N’Guérékata, Gaston M. A Cauchy problem for some fractional abstract differential equation with non local conditions. (English) Zbl 1166.34320 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 5, 1873-1876 (2009). The author discusses the existence and uniqueness of a solution to the Cauchy problem for the fractional differential equation with non local conditions \[ D^q x(t)=f(t,x(t)),\quad t\in [0,T],\quad x(0)+g(x)=x_0, \]where \(0<q<1\) in a Banach space. Here, the fractional derivative is in the sense of Caputo. Reviewer: Li Changpin (Logan) Cited in 2 ReviewsCited in 150 Documents MSC: 34G20 Nonlinear differential equations in abstract spaces 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 26A33 Fractional derivatives and integrals Keywords:Cauchy problem; fractional abstract differential equation; Caputo derivative PDFBibTeX XMLCite \textit{G. M. N'Guérékata}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 5, 1873--1876 (2009; Zbl 1166.34320) Full Text: DOI References: [1] Aizicovici, S.; McKibben, M., Existence results for a class of abstract nonlocal Cauchy problems, Nonlinear Anal. TMA, 39, 649-668 (2000) · Zbl 0954.34055 [2] Byszewski, L., Theorems about the existence and uniqueness of solutions of a semilinear evolution nonloncal Cauchy problem, J. Math. Anal. Appl., 162, 494-505 (1991) · Zbl 0748.34040 [3] Deng, K., Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl., 179, 630-637 (1993) · Zbl 0798.35076 [4] Ezzinbi, K.; Liu, J., Nondensely defined evolution equations with nonlocal conditions, Math. Comput. Modelling, 36, 1027-1038 (2002) · Zbl 1035.34063 [5] Hernández, E., Existence of solutions to a second order partial differential equation with nonlocal condition, Electron. J. Differential Equations, 2003, 51, 1-10 (2003) [6] V. Lakshmikantham, Theory of fractional differential equations, Nonlinear Anal. TMA, in press (doi:10.1016/j.na.2007.09.025); V. Lakshmikantham, Theory of fractional differential equations, Nonlinear Anal. TMA, in press (doi:10.1016/j.na.2007.09.025) [7] V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. TMA, in press (doi:10.1016/j.na.2007.08.042); V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. TMA, in press (doi:10.1016/j.na.2007.08.042) [8] V. Lakshmikantham, A.S. Vatsala, Theory of fractional differential inequalities and applications, Commun. Appl. Anal. (in press); V. Lakshmikantham, A.S. Vatsala, Theory of fractional differential inequalities and applications, Commun. Appl. Anal. (in press) · Zbl 1159.34006 [9] N’Guérékata, G. M., Existence and uniqueness of an integral solution to some Cauchy problem with nonlocal conditions, (Differential and Difference Equations and Applications, vol. 843-849 (2006), Hindawi Publ. Corp: Hindawi Publ. Corp New York) · Zbl 1169.34331 [10] Podlubny, I., Fractional Differential Equations (1999), San Diego Academic Press · Zbl 0918.34010 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.