Iterative Positive Solutions to a Coupled Hadamard-Type Fractional Differential System on Infinite Domain with the Multistrip and Multipoint Mixed Boundary Conditions

Du, Xinran; Meng, Yuan; Pang, Huihui

doi:https://doi.org/10.1155/2020/6508075

Journal of Function Spaces

On this page

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

Research Article | Open Access

Volume 2020 | Article ID 6508075 | https://doi.org/10.1155/2020/6508075

Iterative Positive Solutions to a Coupled Hadamard-Type Fractional Differential System on Infinite Domain with the Multistrip and Multipoint Mixed Boundary Conditions

Xinran Du,¹Yuan Meng,¹and Huihui Pang¹

Guest Editor: Abdon Atangana

Received06 Apr 2020

Accepted08 Jun 2020

Published03 Jul 2020

Abstract

This paper is devoted to the existence of positive solutions for a nonlinear coupled Hadamard fractional differential system, with multistrip and multipoint mixed boundary conditions on an infinite interval. Based on the Arzelá-Ascoli theorem, we establish an important lemma to prove the complete continuity of operators on the infinite interval. Using the monotone iterative technique, the existence criteria for positive extremal solutions can be acquired, and an example is given to illustrate the feasibility of the above study as well.

1. Introduction

Fractional calculus, which is organically united with integral calculus, extends the concept of classical integral calculus to the whole real number line and even the complex plane. In latest researches, it has been fully shown that fractional calculus definitely has some excellent properties. This kind of calculus is nonlocal in nature, which can accurately describe several materials and processes with traits of heredity and memory [1, 2]. There are numerous applications in a variety of disciplinary fields such as characterization of anomalous diffusion [3], random processes [4], viscoelasticity [5], non-Newtonian fluid mechanics [6], and biomathematics [7]. Moreover, differential equations can succinctly establish the relationship between variables and their derivatives. Therefore, the study of differential equations with fractional calculus is of significance theoretically and practically. In fact, fractional differential equations have been widely focused on and studied in depth. For the recent development of the topic, many researchers studied the existence [8–10], uniqueness [11, 12], and multiplicity of the solutions to fractional differential problems [13, 14].

Meanwhile, it is observed that there have been many studies based on Riemann-Liouville- and Caputo-type fractional differential equations. However, there is another kind of fractional differential equation, which is based on the Hadamard-type fractional calculus definition, found in the literature due to Hadamard (see [15]). It is worth mentioning that Hadamard’s construction of fractional integrodifferential, including a logarithmic function of arbitrary exponent, is a fractional power of the type . Therefore, it is well suited to the case of the half-axis and is invariant relative to dilation. As a result, the Hadamard fractional definition will be a strong tool to expand the interval further. From a finite interval to an infinite semi-infinite interval, this is theoretically a high generalization of many existing models, more general and innovative. This model has already been used in applied mathematics and physics. A detailed description about the boundary value problems of the Hadamard fractional derivative and integral on infinite intervals can be seen in [16–20].

In [16], Thiramanus et al. considered the positive solution for Hadamard fractional differential equations on infinite intervals by applying the idea of Leggett-Williams and Guo-Krasnoselskii fixed point theorems where denotes the Hadamard fractional derivative of order , and is the Hadamard fractional integral of order , and are the given constants.

In [19], Pei et al. not only established the existence of positive solutions but also sought the positive minimal and maximal solutions and got two explicit monotone iterative sequences which converge to the extremal solutions where denotes Hadamard fractional derivative, and is the Hadamard fractional integral. are the given constants and satisfy .

Furthermore, the system of fractional differential equations boundary value problems has also received much attention. More and more researchers invest effort in this direction; see [20, 21] and the references therein.

More recently in [20], Yang investigated the extremal iterative solutions for a coupled system of nonlinear Hadamard fractional differential equations with Cauchy initial value conditions by using the comparison principle and the monotone iterative technique combined with the method of upper and lower solutions where , and are the left-sided Hadamard fractional derivative and integral of order , respectively.

In [21], Ahmad and Ntouyas proved the existence and uniqueness of positive solutions of a class of boundary value problems of fractional differential equations by utilizing the Leray-Schauder alternative principle and the Banach contraction mapping principle where and , respectively, denote the Hadamard fractional derivative and Hadamard fractional integral, and are given continuous functions.

Inspired by the aforementioned work, to get more extensive results, we investigate the existence of iterative positive solutions to the following Hadamard fractional differential systems subject to the coupled fractional-order integral and discrete mixed boundary conditions where denotes the Hadamard fractional derivative of order and is the Hadamard fractional integral of order , for , and , for .

The main aim of this paper is to investigate the coupled nonlinear fractional differential system of the Hadamard type on infinite intervals subject to the coupled multistrip and multipoint mixed boundary conditions. This kind of condition is a linear combination of values at the multiple band integrals and the different discrete points, which highly summarizes the characteristics of boundary conditions in the existing study. There are two unknown functions which influence each other in this system. And the nonlinear part, and , contain lower order derivative operators as well. Based on the above model, the present research results are not abundant or even almost blank; therefore, we fill the gap in this paper.

The proofs of our main result are derived by using the monotone iterative method, which gets two significant benefits. Firstly, by selecting the appropriate initial function vector of concise form, the existence of solution can be guaranteed with the effective and succinct process. Moreover, we can seek the approximate positive solution under a different level of precision, which has more practical application value. Through comprehensive consideration, we construct the initial iterative function vector which satisfies the multiconstraints from the cone and the monotonicity of the complete continuous operator and its respective derivatives. More researches on the details of the method can be found in these references [22–27].

This article is structured as follows: Section 2 contains numerous relative definitions and lemmas. Also, there are some main results and their proofs. In Section 3, we present the existence results of monotone iterative positive solutions. Besides, the main results are illustrated by a practical example.

2. Preliminaries

In this section, we will present here the definitions, some lemmas from the theory of fractional calculus, and some auxiliary results for the proof of our main results.

Definition 1 [1]. If , then the Hadamard fractional integral of order for a function is defined as provided the integral exists. Such a space, which we denote by , consists of those complex-valued Lebesgue measurable functions on for which , with In particular, when the space coincides with the -space:

Definition 2 [1]. The Hadamard derivative of fractional order for a function is defined as where denotes the integer part of the real number and .

Lemma 3 [2]. If , then

Lemma 4 [2]. Let and . Then, the Hadamard fractional differential equation has the solutions and the following formula holds: where , and where , and .

For convenience, we denote

In the forthcoming analysis, we always need the following assumptions: (F1)(F2)(F3), where are defined by (15)(F4) is continuous

Subject to BVP (5) and (6), we consider a corresponding linear boundary value problem as follows and establish the expression of the corresponding Green’s functions.

Lemma 5. Assume that (F1)-(F3) hold. For with , the fractional differential system with boundary condition (6) has an integral representation where and for ,

Proof. From Lemma 4, we can reduce (16) and (6) to the following equivalent integral equations: where are constants.

From , we have . Further, we use the second condition of (6) to reduce (21) to

Then, we can get

Combining (15), (23), and (24), it can be seen that where is defined by (15). From (22) and (25), we have where and are introduced by (19). Similarly, we also have where and are also given by (19).

This completes the proof of the lemma.

Moreover, according to (17) and Lemma 4, the fractional order derivative of solution (17) can be expressed as where and for ,

Lemma 6. Assume that (F1) holds. Then, the functions and , for , defined by (20) and (33) have the following properties: (1)(2)

Proof. (1)For , we haveFor , we have (2)According to (33), we can easily getThis completes the proof of the lemma.

For convenience, we denote

Lemma 7. Assume that (F1)-(F3) hold. Then, for , the functions and , for , defined by (18), (19), and (29), satisfy the following results: (1)(2)

Proof. (1)According to (F3), Lemma 6 and the definition of , we obtainwhere is defined by (37). Similarly, we get where is defined by (38). (2)According to (F3), Lemma 6, and the definition of , we infer thatwhere is defined by (39). Analogously, we get where is defined by (40).
This completes the proof of the lemma.

Let , and , and is endowed with the norm where .

Also, let , , and is endowed with the norm where . Then, we introduce the product space endowed with the norm and define a partial order over the product space: while , , , .

Lemma 8. is a Banach space.

Proof. Let be a Cauchy sequence in the space . Then, , such that for n, . Therefore, for any fixed , we have , which are Cauchy sequences in . So we can associate to each for a unique , . Letting , we obtain and for all and . It easily shows that and in X as . In other words, uniformly converge to , also uniformly converge to on .

Now we should prove that , while u, .

For , we have

In view of the convergence of , we obtain that uniformly for . Otherwise, by Lemma 4, we have , for . The two facts imply that

Combining with for , we have

Take the derivative of -order on both sides of equation (53). In consequence, we have

From Lemma 4, it leads to which proves that is a Banach space.

Also, is a Banach space which can be proved in the same way. Moreover, the product space is also a Banach space with the norm

The proof of this lemma is completed.

Further, we define a cone . For all , in view of Lemma 5 and (F4), let be the operator defined by where and for convenience, we set

Also, according to (28), it is easy to see that

Now, we list the following assumptions:

(H₁) For , there exist nonnegative functions , , , and defined on and constants such that and denote that

(H₂) is increasing with respect to the second, third, fourth, and last variables, and , for .

Lemma 9. If holds, for any , we have

Proof. For any , we have Hence, The proof is completed.

Lemma 10. Let be a bounded set. Then is relatively compact in if the following conditions hold: (i)For any and are equicontinuous on any compact interval of (ii)for any there exists a constant such thatfor any and .

Proof. According to the conditions, we only need to prove that is totally bounded. The proof consists of the following two parts. (a)Consider the case that , we can define a Banach space. , with the norm . Arzelá-Ascoli theorem and (i) can ensure that is relatively a compact set, which means is a totally bounded set. In other words, for , there exist finitely many balls such thatwhere Similarly, define a Banach space equipped with norm . That is to say, for , there exist finitely many balls such that where (b)Define .Evidently, . Let us consider . Then, the set can be covered by the balls , where .
From (a), we can get that there exist and such that . Thus, for any , we get Hence, for , inequalities in (i) and (70) indicate that Likely, by inequalities in (i) and (71) we know that It follows from all the above that , which implies that is a totally bounded set.
The proof is finished.
There are similar conclusions for space ; we omit the details here.

Lemma 11 [21]. The operator is completely continuous.

Proof. There are four steps to complete the whole proof. (a) is uniformly boundedLet be any bounded subset of P, then there exists such that for all ; it follows that So, for , is uniformly bounded. Furthermore, we get is also uniformly bounded.
Therefore, it follows from the above inequalities that the operator is uniformly bounded. (b)The operator is equicontinuous on any compact interval of .Let be any compact interval. We assume that , , then For any compact set , and are uniformly continuous as . Notice that this function is only related to when , so it is uniformly continuous on . Above all, for all and , such that if , then Combined with Lemma 9, we get for , Together with (75) and (76), the above inequality implies that is equicontinuous on . Similar conclusion can be drawn in .
Note that Since and do not depend on , we can obtain that is equicontinuous on . Moreover, is also equicontinuous on .
Hence, is equicontinuous on any compact interval of . (c)The operator and are equiconvergent at ∞For any , we have then combining with the conclusion of the , we get which is equiconvergent at .For any , we have then combining with the conclusion of the , which implies that is equiconvergent at . (d)The operator is continuousLet , then , we also let We have, With the Lebesgue-dominated convergence theorem and continuity of , we guarantee that So we have Above all, is continuous. Meanwhile, we can obtain that is continuous. In consequence, T is continuous.
From the above steps, we get the operator is completely continuous. The proof is completed.

3. Existence Results of Monotone Iterative Positive Solutions

Now, based on Lemma 7, Lemma 8, and Lemma 11, in what follows, we show that there exist positive extremal solutions for BVP (5) and (6) by the monotone iterative method.

Theorem 12. Assume that (F1)-(F4) hold, for , , choose Under the conditions (H₁) and (H₂), BVP (5) and (6) have positive solutions and satisfying , which , , which ,

Proof. Denote , where is introduced by (86). In the following, we first prove that . For any , in view of Lemma 7 and Lemma 9, we have further, we can also get .
As a result, we obtain and .

According to (91) and (93), it is obvious that . By using the completely continuous operator , we define the sequences and as ,, for Since , we get that , for

For , according to the definition of the iterative scheme, we have

Similarly,

From (96) and (97), we can get

Next, we discuss the monotonicity of fractional derivative of . From (91), we obtain

Hence, one has

Combining (100) and (101), it is easy to see that

Thus, for , from (98), (102), and , we do the second iteration

By induction, for , one has

Due to the complete continuity of the operator , there exist such that as , which can be obtained through that has a convergent subsequence . This together with (104) implies that . Additionally, is continuous and ; it implies .

For the sequence , we apply a similar argument. For , we have

To sum up, for , we have

Analogously, for , one has

Similarly, we can also get that . Additionally, is continuous and ; it implies that .

Consequently, there exist and in B which are nonnegative extremal solutions of BVP (5) and (6). Since , for , then zero is not a solution of problem (5). It is obvious that . The proof is completed.

Example 13. For , consider the following fractional differential system: with the coupled integral and discrete mixed boundary conditions: In this model, we set It is obvious that (F1) and (F2) hold. By calculation, we get According to the forms of and shown in (108), we have We can show that
similarly, we get And, we have also, , which imply that () holds.

From the expression of function , we can know that is increasing with respect to the second, third, fourth and last variables, so holds.

Set , , and , we get . Thus, by Theorem 12, BVP (108) and (109) have monotone positive solutions and , respectively, in , which can be approximated by the following iterative sequences: and the initial values are

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

All the authors contributed equally and significantly to writing this article. All the authors read and approved the final manuscript.

Acknowledgments

The project is supported by the National Training Program of Innovation (Project No. 202010019061).

References

I. Podlubny, “Fractional differential equations,” in Mathematics in Science and Engineering, Academic Press, Cambridge, MA, USA, 1999.
View at: Google Scholar
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, the Netherlands, 2006.
H. Sun, W. Chen, C. Li, and Y. Q. Chen, “Fractional differential models for anomalous diffusion,” Physica A: Statistical Mechanics and its Applications, vol. 389, no. 14, pp. 2719–2724, 2010.
View at: Publisher Site | Google Scholar
M. Seghier, A. Ouahab, and J. Henderson, “Random solutions to a system of fractional differential equations via the Hadamard fractional derivative,” The European Physical Journal Special Topics, vol. 226, no. 16-18, pp. 3525–3549, 2017.
View at: Publisher Site | Google Scholar
H. H. Sherief and M. A. el-Hagary, “Fractional order theory of thermo-viscoelasticity and application,” Mechanics of Time-Dependent Materials, 2019.
View at: Publisher Site | Google Scholar
G. Alotta, E. Bologna, G. Failla, and M. Zingales, “A fractional approach to non-Newtonian blood rheology in capillary vessels,” Journal of Peridynamics and Nonlocal Modeling, vol. 1, no. 2, pp. 88–96, 2019.
View at: Publisher Site | Google Scholar
E. F. D. Goufo, “A biomathematical view on the fractional dynamics of cellulose degradation,” Fractional Calculus and Applied Analysis, vol. 18, no. 3, pp. 554–564, 2015.
View at: Publisher Site | Google Scholar
X. Zhao, Y. Liu, and H. Pang, “Iterative positive solutions to a coupled fractional differential system with the multistrip and multipoint mixed boundary conditions,” Advances in Difference Equations, vol. 2019, no. 1, 2019.
View at: Publisher Site | Google Scholar
Z. Bai, Y. Cheng, and S. Sun, “On solutions of a class of three-point fractional boundary value problems,” Boundary Value Problems, vol. 2020, no. 1, 2020.
View at: Publisher Site | Google Scholar
B. Ahmad, S. K. Ntouyas, and A. Alsaedi, “Existence of solutions for fractional differential equations with nonlocal and average type integral boundary conditions,” Journal of Applied Mathematics and Computing, vol. 53, no. 1-2, pp. 129–145, 2017.
View at: Publisher Site | Google Scholar
Y. Cui, “Uniqueness of solution for boundary value problems for fractional differential equations,” Applied Mathematics Letters, vol. 51, pp. 48–54, 2016.
View at: Publisher Site | Google Scholar
L. Guo, L. Liu, and Y. Wu, “Uniqueness of iterative positive solutions for the singular fractional differential equations with integral boundary conditions,” Boundary Value Problems, vol. 2016, no. 1, 2016.
View at: Publisher Site | Google Scholar
W. Zhang, Z. Bai, and S. Sun, “Extremal solutions for some periodic fractional differential equations,” Advances in Difference Equations, vol. 2016, no. 1, 2016.
View at: Publisher Site | Google Scholar
H. M. Ahmed and M. M. El-Borai, “Hilfer fractional stochastic integro-differential equations,” Applied Mathematics and Computation, vol. 331, pp. 182–189, 2018.
View at: Publisher Site | Google Scholar
J. Hadamard, “Essai sur ietude des fonctions donnees par leur developpment de taylor,” Journal de Mathématiques Pures et Appliquées, vol. 8, pp. 101–186, 1892.
View at: Google Scholar
P. Thiramanus, S. K. Ntouyas, and J. Tariboon, “Positive solutions for Hadamard fractional differential equations on infinite domain,” Advances in Difference Equations, vol. 2016, no. 1, 2016.
View at: Publisher Site | Google Scholar
S. Aljoudi, B. Ahmad, J. J. Nieto, and A. Alsaedi, “A coupled system of Hadamard type sequential fractional differential equations with coupled strip conditions,” Chaos, Solitons & Fractals, vol. 91, pp. 39–46, 2016.
View at: Publisher Site | Google Scholar
J. Jiang, W. Liu, and H. Wang, “Positive solutions to singular Dirichlet-type boundary value problems of nonlinear fractional differential equations,” Advances in Difference Equations, vol. 2018, no. 1, 2018.
View at: Publisher Site | Google Scholar
K. Pei, G. Wang, and Y. Sun, “Successive iterations and positive extremal solutions for a Hadamard type fractional integro-differential equations on infinite domain,” Applied Mathematics and Computation, vol. 312, pp. 158–168, 2017.
View at: Publisher Site | Google Scholar
W. Yang, “Monotone iterative technique for a coupled system of nonlinear Hadamard fractional differential equations,” Journal of Applied Mathematics and Computing, vol. 59, no. 1-2, pp. 585–596, 2019.
View at: Publisher Site | Google Scholar
B. Ahmad and S. Ntouyas, “A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations,” Fractional Calculus and Applied Analysis, vol. 17, no. 2, pp. 348–360, 2014.
View at: Publisher Site | Google Scholar
Y. Sun and S. Yan, “Positive solutions and monotone iterative sequences for a fractional differential equation with integral boundary conditions,” Advances in Difference Equations, vol. 2014, no. 1, 2014.
View at: Publisher Site | Google Scholar
Y. He, “Existence results and the monotone iterative technique for nonlinear fractional differential systems involving fractional integral boundary conditions,” Advances in Difference Equations, vol. 2017, no. 1, 2017.
View at: Publisher Site | Google Scholar
Y. Cui, Q. Sun, and X. Su, “Monotone iterative technique for nonlinear boundary value problems of fractional order ,” Advances in Difference Equations, vol. 2017, no. 1, 2017.
View at: Publisher Site | Google Scholar
G. Wang, “Explicit iteration and unbounded solutions for fractional integral boundary value problem on an infinite interval,” Applied Mathematics Letters, vol. 47, pp. 1–7, 2015.
View at: Publisher Site | Google Scholar
G. Wang, K. Pei, and Y. Q. Chen, “Stability analysis of nonlinear Hadamard fractional differential system,” Journal of the Franklin Institute-Engineering and Applied Mathematics, vol. 356, no. 12, pp. 6538–6546, 2019.
View at: Publisher Site | Google Scholar
X. Ren, G. Wang, Z. Bai, and A. A. El-Deeb, “Maximum principle and its application to multi-index Hadamard fractional diffusion equation,” Boundary Value Problems, vol. 2019, no. 1, 2019.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2020 Xinran Du 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

315

Downloads

630

Citations