# zbMATH — the first resource for mathematics

Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations. (English) Zbl 1201.34008
Considered are existence and uniqueness of solutions of the following initial value problems
$D^\alpha x(t)=f(t,D^\beta x(t)),\, 0<t\leq 1;\quad x^{(k)}=n_k,\,k=0,1,\dots,m-1,$ where $$m-1<\alpha<m,$$ $$n-1<\beta<n$$ $$(m,n\in \mathbb{N},\,m-1>n)$$, $$D^\alpha$$ stands for the Caputo derivative of order $$\alpha$$ and $$f$$ is a continuous function defined on $$[0,1]\times \mathbb{R}$$. The proofs are achieved by means of the contraction mapping principle.

##### MSC:
 34A08 Fractional ordinary differential equations and fractional differential inclusions 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
Full Text:
##### References:
 [1] Lin, W., Global existence theory and chaos control of fractional differential equations, J. math. anal. appl., 332, 709-726, (2007) · Zbl 1113.37016 [2] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier Amsterdam · Zbl 1092.45003 [3] Diethelm, K.; Freed, A.D., On the solution of nonlinear fractional order differential equations used in the modelling of viscoplasticity, (), 217-224 [4] Lakshmikantham, V., Theory of fractional functional differential equations, Nonlinear anal., 69, 3337-3343, (2008) · Zbl 1162.34344 [5] Lakshmikantham, V.; Vatsala, A.S., Basic theory of fractional differential equations, Nonlinear anal., 69, 2677-2682, (2008) · Zbl 1161.34001 [6] Lakshmikantham, V.; Vatsala, A.S., General uniqueness and monotone iterative technique for fractional differential equations, Appl. math. lett., 21, 828-834, (2008) · Zbl 1161.34031 [7] Benchohra, M.; Henderson, J.; Ntoyuas, S.K.; Ouahab, A., Existence results for fractional order functional differential equations with infinite delay, J. math. anal. appl., 338, 1340-1350, (2008) · Zbl 1209.34096 [8] Caputo, M., Linear models of dissipation whose $$Q$$ is almost frequency independent (part II), Geophys. J. R. astron. soc., 13, 529-539, (1967) [9] Daftardar-Gejji, V.; Jaffari, H., Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives, J. math. anal. appl., 328, 1026-1033, (2007) · Zbl 1115.34006 [10] Delbosco, D.; Rodino, L., Existence and uniqueness for a nonlinear fractional differential equation, J. math. anal. appl., 204, 609-625, (1996) · Zbl 0881.34005 [11] El-Sayed, W.G.; El-Sayed, A.M.A., On the functional integral equations of mixed type and integro-differential equations of fractional orders, Appl. math. comput., 154, 461-467, (2004) · Zbl 1061.45004 [12] Kilbas, A.A.; Marzan, S.A., Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Differ. uravn., 41, 1, 82-86, (2005), (in Russian) · Zbl 1160.34301 [13] Bonilla, B.; Rivero, M.; Rodriguez-Germa, L.; Trujillo, J.J., Fractional differential equations as alternative models to nonlinear differential equations, Appl. math. comput., 187, 79-88, (2007) · Zbl 1120.34323 [14] Jaradat, O.K.; Al-Omari, A.; Momani, S., Existence of the mild solution for fractional semilinear initial value problem, Nonlinear anal., 69, 3153-3159, (2008) · Zbl 1160.34300 [15] Li, C.; Deng, W., Remarks on fractional derivatives, Appl. math. comput., 187, 777-784, (2007) · Zbl 1125.26009 [16] Kosmatov, N., Integral equations and initial value problems for nonlinear differential equations of fractional order, Nonlinear anal., 70, 2521-2529, (2009) · Zbl 1169.34302
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.