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Calculus of variations on time scales with contingent epiderivatives

We propose the use of the contingent epiderivative as a tool of differentiation on the calculus of variations on time scales. By introducing the problem L(y) = ∫ b a L(t, y∗(t), D↑y(t)(u))d↑t −→ min y(a) = ya , y(b) = yb , (1) we are able to unify the two approaches explored so far in the literature of time scales. Indeed, for u = 1 problem (1) is reduced to the delta problem of the calculus of variations studied in [2, 3], ∫ b a L(t, y(t), y(t))∆t −→ min , y(a) = ya , y(b) = yb , while for u = −1 we get the nabla variational problem on time scales [1, 5]: ∫ b a L(t, yρ(t),−y∇(t))∇t −→ min , y(a) = ya , y(b) = yb . Moreover, studying (1) with u = ±1 as a two-objective optimization problem we are able to extend the ideas of [4] and prove new necessary and sufficient Pareto optimality conditions on time scales. Some illustrative examples are given.