In this paper, we study the calculus of variations of the nabla notion on time scales including ∇-derivative, ∇-integral, and ∇-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.
In this paper, we investigate the existence and nonexistence of positive solutions for nonlinear fractional differential equation boundary value problem. By means of fixed-point theorems on a cone and the properties of Green function, some sufficient criteria are established. Our results can be considered as an extension of some previous results. c 2017 all rights reserved.Keywords: Positive solution, fractional differential equation, boundary value problem, the Krasnoselskii fixed point theorem. 2010 MSC: 93B05, 93C05, 93C55.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.