EXISTENCE OF NONTRIVIAL SOLUTIONS FOR AN INTEGRAL BOUNDARY VALUE PROBLEM INVOLVING THE CAPUTO-FABRIZIO-TYPE FRACTIONAL DERIVATIVE

Liang Xue; Qian Sun; Donal O'Regan; Jiafa Xu

doi:10.11948/20240411

2025 Volume 15 Issue 3

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2025 > 15(3): 1786-1802

Next Article Previous Article

Liang Xue, Qian Sun, Donal O'Regan, Jiafa Xu. EXISTENCE OF NONTRIVIAL SOLUTIONS FOR AN INTEGRAL BOUNDARY VALUE PROBLEM INVOLVING THE CAPUTO-FABRIZIO-TYPE FRACTIONAL DERIVATIVE[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1786-1802. doi: 10.11948/20240411

Citation:

Liang Xue, Qian Sun, Donal O'Regan, Jiafa Xu. EXISTENCE OF NONTRIVIAL SOLUTIONS FOR AN INTEGRAL BOUNDARY VALUE PROBLEM INVOLVING THE CAPUTO-FABRIZIO-TYPE FRACTIONAL DERIVATIVE[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1786-1802. doi: 10.11948/20240411

EXISTENCE OF NONTRIVIAL SOLUTIONS FOR AN INTEGRAL BOUNDARY VALUE PROBLEM INVOLVING THE CAPUTO-FABRIZIO-TYPE FRACTIONAL DERIVATIVE

1.
Department of Basic Courses, Anhui Industrial Polytechnic College, Tongling 244000, China
2.
School of Mathematics and Statistics, Ludong University, Yantai 264025, China
3.
School of Mathematical and Statistical Sciences, University of Galway, Galway, Ireland
4.
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China

Author Bio: Email: 39572578@qq.com(L. Xue); Email: donal.oregan@nuigalway.ie(D. O'Regan); Email: xujiafa292@sina.com(J. Xu)
Corresponding author: Email: sunqian_ldu@126.com(Q. Sun)
Fund Project: The authors were supported by Natural Science Research Project of Anhui Educational Committee (2024AH050188) and the Talent Introduction Project of Ludong University (LY2015004)

Abstract

Abstract

In this work we study the existence of nontrivial solutions for a Caputo-Fabrizio-type fractional integral boundary value problem. We first construct a new linear operator, which can include the integral boundary condition, and then under some conditions involving the spectral radius of the linear operator, we use topological degree methods to obtain some existence theorems for our considered problem.
- Caputo-Fabrizio-type fractional-order differential equations /
- integral boundary value problems /
- nontrivial solutions /
- topological degree
MSC: 34B10, 34B15, 34B18

FullText(HTML)

References

[1]	T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Adv. Difference Equ., 2017, 313, 1–11. Google Scholar
[2]	T. Abdeljawad and D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Rep. Math. Phys., 2017, 80, 11–27. doi: 10.1016/S0034-4877(17)30059-9 CrossRef Google Scholar
[3]	K. K. Ali, K. R. Raslan, A. A. Ibrahim and D. Baleanu, The nonlocal coupled system of Caputo-Fabrizio fractional q-integro differential equation, Math. Methods Appl. Sci., 2024, 47, 1764–1780. doi: 10.1002/mma.9646 CrossRef Google Scholar
[4]	K. K. Ali, K. R. Raslan, A. A. Ibrahim and M. S. Mohamed, On study the existence and uniqueness of the solution of the Caputo-Fabrizio coupled system of nonlocal fractional q-integro differential equations, Math. Methods Appl. Sci., 2023, 46, 13226–13242. doi: 10.1002/mma.9246 CrossRef Google Scholar
[5]	M. Alshammari, S. Alshammari and M. S. Abdo, Existence theorems for hybrid fractional differential equations with ψ-weighted Caputo-Fabrizio derivatives, J. Math., 2023, 13, 8843470. Google Scholar
[6]	A. Atangana and S. İ. Araz, A successive midpoint method for nonlinear differential equations with classical and Caputo-Fabrizio derivatives, AIMS Math., 2023, 8, 27309–27327. doi: 10.3934/math.20231397 CrossRef Google Scholar
[7]	V. M. Bulavatsky, Boundary-value problems of fractional-differential consolidation dynamics for the model with the Caputo-Fabrizio derivative, Kibernet. Sistem. Anal., 2023, 59, 159–168. doi: 10.1007/s10559-023-00600-3 CrossRef Google Scholar
[8]	N. Chefnaj, K. Hilal and A. Kajouni, Existence of the solution for hybrid differential equation with Caputo-Fabrizio fractional derivative, Filomat, 2023, 37, 2219–2226. doi: 10.2298/FIL2307219C CrossRef Google Scholar
[9]	A. N. Dalawi, M. Lakestani and E. Ashpazzadeh, Solving fractional optimal control problems involving Caputo-Fabrizio derivative using Hermite spline functions, Iran. J. Sci., 2023, 47, 545–566. doi: 10.1007/s40995-022-01404-4 CrossRef Google Scholar
[10]	D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, volume 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, Inc., Boston, MA, 1988. Google Scholar
[11]	Y. Henka and M. Z. Aissaoui, Hermite wavelets collocation method for solving a Fredholm integro-differential equation with fractional Caputo-Fabrizio derivative, Proyecciones, 2023, 42, 917–930. doi: 10.22199/issn.0717-6279-5542 CrossRef Google Scholar
[12]	S. G. Kebede and A. G. Lakoud, Analysis of mathematical model involving nonlinear systems of Caputo-Fabrizio fractional differential equation, Bound. Value Probl., 2023, 44, 1–17. Google Scholar
[13]	M. G. Kreĭn and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Translation, 1950, 26, 128. Google Scholar
[14]	S. Krim, S. Abbas, M. Benchohra and J. J. Nieto, Implicit Caputo-Fabrizio fractional differential equations with delay, Stud. Univ. Babeş-Bolyai Math., 2023, 68, 727–742. Google Scholar
[15]	W. Lanfang, Existence of positive solutions to two-point boundary value problems of fractional differentional equations involving Caputo-Fabrizio fractional derivatives, J. Jishou University, 2023, 44, 1–5. Google Scholar
[16]	I. Mansouri, M. M. Bekkouche, A. A. Ahmed, F. Yazid and F. S. Djeradi, Analytical and numerical study of a linear coupled system involving Caputo-Fabrizio fractional derivative with boundary conditions, Fract. Differ. Calc., 2023, 13, 171–183. Google Scholar
[17]	K. O. Melha, M. Djilali and V. L. Chinchane, Abstract fractional differential equations with Caputo-Fabrizio derivative, Fract. Differ. Calc., 2023, 13, 149–162. Google Scholar
[18]	J. Nasir, S. Qaisar, A. Qayyum and H. Budak, New results on Hermite-Hadamard type inequalities via Caputo-Fabrizio fractional integral for s-convex function, Filomat, 2023, 37, 4943–4957. Google Scholar
[19]	M. Onitsuka and I. El-Fassi, Generalized Caputo-Fabrizio fractional differential equation, J. Appl. Anal. Comput., 2024, 14, 964–975. Google Scholar
[20]	A. Souigat, Z. Korichi and M. Tayeb Meftah, Solution of the fractional diffusion equation by using Caputo-Fabrizio derivative: Application to intrinsic arsenic diffusion in germanium, Rev. Mexicana Fís., 2024, 70, 010501. Google Scholar
[21]	H. Wang, X. Zhang, Z. Luo and J. Liu, Analysis of numerical method for diffusion equation with time-fractional Caputo-Fabrizio derivative, J. Math., 2023, 11, 7906656. Google Scholar
[22]	W. Wang, J. Ye, J. Xu and D. O'Regan, Positive solutions for a high-order Riemann-Liouville type fractional integral boundary value problem involving fractional derivatives, Symmetry, 2022, 14, 2320. Google Scholar
[23]	H. Yang, S. Qaisar, A. Munir and M. Naeem, New inequalities via Caputo-Fabrizio integral operator with applications, AIMS Math., 2023, 8, 19391–19412. Google Scholar
[24]	W. Zhong, L. Wang and T. Abdeljawad, Separation and stability of solutions to nonlinear systems involving Caputo-Fabrizio derivatives, Adv. Difference Equ., 2020, 166, 1–15. Google Scholar