A Variational Principle For A Class Of Function Without Lower Bounds

In the present paper we investigate the variational proplem of extended real-valued lower semicontinuous functions defined on real Banach spaces that are lower bounded on every bounded set,but may be not lower bounded on the whole space. As we all know, the varational problem about functions almost has the prerequisite having lower bound,however, the varational problem without lower bound has not been studied widely. Luo Daozhong studied a class of variational problems of functions without lower bounds.This paper discusses the variational questions concerning about a lower semicontinuous function without lower bound. First, we prove that adding to a monotonical function, or a continuous convex function, or a differential convex function can turn f into a lower bounded function. Secondly, we prove that the ratio of f andΦwithΦ> 0 is larger than a constant under the condition‖x‖→∞. Then f -αΦhas a lower bound. Using the varational theorem with lower bound, we get the varational theorem of functions without bound.In this paper we first review the developent and some significant result of variational principle. Then we simply introduce the notions that will be used in this paper. Then we introduce the main result of the paper.