Abstract

In this paper, the existence and uniqueness of solutions for a nonlinear fractional differential equation with a two-point boundary condition in a Banach space are investigated by using the contraction mapping principle and the Brouwer fixed-point theorem with Bielecki norm. The iterative scheme of the numerical solution for the nonlinear two-point boundary value problem will be discussed and illustrated by solving some problems. The well-known Ulam-Hyers and Ulam-Hyers-Rassias stability theorems are employed to establish the stability of solutions to the boundary value problem. In the end, we provided a couple of examples to support our results.

1. Introduction

Fractional calculus is an important mathematical topic due to its theoretical foundation and multiple applications in physical, chemical processes, and engineering; for instance, see [16]. The two-point boundary value problem occurs in applied mathematics, theoretical physics, engineering, control, and optimization theory (see [7]). The existence of solutions of initial and boundary value problems of fractional differential equations by the help of different fixed-point theorems has been discussed by many mathematicians, and the readers are referred to see the monographs [821]. In [22], the authors give the existence results for two-point boundary value problem of fractional differential equations at resonance by means of the coincidence degree theory. Mongkolkeha and Gopal [23] proposed new common fixed-point theorems for the Ciric type generalized F-contraction in metric spaces with the w-distance. The fixed points for the monotone -nonexpansive and generalized -nonexpansive mappings in hyperbolic space have been approximated by [24].

In fact, the subject of numerical methods for solving fractional differential equations has gained prominence and has been discussed by several authors, including a series of papers [2531] and references cited therein, which include some recent studies on the approximation method for differential equations of fractional order. El-Ajou et al. [32] extended the application of the homotopy analysis method (HAM) to provide symbolic approximate solution for two-point boundary value problems of fractional order. Lyons et al. [33] prove an extension of Picard’s iterative existence and uniqueness theorem to Caputo fractional ordinary differential equations, when the nonhomogeneous term satisfies the usual Lipschitz’s condition. In [34], the successive approximation method was applied to solve the temperature field based on the given Mittag-Leffler-type.

Nie et al. [35] investigated the existence and numerical method of two-point boundary value problems for fractional differential equations with Caputo’s derivative or Riemann-Liouville derivative. The solutions can be deduced by the contraction mapping principle and fractional Green function. For the Caputo’s derivative case, it has the form

For Riemann-Liouville derivative case, it has the form

More recently, Hyers-Ulam type stability theorems for nonlinear fractional differential equations have attracted a lot of attention as an interesting field and have been investigated in many papers (see [3640]). Murad et al. [41] studied the existence, Ulam-Hyers, and Ulam-Hyers-Rassias theorems of solutions to a differential equation of mixed Caputo-Riemann fractional derivatives.

Dai et al. [42] discussed the existence and Hyers-Ulam and Hyers-Ulam-Rassias stability of solutions for the fractional differential equation with boundary condition. Prasad et al. [43] investigated the existence and Ulam stability of the fractional-order iterative two-point boundary value problem which has the form where is the Caputo fractional derivative and and .

In this paper, we consider the nonlinear fractional differential equation which has the form

with boundary conditions where , , , and are the Caputo fractional derivatives, and are constants. The main objective is to study the existence of a solution to the boundary value problem (5) and (6). The results are based on Brouwer’s fixed-point theorem and Banach contraction mapping principle. The analytical approximate technique to obtain the solution is a part of this work, and some examples are illustrated to explain the algorithm. Furthermore, we discuss the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of the boundary value problem (5) and (6). Some examples are also constructed to illustrate and validate the main results.

2. Preliminaries

Let us give some definitions and lemmas that are basic and needed at various places in this work.

Definition 1 (see [44]). Let be a function which is defined almost everywhere on . If , then provided that this integral (Lebesgue) exists.

Definition 2 (see [5]). For a continuous function , the Caputo derivative of fractional order is defined as provided that exists, where denotes the integer part of the real number .

Lemma 3 (see [1]). Let . If we assume , then the Caputo fractional differential equation has the solution where , and

Lemma 4 (see [1]). Let with fractional derivative of order that belongs to Then, for where is the smallest integer greater than or equal to

Theorem 5 (see [45]) (Brouwer’s fixed-point theorem). Let be a nonempty compact (closed and bounded) convex set in and be a continuous self-mapping. Then, has (at least) one fixed point in .

Theorem 6 (see [45]) (Banach contraction mapping principle). Let be a Banach space. If is a contraction, then has a unique fixed point in .

Lemma 7. Let and , ; then, the solution of the boundary value problem (5) and (6) is given by

Proof. Byapplying the Lemma 4, we may reduce equation (5) to an equivalent equation Using the boundary condition (6), we find that Substituting the values of and in equation (13), the result is The derivative can be written as

3. The Existence of Solution

This section deals with the existence and uniqueness of solution for the fractional differential equation (5) with boundary condition (6). Let be the Banach space endowed with a Bielecki norm

Let the space , equipped with the norm

which is a Banach space, where is a fixed constant. Consider the following assumptions:

(H1) There exists a function such that , where .

(H2) There exists constants such that

for each and all Let us set the following notation for convenience:

Our results are based on the Brouwer’s fixed-point theorem and Banach contraction principle.

Theorem 8. Suppose that (H1) holds. Then, the boundary value problem (5) and (6) has at least one solution on .

Proof. Define the operator as and the operator can be written as Firstly, we will prove that , where , and choose , for , and the following is obtained: and by using (H1), then where Thus, we have Now to establish the bounded of equation (22) by (H1), get where and , and we have . Hence,
According to the Brouwer fixed-point theorem, the boundary value problem (5) and (6) has at least one solution.

Theorem 9. Assume that (H2) holds. If where then the boundary value problem (5) and (6) has a unique solution.

Proof. We prove that is a contraction. Let . Then, and by the condition (H2), the result is and using Holder inequality, the first term of equation (32) becomes and for the second term, let , and it follows By the same way, Then, we have Hence, we conclude that the problem (5) and (6) has a unique solution by the contraction mapping principle.

Example 1. Consider the following fractional boundary value problem: Here, , , and ; then (H2) is satisfied with and , and one can arrive at the following results: Hence, by Theorem 9, the boundary value problem (38) has a unique solution on .

Example 2. Consider the following fractional boundary value problem: Here, , , and , and according to the Lipschitz condition, we have Finally, the following obtained Therefore, from Theorem 9, we conclude that boundary value problem (40) has a unique solution.

4. Iterative Numerical Schema

For the solution of the boundary value problem (5) and (6), an iterative schema is provided. Starting with and , the width , and , the integral equations (21) and (22) are numerically evaluated to obtain the sequence , . The trapezoidal rule is most convenient for estimating , , and the approximation for (21) and (22) becomes where

and the process of iteration may be terminated by setting a criterion where is a constant which is to be taken smaller than the discretization error in for the solution of the boundary value problem (5) and (6).

5. Convergence of and

To prove the convergence of the iteration schema (42) and (43), let us suppose that be continuous function that satisfies the conditions.

(H3) There exists a constant such that ,

for each and all . After simplifying equations (42) and (43), we have

Now, Computing and from equations (47) and (48) and by using (H2) and (H3), the result is

Let

and so on, in general

If then the process of iteration is convergent and the bound of truncation error is given by

6. Numerical Illustrations

To show the efficiency of this method, we will approximate the solutions for some fractional differential equations of order and , using a prepared program in Matlab. To solve these problems, we used equations (42) and (43) to obtain the sequences By using the exact solutions, we computed the error at each pivotal point. The partial output of these error terms is presented in Tables 1 and 2.

Example 3. Consider the fractional boundary value problem The exact solution is

A comparison of the absolute errors, computed by the proposed method with , , and at , is calculated in Table 1. The approximate solutions obtained with the exact solution of corresponding fractional-order equation when , , and are given in Figure 1(a).

Example 4. Consider the fractional boundary value problem The exact solution is

In Figure 1(b), the approximate solutions obtained with together with the exact solution of this problem are plotted. Furthermore, by considering , , and , the absolute errors at some selected points are reported in Table 1.

Example 5. Consider the fractional boundary value problem The exact solution is

Table 1 shows the absolute error between exact and numerical solutions when , , and . Figure 2(a) compares both the exact and numerical solutions for the fractional differential equation (56) with , , and in some points

Example 6. Consider the boundary value problem

. The exact solution is

The numerical results of Example 6 for the values and are shown in Figure 2(b). In addition, the absolute error is presented in Table 2 when , , and

7. Stability Theorems

In this section, we investigate the Ulam-Hyers and Ulam-Hyers-Rassias stability of the boundary value problem (5) and (6). For the definitions of Ulam-Hyers stability and Ulam-Hyers-Rassias stability, see [46]. For , the norm is defined as

Definition 10. Equation (5) is Ulam-Hyers stability if there exists a real number such that for each and for each solution of the inequality there exists a solution of equation (5) with

Definition 11. Equation (5) is Ulam-Hyers-Rassias stability with respect to if there exists a real number