The authors study fractional differential equations with boundary condition given as \[ D^{\alpha}_{0+}x(t)= f(t,x(t)), t\in [0,+\infty), \eqno (1) \] \[ \lim_{t\rightarrow 0+} t^{1-\alpha}x(t)=x_{0}, \eqno (2) \] where \(\alpha\in(0,1), D^{\alpha}_{0+}\) is the Riemann-Liouville fractional derivative, \(f: [0,+\infty)\times \mathbb{R}^{s}\rightarrow \mathbb{R}^{s}\) satisfies a Lipschitz type condition. In order to find solutions of the problem (1) - (2) the authors construct a special function space \(C_{1-\alpha}([0,+\infty); \mathbb{R}^{s}).\) To solve the problem (1) - (2) the authors use Banach’s theorem to prove the existence and uniqueness of a solution of the equivalent integral equation. Then the authors consider the differential equation \[ D^{\alpha}_{0+}x(t)= Ax(t)+Q(t)x(t)+g(t), t\in [0,+\infty), \eqno (3) \] where \(A\in\mathbb{R}^{s\times s}\) and \(Q:[0,+\infty)\rightarrow \mathbb{R}^{s\times s}, g:[0,+\infty)\rightarrow \mathbb{R}^{s}\) are continuous. The authors prove the globally attractivity of equation (3) in the sense of convergence to zero of solutions at infinity.