Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function

Abstract

In this paper, we study the calculus of variations of the nabla notion on time scales including $\nabla$-derivative, $\nabla$-integral, and $\nabla$-derivatives of exponential function. The Euler-Lagrange equations of the first-order both single-variable problem and multivariable problem with nabla derivatives of exponential function on time scales are obtained. In particular, we show that the calculus of variations with multiple variables could solve the problem of conditional extreme value. Moreover, we verify the solution to the multivariable problem is exactly the extremum pair. As applications of these results, an example of conditional extremum is provided.