×

Existence results for a coupled system of fractional integro-differential equations with time-dependent delay. (English) Zbl 1376.34065

Summary: Our purpose in this manuscript is concerned with the existence of solutions for a coupled system of fractional integro-differential equations with time-dependent delay. We use the Krasnoselskii’s fixed-point theorem along with fractional calculus. This work completes recent results on the subject and an example of partial integro-differential equation is presented.

MSC:

34K37 Functional-differential equations with fractional derivatives
34A08 Fractional ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abazari, R., Kilicman, A.: Application of differential transform method on nonlinear integro-differential equations with proportional delay. Neural Comput. Appl. 24, 391-397 (2014) · doi:10.1007/s00521-012-1235-4
[2] Abbas, S., Kavitha, V., Murugesu, R.: Stepanov-like weighted pseudo almost automorphic solutions to fractional order abstract integro-differential equations. Proc. Indian Acad. Sci. (Math. Sci.) 125, 323-351 (2015) · Zbl 1327.35391
[3] Agarwal, P., Rogosin S.V., Trujillo, J.J.: Certain fractional integral operators and the generalized multi-index Mittag-Leffler functions. Proc. Indian Acad. Sci. (Math. Sci.) 125, 291-306 (2015) · Zbl 1323.33020
[4] Agarwal, R.P., Andrade, D.B., Siracusa, G.: On fractional integro-differential equations with state-dependent delay. Comput. Math. Appl. 62, 1143-1149 (2011) · Zbl 1228.35262 · doi:10.1016/j.camwa.2011.02.033
[5] Ahmad, B., Ntouyasb, S.K., Alsaedi, A.: On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions. Chaos Solitons Fractals 83, 234-241 (2016) · Zbl 1355.34012 · doi:10.1016/j.chaos.2015.12.014
[6] Aissani, K., Benchohra, M., Abada, N., Agarwal, R.P.: Existence results for nondensely defined impulsive semilinear functional differential equations with state-dependent delay. Asian-Eur. J. Math. 4, 449-468 (2008) · Zbl 1179.34070
[7] Ali, B., Abbas, M.: Existence and stability of fixed point set of Suzuki-type contractive multivalued operators in b-metric spaces with applications in delay differential equations. J. Fixed Point Theory Appl. (2017). doi:10.1007/s11784-017-0426-0 · Zbl 1490.54030 · doi:10.1007/s11784-017-0426-0
[8] Benchohra, M., Litimein, S., Trujillo, J.J., Velasco, M.P.: Abstract fractional integro-differential equations with state-dependent delay. Int. J. Evol. Equ. 6, 25-38 (2012) · Zbl 1263.26013
[9] Blanco-Cocom, L., Estrella, A.G., Avila-Vales, E.: Solving delay differential systems with history functions by the Adomian decomposition method. Appl. Math. Comput. 218, 5994-6011 (2012) · Zbl 1246.65101 · doi:10.1016/j.amc.2011.11.082
[10] Carvalho dos Santos, J.P., Mallika Arjunan, M., Cuevas, C.: Existence results for fractional neutral integro-differential equations with state-dependent delay. Comput. Math. Appl. 62, 1275-1283 (2011) · Zbl 1228.45014 · doi:10.1016/j.camwa.2011.03.048
[11] Chakraverty, S., Tapaswini, S., Behera, D.: Uncertain Fractional Fornberg-Whitham Equations, in: Fuzzy Arbitrary Order System: Fuzzy Fractional Differential Equations and Applications. Wiley, Hoboken (2016) · Zbl 1381.34001
[12] Coussot, C.: Fractional derivative models and their use in the characterization of hydropolymer and invivo breast tissue viscoelasticity. Master Thesis, University of Illiniois at Urbana-Champain (2008) · Zbl 1231.65179
[13] Dabas, J., Chauhan, A.: Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential equation with infinite delay. Math. Comput. Modelling 57, 754-763 (2013) · Zbl 1305.34132 · doi:10.1016/j.mcm.2012.09.001
[14] Daftardar-Gejji, V., Bhalekar, S., Gade, P.: Dynamics of fractional-ordered Chen system with delay. Pramana J. Phys. 79, 61-69 (2012) · doi:10.1007/s12043-012-0291-8
[15] Deng, J., Qu, H.: New uniqueness results of solutions for fractional differential equations with infinite delay. Comput. Math. Appl. 60, 2253-2259 (2010) · Zbl 1205.34098 · doi:10.1016/j.camwa.2010.08.015
[16] Diethelm, K.: The Analysis of Differential Equations. Springer-Verlag, Berlin, Heidelberg (2010) · Zbl 1215.34001
[17] Ding, Y., Ye, H.: A fractional-order differential equation model of HIV infection of CD \[4^+4\]+ T-cells. Math. Comput. Modelling 50, 386-392 (2009) · Zbl 1185.34005 · doi:10.1016/j.mcm.2009.04.019
[18] Faria, T.: Periodic solutions for a non-monotone family of delayed differential equations with applications to Nicholson systems. J. Differ. Equ. 263, 509-533 (2017) · Zbl 1370.34125 · doi:10.1016/j.jde.2017.02.042
[19] Garg, M., Rao, A.: Fractional extensions of some boundary value problems in oil strata. Proc. Indian Acad. Sci. (Math. Sci.) 117, 267-281 (2007) · Zbl 1210.35138
[20] Ge, Z.M., Jhuang, W.R.: Chaos, control and synchronization of a fractional order rotational mechanical system with a centrifugal governor. Chaos Solitons Fractals 33, 270-289 (2007) · Zbl 1152.34355 · doi:10.1016/j.chaos.2005.12.040
[21] Hale, J., Kato, J.: Phase space for retarded equations with infinite delay. Funkcialaj Ekvacioj 21, 11-41 (1978) · Zbl 0383.34055
[22] Hernandez, E., Mckibben, M.A.: On state-dependent delay partial neutral functional-differential equations. Appl. Math. Comput. 186, 294-301 (2007) · Zbl 1119.35106 · doi:10.1016/j.amc.2006.07.103
[23] Hernandez, E., Mckibben, M.A., Henriquez, H.R.: Existence results for partial neutral functional differential equations with state-dependent delay. Math. Comput. Modelling 49, 1260-1267 (2009) · Zbl 1165.34420 · doi:10.1016/j.mcm.2008.07.011
[24] Hernandez, E., Pierri, M., Goncalves, G.: Existence results for an impulsive abstract partial differential equation with state-dependent delay. Comput. Math. Appl. 52, 411-420 (2006) · Zbl 1153.35396 · doi:10.1016/j.camwa.2006.03.022
[25] Hernandez, E., Prokopczyk, A., Ladeira, L.: A note on partial functional differential equations with state-dependent delay. Nonlinear Anal. Real World Appl. 7, 510-519 (2006) · Zbl 1109.34060 · doi:10.1016/j.nonrwa.2005.03.014
[26] Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) · Zbl 0998.26002 · doi:10.1142/3779
[27] Jafari, H., Khaliquea, C.M., Nazari, M.: An algorithm for the numerical solution of nonlinear fractional-order Van der Pol oscillator equation. Math. Comput. Modelling 55, 1782-1786 (2012) · Zbl 1255.65142 · doi:10.1016/j.mcm.2011.11.029
[28] Jafari, H., Khaliquea, C.M., Nazari, M.: Application of the Laplace decomposition method for solving linear and nonlinear fractional diffusion-wave equations. Appl. Math. Lett. 24, 1799-1805 (2011) · Zbl 1231.65179 · doi:10.1016/j.aml.2011.04.037
[29] Jafari, H., Nazari, M., Baleanu, D., Khaliquea, C.M.: A new approach for solving a system of fractional partial differential equations. Comput. Math. Appl. 66, 838-843 (2013) · Zbl 1381.35221 · doi:10.1016/j.camwa.2012.11.014
[30] Jiao, Z., Chen, Y.Q., Podlubny, I.: Distributed-Order Dynamic Systems. Springer, New York (2012) · Zbl 1401.93005 · doi:10.1007/978-1-4471-2852-6
[31] Kalamani, P., Baleanu, D., Selvarasu S., Mallika-Arjunan, M.: On existence results for impulsive fractional neutral stochastic integro-differential equations with nonlocal and state-dependent delay conditions. Adv. Differ. Equ. (2016). doi:10.1186/s13662-016-0885-4 · Zbl 1342.34107
[32] Killbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. In: North-Holland Mathematics Studies. Elsevier Science, Amsterdam (2006) · Zbl 1092.45003
[33] Krawcewicz, W., Yu J., Xiao, H.: Multiplicity of periodic solutions to symmetric delay differential equations. J. Fixed Point Theory Appl. 13, 103-141 (2013) · Zbl 1288.47052
[34] Maleknejad, K., Nouri, K., Torkzadeh, L.: Operational matrix of fractional integration based on the shifted second kind Chebyshev polynomials for solving fractional differential equations. Mediterr. J. Math. 13, 1377-1390 (2016) · Zbl 1350.26015
[35] Maleknejad, K., Nouri, K., Torkzadeh, L.: Study on multi-order fractional differential equations via operational matrix of hybrid basis functions. Bull. Iranian Math. Soc. 43, 307-318 (2017) · Zbl 1409.34013
[36] Mallika-Arjunan, M., Kavitha, V.: Existence results for impulsive neutral functional differential equations with state-dependent delay. Electron. J. Qual. Theory Differ. Equ. 26, 1-13 (2009) · Zbl 1183.34121 · doi:10.14232/ejqtde.2009.1.26
[37] Nouri, K., Elahi-Mehr, S., Torkzadeh, L.: Investigation of the behavior of the fractional Bagley-Torvik and Basset equations via numerical inverse laplace transform. Romanian Rep. Phys. 68, 503-514 (2016)
[38] Ostoja-Starzewski, M.: Towards thermoelasticity of fractal media. J. Thermal Stresses 30, 889-896 (2007) · Zbl 1126.74004 · doi:10.1080/01495730701495618
[39] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008
[40] Povstenko, Y.Z.: Fractional Thermoelasticity. Springer, New York (2015) · Zbl 1316.74001 · doi:10.1007/978-3-319-15335-3
[41] Ren, Y., Qin, Y., Sakthivel, R.: Existence results for fractional order semilinear integro-differential evolution equations with infinite delay. Integral Equ. Oper. Theory 67, 33-49 (2010) · Zbl 1198.45009 · doi:10.1007/s00020-010-1767-x
[42] Shah, K., Khalil, H., Khan, R.A.: 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) · Zbl 1353.34028 · doi:10.1016/j.chaos.2015.06.008
[43] Su, X.: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 22, 64-69 (2009) · Zbl 1163.34321 · doi:10.1016/j.aml.2008.03.001
[44] Suganya, S., Arjunan, M.M., Trujillo, J.J.: Existence results for an impulsive fractional integro-differential equation with state-dependent delay. Appl. Math. Comput. 266, 54-69 (2015) · Zbl 1410.34242
[45] Suganya, S., Kalamani, P., Arjunan, M.M.: Existence of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces. Comput. Math. Appl. (in Press) · Zbl 1348.34130
[46] Yang, H., Agarwal, R.P., Nashine, H.K., Liang, Y.: Fixed point theorems in partially ordered Banach spaces with applications to nonlinear fractional evolution equations. J. Fixed Point Theory Appl. (2016). doi:10.1007/s11784-016-0316-x · Zbl 1398.35281
[47] Yu, Y., Perdikaris, P., Karniadakis, G.E.: Fractional modelling of viscoelasticity in 3D cerebral arteries and aneurysms. J. Comput. Phys. 323, 219-242 (2016) · Zbl 1415.74049 · doi:10.1016/j.jcp.2016.06.038
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.