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Existence and uniqueness of positive solutions for three-point boundary value problem with fractional q-differences

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Abstract

In this paper, we consider the following nonlinear q-fractional three-point boundary value problem

$$\begin{array}{l}(D_{q}^{\alpha}u)(t) + f(t,u(t))=0, \quad 0 < t < 1, 2 < \alpha< 3,\\ [2pt]u(0) = (D_qu)(0) = 0, \quad(D_qu)(1) = \beta(D_qu)(\eta),\end{array}$$

where 0<βη α-2<1. By using a fixed-point theorem in partially ordered sets, we obtain sufficient conditions for the existence and uniqueness of positive and nondecreasing solutions to the above boundary value problem.

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Acknowledgements

The paper is supported by Research Fundation during the 12st Five-Year Plan Period of Department of Education of Jilin Province, China (Grant [2011] No. 196), Natural Science Foundation of Changchun Normal University.

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Authors and Affiliations

  1. College of Mathematics, Changchun Normal University, Changchun, 130032, Jilin, PR China

    Sihua Liang

  2. Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu, 210046, PR China

    Jihui Zhang

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  1. Sihua Liang
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Correspondence to Sihua Liang.

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Liang, S., Zhang, J. Existence and uniqueness of positive solutions for three-point boundary value problem with fractional q-differences. J. Appl. Math. Comput. 40, 277–288 (2012). https://doi.org/10.1007/s12190-012-0551-2

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  • DOI: https://doi.org/10.1007/s12190-012-0551-2

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