On the Nonlocal Problem for the Equation with the Hilfer Fractional Derivative

Abstract

In the paper, we study the nonlocal problem for a fractional partial differential equation with the Hilfer derivative. The non-local boundary value problem, \(D^{\alpha,\beta}u(t)+Au(t)=f(t)\) (\(0<\alpha<1\), \(0\leq\beta\leq 1\) and \(0<t\leq T\)), \(I^{\delta}u(t)=\gamma I^{\delta}u(+0)+\varphi\) (\(\gamma\) is a constant), in an arbitrary separable Hilbert space H with the strongly positive self-adjoint operator \(A\), is considered. In addition to the forward problem, the article also explores the inverse problem of determining the right-hand side of the equation. Existence and uniqueness theorems are proved to solve the forward and inverse problems.