Calculus of variations on time scales with nabladerivatives of exponential function

Abstract

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