×

S-asymptotically \(\omega\)-periodic mild solutions of neutral fractional differential equations with finite delay in Banach space. (English) Zbl 1368.34096

From the introduction and summary: In this paper, we study the existence of S-asymptotically \(\omega\)-periodic solutions for the following fractional differential equation \[ \begin{aligned} ^cD^q_t(u(t)- g(t,u_t)) &= A(u(t)- g(t, u_t))+ f(t,u_t),\quad t\geq 0,\\ u(t) &= \phi(t),\quad t\in [-\delta, 0],\end{aligned} \] where \(\delta>0\), \(0<q<1\). The fractional derivative is understood here in the Caputo sense. \(A:D(A)\subset X\to X\) is the generator of an analytic semigroup on a Banch space \(X\).
Moreover, we apply the above results to study the S-asymptotically \(\omega\)-periodicity for the fractional-order diffusion equations and the fractional-order autonomous neural networks with delay.

MSC:

34K37 Functional-differential equations with fractional derivatives
34K30 Functional-differential equations in abstract spaces
34K13 Periodic solutions to functional-differential equations
34K40 Neutral functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Tarasov, V.E.: Fractional dynamics: application of fractional calculus to dynamics of particles. Springer, Fields and media. HEP (2010) · Zbl 1214.81004 · doi:10.1007/978-3-642-14003-7
[2] Garrapa, R., Popolizio, M.: On accurate product integration rules for linear fractional differential equations. J. Comput. Appl. Math. 235, 1085-1097 (2011) · Zbl 1206.65176 · doi:10.1016/j.cam.2010.07.008
[3] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations, vol. 204. Elsevier Science B.V, Amsterdam (2006) · Zbl 1092.45003 · doi:10.1016/S0304-0208(06)80001-0
[4] Mainardi, F.; Paradisi, P.; Gorenflo, R.; Kertesz, J. (ed.); Kondor, I. (ed.), Probability distributions generated by fractional diffusion equations (2000), Dordrecht
[5] Zhou, Y., Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal. 11(5), 4465-4475 (2010) · Zbl 1260.34017 · doi:10.1016/j.nonrwa.2010.05.029
[6] El-Borai, M.M.: Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fractals 14, 433-440 (2002) · Zbl 1005.34051 · doi:10.1016/S0960-0779(01)00208-9
[7] Henríquez, H.R., Pierre, M., Taboas, P.: On S-asymptotically \[\omega\] ω-periodic function on Banach spaces and applications. J. Math. Anal. Appl 343, 1119-1130 (2008) · Zbl 1146.43004 · doi:10.1016/j.jmaa.2008.02.023
[8] Dimbour, W., Mophou, G., N’Guérékata, G. M.: S-asymptotically periodic solutions for partial differential equations with finite delay. Electr. J. Differ. Equ., 117, 1-12 (2011) · Zbl 1231.34136
[9] Dimbour, W., N’Guérékata, G.M.: S-asymptotically \[\omega\] ω-periodic solutions to some classes of partial evolution equations. Appl. Math. Comput. 218, 7622-7628 (2012) · Zbl 1251.35182
[10] Wang, J., Fĕckan, M., Zhou, Y.: Nonexistence of periodic solutions and asymptotically periodic solutions for fractional differential equations. Commun. Nonlinear Sci. Numer. Simulat. 18, 246-256 (2013) · Zbl 1253.35204 · doi:10.1016/j.cnsns.2012.07.004
[11] dos Santos, J.P.C., Cuevas, C.: Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations. Appl. Math. Lett. 23, 960-965 (2010) · Zbl 1198.45014 · doi:10.1016/j.aml.2010.04.016
[12] Cuevas, C., de Souza, J.C.: S-asymptotically \[\omega\] ω-periodic solutions of semilinear fractional integro-differential equations. Appl. Math. Lett. 22, 865-870 (2009) · Zbl 1176.47035 · doi:10.1016/j.aml.2008.07.013
[13] Martin, R.H.: Nonlinear operators and differential equations in Banach spaces. Robert E. Krieger Publ. Co, Florida (1987)
[14] Hernández, E.M., Tanaka, S.M.: Global solutions for abstract functional differential equations with nonlocal conditions. Electr. J. Qualti. 50,1-8 (2009) · Zbl 1196.34102
[15] Lunardi, A.: Analytic semigroup and optimal regularity in parabolic problems, in PNLDE vol. 16, Birkhauser Verlag Basel (1995) · Zbl 0816.35001
[16] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York, NY, USA (1983) · Zbl 0516.47023
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.