Existence Of Solutions For Fractional Difference Equation With Fractional Boundary Conditions

In recent years, because of the appearing of mathematical model of fractional differential equations and the need for the approximate calculation of fractional differential equations, the study of differential equations of fractional rises recently. Fractional differential equations have been widely used in natural sciences, biology and ecology, engineering and many other fields. Fractional model can describe the phenomena and react the nature of certain objects more accurately than the integer-order model, so it becomes a concern research. The study of differential equations is to improve the functional equation theory and application of basic work to provide help and support on the theories and methods for the study of other functional equation.We study the existence of solution of two fractional difference equations with fractional boundary conditions, first we introduce the research background and the work. Then we introduce the basic concepts of fractional differences and sums, the related properties and the theorem of functional analysis, we transform the equation and into equivalent expressions of sum to get the expressions of the solutions. Next we obtain the Green’s functions of two fractional difference equations, and prove the characteristics of the corresponding Green’s function. Finally we prove the existence and uniqueness of solution to problem (1),(2) by using contraction mapping theorem, Brouwer fixed point theorem and fixed point theorems of cone expansion and compression; then prove the existence and uniqueness of solution to problem (3),(4) by using contraction mapping theorem, fixed point theorems of cone expansion and compression, and Krasnosel’skii fixed point theorem.