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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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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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