#### Ruixiong Fan, Weixuan Wang, Chengbo Zhai, Positive solutions for Hadamard fractional boundary value problems on an infinite interval, Vol. 2020 (2020), Article ID 2, pp. 1-14

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DOI: 10.23952/jnfa.2020.2

Received March 21, 2019; Accepted January 4, 2020; Published January 19, 2020

Abstract. In this paper, we consider a boundary value problem of Hadamard  fractional differential equations on an infinite interval
$^H\!D^\alpha u(t)+\lambda a(t)F(t,u(t))=0,~1<\alpha<2,~t\in(1,\infty),$
$u(1)=0,~{}^H\!D^{\alpha-1}u(\infty)=\sum_{i=1}^m\gamma_i{}^H\!I^{\beta_i}u(\eta),$
where $^H\!D^\alpha$ is the Hadamard fractional derivative of order $\alpha$, $\eta\in(1,\infty)$, $^H\!I(\cdot)$ denotes the  Hadamard fractional integral, $\lambda>0$ is a  parameter, $\beta_i, \gamma_i\geq0(i=1,2,\ldots,~m)$ are  constants and $\Gamma(\alpha)>\sum_{i=1}^m\frac{\gamma_i\Gamma(\alpha)}{\Gamma(\alpha+\beta_i)}(\log \eta)^{\alpha+\beta_i-1}.$ The existence and uniqueness of positive solutions is given  for each fixed $\lambda>0.$ The relations between the positive solution and the parameter $\lambda$ are presented.  The  results obtained in this paper show that the unique positive solution $u_\lambda^*$ has good properties: continuity, monotonicity, iteration and approximation. The method of this paper is based upon different fixed point theorems and properties for two types of operators: monotone operators and mixed monotone operators. Finally, two examples are also provided.