On the microgravity condition, gravity affects the motion of objects and the flow of fluids, and the continuum assumption is not valid, therefore, a fractal Chaplygin-He gas model is developed by a new fractal derivative in microgravity space. A fractal Variational principle is successfully established via the fractal semiinverse method.

The present study is concerned with the Variational principle and plane wave propagation in double porous thermoelastic infinite medium. Lord-Shulman theory [2] of thermoelasticity with one relaxation time has been used to investigate the problem. It is found that for two dimensional model, there exists four coupled longitudinal waves namely longitudinal wave (P), longitudinal thermal wave (T), longitudinal volume fractional wave corresponding to pores (PV1), and longitudinal volume fractional wave corresponding to fissures (PV2), in addition to, a transverse wave (S) which is not affected by the volume fraction fields and thermal properties. The different characteristics of the wave such as phase velocity and attenuation quality factor are computed numerically and depicted graphically. Some special cases are also deduced from the present investigation.

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].

In this paper, we consider different types of generalized vector Variational-like inequalities and study the relationships between their  olutions . We study the general forms of Stampacchia and Minty type vector Variational inequalities for bifunctions and establish the existence of their solutions in the setting of topological vector spaces. We extend these vector Variational inequalities for the Clarke’s sub differential of non-differentiable locally Lipschitz functions and prove the existence of their solutions. As applications, we establish some existence results for a solution of the vector optimization problem by using Stampacchia and Minty type vector Variational inequalities.

In this paper, we examine the perturbed absolute value Variational inequalities (PAVVI), a new class of Variational inequalities. For the (PAVVI), some new merit functions are established. We develop the error bounds for (PAVVI) using these merit functions. The results presented here recapture a number of previously established findings in the relevant fields because (PAVVI) include Variational inequalities, the absolute value complementarity problem, and systems of absolute value equations as special cases.

In this paper, we propose a split nonconvex Variational inequality problem which is a natural extension of split convex Variational inequality problem in two different Hilbert spaces. Relying on the prox-regularity notion, we introduce and establish the convergence of an iterative method for the new split nonconvex Variational inequality problem. Further, we also establish the convergence of an iterative method for the split convex Variational inequality problem. The results presented in this paper are new and different form the previously known results for nonconvex (convex) Variational inequality problems. These results also generalize, unify, and improve the previously known results of this area.