We study the existence and global asymptotic behavior of positive continuous solutions to the following nonlinear fractional boundary value problem \[ (p_\lambda) \begin{cases} D^\alpha u(t)=\lambda f(t,u(t)),\,\,\,\,\, t\in (0,1),\\ \lim_{t\rightarrow 0^+}t^{2-\alpha} u(t)=\mu, \quad u(1)=\nu, \end{cases} \] where \(1 < \alpha\leq 2; D^\alpha\) is the Riemann-Liouville fractional derivative, and \(\lambda,\mu\) and \(\nu\) are nonnegative constants such that \(\mu + \nu > 0\). Our purpose is to give two existence results for the above problem, where \(f(t; s)\) is a nonnegative continuous function on \((0; 1)\times[0;\infty)\); nondecreasing with respect to the second variable and satisfying some appropriate integrability condition. Some examples are given to illustrate our existence results.