Positive solutions for a class of fractional differential coupled system with integral boundary value conditions

Volume 9, Issue 5, pp 2922--2942 http://dx.doi.org/10.22436/jnsa.009.05.86

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Authors

Daliang Zhao - Department of Mathematics, Shandong Normal University, Jinan, 250014, P. R. China. Yansheng Liu - Department of Mathematics, Shandong Normal University, Jinan, 250014, P. R. China.

Abstract

This paper investigates the existence of positive solutions for the following high-order nonlinear fractional differential boundary value problem (BVP, for short) \[ \begin{cases} D^\alpha_{0^+} u(t) + f(t,v(t))=0,\,\,\,\,\, t\in (0,1),\\ D^\alpha_{0^+} v(t) + g(t,u(t))=0,\,\,\,\,\, t\in (0,1),\\ u^{(j)}(0)=v^{(j)}(0)=0,\,\,\,\,\, 0\leq j\leq n-1, j\neq 1,\\ u'(1)=\lambda \int^1_0 u(t)d(t),\quad v'(1)=\lambda \int^1_0 v(t)d(t), \end{cases} \] where \(n - 1 < \alpha\leq n; n \geq 3; 0 \leq\lambda < 2, D^\alpha_{0^+}\) is the Caputo fractional derivative. By using the monotone method, the theory of fixed point index on cone for differentiable operators and the properties of Green's function, some new uniqueness and existence criteria for the considered fractional BVP are established. As applications, some examples are worked out to demonstrate the main results.

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Keywords

• Fractional differential equations
• differentiable operators
• fixed point index theorems on cone

MSC

•  34A08
•  34B15
•  34B18

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