Mild solutions for fractional non-instantaneous impulses integro-differential equations with nonlocal conditions

Ye Li; Biao Qu; Ye Li; Biao Qu

doi:10.3934/math.2024589

AIMS Mathematics

2024, Volume 9, Issue 5: 12057-12071. doi: 10.3934/math.2024589

Previous Article Next Article

Research article Special Issues

Mild solutions for fractional non-instantaneous impulses integro-differential equations with nonlocal conditions

Ye Li ^{1,2
,
,},
Biao Qu ²

1.
School of Data and Computer Science, Shandong Women's University, China
2.
Institute of Operations Research, Qufu Normal University, China

Received: 26 December 2023 Revised: 12 March 2024 Accepted: 20 March 2024 Published: 27 March 2024
MSC : 46T99, 47H08, 47H10

In this paper, we investigated Caputo fractional integro-differential equations with non-instantaneous impulses and nonlocal conditions. By employing the solution operator, the Mönch fixed point theorem, and the stepwise estimation method, we eliminated the Lipschitz condition of the nonlinear term, while also dispensing with the requirement for the compressibility coefficient condition $ 0 < k < 1 $. The main results presented represented a generalization and enhancement of previous findings. Furthermore, an example was provided to verify the application of our main results.
- fractional integro-differential equations,
- solution operator,
- non-instantaneous impulse,
- measure of non-compactness,
- Mönch fixed point theorem
Citation: Ye Li, Biao Qu. Mild solutions for fractional non-instantaneous impulses integro-differential equations with nonlocal conditions[J]. AIMS Mathematics, 2024, 9(5): 12057-12071. doi: 10.3934/math.2024589

Related Papers:

Abstract

In this paper, we investigated Caputo fractional integro-differential equations with non-instantaneous impulses and nonlocal conditions. By employing the solution operator, the Mönch fixed point theorem, and the stepwise estimation method, we eliminated the Lipschitz condition of the nonlinear term, while also dispensing with the requirement for the compressibility coefficient condition $ 0 < k < 1 $. The main results presented represented a generalization and enhancement of previous findings. Furthermore, an example was provided to verify the application of our main results.

References

[1]	I. Podlubny, Fractional differential equations, San Diego: Academic Press, 1999.
[2]	S. Das, Functional fractional calculus, Berlin, Heidelberg: Springer, 2011. https://doi.org/10.1007/978-3-642-20545-3
[3]	S. Abbas, M. Benchohra, G. M. N'Guérékata, Topics in fractional differential equations, New York: Springer, 2012. https://doi.org/10.1007/978-1-4614-4036-9
[4]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.
[5]	C. Milici, G. Drăgănescu, J. T. Machado, Introduction to fractional differential equations, Cham: Springer, 2019. https://doi.org/10.1007/978-3-030-00895-6
[6]	V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of impulsive differential equations, London: World Scientific, 1989. https://doi.org/10.1142/0906
[7]	K. Deimling, Nonlinear functional analysis, Berlin, Heidelberg: Spinger, 1985. https://doi.org/10.1007/978-3-662-00547-7
[8]	B. Zhu, L. S. Liu, Existence and uniqueness of the mild solutions for a class of fractional non-autonomous evolution equations with impulse, Acta Math. Sinica, 39 (2019), 105–113.
[9]	X. G. Zhang, D. Z. Kong, H. Tian, Y. H. Wu, B. Wiwatanapataphee, An upper-lower solution method for the eigenvalue problem of Hadamard-type singular fractional differential equation, Nonlinear Anal. Model., 27 (2022), 789–802. https://doi.org/10.15388/NAMC.2022.27.27491 doi: 10.15388/NAMC.2022.27.27491
[10]	B. Zhu, L. S. Liu, Y. H. Wu, Existence and uniqueness of global mild solutions for a class of nonlinear fractional reactiondiffusion equations with delay, Comput. Math. Appl., 78 (2019), 1811–1818. https://doi.org/10.1016/j.camwa.2016.01.028 doi: 10.1016/j.camwa.2016.01.028
[11]	P. Y. Chen, Y. X. Li, Q. Y. Chen, B. H. Feng, On the initial value problem of fractional evolution equations with noncompact semigroup, Comput. Math. Appl., 67 (2014), 1108–1115. https://doi.org/10.1016/j.camwa.2014.01.002 doi: 10.1016/j.camwa.2014.01.002
[12]	B. Zhu, Y. B. Han, L. S. Liu, Existence of solutions for a class of mixed fractional order semilinear integro-differential equations, Acta Math. Sinica, 39 (2019), 1334–1341.
[13]	X. A. Hao, L. S. Liu, Mild solution of second-order impulsive integro-differential evolution equations of Volterra type in Banach spaces, Qual. Theory Dyn. Syst., 19 (2020), 5. https://doi.org/10.1007/s12346-020-00345-w doi: 10.1007/s12346-020-00345-w
[14]	L. S. Liu, H. Y. Qin, Existence and uniqueness of mild solutions for fractional impulsive integro-differential evolution equations of order $ 1 < \beta\leq2 $ with nonlocal conditions, Sci. Sin. Math., 50 (2020), 1807–1828.
[15]	Y. Li, B. Qu, Mild solutions for fractional impulsive integro-differential evolution equations with nonlocal conditions in Banach spaces, Symmetry, 14 (2022), 1655. https://doi.org/10.3390/sym14081655 doi: 10.3390/sym14081655
[16]	P. Verma, M. Kumar, An analytical solution with existence and uniqueness conditions for fractional integro differential equations, Int. J. Model. Simul. Sci. Comput., 11 (2020), 2050045. https://doi.org/10.1142/S1793962320500452 doi: 10.1142/S1793962320500452
[17]	P. Verma, M. Kumar, Analytical solution with existence and uniqueness conditions of non-linear initial value multi-order fractional differential equations using Caputo derivative, Eng. Comput., 38 (2022), 661–678. https://doi.org/10.1007/s00366-020-01061-4 doi: 10.1007/s00366-020-01061-4
[18]	E. Hernandez, D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Am. Math. Soc., 141 (2013), 1641–1649. https://doi.org/10.1090/S0002-9939-2012-11613-2 doi: 10.1090/S0002-9939-2012-11613-2
[19]	M. Pierri, D. O'Regan, V. Rolnik, Existence of solutions for semilinear differential equations with not instantaneous impulses, Appl. Math. Comput., 219 (2013), 6743–6749. https://doi.org/10.1016/j.amc.2012.12.084 doi: 10.1016/j.amc.2012.12.084
[20]	M. Pierri, H. R. Hernan, A. Prokopczyk, Global solutions for abstract differential equations with non-instantaneous impulses, Mediterr. J. Math., 13 (2016), 1685–1708. https://doi.org/10.1007/s00009-015-0609-0 doi: 10.1007/s00009-015-0609-0
[21]	B. Zhu, Y. B. Han, Y. H. Wu, Existence of mild solutions for a class of fractional non-autonomous evolution equations with delay, Acta Math. Sinica, 36 (2020), 870–878.
[22]	J. R. Wang, M. Feckan, A. Debbouche, Time optimal controls of system governed by non-instantaneous impulsive differential equations, J. Optim. Theory Appl., 182 (2019), 573–587. https://doi.org/10.1007/s10957-018-1313-6 doi: 10.1007/s10957-018-1313-6
[23]	J. R. Wang, M. M. Li, D. O'Regan, M. Feckan, Robustness for linear evolution equations with non-instantaneous impulsive effects, Bull. Sci. Math., 159 (2020), 102827. https://doi.org/10.1016/j.bulsci.2019.102827 doi: 10.1016/j.bulsci.2019.102827
[24]	D. Araya, C. Lizama, Almost automorphic mild solutions to fractional differential equations, Nonlinear Anal. Theor., 69 (2008), 3692–3705. https://doi.org/10.1016/j.na.2007.10.004 doi: 10.1016/j.na.2007.10.004
[25]	D. Guo, Nonlinear functional analysis, Jinan: Shandong Science and Technology, 2002.
[26]	F. D. Ge, H. C. Zhou, C. H. Kou, Approximate controllability of semilinear evolution equations of fractional order with nonlocal and impulsive conditions via an approximating technique, Appl. Math. Comput., 275 (2016), 107–120. https://doi.org/10.1016/j.amc.2015.11.056 doi: 10.1016/j.amc.2015.11.056
[27]	L. S. Liu, Iterative method for solutions and coupled quasi-solutions of nonlinear integro-differential equations of mixed type in Banach spaces, Nonlinear Anal. Theor., 42 (2000), 583–598. https://doi.org/10.1016/S0362-546X(99)00116-9 doi: 10.1016/S0362-546X(99)00116-9

Reader Comments

Your name:*

Email:*
© 2024 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)