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Existence and uniqueness of solutions for a first-order discrete fractional boundary value problem

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper we use topological degree to establish the existence of nontrivial solutions for a first-order discrete fractional boundary value problem with a sign-changing nonlinearity. Also using the monotone iterative technique we discuss problems with a nonnegative nonlinearity and establish the uniqueness of positive solutions.

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References

  1. Atici, F.M., Eloe, P.W.: A Transform Method in Discrete Fractional Calculus. Int. J. Differ. Equations 2, 165–176 (2007)

    MathSciNet  Google Scholar 

  2. Atici, F.M., Eloe, P.W.: Discrete Fractional Calculus with the Nabla Operator. Electron. J. Qual. Theory Differ. Equations 3, 1–12 (2009)

    Article  MathSciNet  Google Scholar 

  3. Atici, F.M., Eloe, P.W.: Initial Value Problems in Discrete Fractional Calculus. Proc. Am. Math. Soc. 137, 981–989 (2009)

    Article  MathSciNet  Google Scholar 

  4. Atici, F.M., Eloe, P.W.: Two-Point Boundary Value Problems for Finite Fractional Difference Equations. J. Differ. Equ. Appl. 17, 445–456 (2011)

    Article  MathSciNet  Google Scholar 

  5. Reunsumrit, J., Sitthiwirattham, T.: Positive Solutions of Three-Point Fractional Sum Boundary Value Problem for Caputo Fractional Difference Equations via an Argument with a Shift. Positivity 20, 861–876 (2016)

    Article  MathSciNet  Google Scholar 

  6. Lv, Z., Gong, Y., Chen, Y.: Multiplicity and Uniqueness for a Class of Discrete Fractional Boundary Value Problems. Appl. Math. 59, 673–695 (2014)

    Article  MathSciNet  Google Scholar 

  7. Goodrich, C.S.: On Semipositone Discrete Fractional Boundary Value Problems with Non-Local Boundary Conditions. J. Differ. Equations Appl. 19, 1758–1780 (2013)

    Article  MathSciNet  Google Scholar 

  8. Goodrich, C.S.: On a First-Order Semipositone Discrete Fractional Boundary Value Problem. Arch. Math. 99, 509–518 (2012)

    Article  MathSciNet  Google Scholar 

  9. Goodrich, C.S., Peterson, A.C.: Discrete Fractional Calculus. Springer, New York (2015)

    Book  Google Scholar 

  10. Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, Orlando (1988)

    MATH  Google Scholar 

  11. Yang, L., Chen, H., Luo, L., Luo, Z.: Successive Iteration and Positive Solutions for Boundary Value Problem of Nonlinear Fractional \(q\)-Difference Equation. J. Appl. Math. Comput. 42, 89–102 (2013)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, China

    Jiafa Xu

  2. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

    Donal O’Regan

Authors
  1. Jiafa Xu
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  2. Donal O’Regan
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Correspondence to Jiafa Xu.

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The authors declare that they have no competing interests.

Additional information

Research supported by the National Natural Science Foundation of China(Grant No. 11601048), Natural Science Foundation of Chongqing (Grant No. cstc2016jcyjA0181), the Science and Technology Research Program of Chongqing Municipal Education Commission(Grant No. KJ1703050), Natural Science Foundation of Chongqing Normal University (Grant No. 15XLB011).

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Xu, J., O’Regan, D. Existence and uniqueness of solutions for a first-order discrete fractional boundary value problem. RACSAM 112, 1005–1016 (2018). https://doi.org/10.1007/s13398-017-0406-7

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  • DOI: https://doi.org/10.1007/s13398-017-0406-7

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