The following fractional difference boundary value problems▵νyt=-ft+ν-1,yt+ν-1,y(ν-2)=y(ν+b+1)=0are considered, where1<ν≤2is a real number and▵νy(t)is the standard Riemann-Liouville fractional difference. Based on the Krasnosel’skiǐ theorem and the Schauder fixed point theorem, we establish some conditions onfwhich are able to guarantee that this FBVP has at least two positive solutions and one solution, respectively. Our results significantly improve and generalize those in the literature. A number of examples are given to illustrate our main results.

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We consider the existence of positive solutions to the nonlinear fractional differential equation boundary value problemD0+αCut+fut,u't=0,  t∈0,1,  u0=u1=u″0=0, wheref:0,+∞×R→0,+∞is continuous,α∈2,3, andD0+αCis the standard Caputo differentiation. By using fixed point theorems on cone, we give some existence results concerning positive solutions. Here the solutions especially are the interior points of cone.

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We investigate the existence of solutions for a Caputo fractional difference equation boundary value problem. We use Schauder fixed point theorem to deduce the existence of solutions. The proofs are based upon the theory of discrete fractional calculus. We also provide some examples to illustrate our main results.