DOI QR코드

DOI QR Code

COMPARISON OF SMALLEST EIGENVALUES FOR RIGHT FOCAL ATICI-ELOE FRACTIONAL DIFFERENCE EQUATIONS

  • Yang, Aijun (Zhejiang University of Technology, College of Science) ;
  • Zhang, Li (Beijing Union University, Department of Foundation Courses) ;
  • Henderson, Johnny (Baylor University, Department of Mathematics)
  • 투고 : 2017.08.22
  • 심사 : 2017.11.16
  • 발행 : 2017.11.30

초록

The theory of $u_0-positive$ operators is applied to obtain smallest eigenvalue comparison results for right focal boundary value problems of Atici-Eloe fractional difference equations.

키워드

참고문헌

  1. F. Atici & P. W. Eloe: Initial value problems in discrete fractional calculus. Proc. Amer. Math. Soc. 137 (2009), 981-989.
  2. F. Atici & P. W. Eloe: Discrete fractional calculus with the nabla operator. Elect. J. Qual. Theory Differential Equations, Spec. Ed. I No. 3 (2009), 1-12.
  3. F. Atici & P. W. Eloe. Two-point boundary value problems for finite fractional difference equations. J. Difference Equa. Appl. 17 (2011), 445-456. https://doi.org/10.1080/10236190903029241
  4. F. Atici & S. Sengul: Modeling with fractional difference equations. J. Math. Anal. Appl. 369 (2010), 1-9. https://doi.org/10.1016/j.jmaa.2010.02.009
  5. P. W. Eloe & J. T. Neugebauer: Existence and comparison of smallest eigenvalues for a fractional boundary-value problem. Electron. J. Differential Equations 2014 (2014), no. 43, 1-10.
  6. P. W. Eloe & J. T. Neugebauer: Conjugate point for fractional differential equations. Fract. Calc. Appl. Anal. 17 (2014), no. 3, 855-871. https://doi.org/10.2478/s13540-014-0201-5
  7. C. Goodrich & A. Peterson: Discrete Fractional Calculus. Springer, Preliminary Version, 2014.
  8. C. S. Goodrich: Solutions to a discrete right-focal fractional boundary value problem. Int. J. Difference Equ. 5, (2010), no. 2, 195-216.
  9. J. Henderson & R. Luca: Positive solutions for a system of nonlocal fractional boundary value problems. Fract. Calc. Appl. Anal. 16 (2013), no. 4, 985-1008.
  10. B. G. Jia, L. Erbe & A. Peterson: Comparison theorems and asymptotic behavior of solutions of discrete fractional equations. Electron. J. Qual. Theory Differ. Equ. 2015 (2015), no. 89, 1-18. https://doi.org/10.1186/s13662-014-0331-4
  11. M. Keener & C. C. Travis: Positive cones and focal points for a class of nth order differential equations. Trans. Amer. Math. Soc. 237 (1978), 331-351.
  12. 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.
  13. M. Krasnosel'skii: Positive Solutions of Operator Equations. Fizmatgiz, Moscow, 1962; English Translation P. Noordhoff Ltd. Groningen, The Netherlands, 1964.
  14. M. G. Krein & M. A. Rutman: Linear Operators Leaving a Cone Invariant in a Banach Space. in American Mathematical Society Translations, Series 1, Providence, 1962.
  15. F. Mainardi: Fractional calculus: Some basic problems in continuum and statistical mechanics, in Fractals and Fractional Calculus in Continuum Mechanics. A. Carpinteri and F. Mainardi, Eds., Springer-Verlag, Wien, 1997.
  16. F. Metzler, W. Schick, H. G. Kilian & T. F. Nonnenmacher: Relaxation in ¯lled polymers: A fractional calculus approach. J. Chem. Phys. 103 (1995), 7180-7186. https://doi.org/10.1063/1.470346
  17. K. S. Miller & B. Ross: An Introduction to the Fractional Calculus and Differential Equations. John Wiley, New York, 1993.
  18. I. Podlubny: Fractional Differential Equations, Mathematics in Sciences and Engineering. 198, Academic Press, San Diego, 1999.
  19. S. G. Samko, A. A. Kilbas & O. I. Marichev: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon, 1993.
  20. A. J. Yang & H. L. Wang: Positive solutions of two-point boundary value problems of nonlinear fractional differential equation at resonance. E. J. Qualitative Theory of Diff. Equ. 71 (2011), 1-15.
  21. A. J. Yang & H. L. Wang: Positive solutions for higher-order nonlinear fractional differential equation with integral boundary condition. E. J. Qualitative Theory of Diff. Equ. 1 (2011), 1-15.
  22. A. J. Yang, J. Henderson & C. Nelms JR.: Extremal points for a higher-order fractional boundary value problem. Electron. J. Differential Equations 2015 (2015), no. 161, 1-12.
  23. A. J. Yang, J. Henderson & H. L. Wang: Parameter dependence for existence, nonexistence and multiplicity of nontrivial solutions for an Atici-Eloe fractional difference Lidstone BVP. Electron. J. Qual. Theory Differ. Equ. 2017 (2017), no. 38, 1-12. https://doi.org/10.1186/s13662-016-1057-2