The authors investigate solvability of the initial fractional difference problem \[ \nabla^\alpha_{t_0}y(t)=f(t,y(t)),\quad y(t_0)=y_0,\tag{\(*\)} \] for \(t=t_0+1,t_0+2,\dots,t_0+n\). Here, \(\nabla^\alpha_{a}\), \(0<\alpha<1\), is the Riemann-Liouville fractional difference operator. First, conditions on the function \(f\) are given (essentially continuity and \(0<\partial f/\partial y<1\) for \(y\in {\mathbb R}\) and \(t=t_0+j\), \(j=0,\dots,n\)) which guarantee that the lower and upper solutions of (\(*\)) are well ordered and then it is shown that solutions of (\(*\)) stay between lower and upper solutions. In the final part of the paper, using a quasi-linearization method, a sequence of lower and upper solutions is constructed which converges to a solution of (\(*\)). The authors are motivated by the paper of A. Cabada et al. [Comput. Math. Appl. 39, No. 1–2, 21–33 (2000; Zbl 0972.39002)], where the first-order difference equation is considered.