# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A note on the fractional Cauchy problems with nonlocal initial conditions. (English) Zbl 1251.45008
The authors consider the Cauchy problem $$D^\beta_t u(t)+ Au(t)= f(t,u(t))+ \int^t_0 K(t-s) g(s,u(s))\,ds,\quad t\in [0,T],$$ with $u(0)= H(u)$. Here $D^\beta_t$ is a fractional time derivative of order $\beta\in(0, 1)$, $-A$ generates a compact analytic semigroup $T(t)$ on a Banach space $X$, $u\in C([0,T]; X_\alpha)$ ($X_\alpha$ the domain of $A^\alpha$, $0<\alpha< 1$); $f,g: [0,T]\times X_\alpha\to X$, $K\in C[0,T]$ and $H: C([0,T]; X_\alpha)\to X_\alpha$. In the example, $$H(u)= \sum^N_{i=1} \int^\pi_0 K_0(x,y)\cos u(t_i; y)\,dy,\quad 0\le x\le\pi.$$ The authors first give a technical definition of mild solutions of the Cauchy problem, and then -- under conditions too lengthy to be included here -- prove the existence of a mild solution. The proof is by Schauders fixed point theorem.

##### MSC:
 45N05 Abstract integral equations, integral equations in abstract spaces 45J05 Integro-ordinary differential equations 45G10 Nonsingular nonlinear integral equations 26A33 Fractional derivatives and integrals (real functions) 45D05 Volterra integral equations
Full Text:
##### References:
 [1] Agarwal, R. P.; Lakshmikantham, V.; Nieto, J. J.: On the concept of solution for fractional differential equations with uncertainty, Nonlinear analysis 72, 2859-2862 (2010) · Zbl 1188.34005 · doi:10.1016/j.na.2009.11.029 [2] Balachandran, K.; Trujillo, Juan J.: The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces, Nonlinear analysis 72, 4587-4593 (2010) · Zbl 1196.34007 · doi:10.1016/j.na.2010.02.035 [3] Cuevas, C.; De Souza, Julio César: S-asymptotically ${\omega}$-periodic solutions of semilinear fractional integro-differential equations, Applied mathematics letters 22, 865-870 (2009) · Zbl 1176.47035 · doi:10.1016/j.aml.2008.07.013 [4] Cuevas, C.; Lizama, C.: Almost automorphic solutions to a class of semilinear fractional differential equations, Applied mathematics letters 21, 1315-1319 (2008) · Zbl 1192.34006 · doi:10.1016/j.aml.2008.02.001 [5] Diagana, T.; Mophou, Gisèle M.; N’guérékata, G. M.: On the existence of mild solutions to some semilinear fractional integro-differential equations, Electronic journal of qualitative theory of differential equations 58, 1-17 (2010) · Zbl 1211.34094 · emis:journals/EJQTDE/2010/201058.html [6] Hilfer, H.: Applications of fractional calculus in physics, (2000) · Zbl 0998.26002 [7] Kilbas, A. A.; Srivastava, Hari M.; Trujillo, J. Juan: Theory and applications of fractional differential equations, North-holland mathematics studies 204 (2006) · Zbl 1092.45003 [8] Kolmanovskii, V.; Myshkis, A.: Introduction to the theory and applications of functional differential equations, (1999) · Zbl 0917.34001 [9] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and differential equations, (1993) · Zbl 0789.26002 [10] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008 [11] Aizicovici, S.; Mckibben, M.: Existence results for a class of abstract nonlocal Cauchy problems, Nonlinear analysis 39, 649-668 (2000) · Zbl 0954.34055 · doi:10.1016/S0362-546X(98)00227-2 [12] Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, Journal of mathematical analysis and applications 162, 494-505 (1991) · Zbl 0748.34040 · doi:10.1016/0022-247X(91)90164-U [13] Chen, P. J.; Gurtin, M. E.: On a theory of heat conduction involving two temperatures, Zeitschrift für angewandte Mathematik und physik 19, 614-627 (1968) · Zbl 0159.15103 · doi:10.1007/BF01594969 [14] Liang, J.; Van Casteren, J.; Xiao, T. J.: Nonlocal Cauchy problems for semilinear evolution equations, Nonlinear analysis 50, 173-189 (2002) · Zbl 1009.34052 · doi:10.1016/S0362-546X(01)00743-X [15] Liang, J.; Liu, J. H.; Xiao, T. J.: Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Mathematical and computer modelling 49, 798-804 (2009) · Zbl 1173.34048 · doi:10.1016/j.mcm.2008.05.046 [16] Liang, J.; Xiao, T. J.: Semilinear integrodifferential equations with nonlocal initial conditions, Computers mathematics with applications 47, 863-875 (2004) · Zbl 1068.45014 · doi:10.1016/S0898-1221(04)90071-5 [17] Xiao, T. J.; Liang, J.: Existence of classical solutions to nonautonomous nonlocal parabolic problems, Nonlinear analysis 63, e225-e232 (2005) · Zbl 1159.35383 · doi:10.1016/j.na.2005.02.067 [18] Pazy, A.: Semigroups of linear operators and applications to partial differential equations, (1983) · Zbl 0516.47023 [19] Liu, H.; Chang, J. C.: Existence for a class of partial differential equations with nonlocal conditions, Nonlinear analysis 70, 3076-3083 (2009) · Zbl 1170.34346 · doi:10.1016/j.na.2008.04.009 [20] El-Borai, M. M.: Some probability densities and fundamental solutions of fractional evolution equations, Chaos, solitons and fractals 14, 433-440 (2002) · Zbl 1005.34051 · doi:10.1016/S0960-0779(01)00208-9 [21] El-Borai, M. M.: On some stochastic fractional integro-differential equations, Advances in dynamical systems and applications 1, 49-57 (2006) · Zbl 1126.45005