Stieltjes integral boundary value problem involving a nonlinear multi-term Caputo-type sequential fractional integro-differential equation

Jiqiang Zhang; Siraj Ul Haq; Akbar Zada; Ioan-Lucian Popa; Jiqiang Zhang; Siraj Ul Haq; Akbar Zada; Ioan-Lucian Popa

doi:10.3934/math.20231454

AIMS Mathematics

2023, Volume 8, Issue 12: 28413-28434. doi: 10.3934/math.20231454

Previous Article Next Article

Research article Special Issues

Stieltjes integral boundary value problem involving a nonlinear multi-term Caputo-type sequential fractional integro-differential equation

1.
Department of Basic, Anhui Sanlian University, China
2.
Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan
3.
Department of Computing, Mathematics and Electronics, "1 Decembrie 1918" University of Alba Iulia, Alba Iulia, 510 0 09, Romania
4.
Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania

Received: 12 May 2023 Revised: 27 September 2023 Accepted: 29 September 2023 Published: 18 October 2023
MSC : 26A33, 34A08, 34B15

In this article, we analyze the existence and uniqueness of mild solution to the Stieltjes integral boundary value problem involving a nonlinear multi-term, Caputo-type sequential fractional integro-differential equation. Krasnoselskii's fixed-point theorem and the Banach contraction principle are utilized to obtain the existence and uniqueness of the mild solution of the aforementioned problem. Furthermore, the Hyers-Ulam stability is obtained with the help of established methods. Our proposed model contains both the integer order and fractional order derivatives. As a result, the exponential function appears in the solution of the model, which is a fundamental and naturally important function for integer order differential equations and its many properties. Finally, two examples are provided to illustrate the key findings.
- existence,
- mild solution,
- fixed point,
- Hyers-Ulam stability,
- Caputo derivative,
- Stieltjes boundary conditions
Citation: Jiqiang Zhang, Siraj Ul Haq, Akbar Zada, Ioan-Lucian Popa. Stieltjes integral boundary value problem involving a nonlinear multi-term Caputo-type sequential fractional integro-differential equation[J]. AIMS Mathematics, 2023, 8(12): 28413-28434. doi: 10.3934/math.20231454

Related Papers:

Abstract

In this article, we analyze the existence and uniqueness of mild solution to the Stieltjes integral boundary value problem involving a nonlinear multi-term, Caputo-type sequential fractional integro-differential equation. Krasnoselskii's fixed-point theorem and the Banach contraction principle are utilized to obtain the existence and uniqueness of the mild solution of the aforementioned problem. Furthermore, the Hyers-Ulam stability is obtained with the help of established methods. Our proposed model contains both the integer order and fractional order derivatives. As a result, the exponential function appears in the solution of the model, which is a fundamental and naturally important function for integer order differential equations and its many properties. Finally, two examples are provided to illustrate the key findings.

References

[1]	S. Abbas, M. Benchohra, G. M. N'Guerekata, Topics in fractional differential equations, New York: Springer, 2012. https://doi.org/10.1007/978-1-4614-4036-9
[2]	B. Ahmad, A. Alsaedi, S. K. Ntouyas, J. Tariboon, Hadamard-type fractional differential equations, inclusions and inequalities, Springer, 2017. https://doi.org/10.1007/978-3-319-52141-1
[3]	B. Ahmad, J. J. Nieto, Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions, Bound. Value Probl., 2009 (2009), 708576. https://doi.org/10.1155/2009/708576 doi: 10.1155/2009/708576
[4]	B. Ahmad, J. J. Nieto, Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations, Abstr. Appl. Anal., 2009 (2009), 494720. https://doi.org/10.1155/2009/494720 doi: 10.1155/2009/494720
[5]	B. Ahmad, R. P. Agarwal, M. Alblewi, A. Alsaedi, On Nonlinear multi-term fractional integro-differential equations with Anti-periodic boundary conditions, Prog. Fract. Differ. Appl., 8 (2022), 349–356. https://doi.org/10.18576/pfda/080301 doi: 10.18576/pfda/080301
[6]	M. Alqhtani, K. M. Owolabi, K. M. Saad, E. Pindza, Spatiotemporal chaos in spatially extended fractional dynamical systems, Commun. Nonlinear Sci., 119 (2023), 107118. https://doi.org/10.1016/j.cnsns.2023.107118 doi: 10.1016/j.cnsns.2023.107118
[7]	M. Alqhtani, K. M. Saad, Numerical solutions of space-fractional diffusion equations via the exponential decay kernel, AIMS Math., 7 (2022), 6535–6549. https://doi.org/10.3934/math.2022364 doi: 10.3934/math.2022364
[8]	D. Baleanu, J. A. T. Machado, A. C. J. Luo, Fractional dynamics and control, New York: Springer, 2011. https://doi.org/10.1007/978-1-4614-0457-6
[9]	M. Benchora, S. Hamani, J. Henderson, Functional differential inclusions with integral boundary conditions, Electron. J. Qual. Theory Differ. Equ., 15 (2007), 1–13. https://doi.org/10.14232/ejqtde.2007.1.15 doi: 10.14232/ejqtde.2007.1.15
[10]	M. Benchohra, S. Hamani, S. K. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal. Theor., 71 (2009), 2391–2396. https://doi.org/10.1016/j.na.2009.01.073 doi: 10.1016/j.na.2009.01.073
[11]	T. A. Burton, T. Furumochi, Krasnoselskii's fixed point theorem and stability, Nonlinear Anal. Theor., 49 (2002), 445–454. https://doi.org/10.1016/S0362-546X(01)00111-0 doi: 10.1016/S0362-546X(01)00111-0
[12]	A. Carpinteri, F. Mainardi, Fractals and fractional calculus in continuum mechanics, Springer, 1997. https://doi.org/10.1007/978-3-7091-2664-6
[13]	M. Cichon, H. A. H. Salem, On the lack of equivalence between differential and integral forms of the Caputo-type fractional problems, J. Pseudo-Differ. Oper. Appl., 11 (2020), 1869–1895. https://doi.org/10.1007/s11868-020-00345-z doi: 10.1007/s11868-020-00345-z
[14]	K. Diethelm, The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type, Berlin: Springer, 2010. https://doi.org/10.1007/978-3-642-14574-2
[15]	H. Fallahgoul, S. Focardi, F. Fabozzi, Fractional calculus and fractional processes with applications to financial economics: Theory and application, Academic Press, 2016.
[16]	J. Graef, L. Kong, M. Wang, Existence and uniqueness of solutions for a fractional boundary value problem on a graph, Fract. Calc. Appl. Anal., 17 (2014), 499–510. https://doi.org/10.2478/s13540-014-0182-4 doi: 10.2478/s13540-014-0182-4
[17]	J. Henderson, R. Luca, Nonexistence of positive solutions for a system of coupled fractional boundary value problems, Boundary Value Probl., 2015 (2015), 138. https://doi.org/10.1186/s13661-015-0403-8 doi: 10.1186/s13661-015-0403-8
[18]	D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci., 27 (1941), 222–224. https://doi.org/10.1073/pnas.27.4.222 doi: 10.1073/pnas.27.4.222
[19]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
[20]	R. Magin, Fractional calculus in Bioengineering Begell House Publishers, 2006.
[21]	M. Obloza, Hyers stability of the linear differential equation, 1993.
[22]	H. A. H. Salem, M.Cichon, W. Shammakh, Generalized fractional calculus in Banach spaces and applications to existence results for boundary value problems, Bound. Value Probl., 2023 (2023), 57. https://doi.org/10.1186/s13661-023-01745-y doi: 10.1186/s13661-023-01745-y
[23]	H. A. H. Salem, M. Cichon, Analysis of tempered fractional calculus in H$\ddot{o}$lder and Orlicz spaces, Symmetry, 14 (2022), 1581. https://doi.org/10.3390/sym14081581 doi: 10.3390/sym14081581
[24]	H. M. Srivastava, K. M. Saad, W. M. Hamanah, Certain new models of the multi-space fractal-fractional Kuramoto-Sivashinsky and Korteweg-de Vries equations, Mathematics, 10 (2022), 1089. https://doi.org/10.3390/math10071089 doi: 10.3390/math10071089
[25]	S. M. Ulam, A collection of mathematical problems, 1960.
[26]	J. Vanterler da C. Sousa, E. Capelas de Oliveira, On the $\psi$-Hilfer fractional derivatives, Commun. Nonlinear Sci., 60 (2018), 72–91. https://doi.org/10.1016/j.cnsns.2018.01.005 doi: 10.1016/j.cnsns.2018.01.005
[27]	J. Vanterler da C. Sousa, D. Santos de Oliveira, E. Capelas de Oliveira, On the existence and stability for non-instantaneous impulsive fractional integrodifferential equation, Math. Method. Appl. Sci., 42 (2019), 1249–1261. https://doi.org/10.1002/mma.5430 doi: 10.1002/mma.5430
[28]	L. Vangipuram, L. Srinivasa, D. J. Vasundhara, Theory of fractional dynamic systems, 2009.
[29]	G. Wang, S. Liu, L. Zhang, Eigenvalue problem for nonlinear fractional differential equations with integral boundary conditions, Abstr. Appl. Anal., 2014 (2014), 916260. https://doi.org/10.1155/2014/916260 doi: 10.1155/2014/916260
[30]	R. L. Wheeden, Measure and Integral: An Introduction to Real Analysis, CRC Press, 2015.
[31]	A. Zada, M. Alam, U. Riaz, Analysis of q-fractional implicit boundary value problems having Stieltjes integral conditions, Math. Method. Appl. Sci., 44 (2021), 4381–4413. https://doi.org/10.1002/mma.7038 doi: 10.1002/mma.7038
[32]	A. Zada, S. Ali, T. Li, Analysis of a new class of impulsive implicit sequential fractional differential equations, Int. J. Nonlinear Sci. Num., 21 (2020), 571–587. https://doi.org/10.1515/ijnsns-2019-0030 doi: 10.1515/ijnsns-2019-0030
[33]	A. Zada, S. Ali, Stability of integral Caputo-type boundary value problem with noninstantaneous impulses, Int. J. Appl. Comput. Math., 5 (2019), 55. https://doi.org/10.1007/s40819-019-0640-0 doi: 10.1007/s40819-019-0640-0
[34]	A. Zada, W. Ali, C. Park, Ulam's type stability of higher order nonlinear delay differential equations via integral inequality of Gr$\ddot{o}$nwall-Bellman-Bihari's type, Appl. Math. Comput., 350 (2019), 60–65. https://doi.org/10.1016/j.amc.2019.01.014 doi: 10.1016/j.amc.2019.01.014
[35]	A. Zada, S. O. Shah, Hyers-Ulam stability of first-order non-linear delay differential equations with fractional integrable impulses, Hacet. J. Math. Stat., 47 (2018), 1196–1205.
[36]	A. Zada, S. Shaleena, T. Li, Stability analysis of higher order nonlinear differential equations in $\beta$-normed spaces, Math. Method. Appl. Sci., 42 (2019), 1151–1166. https://doi.org/10.1002/mma.5419 doi: 10.1002/mma.5419
[37]	L. Zhang, B. Ahmadr, G. Wang, Successive iterations for positive extremal solutions of nonlinear fractional differential equations on a half-line, B. Aust. Math. Soc., 91 (2015), 116–128. https://doi.org/10.1017/S0004972714000550 doi: 10.1017/S0004972714000550
[38]	B. Zhang, R. Majeed, M. Alam, On fractional Langevin equations with stieltjes integral conditions, Mathematics, 10 (2022), 3877. https://doi.org/10.3390/math10203877 doi: 10.3390/math10203877
[39]	G. M. Zaslavsky, Hamiltonian chaos and fractional dynamics, Oxford University Press, 2005.

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)