First order dynamic inclusions on time scales (Q1826773)

The authors deal with the multi-valued boundary value problem \[ y^\nabla(t)\in F(t, y(t))\quad\text{a.e. on }[a, b]_x,\qquad L(y(a), y(b))= 0,\tag{1} \] on a time scale \(\mathbb{T}\), where \([a,b]_x= \{t\in\mathbb{T}\mid a\leq t\leq b\}\), \(F: [a,b]_x\times \mathbb{R}\to \mathbb{R}\setminus\{0\}\) is a multi-valued map with compact and convex values and \(L: \mathbb{R}^2\to \mathbb{R}^2\) is a continuous single-valued map. The proof for the existence of a solution of (1) is based on the method of upper and lower solutions. The authors present some examples to illustrate that if one replaces the \(\nabla\)-derivative by the \(\Delta\)-derivative in the dynamic inclusion (1) then such a result holds only under more restrictive assumptions on \(F\).