Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions
In this paper, we consider a discrete fractional boundary value problem of the form ???y(t)=f(t+??1,y(t+??1)) ? ? ? y ( t ) = f ( t + ? ? 1 , y ( t + ? ? 1 ) ) , y(??2)=g(y) y ( ? ? 2 ) = g ( y ) , y(?+b)=0 y ( ? + b ) = 0 , where f:[??1,,?+b?1]N??2R?R f : [ ? ? 1 ,  , ? + b ? 1 ] N ? ? 2  R ? R is continuous, g:C([??2,?+b]N??2,R) g : C ( [ ? ? 2 , ? + b ] N ? ? 2 , R ) is a given functional, and 1<??2 1 < ? ? 2 . We give a representation for the solution to this problem. Finally, we prove the existence and uniqueness of solution to this problem by using a variety of tools from nonlinear functional analysis including the contraction mapping theorem, the Brouwer theorem, and the Krasnoselskii theorem.
existence and uniqueness of solution boundary value problem discrete fractional calculus nonlocal boundary conditions positive solution