A Vectorial Ekeland's Variational Principle In Fréchet Spaces And Its Equivalent Theorems

In the framework of Fr(?)chet spaces,by improving the methods of Phelps and using Gerstewiz function,we give two versions of vectorial Ekeland's variational principle concerning vector-valued functions,which take values in partial ordered locally convex spaces and the perturbation of the principle contains countable generating semi-norms. As their application,we give a result of sharp efficient solution in vector optimization. From the vectorial variational principle,we deduce the vectorial Caristi's fixed points theorem and vectorial Takahashi's minimization theorem on Fr(?)chet spaces,and prove the equivalence among the three theorems.Then we apply the above results to discuss the density of extremal points in the variational principle.By modifying and developing the method of Cammaroto and Chinni,we obtain a number of density results on extremal points of the vector-valued variational principle,which extend and improve the related known results.Futhermore,we deduce the density results of corresponding Caristi's fixed points and Takahashi's minimization points.Finally,using the sequence of functions(φ_n)_n∈N(hereφ_n:[0,∞)→[0,∞) is a subadditive,nondecreasing and lower semicontinuous function),we obtain more general extensions of the above results.