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In this paper we consider a fractional difference inclusion of the form(α  -  1Δ  α   y  )    (  t  )   ∈  λ  y   (  t  +  α  -  1  )   +  f   (  t  +  α  -  1  ,  y   (  t  +  α  -  1  )   )$({\,}_{\alpha -1}\Delta ^{\alpha }y)(t) \in \lambda y(t+\alpha -1)+f(t+\alpha -1,y(t+\alpha -1))$   ,y   (  α  -  1  )   =  y   (  α  -  1  +  T  )   +   ∑   j  =  1   N    F  j    (  y   (   t  j   )   )$y(\alpha -1)=y(\alpha -1+T)+\sum _{j=1}^{N}F_j(y(t_j))$   , where0  &lt;  α  &lt;  1    ${0&amp;lt;\alpha &amp;lt;1}$   . Consequently, we treat the situation in which the boundary condition is both nonlocal and nonlinear. Thus, the boundary conditions under study can be very general. We provide conditions under which this problem has at least one positive solution and then discuss the application of the results. Finally, since we allowF  j   ≡  0    ${F_j\equiv 0}$   , for each  j , our results are also new for the case of periodic boundary conditions.

On positive solutions to fractional difference inclusions