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Existence and uniqueness of mild solutions to impulsive fractional differential equations. (English) Zbl 1187.34108
Summary: Our aim in this paper is to study the existence and the uniqueness of the solution for the fractional semilinear differential equation:
\[ \begin{cases} D^\alpha_t x(t)=Ax(t)+f(t,x(t)),\quad t\in I=[0,T],\;t\neq t_k,\\ x(0)=x_0\in X,\\ \Delta x|_{t=t_k}=l_k(x(t^-_k)),\quad k=1,\dots,m,\end{cases}\tag{1} \]
where \(0<\alpha<1\), the operator \(A:D(A)\subset X\to X\) is a generator of \({\mathcal C}_0\)-semigroup \((T(t))_{t\geq 0}\) on a Banach space \(\mathbb X\), \(D^\alpha_t\) is the Caputo fractional derivative, \(f:I\times \mathbb X\to \mathbb X\) is a given continuous function \(I_k:\mathbb X\to \mathbb X\), \(0=t_0<t_1<\cdots<t_m<t_{m+1}=T\). \(\Delta x|_{t=t_k}=x(t^+_k)-x(t^-_k)\), \(x(t^+_k)=\lim_{h\to0^+}x(t_k+h)\) and \(x(t^-_k)=\lim_{h\to 0}-x(t_k+h)\) represent respectively the right and left limits of \(x(t)\) at \(t=t_k\).

MSC:
34K30 Functional-differential equations in abstract spaces
34K05 General theory of functional-differential equations
34K37 Functional-differential equations with fractional derivatives
34K45 Functional-differential equations with impulses
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[1] Benchohra, M.; Henderson, J.; Ntouyas, S.K., Impulsive differential equations and inclusions, vol. 2, (2006), Hindawi Publishing Corporation New York · Zbl 1130.34003
[2] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differntial equations, (1989), World Scientific Singapore · Zbl 0719.34002
[3] Samoilenko, A.M.; Perestyuk, N.A., Impulsive differential equations, (1995), World Scientific Singapore · Zbl 0837.34003
[4] Lakshmikantham, V., Theory of fractional differential equations, Nonlinear analysis. theory methods and applications, 60, 10, 3337-3343, (2008) · Zbl 1162.34344
[5] Lakshmikantham, V.; Vatsala, A.S., Basic theory of fractional differential equations, Nonlinear analysis. theory methods and applications, 69, 8, 2677-2682, (2008) · Zbl 1161.34001
[6] Lakshmikantham, V.; Vatsala, A.S., Theory of fractional differential inequalities and applications, Communications in applied analysis, 11, 3&4, 395-402, (2007) · Zbl 1159.34006
[7] G.M. Mophou, O. Nakoulima, G.M. N’Guérékata, Existence results for some fractional differential equations with nonlocal conditions, Nonlinear Studies (To appear) · Zbl 1204.34010
[8] Mophou, G.M.; N’Guérékata, G.M., Existence of mild solution for some fractional differential equations with nonlocal conditions, Semigroup forum, 79, 2, 322-335, (2009) · Zbl 1180.34006
[9] N’Guérékata, G.M., A Cauchy problem for some fractional abstract differential equation with nonlocal conditions, Nonlinear analysis. theory methods and applications, 70, 5, 1873-1876, (2009) · Zbl 1166.34320
[10] Podlubny, I., Fractional differential equations, (1999), San Diego Academic Press · Zbl 0918.34010
[11] Lin, Wei, Global existence and chaos control of fractional differential equations, Journal of mathematical analysis and applications, 332, 709-726, (2007) · Zbl 1113.37016
[12] Zhang, S., Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electronic journal of differential equations, 2006, 36, 1-12, (2006)
[13] Agarwal, R.P.; Benchohra, M.; Slimani, B.A., Existence results for differential equations with fractional order and impulses, Memors on differential equations and mathemaical physics, 44, 1-21, (2008) · Zbl 1178.26006
[14] Benchohra, M.; Slimani, B.A., Existence and uniqueness of solutions to impulsive fractional differential equations, Electronic journal of differential equations, 2009, 10, 1-11, (2009) · Zbl 1178.34004
[15] Ahmad, Bashir; Sivasundaram, S., Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear analysis: hybrid systems, 3, 3, 251-258, (2009) · Zbl 1193.34056
[16] Jaradat, O.K.; Al-Omari, A.; Momani, S., Existence of the mild solution for fractional semilinear initial calue problems, Nonlinear analysis, 69, 3153-3159, (2008) · Zbl 1160.34300
[17] Brezis, H., Analyse fonctionnelle. théorie et applications, (1983), Masson Paris · Zbl 0511.46001
[18] Deng, K., Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, Journal of mathematical analysis and applications, 179, 630-637, (1993) · Zbl 0798.35076
[19] Aizicovici, S.; McKibben, M., Existence results for a class of abstract nonlocal Cauchy problems, Nonlinear analysis. theory methods and applications, 39, 649-668, (2000) · Zbl 0954.34055
[20] Anguraj, A.; Karthikeyan, P.; N’Guérékata, G.M., Nonlocal Cauchy problem for some fractional abstract differential equations in Banach spaces, Communications in mathematical analysis, 6, 1, 31-35, (2009) · Zbl 1167.34387
[21] Byszewski, L., Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. math. anal. appl., 162, 494-505, (1991) · Zbl 0748.34040
[22] Liu, Hsiang; Chang, Jung-Chan, Existence for a class of partial differential equations with nonlocal conditions, Nonlinear analysis, TMA, 70, 9, 3076-3083, (2009) · Zbl 1170.34346
[23] Fan, Z., Existence of nondensely defined evolution equations with nonlocal conditions, Nonlinear analysis, 70, 11, 3829-3836, (2009) · Zbl 1170.34345
[24] Ezzinbi, K.; Liu, J., Nondensely defined evolution equations with nonlocal conditions, Mathematical and computer modelling, 36, 1027-1038, (2002) · Zbl 1035.34063
[25] Hernández, E., Existence of solutions to a second order partial differential equation with nonlocal condition, Electronic journal of differential equations, 2003, 51, 1-10, (2003)
[26] N’Guérékata, G.M., Existence and uniqueness of an integral solution to some Cauchy problem with nonlocal conditions, (), 843-849 · Zbl 1147.35329
[27] Krasnoselskii, M.A., Topological methods in the theory of nonlinear integral equations, (1964), Pergamon Press New York
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