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Positive solutions for a class of nonlinear fractional differential equations with nonlocal boundary value conditions

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Abstract

We use the fixed point index theory of condensing mapping in cones discuss the existence of positive solutions for the following boundary value problem of fractional differential equations in a Banach space E

$$\begin{aligned} \left\{ \begin{array}{ll} -D^{\,\beta }_{0^{+}}u(t)=f(t,u(t)),\quad t\in J, \\ u(0)=u^{\prime }(0)=\theta ,\quad u(1)=\rho \int _{0}^{1}u(t)dt,\\ \end{array} \right. \end{aligned}$$

where both \(2<\beta \le 3\) and \(0<\rho <\beta \) are real numbers, \(J=[0,1]\), \(D^{\,\beta }_{0^{+}}\) is the Riemann–Liouville fractional derivative, \(f : J\times K \rightarrow K\) is continuous, K is a normal cone in Banach space E, \(\theta \) is the zero element of E. Under more general conditions of growth and noncompactness measure about nonlinearity f, we obtain the existence of positive solutions.

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Authors and Affiliations

  1. Department of Mathematics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China

    Pengyu Chen & Yabing Gao

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  1. Pengyu Chen
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  2. Yabing Gao
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Correspondence to Pengyu Chen.

Additional information

Research supported by NNSFs of China (11501455, 11661071), Key project of Gansu Provincial National Science Foundation (1606RJZA015) and Project of NWNU-LKQN-14-3.

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Chen, P., Gao, Y. Positive solutions for a class of nonlinear fractional differential equations with nonlocal boundary value conditions. Positivity 22, 761–772 (2018). https://doi.org/10.1007/s11117-017-0542-5

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  • DOI: https://doi.org/10.1007/s11117-017-0542-5

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