×
Agarwal, Ravi P.; Ahmad, Bashir

Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions. (English) Zbl 1228.34009

Comput. Math. Appl. 62, No. 3, 1200-1214 (2011).
Summary: This paper studies the existence of solutions for nonlinear fractional differential equations and inclusions of order \(q\in (3,4]\) with anti-periodic boundary conditions. In the case of inclusion problem, the existence results are established for convex as well as nonconvex multivalued maps. Our results are based on some fixed point theorems, Leray-Schauder degree theory, and nonlinear alternative of Leray-Schauder type. Some illustrative examples are discussed.

Cited in 89 Documents

MSC:

34A08 Fractional ordinary differential equations
34B99 Boundary value problems for ordinary differential equations

Keywords:

fractional differential equations; inclusions; anti-periodic boundary conditions; existence; multivalued maps; fixed point theorems
PDF BibTeX XML Cite
\textit{R. P. Agarwal} and \textit{B. Ahmad}, Comput. Math. Appl. 62, No. 3, 1200--1214 (2011; Zbl 1228.34009)
Full Text: DOI
  • logo of hbz
  • logo of WorldCat

References:

[1] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives, theory and applications, (1993), Gordon and Breach Yverdon · Zbl 0818.26003
[2] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[3] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., ()
[4] ()
[5] Chang, Y.-K.; Nieto, J.J., Some new existence results for fractional differential inclusions with boundary conditions, Math. comput. modelling, 49, 605-609, (2009) · Zbl 1165.34313
[6] Ahmad, B.; Nieto, J.J., Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. math. appl., 58, 1838-1843, (2009) · Zbl 1205.34003
[7] Agarwal, R.P.; Belmekki, M.; Benchohra, M., A survey on semilinear differential equations and inclusions involving riemann – liouville fractional derivative, Adv. difference equ., 47, (2009), Art. ID 981728 · Zbl 1182.34103
[8] Henderson, J.; Ouahab, A., Fractional functional differential inclusions with finite delay, Nonlinear anal., 70, 2091-2105, (2009) · Zbl 1159.34010
[9] Gafiychuk, V.; Datsko, B.; Meleshko, V., Mathematical modeling of different types of instabilities in time fractional reaction – diffusion systems, Comput. math. appl., 59, 1101-1107, (2010) · Zbl 1189.35151
[10] Agarwal, R.P.; Lakshmikantham, V.; Nieto, J.J., On the concept of solution for fractional differential equations with uncertainty, Nonlinear anal., 72, 2859-2862, (2010) · Zbl 1188.34005
[11] Ahmad, B., Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations, Appl. math. lett., 23, 390-394, (2010) · Zbl 1198.34007
[12] Nieto, J.J., Maximum principles for fractional differential equations derived from Mittag-Leffler functions, Appl. math. lett., 23, 1248-1251, (2010) · Zbl 1202.34019
[13] Darwish, M.A.; Ntouyas, S.K., On initial and boundary value problems for fractional order mixed type functional differential inclusions, Comput. math. appl., 59, 1253-1265, (2010) · Zbl 1189.34029
[14] Ahmad, B.; Otero-Espinar, V., Existence of solutions for fractional differential inclusions with anti-periodic boundary conditions, Bound. value probl., 11, (2009), Art. ID 625347 · Zbl 1172.34004
[15] Ahmad, B.; Nieto, J.J., Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory, Topol. methods nonlinear anal., 35, 295-304, (2010) · Zbl 1245.34008
[16] Ahmad, B., Existence of solutions for fractional differential equations of order \(q \in(2, 3]\) with anti-periodic boundary conditions, J. appl. math. comput., 34, 385-391, (2010) · Zbl 1216.34003
[17] R.P. Agarwal, B. Ahmad, Existence of solutions for impulsive anti-periodic boundary value problems of fractional semilinear evolution equations, Dyna. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal., (in press). · Zbl 1226.26005
[18] Wang, G.; Ahmad, B.; Zhang, L., Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order, Nonlinear anal., 74, 792-804, (2011) · Zbl 1214.34009
[19] Smart, D.R., Fixed point theorems, (1980), Cambridge University Press · Zbl 0427.47036
[20] Hu, Sh.; Papageorgiou, N., Handbook of multivalued analysis, theory I, (1997), Kluwer Dordrecht · Zbl 0887.47001
[21] Deimling, K., Multivalued differential equations, (1992), Walter De Gruyter Berlin, New York · Zbl 0760.34002
[22] Kisielewicz, M., Differential inclusions and optimal control, (1991), Kluwer Dordrecht, The Netherlands · Zbl 0731.49001
[23] Lasota, A.; Opial, Z., An application of the kakutani – ky Fan theorem in the theory of ordinary differential equations, Bull. acad. Pol. sci., ser. sci. math. astron. phys., 13, 781-786, (1965) · Zbl 0151.10703
[24] Bressan, A.; Colombo, G., Extensions and selections of maps with decomposable values, Studia math., 90, 69-86, (1988) · Zbl 0677.54013
[25] Covitz, H.; Nadler, S.B., Multivalued contraction mappings in generalized metric spaces, Israel J. math., 8, 5-11, (1970) · Zbl 0192.59802
[26] Granas, A.; Dugundji, J., Fixed point theory, (2005), Springer-Verlag New York
[27] Castaing, C.; Valadier, M., ()
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.