Calculus of variations on time scales with nabla derivatives

Abstract

We prove a necessary optimality condition of Euler-Lagrange type for variational problems on time scales involving nabla derivatives of higher-order. The proof is done using a new and more general fundamental lemma of the calculus of variations on time scales.

The variational calculus on time scales

“…The following fundamental lemma of the calculus of variations on time scales, involving a nabla derivative and a nabla integral, has been proved in [43].…”

“…Ifŷ ∈ C 2 ld , then nabla-differentiating (21) we obtain the Euler-Lagrange differential equation (4) as proved in [43]:…”

“…Motivated by applications in economics [4,7], a different formulation for the problems of the calculus of variations on time scales has been considered, which involve a functional with a nabla derivative and a nabla integral [3,6,43]:…”

“…An example of this is given by the time scale versions of the Euler-Lagrange equations: if y ∈ C 2 rd is an extremizer of (1), then y satisfies the deltadifferential equation ∆ ∆t ∂ 3 L t, y σ (t), y ∆ (t) = ∂ 2 L t, y σ (t), y ∆ (t) (3) for all t ∈ [a, b] κ 2 [12]; if y ∈ C 2 ld is an extremizer of (2), then y satisfies the nabla-differential equation ∇ ∇t ∂ 3 L t, y ρ (t), y ∇ (t) = ∂ 2 L t, y ρ (t), y ∇ (t) (4) for all t ∈ [a, b] κ 2 [43], where we use ∂ i L to denote the standard partial derivative of L(·, ·, ·) with respect to its ith variable, i = 1, 2, 3. In the classical context T = R one has J ∆ (y) = J ∇ (y) = b a L (t, y(t), y (t)) dt (5) and both (3) and (4) coincide with the standard EulerLagrange equation: if y ∈ C 2 is an extremizer of the integral functional (5), then d dt ∂ 3 L (t, y(t), y (t)) = ∂ 2 L (t, y(t), y (t)) for all t ∈ [a, b].…”

“…Theorem 30. Ifŷ ∈ C 1 is an extremizer for the isoperimetric problem (41)- (43), then there exist two constants λ 0 and λ, not both zero, such thatŷ satisfies the following delta-nabla integral equations:…”