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Anti-periodic fractional boundary value problems with nonlinear term depending on lower order derivative. (English) Zbl 1281.34005

The authors are concerned with a new class of anti-periodic fractional-order boundary value problems where the nonlinear term involves a lower-order fractional derivative. They establish existence and uniqueness results using some standard fixed point theorems.

MSC:

34A08 Fractional ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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[1] S. Abbas, M. Benchohra, A.N. Vityuk, On fractional order derivatives and Darboux problem for implicit differential equations. Fract. Calc. Appl. Anal. 15,No 2 (2012), 168-182; DOI: 10.2478/s13540-012-0012-5; http://www.springerlink.com/content/1311-0454/15/2/. · Zbl 1302.35395
[2] R.P. Agarwal, B. Ahmad, Existence of solutions for impulsive antiperiodic boundary value problems of fractional semilinear evolution equations. Dynam. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 18 (2011), 457-470. · Zbl 1226.26005
[3] R.P. Agarwal, M. Belmekki, M. Benchohra, A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative. Adv. Difference Equ. 2009 (2009), Art. ID 981728, 47 pp. · Zbl 1182.34103
[4] R.P. Agarwal, V. Lakshmikantham, J.J. Nieto, On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. 72, No 6 (2010), 2859-2862. http://dx.doi.org/10.1016/j.na.2009.11.029 · Zbl 1188.34005
[5] A. Aghajani, Y. Jalilian, J.J. Trujillo, On the existence of solutions of fractional integro-differential equations. Fract. Calc. Appl. Anal. 15, No 1 (2012), 44-69; DOI: 10.2478/s13540-012-0005-4; http://www.springerlink.com/content/1311-0454/15/1/.
[6] B. Ahmad, Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations. Appl. Math. Lett. 23, No 4 (2010), 390-394. http://dx.doi.org/10.1016/j.aml.2009.11.004 · Zbl 1198.34007
[7] B. Ahmad, Existence of solutions for fractional differential equations of order q ∈ (2, 3] with anti-periodic boundary conditions. J. Appl. Math. Comput. 34, No 1-2 (2010), 385-391. http://dx.doi.org/10.1007/s12190-009-0328-4 · Zbl 1216.34003
[8] B. Ahmad, J.J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 58, No 9 (2009), 1838-1843. http://dx.doi.org/10.1016/j.camwa.2009.07.091 · Zbl 1205.34003
[9] B. Ahmad, J.J. Nieto, Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory. Topol. Methods Nonlinear Anal. 35, No 2 (2010), 295-304. · Zbl 1245.34008
[10] B. Ahmad, J.J. Nieto, Anti-periodic fractional boundary value problems, Comput. Math. Appl. 62, No 3 (2011), 1150-1156. http://dx.doi.org/10.1016/j.camwa.2011.02.034 · Zbl 1228.34010
[11] B. Ahmad, V. Otero-Espinar, Existence of solutions for fractional differential inclusions with anti-periodic boundary conditions. Bound. Value Probl. 2009 (2009), Art. ID 625347, 11 pages. · Zbl 1172.34004
[12] A. Cernea, A note on the existence of solutions for some boundary value problems of fractional differential inclusions. Fract. Calc. Appl. Anal. 15, No 2 (2012), 183-194; DOI: 10.2478/s13540-012-0013-4; http://www.springerlink.com/content/1311-0454/15/2/. · Zbl 1281.34019
[13] Y.-K. Chang, J.J. Nieto, Some new existence results for fractional differential inclusions with boundary conditions. Math. Comput. Modelling 49, No 9-10 (2009), 605-609. http://dx.doi.org/10.1016/j.mcm.2008.03.014 · Zbl 1165.34313
[14] Y. Chen, J.J. Nieto, D. O’Regan, Anti-periodic solutions for evolution equations associated with maximal monotone mappings. Appl. Math. Lett. 24, No 3 (2011), 302-307. http://dx.doi.org/10.1016/j.aml.2010.10.010 · Zbl 1215.34069
[15] V. Gafiychuk, B. Datsko, V. Meleshko, Mathematical modeling of different types of instabilities in time fractional reaction-diffusion systems. Comput. Math. Appl. 59, No 3 (2010), 1101-1107. http://dx.doi.org/10.1016/j.camwa.2009.05.013 · Zbl 1189.35151
[16] J. Henderson, A. Ouahab, Fractional functional differential inclusions with finite delay. Nonlinear Anal. 70 (2009), 2091-2105. http://dx.doi.org/10.1016/j.na.2008.02.111 · Zbl 1159.34010
[17] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam (2006). http://dx.doi.org/10.1016/S0304-0208(06)80001-0
[18] J.J. Nieto, Maximum principles for fractional differential equations derived from Mittag-Leffler functions. Appl. Math. Lett. 23, No 10 (2010), 1248-1251. http://dx.doi.org/10.1016/j.aml.2010.06.007 · Zbl 1202.34019
[19] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999). · Zbl 0924.34008
[20] J. Sabatier, O.P. Agrawal, J.A.T. Machado (Eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007). · Zbl 1116.00014
[21] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, Yverdon (1993). · Zbl 0818.26003
[22] D.R. Smart, Fixed Point Theorems. Cambridge Univ. Press, Cambridge (1980). · Zbl 0427.47036
[23] G. Wang, B. Ahmad, L. Zhang, Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order. Nonlinear Anal. 74, No 3 (2011), 792-804. http://dx.doi.org/10.1016/j.na.2010.09.030 · Zbl 1214.34009
[24] G. Wang, D. Baleanu, L. Zhang, Monotone iterative method for a class of nonlinear fractional differential equations. Fract. Calc. Appl. Anal. 15, No 2 (2012), 244-252; DOI: 10.2478/s13540-012-0018-z; http://www.springerlink.com/content/1311-0454/15/2/. · Zbl 1273.34021
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