Chen, Anping; Chen, Fulai; Deng, Siqing On almost automorphic mild solutions for fractional semilinear initial value problems. (English) Zbl 1189.34079 Comput. Math. Appl. 59, No. 3, 1318-1325 (2010). Summary: This paper investigates almost automorphic mild solutions of the fractional semilinear equation \(D\alpha x(t)=Ax(t)+f(t,x(t))\), \(0<\alpha <1\), considered in a Banach space \(X\), where \(A\) is a linear operator of sectorial type \(\omega <0\). Some sufficient conditions are given for the existence, uniqueness and uniform stability of almost automorphic mild solutions to this semilinear equation. Cited in 17 Documents MSC: 34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations 26A33 Fractional derivatives and integrals 34A08 Fractional ordinary differential equations 34G20 Nonlinear differential equations in abstract spaces 45J05 Integro-ordinary differential equations Keywords:fractional differential equation; almost automorphic function; mild solution PDFBibTeX XMLCite \textit{A. Chen} et al., Comput. Math. Appl. 59, No. 3, 1318--1325 (2010; Zbl 1189.34079) Full Text: DOI References: [1] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010 [2] Diethelm, K.; Ford, N. J., Analysis of fractional differential equations, J. Math. Anal. Appl., 265, 229-248 (2002) · Zbl 1014.34003 [3] Yu, C.; Gao, G., On the solution of nonlinear fractional order differential equation, Nonlinear Anal., 63, e971-e976 (2005) · Zbl 1224.34005 [4] Delbosco, D.; Rodino, L., Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl., 204, 609-625 (1996) · Zbl 0881.34005 [5] Zhang, S., The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl., 252, 804-812 (2000) · Zbl 0972.34004 [6] Daftardar-Gejji, V., Positive solutions of a system of non-autonomous fractional differential equations, J. Math. Anal. Appl., 302, 56-64 (2005) · Zbl 1064.34004 [7] Cuevas, C.; Lizama, C., Almost automorphic mild solutions to a class of fractional differential equations, Appl. Math. Lett., 21, 1315-1319 (2008) · Zbl 1192.34006 [8] Fu, X.; Ezzinbi, K., Existence of solutions for neutral functional differential evolution equations with nonlocal conditions, Nonlinear Anal., 54, 215-227 (2003) · Zbl 1034.34096 [9] Hernández, E., Existence results for partial neutral functional differential equations with nonlocal conditions, Cadenos De Math., 02, 239-250 (2001) [10] Araya, D.; Lizama, C., Almost automorphic mild solutions to fractional differential equations, Nonlinear Anal., 69, 3692-3705 (2008) · Zbl 1166.34033 [11] Jaradat, O. K.; Al-Omari, A.; Momani, S., Existence of the mild solution for fractional semilinear initial value problems, Nonlinear Anal., 69, 3153-3159 (2008) · Zbl 1160.34300 [12] Cuesta, E., Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discrete Contin. Dyn. Syst., Suppl., 277-285 (2007) · Zbl 1163.45306 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.