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DIFFERENTIATING SOLUTIONS OF A BOUNDARY VALUE PROBLEM ON A TIME SCALE

  • LEE H. BAXTER (a1), JEFFREY W. LYONS (a2) and JEFFREY T. NEUGEBAUER (a3)

Abstract

We show that the solution of the dynamic boundary value problem $y^{{\rm\Delta}{\rm\Delta}}=f(t,y,y^{{\rm\Delta}})$ , $y(t_{1})=y_{1}$ , $y(t_{2})=y_{2}$ , on a general time scale, may be delta-differentiated with respect to $y_{1},~y_{2},~t_{1}$ and $t_{2}$ . By utilising an analogue of a theorem of Peano, we show that the delta derivative of the solution solves the boundary value problem consisting of either the variational equation or its dynamic analogue along with interesting boundary conditions.

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