Boundary value problem for linear and nonlinear fractional differential equations

In this paper, we are interested in the boundary conditions u(0)=au(1),u′(0)=bu′(1) for linear fractional differential equation 0cDtαu(t)+λu(t)=0. Via Laplace transform and inverse Laplace transform, we obtain eigenvalues and eigenfunctions. Furthermore, we study the same boundary conditions for nonlinear fractional differential equation 0cDtαu(t)+f(t,u(t))=0. Combining the obtained eigenvalues and eigenfunctions and the improved Leray-Schauder degree, we prove that there exists at least one nontrivial solution for nonlinear boundary value problem.