In this paper, a new version of Ekeland's variational principle by using the concept of $\tau$-distance for bounded from below functions which are not necessarily lower semicontinuous is provided. This new version of Ekeland's variational principle will be applied to establish an existence theorem for a solution to the equilibrium problem in the setting of complete metric spaces.

In this paper we prove that if X is a Banach space, then for every lower semi-continuous bounded below function f; there exists a ('1; '2)-convex function g; with arbitrarily small norm, such that f + g attains its strong minimum on X: This result extends some of the well-known varitional principles as that of Ekeland [On the variational principle, J. Math. Anal. Appl. 47 (1974) 323{ 353], that of Borwein-Preiss [A smooth variational principle with applications to subdi erentiability and to di erentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987) 517{527] and that of Deville-Godefroy-Zizler [Un principe variationel utilisant des fonctions bosses, C. R. Acad. Sci. (Paris). Ser. I 312 (1991) 281{286] and [A smooth variational principle with applications to Hamilton-Jacobi equations in in nite dimensions, J. Funct. Anal. 111 (1993) 197{212].

The Kaup-Newell system describes the pulse propagation in optical fibers. This paper applies the semi-inverse method to construct the system’s variational formulations, two trial-variational formulations with an unknown function are established, and a detailed derivation is given to determine the unknown function. Finally, the Kaup-Newell system with two variables is converted to a partial differential equation with only one variable.

In this paper, we use the auxiliary principle technique to study the existence of a solution of the extended general variational inequalities, which were introduced and studied by the author. Several special cases are also discussed.

The variational theory is an inextricable part of both continuum mechanics and physics, and plays an important role in mathematics and nonlinear science, however it is difficult to find a variational formulation for a nonlinear system, and it is more difficult for a fractional differential system. This paper is to search for a variational formulation for the Schrödinger-KdV system with M-fractional derivatives. The fractional complex transformation is used to convert the system into a traditional differential system, and the semi-inverse method is further applied to establish a needed variational principle.

A nonlinear oscillator arising in the microelectromechanical system is complex and it is difficult to obtain a variational principle. This paper begins with a wrong variational formulation and uses the semi-inverse method to obtain a genuine variational principle. Additionally, this paper gives simple formula for the fast frequency estimation of the nonlinear oscillator. Only simple calculation is needed to have a relatively high accuracy results when compared with the other methods.

The basic governing equations in anisotropic elastic material under the effect of porous piezothermoelastic are presented. Biot [1], Lord & Shulman [4] and Sherief et al. [5] theories are used to develop the basic equations for porous piezothermoelastic with mass diffusion material. The variational principle, uniqueness theorem and theorem of reciprocity in this model are established under the assumption of positive definiteness of elastic, porousthermal, chemical potential and electric field.

A double ambiguity has been charged against Rawls’s difference principle (DP). Is it Maximin, Leximin, or something else? Usually, following A. Sen, scholars identify DP with the so-called Leximin. One argues here that one has to distinguish 1° the Leximin, 2° the Maximin (as rule of justice formally analogous to the maximin rule of decision), represented by the figure in L of the perfectly substitutable goods, and 3° the genuine DP. When the augmentation of inequality benefits the worse off, only Pareto-strong improvements are permitted. Leximin would also permit Pareto-weak improvements too (after the first maximum D), where only the richest improves: from (2, 3) to (2, 5), say. This is forbidden by DP. With two classes, unlike Maximin, DP has no curve of indifference and is always decisive, as Leximin is. For undecisive Rules of Justice, which admit indifferent curves, I propose to add a lexically secondary rule, to break ties. That move is able to clarify the links and the differences between on the one hand Maximin alone, with its typical indifference curves in L, and on the other hand, the DP properly understood and the Leximin, which both have no indifferent curves. With two classes of persons (best off/worse off), DP appears more egalitarian than Leximin, because it's secondary rule is MinIn (Minimization of Inequality). But the intuition behind the distinction is that it cannot possible “fair” that only the best off improves in a productive social cooperation.