Boundary Value Problem of Nonlinear Fractional Differential Equations of Mixed Volterra-Fredohlm Integral Equations in Banach Space

Ala Eddine Taier; Ranchao Wu; Naveed Iqbal

doi:doi:10.11648/j.ajam.20241201.11

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Boundary Value Problem of Nonlinear Fractional Differential Equations of Mixed Volterra-Fredohlm Integral Equations in Banach Space

Ala Eddine Taier^*, Ranchao Wu, Naveed Iqbal

Published in American Journal of Applied Mathematics (Volume 12, Issue 1)

Received: 6 January 2024 Accepted: 29 January 2024 Published: 28 February 2024

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Abstract

This paper is dedicated to study the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional differential equations of mixed Volterra-Fredohlm integral equations in Banach space, the recent researches considered the study of differential equations of mixed Volterra-Fredholm integral 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 differential equations with mixed Volterra-Fredholm integral equations using fixed point theory. This work have two important results, the first result was the discussion of the existence of solutions using the Krasnoselskii fixed point theorem after transforming the problem into integral equation firstly then into operator problem suitable for the fixed point theory. The second result will be the existence and uniqueness of solution, this result was obtained by the use of Banach fixed point theorem after the same transformation used in the first one. This work give as conclusion that the boundary value problem of nonlinear fractional differential equations of mixed Volterra-Fredholm integral equation has a unique solution in Banach space. Finally, this work was ended with an example to illustrate the results obtained.

Published in	American Journal of Applied Mathematics (Volume 12, Issue 1)
DOI	10.11648/j.ajam.20241201.11
Page(s)	1-8
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

Mixed Volterra-Fredohlm Integral Equation, Existence and Uniqueness, Fixed Point Theory

References

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[4]	J. Graef, L. Kong Existence of positive solutions to a higher order singular boundary value problem with fractional Q- derivatives, Fract. Calc. Appl. Anal. 16 (2013), 695-708.
[5]	D.O’Regan, S,Stanek, Fractionalboundaryvalueproblemswith singularities in space variables, Nonlinear Dynam. 71 (2013), 641-652.
[6]	F. Mirzaee, S. F. Hoseini, Application of Fibonacci collocation method for solving Volterra-Fredholm integral equations, Appl. Math. Comput. 273 (2016), 637-644.
[7]	F. Mirzaee, E. Hadadiyan, Numerical solution of Volterra- Fredholm integral equations via modification of hat functions, Appl. Math. Comput. 280 (2016), 110-123.
[8]	P. M. A. Hasan, N. A. Sulaiman, Existence and Uniqueness of Solution for Linear Mixed Volterra-Fredholm Integral Equations in Banach Space, Am. J. Comput. Appl. Math. 9 (2019), 1-5.
[9]	L. Mei, Y. Lin, Simplified reproducing kernel method and convergence order for linear Volterra integral equations with variable coefficients, J. Comput. Appl. Math. 346 (2019), 390- 398.
[10]	S. Micula, On some iterative numerical methods for mixed Volterra? Fredholm integral equations, Symmetry 11 (2019), 1200.
[11]	S. Deniz S, N. Bildik, Optimal perturbation iteration method for Bratu-type problems, Journal of King Saud University - Science 30 (2018), 91-99.
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[17]	Ahmad B, Ntouyas S K., A four-point nonlocal integral boundary value problem for fractional differential equations of arbitraryorder, Electron. J.Qual. TheoryDiffer. Equ. 22(2011), 15.

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APA Style

Taier, A. E., Wu, R., Iqbal, N. (2024). Boundary Value Problem of Nonlinear Fractional Differential Equations of Mixed Volterra-Fredohlm Integral Equations in Banach Space. American Journal of Applied Mathematics, 12(1), 1-8. https://doi.org/10.11648/j.ajam.20241201.11

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ACS Style

Taier, A. E.; Wu, R.; Iqbal, N. Boundary Value Problem of Nonlinear Fractional Differential Equations of Mixed Volterra-Fredohlm Integral Equations in Banach Space. Am. J. Appl. Math. 2024, 12(1), 1-8. doi: 10.11648/j.ajam.20241201.11

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AMA Style

Taier AE, Wu R, Iqbal N. Boundary Value Problem of Nonlinear Fractional Differential Equations of Mixed Volterra-Fredohlm Integral Equations in Banach Space. Am J Appl Math. 2024;12(1):1-8. doi: 10.11648/j.ajam.20241201.11

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@article{10.11648/j.ajam.20241201.11,
  author = {Ala Eddine Taier and Ranchao Wu and Naveed Iqbal},
  title = {Boundary Value Problem of Nonlinear Fractional Differential Equations of Mixed Volterra-Fredohlm Integral Equations in Banach Space},
  journal = {American Journal of Applied Mathematics},
  volume = {12},
  number = {1},
  pages = {1-8},
  doi = {10.11648/j.ajam.20241201.11},
  url = {https://doi.org/10.11648/j.ajam.20241201.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20241201.11},
  abstract = {This paper is dedicated to study the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional differential equations of mixed Volterra-Fredohlm integral equations in Banach space, the recent researches considered the study of differential equations of mixed Volterra-Fredholm integral 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 differential equations with mixed Volterra-Fredholm integral equations using fixed point theory. This work have two important results, the first result was the discussion of the existence of solutions using the Krasnoselskii fixed point theorem after transforming the problem into integral equation firstly then into operator problem suitable for the fixed point theory. The second result will be the existence and uniqueness of solution, this result was obtained by the use of Banach fixed point theorem after the same transformation used in the first one. This work give as conclusion that the boundary value problem of nonlinear fractional differential equations of mixed Volterra-Fredholm integral equation has a unique solution in Banach space. Finally, this work was ended with an example to illustrate the results obtained.},
 year = {2024}
}

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TY  - JOUR
T1  - Boundary Value Problem of Nonlinear Fractional Differential Equations of Mixed Volterra-Fredohlm Integral Equations in Banach Space
AU  - Ala Eddine Taier
AU  - Ranchao Wu
AU  - Naveed Iqbal
Y1  - 2024/02/28
PY  - 2024
N1  - https://doi.org/10.11648/j.ajam.20241201.11
DO  - 10.11648/j.ajam.20241201.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
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AB  - This paper is dedicated to study the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional differential equations of mixed Volterra-Fredohlm integral equations in Banach space, the recent researches considered the study of differential equations of mixed Volterra-Fredholm integral 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 differential equations with mixed Volterra-Fredholm integral equations using fixed point theory. This work have two important results, the first result was the discussion of the existence of solutions using the Krasnoselskii fixed point theorem after transforming the problem into integral equation firstly then into operator problem suitable for the fixed point theory. The second result will be the existence and uniqueness of solution, this result was obtained by the use of Banach fixed point theorem after the same transformation used in the first one. This work give as conclusion that the boundary value problem of nonlinear fractional differential equations of mixed Volterra-Fredholm integral equation has a unique solution in Banach space. Finally, this work was ended with an example to illustrate the results obtained.
VL  - 12
IS  - 1
ER  -

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Author Information

Ala Eddine Taier

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

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Ranchao Wu

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

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Naveed Iqbal

Department of Mathematics , College of Science, University of Ha’il, Ha’il, Saudi Arabia

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APA Style

Taier, A. E., Wu, R., Iqbal, N. (2024). Boundary Value Problem of Nonlinear Fractional Differential Equations of Mixed Volterra-Fredohlm Integral Equations in Banach Space. American Journal of Applied Mathematics, 12(1), 1-8. https://doi.org/10.11648/j.ajam.20241201.11

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ACS Style

Taier, A. E.; Wu, R.; Iqbal, N. Boundary Value Problem of Nonlinear Fractional Differential Equations of Mixed Volterra-Fredohlm Integral Equations in Banach Space. Am. J. Appl. Math. 2024, 12(1), 1-8. doi: 10.11648/j.ajam.20241201.11

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AMA Style

Taier AE, Wu R, Iqbal N. Boundary Value Problem of Nonlinear Fractional Differential Equations of Mixed Volterra-Fredohlm Integral Equations in Banach Space. Am J Appl Math. 2024;12(1):1-8. doi: 10.11648/j.ajam.20241201.11

Copy | Download

@article{10.11648/j.ajam.20241201.11,
  author = {Ala Eddine Taier and Ranchao Wu and Naveed Iqbal},
  title = {Boundary Value Problem of Nonlinear Fractional Differential Equations of Mixed Volterra-Fredohlm Integral Equations in Banach Space},
  journal = {American Journal of Applied Mathematics},
  volume = {12},
  number = {1},
  pages = {1-8},
  doi = {10.11648/j.ajam.20241201.11},
  url = {https://doi.org/10.11648/j.ajam.20241201.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20241201.11},
  abstract = {This paper is dedicated to study the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional differential equations of mixed Volterra-Fredohlm integral equations in Banach space, the recent researches considered the study of differential equations of mixed Volterra-Fredholm integral 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 differential equations with mixed Volterra-Fredholm integral equations using fixed point theory. This work have two important results, the first result was the discussion of the existence of solutions using the Krasnoselskii fixed point theorem after transforming the problem into integral equation firstly then into operator problem suitable for the fixed point theory. The second result will be the existence and uniqueness of solution, this result was obtained by the use of Banach fixed point theorem after the same transformation used in the first one. This work give as conclusion that the boundary value problem of nonlinear fractional differential equations of mixed Volterra-Fredholm integral equation has a unique solution in Banach space. Finally, this work was ended with an example to illustrate the results obtained.},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - Boundary Value Problem of Nonlinear Fractional Differential Equations of Mixed Volterra-Fredohlm Integral Equations in Banach Space
AU  - Ala Eddine Taier
AU  - Ranchao Wu
AU  - Naveed Iqbal
Y1  - 2024/02/28
PY  - 2024
N1  - https://doi.org/10.11648/j.ajam.20241201.11
DO  - 10.11648/j.ajam.20241201.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 1
EP  - 8
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20241201.11
AB  - This paper is dedicated to study the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional differential equations of mixed Volterra-Fredohlm integral equations in Banach space, the recent researches considered the study of differential equations of mixed Volterra-Fredholm integral 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 differential equations with mixed Volterra-Fredholm integral equations using fixed point theory. This work have two important results, the first result was the discussion of the existence of solutions using the Krasnoselskii fixed point theorem after transforming the problem into integral equation firstly then into operator problem suitable for the fixed point theory. The second result will be the existence and uniqueness of solution, this result was obtained by the use of Banach fixed point theorem after the same transformation used in the first one. This work give as conclusion that the boundary value problem of nonlinear fractional differential equations of mixed Volterra-Fredholm integral equation has a unique solution in Banach space. Finally, this work was ended with an example to illustrate the results obtained.
VL  - 12
IS  - 1
ER  -

Copy | Download