Existence and approximate controllability of Hilfer fractional evolution equations with almost sectorial operators. (English) Zbl 1486.34018
Summary: In this article, we are concerned with the existence of mild solutions and approximate controllability of Hilfer fractional evolution equations with almost sectorial operators and nonlocal conditions. The existence results are obtained by first defining Green’s function and approximate controllability by specifying a suitable control function. These results are established with the help of Schauder’s fixed point theorem and theory of almost sectorial operators in a Banach space. An example is also presented for the demonstration of obtained results.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 93B05 | Controllability |
| 47N20 | Applications of operator theory to differential and integral equations |
| 26A33 | Fractional derivatives and integrals |
Keywords:
Hilfer fractional derivative; almost sectorial operator; approximate controllability; nonlocal conditionsReferences:
| [1] | Alkhazzan, A.; Jiang, P.; Baleanu, D.; Khan, H.; Khan, A., Stability and existence results for a class of nonlinear fractional differential equations with singularity, Math. Methods Appl. Sci., 41, 18, 9321-9334 (2018) · Zbl 1406.34006 · doi:10.1002/mma.5263 |
| [2] | Bedi, P.; Kumar, A.; Abdeljawad, T.; Khan, A., Existence of mild solutions for impulsive neutral Hilfer fractional evolution equations, Adv. Differ. Equ., 2020 (2020) · Zbl 1482.34184 · doi:10.1186/s13662-020-02615-y |
| [3] | Bedi, P.; Kumar, A.; Abdeljawad, T.; Khan, A., S-asymptotically ω-periodic mild solutions and stability analysis of Hilfer fractional evolution equations, Evol. Equ. Control Theory (2020) · Zbl 1501.34006 · doi:10.3934/eect.2020089 |
| [4] | Chang, Y. K.; Pereira, A.; Ponce, R., Approximate controllability for fractional differential equations of Sobolev type via properties on resolvent operators, Fract. Calc. Appl. Anal., 20, 4, 963-987 (2017) · Zbl 1369.93081 · doi:10.1515/fca-2017-0050 |
| [5] | Chen, P.; Zhang, X., Approximate controllability of nonlocal problem for non-autonomous stochastic evolution equations, Evol. Equ. Control Theory (2019) · Zbl 1469.93008 · doi:10.3934/eect.2020076 |
| [6] | Chen, P.; Zhang, X.; Li, Y., Approximation technique for fractional evolution equations with nonlocal integral conditions, Mediterr. J. Math., 14, 6 (2017) · Zbl 1387.35604 · doi:10.1007/s00009-017-1029-0 |
| [7] | Chen, P.; Zhang, X.; Li, Y., A blowup alternative result for fractional nonautonomous evolution equation of Volterra type, Commun. Pure Appl. Anal., 17, 5, 1975-1992 (2018) · Zbl 1397.35331 · doi:10.3934/cpaa.2018094 |
| [8] | Chen, P.; Zhang, X.; Li, Y., Non-autonomous parabolic evolution equations with non-instantaneous impulses governed by noncompact evolution families, J. Fixed Point Theory Appl., 21, 3 (2019) · Zbl 1423.35425 · doi:10.1007/s11784-019-0719-6 |
| [9] | Chen, P.; Zhang, X.; Li, Y., Non-autonomous evolution equations of parabolic type with non-instantaneous impulses, Mediterr. J. Math., 16, 5 (2019) · Zbl 1483.35333 · doi:10.1007/s00009-019-1384-0 |
| [10] | Chen, P.; Zhang, X.; Li, Y., Fractional non-autonomous evolution equation with nonlocal conditions, J. Pseudo-Differ. Oper. Appl., 10, 4, 955-973 (2019) · Zbl 1427.34006 · doi:10.1007/s11868-018-0257-9 |
| [11] | Chen, P.; Zhang, X.; Li, Y., Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators, Fract. Calc. Appl. Anal., 23, 1, 268-291 (2020) · Zbl 1441.34006 · doi:10.1515/fca-2020-0011 |
| [12] | Chen, P.; Zhang, X.; Li, Y., Approximate controllability of non-autonomous evolution system with nonlocal conditions, J. Dyn. Control Syst., 26, 1, 1-16 (2020) · Zbl 1439.34065 · doi:10.1007/s10883-018-9423-x |
| [13] | Chen, P.; Zhang, X.; Li, Y., Cauchy problem for fractional non-autonomous evolution equations, Banach J. Math. Anal., 14, 2, 559-584 (2020) · Zbl 1452.35236 · doi:10.1007/s43037-019-00008-2 |
| [14] | Debbouche, A.; Antonov, V., Approximate controllability of semilinear Hilfer fractional differential inclusions with impulsive control inclusion conditions in Banach spaces, Chaos Solitons Fractals, 102, 140-148 (2017) · Zbl 1374.93048 · doi:10.1016/j.chaos.2017.03.023 |
| [15] | Devi, A.; Kumar, A.; Abdeljawad, T.; Khan, A., Existence and stability analysis of solutions for fractional Langevin equation with nonlocal integral and anti-periodic type boundary conditions, Fractals (2020) · Zbl 1487.34013 · doi:10.1142/S0218348X2040006X |
| [16] | Ding, X. L.; Ahmad, B., Analytical solutions to fractional evolution equations with almost sectorial operators, Adv. Differ. Equ., 2016, 1 (2016) · Zbl 1419.35204 · doi:10.1186/s13662-016-0927-y |
| [17] | Du, J.; Jiang, W.; Niazi, A. U.K., Approximate controllability of impulsive Hilfer fractional differential inclusions, J. Nonlinear Sci. Appl., 10, 2, 595-611 (2017) · Zbl 1412.34022 · doi:10.22436/jnsa.010.02.23 |
| [18] | Fu, X., Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay, Evol. Equ. Control Theory, 6, 4, 517-534 (2017) · Zbl 1381.34097 · doi:10.3934/eect.2017026 |
| [19] | Gomez-Aguilar, J. F.; Cordova-Fraga, T.; Abdeljawad, T.; Khan, A.; Khan, H., Analysis of fractal-fractional malaria transmission model, Fractals (2020) · Zbl 1482.92097 · doi:10.1142/S0218348X20400411 |
| [20] | Gu, H.; Trujillo, J. J., Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput., 257, 344-354 (2015) · Zbl 1338.34014 · doi:10.1016/j.amc.2014.10.083 |
| [21] | Jaiswal, A.; Bahuguna, D., Hilfer fractional differential equations with almost sectorial operators, Differ. Equ. Dyn. Syst. (2020) · Zbl 1519.34068 · doi:10.1007/s12591-020-00514-y |
| [22] | Khan, A.; Khan, T. S.; Syam, M. I.; Khan, H., Analytical solutions of time-fractional wave equation by double Laplace transform method, Eur. Phys. J. Plus, 134, 4 (2019) · doi:10.1140/epjp/i2019-12499-y |
| [23] | Khan, A.; Shah, K.; Li, Y.; Khan, T. S., Ulam type stability for a coupled system of boundary value problems of nonlinear fractional differential equations, J. Funct. Spaces (2017) · Zbl 1377.34012 · doi:10.1155/2017/3046013 |
| [24] | Khan, A.; Syam, M. I.; Zada, A.; Khan, H., Stability analysis of nonlinear fractional differential equations with Caputo and Riemann-Liouville derivatives, Eur. Phys. J. Plus, 133, 7 (2018) · doi:10.1140/epjp/i2018-12119-6 |
| [25] | Khan, H.; Alipour, M.; Khan, R. A.; Tajadodi, H.; Khan, A., On approximate solution of fractional order logistic equations by operational matrices of Bernstein polynomials, J. Math. Comput. Sci., 14, 222-232 (2015) · doi:10.22436/jmcs.014.03.05 |
| [26] | Khan, H.; Chen, W.; Khan, A.; Khan, T. S.; Al-Madlal, Q. M., Hyers-Ulam stability and existence criteria for coupled fractional differential equations involving p-Laplacian operator, Adv. Differ. Equ., 2018, 1 (2018) · Zbl 1448.34019 · doi:10.1186/s13662-018-1899-x |
| [27] | Khan, H.; Gomez Aguilar, J. F.; Abdeljawad, T.; Khan, A., Existence results and stability criteria for ABC-fuzzy-Volterra integro-differential equation, Fractals (2020) · Zbl 1482.45005 · doi:10.1142/S0218348X20400484 |
| [28] | Khan, H.; Gómez-Aguilar, J. F.; Alkhazzan, A.; Khan, A., A fractional order HIV-TB coinfection model with nonsingular Mittag-Leffler law, Math. Methods Appl. Sci., 43, 6, 3786-3806 (2020) · Zbl 1452.92039 · doi:10.1002/mma.6155 |
| [29] | Khan, H.; Tunç, C.; Alkhazan, A.; Ameen, B.; Khan, A., A generalization of Minkowski’s inequality by Hahn integral operator, J. Taibah Univ. Sci., 12, 5, 506-513 (2018) · doi:10.1080/16583655.2018.1493859 |
| [30] | Khan, Z. A., Integral inequality of Gronwall type with an application, J. Math. Comput. Sci., 5, 1, 34-41 (2015) |
| [31] | Khan, Z. A., Solvability for a class of integral inequalities with maxima on the theory of time scales and their applications, Bound. Value Probl., 2019 (2019) · Zbl 1513.26055 · doi:10.1186/s13661-019-1259-0 |
| [32] | Khan, Z. A., Analysis on some powered integral inequalities with retarded argument and application, J. Taibah Univ. Sci., 14, 1, 488-495 (2020) · doi:10.1080/16583655.2020.1747218 |
| [33] | Li, F., Mild solutions for abstract fractional differential equations with almost sectorial operators and infinite delay, Adv. Differ. Equ., 2013, 1 (2013) · Zbl 1391.34013 · doi:10.1186/1687-1847-2013-327 |
| [34] | Lv, J.; Yang, X., Approximate controllability of Hilfer fractional neutral stochastic differential equations, Dyn. Syst. Appl., 27, 4, 691-713 (2018) · doi:10.12732/dsa.v27i4.1 |
| [35] | Lv, J.; Yang, X., Approximate controllability of Hilfer fractional differential equations, Math. Methods Appl. Sci., 43, 1, 242-254 (2020) · Zbl 1451.93030 · doi:10.1002/mma.5862 |
| [36] | Mahmudov, N. I.; McKibben, M. A., On the approximate controllability of fractional evolution equations with generalized Riemann-Liouville fractional derivative, J. Funct. Spaces, 2015 (2015) · Zbl 1319.93013 · doi:10.1155/2015/263823 |
| [37] | Periago, F.; Straub, B., A functional calculus for almost sectorial operators and applications to abstract evolution equations, J. Evol. Equ., 2, 1, 41-68 (2002) · Zbl 1005.47015 · doi:10.1007/s00028-002-8079-9 |
| [38] | Podlubny, I., Fractional Differential Equations (1999), San Diego: Academic Press, San Diego · Zbl 0918.34010 |
| [39] | Ullah, M.; Sarwar, M.; Khan, H.; Abdeljawad, T.; Khan, A., Near-coincidence point results in metric interval space and hyperspace via simulation functions, Adv. Differ. Equ., 2020, 1 (2020) · Zbl 1485.54061 · doi:10.1186/s13662-020-02734-6 |
| [40] | Wang, J.; Fečkan, M.; Zhou, Y., Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions, Evol. Equ. Control Theory, 6, 3, 471 (2017) · Zbl 06744040 · doi:10.3934/eect.2017024 |
| [41] | Yang, M.; Wang, Q. R., Approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions, Math. Methods Appl. Sci., 40, 4, 1126-1138 (2017) · Zbl 1366.34012 · doi:10.1002/mma.4040 |
| [42] | Yong, Z.; Jinrong, W.; Lu, Z., Basic Theory of Fractional Differential Equations (2016), Singapore: World Scientific, Singapore |
| [43] | Zhang, L.; Zhou, Y., Fractional Cauchy problems with almost sectorial operators, Appl. Math. Comput., 257, 145-157 (2015) · Zbl 1338.34030 · doi:10.1016/j.amc.2014.07.024 |
| [44] | Zhou, Y.; Vijayakumar, V.; Ravichandran, C.; Murugesu, R., Controllability results for fractional order neutral functional differential inclusions with infinite delay, Fixed Point Theory, 18, 2, 773-798 (2017) · Zbl 1383.34098 · doi:10.24193/fpt-ro.2017.2.62 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
