In this paper, we study the existence and uniqueness of solutions for the boundary value problem of fractional difference equations { ? Δ ν y ( t ) = f ( t + ν ? 1 , y ( t + ν ? 1 ) ) , y ( ν ? 3 ) = 0 , Δ y ( ν ? 3 ) = 0 , y ( ν + b ) = g ( y ) and { ? Δ ν y ( t ) = λ f ( t + ν ? 1 , y ( t + ν ? 1 ) ) , y ( ν ? 3 ) = 0 , Δ y ( ν ? 3 ) = 0 , y ( ν + b ) = g ( y ) , respectively, where t = 1 , 2 , … , b , 2 < ν ≤ 3 , f : { ν ? 1 , … , ν + b } × R → R is a continuous function and g ∈ C ( [ ν ? 3 , ν + b ] Z ν ? 3 , R ) is a continuous functional. We prove the existence and uniqueness of a solution to the first problem by the contraction mapping theorem and the Brouwer theorem. Moreover, we present the existence and nonexistence of a solution to the second problem in terms of the parameter λ by the properties of the Green function and the Guo-Krasnosel’skii theorem. Finally, we present some examples to illustrate the main results. MSC:34A08, 34B18, 39A12.
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