On the nonlinear impulsive Ψ–Hilfer fractional differential equations
Abstract
In this paper, we consider the nonlinear Ψ-Hilfer impulsive fractional differential equation. Our main objective is to derive the formula for the solution and examine the existence and uniqueness of solutions. The acquired results are extended to the nonlocal Ψ-Hilfer impulsive fractional differential equation. We gave an applications to the outcomes we obtained. Further, examples are provided in support of the results we got.
Keyword : Ψ–Hilfer fractional derivative, fractional differential equations, impulsive, nonlocal, existence and uniqueness, fixed point theorem
How to Cite
Kucche, K. D., Kharade, J. P., & Sousa, J. V. da C. (2020). On the nonlinear impulsive Ψ–Hilfer fractional differential equations. Mathematical Modelling and Analysis, 25(4), 642-660. https://doi.org/10.3846/mma.2020.11445
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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S.G. Samko, A.A. Kilbas and O.I. Marichev. Fractional integrals and derivatives, theory and applications. Gordon and Breach, Yverdon, 1993.
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K. Shah, H. Khalil and R.A. Khan. Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations. Chaos, Solitons & Fractals, 77:240–246, 2015. https://doi.org/10.1016/j.chaos.2015.06.008
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J.V.C. Sousa and E. Capelas de Oliveira. On the Ulam–Hyers–Rassias stability for nonlinear fractional differential equations using the ψ–Hilfer operator. Fixed Point Theory and Appl., 20(3):96, 2018. https://doi.org/10.1007/s11784-018-0587-5
J.V.C. Sousa and E. Capelas de Oliveira. Ulam–Hyers stability of a nonlinear fractional Volterra integro-differential equation. Appl. Math. Lett., 81:50–56, 2018. https://doi.org/10.1016/j.aml.2018.01.016
J.V.C. Sousa, E. Capelas de Oliveira and K.D. Kucche. On the fractional functional differential equation with abstract Volterra operator. Bull. Braz. Math. Soc., New Series, pp. 1–20, 2018.
J.V.C. Sousa, E. Capelas de Oliveira and L.A. Magna. Fractional calculus and the ESR test. AIMS Math, 2(4):692–705, 2017. https://doi.org/10.3934/Math.2017.4.692
J.V.C. Sousa, K.D. Kucche and E. Capelas de Oliveira. Stability of ψ-Hilfer impulsive fractional differential equations. Appl. Math. Lett., 88:73–80, 2019. https://doi.org/10.1016/j.aml.2018.08.013
J.V.C. Sousa and E. Capelas Oliveira. A Gronwall inequality and the Cauchytype problem by means of ψ–Hilfer operator. Diff. Equ. & Appl., 11(1):87–106, 2019. https://doi.org/10.7153/dea-2019-11-02
J. Wang, K. Shah and A. Ali. Existence and Hyers–Ulam stability of fractional nonlinear impulsive switched coupled evolution equations. Math. Meth. Appl. Sci., 41(6):2392–2402, 2018. https://doi.org/10.1002/mma.4748
J. Wang, Y. Zhou and M. Fečkan. On recent developments in the theory of boundary value problems for impulsive fractional differential equations. Comput. Math. Appl., 64(10):3008–3020, 2012. https://doi.org/10.1016/j.camwa.2011.12.064
J. Wang, Y. Zhou and Z. Lin. On a new class of impulsive fractional differential equations. Appl. Math. Comput., 242:649–657, 2014. https://doi.org/10.1016/j.amc.2014.06.002
J.R. Wang, Y. Zhou and Fečkan. Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput. Math. Appl., 64(10):3389– 3405, 2012. https://doi.org/10.1016/j.camwa.2012.02.021
W. Wei, X. Xiang and Y. Peng. Nonlinear impulsive integrodifferential equation of mixed type and optimal controls. Optimization, 55(1-2):141–156, 2006. https://doi.org/10.1080/02331930500530401
Y. Zhou. Basic theory of fractional differential equations. World scientific, 2014. https://doi.org/10.1142/9069
R. Agarwal, S. Hristova and D. ORegan. Some stability properties related to initial time difference for caputo fractional differential equations. Frac. Cal. Appl. Anal., 21(1):72–93, 2018. https://doi.org/10.1515/fca-2018-0005
R.P. Agarwal, M. Benchohra and B.A. Slimani. Existence results for differential equations with fractional order and impulses. Mem. Diff. Equ. Math. Phys, 44(1):1–21, 2008.
B. Ahmad and S. Sivasundaram. Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Analysis: Hybrid Systems, 4(1):134–141, 2010. https://doi.org/10.1016/j.nahs.2009.09.002
A. Ali, K. Shah and D. Baleanu. Ulam stability results to a class of nonlinear implicit boundary value problems of impulsive fractional differential equations. Advances in Difference Equations, 2019(5):1–21, 2019. https://doi.org/10.1186/s13662-018-1940-0
Z. Bai, X. Dong and C. Yin. Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions. Boundary Value Problems, 2016(63), 2016. https://doi.org/10.1186/s13661-016-0573-z
D.D. Bainov and P.S. Simeonov. Impulsive differential equations: periodic solutions and applications. Longman Scientific and Technical Group Limited,New York, 1993.
K. Balachandran and J.J. Trujillo. The nonlocal cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces. Nonlinear Analysis: Theo. Meth. & Appl., 72(12):4587–4593, 2010. https://doi.org/10.1016/j.na.2010.02.035
M. Benchohra and F. Berhoun. Impulsive fractional differential equations with variable times. Comput. Math. Appl., 59(3):1245–1252, 2010. https://doi.org/10.1016/j.camwa.2009.05.016
M. Benchohra, J. Henderson and S.K. Ntouyas. Impulsive differential equations and inclusions. Hindawi Publishing Corporation, Vol. 2, New York, 2006.
M. Benchohra and D. Seba. Impulsive fractional differential equations in Banach spaces. Elect. J. Qual. Theory Differ. Equ., 8(1), 2009. https://doi.org/10.14232/ejqtde.2009.4.8
M. Benchohra and B.A. Slimani. Existence and uniqueness of solutions to impulsive fractional differential equations. Elect. J. Diff. Equ., 2009(10):1–11, 2009.
S. Das and P.K. Gupta. A mathematical model on fractional Lotka–Volterra equations. Theor. Biol., 277(1):1–6, 2011. https://doi.org/10.1016/j.jtbi.2011.01.034
E. Capelas de Oliveira and J.V.C. Sousa. Ulam–Hyers–Rassias stability for a class of fractional integro-differential equations. Results Math., 73(3):111, 2018.
K. Diethelm. The analysis of fractional differential equation, Lecture notes in Mathematics. Springer, New York, 2010. https://doi.org/10.1007/978-3-642-14574-2
S. Dipierro and E. Valdinoci. A simple mathematical model inspired by the Purkinje cells: from delayed travelling waves to fractional diffusion. Bull. Math. Biol., 80(7):1849–1870, 2018. https://doi.org/10.1007/s11538-018-0437-z
M. Feckan, Y. Zhou and J. Wang. On the concept and existence of solution for impulsive fractional differential equations. Commun Nonlinear Sci Numer Simulat., 17(7):3050–3060, 2012. https://doi.org/10.1016/j.cnsns.2011.11.017
T.L. Guo and W. Jiang. Impulsive fractional functional differential equations. Comput. Math. Appl., 64(10):3414–3424, 2012. https://doi.org/10.1016/j.camwa.2011.12.054
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo. Theory and applications of fractional differential equations. Elsevier. Science, B.V., Amsterdam, 2006.
K.D. Kucche, A.D. Mali and J.V.C. Sousa. On the nonlinear ψ–Hilfer fractional differential equations. Comp. Appl. Math., 38(73), 2019. https://doi.org/10.1007/s40314-019-0833-5
S. Kumar, A. Kumar and M.O. Zaid. A nonlinear fractional model to describe the population dynamics of two interacting species. Math. Meth. Appl. Sci., 40(11):4134–4148, 2017. https://doi.org/10.1002/mma.4293
Z. Liu and X. Li. Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simulat., 18(6):1362–1373, 2013. https://doi.org/10.1016/j.cnsns.2012.10.010
G.M. Mophou. Existence and uniqueness of mild solutions to impulsive fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications, 72(3-4):1604–1615, 2010. https://doi.org/10.1016/j.na.2009.08.046
S.G. Samko, A.A. Kilbas and O.I. Marichev. Fractional integrals and derivatives, theory and applications. Gordon and Breach, Yverdon, 1993.
A.M. Samoilenko and N.A. Perestyuk. Impulsive differential equations. World Scientific, Singapore, 1995. https://doi.org/10.1142/2892
K. Shah, H. Khalil and R.A. Khan. Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations. Chaos, Solitons & Fractals, 77:240–246, 2015. https://doi.org/10.1016/j.chaos.2015.06.008
J.V.C. Sousa and E. Capelas de Oliveira. On the ψ–Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simulat., 60:72–91, 2018. https://doi.org/10.1016/j.cnsns.2018.01.005
J.V.C. Sousa and E. Capelas de Oliveira. On the Ulam–Hyers–Rassias stability for nonlinear fractional differential equations using the ψ–Hilfer operator. Fixed Point Theory and Appl., 20(3):96, 2018. https://doi.org/10.1007/s11784-018-0587-5
J.V.C. Sousa and E. Capelas de Oliveira. Ulam–Hyers stability of a nonlinear fractional Volterra integro-differential equation. Appl. Math. Lett., 81:50–56, 2018. https://doi.org/10.1016/j.aml.2018.01.016
J.V.C. Sousa, E. Capelas de Oliveira and K.D. Kucche. On the fractional functional differential equation with abstract Volterra operator. Bull. Braz. Math. Soc., New Series, pp. 1–20, 2018.
J.V.C. Sousa, E. Capelas de Oliveira and L.A. Magna. Fractional calculus and the ESR test. AIMS Math, 2(4):692–705, 2017. https://doi.org/10.3934/Math.2017.4.692
J.V.C. Sousa, K.D. Kucche and E. Capelas de Oliveira. Stability of ψ-Hilfer impulsive fractional differential equations. Appl. Math. Lett., 88:73–80, 2019. https://doi.org/10.1016/j.aml.2018.08.013
J.V.C. Sousa and E. Capelas Oliveira. A Gronwall inequality and the Cauchytype problem by means of ψ–Hilfer operator. Diff. Equ. & Appl., 11(1):87–106, 2019. https://doi.org/10.7153/dea-2019-11-02
J. Wang, K. Shah and A. Ali. Existence and Hyers–Ulam stability of fractional nonlinear impulsive switched coupled evolution equations. Math. Meth. Appl. Sci., 41(6):2392–2402, 2018. https://doi.org/10.1002/mma.4748
J. Wang, Y. Zhou and M. Fečkan. On recent developments in the theory of boundary value problems for impulsive fractional differential equations. Comput. Math. Appl., 64(10):3008–3020, 2012. https://doi.org/10.1016/j.camwa.2011.12.064
J. Wang, Y. Zhou and Z. Lin. On a new class of impulsive fractional differential equations. Appl. Math. Comput., 242:649–657, 2014. https://doi.org/10.1016/j.amc.2014.06.002
J.R. Wang, Y. Zhou and Fečkan. Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput. Math. Appl., 64(10):3389– 3405, 2012. https://doi.org/10.1016/j.camwa.2012.02.021
W. Wei, X. Xiang and Y. Peng. Nonlinear impulsive integrodifferential equation of mixed type and optimal controls. Optimization, 55(1-2):141–156, 2006. https://doi.org/10.1080/02331930500530401
Y. Zhou. Basic theory of fractional differential equations. World scientific, 2014. https://doi.org/10.1142/9069