-metric space $(\mathcal{E},\mathcal{M})$.

Remark 8. 

If the metric were purely quadratic, $\mathcal{M}(\mathfrak{p},\mathfrak{q})={\parallel \mathfrak{p}-\mathfrak{q}\parallel }_{\infty }^2$, then inequality (2) would reduce to $\mathcal{M}(\mathrm{\Theta }\mathfrak{p},\mathrm{\Theta }\mathfrak{q})\le {\left(LT\right)}^2\mathcal{M}(\mathfrak{p},\mathfrak{q})$, and the Banach contraction principle would suffice. However, the additional nonlinear term ${\left[\mathcal{M}(\mathfrak{p},\mathfrak{q})\right]}^{\alpha }$ introduces a superlinear correction that cannot be absorbed into the linear part. Thus, Theorem 1 in the $(\alpha ,c,s)$-interpolative b-metric framework is essential for establishing convergence in this setting.

6. Conclusions

In this paper we introduced the notion of $(\alpha ,c,s)$-interpolative b-metric spaces, a new class of generalized metric structures that incorporates both a multiplicative scaling factor and a nonlinear interpolative correction term with exponent $\alpha \in (\frac{1}{2},1)$. This framework properly extends and unifies the theories of classical metric spaces, b-metric spaces, and interpolative metric spaces.

We established fundamental properties of interpolative b-metric spaces and proved a Ćirić-type fixed point theorem within this setting, together with analogues of the classical results of Banach, Kannan, and Bianchini. These results demonstrate that interpolative b-metrics provide a natural and flexible environment in which contractive mappings retain fixed points with uniqueness.

Finally, we applied the theory to a nonlinear Volterra-type integral equation, showing that the interpolative b-metric framework can effectively guarantee the existence and uniqueness of solutions under generalized contractive conditions. This illustrates both the breadth and applicability of the theory, and opens the way for further extensions of fixed point results in more complex analytical and applied contexts.

Our results contribute to the growing literature on fixed point theory in generalized metric environments and pave the way for further future applications in the following branches:

Differential equations: The generalized contraction principles obtained here can be used to prove the existence and uniqueness of solutions to nonlinear boundary value problems and evolution equations, especially when the underlying functional spaces exhibit non-Euclidean geometry [21,22].

Optimization: Many iterative optimization algorithms generate sequences in spaces where classical metric assumptions fail; our framework can model such settings and ensure convergence under weaker contractive assumptions [23].

Data analysis and applied modeling: In high-dimensional data spaces or in similarity-based machine learning models, distance measures often deviate from strict metric properties; interpolative b-metric spaces can effectively accommodate such distortions while preserving fixed point existence and uniqueness [24].

Other contractions: Other important contractions, such as Hardy-Rogers or generalized Fisher contractions, may be studied in the framework of interpolative b-metric spaces, considering the works of Rhoades [25] and Rahimi and Rad [26].