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First order dynamic inclusions on time scales. (English) Zbl 1064.34009
The authors deal with the multi-valued boundary value problem $$y^\nabla(t)\in F(t, y(t))\quad\text{a.e. on }[a, b]_x,\qquad L(y(a), y(b))= 0,\tag1$$ on a time scale $\bbfT$, where $[a,b]_x= \{t\in\bbfT\mid a\le t\le b\}$, $F: [a,b]_x\times \bbfR\to \bbfR\setminus\{0\}$ is a multi-valued map with compact and convex values and $L: \bbfR^2\to \bbfR^2$ 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 $F$.

MSC:
34A60Differential inclusions
34B15Nonlinear boundary value problems for ODE
34A45Theoretical approximation of solutions of ODE
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References:
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