Recently, initial-boundary-value problems with the Caputo fractional derivative for ordinary differential equations have been intensively studied. This paper studies a nonlinear integro-differential equation of fractional order, containing a composition of fractional derivatives with different origins and mixed conditions. The equation under consideration acts as a model equation of motion in a fractal medium. First, we use three fixed-point theorems to prove the existence and uniqueness results. Then, the Ulam stability criterion of the solution is given. The main results will be illustrated by a proposed example.

In the present paper, we examine the existence of solutions to fourth-order differential equation systems when the $L^1$-Carathéodory function is on the right-hand side. A concept of solution-tube for these issues is presented. The concepts of upper and lower solutions for fourth-order differential equations are extended to systems owing to this idea.