ON A CLASS OF CAUCHY-TYPE PROBLEMS ASSOCIATED WITH WEIGHTED FRACTIONAL DERIVATIVES

Soumia Bourchi; Yassine Adjabi; Hamid Boulares; Fahd Jarad; Sumati Kumari Panda; Abdelkader Moumen; Mohamed Bouye

doi:10.11948/20250368

2027 Volume 17 Issue 1

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Soumia Bourchi, Yassine Adjabi, Hamid Boulares, Fahd Jarad, Sumati Kumari Panda, Abdelkader Moumen, Mohamed Bouye. ON A CLASS OF CAUCHY-TYPE PROBLEMS ASSOCIATED WITH WEIGHTED FRACTIONAL DERIVATIVES[J]. Journal of Applied Analysis & Computation, 2027, 17(1): 184-211. doi: 10.11948/20250368

Citation:

Soumia Bourchi, Yassine Adjabi, Hamid Boulares, Fahd Jarad, Sumati Kumari Panda, Abdelkader Moumen, Mohamed Bouye. ON A CLASS OF CAUCHY-TYPE PROBLEMS ASSOCIATED WITH WEIGHTED FRACTIONAL DERIVATIVES[J]. Journal of Applied Analysis & Computation, 2027, 17(1): 184-211. doi: 10.11948/20250368

ON A CLASS OF CAUCHY-TYPE PROBLEMS ASSOCIATED WITH WEIGHTED FRACTIONAL DERIVATIVES

1.
Laboratory of Mathematics, Computer Science and Applications, Djilali Bounaama University, Khemis Miliana 44001, Algeria
2.
Department of Mathematics, Faculty of Sciences, University of M'hamed Bougara, Boumerdes 35000, and Dynamic Systems Laboratory, U.S.T.H.B, Algiers, Algeria
3.
Laboratory of Analysis and Control of Differential Equations “ACED”, Faculty MISM, Department of Mathematics, University of 8 May 1945 Guelma, P.O. Box 401, Guelma 24000, Algeria
4.
Department of Mathematics, Faculty of Arts and Sciences, Çankaya University, Ankara 06790, Turkey
5.
Department of Engineering Mathematics and Artificial Intelligence, Azerbaijan Technical University, Baku, Azerbaijan
6.
Department of Mathematics, GMR Institute of Technology, Rajam-532127, Andhra Pradesh, India
7.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 55473, Saudi Arabia
8.
Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia

Author Bio: Email: s.bourchi@univ-dbkm.dz(S. Bourchi); Email: adjabiy@univ-boumerdes.dz(Y. Adjabi); Email: boulareshamid@gmail.com(H. Boulares); Email: fahd@cankaya.edu.tr(F. Jarad); Email: sumatikumari.p@gmrit.edu.in(S. K. Panda); Email: mbmahmad@kku.edu.sa(M. Bouye)
Corresponding author: Email: mo.abdelkader@uoh.edu.sa(A. Moumen)

Abstract

Abstract

This paper investigates the existence and uniqueness of mild solutions for a class of nonlocal Cauchy–type problems involving integro–differential equations with weighted $ \varsigma $–Caputo fractional derivatives. The equations incorporate generalized Volterra and Fredholm integral operators with singular kernels of order $ \ell \in (0, 1) $, within the framework of an ordered Banach space. The main results are established by combining semigroup theory, monotone iterative techniques, upper and lower solution methods, Kuratowski's measure of noncompactness, generalized Gronwall inequalities, and fixed point theorems. Continuous dependence of mild solutions on initial data is also proved. Two concrete examples are provided to illustrate the applicability of the abstract results.
- Weighted $\varsigma$-Caputo fractional derivative /
- abstract Cauchy problem /
- Volterra integral operator /
- Fredholm integral operator /
- Kuratowski measure of noncompactness /
- $C_0$-semigroup /
- monotone iterative technique /
- upper and lower solutions /
- mild solution /
- existence and uniqueness /
- cone
MSC: 34A08, 34A12, 34B40, 26A33, 35R11, 35K90, 47D06

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References

[1]	Y. Adjabi, F. Jarad and T. Abdeljawad, On generalized fractional operators and a Gronwall type inequality with applications, Filomat, 2017, 31, 5457–5473. doi: 10.2298/FIL1717457A CrossRef Google Scholar
[2]	O. Agrawal, Some generalized fractional calculus operators and their applications in integral equations, Fract. Calc. Anal. Appl., 2012, 15(4), 700–711. doi: 10.2478/s13540-012-0047-7 CrossRef Google Scholar
[3]	K. Ali, A. El-Sayed and M. Abdelrahman, Existence and uniqueness of solutions for a fractional $q$-integro-differential equation involving the Caputo–Fabrizio fractional derivative with nonlocal conditions, J. Math. Comput. Sci., 2024, 29, 1–18. $q$-integro-differential equation involving the Caputo-Fabrizio fractional derivative with nonlocal conditions" target="_blank">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]	A. Anguraj, P. Karthikeyan and J. J. Trujillo, Existence of solutions to fractional mixed integro-differential equations with nonlocal initial condition, Adv. Differ. Equ., 2011, Article ID 690653, 12 pp. Google Scholar
[6]	J. Banas and K. Goebel, Measures of noncompactness in Banach spaces, Commentationes Mathematicae Universitatis Carolinae, 1980, 21(1). Google Scholar
[7]	J. Banas and M. Mursaleen, Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations, Springer, New Delhi, 2014. Google Scholar
[8]	E. Bazhlekova, Evolution Equations in Fractional Banach Spaces, PhD thesis, Eindhoven University of Technology, 2001. Google Scholar
[9]	A. Ben Brahim and D. F. M. Torres, Existence and uniqueness of mild solutions for a class of $\psi$-Caputo time-fractional systems of order from one to two, Fract. Calc. Appl. Anal., 2025, 28(1), 1–27. doi: 10.1007/s13540-024-00358-1 CrossRef $\psi$-Caputo time-fractional systems of order from one to two" target="_blank">Google Scholar
[10]	L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 1991, 162, 494–505. doi: 10.1016/0022-247X(91)90164-U CrossRef Google Scholar
[11]	K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. Google Scholar
[12]	S. W. Du and V. Lakshmikantham, Monotone iterative technique for differential equations in Banach spaces, J. Math. Anal. Appl., 87, (1982), 454–459. doi: 10.1016/0022-247X(82)90134-2 CrossRef Google Scholar
[13]	H. Gou and Y. Li, The method of lower and upper solutions for impulsive fractional evolution equations, Ann. Funct. Anal., 11, (2020), 350. doi: 10.1007/s43034-019-00007-2 CrossRef Google Scholar
[14]	S. Hasan, A. Jleli and B. Samet, Existence and uniqueness results for a class of fractional integro-stochastic differential equations, Fractal and Fractional, 2025, 9(1), Article 42. doi: 10.3390/fractalfract9010042 CrossRef Google Scholar
[15]	H. P. Heinz, On the behaviour of measures of noncompactness with respect to differentiation and integration of vector valued functions, Nonlinear Anal., 1983, 7(12), 1351–1371. doi: 10.1016/0362-546X(83)90006-8 CrossRef Google Scholar
[16]	R. W. Ibrahim and S. Momani, Upper and lower bounds of solutions for fractional integral equations, Surveys in Mathematics and its Applications, 2007, 2, 145–156. Google Scholar
[17]	T. Jankowski, Fractional equations of Volterra type involving a Riemann-Liouville derivative, Appl. Math. Lett., 2013, 26, 344–350. doi: 10.1016/j.aml.2012.10.002 CrossRef Google Scholar
[18]	F. Jarad, T. Abdeljawad and K. Shah, On the weighted fractional operators of a function with respect to another function, Fractals, 2020, 28(8), 2040011. doi: 10.1142/S0218348X20400113 CrossRef Google Scholar
[19]	O. K. Jaradat, A. Al-Omari and S. Momani, Existence of the mild solution for fractional semilinear initial value problems, Nonlinear Anal., 2008, 69, 3153–3159. doi: 10.1016/j.na.2007.09.008 CrossRef Google Scholar
[20]	H. Jian, B. Liu and S. Xie, Monotone iterative solutions for nonlinear fractional differential systems with deviating arguments, Appl. Math. Comput., 2015, 262, 1–14. Google Scholar
[21]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006. Google Scholar
[22]	V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal., 2008, 69(8), 2677–2682. doi: 10.1016/j.na.2007.08.042 CrossRef Google Scholar
[23]	B. Li and H. Gou, Monotone iterative method for the periodic boundary value problems of impulsive evolution equations in Banach spaces, Chaos, Solitons and Fractals, 2018, 110, 209–215. doi: 10.1016/j.chaos.2018.03.027 CrossRef Google Scholar
[24]	K. Li and J. Jia, Existence and uniqueness of mild solutions for abstract delay fractional differential equations, Comput. Math. Appl., 2011, 62, 1398–1404. doi: 10.1016/j.camwa.2011.02.038 CrossRef Google Scholar
[25]	Y. Li, The positive solutions of abstract semilinear evolution equations and their applications, Acta Math. Sin., 1996, 39(5), 666–672. Google Scholar
[26]	H. Lmou, A. Boutarfa and S. Muthaiah, Existence theory on the Caputo-type fractional differential Langevin hybrid inclusion with variable coefficient, Bound. Value Probl., 2024, Article ID 48. Google Scholar
[27]	E. Malkowsky and V. Rakocevic, An introduction into the theory of sequence spaces and measures of noncompactness, Zb. rad. Beogr., 2000, 17, 143–234. Google Scholar
[28]	H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal., TMA, 1980, 4, 985–999. Google Scholar
[29]	S. K. Panda, A. Atangana and T. Abdeljawad, Existence results and numerical study on novel coronavirus 2019-Ncov/Sars-Cov-2 model using differential operators based on the generalized Mittag-Leffler kernel and fixed points, Fractals, 2022, 30(8), 2240214. doi: 10.1142/S0218348X22402149 CrossRef Google Scholar
[30]	D. N. Pandey, A. Ujlayan and D. Bahuguna, On a solution to fractional order integro-differential equations with analytic semigroups, Nonlinear Anal., 2009, 71, 3690–3698. doi: 10.1016/j.na.2009.02.018 CrossRef Google Scholar
[31]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin, 1983. Google Scholar
[32]	S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon & Breach Science Publishers, Yverdon, 1993. Google Scholar
[33]	A. Suechoei and P. Sa Ngiamsunthorn, Extremal solutions of $\varphi$-Caputo fractional evolution equations involving integral kernels, AIMS Mathematics, 2021, 6(5), 4734–4757. doi: 10.3934/math.2021278 CrossRef $\varphi$-Caputo fractional evolution equations involving integral kernels" target="_blank">Google Scholar
[34]	J. X. Sun and Z. Q. Zhao, Extremal solutions initial value problem for integro-differential of mixed type in Banach spaces, Ann. Diff. Eqs., 1992, 8, 469–475. Google Scholar
[35]	J. Vanterler da, C. Sousa and E. Capelas de Oliveira, A Gronwall inequality and the Cauchy-type problem by means of $\psi$ -Hilfer operator, Differ. Equ. Appl., 2019, 11, 87–106. $\psi$ -Hilfer operator" target="_blank">Google Scholar
[36]	G. Wang, R. P. Agarwal and A. Cabada, Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations, Appl. Math. Lett., 2012, 25, 1019–1024. doi: 10.1016/j.aml.2011.09.078 CrossRef Google Scholar
[37]	X. Zhang and Y. Li, Monotone iterative technique for fractional partial differential equations with impulses, J. Comput. Anal. Appl., 2018, 25(3). Google Scholar
[38]	M. Zhou and Y. Zhou, Existence of mild solutions for fractional integro-differential equations with Hilfer derivatives, Mathematics, 2024, 12(18), Article 2876. doi: 10.3390/math12182876 CrossRef Google Scholar
[39]	Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal., RWA, 2010, 11, 4465–4475. doi: 10.1016/j.nonrwa.2010.05.029 CrossRef Google Scholar