THEORETICAL STUDY OF A CLASS OF $\zeta$-CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS IN A BANACH SPACE

Oualid Zentar; Mohamed Ziane; Mohammed Al Horani; Ismail Zitouni

doi:10.11948/20230436

2024 Volume 14 Issue 5

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2024 > 14(5): 2808-2821

Next Article Previous Article

Oualid Zentar, Mohamed Ziane, Mohammed Al Horani, Ismail Zitouni. THEORETICAL STUDY OF A CLASS OF $\zeta$-CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS IN A BANACH SPACE[J]. Journal of Applied Analysis & Computation, 2024, 14(5): 2808-2821. doi: 10.11948/20230436

Citation:

Oualid Zentar, Mohamed Ziane, Mohammed Al Horani, Ismail Zitouni. THEORETICAL STUDY OF A CLASS OF $\zeta$-CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS IN A BANACH SPACE[J]. Journal of Applied Analysis & Computation, 2024, 14(5): 2808-2821. doi: 10.11948/20230436

THEORETICAL STUDY OF A CLASS OF $\zeta$-CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS IN A BANACH SPACE

1.
Department of Computer Science, University of Tiaret, Tiaret, Algeria
2.
Department of Mathematics, University of Tiaret, Tiaret, Algeria
3.
Department of Mathematics, The University of Jordan, Amman, 11942, Jordan
4.
Laboratory of Research in Artificial Intelligence and Systems (LRAIS), University of Tiaret, Algeria

Author Bio: Email: oualid.zentar@univ-tiaret.dz (O. Zentar); Email: mohamed.ziane@univ-tiaret.dz (M. Ziane); Email: zitouni.ism@gmail.com (I. Zitouni)
Corresponding author: Email: horani@ju.edu.jo (M. Al Horani)

Abstract

Abstract

A study of an important class of nonlinear fractional differential equations driven by $ \zeta $-Caputo type derivative in a Banach space framework is presented. The classical Banach contraction principle associated with the Bielecki-type norm and a fixed-point theorem with respect to convex-power condensing operators are used to achieve some existence results. Two illustrative examples are provided to justify the theoretical results.
- $\zeta$-Caputo derivative /
- fixed point theorem /
- Hausdorff measure of noncompactness
MSC: 34A08, 26A33, 47H08

FullText(HTML)

References

[1]	S. Abbas, M. Benchohra and G. M. N'Guérékata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015. Google Scholar
[2]	A. Aghajani, E. Pourhadi and J. Trujillo, Application of measure of noncompactness to a Cauchy problem for fractional differential equations in Banach spaces, Fract. Calc. Appl. Anal., 2013, 16, 962–977. doi: 10.2478/s13540-013-0059-y CrossRef Google Scholar
[3]	B. Ahmad, A. F. Albideewi, S. K. Ntouyas and A. Alsaedi, Existence results for a multipoint boundary value problem of nonlinear sequential Hadamard fractional differential equations, Cubo (Temuco), 2021, 23, 225–237. doi: 10.4067/S0719-06462021000200225 CrossRef Google Scholar
[4]	R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 2017, 44, 460–481. doi: 10.1016/j.cnsns.2016.09.006 CrossRef Google Scholar
[5]	T. V. An, N. D. Phu and N. V. Hoa, A survey on non-instantaneous impulsive fuzzy differential equations involving the generalized Caputo fractional derivative in the short memory case, Fuzzy Sets and Systems., 2022, 443, 160–197. doi: 10.1016/j.fss.2021.10.008 CrossRef Google Scholar
[6]	H. Arfaoui, New numerical method for solving a new generalized American options under $\Psi$-Caputo time-fractional derivative Heston model, to appear in Rocky Mountain J. Math. Google Scholar
[7]	M. Awadalla, N. Yameni, Y. Yves and K. Asbeh, $\Psi$-Caputo logistic population growth model, J. Math., 2021, 2021, 1–9. $\Psi$-Caputo logistic population growth model" target="_blank">Google Scholar
[8]	Z. Baitiche, C. Derbazi, J. Alzabut, M. E. Samei, M. K. Kaabar and Z. Siri, Monotone iterative method for $\Psi$-Caputo fractional differential equation with nonlinear boundary conditions, Fractal Fract., 2021, 5(3), 81. doi: 10.3390/fractalfract5030081 CrossRef $\Psi$-Caputo fractional differential equation with nonlinear boundary conditions" target="_blank">Google Scholar
[9]	Z. Baitiche, C. Derbazi and M. Matar, Ulam-stability results for a new form of nonlinear fractional Langevin differential equations involving two fractional orders in the $\psi$–Caputo sense, Applicable Analysis., 2021. Google Scholar
[10]	J. Banas and K. Goebel, Measure of Noncompactness in Banach Spaces, Lectures Notes in Pure and Applied Mathematics, 50, Marcel Dekker, New York, 1980. Google Scholar
[11]	K. Diethelm and N. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 2002, 265, 229–248. doi: 10.1006/jmaa.2000.7194 CrossRef Google Scholar
[12]	A. El Mfadel, S. Melliani and M. Elomari, Existence results for nonlocal Cauchy problem of nonlinear $\Psi$-Caputo type fractional differential equations via topological degree methods, Advances in the Theory of Nonlinear Analysis and its Application, 2022, 6(2), 270–279. doi: 10.31197/atnaa.1059793 CrossRef $\Psi$-Caputo type fractional differential equations via topological degree methods" target="_blank">Google Scholar
[13]	Q. Fan, G. -C. Wu and H. Fu, A note on function space and boundedness of the general fractional integral in continuous time random walk, J. Nonlin. Math. Phys., 2022, 29(1), 95–102. doi: 10.1007/s44198-021-00021-w CrossRef Google Scholar
[14]	A. Granas and J. Dugundji, Fixed Point Theory, New York (NY), Springer, 2003. Google Scholar
[15]	M. A. Hammad, Conformable fractional martingales and some convergence theorems, Mathematics, 2022, 10, 6. Google Scholar
[16]	M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter, Berlin, 2001. Google Scholar
[17]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier, Amsterdam, Netherlands, 2006, 204. Google Scholar
[18]	T. Kosztołowicz and A. Dutkiewicz, Subdiffusion equation with Caputo fractional derivative with respect to another function, Phys. Rev. E, 2021, 104(1), 014118. doi: 10.1103/PhysRevE.104.014118 CrossRef Google Scholar
[19]	F. Norouzi and G. N'Guérékata, A study of $\psi$-Hilfer fractional differential system with application in financial crisis, Chaos Solitons Fractals: X, 2021, 6, 1–15. Google Scholar
[20]	J. Sousa and E. Oliveira, Existence, uniqueness, estimation and continuous dependence of the solutions of a nonlinear integral and an integrodifferential equations of fractional order, ArXiv Preprint ArXiv: 1806.01441, 2018. Google Scholar
[21]	J. Sun and X. Zhang, The fixed point theorem of convex-power condensing operator and applications to abstract semilinear evolution equations, Acta Math. Sin., 2005, 48, 439–446. Google Scholar
[22]	M. Tayeb, H. Boulares, A. Moumen and M. Imsatfia, Processing fractional differential equations using $\psi$-Caputo derivative, Symmetry, 2023, 15(4), 955. doi: 10.3390/sym15040955 CrossRef $\psi$-Caputo derivative" target="_blank">Google Scholar
[23]	F. Tricomi and A. Erdélyi, The asymptotic expansion of a ratio of Gamma functions, Pacific J. Math., 1951, 1, 133–142. doi: 10.2140/pjm.1951.1.133 CrossRef Google Scholar
[24]	J. Vanterler and C. Sousa, Existence results and continuity dependence of solutions for fractional equations, Differ Equ Appl., 2020, 12, 377–396. Google Scholar