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Derivative Type Mapping Theorem for the Interpolative Berinde Weak Contraction in Metric Spaces with Application

This paper establishes a fixed point theorem of the derivative type for interpolative Berinde weak contractive mappings in metric spaces, provides an illustrative example, and applies the result to solve a Fredholm integral equation.

Original authors: Clement Boateng Ampadu

Published 2026-04-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Clement Boateng Ampadu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to find a specific, magical "resting spot" in a vast, bumpy landscape. In mathematics, this landscape is called a Metric Space (a place where you can measure distances between points), and the "resting spot" is called a Fixed Point. A fixed point is a place where, if you apply a certain rule or transformation (let's call it a "machine" $T$ ), you end up exactly where you started. If you put a rock into the machine, and it spits out the same rock, that rock is the fixed point.

This paper is about proving that under very specific, clever rules, such a resting spot always exists and is unique (there's only one), even if the landscape is tricky.

Here is the breakdown of the paper's ideas using simple analogies:

1. The Old Rules vs. The New Rules

The Classic Rule (Banach Contraction): Imagine a machine that always shrinks the distance between two objects. If you put two balls 10 meters apart, the machine makes them 5 meters apart. If you keep running them through the machine, they eventually crash into the exact same spot. This is the famous "Banach Contraction Principle."
The "Interpolative" Rule (Berinde Weak Contraction): The authors look at a more complex machine. Instead of just shrinking the distance between two points, this machine looks at the distance between the points and how far those points are from their own "next steps." It's like a GPS that says, "To get to the destination, you don't just look at where you are; you also look at how far you've already traveled."
The "Derivative Type" Twist: This is the paper's main innovation. Usually, we measure distance with a ruler. But here, the authors say, "Let's not just measure the distance; let's measure how fast the distance is changing."
- Analogy: Imagine you are driving. A standard contraction says, "You are getting closer to the city." A Derivative Type contraction says, "The rate at which you are getting closer is slowing down in a very specific, controlled way." They use a special function (called $\phi$ ) to measure this "speed of shrinking" rather than just the distance itself.

2. The Main Discovery (The Theorem)

The authors prove a new theorem: If your landscape is "bounded" (it's not infinitely large) and your machine follows these specific "Derivative Type Interpolative" rules, then no matter where you start, you will eventually be pulled to that one unique resting spot.

The "Bounded" Condition: Think of the landscape as a giant, but finite, trampoline. If the trampoline were infinite, you might run forever. But because it has edges (it's bounded), you are guaranteed to eventually stop bouncing and settle down.
The Proof Logic:
1. They start at a random point.
2. They apply the machine over and over, creating a chain of points.
3. They show that the "speed of shrinking" (the derivative) gets smaller and smaller, like a ball losing energy.
4. Because the speed of shrinking hits zero, the points stop moving and merge into one single point.
5. They prove there can't be two different resting spots; if there were, the rules would break.

3. A Real-World Example

To show this isn't just math magic, they give a simple example:

The Landscape: The number line from 0 to 1.
The Machine: A rule that takes any number $x$ and turns it into $x^2 / 3$ .
The Result: No matter what number you start with (say, 0.9), if you keep applying this rule, the numbers get smaller and smaller until they hit 0. Zero is the fixed point. The paper proves that even with their complex "derivative" rules, this behavior holds true.

4. The Big Application: Solving Integral Equations

Why do we care about this? The authors apply this math to solve a Fredholm Integral Equation.

What is that? Imagine you are trying to predict the temperature of a metal rod over time. The temperature at any point depends on the temperature of every other point on the rod, plus some external heat source. This is a massive, complicated puzzle where the answer depends on itself.
The Solution: These equations are hard to solve because they are "non-linear" (messy and unpredictable).
How the Paper Helps: The authors show that if the "heat source" and the "rod's behavior" follow their specific "Derivative Type" rules, then there is exactly one correct answer to the temperature puzzle. It guarantees that the physics problem has a solution and that you won't get two different, conflicting answers.

Summary

Think of this paper as a new, ultra-precise navigation system.

Old systems said, "Keep walking toward the goal."
This new system says, "Check your speed, check your past steps, and ensure your rate of slowing down follows this specific curve."
If you follow these rules in a finite world, the system guarantees you will arrive at the one and only destination, and it proves this works even for complex, real-world problems like predicting heat flow or fluid dynamics.

The paper essentially says: "We found a new, stricter set of rules for shrinking distances. If you follow them, you are guaranteed to find a unique solution to your problem."

1. Problem Statement

The paper addresses the problem of establishing fixed point theorems for a specific class of nonlinear mappings within metric spaces. It seeks to unify and extend two existing concepts in fixed point theory:

Interpolative Berinde Weak Contractions: A generalization of the Banach contraction principle introduced by Berinde and extended by Olatinwo, where the contraction condition involves an interpolative combination of distances $d(x, y)$ and $d(x, Tx)$.
Derivative Type Contractions: A concept introduced by Olatinwo where the contractive condition is defined not on the distance $d(x,y)$ directly, but on the derivative of a function $\phi$ evaluated at that distance.

The core problem is to formulate a Derivative Type Interpolative Berinde Weak Contraction and prove the existence and uniqueness of a fixed point for such mappings in a complete metric space, subsequently applying this theoretical framework to solve non-linear Fredholm integral equations.

2. Methodology

The author employs a rigorous analytical approach combining metric space topology and functional analysis:

Definition Extension: The paper defines a new class of operators, the Interpolative Berinde Weak Operator of the Derivative Type. A mapping $T: X \to X$ satisfies this if there exist $\lambda \in [0, 1)$ and $\alpha \in (0, 1)$ such that for all $x, y \in X \setminus \text{Fix}(T)$ :
$\left. \frac{d\phi}{dt} \right|_{t=d(Tx,Ty)} \leq \lambda \left[ \left. \frac{d\phi}{dt} \right|_{t=d(x,y)} \right]^\alpha \left[ \left. \frac{d\phi}{dt} \right|_{t=d(x,Tx)} \right]^{1-\alpha}$
Here, $\phi: \mathbb{R}^+ \to \mathbb{R}^+$ is a function with a strictly positive derivative for $\epsilon > 0$ .
Constructive Proof (Picard Iteration):
- An arbitrary sequence $\{x_n\}$ is generated via $x_{n+1} = Tx_n$ .
- The contractive condition is applied recursively to derive an inequality for the derivative term at the distance between successive terms: $\left. \frac{d\phi}{dt} \right|_{t=d(x_n, x_{n+1})} \leq \lambda^n \left. \frac{d\phi}{dt} \right|_{t=d(x_0, x_1)}$ .
- Using the triangle inequality and the geometric series sum (since $\lambda < 1$ ), the author demonstrates that the sequence is Cauchy.
- The assumption that the space is $d$ -bounded (finite diameter) is crucial to bound the initial terms and ensure convergence.
Limit Analysis:
- The limit of the sequence is shown to be a fixed point by substituting the limit $x^*$ into the contractive inequality and utilizing the property that $\left. \frac{d\phi}{dt} \right|_{t=0} = 0$ .
- Uniqueness is proven by contradiction, assuming two distinct fixed points and showing the contractive condition forces their distance to be zero.
Application to Integral Equations:
- The theoretical result is applied to the space $Y = C[0, 1]$ (continuous functions on $[0, 1]$ ) equipped with the supremum metric.
- A non-linear Fredholm integral equation is reformulated as a fixed point problem $u = Fu$.
- The contractive condition is translated into a condition on the kernel $K(t, s, u(s))$ to satisfy the derivative-type inequality.

3. Key Contributions

Novel Theoretical Framework: The paper introduces the "Derivative Type Interpolative Berinde Weak Contraction," effectively merging the derivative-type contraction concept with the interpolative Berinde weak contraction. This creates a more flexible condition for proving fixed points than standard contractions.
Generalization: The main theorem (Theorem 2.3) generalizes previous results by Olatinwo and Berinde. As noted in Remark 2.2, if the derivative function $\phi(t) = t$ , the new definition reduces exactly to the standard interpolative Berinde weak contraction, proving the new result is a strict generalization.
Application to Integral Equations: The paper successfully bridges abstract fixed point theory and applied analysis by providing sufficient conditions for the existence and uniqueness of solutions to non-linear Fredholm integral equations, a common problem in physics and engineering.
Illustrative Example: The paper provides a concrete example ( $X=[0,1]$ , $T(x)=x^2/3$ , $\phi(x)=x^3$ ) to demonstrate the validity of the theorem and the behavior of the derivative inequality.

4. Main Results

Theorem 2.3 (Main Fixed Point Theorem): In a complete, $d$ -bounded metric space $(X, d)$ , any interpolative Berinde weak operator of the derivative type possesses a unique fixed point $x^*$ . Furthermore, for any starting point $x \in X$ , the iterative sequence $T^n(x)$ converges to $x^*$ .
Theorem 3.1 (Application): The non-linear Fredholm integral equation $u(t) = v(t) + \int_0^1 K(t, s, u(s))ds$ has a unique solution in $C[0, 1]$ provided the kernel $K$ satisfies the specific derivative-type interpolative inequality involving $\lambda$ and $\alpha$ .

5. Significance

Theoretical Advancement: By introducing the derivative operator into the interpolative framework, the paper expands the toolkit available for analyzing nonlinear operators that may not satisfy standard Lipschitz or Banach contraction conditions but do satisfy these derivative-based conditions.
Broad Applicability: The requirement of the space being $d$ -bounded is a specific constraint, but the result is significant for bounded domains often found in practical applications (like the interval $[0,1]$ in the integral equation application).
Problem Solving: The application to Fredholm integral equations demonstrates the practical utility of the abstract theorem. It offers a new method for proving existence and uniqueness for integral equations where traditional contraction mapping principles might fail or require stricter assumptions on the kernel.
Future Research: The paper opens avenues for exploring derivative-type contractions in other generalized metric spaces (e.g., multiplicative metric spaces, as hinted in the introduction) and for other types of integral or differential equations.

In summary, this paper successfully extends fixed point theory by integrating derivative concepts into interpolative weak contractions, providing a robust theoretical foundation with direct applications to solving non-linear integral equations.