FRACTIONAL ANALYTIC SOLUTIONS AND FIXED POINT RESULTS WITH SOME APPLICATIONS

. This study explores a collection of theorems that provide valuable insights into the existence and properties of ﬁxed points in mathematical and real-life problems. The ﬁrst theorem establishes the existence of ﬁxed points for contractive mappings, guaranteeing the convergence of iterative sequences. Building upon this result, the second theorem extends the concept to complete metric spaces, enabling the convergence analysis of sequences generated by repeated application of the mapping. To demonstrate the practical relevance of these theorems, a real-life example is presented in the context of population dynamics. By formulating the dynamics as a system of equations, the theorems are applied to determine equilibrium points and analyze the long-term behavior of populations. Numerical solutions and graphical representations shed light on the stability and coexistence of species, showcasing the applicability of the theorems in ecological, economic, and engineering contexts. Moreover, the introduction of fractional calculus in the third theorem enriches the analysis by considering fractional derivatives in self-mappings. This theorem establishes a connection between sequence convergence and the existence of ﬁxed points, providing a powerful tool for studying complex systems with fractional dynamics.


INTRODUCTION
In various branches of mathematics and applied sciences, the study of mathematical theorems and their practical applications is of great significance.The development of powerful results and their subsequent application in real-world scenarios has led to numerous advancements and insights in diverse fields.In this paper, we explore some key theorems and their applications, highlighting their relevance and impact.
The Banach Fixed Point Theorem, also known as the Contraction Mapping Theorem, stands as a fundamental result in fixed point theory.This theorem guarantees the existence and uniqueness of fixed points for certain types of mappings in complete metric spaces [4].The theorem's implications extend beyond pure mathematics and find applications in various disciplines, such as physics, computer science, and economics.The convergence properties provided by the theorem enable the analysis of iterative processes and the study of equilibrium states in dynamical systems.For details see [9,12,13,14].
Expanding upon the Banach Fixed Point Theorem, several generalizations and variations have been developed to address different scenarios and mathematical structures.One such extension is the concept of metric completeness, which allows the examination of fixed points in more general spaces beyond Euclidean settings [11].Complete metric spaces offer a broader framework for studying convergence and fixed point properties in function spaces, topological spaces, and other mathematical domains.The application of these extended theorems aids in understanding the behavior of systems with complex dynamics.
Real-life applications of fixed point theorems can be found in diverse fields.For instance, in economics, fixed point theory provides a foundation for analyzing general equilibrium models, where multiple economic variables interact to reach equilibrium states [5].By formulating the problem as a fixed point equation, economists can study the stability and convergence properties of the system.This analysis aids in predicting market dynamics, determining optimal resource allocation, and understanding macroeconomic phenomena.
In the field of computer science, fixed point theorems find application in algorithm design and optimization.Algorithms that rely on iterative processes can be analyzed using fixed point theory to ensure convergence and assess computational efficiency.Furthermore, fixed point theorems have been employed in graph theory and network analysis to identify critical points and steady states in complex networks [15].These findings have implications for understanding social networks, transportation systems, and communication networks.
Recent developments in fractional calculus have introduced new dimensions to fixed point theory.fractional calculus incorporates non-local and non-integer order derivatives, leading to the fractional fixed point theorem.This theorem establishes connections between fractional dynamics and the existence of fixed points in fractional spaces.The applications of this extension can be found in the modeling of anomalous transport phenomena, fractional differential equations, and fractional-order control systems [1,2,3].
In this paper, we present a comprehensive exploration of these theorems and their applications in various scientific domains.We provide illustrative examples, numerical simulations, and graphical representations to demonstrate their practical utility and impact.By understanding the theoretical results and leveraging computational techniques, researchers and practitioners can gain valuable insights into the behavior of complex systems, make informed decisions, and develop innovative solutions.

PRELIMINARIES
In this paper, we consider the concept The revised version of the definition of a metric space, which is widely accepted and introduced by Rudin [17], is as follows: Definition 2.1.Let X be a non-empty set.A metric space is a set X equipped with a distance function d : X × X → R that satisfies the following properties: (1) Non-negativity: d(x, y) ≥ 0 for all x, y ∈ X, and d(x, y) = 0 if and only if x = y.
Metric spaces provide a framework for studying the concept of distance and convergence.
They serve as a fundamental setting for analyzing fixed points and their properties in various mathematical contexts.
In the realm of mathematical analysis and its applications, fixed points hold immense significance.Inspired by Rudin [16], we define a fixed point in the following manner: Definition 2.2.Let X be a metric space, and consider a mapping T : X → X.A point x ∈ X is termed a fixed point of T if it satisfies the equation T (x) = x.
Fixed points offer invaluable insights into the characteristics and behaviors of mappings and their associated iterative algorithms.The existence and uniqueness of fixed points have profound implications across various mathematical and applied disciplines, providing a foundation for profound discoveries and practical advancements.
In the realm of metric spaces, the concept of a Cauchy sequence holds great significance.
Inspired by Rudin [16], we provide the following definition: Definition 2.3.Consider a metric space (X, d).A sequence x n in X is termed a Cauchy sequence if, for every ε > 0, there exists N ∈ N such that for all m, n ≥ N, we have d(x m , x n ) < ε.
The notion of a Cauchy sequence plays a pivotal role in the analysis of metric spaces, as it characterizes sequences where the terms become arbitrarily close to each other as the sequence progresses.By capturing the idea of convergence within a sequence, Cauchy sequences lay the groundwork for comprehending the concepts of convergence and completeness within the realm of metric spaces.

Definition 2.4. [10]
A function f : X → Y , where X and Y are metric spaces, satisfies the Lipschitz condition if there exists a constant L ≥ 0 such that for all x 1 , x 2 ∈ X, we have: where d X and d Y denote the distance functions in X and Y , respectively.
The Lipschitz condition ensures that the function does not exhibit abrupt or excessive changes, guaranteeing a certain level of smoothness and control over its behavior.Functions that satisfy this condition are known as Lipschitz continuous functions.The importance of the Lipschitz condition lies in its wide range of applications across different branches of mathematics and beyond.It is a key concept in the analysis of differential equations, optimization problems, and numerical methods.The Lipschitz continuity of a function plays a crucial role in establishing existence and uniqueness of solutions, convergence properties of iterative algorithms, and stability of dynamical systems.Definition 2.5.[6] The Caputo fractional derivative of order α ∈ (0, 1) for a function f (t), denoted by D α t f (t), is defined as: where Γ(•) denotes the gamma function.
The Caputo fractional derivative of order α provides a powerful tool for analyzing and modeling various phenomena exhibiting non-local and memory-dependent behavior.It extends the concept of differentiation to fractional orders, capturing the fractional-order dynamics of a function.The definition involves an integral representation that accounts for the non-locality of the derivative.The Caputo fractional derivative has found applications in various fields, including physics, engineering, and finance, offering a versatile approach to describing complex systems with fractional dynamics.
Fractional analytic solutions refer to a special class of solutions in fractional calculus that possess desirable properties and exhibit analytic behavior.In particular, they are solutions to fractional differential equations that can be represented by convergent power series expansions.
A fractional differential equation involving fractional analytic solutions can be written as: where D α denotes the fractional derivative operator of order α, f (x) is the unknown function, and g(x) is a given function.The goal is to find a fractional analytic solution f (x) that satisfies the equation.
What distinguishes fractional analytic solutions is their ability to be expressed as power series expansions that converge within a certain domain.These power series have the form: where the coefficients a n are determined based on the given equation and initial/boundary conditions.The convergence of the power series allows for the analytic representation of the solution, facilitating further analysis and computations.
The numerical solution of fractional differential equations is an essential aspect of fractional calculus, enabling the study of complex systems and phenomena involving fractional derivatives.Two commonly used methods for numerical approximation are the Grünwald-Letnikov approximation and the Euler method.
The Grünwald-Letnikov approximation is based on the idea of approximating the fractional derivative by a finite difference quotient.Given a fractional derivative of order α and a function f (x), the Grünwald-Letnikov approximation can be defined as: where h is the step size and N is the number of terms in the summation.By discretizing the domain and applying this approximation, one can numerically solve fractional differential equations.
The Euler method, on the other hand, is a well-known numerical method for solving ordinary differential equations.It can also be adapted to solve fractional differential equations.The basic idea is to approximate the fractional derivative by a finite difference quotient and update the solution iteratively.For a fractional derivative of order α and a function f (x), the Euler method can be expressed as: where h is the step size.By discretizing the domain and applying this iteration, the solution of the fractional differential equation can be approximated.
Both the Grünwald-Letnikov approximation and the Euler method have their advantages and limitations.The Grünwald-Letnikov approximation offers higher accuracy but requires more computational effort due to the summation involved.On the other hand, the Euler method is simpler and computationally efficient but may suffer from numerical stability issues for certain types of fractional differential equations.These numerical methods play a significant role in the analysis and simulation of fractional differential equations.They enable the investigation of the behavior and properties of fractional systems, allowing researchers to gain insights into complex phenomena involving fractional derivatives.

MAIN RESULTS
In the present study, we introduce several theorems: Theorem 3.1.Let D α denote the Caputo fractional derivative of order α ∈ (0, 1), and let f : [a, b] × R → R be a continuous function satisfying the following conditions: (1) Lipschitz condition: There exists a constant (2) Uniform boundedness: There exists a constant Then, the initial value problem Let Applying the Grünwald-Letnikov approximation of the Caputo fractional derivative, we obtain where Therefore, we have Now, using the Lipschitz condition and the uniform boundedness of f , we can show that T is a contraction mapping: This shows that T is a contraction mapping.
By the Banach fixed point theorem, there exists a unique fixed point Moreover, ϕ * satisfies the integral equation which is equivalent to the fractional differential equation with the initial condition.Therefore, ϕ * is the desired fractional analytic solution.
Hence, the initial value problem has a unique fractional analytic solution on [a, b].
This theorem provides a framework for establishing the existence and uniqueness of fractional analytic solutions for fractional differential equations.The proof relies on the properties of the Caputo fractional derivative, the Lipschitz condition, and the uniform boundedness of the function f .The resulting fractional analytic solution can be employed to study the dynamics and behavior of fractional systems in various scientific and engineering applications.
To find the fractional analytic solution, we can apply the fixed point theorem.Let T : ]) be defined by We will show that T is a contraction mapping.Let ϕ 1 , ϕ 2 ∈ C([0, 1]).By the Lipschitz condition of the function −2ϕ + 3t α , we have where shows that T is a contraction mapping.
By the Banach fixed point theorem, there exists a unique fixed point ϕ * of T in C([0, 1]).
Moreover, ϕ * satisfies the integral equation which is equivalent to the fractional differential equation with the initial condition.Therefore, ϕ * is the desired fractional analytic solution.
Hence, the initial value problem has a unique fractional analytic solution on [0, 1].
Example 3.2.Consider a real-life scenario where fractional analytic solutions and fixed point theory can be applied.Suppose we have a population dynamics model that describes the growth of a certain species over time.The model is given by the following fractional differential equation: where N(t) represents the population size at time t, r is the growth rate, K is the carrying capacity, and D α denotes the Caputo fractional derivative of order α ∈ (0, 1).
To analyze the population dynamics, we can seek both the numerical and analytic solutions of the fractional differential equation.By applying numerical methods and the fixed point theory, we can gain insights into the behavior of the population over time.
Numerical Solution: Let's consider a specific scenario with the following parameter values: r = 0.1 (10% growth rate) and K = 1000 (carrying capacity).We want to determine the population size over a period of 20 years.
Using numerical methods, such as the Grünwald-Letnikov approximation, we can approximate the fractional derivative and solve the fractional differential equation numerically.We can discretize the time interval into small time steps and calculate the population size at each time step.
For example, let's consider a time step of ∆t = 0.1 (corresponding to 1 month).We can start with an initial population size of N(0) = 100 and use the numerical approximation method to calculate the population size at each time step over the 20-year period.
Analytic Solution: In addition to the numerical solution, we can also find the analytic solution of the fractional differential equation.The analytic solution for this model is given by: where N 0 is the initial population size.
Graphical Comparison: We can compare the numerical solution with the analytic solution by plotting them on a graph.
The plot will show the population size over the 20-year period, demonstrating the agreement between the numerical and analytic solutions.The table presents the numerical approximation of the population size at different time points using the Grünwald-Letnikov method.The analytic solution provides an exact mathematical expression for the population size.The graph visually compares the numerical solution (blue line) with the analytic solution (red line) over the 20-year period, demonstrating the agreement between the two approaches.This real-life example showcases the practical application of fractional analytic solutions and fixed point theory in population dynamics modeling, providing insights into the growth and behavior of a species over time.Theorem 3.2.Let X be a complete metric space, and let T : X → X be a contraction mapping with a contraction constant 0 ≤ k < 1.Consider the fractional differential equation where D α denotes the Caputo fractional derivative of order α ∈ (0, 1).
for all t ∈ [a, b] and x 1 , x 2 ∈ X, where L > 0 is a Lipschitz constant, then the fractional differential equation has a unique solution y(t) in X.
Proof.We will prove the theorem by utilizing the Banach fixed-point theorem.Let X C be the space of continuous functions on [a, b] equipped with the metric d C , defined as where d is the metric on X.It can be shown that (X C , d C ) is a complete metric space.
Consider the operator T : X C → X C defined by We will show that T is a contraction mapping.
Let ϕ 1 , ϕ 2 ∈ X C .By the Lipschitz condition of f , we have Hence, the fractional differential equation has a unique solution in X.
Example 3.3.Consider a population of rabbits in a controlled environment.The population growth can be modeled by a fractional differential equation of the form: where D α represents the Caputo fractional derivative, P(t) is the population at time t, k is the growth rate, and K is the carrying capacity of the environment.
To illustrate this, let's consider the following parameters: k = 0.3, K = 100, P(0) = 10, and α = 0.8.We will solve the fractional differential equation numerically using a suitable numerical method and compare it with the analytic solution.
Numerical Solution: We will use the Euler method to numerically solve the fractional differential equation.The population will be evaluated at discrete time steps, t i = ih, where i is the index and h is the time step size.
Using the Euler method, the numerical solution is given by the recursion formula: where D α P i is the fractional derivative approximation.
Let's compute the numerical solution for 0 ≤ t ≤ 5 with a time step size of h = 0.1.The analytic solution of the fractional differential equation can be obtained using the Laplace transform method.For the given parameters, the analytic solution is given by: The table and graph above demonstrate the population growth of rabbits over time.The numerical solution is obtained using the Euler method, while the analytic solution is derived using the Laplace transform method.As shown, both solutions exhibit similar growth patterns, confirming the accuracy of the numerical method.Theorem 3.3.Let X be a complete metric space, and let T : X → X be a self-mapping.Assume that there exists a constant α ∈ (0, 1) and a function φ : X → R + such that for all x, y ∈ X, represents the fractional derivative of order α.
If there exists x 0 ∈ X such that the sequence {x n } defined by x n = T n x 0 converges to a point x * ∈ X, then x * is a fixed point of T .
Proof.We utilize the properties of the fractional derivative and establish the conditions under which the convergence of the sequence implies the existence of a fixed point.Assume that {x n } converges to x * .We need to show that x * is a fixed point of T , i.e., T x * = x * .
Since {x n } converges to x * , by the definition of convergence, for any ε > 0, there exists N ∈ N such that for all n ≥ N, we have d(x n , x * ) < ε.Consider n ≥ N. Using the property of T and the fractional derivative D α 1 2 , we have is a non-decreasing operator, we have .
Using the triangle inequality, we get Substituting this back into the inequality, we have Taking the limit as n approaches infinity, we have Since {x n } converges to x * , we have lim n→∞ φ (x n ) = φ (x * ) and lim n→∞ d(x n , x * ) = 0. Therefore, we obtain This implies d(T x n , T x * ) = 0, which in turn implies T x * = x * .Thus, x * is a fixed point of T .
To prove the uniqueness, assume there are two fixed points x 1 and x 2 of the operator T , i.e., T x 1 = x 1 and T x 2 = x 2 .We aim to show that Consider the distance between x 1 and x 2 : Using the properties of the operator T and the fractional derivative D α 1 2 , we have: is a non-decreasing operator, we have: .
Substituting this back into the inequality, we have: Since 0 ≤ φ (x 1 ) α ≤ φ (x 1 ) α + φ (x 2 ) α , we can rewrite the inequality as: ) α }, we can further simplify the inequality as: Dividing both sides of the inequality by d(x 1 , x 2 ) α (since d(x 1 , x 2 ) > 0), we have: Since α ∈ (0, 1], we have 1 − α ≥ 0, and thus we can further simplify the inequality as: Now, let's assume that x 1 = x 2 .Without loss of generality, suppose φ (x 1 ) α > φ (x 2 ) α .Then, we have: Let M = 2φ (x 1 ) α .Since φ : X → R + is continuous, M is a positive constant.Thus, we can rewrite the inequality as: from both sides of the inequality, we have: Since d(x 1 , x 2 ) > 0, we can multiply both sides of the inequality by d(x 1 , x 2 ) to obtain: Since K(φ (x 1 ) + φ (x 2 )) is a constant and Md(x 1 , x 2 ) is non-negative, we have: . Then, we have: Since d(x 1 , x 2 ) > 0, we can divide both sides of the inequality by d(x 1 , x 2 ) to obtain: But this contradicts our assumption that x 1 = x 2 .Therefore, we must have x 1 = x 2 .Thus, the fixed point of T is unique.
Corollary 3.1.Let X be a complete metric space, and let T : X → X be a self-mapping.Suppose there exists a constant α ∈ (0, 1) and a function φ : X → R + such that for all x, y ∈ X, represents the fractional derivative of order α.
If there exists x 0 ∈ X such that the sequence {x n } defined by x n = T n x 0 converges to a point x * ∈ X and φ (x * ) = 0, then x * is a fixed point of T .
Proof.Consider the sequence {x n } defined by x n = T n x 0 .Since {x n } converges to x * , by the uniqueness part of Theorem ??, we know that x * is a fixed point of T .Now, let's prove that φ (x * ) = 0. Suppose, for the sake of contradiction, that φ (x * ) > 0.Then, by the inequality in the corollary, we have Since {x n } converges to x * , we can take the limit as n approaches infinity, which gives us which is a contradiction.Hence, we must have φ (x * ) = 0, and therefore, x * is a fixed point of T .
Example 3.4.Consider a population of organisms whose growth is modeled by the following equation: where P(t) is the population size at time t, P 0 is the initial population size, r is the growth rate, and e is the base of the natural logarithm.
We can rewrite this equation in the form of a self-mapping T : X → X by letting T (P) = P 0 e r∆t , where ∆t is a small time interval.In this case, X represents the space of possible population sizes.
Let's assume that α = 1 2 , and φ (P) = √ P. To demonstrate the convergence of the sequence {x n } defined by x n = T n x 0 , let's consider an example with the following parameters: We can compute the numerical and analytic solutions and present them in a combined table.Graphical Representation: Let's visualize the convergence of the numerical solution and compare it with the analytic solution using a graph.4, the numerical and analytic solutions converge to similar values.This convergence is further illustrated in Figure 4, where the blue line represents the numerical solution and the red line represents the analytic solution.
By applying Theorem 3.3, we can conclude that the limiting value of the sequence {x n }, which is approximately 159.402, represents a fixed point of the self-mapping T .In the context of population growth, this corresponds to a stable population size under the given growth dynamics.

APPLICATIONS
In this section, we shall leverage the theoretical insights garnered from the preceding section to elucidate the existence and uniqueness of solutions for fractional differential equations falling under the Caputo class and others.By delving into the theoretical underpinnings of these equations, we can gain a deeper comprehension of their origins and devise strategies to solve them.To delve further into this fascinating topic and applications, we recommend consulting contemporary publications such as ( [1], [2], [3], [18]).
4.1.Future Value of an Investment.Suppose you want to invest a certain amount of money in a savings account that offers compound interest.The formula to calculate the future value of the investment is given by: where FV is the future value, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.
Let's consider an example where you invest $5000 at an annual interest rate of 5%, compounded quarterly over a period of 10 years.By plugging in the values into the formula, we can calculate the future value using both numerical and analytic solutions.The numerical solution involves directly evaluating the formula using the given values.The analytic solution, on the other hand, involves evaluating the formula using the limits and properties of exponential functions.4.2.Investment Growth.Let's consider an investment portfolio with an initial value of P 0 dollars.The portfolio's value grows over time with an annual interest rate of r.We can model the growth of the portfolio using the equation: where P(t) represents the portfolio value at time t.
To analyze the convergence of the portfolio value, we can define a self-mapping T : X → X as T (P) = P 0 1 + r 100 ∆t , where ∆t is the time interval between investments.In this case, X represents the space of possible portfolio values.
Let's assume that α = 1 2 , and φ (P) = √ P. To demonstrate the convergence of the sequence {x n } defined by x n = T n x 0 , let's consider an example with the following parameters: We can compute the numerical and analytic solutions and present them in a combined table.Graphical Representation: We can also visualize the convergence of the numerical and analytic solutions using a graph.4.3.Population Dynamics.Consider a population of three species: Species 1, Species 2, and Species 3. The population sizes of these species can be described by a system of equations representing their dynamics over time.
Let's assume that the population sizes at time t are represented by the variables x 1 (t), x 2 (t), and x 3 (t), respectively.We can model the population dynamics using the following system of equations: where f 1 , f 2 , and f 3 represent the growth rates or interaction functions for each species.
To find the equilibrium points of the system, we can solve the equations dx 1 dt = 0, dx 2 dt = 0, and dx 3 dt = 0. Using Theorem 3.3, we can find the fixed points of the system by mapping it to a self-mapping problem.Let X be the complete metric space representing the possible population states, and let T : X → X be the self-mapping defined as T (x 1 , x 2 , x 3 ) = ( f 1 (x 1 , x 2 , x 3 ), f 2 (x 1 , x 2 , x 3 ), f 3 (x 1 , x 2 , x 3 )).
To find the equilibrium points, we solve the equations dx 1 dt = 0, dx 2 dt = 0, and dx 3 dt = 0. (2, 1, 3) Table 7 shows the equilibrium points and their corresponding population sizes.P 1 represents an extinct population, P 2 represents a stable coexistence of all three species, and P 3 represents another stable coexistence with different population sizes.This application demonstrates how Theorem 3.3 can be applied to analyze and solve a system of equations representing population dynamics.The numerical solutions in the table and the graph provide insights into the equilibrium points and the behavior of the populations over time.

CONCLUSION
The theorems presented in this study provide powerful tools for analyzing various mathematical and real-life problems.Theorem 3.1 establishes the existence of fixed points for mappings satisfying certain contractive conditions, guaranteeing the convergence of iterative sequences.
Theorem 3.2 extends this concept to mappings defined on complete metric spaces, allowing for the convergence of sequences generated by repeated application of the mapping.Theorem 3.3 introduces the notion of fractional calculus, enabling the study of self-mappings with fractional derivatives.This theorem establishes a relationship between the convergence of a sequence and the existence of fixed points for the mapping.To illustrate the practical relevance

= y 0 ,
has a unique fractional analytic solution y(t) on [a, b].Proof.Consider the operator T : C([a, b]) → C([a, b]) defined by (T ϕ)(t) = y 0 + t a f (s, ϕ(s))ds, where C([a, b]) denotes the space of continuous functions on [a, b].We will show that T is a contraction mapping.

Example 3 . 1 .
Consider the initial value problem

FIGURE 1 .
FIGURE 1. Population Dynamics over 20 Years where L(b − a) is the Lipschitz constant.Since 0 ≤ L(b − a) < 1, we have shown that T is a contraction mapping.By the Banach fixed-point theorem, there exists a unique fixed point ϕ * of T in X C .Moreover, ϕ * satisfies the integral equation ϕ * (t) = y 0 + t a f (s, ϕ * (s))ds, which is equivalent to the fractional differential equation with the initial condition.Therefore, ϕ * is the desired unique solution in X.

100 e −0.3t 0. 8 .Figure 2 .
Figure 2. The numerical and analytic solutions by plotting the population growth over time

Figure 4 .
Figure 4. Convergence of the numerical and analytic solutions

Figure 8 .
Figure 7. Convergence of the solutions

TABLE 1 .
Numerical Approximation using Grünwald-Letnikov Method Tabel 2. Numerical solution for 0 ≤ t ≤ 5 with a time step size of h = 0.1

Table 4 .
Numerical and analytic solutions for the population growth

Table 5 .
Numerical and analytic solutions for the future value of an investment

Table 6 .
Numerical and analytic solutions for investment growth

Table 7 .
Equilibrium Points and Corresponding Population Sizes