Cauchy problem for non-autonomous fractional evolution equations with nonlocal conditions of order $ (1, 2) $

Naveed Iqbal; Azmat Ullah Khan Niazi; Ikram Ullah Khan; Rasool Shah; Thongchai Botmart; Naveed Iqbal; Azmat Ullah Khan Niazi; Ikram Ullah Khan; Rasool Shah; Thongchai Botmart

doi:10.3934/math.2022496

AIMS Mathematics

2022, Volume 7, Issue 5: 8891-8913. doi: 10.3934/math.2022496

Previous Article Next Article

Research article Special Issues

Cauchy problem for non-autonomous fractional evolution equations with nonlocal conditions of order $ (1, 2) $

1.
Department of Mathematics, Faculty of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Mathematics and Statistics, University of Lahore, Sargodha, Pakistan
3.
Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan
4.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received: 18 November 2021 Revised: 06 February 2022 Accepted: 13 February 2022 Published: 07 March 2022
MSC : 26A33, 4K37

This article contracts through Cauchy problems in infinite-dimensional Banach spaces towards a system of nonlinear non-autonomous mixed type integro-differential fractional evolution equation by nonlocal conditions through noncompactness measure (MNC). We demonstrate the existence of novel mild solutions in the condition that the nonlinear function mollifies generally adequate, an MNC form and local growth form, using evolution families and fractional calculus theory, as well as the fixed-point theorem w.r.t. K-set-contractive operator and another MNC assessment procedure. Our findings simplify and improve upon past findings in this area. Finally, towards the end of this article, as an example of submissions, we use a fractional non-autonomous partial differential equation (PDE) with nonlocal conditions and a homogeneous Dirichlet boundary condition.
- initial value problem,
- non-autonomous fractional evolution equations,
- measure of noncompactness,
- nonlocal conditions,
- mild solution,
- analytic semigroup
Citation: Naveed Iqbal, Azmat Ullah Khan Niazi, Ikram Ullah Khan, Rasool Shah, Thongchai Botmart. Cauchy problem for non-autonomous fractional evolution equations with nonlocal conditions of order $ (1, 2) $[J]. AIMS Mathematics, 2022, 7(5): 8891-8913. doi: 10.3934/math.2022496

Related Papers:

Abstract

This article contracts through Cauchy problems in infinite-dimensional Banach spaces towards a system of nonlinear non-autonomous mixed type integro-differential fractional evolution equation by nonlocal conditions through noncompactness measure (MNC). We demonstrate the existence of novel mild solutions in the condition that the nonlinear function mollifies generally adequate, an MNC form and local growth form, using evolution families and fractional calculus theory, as well as the fixed-point theorem w.r.t. K-set-contractive operator and another MNC assessment procedure. Our findings simplify and improve upon past findings in this area. Finally, towards the end of this article, as an example of submissions, we use a fractional non-autonomous partial differential equation (PDE) with nonlocal conditions and a homogeneous Dirichlet boundary condition.

References

[1]	E. G. Bajlekova, Fractional evolution equations in Banach spaces, Ph.D. thesis, Department of Mathematics, Eindhoven University of Technology, 2001.
[2]	L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162 (1991), 494–505. https://doi.org/10.1016/0022-247X(91)90164-U doi: 10.1016/0022-247X(91)90164-U
[3]	L. Byszewski, Application of properties of the right hand sides of evolution equations to an investigation of nonlocal evolution problems, Nonlinear Anal., 33 (1998), 413–426. https://doi.org/10.1016/S0362-546X(97)00594-4 doi: 10.1016/S0362-546X(97)00594-4
[4]	P. Chen, Y. Li, Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math., 63 (2013), 731–744. https://doi.org/10.1007/s00025-012-0230-5 doi: 10.1007/s00025-012-0230-5
[5]	P. Chen, X. Zhang, Y. Li, Approximation technique for fractional evolution equations with nonlocal integral conditions, Mediterr. J. Math., 14 (2017), 226. https://doi.org/10.1007/s00009-017-1029-0 doi: 10.1007/s00009-017-1029-0
[6]	P. Chen, X. Zhang, Y. Li, Fractional non-autonomous evolution equation with nonlocal conditions, J. Pseudo-Differ. Oper. Appl., 10 (2019), 955–973. https://doi.org/10.1007/s11868-018-0257-9 doi: 10.1007/s11868-018-0257-9
[7]	H. Khan, R. Shah, P. Kumam, D. Baleanu, & M. Arif, Laplace decomposition for solving nonlinear system of fractional order partial differential equations. Adva. in Diff. Eqs., 1 (2020), 1–18. https://doi.org/10.1186/s13662-020-02839-y
[8]	C. Corduneanu, Principles of differential and integral equations, Boston: Allyn & Bacon, 1971.
[9]	K. Deimling, Nonlinear functional analysis, Springer, 1985. https://doi.org/10.1007/978-3-662-00547-7
[10]	M. M. El-Borai, The fundamental solutions for fractional evolution equations of parabolic type, J. Appl. Math. Stoch. Anal., 2004 (2004), 197–211. http://dx.doi.org/10.1155/S1048953304311020 doi: 10.1155/S1048953304311020
[11]	M. M. El-Borai, K. E. El-Nadi, E. G. El-Akabawy, On some fractional evolution equations, Comput. Math. Appl., 59 (2010), 1352–1355. https://doi.org/10.1016/j.camwa.2009.05.005 doi: 10.1016/j.camwa.2009.05.005
[12]	K. Ezzinbi, X. Fu, K. Hilal, Existence and regularity in the $\alpha$-norm for some neutral partial differential equations with nonlocal conditions, Nonlinear Anal., 67 (2007), 1613–1622. https://doi.org/10.1016/j.na.2006.08.003 doi: 10.1016/j.na.2006.08.003
[13]	R. Shah, H. Khan, D. Baleanu, Fractional Whitham-Broer-Kaup equations within modified analytical approaches. Axioms, 8 (2019), 125. https://doi.org/10.3390/axioms8040125
[14]	R. Gorenflo, F. Mainardi, Fractional calculus and stable probability distributions, Arch. Mech., 50 (1998), 377–388.
[15]	H. Gou, B. Li, Local and global existence of mild solution to impulsive fractional semilinear integro differential equation with noncompact semigroup, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 204–214. https://doi.org/10.1016/j.cnsns.2016.05.021 doi: 10.1016/j.cnsns.2016.05.021
[16]	D. Guo, Solutions of nonlinear integro-differential equations of mixed type in Banach spaces, J. Appl. Math. Simul., 2 (1989), 1–11.
[17]	H. P. Heinz, On the behaviour of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., 7 (1983), 1351–1371. https://doi.org/10.1016/0362-546X(83)90006-8 doi: 10.1016/0362-546X(83)90006-8
[18]	D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math, Springer, 1981. https://doi.org/10.1007/BFb0089647
[19]	N. Iqbal, A. U. K. Niazi, R. Shafqat, S. Zaland, Existence and uniqueness of mild solution for fractional order controlled fuzzy evolution equation, J. Funct. Spaces, 2021 (2021), 5795065. https://doi.org/10.1155/2021/5795065 doi: 10.1155/2021/5795065
[20]	N. Iqbal, R. Wu, Pattern formation by fractional cross-diffusion in a predator-prey model with Beddington-DeAngelis type functional response, Int. J. Mod. Phys., 33 (2019), 1950296. https://doi.org/10.1142/S0217979219502965 doi: 10.1142/S0217979219502965
[21]	N. Iqbal, H. Yasmin, A. Ali, A. Bariq, M. M. Al-Sawalha, W. W. Mohammed, Numerical methods for fractional-order Fornberg-Whitham equations in the sense of Atangana-Baleanu derivative, J. Funct. Spaces, 2021 (2021), 2197247. https://doi.org/10.1155/2021/2197247 doi: 10.1155/2021/2197247
[22]	N. Iqbal, H. Yasmin, A. Rezaiguia, J. Kafle, A. O. Almatroud, T. S. Hassan, Analysis of the fractional-order Kaup-Kupershmidt equation via novel transforms, J. Math., 2021 (2021), 2567927. https://doi.org/10.1155/2021/2567927 doi: 10.1155/2021/2567927
[23]	K. Kavitha, V. Vijayakumar, R. Udhayakumar, N. Sakthivel, K. S. Nisar, A note on approximate controllability of the Hilfer fractional neutral differential inclusions with infinite delay, Math. Methods Appl. Sci., 44 (2021), 4428–4447. https://doi.org/10.1002/mma.7040 doi: 10.1002/mma.7040
[24]	K. Kavitha, V. Vijayakumar, A. Shukla, K. S. Nisar, R. Udhayakumar, Results on approximate controllability of Sobolev-type fractional neutral differential inclusions of Clarke subdifferential type, Chaos Solitons Fract., 151 (2021), 111264. https://doi.org/10.1016/j.chaos.2021.111264 doi: 10.1016/j.chaos.2021.111264
[25]	W. K. Williams, V. Vijayakumar, R. Udhayakumar, S. K. Panda, K. S. Nisar, Existence and controllability of nonlocal mixed Volterra‐Fredholm type fractional delay integro‐differential equations of order $1 < r < 2$, Numer. Meth. Part. Differ. Eq., 2020. https://doi.org/10.1002/num.22697
[26]	V. Lakshmikantham, S. Leela, Nonlinear differential equations in abstract spaces, New York: Pergamon Press, 1981.
[27]	J. Liang, J. Liu, T. J. Xiao, Nonlocal Cauchy problems governed by compact operator families, Nonlinear Anal., 57 (2004), 183–189. https://doi.org/10.1016/j.na.2004.02.007 doi: 10.1016/j.na.2004.02.007
[28]	J. Liang, J. H. Liu, T. J. Xiao, Nonlocal impulsive problems for integro-differential equations, Math. Comput. Model., 49 (2009), 798–804.
[29]	L. Liu, C. Wu, F. Guo, Existence theorems of global solutions of initial value problems for nonlinear integro-differential equations of mixed type in Banach spaces and applications, Comput. Math. Appl., 47 (2004), 13–22. https://doi.org/10.1016/S0898-1221(04)90002-8 doi: 10.1016/S0898-1221(04)90002-8
[30]	L. Liu, F. Guo, C. Wu, Y. Wu, Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl., 309 (2005), 638–649. https://doi.org/10.1016/j.jmaa.2004.10.069 doi: 10.1016/j.jmaa.2004.10.069
[31]	M. McKibben, Discovering evolution equations with applications: Volume 1-Deterministic equations, Chapman and Hall/CRC, 2010.
[32]	A. U. K. Niazi, N. Iqbal, W. W. Mohammed, Optimal control of nonlocal fractional evolution equations in the $\alpha$-norm of order $(1, 2)$, Adv. Differ. Equ., 2021 (2021), 142. https://doi.org/10.1186/s13662-021-03312-0 doi: 10.1186/s13662-021-03312-0
[33]	A. U. K. Niazi, N. Iqbal, R. Shah, F. Wannalookkhee, K. Nonlaopon, Controllability for fuzzy fractional evolution equations in credibility space, Fractal Fract., 5 (2021), 112 https://doi.org/10.3390/fractalfract5030112 doi: 10.3390/fractalfract5030112
[34]	A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer, 1983. https://doi.org/10.1007/978-1-4612-5561-1
[35]	M. M. Raja, V. Vijayakumar, R. Udhayakumar, Y. Zhou, A new approach on the approximate controllability of fractional differential evolution equations of order $1 < r < 2$ in Hilbert spaces, Chaos Solitons Fract., 141 (2020), 110310.
[36]	M. M. Raja, V. Vijayakumar, R. Udhayakumar, Y. Zhou, A new approach on approximate controllability of fractional evolution inclusions of order $1 < r < 2$ with infinite delay, Chaos Solitons Fract., 141 (2020), 110343. https://doi.org/10.1016/j.chaos.2020.110310 doi: 10.1016/j.chaos.2020.110310
[37]	H. B. Shi, W. T. Li, H. R. Sun, Existence of mild solutions for abstract mixed type semilinear evolution equations, Turk. J. Math., 35 (2011), 457–472.
[38]	J. Sun, X. Zhang, The fixed point theorem of convex-power condensing operator and applications to abstract semilinear evolution equations, Acta Math. Sin., 48 (2005), 439–446.
[39]	R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, 2 Eds., Springer, 1997. https://doi.org/10.1007/978-1-4612-0645-3
[40]	W. W. Mohammed, S. Albosaily, N. Iqbal, M. El-Morshedy, The effect of multiplicative noise on the exact solutions of the stochastic Burgers' equation, Waves Random Complex Media, 2021, 1–13. https://doi.org/10.1080/17455030.2021.1905914
[41]	L. Zhu, G. Li, Existence results of semilinear differential equations with nonlocal initial conditions in Banach spaces, Nonlinear Anal., 74 (2011), 5133–5140. https://doi.org/10.1016/j.na.2011.05.007 doi: 10.1016/j.na.2011.05.007

Reader Comments

Your name:*

Email:*
© 2022 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)