Lakshmikantham Monotone Iterative Principle for Hybrid Atangana-Baleanu-Caputo Fractional Differential Equations

Nadia Benkhettou; Abdelkrim Salim; Jamal Eddine Lazreg; Saïd Abbas; Mouffak Benchohra

doi:10.2478/awutm-2023-0007

.blurhash-client-img { display: none !important; }

Lakshmikantham Monotone Iterative Principle for Hybrid Atangana-Baleanu-Caputo Fractional Differential Equations

Annals of West University of Timisoara - Mathematics and Computer Science

Volume 59 (2023): Issue 1 (June 2023)

By: Nadia Benkhettou, Abdelkrim Salim, Jamal Eddine Lazreg, Saïd Abbas and Mouffak Benchohra

Open Access

|May 2023

Abstract

In this paper, we study the following fractional differential equation involving the Atangana-Baleanu-Caputo fractional derivative: ${\begin{array}{l} A B C_{a} D_{τ}^{θ} [x (ϑ) - F (ϑ, x (ϑ))] = G (ϑ, x (ϑ)), ϑ \in J : = [a, b], \\ x (a) = φ_{a} \in ℝ . \end{array}$ $$\left\{ {\matrix{ {AB{C_a}D_\tau ^\theta [x(\vartheta ) - F(\vartheta ,x(\vartheta ))] = G(\vartheta ,x(\vartheta )),\;\;\;{\kern 1pt} \vartheta \in J: = [a,b],} \hfill \cr {x(a) = {\varphi _a} \in .} \hfill \cr } } \right.$$

The result is based on a Dhage fixed point theorem. Further, an example is provided for the justification of our main result.

References

S. Abbas, M. Benchohra, J.R. Graef, J. Henderson, Implicit Fractional Differential and Integral Equations: Existence and Stability, De Gruyter, Berlin, 2018.
Search in Google Scholar Back to article
S. Abbas, M. Benchohra, G. M. N’Guérékata, Topics in Fractional Differential Equations, Springer, New York, 2012.
Search in Google Scholar Back to article
S. Abbas, M. Benchohra, G. M. N’Guérékata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015.
Search in Google Scholar Back to article
M. S. Abdo, T. Abdeljawad, K. D. Kishor, M. A. Alqudah, M. A. Saeed, M. B. Jeelani, On nonlinear pantograph fractional differential equations with Atangana-Baleanu-Caputo derivative, Adv. Difference Equ. (2021), Paper No. 65, 17 pp
Search in Google Scholar Back to article
R. S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On the solution of a boundary value problem associated with a fractional differential equation, Math. Methods Appl. Sci., https://doi.org/10.1002/mma.6652
Search in Google Scholar Back to article
R. S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions, RACSAM (2021), 115-155, https://doi.org/10.1007/s13398-021-01095-3
Search in Google Scholar Back to article
R. S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On the solutions of fractional differential equations via Geraghty type hybrid contractions, Appl. Comput. Math. 20 (2021), 313-333.
Search in Google Scholar Back to article
H. Akhadkulov, F. Alsharari, T. Y. Ying, Applications of Krasnoselskii-Dhage type fixed-point Theorems to fractional hybrid differential equations, Tamkang J. Math. 52 (2) (2021), 281-292.
Search in Google Scholar Back to article
B. Alqahtani, H. Aydi, E. Karapinar, V. Rakocevic, A solution for Volterra fractional integral equations by hybrid contractions, Mathematics 7 (2019), 694, https://doi.org/10.3390/math7080694
Search in Google Scholar Back to article
A. Atangana, On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation, Appl. Math. Comput. 273 (2016), 948-956.
Search in Google Scholar Back to article
A. Atangana, B. S. Alkahtani, Analysis of the Keller-Segel model with a fractional derivative without singular kernel, Entropy 17 (2015), 4439-4453.
Search in Google Scholar Back to article
A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci. 20 (2016), 763-769.
Search in Google Scholar Back to article
A. Atangana, D. Baleanu, Caputo-Fabrizio derivative applied to groundwater flow with in confined aquifer, J. Eng. Mech. (2016), 5 pages.
Search in Google Scholar Back to article
M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl. 1 (2) (2015), 73-85.
Search in Google Scholar Back to article
B.C. Dhage, Some variants of two basic hybrid fixed point theorems of Krasnoselskii and Dhage with applications, Nonlinear Stud. 25 (2018), 559-573.
Search in Google Scholar Back to article
B. C. Dhage, Hybrid fixed point theory in partially ordered normed linear spaces and applications to fractional integral equations, Differ. Equ. Appl. 5 (2013), 155-184.
Search in Google Scholar Back to article
B. C. Dhage, Global attractivity results for comparable solutions of nonlinear hybrid fractional integral equations, Differ. Equ. Appl. 6 (2014), 165-186.
Search in Google Scholar Back to article
B. C. Dhage, Partially condensing mappings in ordered normed linear spaces and applications to functional integral equations, Tamkang J. Math. 45 (4) (2014), 397-426.
Search in Google Scholar Back to article
B. C. Dhage, Nonlinear D-set-contraction mappings in partially ordered normed linear spaces and applications to functional hybrid integral equations, Malaya J. Mat. 3 (1) (2015), 62-85.
Search in Google Scholar Back to article
B. C. Dhage, N. S. Jadhav, Basic results in the theory of hybrid differential equations with linear perturbations of second type, Tamkang J. Math. 44 (2013), 171-186.
Search in Google Scholar Back to article
B. D. Karande, S. N. Kondekar, Existence of solution to a quadratic functional integrodifferential fractional equation, Commun. Math. Appl. 11 (4) (2020), 635-650.
Search in Google Scholar Back to article
E. Karapinar, H. D. Binh, N. H. Luc, N. H. Can, On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems, Adv. Differ. Equ. 2021 (2021), 70, https://doi.org/10.1186/s13662-021-03232-z
Search in Google Scholar Back to article
E. Karapinar, A. Fulga, M. Rashid, L. Shahid, H. Aydi, Large contractions on quasi-metric spaces with an application to nonlinear fractional differential-equations, Mathematics 7 (2019), 444, https://doi.org/10.3390/math7050444
Search in Google Scholar Back to article
K. D. Kucche, S. T. Sutar, Analysis of nonlinear fractional differential equations involving Atangana-Baleanu-Caputo derivative, Chaos Solitons Fractals 143 (2021), 9 pp.
Search in Google Scholar Back to article
K. D. Kucche, S. T. Sutar, On nonlinear hybrid fractional differential equations with Atangana-Baleanu-Caputo derivative, Chaos Solitons Fractals 143 (2021), 11 pp.
Search in Google Scholar Back to article
J. E. Lazreg, S. Abbas, M. Benchohra, E. Karapinar, Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces, Open Math. 19 (2021), 363-372, https://doi.org/10.1515/math-2021-0040
Search in Google Scholar Back to article
H. Lu, S. Sun, D. Yang, H. Teng, Theory of fractional hybrid differential equations with linear perturbations of second type, Bound. Value Probl. 23 (2013) 1-16.
Search in Google Scholar Back to article
Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/awutm-2023-0007 | Journal eISSN: 1841-3307 | Journal ISSN: 1841-3293

Journal RSS Feed

Language: English

Page range: 79 - 91

Published on: May 3, 2023

Published by: West University of Timisoara

In partnership with: Paradigm Publishing Services

Publication frequency: Volume open

Keywords:

Atangana-Baleanu-Caputo fractional derivative,

existence,

uniqueness,

lower solution,

Dhage iteration principle,

partially ordered Banach space,

nonlinear D-contraction operator

Related subjects:

Mathematics,

General mathematics

© 2023 Nadia Benkhettou, Abdelkrim Salim, Jamal Eddine Lazreg, Saïd Abbas, Mouffak Benchohra, published by West University of Timisoara
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 59 (2023): Issue 1 (June 2023)