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Positive Solutions for a Class of Fractional Boundary Value Problem with q-Derivatives

  • Furi GuoEmail author
  • Shugui Kang
Article
  • 18 Downloads

Abstract

This work is concerned with the existence and uniqueness of positive solutions to a nonlinear boundary value problem with fractional q-derivative:
$$\begin{aligned}&D_q^\alpha u(t)+f(t,u(t),u(t))+g(t,u(t))=0, \quad 0<t<1 ,\\&u(0)=0 ,\quad u(1)=\beta u(\eta ), \end{aligned}$$
where \(D_q^\alpha \) is the fractional q-derivative of Riemann–Liouville type, \(0<q<1\)\(1<\alpha \le 2\)\(0<\eta <1,\)  \(0<\beta \eta ^{\alpha -1}<1.\) By virtue of fixed-point theorems for mixed monotone operator, some existence and uniqueness results of positive solutions are obtained. An example to illustrate our results is given.

Keywords

Positive solution mixed monotone operator fractional q-difference equation boundary value problem 

Mathematics Subject Classification

34B18 33D05 39A13 

Notes

Acknowledgements

This work is supported by the National Nature Science Foundation under Grant (11271235).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and Computer ScienceShanxi Datong UniversityDatongPeople’s Republic of China

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