Langevin equation involving two fractional orders with three-point boundary conditions.
The authors give existence and uniqueness results for the equation \[D^{\beta}(D^{\alpha}+\lambda)u(t)=f(t,u(t))\] where \(t\in[0,1]\) and the boundary conditions are \(u(0)=0\), \(D^{\alpha}u(0)=0\) and \(u(1)=\mu u(\eta)\). \(D^{\alpha}\) and \(D^{\beta}\) are fractional derivatives of orders \(0<\alpha\leq 1\) and \(1<\beta\leq 2\), \(\mu\), \(\lambda\) are real numbers and \(0<\eta<1\) with \(1-\mu\eta^{\alpha+1}\) not zero. First, the authors give some elementary properties of fractional integrals and fractional derivatives. In the first result it is assumed that \(f\) is globally Lipschitz in the second variable. Under certain smallness conditions on the Lipschiz constant and \(\lambda\) the contraction principle can be applied to the equation. If the function \(f\) is assumed to be uniformly bounded in the second variable the Krasnoselskii fixed point theorem can be applied. Under rather strong assumptions on the nonlinearity \(f\) the Leray-Schauder degree theory can be applied. All these applications give existence results (some give also uniqueness) for the equation above.
scientific article; zbMATH DE number 7217168