Abstract
We study the existence and monotone iterative approximation of mild solutions of fractional-order neutral differential equations involving a generalized fractional derivative of order which can be reduced to Riemann–Liouville or Hadamard fractional derivatives. The existence of mild solutions is obtained via fixed point techniques in a partially ordered space. The approach is constructive and can be applied numerically. In particular, we construct a monotone sequence of functions converging to a solution which is illustrated by a numerical example.
1. Introduction
Differential equations with fractional order, commonly referred to as fractional differential equations, have important role for describing the dynamics of economic models [1], finance [2], engineering [3], and scientific systems [4–8]. In recent years, various types of the fractional derivatives are defined via fractional integrals and were studied including Riemann–Liouville, Caputo, Hadamard, Caputo–Hadamard, Hilfer, Hilfer–Hadamard, Caputo-type, and Liouville fractional derivatives [9–14].
Fractional calculus has triggered several research interests in various aspects including existence and uniqueness of solutions, stability and dynamical analysis, numerical methods for solutions, and modelling. Some authors investigated the nonlocal fractional operator based on a generalized Mittag-Leffler kernel [15, 16]. Dynamics of the fractional differential system is also the main research focus in many applications, for example, HIV modelling [8, 17, 18], COVID-19 modelling [7], and integrator circuit model [19].
In many applications, differential equations involve functions and derivatives that contain unknown functions at a shifted time. This class of differential equation is called a delay differential equation and is commonly used to describe systems when the rate of change depends on the stages in the previous times. Delay differential equations have become important for mathematical models of phenomena in physical, engineering, and biomedical sciences. Especially, many authors have studied epidemic models with time delays [20–22].
One special type of delay differential equation is the neutral differential equation which has been extensively studied by many researchers. Agarwal and Bahuguna [23] studied the existence and solution of the first-order neutral differential equation of Sobolev type with a nonlocal history condition:where in a real Banach space . The results were proved by using Schauder’s fixed point theorem. Agarwal et al. [24] studied the existence of fractional neural functional differential equations:where is the Caputo fractional derivative of order . The exisence results of solution of problem (2) were obtained by using Krasnoselskii’s fixed point theorem.
In addition, Ren et al. [25] investigated periodic solutions for a second-order neutral differential equation of Rayleigh type:
Sufficient conditions for the existence periodic solutions were obtained from the fixed point index theorem. In recent years, Boularesy et al. [26] studied the existence and uniqueness of solutions to nonlinear fractional-order neutral differential equation:where is the standard Caputo fractional derivative of order , by using Krasnoselskii’s fixed point theorem and the Banach contraction principle.
Motivated by previous results in the literature and our previous work in [27], we extend problem (4) to study the existence result for mild solutions of fractional-order neutral differential equation with a more general class of fractional derivative:subject to the initial conditionwith and , where and the initial . Here, is the standard generalized fractional derivative of order .
The main contribution in this paper is that we consider the constructive approach for the existence of solutions for generalized fractional differential equations which can be applied numerically. This generalized fractional derivative can be reduced into Riemann–Liouville fractional derivative, Caputo fractional derivative, and Hadamard fractional derivative as follows. If and , our existence result reduces to the existence of mild solutions with respect to the Riemann–Liouville fractional derivative. On the other hand, if and , our result implies the existence of mild solution when the Caputo fractional derivative is considered. In addition, if and , we obtain the existence of mild solutions for the problem with Hadamard fractional derivative. Furthermore, based on the monotone iterative approach, our result gives a constructive method to approximate the mild solutions. By constructing a monotone iterative sequence of functions from upper or lower solutions, we obtain a sequence of functions converging to the mild solutions.
This paper can be outlined as follows. We introduce some preliminary background in Section 2. We present the existence and monotone iterative approximation result for mild solutions of fractional-order neutral differential equation with delay in the sense of generalized fractional derivative for the Riemann–Liouville and the Hadamard derivatives in Section 3. Finally, in Section 4, we give an example to demonstrate the existence of mild solution of IVP (5).
2. Preliminary Results for Neutral Differential Equations with Generalized Fractional Derivatives
Definition 1. (see [28]). Let and be real numbers with . Let be finite interval on the nonnegative numbers . For and , we define the space of functions as follows.(1) is the space of continuous functions on with the norm(2)The weighted space is the set equipped with the norm and .(3)The weighted space is the set and .
Definition 2. Let and be real numbers with and , where denotes the space of Lebesgue mensurable functions. The left-sided generalized factional integral is given byIt should mentioned that once , the integral in (11) becomes the Riemann–Liouville fractional integral. In case that one takes the limit as in (5), it becomes the Hadamard fractional integral.
It can be seen that the operator can be expressed as
Definition 3. Let be a positive real number such that and , where is the integer part of . The left-sided generalized fractional derivative is defined bywhere and .
It should be noted that once , the derivative in (13) becomes the Riemann–Liouville fractional derivative. Also by taking the limit as , the derivative in (13) becomes the Hadamard fractional derivative.
Definition 4. Let and be real numbers such that and . If and , thenfor all .
Definition 5. The function is a mild solution of the initial value problem for the neutral fractional differential equations with generalized fractional derivative (5) if the following integral equation is satisfied:and for , where .
Consider the Banach space equipped with partially order relation. For any we define the order relation if and only if with respect to on . This defines a partial ordering on . We next outline the preliminary results on partially ordered space. Let be a normed linear space with a partially ordered relation . The space is called regular if for any nondecreasing sequence in such that converges to as , we have for all . In particular, we see that the space is regular.
Definition 6. The operator is called nondecreasing if for any order in have relation preserved under , that is, implies for all .
Definition 7. (see [29]). Let be a partially ordered set. The operator is called partially continuous at if for any , there is such that under the supremum norm, we have and for all comparable to in . is called partially continuous on when the operator is partially continuous at every in .
Definition 8. (see [29]). Let be a partially ordered set. The operator is called partially bounded if the set is bounded for all chains . In particular, it is called uniformly partially bounded if the set is bounded with the same constant for every chain in .
Definition 9. (see [29]). Let be a partially ordered set. The operator is called partially compact if the set is relatively compact in for every chain in .
Definition 10. (see [29]). Let . The order relations and the metric induced by the norm are said to be compatible if the following condition holds: if is a subsequence of a monotone sequence converging to , then all of the sequence converges to .
Definition 11. (see [29]). Let be an upper semi-continuous and nondecreasing function. Then, is called a function if .
Definition 12. (see [29]). Let be a normed space with a partially ordered relation . The operator is called partially nonlinear -contraction if there is such thatfor all comparable pairs with for all .
Theorem 1 (see [30]). Let be a compatible and regular partially ordered linear space . and are nondecreasing and satisfy(a)An operator is partially nonlinear contraction.(b)An operator is partially continuous and partially compact.(c)There is a point in with .Then, there exists at least one solution of the operator equation . Furthermore, a monotone sequence given byconverges monotonically to .
To establish the existence result for IVP (5), we consider the following assumptions:(i)The functions and are continuous.(ii)The function is nondecreasing in for each and .(iii)There is a parameter such that with respect to on and .(iv)There exists a -contraction such that for , and with .(v)The function is nondecreasing in two variables, that is, if and , for .(vi)There exists a parameter such that for .(vii)There exist functions and such that is a lower solution of IVP (5), that is,
3. Existence and Monotone Iterative Approximation Results
In this section, we present a result on the existence and monotone iterative approximation of a mild solution of fractional-order neutral differential equations involving a generalized fractional derivative.
Theorem 2. Suppose that all seven hypotheses (i)–(vii) are satisfied. Then, IVP (5) has a mild solution in which for andcan be monotonically approximated by sequence defined bywhere is the lower solution and .
Proof. We denote for the partially ordered Banach space. A mild solution to IVP (5) can be considered as the following operator equation:wherefor . We require to show that the operators and satisfy all conditions in Theorem 1 which will be divided into 5 parts. First Step. By Theorem 1, we shall prove that the two operators and are nondecreasing. For all such that , by consideration of hypothesis (ii), we obtain This illustrates that the operator is a nondecreasing operator in . For the operator , we obtain from hypothesis () that for any in . Clearly, is a nondecreasing operator in . Second Step. For this step, we will prove that the operator satisfies the following properties:(i)The operator is partially bounded on .(ii)The operator is a partially nonlinear contraction on . For this moment, let . By the property of in hypothesis (iii), we get for all . Therefore, an operator is partially bounded on . Then, we show that the operator is contraction. For any such that , we see from assumption (iv) that for each . Hence, we take the supremum norm for all with . This implies that is a partially nonlinear contraction in the space . Third Step. We shall show that the operator satisfies condition (b) of Theorem 1, that is, is partially continuous on . We first show that is pointwise convergent. Let be a sequence in a chain in satisfying as . Using the boundedness and continuity of from assumptions (i) and (vi) together with the dominated convergence theorem, we get for each . This implies that converges to pointwise on . Next, we prove the equicontinuity of in . Let with . We have as uniformly for all . So, uniformly. Since converges to pointwise and uniformly, this implies that operator is partially continuous on . Fourth Step. We show that is partially compact. Let where is a chain in . We have for some . By condition (vi), we get for all . By taking the supremum in the last assertion, we get for all . Clearly, the operator is uniformly bounded on every chain in . It remains to prove that is equicontinuous on every chain in . Let and take with . We have as uniformly for . Thus, we get that is also relatively compact and we can conclude that the operator is partially compact on every chain in . Fifth Step. Finally, we prove that there exists an element which is a lower solution of IVP (5). By hypothesis (vii), there exist functions and satisfying Then, it follows that where for . This means that satisfies . As a consequence, we conclude that the operator equation has a solution since the two operators and satisfy all conditions in Theorem 1. In addition, the solution of IVP (5) can be monotonically approximated by a sequence as .
4. Example
Consider the neutral fractional differential equation with generalized fractional derivatives
Here,and
Let ; we get . Clearly, both functions and are continuous and nondecreasing. Considering functions and in equation (31), we see thatand
Next, we verify that the function is a -contraction with . We havefor all . Hence, the functions and satisfy assumptions (i)–(vi). Finally, for assumption (vii), we claim that a lower solution of equation (24) is given by for all . We see that for all . Let for . It can be seen fromfor thatfor . Then, we see that all conditions of Theorem 2 are satisfied. Hence, equation (31) has a solution in which can be obtained from the monotone iterative sequence defined bywith for all .
We numerically demonstrate the monotone iterative approximation of solutions corresponding to various fractional derivatives including Riemann–Liouville fractional derivative and Hadamard fractional derivative in Figures 1–3.
In Figure 1, we numerically compute a monotone iterative increasing sequence converging to the mild solution of equation (31) for and which corresponds to the fractional differential equation under the Riemann–Liouville fractional derivative for with time delay . The lower solution is chosen to be on . It can be seen that sequence increases at and then gradually decreases on which exhibits the behavior of the mild solution.
In Figure 2, we numerically compute a monotone iterative increasing sequence converging to the mild solution of equation (31) for and which corresponds to the fractional differential equation under the generalized fractional derivative for with time delay . The lower solution is chosen to be on . It can be seen that sequence increases rapidly at and then becomes constant on which exhibits the behavior of the mild solution.
In Figure 3, we numerically compute a monotone iterative increasing sequence converging to the mild solution of equation (31) for and which corresponds to the fractional differential equation under the Hadamard fractional derivative for with time delay . Similar to Figure 1, the lower solution is chosen to be on . It can be seen that sequence increases at and then gradually decreases on which exhibits the behavior of the mild solution.
5. Conclusions
We establish the existence and monotone iterative approximation of mild solutions to the neutral fractional differential equations under a generalized fractional derivative which can be applied numerically. The monotone iterative sequence for the mild solution of equation (31) for admits the following particular cases. If and , then we have and we obtain the existence and approximation of solution under the Riemann–Liouville fractional derivative. If and , then we get the existence and approximation of solution under Hadamard fractional derivative.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The first author was supported by Kanchanaburi Rajabhat University. The second author was supported by King Mongkut’s University of Technology Thonburi. This research project is supported by Thailand Science Research and Innovation (TSRI), Basic Research Fund: Fiscal year 2022 under project number FRB650048/0164.