×

zbMATH — the first resource for mathematics

The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces. (English) Zbl 1196.34007
Summary: We prove the existence of solutions of certain classes of nonlinear fractional integrodifferential equations in Banach spaces. Further, Cauchy problems with nonlocal initial conditions are discussed for the aforementioned fractional integrodifferential equations. At the end, an example is presented.

MSC:
34A08 Fractional ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bonilla, B.; Rivero, M.; Rodríguez-Germa, L.; Trujillo, J.J., Fractional differential equations as alternative models to nonlinear differential equations, Applied mathematics and computation, 187, 79-88, (2007) · Zbl 1120.34323
[2] Jumarie, G., An approach via fractional analysis to nonlinearity induced by coarse-graining in space, Nonlinear analysis: real world applications, 11, 535-546, (2010) · Zbl 1195.37054
[3] Kosmatov, N., Integral equations and initial value problems for nonlinear differential equations of fractional order, Nonlinear analysis, 70, 2521-2529, (2009) · Zbl 1169.34302
[4] Luchko, Y.F.; Rivero, M.; Trujillo, J.J.; Velasco, M.P., Fractional models, non-locality and complex systems, Computers and mathematics with applications, 59, 1048-1056, (2010) · Zbl 1189.37095
[5] Ahmad, B.; Nieto, J.J., Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions, Boundary value problems, 2009, (2009), 11 pages. Article ID 708576 · Zbl 1167.45003
[6] 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
[7] Delbosco, D.; Rodino, L., Existence and uniqueness for a fractional differential equation, Journal of mathematical analysis and applications, 204, 609-625, (1996) · Zbl 0881.34005
[8] Lakshmikantham, V., Theory of fractional functional differential equations, Nonlinear analysis, 69, 3337-3343, (2008) · Zbl 1162.34344
[9] Lakshmikantham, V.; Vatsala, A.S., Basic theory of fractional differential equations, Nonlinear analysis, 69, 2677-2682, (2008) · Zbl 1161.34001
[10] Lakshmikantham, V.; Vatsala, A.S., General uniqueness and monotone iterative technique for fractional differential equations, Applied mathematics letters, 21, 828-834, (2008) · Zbl 1161.34031
[11] Lakshmikantham, V.; Vasundhara Devi, J., Theory of fractional differential equations in Banach spaces, European journal of pure and applied mathematics, 1, 38-45, (2008) · Zbl 1146.34042
[12] Wei, Z.; Che, J., Initial value problems for fractional differential equations involving riemann – liouville sequential fractional derivative, Journal of mathematical analysis and applications, 367, 260-272, (2010) · Zbl 1191.34008
[13] El-Borai, M.M., Semigroups and some nonlinear fractional differential equations, Applied mathematics and computation, 149, 823-831, (2004) · Zbl 1046.34079
[14] Rashid, M.H.M.; El-Qaderi, Y., Semilinear fractional integrodifferential equations with compact semigroup, Nonlinear analysis, 71, 6276-6282, (2009) · Zbl 1184.45007
[15] El-Sayeed, M.A.A., Fractional order diffusion wave equation, International journal of theoretical physics, 35, 311-322, (1996) · Zbl 0846.35001
[16] 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
[17] Balachandran, K.; Chandrasekaran, M., Nonlocal Cauchy problem for semilinear integrodifferential equations with deviating argument, Proceedings of the Edinburgh mathematical society, 44, 63-70, (2001) · Zbl 0985.45010
[18] Byszewski, L.; Acka, H., Existence of solutions of a semilinear functional differential evolution nonlocal problems, Nonlinear analysis, 34, 65-72, (1998)
[19] Liang, J.; Liu, J.H.; Xiao, T.J., Nonlocal Cauchy problems governed by compact operator families, Nonlinear analysis, 57, 183-189, (2004) · Zbl 1083.34045
[20] Balachandran, K.; Park, D.G., Existence of solutions of quasilinear integrodifferential evolution equations in Banach spaces, Bulletin of the Korean mathematical society, 46, 691-700, (2009) · Zbl 1188.34076
[21] Balachandran, K.; Paul Samuel, F., Existence of solutions for quasilinear delay integrodifferential equations with nonlocal condition, Electronic journal of differential equations, 2009, 6, 1-7, (2009) · Zbl 1173.34353
[22] Balachandran, K.; Uchiyama, K., Existence of solutions of quasilinear integrodifferential equations with nonlocal condition, Tokyo journal of mathematics, 23, 203-210, (2000) · Zbl 0976.45011
[23] N’Guerekata, G.M., A Cauchy problem for some fractional abstract differential equation with nonlocal condition, Nonlinear analysis, 70, 1873-1876, (2009) · Zbl 1166.34320
[24] Balachandran, K.; Park, J.Y., Nonlocal Cauchy problem for abstract fractional semilinear evolution equations, Nonlinear analysis, 71, 4471-4475, (2009) · Zbl 1213.34008
[25] Benchohra, M.; Seba, D., Impulsive fractional differential equations in Banach spaces, Electronic journal of qualitative theory of differential equations, 8, 1-14, (2009), Spec. Ed. I · Zbl 1189.26005
[26] Balachandran, K.; Kiruthika, S., Existence of solutions of abstract fractional impulsive semilinear evolution equations, Electronic journal of qualitative theory of differential equations, 4, 1-12, (2010) · Zbl 1201.34091
[27] Chang, Y.K.; Nieto, J.J., Existence of solutions for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators, Numerical functional analysis and optimization, 30, 227-244, (2009) · Zbl 1176.34096
[28] Belmekki, M.; Nieto, J.J.; Rodríguez-Lopez, R., Existence of periodic solutions for a nonlinear fractional differential equation, Boundary value problems, 2009, (2009), Article ID. 324561 · Zbl 1181.34006
[29] Cuevas, C.; Cesar de Souza, J., Existence of \(S\)-asymptotically \(\omega\)-periodic solutions for fractional order functional integrodifferential equations with infinite delay, Nonlinear analysis, 72, 1683-1689, (2010) · Zbl 1197.47063
[30] Caputo, M., Linear model of dissipation whose \(Q\) is almost frequency independent. part II, Geophysical journal of royal astronomical society, 13, 529-539, (1967)
[31] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier Amsterdam · Zbl 1092.45003
[32] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives: theory and applications, (1993), Gordan and Breach Amsterdam · Zbl 0818.26003
[33] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley New York · Zbl 0789.26002
[34] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · 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. 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.