On Local Generalized Ulam–Hyers Stability for Nonlinear Fractional Functional Differential Equation

Nie, Dongming; Khan Niazi, Azmat Ullah; Ahmed, Bilal

doi:https://doi.org/10.1155/2020/3276873

Mathematical Problems in Engineering

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Volume 2020 | Article ID 3276873 | https://doi.org/10.1155/2020/3276873

On Local Generalized Ulam–Hyers Stability for Nonlinear Fractional Functional Differential Equation

Dongming Nie,¹Azmat Ullah Khan Niazi,^2,3and Bilal Ahmed³

Guest Editor: Shaohui Wang

Received27 Apr 2020

Accepted27 May 2020

Published18 Jun 2020

Abstract

We discuss the existence of positive solution for a class of nonlinear fractional differential equations with delay involving Caputo derivative. Well-known Leray–Schauder theorem, Arzela–Ascoli theorem, and Banach contraction principle are used for the fixed point property and existence of a solution. We establish local generalized Ulam–Hyers stability and local generalized Ulam–Hyers–Rassias stability for the same class of nonlinear fractional neutral differential equations. The simulation of an example is also given to show the applicability of our results.

1. Introduction

Fractional differential equations (FDEs) have boosted considerably due to their application in various fields of sciences, such as engineering, chemistry, mechanics, and physics. Recently, fractional ordinary differential equations (ODEs) and partial differential equations (PDEs) have been developed significantly. Indeed, we can also find applications in control, electromagnetism, and electrochemistry (see [1, 2]). For more details in this area, one can also see the monographs of Kilbas et al. [3], Li et al. [4, 5], Liu et al. [6–9], Miller and Ross [10], Podulbny [11], Rehman et al. [12], and the references therein.

Positive solution of fractional ODEs have already discussed in [13–15]. However, until now research on existence of positive solution for fractional functional differential equations (FFDEs) is rare. Research in different fields including engineering, physics, and biosciences have proved that numerous system structures explain more exactly with the help of FDEs [16–21] and similarly FDEs with delay [22–27] are more accurate to illustrate the real world problems compared to FDEs without delay. So, in [28], Miller and Ross mentioned that this field has been touched by almost all fields of engineering and science.

Idea of Ulam stability started in 1940, when during the talk at Wisconsin University Ulam posed a problem that “When can we assert that an approximate solution of a functional equation can be approximated by a solution of the corresponding equation?” (for more details, see [29]). After one year, Hyers first gave the answer to the Ulam’s question [30] in case of Banach spaces. After that, stability of this type is called the Ulam–Hyers stability. Rassias [31] in 1978 provided an outstanding generalization of the Ulam–Hyers stability of mappings by considering variables.

Usually, the discussion of the existence and uniqueness of a solution for nonlinear FDEs normally fixed point theory has been used [16, 32, 33]. Motivated by these and [34–36], in this work, we have discussed existence and uniqueness of solution also after applying some sufficient conditions, obtained positive solution, and at the end established local generalized Ulam–Hyers stability and local generalized Ulam–Hyers–Rassias stability and presented stability results graphically for the class of nonlinear FDEs with delay given bywhere is the Caputo derivative and , while and where is space of continuous functions, is a nonnegative real number also are continuous functions, and is defined as .

Remaining paper is arranged as follows. Section 2 includes some basic definitions and lemmas used throughout this paper. In Section 3, we have the main results. In Section 4, we establish local generalized Ulam–Hyers stability and local generalized Ulam–Hyers–Rassias stability for problem (1), and in Section 5, simulation of an example is given to show the applicability of our results.

2. Preliminaries

Let be a cone in a real Banach space and partial order introduced by in is as

Definition 1. Let be the order interval defined as

Definition 2. The functional is said to be nondecreasing w.r.t. on such thatimplies thatfor any .

Definition 3. (see [11]). The fractional integral of order for a function with lower limit 0 can be defined aswhere is the gamma function and right-hand side of upper equality is defined pointwise on .

Definition 4. (see [11]). The left Caputo fractional derivative of order is given bywhere ( stands for the bracket function of ).

Lemma 1. (Leray Schauder fixed point theorem, see [11]). Let be a nonempty, closed, bounded, and convex subset of Banach space and is a compact and continuous map; then, has a fixed point in .

Theorem 1. If , where is a cone of space of partial order , be nondecreasing. If there exists such that , , and is lower solution and is an upper solution of , then has minimum solution and maximum solution in such that , and if anyone of these conditions holds(i) is reflexive, is weak continuous or continuous, and is normal(ii) is completely continuous and is normal(iii) is continuous and is regular

3. Main Results

In this section, we have discussed existence and uniqueness of solution and some conditions for positive solution of equation (1). Consider is a Banach space endowed with the supremum norm and cone is defined as . From equation (1), we haveand we can easily get that

Let be the function defined by

So, and for every with , and we defined as

Let satisfy equation (9), and we can decompose as so that , for every , where is such that and the function satisfies

Set , where is a cone and and is a Banach space with Sup norm . Let the operator be defined by

Before proving main results, we introduce the following conditions:(C1)Let us take with , where function is nondecreasing and and satisfy Lipchitz condition that is, and there exists nonnegative constants such that(C2)If be bounded, is nondecreasing andwhere . Let . Define the operator by

Theorem 2. Assume that and are nonnegative and continuous; then, operator is completely continuous.

Proof. As we know that and are nonnegative and continuous, so operator is continuous. Let is bounded, i.e., there exists a constant such that . Here, we have to keep in mind the condition that . If , and for all , we get thatHence, is bounded. Now, we show that is equicontinuous, and the proof is divided into three possible steps. Step 1: for every and such that , thenAs , the right-hand side of above inequality tends to zero. Step 2: for every , as and while , there exists such that when , .If , then for , we get that Step 3: for every , , and where . As is continuous, we haveHence, is equicontinuous. So, is compact by Arzela–Ascoli Theorem.

Theorem 3. Suppose that C1 and C2 hold. Let and . Then, there exists a unique solution for IVP (1) on the interval .

Proof. Consider the operator defined asWe will show that is a contraction. Let , then we obtain the following sequence of inequalities:Therefore,So, is a contraction, and hence by Banach’s contraction principle has a unique fixed point.

Definition 5. The function is said to be a lower solution of equation (1) (operator ) ifand the function is said to be an upper solution of equation (1) ifIf inequalities are strict, then and are strict lower and upper solutions.

Theorem 4. Let(H1) and from to are continuous, and and are nondecreasing in for each (H2) is lower and is an upper solution of equation (1), satisfying condition , and Then, equation (1) has at least a positive solution.

Proof. We only take fixed point operator . By Theorem 2, is completely continuous and obviously by equation (16), and are lower and upper solutions of , respectively. By (H1), , we haveSo, the operator is nondecreasing. Also, we haveAs is a lower solution, therefore we can say that , similarly from the definition of upper solution of .
Hence, is a completely continuous operator. Since is a normal cone, Theorem 1 implies that has a fixed point .(H3)Here, we suppose that there exists such that .

Theorem 5. Assume conditions (H1) and (H3) holds; then, equation (1) has a positive solution.

Proof. Consider the equationObviously, equation (28) has a solution , which is a lower solution of equation (1). Similarly, consider the equationWe know that , is an upper solution of equation (1) and , so the above definition proves that equation (1) has minimum one positive solution.
For getting another result for a positive solution of equation (1), now we take the more general case of (1) and find the existence of a positive solution for the equations:

Theorem 6. Equation (30) has a least positive solution.

Proof. Equation (30) is equivalent toLet be an operator defined as is completely continuous by Theorem 2.
Let us take a bounded, closed, and convex subset of the Banach space , where and . The possible two cases are Case 1: for all , we obtainSince , henceHere, . Case 2: if and , thenHence, . The Leray–Schauder fixed point theorem guarantees that operator has at least one fixed point and so equation (30) has at least one positive solution.

4. Stability

In this section, we will discuss local generalized Ulam–Hyers stability and local generalized Ulam–Hyers–Rassias stability for a class of fractional neutral differential equations. For the case with are continuous functions on a closed interval or more generally compact sets, then they are bounded so we can replace the supremum by the maximum. In this case, norm is also called the maximum norm. Let be a Banach space with , where is a nonnegative real number and . We consider the following differential equation:where is a Caputo derivative with and . We focus on the following inequalities:

Definition 6. (see [36]). Equation (36) is local generalized Ulam–Hyers stable if there exists a positive real number such that, for each positive and for every solution of (37), there exists a solution of (36) with .

Definition 7. (see [36]). Equation (36) is local generalized Ulam–Hyers–Rassias stable with respect to if there exists such that, for every solution of (38), there exists a solution of (36) with .

Remark 1. A solution of differential equation is stable (asymptotically stable) if it attracts all other solutions with sufficiently close initial values. On the contrary, in Hyers–Ulam stability, we compare solution of given differential equation with the solution of differential inequality. We say solution of differential equation is stable if it stays close to solution of differential inequality. Hyers–Ulam stability may not imply the asymptotic stability.

Remark 2. (see [37]). A function is a solution of (37) if and only if there exists such that(i)(ii)Similarly, a function is a solution of (38) if and only if there exists such that(i)(ii)

Remark 3. Let and be a solution of inequality (37); then, is a solution of the following inequality:From Remark 2, we havethenTherefore,By using the same process as in Section 3, i.e., let be the function defined byTherefore, and for every with . We defined asLet satisfies equation (42). We can decompose as so that , for every , where is such that and the function satisfiesBy following the conditions in Section 3, we get from (42) thatHence,If is a solution of inequality (38) then is a solution of the following inequality:Before stating stability results, let us take condition (H4) as(H4)

Theorem 7. Suppose that is true and other two conditions(a), , and , .(b) and satisfy Lipchitz conditions:where and are nonnegative real numbers holds, then(i)Equation (36) has a unique solution(ii)Equation (36) is local generalized Ulam–Hyers stable

Proof. Let be a unique solution of equation (37). Denote as the unique solution of the equationThen, we haveWe can see that , for . For , we obtainFor , by usingThus,So,Hence, equation (36) is local generalized Ulam–Hyers stable.

Theorem 8. If is true and(a), , and , .(b) and satisfy Lipchitz conditions:where and are nonnegative real numbers.(c)The function is an increasing function and there exists such thatholds, then equation (36) is local generalized Ulam–Hyers–Rassias stable with respect to .

Proof. Following the same steps as in Theorem 7, we can find the result, i.e., here we will obtainTherefore,Hence, equation (36) is local generalized Ulam–Hyers–Rassias stable.

5. Example

In this section, we present an example to explain the applicability of main results.

Example 1. Consider the problemOne can easily prove that this problem has a unique fixed point and that it has a positive solution, i.e.,Define the operator asFor stability, we take (60) asand the inequalitiesTo prove that equation (63) is local generalized Ulam–Hyers stable, we take the conditions as in Theorem 7, i.e., a function is a solution of inequality (63) if and only if there exists such thatApplying on both sides, we obtainHere, we take unique solution of equation (63) asBy further simplification, we obtainHence, (63) is local generalized Ulam–Hyers stable.

Remark 4. If we replace equation (66) by the inequality,By repeating the same process as in the above example, one can easily verify the main results of Theorem 8.

6. Graphical Presentation

For graphical representation of the solution of the problem given in equation (60), we adopted Adams–Bashforth–Moulton scheme [38] to obtain the numerical solution for this fractional differential equation. To analyze the effect and contribution of time delayed factor, the modified predictor-corrector scheme [39] is incorporated for simulation. To check and demonstrate the stability of the consider model, graphical representation of the solution with different variations of the time delay factor along with other parameters is made. From the numerical results, we could note the UH-stability of the system with varying orders and delays. With the higher orders, the system will achieve the Ulam–Hyers stability more rapidly. It also holds true with increasing delays. One can see the stability of system (60) in Figures (1)–(4) for various values of time delays and fractional derivatives.

7. Conclusions

We present some new results about the existence of positive solution for a class of nonlinear fractional differential equations with delay involving Caputo derivative. Leray–Schauder theorem, Arzela–Ascoli theorem, and Banach contraction principle are used for the fixed point property and existence of a solution. We also establish local generalized Ulam–Hyers stability and local generalized Ulam–Hyers–Rassias stability for the same class of nonlinear fractional neutral differential equations. The simulation of an example is also given to show the applicability of our results. The current concepts have significant applications since it means that if we are studying local generalized Hyers–Ulam–Rassias stable (or Hyers–Ulam stable) system then one does not have to reach the exact solution. We just need to get a function which satisfies a suitable approximation inequality. In other words, local generalized Hyers–Ulam–Rassias stability (or Hyers–Ulam stability) guarantees that there exists a close exact solution. This is altogether useful in many applications where finding the exact solution is quite difficult such as optimization, numerical analysis, biology, and economics. It also helps, if the stochastic effects are small, to use deterministic model to approximate a stochastic one.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

D. Nie, A. U. K. Niazi, and B. Ahmed contributed to and designed the study. A.U.K. Niazi gave the main idea of the manuscript. D. Nie wrote the first draft, and B. Ahmed performed the design of figures. All authors contributed to the manuscript revision, read, and approved the submitted version.

Acknowledgments

This research is supported by Natural Science Research Project of Anhui Education Department (No. KJ2019A0875); School Level Scientific Research Project of Anhui Xinhua University (No. 2019zr005).

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Copyright © 2020 Dongming Nie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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