Cauchy problem for a class of fractional differential equations on time scales

References

• T. Abdeljawad and D. Baleanu, Caputo q-fractional initial value problems and a q-analogue Mittag-Leffler function, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), pp. 4682–4688 doi: 10.1016/j.cnsns.2011.01.026.
   Web of Science ®Google Scholar
• R. P. Agarwal, M. Bohner, D. O'Regan and A. Peterson, Dynamic equations on time scales: A survey, J. Comput. Appl. Math. 141 (2002), pp. 1–26 doi: 10.1016/S0377-0427(01)00432-0.
   Web of Science ®Google Scholar
• B. Ahmada and J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl. 58 (2009), pp. 1838–1843 doi: 10.1016/j.camwa.2009.07.091.
   Web of Science ®Google Scholar
• G. A. Anastassiou, Principles of delta fractional calculus on time scales and inequalities, Math. Comput. Model. 52 (2010), pp. 556–566 doi: 10.1016/j.mcm.2010.03.055.
   Web of Science ®Google Scholar
• G. A. Anastassiou, Foundations of nabla fractional calculus on time scales and inequalities, Comput. Math. Appl. 59 (2010), pp. 3750–3762 doi: 10.1016/j.camwa.2010.03.072.
   Web of Science ®Google Scholar
• F. M. Atici and P. W. Eloe, A transform method in discrete fractional calculus, Int. J. Differ. Equ. 2(2) (2007), pp. 165–176.
   Google Scholar
• F. M. Atici and P. W. Eloe, Fractional q-calculus on a time scale, J. Nonlinear Math. Phys. 14(3) (2007), pp. 341–352 doi: 10.2991/jnmp.2007.14.3.4.
   Web of Science ®Google Scholar
• F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc. 137 (2009), pp. 981–989 doi: 10.1090/S0002-9939-08-09626-3.
   Web of Science ®Google Scholar
• F. M. Atici and P. W. Eloe, Two-point boundary value problems for finite fractional difference equations, J. Difference Equ. Appl. 17 (2011), pp. 445–456 doi: 10.1080/10236190903029241.
   Web of Science ®Google Scholar
• F. M. Atici and S. Sengül, Modeling with fractional difference equations, J. Math. Anal. Appl. 369 (2010), pp. 1–9 doi: 10.1016/j.jmaa.2010.02.009.
   Web of Science ®Google Scholar
• N. R.O. Bastos, D. Mozyrska and D. F.M. Torres, Fractional derivatives and integrals on time scales via the inverse generalized Laplace transform, Int. J. Math. Comput. 11 (2011), pp. 1–9.
   Google Scholar
• M. Bohner and A. Peterson, Dynamic equations on time scales: An Introduction with Applications, Birkhäuser, Boston, MA, 2001.
   Google Scholar
• A. P. Chen and Y. Chen, Existence of solutions to anti-periodic boundary value problem for nonlinear fractional differential equations with impulses, Adv. Difference Equ. 2011 (2011), pp. 17 doi: 10.1186/1687-1847-2011-17.
   Web of Science ®Google Scholar
• S. Das, Functional Fractional Calculus, 2.Springer, HeidelbergGermany, 2011.
   Google Scholar
• M. El-Shahed, Positive solutions for boundary value problem of nonlinear fractional differential equation, Abstr. Appl. Anal. 2007 (2007), pp. 8 doi: 10.1155/2007/10368.
   Web of Science ®Google Scholar
• R. A.C. Ferreira, Positive solutions for a class of boundary value problems with fractional q-differences, Comput. Math. Appl. 61 (2011), pp. 367–373 doi: 10.1016/j.camwa.2010.11.012.
   Web of Science ®Google Scholar
• G. Frederico and D. Torres, A formulation of Noether's theorem for fractional problems of the calculus of variations, J. Math. Anal. Appl. 334 (2007), pp. 834–846 doi: 10.1016/j.jmaa.2007.01.013.
   Web of Science ®Google Scholar
• G. Frederico and D. Torres, Fractional conservation laws in optimal control theory, Nonlinear Dynam. 53 (2008), pp. 215–222 doi: 10.1007/s11071-007-9309-z.
   Web of Science ®Google Scholar
• C. S. Goodrich, Existence of a positive solution to a system of discrete fractional boundary value problems, Appl. Math. Comput. 217 (2011), pp. 4740–4753 doi: 10.1016/j.amc.2010.11.029.
   Web of Science ®Google Scholar
• C. S. Goodrich, On discrete sequential fractional boundary value problems, J. Math. Anal. Appl. 385 (2012), pp. 111–124 doi: 10.1016/j.jmaa.2011.06.022.
   Web of Science ®Google Scholar
• D. J. Guo, Nonlinear Functional Analysis, Shandong Science and Technology Press, Jinan, 2005.
   Google Scholar
• M. A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.
   Google Scholar
• V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan, Dynamic Systems on Measure Chains, Kluwer, Dordrecht, 1996.
   Google Scholar
• R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Redding, CT, 2006.
   Google Scholar
• K. S. Miller and B. Ross, Fractional difference calculus, Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, May 1988, Ellis Horwood Series in Mathematics & Its Applications. Horwood, Chichester, 1989 139–152.
   Google Scholar
• K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
   Google Scholar
• J. Sabatier, O. P. Agrawal and J. A. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, HeidelbergGermany, 2007.
   Google Scholar
• S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integral and Derivative: Theory and Applications, Gordon and Breach Science, Yverdon, 1993.
   Google Scholar
• J. X. Sun, Nonlinear Functional Analysis and Its Application, Science Press, Beijing, 2008.
   Google Scholar
• Y. S. Tian and Z. B. Bai, Existence results for the three-point impulsive boundary value problem involving fractional differential equations, Comput. Math. Appl. 59 (2010), pp. 2601–2609 doi: 10.1016/j.camwa.2010.01.028.
   Web of Science ®Google Scholar
• A. K. Tripathy and S. Panigrahi, On the oscillatory behaviour of a class of nonlinear delay difference equations of second order, Indian J. Pure Appl. Math. 42127–40.
   Web of Science ®Google Scholar
• G. T. Wang, B. Ahmad and L. H. Zhang, Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions, Comput. Math. Appl. 62 (2011), pp. 1389–1397 doi: 10.1016/j.camwa.2011.04.004.
   Web of Science ®Google Scholar
• P. A. Williams, Fractional calculus on time scales with Taylor's theorem, Fract. Calc. Appl. Anal. 15(4) (2007), pp. 616–638 doi: 10.2478/s13540-012-0043-y.
   Web of Science ®Google Scholar
• S. A. Yousefi, M. Dehghan and A. Lotfi, Generalized Euler–Lagrange equations for fractional variational problems with free boundary conditions, Comput. Math. Appl. 62 (2011), pp. 987–995 doi: 10.1016/j.camwa.2011.03.064.
   Web of Science ®Google Scholar
• X. Z. Zhang and C. X. Zhu, Periodic boundary value problems for first order dynamic equations on time scales, Adv. Difference Equ. 2012(76) (2012), pp. 1–16.
   Google Scholar
• X. Z. Zhang, C. X. Zhu and Z. Q. Wu, The Cauchy problem for a class of fractional impulsive differential equations with delay, Electron. J. Qual. Theory Differ. Equ. 37 (2012), pp. 1–13.
   Web of Science ®Google Scholar
• Y. G. Zhao, S. R. Sun, Z. L. Han and Q. P. Li, Positive solutions to boundary value problems of nonlinear fractional differential equations, Abstr. Appl. Anal. 2011 (2011), pp. 16 Article ID 390543..
   Google Scholar
• C. X. Zhu, Research on some problems for nonlinear operators, Nonlinear Anal, 71 (2009), pp. 4568–4571.
   Google Scholar
• C. X. Zhu and Z. B. Xu, Random ambiguous point of random k(ω)-set-contractive operator, J. Math. Anal. Appl. 328(1) (2007), pp. 2–6 doi: 10.1016/j.jmaa.2006.04.044.
   Web of Science ®Google Scholar