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)
Existence results for fractional order semilinear integro-differential evolution equations with infinite delay. (English) Zbl 1198.45009
This paper concerns some existence results for the mild solutions of fractional order semilinear integro-differential evolution equations with infinite delay. The approach is based on Banach’s contraction principle, a nonlinear alternative of the Leray-Schauder type and Krasnoselskii-Schaefer type fixed point theorems.

MSC:
45J05Integro-ordinary differential equations
26A33Fractional derivatives and integrals (real functions)
45G10Nonsingular nonlinear integral equations
WorldCat.org
Full Text: DOI
References:
[1] Anguraj A., Karthikeyan P., N’Guérékata G.M.: Nonlocal Cauchy problem for some fractional abstract differential equations in Banach spaces. Comm. Math. Anal. 6, 31--35 (2009) · Zbl 1167.34387
[2] Balachandran, K., Park, J.Y.: Nonlocal Cauchy problem for abstract fractional semilinear evolution equations. Nonlinear Anal. (2009). doi: 10.1016/j.na.2009.03.005 · Zbl 1213.34008
[3] Benchohra M., Henderson J., Ntouyas S.K., Ouahaba A.: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338, 1340--1350 (2008) · Zbl 1209.34096 · doi:10.1016/j.jmaa.2007.06.021
[4] Burton T.A., Kirk C.: A fixed point theorem of Krasnoselskii-Schaefer type. Math. Nachr. 189, 23--31 (1998) · Zbl 0896.47042 · doi:10.1002/mana.19981890103
[5] Chang, Y.K., Kavitha, V., Mallika Arjunan, M.: Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order. Nonlinear Anal. (2009). doi: 10.1016/j.na.2009.04.058 · Zbl 1179.45010
[6] Granas A., Dugundji J.: Fixed Point Theory. Springer-Verlag, New York (2003) · Zbl 1025.47002
[7] Hale J., Kato J.: Phase spaces for retarded equations with infinite delay. Funkcial Ekvac. 21, 11--41 (1978) · Zbl 0383.34055
[8] Hernández E.: Existence results for partial neutral functional integro-differential equations with unbounded delay. J. Math. Anal. Appl. 292, 194--210 (2004) · Zbl 1056.45012 · doi:10.1016/j.jmaa.2003.11.052
[9] Hino, Y., Murakami, S., Naito, T.: Functional-differential equations with infinite delay. In: Lecture Notes in Mathematics, vol. 1473. Springer, Berlin (1991) · Zbl 0732.34051
[10] Hu L., Ren Y., Sakthivel R.: Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays. Semigroup Forum 79, 507--514 (2009) · Zbl 1184.45006 · doi:10.1007/s00233-009-9164-y
[11] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. In: North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) · Zbl 1092.45003
[12] Kolmanovskii V., Myshkis A.: Applied Theory of Functional Differential Equations. Kluwer, Dordrecht (1992) · Zbl 0917.34001
[13] Kolmanovskii V., Myshkis A.: Introduction to the Theory and Applications of Functional Differential Equations. Kluwer, Dordrecht (1999) · Zbl 0917.34001
[14] Lakshmikantham V.: Theory of fractional differential equations. Nonlinear Anal. 60, 3337--3343 (2008) · Zbl 1162.34344
[15] Lakshmikantham V., Vatsala A.S.: Basic theory of fractional differential equations. Nonlinear Anal. 69, 2677--2682 (2008) · Zbl 1161.34001 · doi:10.1016/j.na.2007.08.042
[16] Lin W.: Global existence and chaos control of fractional differential equations. J. Math. Anal. Appl. 332, 709--726 (2007) · Zbl 1113.37016 · doi:10.1016/j.jmaa.2006.10.040
[17] Metzler F., Schick W., Kilian H.G., Nonnemacher T.F.: Relaxation in filled polymers: a fractional calculus approach. J. Chem. Phys. 103, 7180--7186 (1995) · doi:10.1063/1.470346
[18] Mophou G.M., N’Guérékata G.M.: Mild solutions for semilinear fractional differential equations. Electron. J. Differ. Equ. 21, 1--9 (2009)
[19] Mophou G.M., N’Guérékata G.M.: Existence of mild solution for some fractional differential equations with nonlocal conditions. Semigroup Forum 79, 315--322 (2009) · Zbl 1180.34006 · doi:10.1007/s00233-008-9117-x
[20] N’Guérékata, G.M.: Existence and uniqueness of an integral solution to some Cauchy problem with nonlocal conditions. In: Differential and Difference Equations and Applications, pp. 843--849. Hindawi Publ. Corp., New York (2006) · Zbl 1147.35329
[21] N’Guérékata G.M.: A Cauchy problem for some fractional abstract differential equation with nonlocal conditions. Nonlinear Anal. 70, 1873--1876 (2009) · Zbl 1166.34320 · doi:10.1016/j.na.2008.02.087
[22] Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1963) · Zbl 0516.47023
[23] Podlubny I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008
[24] Wu J.: Theory and Applications of Partial Functional Differential Equations. Springer, New York (1996) · Zbl 0870.35116
[25] Zhang S.: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ. 36, 1--12 (2006) · Zbl 1134.39008 · doi:10.1155/ADE/2006/90479