Nonlocal problem for a nonlinear system of fractional order impulsive integro-differential equations\\ with maxima

T. K. Yuldashev ; Kh. Kh. Saburov ; T. A. Abduvahobov

Čelâbinskij fiziko-matematičeskij žurnal, Tome 7 (2022) no. 1, pp. 113-122.

Résumé

A nonlocal boundary value problem for a system of ordinary integro-differential equations with impulsive effects, maxima and fractional Gerasimov — Caputo operator is investigated. The boundary value condition is given in the integral form. The method of successive approximations in combination with the method of compressing mapping is used. The existence and uniqueness of a solution of the boundary value problem are proved.

Détail

Keywords: impulsive integro-differential equation, Gerasimov — Caputo operator, nonlocal boundary condition, successive approximations, unique solvability.

Bibliographie

[1] Halanay A., Wexler D., Teoria calitativa a sistemelor cu impulsuri, Editura Academiei Republicii Socialiste Romania, Bucuresti, 1968 | MR | Zbl

[2] Lakshmikantham V., Bainov D.D., Simeonov P.S., Theory of impulsive differential equations, World Scientific, Singapore, 1989 | MR | Zbl

[3] Samoilenko A.M., Perestyk N.A., Impulsive differential equations, World Sci., Singapore, 1995 | MR | Zbl

[4] Benchohra M., Henderson J., Ntouyas S.K., Impulsive differential equations and inclusions. Contemporary mathematics and its application, Hindawi Publishing Corporation, New York, 2006 | MR

[5] Boichuk A.A., Samoilenko A.M., Generalized inverse operators and Fredholm boundary-value problems, 2nd ed., Walter de Gruyter GmbH, Berlin, Boston, 2016 | MR

[6] Yuldashev T.K., “Nonlocal mixed-value problem for a Boussinesq-type integrodifferential equation with degenerate kernel”, Ukrainian Mathematical Journal, 68:8 (2016), 1278–1296 | DOI | MR

[7] Yuldashev T.K., “Determination of the coefficient and boundary regime in boundary value problem for integro-differential equation with degenerate kernel”, Lobachevskii Journal of Mathematics, 38:3 (2017), 547–553 | DOI | MR | Zbl

[8] Yuldashev T.K., “Nonlocal boundary value problem for a nonlinear Fredholm integro-differential equation with degenerate kernel”, Differential Equations, 54:12 (2018), 1646–1653 | DOI | MR | Zbl

[9] Assanova A., “An integral-boundary value problem for a partial differential equation of second order”, Turkish Journal of Mathematics, 43:4 (2019), 1967–1978 | DOI | MR | Zbl

[10] Yuldashev T.K., “Spectral features of the solving of a Fredholm homogeneous integro-differential equation with integral conditions and reflecting deviation”, Lobachevskii Journal of Mathematics, 40:12 (2019), 2116–2123 | DOI | MR | Zbl

[11] Yuldashev T.K., “On inverse boundary value problem for a Fredholm integro-differential equation with degenerate kernel and spectral parameter”, Lobachevskii Journal of Mathematics, 40:2 (2019), 230–239 | DOI | MR | Zbl

[12] Yuldashev T.K., “On the solvability of a boundary value problem for the ordinary Fredholm integrodifferential equation with a degenerate kernel”, Computational Mathematics and Mathematical Physics, 59:2 (2019), 241–252 | DOI | MR | Zbl

[13] Assanova A.T., Imanchiyev A.E., Kadirbayeva Zh.M., “A nonlocal problem for loaded partial differential equations of fourth order”, Bulletin of the Karaganda University. Mathematics, 97:1 (2020), 6–16 | DOI | MR

[14] Assanova A.T., Tokmurzin Z.S., “A nonlocal multipoint problem for a system of fourth-order partial differential equations”, Eurasian Mathematical Journal, 11:3 (2020), 8–20 | DOI | MR | Zbl

[15] Yuldashev T.K., “Nonlinear optimal control of thermal processes in a nonlinear inverse problem”, Lobachevskii Journal of Mathematics, 41:1 (2020), 124–136 | DOI | MR | Zbl

[16] Assanova A.T., “On the solvability of nonlocal problem for the system of Sobolev-type differential equations with integral condition”, Georgian Mathematical Journal, 28:1 (2021), 49–57 | DOI | MR | Zbl

[17] Minglibayeva A.B., Assanova A.T., “An existence of an isolated solution to nonlinear two-point boundary value problem with parameter”, Lobachevskii Journal of Mathematics, 42:3 (2021), 587–597 | DOI | MR | Zbl

[18] Yuldashev T.K., “On a Volterra type fractional integro-differential equation with degenerate kernel”, AIP Conference Proceedings, 2365:020016 (2021), 12 pp.

[19] Anguraj A., Arjunan M.M., “Existence and uniqueness of mild and classical solutions of impulsive evolution equations”, Electonic Journal of Differential Equations, 111 (2005), 1–8 | MR

[20] Bin L., Xinzhi L., Xiaoxin L., “Robust global exponential stability of uncertain impulsive systems”, Acta Mathematika Scientia, 25:1 (2005), 161–169 | DOI | MR | Zbl

[21] Li M., Han M., “Existence for neutral impulsive functional differential equations with nonlocal conditions”, Indagationes Mathematicae, 20:3 (2009), 435–451 | DOI | MR | Zbl

[22] Ji Sh., Wen Sh., “Nonlocal Cauchy problem for impulsive differential equations in Banach spaces”, International Journal of Nonlinear Science, 10:1 (2010), 88–95 | MR | Zbl

[23] Sharifov Y.A., “Conditions optimality in problems control with systems impulsive differential equations with nonlocal boundary conditions”, Ukrainain Mathematical Journal, 64:6 (2012), 836–847 | MR | Zbl

[24] Ashyralyev A., Sharifov Y.A., “Existence and uniqueness of solutions for nonlinear impulsive differential equations with two-point and integral boundary conditions”, Advances in Difference Equations, 2013 (2013), 173 | DOI | MR | Zbl

[25] Gao Zh., Yang L., Liu G., “Existence and uniqueness of solution of impulsive fractional integro-differential equations with nonlocal conditions”, Applied Mathematics, 4 (2013), 859–863 | DOI | MR

[26] Sharifov Ya.A., “Optimal control for systems with impulsive actions under nonlocal boundary conditions”, Russian Mathematics, 57:2 (2013), 65–72 | DOI | MR | Zbl

[27] Mardanov M.J., Sharifov Ya.A., Molaei Habib H., “Existence and uniqueness of solutions for first-order nonlinear differential equations with two-point and integral boundary conditions”, Electronic Journal of Differential Equations, 2014, no. 259, 1–8 | MR

[28] Yuldashev T.K., Ovsyanikov S.M., “An approximate solution of a system of nonlinear integral equations with a delay argument and an approximate calculation of the quality functional”, Journal of the Middle Volga Mathematical Society, 17:2 (2015), 85–95 (In Russ.) | Zbl

[29] Yuldashev T.K., “Limit value problem for a system of integro-differential equations with two mixed maxima”, Bulletin of Samara State Technical University. Series: Physical and Mathematical Sciences, 1:16 (2008), 15–22 (In Russ.) | MR | Zbl

[30] Samko S.G., Kilbas A.A., Marichev O.I., Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993 | MR | Zbl

[31] Mainardi F., “Fractional calculus: some basic problems in continuum and statistical mechanics”, Fractals and Fractional Calculus in Continuum Mechanics, eds. Carpinteri A., Mainardi F., Springer, Wien, 1997 | MR

[32] Area I., Batarfi H., Losada J., Nieto J.J., Shammakh W.M., Torres A., “On a fractional order Ebola epidemic model”, Advances in Differerence Equations, 2015, no. 278 | MR

[33] Hussain A., Baleanu D., Adeel M., “Existence of solution and stability for the fractional order novel coronavirus (nCoV-2019) model”, Advances in Differerence Equations, 2020, no. 384 | MR

[34] Ullah S., Khan M.A., Farooq M., Hammouch Z., Baleanu D., “A fractional model for the dynamics of tuberculosis infection using Caputo — Fabrizio derivative”, Discrete and Continuous Dynamical Systems. Series S, 13:3 (2020), 975–993 | DOI | MR | Zbl

[35] Handbook of Fractional Calculus with Applications, v. 1–8, eds. Tenreiro Machado J.A., Walter de Gruyter GmbH,, Berlin, Boston, 2019

[36] Kumar D., Baleanu D., “Editorial: Fractional calculus and its applications in physics”, Frontiers in Physics, 7:6 (2019)

[37] Sun H., Chang A., Zhang Y., Chen W., “A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications”, Fractional Calculus and Applied Analysis, 22:1 (2019), 27–59 | DOI | MR | Zbl

[38] Patnaik S., Hollkamp J.P., Semperlotti F., “Applications of variable-order fractional operators: a review”, Proceedings of the Royal Society A., 476:2234 (2020), 1–32 | MR

[39] Yuldashev T.K., Kadirkulov B.J., “Boundary value problem for weak nonlinear partial differential equations of mixed type with fractional Hilfer operator”, Axioms, 9:2 (2020), 68 | DOI | MR

[40] Yuldashev T.K., Kadirkulov B.J., “Nonlocal problem for a mixed type fourth-order differential equation with Hilfer fractional operator”, Ural Mathematical Journal, 6:1 (2020), 153–167 | DOI | MR | Zbl

[41] Yuldashev T.K., Karimov E.T., “Inverse problem for a mixed type integro-differential equation with fractional order Caputo operators and spectral parameters”, Axioms, 9:4 (2020), 121 | DOI | MR

[42] Fedorov V.E., Plekhanova M.V., Izhberdeeva E.M., “Initial value problems of linear equations with the Dzhrbashyan — Nersesyan derivative in Banach spaces”, Symmetry, 13:1058 (2021) | Zbl

[43] Volkova A.R., Izhberdeeva E.M., Fedorov V.E., “Initial value problem for linear equations with composition of fractional derivatives”, Chelyabinsk Physical and Mathematical Journal, 6:3 (2021), 269–277 | MR | Zbl

[44] Yuldashev T.K., Kadirkulov B.J., “Inverse boundary value problem for a fractional differential equations of mixed type with integral redefinition conditions”, Lobachevskii Journal of Mathematics, 42:3 (2021), 649–662 | DOI | MR | Zbl