Fixed point theorems on perturbed metric space with an application

This paper establishes fixed point theorems for $F$-perturbed mappings in complete perturbed metric spaces, validates the results with a counterexample, and demonstrates their application to proving the existence of solutions for second-order boundary value problems.

The Gist

The Big Picture: Measuring the World with "Noisy" Rulers

Imagine you are trying to measure the distance between two cities. In a perfect world (what mathematicians call a Metric Space), your ruler is flawless. If you measure from City A to City B, you get an exact number. If you measure A to C and then C to B, the sum is exactly the same as A to B. This is the "Triangle Inequality," the golden rule of geometry.

But in the real world, nothing is perfect. Your ruler might be slightly bent, your eyes might be tired, or the ground might be uneven. Every time you measure, there is a tiny bit of error or noise.

This paper is about what happens when we stop pretending our rulers are perfect and admit that every measurement has a little bit of "static" or "perturbation" attached to it. The authors ask: If our measuring tools are flawed, can we still find a "fixed point"?

What is a "Fixed Point"?

Before we dive into the math, let's understand the goal. A Fixed Point is like a "sweet spot" where things stop changing.

• The Analogy: Imagine you are folding a map of a city and placing it back down on the actual city. No matter how you crumple or fold the map, there is always at least one point on the map that lies directly on top of the exact location it represents on the ground. That point is the "fixed point."
• In Math: It's a point $x$ where if you apply a rule (a function) to it, you get the same point back ($f(x) = x$).

The Problem: The "Noisy" Map

The authors introduce a concept called a Perturbed Metric Space.

• The Exact Metric ($d$): This is the "true" distance, the perfect ruler.
• The Perturbed Metric ($D$): This is the "noisy" ruler. It includes the true distance plus some extra error term ($P$).
  • Formula: $D = \text{True Distance} + \text{Error}$.

The paper asks: If we have a rule (a mapping) that shrinks distances in this "noisy" world, will it still lead us to a unique fixed point?

The Solution: The "F-Perturbed" Rule

The authors introduce a new type of rule called an F-perturbed mapping.

The Metaphor: The Magic Shrinking Machine
Imagine a machine that takes an object and shrinks it.

1. Old Rule (Standard Contraction): The machine must shrink the object by a fixed percentage every time (e.g., cut it in half).
2. New Rule (F-contraction): The machine is smarter. It doesn't just shrink by a percentage; it shrinks based on a special "logarithmic" scale. It says, "If the object is huge, I'll shrink it a lot. If it's tiny, I'll shrink it just enough to make it vanish."

The authors prove that even if you are using this "noisy" ruler ($D$) instead of a perfect one, as long as your machine follows this special "F-rule," it will eventually squeeze everything down to a single, unique point.

Key Takeaway from the Proof:
They show that even with the "noise" (the error term), if you keep applying the rule, the "noise" eventually becomes irrelevant, and the sequence of points converges to a single, stable destination.

The Application: Solving a Physics Puzzle

Why does this matter? The authors apply this math to a Second-Order Boundary Value Problem.

The Analogy: The Hanging Rope
Imagine a rope hanging between two poles. Gravity pulls it down, but the poles hold it up. The shape of the rope depends on how heavy it is at every point.

• The Problem: We want to find the exact shape of the rope.
• The Math: This is a differential equation (a rule describing how the rope bends).
• The Connection: The authors show that finding the shape of the rope is the same as finding a "fixed point" in their noisy metric space.

They prove that under certain conditions (the rope isn't too heavy or chaotic), there is one and only one correct shape for the rope. They then use a computer to simulate this, showing that if you start with a guess and keep refining it (iteration), the computer's answer quickly settles on the correct shape.

The Second Discovery: The "Banach" Connection

In the final section, the authors show that their new, fancy theorem is actually a "super-version" of an old, famous theorem called the Banach Fixed Point Theorem.

• The Metaphor: Think of the Banach theorem as a standard bicycle. It works great on flat roads. The new theorem in this paper is like a mountain bike with suspension. It works on flat roads and on bumpy, noisy terrain (perturbed spaces).
• They prove that if you take their new theorem and remove the "noise," it turns back into the standard bicycle theorem. This validates their work and shows it's a natural extension of existing math.

Summary: What Did They Actually Do?

1. Acknowledged Reality: They admitted that in the real world, measurements are never perfect (they have "perturbations").
2. Created a New Tool: They defined a new mathematical rule (F-perturbed mapping) that works even when measurements are flawed.
3. Proved it Works: They used logic to show that this rule always leads to a single, unique solution.
4. Tested it: They used a computer to solve a physics problem (the shape of a rope/heat flow) and showed that their method finds the answer quickly and accurately.

In a nutshell: The authors built a mathematical bridge that allows us to find perfect solutions even when our data is imperfect and noisy. It's a way of saying, "Even with a broken ruler, we can still find the exact center."

Technical Summary

1. Problem Statement

The paper addresses the need to extend fixed point theory to perturbed metric spaces, a generalization of standard metric spaces where the distance measurement is subject to small errors or perturbations. While fixed point theorems (like Banach's) are well-established in standard and generalized metric spaces (e.g., $b$-metric, $G$-metric), the specific application of Wardowski's $F$-contraction concept to perturbed metric spaces had not been fully explored.

The authors aim to:

1. Define a new class of mappings called $F$-perturbed mappings.
2. Establish existence and uniqueness theorems for fixed points in complete perturbed metric spaces.
3. Apply these theoretical results to prove the existence of a unique solution for a second-order boundary value problem (BVP).
4. Provide a secondary theorem that generalizes the Banach fixed point theorem within this framework.

2. Methodology and Definitions

Perturbed Metric Space

The authors utilize the definition introduced by Jleli and Samet [14]. A triple $(X, D, P)$ is a perturbed metric space if:

• $D: X \times X \to [0, \infty)$ is the perturbed distance.
• $P: X \times X \to [0, \infty)$ is the perturbation function.
• The "exact metric" $d(x, y) = D(x, y) - P(x, y)$ satisfies the standard metric axioms (non-negativity, identity of indiscernibles, symmetry, and triangle inequality).

$F$-Perturbed Mappings

Inspired by Wardowski's $F$-contractions, the authors define an $F$-perturbed mapping $T: X \to X$. A mapping is an $F$-perturbed mapping if there exists $\tau > 0$ and a function $F: \mathbb{R}^+ \to \mathbb{R}$ satisfying specific conditions (strictly increasing, limit behavior at 0, and a specific limit involving a power $k$) such that:
$$D(Tx, Ty) > 0 \implies \tau + F(D(Tx, Ty)) \leq F(D(x, y))$$
This inequality replaces the standard contraction condition $d(Tx, Ty) \leq \alpha d(x, y)$.

Topological Concepts

The paper defines convergence and Cauchy sequences in the context of the exact metric $d$. A sequence is "perturbed convergent" if it converges in $(X, d)$. Completeness is defined similarly: every perturbed Cauchy sequence must converge in the space.

3. Key Contributions and Results

Main Theorem (Theorem 3.3)

Statement: Let $(X, D, P)$ be a complete perturbed metric space and $T: X \to X$ be a perturbed continuous $F$-perturbed mapping. Then $T$ has a unique fixed point in $X$.

Proof Strategy:

1. Construct an iterative sequence $x_{n+1} = Tx_n$.
2. Show that the sequence of distances $\gamma_n = D(x_{n+1}, x_n)$ decreases rapidly such that $\lim_{n \to \infty} F(\gamma_n) = -\infty$, implying $\lim_{n \to \infty} \gamma_n = 0$.
3. Prove that $\{x_n\}$ is a Cauchy sequence in the exact metric $d$.
4. Use the completeness of the space to establish the existence of a limit $x^*$.
5. Use the perturbed continuity of $T$ to show $Tx^* = x^*$.
6. Prove uniqueness via contradiction using the $\tau$ condition.

Application to Boundary Value Problems (Theorem 4.1)

The authors apply the main theorem to the second-order BVP:
$$-u''(t) = f(t, u(t)), \quad t \in (0, 1), \quad u(0) = u(1) = 0$$

• Transformation: The BVP is converted into an integral equation using Green's function $G(t, s)$.
• Space Definition: The space $X = C[0, 1]$ is equipped with a specific perturbed metric $D(u_1, u_2) = \sup |u_1 - u_2| + |u_1(0) - u_2(0)|$.
• Condition: If the non-linear term $f$ satisfies a specific exponential contraction condition involving $\tau$, the integral operator becomes an $F$-perturbed mapping.
• Result: The BVP has a unique solution.

Generalized Theorem (Theorem 6.1)

The paper presents a second theorem generalizing the Banach Fixed Point Theorem. It states that if $D(T^n x, T^n y) \leq a_n D(x, y)$ where $\sum a_n$ is convergent, then $T$ has a unique fixed point. This serves as a bridge to show that the standard Banach contraction is a special case of this framework.

4. Numerical Validation

To demonstrate the practical utility of the theory, the authors provide a numerical example:

• Problem: Solving the integral equation derived from the BVP with $f(s, u) = \frac{s+0.5}{2} \sin(u)$.
• Method: An iterative scheme $u_{n+1} = T u_n$ was implemented starting with an initial guess $u_0(t) = t$.
• Results:
  • The sequence $u_n(t)$ was shown to converge to the exact solution $u(t) = t$.
  • Graphs (Figure 3) illustrate the convergence of the iteration and the decay of the sup-norm error $\|u_{n+1} - u_n\|_\infty$.
  • This confirms that the theoretical fixed point approach yields a valid and computable solution.

5. Significance and Conclusion

• Theoretical Advancement: The paper successfully extends the powerful $F$-contraction framework (Wardowski) to the more complex setting of perturbed metric spaces, acknowledging that real-world measurements often contain errors.
• Practical Application: It provides a rigorous mathematical tool for solving differential equations where the underlying space or measurement is "noisy" or perturbed, specifically validating the existence of solutions for second-order BVPs.
• Computational Relevance: The inclusion of a numerical experiment with convergence analysis bridges the gap between abstract fixed point theory and numerical analysis, showing that these theorems can guide the development of stable iterative algorithms.
• Future Scope: The authors suggest that these results open avenues for developing further fixed point results for various types of contraction mappings in perturbed environments, which could be beneficial for researchers in functional analysis and applied mathematics.