Fixed Point Results in Non-Linear (Θ, ℜm)-Rational type Contractive Mappings with Application in Fractional Differential Equations

Mathematics often feels like a search for order in a chaotic world, a way to predict where things will settle when they are pushed, pulled, or changed. At the heart of this search lies a powerful idea known as a "fixed point." Imagine a map of a city laid out on the actual ground it represents. No matter how you crumple the paper or shift it slightly, there is always at least one point on the map that sits directly above the exact location it describes. That point is a fixed point. In the real world, this concept helps scientists understand everything from the stability of bridges to the behavior of economies. However, many real-world problems, particularly those involving change over time that isn't perfectly smooth or linear, are too complex for the standard mathematical tools used to find these points. These tools often rely on rigid rules about distance and shape that break down when the situation becomes messy or irregular.

To tackle these difficult problems, researchers have spent decades inventing new kinds of mathematical spaces—abstract environments with their own unique rules for measuring distance and connection. One such innovation is the "metric space," a framework that generalizes how we measure the gap between two things. More recently, mathematicians have developed even more flexible versions, such as "m-metric spaces," which allow for a bit more wiggle room in how distance is calculated, and "relation-based" spaces, which focus on whether two things are connected in a specific way rather than just how far apart they are. These advanced frameworks are essential for modeling complex systems where the usual rules of geometry do not apply, such as in the study of fractional differential equations. These equations describe processes that evolve in a "fractional" manner, meaning their change is neither entirely instantaneous nor entirely gradual, but something in between, like the way heat spreads through a material or how a virus might diffuse through a population.

In a recent study, a team of mathematicians from universities in Pakistan, South Africa, and Saudi Arabia set out to expand the toolkit available for solving these complex equations. Their work focuses on a specific type of mathematical mapping called a "non-linear (Θ, ℜm)-rational type contractive mapping." While the name is a mouthful, the concept is about finding a way to guarantee that a solution exists and is unique, even when the rules of the mathematical space are relaxed and the relationships between points are defined by a specific set of connections rather than just distance. The researchers introduced a new way of thinking about these mappings within "relation theoretic m-metric spaces." In these spaces, the environment is complete enough to ensure a solution can be found, but it is not necessarily complete in the traditional, stricter sense. This distinction is crucial because it allows the mathematics to handle scenarios that were previously too irregular to solve.

The team proved that under these new, more flexible conditions, a fixed point is guaranteed to exist. They showed that if you start with a specific point and follow the rules of this new mapping, the sequence of points you generate will eventually settle down to a single, stable location. This result is significant because it unifies and extends several previous findings, bringing together different approaches that were previously treated as separate. The researchers did not stop at theory; they applied their new findings to a practical problem: finding solutions for nonlinear fractional differential equations with integral boundary conditions. These are equations that describe systems where the state at one point depends on the sum of states over a range, a common feature in physics and engineering. By using their new fixed point results, the authors demonstrated that a solution to these difficult equations not only exists but is also unique, provided certain conditions are met.

To ensure their theoretical results were not just abstract exercises, the researchers tested their methods on linear fractional differential equations using two distinct numerical techniques: the shooting method and the homotopy perturbation method combined with a matrix approach. The shooting method is a strategy where a problem with fixed endpoints is turned into an initial value problem, and the starting conditions are adjusted until the solution hits the correct target. The homotopy perturbation method, on the other hand, builds a solution by starting with a simple approximation and gradually refining it. The team applied these methods to a specific equation where the exact solution was already known. They ran numerical experiments with a specific fractional order of 2.9 and compared their calculated results against the known exact solution. The results were strikingly close. At various points along the interval, the difference between their calculated approximation and the true answer was incredibly small, often in the range of millionths or billionths. For instance, at a specific point in the calculation, the error was less than two millionths.

The visual data presented in the study confirms that the curves generated by their numerical methods align almost perfectly with the curve of the exact solution. The graphs show the two lines tracing the same path, with the error between them remaining negligible throughout the entire range. This high level of accuracy suggests that the new mathematical framework is not only theoretically sound but also practically effective for solving real-world problems involving fractional calculus. The researchers concluded that their approach successfully extends the scope of fixed point theory, offering a robust way to handle equations that were previously difficult to solve. By proving that solutions exist and providing reliable methods to find them, this work opens the door for more accurate modeling of complex, non-linear systems in science and engineering. The study stands as a testament to how refining the fundamental rules of mathematics can lead to better tools for understanding the intricate dynamics of the world around us.