Monotone iterative technique with initial time difference for fractional differential equations

Monotone iterative technique with initial time difference for fractional differential equations

In this paper, we employe the monotone iterative technique for fractional differential equations of Riemann-Liouville type by choosing upper and lower solutions that start at different initial times.

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  • [1] Deekshitulu, Gvsr. Generalized monotone iterative technique for fractional R-L differential equations, Nonlinear Studies 16 (1), Pages 85–94, 2009.
  • [2] Hu, T.C., Qian, D. L. and Li C.P. Comparison theorems of fractional differential equations, Comm. Appl. Math. Comput. 23 (1), 97–103, 2009.
  • [3] Köksal, S. and Yakar, C. Generalized quasilinearization method with initial time difference, Simulation, an International Journal of Electrical, Electronic and other Physical Systems 24 (5), 2002.
  • [4] Ladde, G. S, Lakshmikantham, V. and Vatsala A. S. Monotone Iterative Technique for Non- linear Differential Equations (Pitman Publishing Inc., Boston, 1985).
  • [5] Lakshmikantham, V., Leela, S. and Vasundhara, Devi J. Theory of Fractional Dynamic Systems (Cambridge Academic Publishers, Cambridge, 2009).
  • [6] Lakshmikantham, V. and Vatsala, A. S. General uniqueness and monotone iterative tech- nique for fractional differential equations, Applied Mathematics Letters 21 (8), 828–834, 2008.
  • [7] Lakshmikantham, V. and Vatsala, A. S. Basic theory of fractional differential equations, Nonlinear Analysis: Theory, Methods and Applications 69 (8), 2677–2682, 2008.
  • [8] McRae, F.A. Monotone iterative technique and existence results for fractional differential Equations, Nonlinear Analysis: Theory, Methods and Applications 71 (12), 6093–6096, 2009.
  • [9] McRae, F.A. Monotone iterative technique for PBVP of Caputo fractional differential equa- tions, to appear.
  • [10] Oldham, K.B. and Spanier, J. The Fractional Calculus (Academic Press, New York, 1974).
  • [11] Podlubny, I. Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applica- tions (Mathematics in Science and Engineering, 198, Academic Press, San Diego, 1999).
  • [12] Vasundhara Devi, J. Generalized monotone technique for periodic boundary value problems of fractional differential equations, Communications in Applied Analysis 12 (4), 399–406, 2008.
  • [13] Yakar, C. and Yakar, A. An extension of the quasilinearization method with initial time dif- ference, Dynamics of Continuous, Discrete and Impulsive Systems (Series A: Mathematical Analysis) DCDIS 14 (S2) 1-305, 275–279, 2007.
  • [14] Yakar, C. and Yakar, A. Further generalization of quasilinearization method with initial time difference J. of Appl. Funct. Anal. 4 (4), 714–727, 2009.
  • [15] Yakar, C. and Yakar. A. A refinement of quasilinearization method for Caputo sense frac- tional order differential equations, Abstract and Applied Analysis 2010, Article ID 704367, 10 pages, doi:10.1155/2010/704367