The Existence and Uniqueness Results for a Nonlocal Bounbary Value Problem of Caputo-type Hadamard Hybrid Fractional Integro-differential Equations

Ala Eddine Taier; Ranchao Wu; Fatima Zohra Benyoub

doi:doi:10.11648/j.ajam.20241206.14

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The Existence and Uniqueness Results for a Nonlocal Bounbary Value Problem of Caputo-type Hadamard Hybrid Fractional Integro-differential Equations

Ala Eddine Taier^*

, Ranchao Wu

, Fatima Zohra Benyoub

Published in American Journal of Applied Mathematics (Volume 12, Issue 6)

Received: 8 August 2024 Accepted: 12 November 2024 Published: 18 December 2024

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Abstract

This article is dedicated to study the existence and uniqueness of solutions for a non local bounbary value problem of Caputo-type Hadamard hybrid fractional integro-differential equations in Banach space, the recent researches considered the study of differential equations of Caputo-type Hadamard hybrid fractional integro-differential equations with classical order and the study of existence and uniqueness of solutions using approched numerical methodes, the objective of this paper is the study of the existence and uniqueness of fractional order of integro-differential equations involving the Caputo-type Hadamard derivative using fixed point theory. This work have two important results, the first result was the discussion of a new results owing to the fixed point theorem. Before the prove of results the problem was trandformed to Hadamard type problem. The first result based on Dhage fixed point theorem, after transforming our nonlocal boundary value problem into integral equation we defined operator equation, then we applied the fixed point theorem to get the existence resutl. The second result was the existence and uniqueness of solution for our nonlocal boundary value problem, we get this result using the Banach fixed point theorem. We illustrate our results by example to ending our theorical study.

Published in	American Journal of Applied Mathematics (Volume 12, Issue 6)
DOI	10.11648/j.ajam.20241206.14
Page(s)	246-257
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Caputo-type Hadamard Derivative, Existence Results, Existence and Uniqueness Result, Non Local Boundary Conditions, Fixed Point Theorem

References

[1]	B. C. Dhage, V. Lakshmikantham, Basic results on hybrid differential equations. Nonlinear Anal. Hybrid Syst. 4, 414-424 (2010) https://doi.org/10.1016/j.nahs.2009.10.005
[2]	J. Hadamard , Essai sur I’etude des fonctions donnees par leur developpment de taylor, J. Mat. Pure. Appl. Ser. 8 (1892) 101-186.
[3]	Huang, H., Liu, W.: Positive solutions for a class of nonlinear Hadamard fractional differential equations with a parameter. Adv. Differ. Equ. 2018, 96 (2018) https://doi.org/10.1186/s13662-018-1551-9
[4]	Zhai, C., Wang, W., Li, H.: A uniqueness method to a new Hadamard fractional differential system with four- point boundary conditions. J. Inequal. Appl. 2018, 207 (2018) https://doi.org/10.1186/s13660-018-1801-0
[5]	Yang, W.: Positive solutions for singular coupled integral boundary value problems of nonlinear Hadamard fractional differential equations. J. Nonlinear Sci. Appl. 8, 110-129 (2015) https://doi.org/10.22436/jnsa.008.02.04
[6]	Yang, W.: Positive solutions for singular Hadamard fractional differential system with four-point coupled boundary conditions. J. Appl. Math. Comput. 49, 357- 381 (2015) https://doi.org/10.1007/s12190-014-0843-9
[7]	Li, Y. L., Lin, S. Y.: Positive solution for the nonlinear Hadamard type fractional differential equation with p- Laplacian. J. Funct. Spaces Appl. 2013, Article ID 951643 (2013) https://doi.org/10.1155/2013/05/643
[8]	Wang, G., Wang, T.: On a nonlinear Hadamard type fractional differential equation with p-Laplacian operator and strip condition. J. Nonlinear Sci. Appl. 9, 5073-5081 (2016) https://doi.org/10.22436/jnsa.009.07.10
[9]	Zhang, K., Wang, J., Ma, W.: Solutions for integral boundary value problems of nonlinear Hadamard fractional differential equations. J. Funct. Spaces 2018, Article ID 2193234 (2018) https://doi.org/10.1155/2018/2193234
[10]	Li, S., Zhai, C.: Positive solutions for a new class of Hadamard fractional differential equations on infinite intervals. J. Inequal. Appl. 2019, 50 (2019) https://doi.org/10.1186/s13660-019-2102-y
[11]	Zhang, W., Liu, W.: Existence of solutions for several higher-order Hadamard-type fractional differential equations with integral boundary conditions on infinite interval. Bound. Value Probl. 2018, 134 (2018) https://doi.org/10.1186/s13661-018-1053-4
[12]	Ahmad, B., Ntouyas, S. K.: A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations. Fract. Calc. Appl. Anal. 17, 348-360 (2014) https://doi.org/10.2478/s13540-014-0173-5
[13]	Aljoudi, S., Ahmad, B., Nieto, J. J., Alsaedi, A.: On coupled Hadamard type sequential fractional differential equations with variable coefficients and nonlocal integral boundary conditions. Filomat 31, 6041-6049 (2017) https://doi.org/10.2298/FIL1719041A
[14]	Baitiche, Z. Guerbati, K. Benchohra, M. and Zhou, Y. Boundary Value Problems for Hybrid Caputo Fractional Differential Equations, Mathematics 7. 282 (1019) https://doi.org/10.3390/math7030282
[15]	Jiang, J., O’Regan, J., Xu, J., Fu, Z.: Positive solutions for a system of nonlinear Hadamard fractional differential equations involving coupled integral boundary conditions. J. Inequal. Appl. 2019, 204 (2019) https://doi.org/10.1186/s13660-019-2156-x
[16]	Zhai, C., Wang, W.: Solutions for a system of Hadamard fractional differential equations with integral conditions. Numer. Funct. Anal. Optim. 41(7), 1-21 (2019) https://doi.org/10.1080/01630563.2019.1620771
[17]	Z. Baitiche D. Choukri. ON THE SOLVABILITY OF A FRACTIONAL HYBRID DIFFERENTIAL EQUATION OF HADAMARD TYPE WITH DIRICHLET BOUNDARY CONDITIONS IN BANACH ALGEBRAS, Commun. Optim. Theory 2020 (2020) 10.23952/cot.2020.9
[18]	Alesemi, M.: Solvability for a class of nonlinear Hadamard fractional differential equations with parameters. Bound. Value Probl. 2019, 101 (2019) https://doi.org/10.1186/s13661-019-1213-1
[19]	V. Lakshmikantham, Theory of fractional functional differential equations. Nonlinear Anal. 69(10), 3337- 3343 (2008) https://doi.org/10.1016/j.na.2007.09.025
[20]	I. Podlubny, Fractional Differential Equations (Academic Press, New York, 1993).
[21]	A. A. Kilbas, H.M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204 (Elsevier, Amsterdam, 2006).

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APA Style

Taier, A. E., Wu, R., Benyoub, F. Z. (2024). The Existence and Uniqueness Results for a Nonlocal Bounbary Value Problem of Caputo-type Hadamard Hybrid Fractional Integro-differential Equations. American Journal of Applied Mathematics, 12(6), 246-257. https://doi.org/10.11648/j.ajam.20241206.14

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ACS Style

Taier, A. E.; Wu, R.; Benyoub, F. Z. The Existence and Uniqueness Results for a Nonlocal Bounbary Value Problem of Caputo-type Hadamard Hybrid Fractional Integro-differential Equations. Am. J. Appl. Math. 2024, 12(6), 246-257. doi: 10.11648/j.ajam.20241206.14

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AMA Style

Taier AE, Wu R, Benyoub FZ. The Existence and Uniqueness Results for a Nonlocal Bounbary Value Problem of Caputo-type Hadamard Hybrid Fractional Integro-differential Equations. Am J Appl Math. 2024;12(6):246-257. doi: 10.11648/j.ajam.20241206.14

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@article{10.11648/j.ajam.20241206.14,
  author = {Ala Eddine Taier and Ranchao Wu and Fatima Zohra Benyoub},
  title = {The Existence and Uniqueness Results for a Nonlocal Bounbary Value Problem of Caputo-type Hadamard Hybrid Fractional Integro-differential Equations},
  journal = {American Journal of Applied Mathematics},
  volume = {12},
  number = {6},
  pages = {246-257},
  doi = {10.11648/j.ajam.20241206.14},
  url = {https://doi.org/10.11648/j.ajam.20241206.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20241206.14},
  abstract = {This article is dedicated to study the existence and uniqueness of solutions for a non local bounbary value problem of Caputo-type Hadamard hybrid fractional integro-differential equations in Banach space, the recent researches considered the study of differential equations of Caputo-type Hadamard hybrid fractional integro-differential equations with classical order and the study of existence and uniqueness of solutions using approched numerical methodes, the objective of this paper is the study of the existence and uniqueness of fractional order of integro-differential equations involving the Caputo-type Hadamard derivative using fixed point theory. This work have two important results, the first result was the discussion of a new results owing to the fixed point theorem. Before the prove of results the problem was trandformed to Hadamard type problem. The first result based on Dhage fixed point theorem, after transforming our nonlocal boundary value problem into integral equation we defined operator equation, then we applied the fixed point theorem to get the existence resutl. The second result was the existence and uniqueness of solution for our nonlocal boundary value problem, we get this result using the Banach fixed point theorem. We illustrate our results by example to ending our theorical study.},
 year = {2024}
}

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TY  - JOUR
T1  - The Existence and Uniqueness Results for a Nonlocal Bounbary Value Problem of Caputo-type Hadamard Hybrid Fractional Integro-differential Equations
AU  - Ala Eddine Taier
AU  - Ranchao Wu
AU  - Fatima Zohra Benyoub
Y1  - 2024/12/18
PY  - 2024
N1  - https://doi.org/10.11648/j.ajam.20241206.14
DO  - 10.11648/j.ajam.20241206.14
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 246
EP  - 257
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20241206.14
AB  - This article is dedicated to study the existence and uniqueness of solutions for a non local bounbary value problem of Caputo-type Hadamard hybrid fractional integro-differential equations in Banach space, the recent researches considered the study of differential equations of Caputo-type Hadamard hybrid fractional integro-differential equations with classical order and the study of existence and uniqueness of solutions using approched numerical methodes, the objective of this paper is the study of the existence and uniqueness of fractional order of integro-differential equations involving the Caputo-type Hadamard derivative using fixed point theory. This work have two important results, the first result was the discussion of a new results owing to the fixed point theorem. Before the prove of results the problem was trandformed to Hadamard type problem. The first result based on Dhage fixed point theorem, after transforming our nonlocal boundary value problem into integral equation we defined operator equation, then we applied the fixed point theorem to get the existence resutl. The second result was the existence and uniqueness of solution for our nonlocal boundary value problem, we get this result using the Banach fixed point theorem. We illustrate our results by example to ending our theorical study.
VL  - 12
IS  - 6
ER  -

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Author Information

Ala Eddine Taier

Department of Applied Mathematics, School of Mathematical Sciences, Anhui University, Hefei, China

Contact Email

http://orcid.org/0000-0003-1983-6353
Ranchao Wu

Department of Applied Mathematics, School of Mathematical Sciences, Anhui University, Hefei, China

Contact Email

http://orcid.org/0000-0002-7780-0861
Fatima Zohra Benyoub

School of Automation and Electrical Engineering, Beihang University, Beijing, China

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Plain Text BibTeX RIS

APA Style

Taier, A. E., Wu, R., Benyoub, F. Z. (2024). The Existence and Uniqueness Results for a Nonlocal Bounbary Value Problem of Caputo-type Hadamard Hybrid Fractional Integro-differential Equations. American Journal of Applied Mathematics, 12(6), 246-257. https://doi.org/10.11648/j.ajam.20241206.14

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ACS Style

Taier, A. E.; Wu, R.; Benyoub, F. Z. The Existence and Uniqueness Results for a Nonlocal Bounbary Value Problem of Caputo-type Hadamard Hybrid Fractional Integro-differential Equations. Am. J. Appl. Math. 2024, 12(6), 246-257. doi: 10.11648/j.ajam.20241206.14

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AMA Style

Taier AE, Wu R, Benyoub FZ. The Existence and Uniqueness Results for a Nonlocal Bounbary Value Problem of Caputo-type Hadamard Hybrid Fractional Integro-differential Equations. Am J Appl Math. 2024;12(6):246-257. doi: 10.11648/j.ajam.20241206.14

Copy | Download

@article{10.11648/j.ajam.20241206.14,
  author = {Ala Eddine Taier and Ranchao Wu and Fatima Zohra Benyoub},
  title = {The Existence and Uniqueness Results for a Nonlocal Bounbary Value Problem of Caputo-type Hadamard Hybrid Fractional Integro-differential Equations},
  journal = {American Journal of Applied Mathematics},
  volume = {12},
  number = {6},
  pages = {246-257},
  doi = {10.11648/j.ajam.20241206.14},
  url = {https://doi.org/10.11648/j.ajam.20241206.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20241206.14},
  abstract = {This article is dedicated to study the existence and uniqueness of solutions for a non local bounbary value problem of Caputo-type Hadamard hybrid fractional integro-differential equations in Banach space, the recent researches considered the study of differential equations of Caputo-type Hadamard hybrid fractional integro-differential equations with classical order and the study of existence and uniqueness of solutions using approched numerical methodes, the objective of this paper is the study of the existence and uniqueness of fractional order of integro-differential equations involving the Caputo-type Hadamard derivative using fixed point theory. This work have two important results, the first result was the discussion of a new results owing to the fixed point theorem. Before the prove of results the problem was trandformed to Hadamard type problem. The first result based on Dhage fixed point theorem, after transforming our nonlocal boundary value problem into integral equation we defined operator equation, then we applied the fixed point theorem to get the existence resutl. The second result was the existence and uniqueness of solution for our nonlocal boundary value problem, we get this result using the Banach fixed point theorem. We illustrate our results by example to ending our theorical study.},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - The Existence and Uniqueness Results for a Nonlocal Bounbary Value Problem of Caputo-type Hadamard Hybrid Fractional Integro-differential Equations
AU  - Ala Eddine Taier
AU  - Ranchao Wu
AU  - Fatima Zohra Benyoub
Y1  - 2024/12/18
PY  - 2024
N1  - https://doi.org/10.11648/j.ajam.20241206.14
DO  - 10.11648/j.ajam.20241206.14
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 246
EP  - 257
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20241206.14
AB  - This article is dedicated to study the existence and uniqueness of solutions for a non local bounbary value problem of Caputo-type Hadamard hybrid fractional integro-differential equations in Banach space, the recent researches considered the study of differential equations of Caputo-type Hadamard hybrid fractional integro-differential equations with classical order and the study of existence and uniqueness of solutions using approched numerical methodes, the objective of this paper is the study of the existence and uniqueness of fractional order of integro-differential equations involving the Caputo-type Hadamard derivative using fixed point theory. This work have two important results, the first result was the discussion of a new results owing to the fixed point theorem. Before the prove of results the problem was trandformed to Hadamard type problem. The first result based on Dhage fixed point theorem, after transforming our nonlocal boundary value problem into integral equation we defined operator equation, then we applied the fixed point theorem to get the existence resutl. The second result was the existence and uniqueness of solution for our nonlocal boundary value problem, we get this result using the Banach fixed point theorem. We illustrate our results by example to ending our theorical study.
VL  - 12
IS  - 6
ER  -

Copy | Download