Existence and Stability of Implicit Fractional Differential Equations with Stieltjes Boundary Conditions Involving Hadamard Derivatives

Luo, Danfeng; Alam, Mehboob; Zada, Akbar; Riaz, Usman; Luo, Zhiguo

doi:https://doi.org/10.1155/2021/8824935

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Abstract Introduction Preliminaries Examples Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 8824935 | https://doi.org/10.1155/2021/8824935

Existence and Stability of Implicit Fractional Differential Equations with Stieltjes Boundary Conditions Involving Hadamard Derivatives

Danfeng Luo,¹Mehboob Alam,²Akbar Zada,²Usman Riaz,²and Zhiguo Luo³

Academic Editor: Peter Giesl

Received04 Sept 2020

Revised07 Nov 2020

Accepted22 Feb 2021

Published23 Mar 2021

Abstract

In this article, we make analysis of the implicit fractional differential equations involving integral boundary conditions associated with Stieltjes integral and its corresponding coupled system. We use some sufficient conditions to achieve the existence and uniqueness results for the given problems by applying the Banach contraction principle, Schaefer’s fixed point theorem, and Leray–Schauder result of the cone type. Moreover, we present different kinds of stability such as Hyers–Ulam stability, generalized Hyers–Ulam stability, Hyers–Ulam–Rassias stability, and generalized Hyers–Ulam–Rassias stability by using the classical technique of functional analysis. At the end, the results are verified with the help of examples.

1. Introduction

Fractional-order derivatives are the generalization of integer-order derivatives. The idea of fractional-order derivatives has been introduced at the end of the sixteen century when Leibniz used the notation for -order derivative. It is well known that if , then and , respectively. But, if the order is in the fractional form, i.e., , and so on, then what would be the result? This question was mentioned in a letter to L’Hospital by Leibniz in (1695) [1]. Since then, several mathematicians such as Fourier, Laplace, and Letnikov contributed to the development of the fractional calculus. Riemann and Liouville had worked on this problem and introduced the Riemann–Liouville fractional derivative, which was further generalized by Caputo and can be considered as the fundamental concept in fractional calculus. Fractional calculus has many applications in many scientific disciplines, e.g., in the fields of signal and image processing [2], mathematical biological systems [3], electronics, economics, control theory [4], chemistry, biophysics, and blood flow phenomena. For more applications of fractional differential equations, we refer the reader to [5, 6] and the references cited therein.

In 1892 [7], Hadamard presented a concept of fractional derivative, which is different from Caputo- and Riemann–Liouville-type fractional derivatives. An important feature of the Hadamard fractional derivative is that it contains a logarithmic function of an arbitrary exponent in its definition. Here, we stress that the studies about Hadamard fractional differential equations are still at the early stage and need additional analysis. For more details and recent contributions to the topic, see [8–11].

In the field of fractional differential equations, the area which received great attention from researchers is the existence of solutions of different boundary-value problems. Many researchers used different fixed-point theorems and developed different approaches for the existence of solutions of complicated boundary-value problems. For details, we refer the reader to [12–14]. The study of coupled systems of differential equations is also very significant because these types of systems appear naturally in many problems of applied nature. For details and examples, the reader may see [15–18] and the references cited therein.

Another area of research which has received great attention from researchers in the field of fractional differential equations is the notion of stability in the sense of Ulam. There are different kinds of Ulam’s stabilities, i.e., Hyers–Ulam stability , generalized Hyers–Ulam stability , Hyers–Ulam–Rassias stability , and generalized Hyers–Ulam–Rassias stability . The Ulam-type stability was first introduced by Ulam [19] in 1940 and then was studied and generalized by many mathematicians with different approaches [10, 20–29].

Ren and Zhai [30] discussed the existence of a unique solution and multiple positive solutions with nonlocal boundary conditions for the system involving standard Riemann–Liouville fractional q-derivatives:where and denote the standard Riemann–Liouville fractional q-derivatives of order , with being nonnegative.

Benchohra and Lazreg [8] studied the existence and Hyers–Ulam stability of the following implicit fractional differential equations involving Hadamard derivatives:where and denotes the Hadamard fractional derivative of order .

Riaz et al. [10] studied the existence, uniqueness, and stability of the solution of a nonlinear coupled system of impulsive using Hadamard derivatives of the following form:where and are continuous functions.

Riaz et al. [31] studied the existence, uniqueness, and stability of solution of coupled implicit impulsive fractional differential equations using Hadamard derivatives:where and are continuous functions.

In this article, we extend the system of Ren and Zhai [30] to implicit Hadamard fractional derivatives having the Stieltjes integral condition instead of the Riemann–Liouville fractional q-derivative and present its existence, uniqueness, at least one solution, and different kinds of Ulam’s stabilities. We study the following system:where is the continuous function and are the Hadamard fractional derivatives of order , with being non-negative, and is a linear function given byinvolving the Stieltjes integral with respect to the function is right continuous on [1, 2), left continuous at . Particularly, is a nondecreasing function with ; then, is a positive Stieltjes measure.

We also extend system (5) to the coupled system and discuss its existence, uniqueness, at least one solution, and different kinds of Ulam’s stabilities of the problem:where are the continuous functions and , and are the Hadamard fractional derivatives of orders , respectively with being nonnegative, and are linear functions given byinvolving the Stieltjes integral with respect to the function is right continuous on [1, 2), left continuous at . Particularly, is a nondecreasing function with ; then, is a positive Stieltjes measure.

The rest of the article is arranged as follows: In Section 2, we present some basic definitions, lemmas, and theorems that are used in our main results. In Section 3, we use different conditions and some standard fixed-point theorems for the existence and uniqueness of solutions to the given system (5) and its corresponding coupled system (7). In Section 4, we present Ulam’s stabilities for the given systems (5) and (7) under some specific conditions. At the end, examples are given to illustrate the main results.

Throughout the paper, we assume that : . : is continuous. : for and , there are such that with , and . : for all and for each , there exist constants such that : for , there exists a constant such that : let be an increasing function; then, there is such that, for each , the inequality holds. : , and : are continuous. : for and , there are such that with , and . Similarly, for and , there are such that with , and . : for all and for each , there exist constants such that : for , there exist constants such that : let be increasing functions, and there exist such that, for each , the inequalitieshold.

2. Preliminaries

In this section, we present some useful definitions, lemmas, and theorems, which will be used throughout the manuscript\enleadertwodots.

Definition 1 (see [31]). The Hadamard fractional integral of order, of function , is defined bywhere is the gamma function.

Definition 2 (see [31]). The Hadamard fractional derivative of order , of function is defined bywhere is the gamma function.

Lemma 1 (see [31]). Let and be any function; then, the homogeneous differential equation along with Hadamard fractional order has a solutionand the following formula holds:where , and .

Theorem 1 (see [32]) (Arzela-Ascoli’s Theorem). Let be relatively compact, and(A) is a uniformly bounded set such that there exists with(B) is an equicontinuous set, i.e., for every , there exists such that, for any

Theorem 2 (see [33]) (Banach Fixed-Point Theorem). Let be a nonempty closed subset of a Banach space . Then, any contraction mapping from into itself has a unique fixed point.

Theorem 3 (see [33]) (Schaefer’s Fixed-Point Theorem). Let be a Banach space. Suppose the operator is a continuous compact mapping (or completely continuous). Moreover, supposeis a bounded set. Then, has at least one fixed point in

Lemma 2. Let be a Banach space endowed with the norm . Similarly, the norm defined on the product space is . Obviously, is a Banach space. Also, the cone is defined as

Theorem 4 (see [34]). Let be a Banach space containing a cone . with is a relatively open set, and the operator is completely continuous. Then, one of the following conditions holds:(1)There exist and such that (2) has a fixed point in

3. Existence and Uniqueness

In this section, we give the existence and uniqueness of solutions of (5) and its coupled system (7).

3.1. Existence and Uniqueness Solution for System (5)

Our first result is stated as follows.

Lemma 3. Let us assume that and ; then, the fractional differential equationhas a solutionwhere

Proof. ConsiderFor , Lemma 1 givesfor some . By the condition , we have . Hence,Now, considering the condition , we getin view of as , so we must set . Then, we haveFurthermore, we obtainThen, we obtainApplying the 2nd condition, we getSubstituting (37) into (34), we getBy replacing in Lemma 3, we get the integral equation of problem (5) as

Lemma 4. Green’s function , which is obtained in (39), has the following properties:(1) for all (2) is continuous over (3),where

Proof. It is very easy to prove and , so we leave it.
Since we know that Green’s function of the considered problem is in the formusing and mean-value theorem [35] with , we getHence, the proof of is complete.
If is the solution of the given system (5) and , thenwhereWe define an operator aswhere such that

Theorem 5. Let hold. Then, the operator , defined in (45), is compact.

Proof. To show that the operator is compact, we follow several steps. Step 1: we consider a sequence such that in ; then, for each , we haveFrom (46), we can writeNow, by , we havewhich impliesSince we supposed that , as for each . So, by Lebesgue dominated convergence theorem [36], (47) givesThis impliesHence, is continuous. Step 2: now, we are going to prove that the operator is bounded in set . For this, we show that, for any , there exists such that, for eachwe haveFrom (45), for each , we haveNow, by and (46), we haveTaking , we getThus, (55) becomesHence, is uniformly bounded. Step 3: now, to show that the operator is equicontinuous in . For this, let with , since is a bounded set in , and let . Then,The right-hand side of (59) approaches to zero as . Hence, is equicontinuous. As a consequence of Step 1 to 3, the operator is completely continuous. Therefore, in view of the Arzelà-Ascoli theorem, the operator is compact.

Theorem 6. Let the hypotheses and hold, and if , the given problem (5) has at least one solution in .

Proof. For the proof of this theorem, we are considering a set which is defined in the following form:We have to show that the set is bounded. Let such thatThen, for each , we haveNow, by , we haveSo, we getPutting (64) in (62) and taking , we getFor simplicity, letSo, (65) becomeswhich impliesThis shows that the set is bounded. So, by Theorems 3 and 5, we get that the operator has at least one fixed point. Therefore, the given problem (5) has at least one solution in .

Theorem 7. Suppose that the hypothesis , , and hold. Then, the given problem (5) has a unique solution in if

Proof. We shall use the Banach contraction principle to prove that the operator has a unique fixed point, which will be the unique solution of the given system (5), by considering the operator defined in (45).
Let be the solution of (5), and for , we havewhere such thatNow, by , we havewhich impliesSo, (70) becomesUsing and taking on both sides, we getThis implies thatHence, the operator is a contraction. Thus, by the Banach contraction principle, we get that has a unique fixed point, which is a unique solution of the given problem (5).

3.2. Existence and Uniqueness Solution for System (7)

In this section, we show the existence and uniqueness of the solution of the system (7). First, we have the following:

Lemma 5. The systemhas a solution (u, v) if and only ifwhere

Proof. The proof is similar to that given in Lemma 3 and, hence, is not included here.We use the following notations for convenience:Hence, for , (78) becomeswhere satisfying the functional equation.

Lemma 6. Green’s function of the system (7) have the following properties:(1) is continuous over (2),where

Proof. (1)It is easy to prove that is continuous, so we leave it.(2)Since we know that Green’s functions of the considered problem (7) is in the formusing and the mean value theorem [35] with , we getSimilarly, we can obtainHence, the proof of 2 is complete.
If are the solutions of the given system (7) and ; then,Now, we transform the given system (7) into a fixed-point problem. Let an operator be defined asThen, the solution of (7) coincides with the fixed point of , where

Theorem 8. Let and , hold. Then, the operator defined in (87) is completely continuous.

Proof. In view of continuity of and , is also continuous for all . Suppose is a bounded set. So, for every , we haveNow, by , we haveSo, we obtainNow, using 2 of Lemma 6, , and (91) in (89), we getIn the same way, we obtainThus, from (92) and (93), we getThus, is uniformly bounded. Now, we prove the operator is equicontinuous. For this, suppose and ; then,In the same way, we can show thatThe right-hand sides of (94) and (96) approache to zero as . Hence, by the Arzelà-Ascoli theorem, is equicontinuous and uniformly equicontinuous. Also, it is very easy to prove that . Therefore, is completely continuous.

Theorem 9. Under the hypothesis , andThe coupled system (7) has a unique solution.

Proof. Let , and we considerwhereNow, using ,Substituting (100) in (98) and taking , we getIn the same way, we can obtainSo, from (101) and (102), we getThus, is a contraction. Therefore, by the Banach contraction principle, has a fixed point. So, we infer that the given coupled system (7) has a unique solution.

Theorem 10. In view of the continuity of the functions and supposing and withhold, the coupled system (7) has at least one solution.

Proof. Let a set be defined aswhere . Furthermore, the operator defined by in (87) is completely continuous. Suppose ; then, by definition of , we have ;Similarly,Therefore,so . Thus, in view of Theorem 8, is completely continuous.
Now, we consider an eigenvalue problem defined asSo, in view of the solution (u, v) of (109), we obtainSimilarly,Thus,From equation (112), we get . So, in view of Theorem 4, has at least one fixed point which lies in . This shows there is at least one solution of the coupled system (7).

4. Hyers–Ulam Stability

In this section, we provide novel characterizations of the Hyers–Ulam stability for systems (5) and (7). For the various concepts of Hyers–Ulam stability, see, for example, [37].

4.1. Hyers–Ulam Stability Concepts for System (5)

Definition 3. The problem (5) is said to be Hyers–Ulam stable if there exists some constant such that, for any and for any solution of the inequalitythere exists a solution of (5) with

Definition 4. The problem (5) is said to be generalized Hyers–Ulam stable if there exists with such that, for any solution of the inequality (113), there exists a solution of (5) satisfying

Definition 5. The problem (5) is said to be Hyers–Ulam–Rassias stable with respect to if there exists some constant such that, for any and for any solution of the inequalitythere exists a solution of (5) with

Definition 6. The problem (5) is said to be generalized Hyers–Ulam–Rassias stable with respect to if there exists some constant such that, for any solution of the inequality (116), there exists a solution of (5) satisfying

Remark 1. Clearly,(1)Definition 3 Definition 4(2)Definition 5 Definition 6

Remark 2. Let be a solution of the inequality (113); then, there exists a function depending on such that(1)(2)

Lemma 7. Let ; if is the solution of the inequality (113), then will be the solution of the following integral inequality:

Proof. Let be the solution of the inequality (113). So, in view 2 of Remark 2, we haveSo, for , the solution of (120) will be in the formFrom equation (121), we haveFor computational convenience, we use for the sum of terms which are free of , so we haveFrom above, we haveUsing (3) of Lemma 4 and 1 of Remark 2, we get

Theorem 11. Under the hypothesis , , and and ifholds, then the given system (5) is and, consequently, .

Proof. Let be the solution of (113) and be the unique solution of the system given byThen, for , the solution of (127) isConsiderUsing Lemma 7 in (129), we havewhere are of the formBy , we getwhich impliesUsing (3) of Lemma 4 and (133) in (130), we getwheresuch thatThus, problem (5) is .

Remark 3. By setting , in (134), by Definition 4, the given system (5) is .

Lemma 8. Let the hypothesis hold, and suppose is the solution of the inequality (116); then, is a solution of the following inequality:

Proof. From Lemma 7, we haveBy using (3) of Lemma 4, 1 of Remark 2, and , we get

Theorem 12. Under the hypothesis and – and if the inequalityholds, then the given system (5) is stable in the sense of .

Proof. Let be the solution of (113) and be the unique solution of the system given byThen, for , the solution of (141) isConsiderUsing in a similar way as used in Theorem 11, we getNow, by Lemma 8 and by (144), (143) becomeswhich impliesThus, we havewhereHence, the given system (5) is stable.

Remark 4. If in (147), then by Definition 6, the given system (5) is .

4.2. Hyers–Ulam Stability Concepts for System (7)

Definition 7 (see [38]). The given system (7) has if there exists such that there exist some , and for every solution of the inequalitythere exists a solution with

Definition 8 (see [38]). The given system (7) has if there exists with such that, for any solution of the inequality (149), there exists a solution of (7) satisfying

Definition 9 (see [38]). The given system (7) has with respect to with if there exists some constant such that, for any and for any solution of the inequalitiesthere exists a solution with

Definition 10 (see [38]). The given system (7) has with respect to with if there exists some constant such that, for any solution of the inequality (152), there exists a solution of (7) satisfying

Remark 5. Let be a solution of the inequality (149), if there exist functions depending on , respectively, such that(1)(2)

Lemma 9. Let be the solution of (149); then, for , we have

Proof. By 2 of Remark 5 and for , we haveSo, for , the solution of (156) will be in the formFrom the first equation of system (157), we haveFor computational convenience, we use for the sum of terms which are free of , so we haveSo, from the above and taking the absolute value, (158) becomesUsing of Lemma 6 and 1 of Remark 5, we getIn the same way, we have

Theorem 13. Under the hypothesis and ifholds, then the given system (7) is stable in the sense of .

Proof. Let be the solution of (149) and be the solution to the systemThen, for , is the solution of (164), i.e.,Considerwhere are of the formBy , we getand we obtainUsing 2 of Lemma 6 and (168) in (166), we getSimilarly, we havewhere in the formWe write (170) and (171) asrespectively. LetIn the matrix form, the abovementioned inequalities can be written asSolving the abovementioned inequality, we haveFurther simplification givesfrom which we haveLet ; then, from (178), we havewhere

Remark 6. By setting , in (179), by Definition 8, the given system (7) is .

Remark 7. Under the hypothesis and (163) and by using Definitions 9 and 10, one can repeat the process of Lemma 9 and the Theorem 13, and the system (7) will be and .

5. Examples

In this section, we present three examples to demonstrate the existence and stability of the obtained results.

Example 1. where , for . Moreover, .
SetNow, for any and , we haveHence, is satisfied with .
Also, for any , we haveHence, is satisfied withwhere and .
From Theorem 7, we use the inequality which are found asHence, (181) has a unique solution.
Furthermore, with condition (126) holds, andHence, with the help of Theorem 11, the given system (181) is and, hence, . Also, by checking the conditions of Theorem 12, it can be easily verified that the considered problem (181) is and .

Example 2. where , for . Moreover, .
SetNow, for any and , we haveHence, is satisfied with .
Also, for any , we haveHence, is satisfied withwhere .
From Theorem 7, we use the inequality which are found asHence, (188) has a unique solution.
Furthermore, with condition (126) holds, andHence, with the help of Theorem 11, the given system (188) is and, hence, . Also, by checking the conditions of Theorem 11, we can find that the considered problem (188) is and .

Example 3. where , for . Moreover, .
SetBy simple computations, we found thatFrom Theorem 9, we use the inequality which are found asHence, (195) has a unique solution, andHence. with the help of Theorem 13, the given system (195) is . Also, by using and Remark 6 and 7, the given system is , , and .

6. Conclusions

We have obtained some appropriate conditions for the existence, uniqueness, and Ulam’s stabilities of the system (5) and its corresponding coupled system (7). The required results are obtained using the Banach contraction principle, Schaefer’s fixed point theorem, Arzela-Ascoli theorem, and Leray–Schauder of the cone type. Also, with the help of some sufficient conditions, we have derived different kinds of Ulam’s stabilities for the solution of the problem (5) and the coupled problem (7). Additionally, examples are given to support the main results.

Data Availability

No data were used to support this work.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province [(2016)4006], and Major Research Project of Innovative Group in Guizhou Education Department [(2018)012].

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Copyright © 2021 Danfeng Luo 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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