Boundary Value Problem for Nonlinear Implicit Generalized Hilfer-Type Fractional Differential Equations with Impulses

Salim, Abdelkrim; Benchohra, Mouffak; Lazreg, Jamal Eddine; N’Guérékata, Gaston

doi:https://doi.org/10.1155/2021/5592010

Abstract and Applied Analysis

On this page

Abstract Introduction Preliminaries Examples Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 5592010 | https://doi.org/10.1155/2021/5592010

Boundary Value Problem for Nonlinear Implicit Generalized Hilfer-Type Fractional Differential Equations with Impulses

Abdelkrim Salim,¹Mouffak Benchohra,¹Jamal Eddine Lazreg,¹and Gaston N’Guérékata²

Academic Editor: Nasser-Eddine Tatar

Received05 Jan 2021

Revised02 Mar 2021

Accepted08 Apr 2021

Published21 Apr 2021

Abstract

This article deals with some existence, uniqueness, and Ulam-Hyers-Rassias stability results for a class of boundary value problem for nonlinear implicit fractional differential equations with impulses and generalized Hilfer Fractional derivative. The results are obtained using the Banach contraction principle and Krasnoselskii’s and Schaefer’s fixed-point theorems.

1. Introduction

Differential equations of fractional order have been recently proved to be a powerful tool to study many phenomena in various fields of science and engineering such as electrochemistry, finance, hydrology, electromagnetics, and viscoelasticity. There are numerous books and articles focused on linear and nonlinear initial and boundary value problems for fractional differential equations involving different kinds of fractional derivatives, see, for example, [1–6]. Impulsive fractional differential equations have been considered by many authors (see, for instance, [7–12]). Recent results involving different fractional derivatives can be found in [13–21] and the references therein.

Ulam was the first who raise the concept of stability of functional equations [22]. In 1941, Hyers [23] provided the first answer to Ulam’s question. Thereafter, this type of stability is called the Ulam-Hyers stability. In 1978, Rassias [24] was able to make a remarkable generalization of Ulam-Hyers stability of mappings by considering variables.

Considerable attention has been given to the study of the Ulam-Hyers and Ulam-Hyers-Rassias stability of all kinds of functional equations; one can see the monograph of Abbas et al. [3] and the paper by Rus [25] who discussed the Ulam-Hyers stability for operational equations (see also [26–29]).

Recently, in [30], Harikrishnan et al. investigated existence theory and different kinds of stability in the sense of Ulam, for the following initial value problem with nonlinear generalized Hilfer-type fractional differential equation and impulses: where and are a generalized Hilfer fractional derivative of order and type and generalized fractional integral of order , respectively; , , and represent the right and left hand limits of at ; ; is a given function; and are given continuous functions.

Motivated by the works mentioned above, in this paper, we establish existence and uniqueness results to the boundary value problem with nonlinear implicit generalized Hilfer-type fractional differential equation and impulses: where are a generalized Hilfer fractional derivative of order and type and generalized fractional integral of order , respectively; are real with ; ; ; and represent the right and left hand limits of at ; is a given function; and , are given continuous functions.

The present paper is organized as follows. In Section 2, some notations are introduced and we recall some preliminaries about generalized Hilfer fractional derivative and auxiliary results. In Section 3, three results for problems (2)–(4) are presented which are based on the Banach contraction principle and Krasnoselskii’s and Schaefer’s fixed-point theorems. In Section 4, we discuss the Ulam-Hyers-Rassias stability for problems (2)–(4). Finally, we give examples to illustrate the applicability of our main results.

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. Let . By , we denote the Banach space of all continuous functions from into with the norm

We consider the weighted spaces of continuous functions with the norms

Consider the Banach space , and there exist and , with },

Also, we consider the weighted space with the norm

Consider the space of those complex-valued Lebesgue measurable functions on for which , where the norm is defined by

In particular, when , the space coincides with the space: .

Definition 1 [31]. Generalized Hilfer fractional integral.
Let , and . A generalized Hilfer fractional integral of order is defined by where is the Euler gamma function defined by .

Definition 2 [31]. Generalized Hilfer fractional derivative.
Let and . A generalized Hilfer fractional derivative of order α is defined by where and .

Theorem 3 [31]. Let , , and . Then, for , the semigroup property is valid, i.e.,

Lemma 4 [31, 32]. Let , and . Then, is bounded from into

Lemma 5 [32]. Let , , , and . If , then is continuous on and

Lemma 6 [33]. Let . Then, for and , we have

Lemma 7 [32]. Let , , and . Then,

Lemma 8 [32]. Let , . If and , then

Definition 9 [32]. Let order α and type β satisfy and , with . A generalized Hilfer fractional derivative to , with of a function , is defined by

In this paper, we consider the case only, because .

Property 10 [32]. The operator can be written as

Property 11 [32]. The fractional derivative is an interpolator of the following fractional derivatives: Hilfer , Hilfer–Hadamard , generalized , Caputo–type , Riemann–Liouville , Hadamard , Caputo , Caputo–Hadamard , Liouville , and Weyl .
Consider the following parameters satisfying

We define the spaces where .

Since , it follows from Lemma 4 that

Lemma 12 [32]. Let , and . If , then

Lemma 13 (Theorem 25 [32]). Let be a function such that , for any . Then, is a solution of the differential equation: if and only if satisfies the following Volterra integral equation: where .

Theorem 14 ([34]) (Banach’s fixed-point theorem). Let be a nonempty closed subset of a Banach space ; then, any contraction mapping of into itself has a unique fixed point.

Theorem 15 ([34]) (Schaefer’s fixed-point theorem). Let be a Banach space and be a completely continuous operator. If the set is bounded, then has a fixed point.

Theorem 16 ([34]) (Krasnoselskii’s fixed-point theorem). Let be a closed, convex, and nonempty subset of a Banach space and the operators such that (1), for all (2) is compact and continuous(3) is a contraction mapping

Then, there exists such that .

Now, we consider the Ulam stability for problems ((2))–((4)) that will be used in Section 4. Let , , , and be a continuous function.

We consider the following inequality:

Definition 17 ([35]). Problems ((2))–((4)) are Ulam-Hyers-Rassias (U-H-R) stable with respect to if there exists a real number such that for each and for each solution of inequality (27), there exists a solution of (1)–(3) with

Remark 18 ([35]). A function is a solution of inequality (27) if and only if there exist and a sequence , such that (1) and (2)(3)

3. Existence of Solutions

We consider the following linear fractional differential equation: where , , and , with the conditions and where and with and

The following theorem shows that problems ((29))–((31)) have a unique solution given by

Theorem 19. Let , where and . If is a function such that , then satisfies problems ((29))–((31)) if and only if it satisfies (34).

Proof. Assume u satisfies (29)–(31). If , then

Lemma 20. implies that

If , then Lemma 13 implies

Repeating the process in this way, the solution for , , can be written as

Applying on both sides of (39), using Lemma 6 and taking , we obtain

Multiplying both sides of (40) by and using condition (31), we obtain which implies that

Substituting (42) into (39) and (36), we obtain (34).

Reciprocally, applying on both sides of (34) and using Lemma 6 and Theorem 3, we get

Next, taking the limit of (43) and using Lemma 5, with , we obtain

Now, taking in (43), we get

From (44) and (45), we find that which shows that the boundary condition is satisfied. Next, apply operator on both sides of (34), where . Then, from Lemma 6 and Lemma 12, we obtain

Since and by definition of , we have , then (47) implies that

As and from Lemma 4, it follows that

From (48) and (49) and by the definition of the space , we obtain

Applying operator on both sides of (47) and using Lemma 8, Lemma 5, and Property 10, we have that is, (29) holds. Also, we can easily show that

This completes the proof.

As a consequence of Theorem 19, we have the following result.

Lemma 21. Let , where and ; let be a function such that for any . If , then satisfies problems ((2))–((4)) if and only if is the fixed point of the operator defined by where is a function satisfying the functional equation

Assume that the function is continuous and satisfies the following conditions:

(H1). The function be such that

(H2). There exist constants and such that for any and .

(H3). There exists a constant such that for any and

(H4). There exist functions with such that

(H5). The functions are continuous, and there exist constants such that

We are now in a position to state and prove our existence result for problems ((2))–((4)) based on Banach’s fixed point.

Theorem 22. Assume (H1)–(H3) hold. If then problems ((2))–((4)) have a unique solution in .

Proof. The proof will be given in two steps.

Step 1. We show that the operator defined in (53) has a unique fixed point in . Let and ; then, we have where such that

By (H2), we have

Then,

Therefore, for each ,

Thus,

By Lemma 6, we have

Hence, which implies that

By (61), the operator is a contraction. Hence, by Theorem 14, has a unique fixed point .

Step 2. We show that such a fixed point is actually in .
Since is the unique fixed point of operator in , then for each , with , we have where such that

Applying to both sides and by Lemma 6 and Lemma 12, we get

Since , by (H1), the right-hand side is in and thus, which implies that . As a consequence of Steps 1 and 2 together with Theorem 22, we can conclude that problems ((2))–((4)) have a unique solution in .

Our second result is based on Schaefer’s fixed-point theorem.

Theorem 23. Assume (H1), (H4), and (H5) hold. If then problems ((2))–((4)) have at least one solution in .

Proof. We shall use Schaefer’s fixed-point theorem to prove in several steps that the operator defined in (53) has a fixed point.

Step 1. is continuous. Let be a sequence such that in . Then, for each , we have where such that

Since , then we get as , for each , and since and are continuous, then we have

Step 2. We show that maps bounded set into bounded set of . For , there exists a positive constant such that for , we have .

By (H4) and from (53), we have for each , which implies that

Then,

Thus, (53) implies

By Lemma 6, we have

Step 3. maps bounded sets into equicontinuous sets of .

Let , be a bounded set of as in Step 2, and let .

Then,

As , the right-hand side of the above inequality tends to zero. From Steps 1 to 3 with Arzela-Ascoli theorem, we conclude that is continuous and completely continuous.

Step 4. A priori bound. Now it remains to show that the set is bounded. Let , then for some

By (H4), we have for eachwhich implies that then

This implies, by (53) and (H5) and by letting the estimation of Step 2, that for each , we have

By (74), we have

As a consequence of Theorem 15 and using Step 2, we deduce that has a fixed point which is a solution of problems (2)–(4).

Our third result is based on Krasnoselskii’s fixed-point theorem.

Theorem 24. Assume (H1), (H4), and (H5) hold. If then problems ((2))–((4)) have at least one solution in .

Proof. Consider the set where

We define the operators Q₁ and Q₂ on by where , and be a function satisfying the functional equation

Then, the fractional integral equation (53) can be written as operator equation

The proof will be given in several steps.

Step 1. We prove that , for any .
By (H4), (H5), and Lemma 6, for each , we have

Since we have which infers that .

Step 2. is a contraction.
Let and .
By (H4), we have

Then, where , , and such that

Then, by (H5), we have

Hence,

By (90), the operator is a contraction.

Step 3. is continuous and compact.
The continuity of follows from the continuity of . Next, we prove that is uniformly bounded on . Let any . (94) imply where , and is a function satisfying the functional equation

By Lemma 6, we have

This means that is uniformly bounded on . Next, we show that is equicontinuous. Let any and . Then,

Note that

This shows that is equicontinuous. Therefore, is relatively compact. By -type Arzela-Ascoli theorem, is compact. As a consequence of Theorem 16, we deduce that has at least a fixed point and by the same way of the proof of Theorem 22, we can easily show that . Using Lemma 21, we conclude that problems ((2))–((4)) have at least one solution in the space .

4. Ulam-Hyers-Rassias Stability

Now, we are concerned with the Ulam-Hyers-Rassias stability of our problems (2)–(4).

Theorem 25. Assume that in addition to (H1)–(H3) and (61), the following hypothesis holds:

(H6). There exist a nondecreasing function and such that for each , we have and

Then, Equation (2) is U-H-R stable with respect to .

Proof. Consider the operator defined in (53). Let be a solution of inequality (27), and let us assume that is the unique solution of the problem

By Lemma 21, we obtain for each where is a function satisfying the functional equation

Since is a solution of inequality (27), by Remark 2.18, we have

Clearly, the solution of (115) is given by where is a function satisfying the functional equation

Hence, for each , we have

Thus,

By (H2) and Lemma 6, for , we have

Thus,

Then, by (61), we have where

Hence, Equation (2) is U-H-R stable with respect to .

5. Examples

Example 1. Consider the following impulsive BVP of generalized Hilfer Fractional differential equation where , and .

Set

We have with , , , and . Clearly, the continuous function .

Hence, the condition (H1) is satisfied.

For each and ,

Hence, condition (H2) is satisfied with .

And let

Let . Then, we have and so the condition (H3) is satisfied and .

A simple computation shows that condition (61) of Theorem 22 is satisfied, for

Then, problems ((124))–((126)) have a unique solution in .

Also, hypothesis (H6) is satisfied with , , and . Indeed, for each , we get

Consequently, Theorem 25 implies that Equation (124) is U-H-R stable.

Example 2. Consider the following impulsive initial value problem of generalized Hilfer fractional differential equation: where , and .

Set

We have with , , , and . Clearly, the continuous function .

Hence, the condition (H1) is satisfied. For each and ,

Hence, condition (H4) is satisfied with , , , and .

And let

Let . Then, we have and so the condition (H5) is satisfied with and .

The condition (74) of Theorem 23 is satisfied, for

Then, problems ((134))–((136)) have at least one solution in . Also, hypothesis (H6) is satisfied with , and . Indeed, for each , we get

Consequently, by a simple change of the constants , , and from (H1) and (H2) to , and from (H4) and (H5), Theorem 25 implies that Equation (134) is G.U-H-R stable.

Example 3. Consider the following impulsive antiperiodic boundary value problem of generalized Hilfer Fractional differential equation: where , , for , , and .

Set

We have with , and . Clearly, the continuous function .

Hence, the condition (H1) is satisfied. For each and t ∈ (1,2],

Hence, condition (H4) is satisfied with , , and .

And let

Let . Then, we have and so the condition (H5) is satisfied with and .

The condition (90) of Theorem 24 is satisfied, for

Then, problems ((144))–((146)) have at least one solution in . Also, hypothesis (H6) is satisfied with , and . Indeed, for each , we get

The same as Example 2, Theorem 25 implies that Equation (144) is U-H-R stable.

6. Conclusion

In this paper, we are concerned with existence, uniqueness, and Ulam-Hyers-Rassias stability results for a class of boundary value problem for nonlinear implicit fractional differential equations with impulses and generalized Hilfer fractional derivative. We gave three results using the Banach contraction principle and Krasnoselskii’s and Schaefer’s fixed-point theorems. Two illustrative examples about the uniqueness and the stability were presented. An interesting extension of our research would be to address Ulam-Hyers-Mittag-Leffler stability for a class and coupled system of implicit fractional differential equations. This subject will be discussed in the forthcoming papers.

Data Availability

There is no data used during this research.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

S. Abbas, M. Benchohra, J. R. Graef, and J. Henderson, Impli cit Fractional Differential and Integral Equations: Existence and Stability, De Gruyter, London, 2018.
View at: Publisher Site
S. Abbas, M. Benchohra, and G. M. N'Guérékata, Topics in Fractional Differential Equations, vol. 27 of Developments in Mathematics, Springer New York, New York, 2012.
View at: Publisher Site
S. Abbas, M. Benchohra, and G. M. N'Guerekata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2014.
B. Ahmad, A. Alsaedi, S. K. Ntouyas, and J. Tariboon, Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities, Springer, Switzerland, 2017.
View at: Publisher Site
B. Ahmad and S. K. Ntouyas, “Fractional differential inclusions with fractional separated boundary conditions,” Fractional Calculus and Applied Analysis, vol. 15, no. 3, pp. 362–382, 2012.
View at: Publisher Site | Google Scholar
D. Baleanu, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, Dordrecht, 2010.
View at: Publisher Site
M. Benchohra, S. Bouriah, and J. R. Graef, “Boundary value problems for nonlinear implicit Caputo-Hadamard-type fractional differential equations with impulses,” Mediterranean Journal of Mathematics, vol. 14, no. 5, p. 206, 2017.
View at: Publisher Site | Google Scholar
M. Benchohra, J. Henderson, and S. K. Ntouyas, Impulsive Differential Equations and Inclusions, vol. 2, Hindawi Publishing Corporation, New York, 2006.
View at: Publisher Site
M. Benchohra and J. E. Lazreg, “Existence results for nonlinear implicit fractional differential equations with impulse,” Communications in Applied Analysis, vol. 19, pp. 413–426, 2015.
View at: Google Scholar
A. Salim, M. Benchohra, E. Karapinar, and J. E. Lazreg, “Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations,” Advances in Difference Equations, vol. 2020, no. 1, Article ID 3063, 21 pages, 2020.
View at: Publisher Site | Google Scholar
A. Salim, M. Benchohra, J. E. Lazreg, and J. Henderson, “Nonlinear implicit generalized Hilfer-type fractional differential equations with non-instantaneous impulses in Banach spaces,” Advances in the Theory of Nonlinear Analysis and its Application, vol. 4, no. 4, pp. 332–348, 2020.
View at: Publisher Site | Google Scholar
A. Salim, M. Benchohra, J. R. Graef, and J. E. Lazreg, “Boundary value problem for fractional generalised Hilfer-type fractional derivative with non-instantaneous impulses,” Fractal and Fractional, vol. 5, no. 1, pp. 1–21, 2021.
View at: Google Scholar
A. M. AHMAD and N. E. TATAR, “Nonexistence of global solutions for a fractional system of strongly coupledintegro-differential equations,” Turkish Journal of Mathematics, vol. 43, no. 6, pp. 2715–2730, 2019.
View at: Publisher Site | Google Scholar
M. D. Kassim and N. Tatar, “Convergence of solutions of fractional differential equations to power-type functions,” Electronic Journal of Differential Equations, 2020.
View at: Google Scholar
C. Li and S. Sarwar, “Linearization of nonlinear fractional differential systems with Riemann-Liouville and Hadamard derivatives,” Progress in Fractional Differentiation and Applications, vol. 6, pp. 11–22, 2020.
View at: Google Scholar
C. Li and S. Sarwar, “Existence and continuation of solutions for Caputo type fractional differential equations,” Electronic Journal of Differential Equations, vol. 207, pp. 1–14, 2016.
View at: Google Scholar
A. Seemab, M. Ur Rehman, J. Alzabut, and A. Hamdi, “On the existence of positive solutions for generalized fractional boundary value problems,” Boundary Value Problems, vol. 2019, no. 1, Article ID 186, 2019.
View at: Publisher Site | Google Scholar
J. V. C. Sousa, F. Jarad, and T. Abdeljawad, “Existence of mild solutions to Hilfer fractional evolution equations in Banach space,” Annals of Functional Analysis, vol. 12, no. 1, pp. 1–16, 2021.
View at: Publisher Site | Google Scholar
R. Subashini and C. Ravichandran, “On the results of nonlocal Hilfer fractional semilinear differential inclusions,” Proceedings of the Jangjeon Mathematical Society, vol. 22, pp. 249–267, 2019.
View at: Google Scholar
A. Zada, J. Alzabut, H. Waheed, and I. L. Popa, “Ulam-Hyers stability of impulsive integrodifferential equations with Riemann-Liouville boundary conditions,” Advances in Difference Equations, vol. 2020, no. 1, Article ID 64, 2020.
View at: Publisher Site | Google Scholar
A. Zada, H. Waheed, J. Alzabut, and X. Wang, “Existence and stability of impulsive coupled system of fractional integrodifferential equations,” Demonstratio Mathematica, vol. 52, no. 1, pp. 296–335, 2019.
View at: Publisher Site | Google Scholar
S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 1968.
D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, no. 4, pp. 222–224, 1941.
View at: Publisher Site | Google Scholar
T. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American mathematical society, vol. 72, no. 2, pp. 297–300, 1978.
View at: Publisher Site | Google Scholar
I. A. Rus, “Ulam stability of ordinary differential equations,” Studia Universitatis Babes-Bolyai, Mathematica, vol. 25, no. 4, pp. 125–133, 2009.
View at: Google Scholar
S. Abbas, M. Benchohra, J. E. Lazreg, and Y. Zhou, “A survey on Hadamard and Hilfer fractional differential equations: analysis and stability,” Chaos, Solitons and Fractals, vol. 102, pp. 47–71, 2017.
View at: Publisher Site | Google Scholar
S. Abbas, M. Benchohra, J. E. Lagreg, A. Alsaedi, and Y. Zhou, “Existence and Ulam stability for fractional differential equations of Hilfer-Hadamard type,” Advances in Difference Equations, vol. 2017, no. 1, Article ID 1231, 2017.
View at: Publisher Site | Google Scholar
M. Benchohra and J. E. Lazreg, “Existence and Ulam stability for nonlinear implicit fractional differential equations with Hadamard derivative,” Studia Universitatis Babeș-Bolyai Mathematica, vol. 62, no. 1, pp. 27–38, 2017.
View at: Publisher Site | Google Scholar
M. Benchohra and J. E. Lazreg, “On stability for nonlinear implicit fractional differential equations,” Le Matematiche, vol. 70, no. 2, pp. 49–61, 2015.
View at: Google Scholar
S. Harikrishnan, R. W. Ibrahim, and K. Kanagarajan, “Fractional Ulam-stability of fractional impulsive differential equation involving Hilfer-Katugampola fractional differential operator,” Universal Journal of Mathematics and Applications, vol. 1, pp. 106–112, 2018.
View at: Google Scholar
U. Katugampola, “New approach to a generalized fractional integral,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 860–865, 2011.
View at: Publisher Site | Google Scholar
D. S. Oliveira and E. C. de Oliveira, “Hilfer–Katugampola fractional derivatives,” Computational and Applied Mathematics, vol. 37, no. 3, pp. 3672–3690, 2018.
View at: Publisher Site | Google Scholar
R. Almeida, A. B. Malinowska, and T. Odzijewicz, “Fractional differential equations with dependence on the Caputo–Katugampola derivative,” Journal of Computational and Nonlinear Dynamics, vol. 11, no. 6, pp. 1–11, 2016.
View at: Google Scholar
A. Granas and J. Dugundji, Fixed Point Theory, Springer New York, New York, 2003.
View at: Publisher Site
J. Wang, M. Fec˘kan, and Y. Zhou, “Ulam's type stability of impulsive ordinary differential equations,” Journal of Mathematical Analysis and Applications, vol. 395, no. 1, pp. 258–264, 2012.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2021 Abdelkrim Salim 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.

PDF Download Citation

Download other formats

Order printed copies

Views

598

Downloads

1021

Citations