Topological Methods in Nonlinear Analysis

Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory

Bashir Ahmad and Juan J. Nieto

Full-text: Open access

Abstract

In this paper, some existence results for a differential equation of fractional order with anti-periodic boundary conditions are presented. The main tool of study is Leray-Schauder degree theory.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 35, Number 2 (2010), 295-304.

Dates
First available in Project Euclid: 21 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461251008

Mathematical Reviews number (MathSciNet)
MR2676818

Zentralblatt MATH identifier
1245.34008

Citation

Ahmad, Bashir; Nieto, Juan J. Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory. Topol. Methods Nonlinear Anal. 35 (2010), no. 2, 295--304. https://projecteuclid.org/euclid.tmna/1461251008


Export citation

References

  • B. Ahmad and J. J. Nieto, Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions , Nonlinear Anal., 69 (2008), 3291–3298 \ref\key 2 ––––, Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions , Bound. Value Probl. (2009). Art. ID 708576, 11 pp \ref\key 3 ––––, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions , Comput math. Appl., 58 (2009), 1838–1848 \ref\key 4
  • B. Ahmad and S. Sivasundaram, Existence and uniqueness results for nonlinear boundary value problems of fractional differential equations with separated boundary conditions , Comm. Appl. Anal., 13 (2009), 121–128 \ref\key 5
  • M. Altman, A fixed point theorem in Banach space , Bull. Polish Acad. Sci., 5 (1957), 19–22 \ref\key 6
  • M. Belmekki, J. J. Nieto and R. Rodríguez-López, Existence of periodic solution for a nonlinear fractional differential equation , Bound. Value Probl. (2009). Art. ID 324561, 18 pp \ref\key 7
  • B. Bonilla, M. Rivero, L. Rodriguez-Germa and J. J. Trujillo, Fractional differential equations as alternative models to nonlinear differential equations , Appl. Math. Comput., 187 (2007), 79-88. \ref\key 8
  • A. Cabada and D. R. Vivero, Existence and uniqueness of solutions of higher-order antiperiodic dynamic equations , Adv. Differential Equations, 4 (2004), 291–310 \ref\key 9
  • Y.-K. Chang and J. J. Nieto, Some new existence results for fractional differential inclusions with boundary conditions , Math. Comput. Modelling, 49 (2009), 605–609 \ref\key 10 ––––, Existence of solutions for impulsive neutral integro-differential inclusions with nonlocal initial conditions via fractional operators , Numer. Funct. Anal. Optim., 30 (2009), 227–224 \ref\key 11
  • H. L. Chen, Antiperiodic wavelets , J. Comput. Math., 14 (1996), 32–39 \ref\key 12
  • Y. Chen, J. J. Nieto and D. O'Regan, Antiperiodic solutions for fully nonlinear first-order differential equations , Math. Comput. Modelling, 46 (2007), 1183–1190 \ref\key 13
  • V. Daftardar-Gejji and S. Bhalekar, Boundary value problems for multi-term fractional differential equations , J. Math. Anal. Appl., 345 (2008), 754–765 \ref\key 14
  • F. J. Delvos and L. Knoche, Lacunary interpolation by antiperiodic trigonometric polynomials , BIT, 39 (1999), 439–450 \ref\key 15
  • W. Ding, Y. Xing and M. Han, Antiperiodic boundary value problems for first order impulsive functional differential equations , Appl. Math. Comput., 186 (2007), 45–53 \ref\key 16
  • D. Franco and J. J. Nieto, Existence of solutions for first order ordinary differential equations with nonlinear boundary conditions , Appl. Math. Comput., 153 (2004), 793–802 \ref\key 17
  • D. Franco, J. J. Nieto and D. O'Regan, Anti-periodic boundary value problem for nonlinear first order ordinary differential equations , Math. Inequal. Appl., 6 (2003), 477–485 \ref\key 18
  • V. Gafiychuk, B. Datsko and V. Meleshko, Mathematical modeling of time fractional reaction-diffusion systems , J. Comput. Appl. Math., 220(2008), 215–225 \ref\key 19
  • R. W. Ibrahim and M. Darus, Subordination and superordination for univalent solutions for fractional differential equations , J. Math. Anal. Appl., 345 (2008), 871–879 \ref\key 20
  • H. Jafari and S. Seifi, Solving a system of nonlinear fractional partial differential equations using homotopy analysis method , Comm. Nonlinear Sci. Numer. Simul., 14 (2009), 1962–1969 \ref\key 21
  • A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204 , Elsevier Science B.V. Amsterdam (2006) \ref\key 22
  • S. Ladaci, J. L. Loiseau and A. Charef, Fractional order adaptive high-gain controllers for a class of linear systems , Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 707–714 \ref\key 23
  • M. P. Lazarević, Finite time stability analysis of PD$^\alpha$ fractional control of robotic time-delay systems , Mech. Res. Comm. 33(2006), 269–279 \ref\key 24
  • B. Liu, An anti-periodic LaSalle oscillation theorem for a class of functional differential equations , J. Comput. Appl. Math., 223 (2009), 1081–1086 \ref\key 25
  • J. Liu and M. Xu, Some exact solutions to Stefan problems with fractional differential equations , J. Math. Anal. Appl., 351 (2009), 536–542 \ref\key 26
  • Z. Luo, J. Shen and J. J. Nieto, Antiperiodic boundary value problem for first-order impulsive ordinary differential equations , Comput. Math. Appl., 49 (2005), 253–261
  • \ref\key 27M. Nakao, Existence of an anti-periodic solution for the quasilinear wave equation with viscosity , J. Math. Anal. Appl., 204 (1996), 754-764 \ref\key 28
  • T. Pierantozzi, Fractional evolution Dirac-like equations: Some properties and a discrete Von Neumann-type analysis , J. Comput. Appl. Math., 224 (2009), 284–295 \ref\key 29
  • I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999) \ref\key 30
  • S. Z. Rida, H. M. El-Sherbiny and A. A. M. Arafa, On the solution of the fractional nonlinear Schrödinger equation , Phys. Lett. A, 372 (2008), 553–558 \ref\key 31
  • S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon (1993) \ref\key 32
  • J. Shao, Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying delays , Phys. Lett. A, 372 (2008), 5011–5016 \ref\key 33
  • P. Souplet, Optimal uniqueness condition for the antiperiodic solutions of some nonlinear parabolic equations , Nonlinear Anal., 32 (1998), 279–286
  • \ref\key 34Y. Wang, Y.M. Shi, Eigenvalues of second-order difference equations with periodic and antiperiodic boundary conditions. J. Math. Anal. Appl. 309 (2005), 56-69 \ref\key 35
  • K. Wang and Y. Li, A note on existence of \rom(anti-\rom)periodic and heteroclinic solutions for a class of second-order odes, Nonlinear Anal., 70 (2009), 1711–1724 \ref\key 36
  • S. Wang, M. Xu and X. Li, Green's function of time fractional diffusion equation and its applications in fractional quantum mechanics , Nonlinear Anal. Real World Appl., 10 (2009), 1081–1086
  • \ref\key 37Y. Yin, Anti-periodic solutions of some semilinear parabolic boundary value problems , Dynam. Contin. Discrete Impuls. Systems, 1 (1995), 283–297 \ref\key 38
  • S. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equations , Electronic J. Differential Equations, 2006 (2006), 1–12