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The aim of the present paper is to establish a variational principle in metric spaces without assumption of completeness when the involved function is not lower semicontinuous. As consequences, we derive many fixed point results, nonconvex minimization theorem, a nonconvex minimax theorem, a nonconvex equilibrium theorem in noncomplete metric spaces. Examples are also given to illustrate and to show that obtained results are proper generalizations.

We study a Neumann problem with the Hardy–Sobolev nonlinearity. In boundary singularity case, the impact of the mean curvature at singularity on existence of least-energy solution is well known. Existence and nonexistence of least-energy solution is studied by Hashizume (2017) except for lower dimension case. In this paper, we improve this previous work. More precisely, we study four dimensional case and show existence of minimizer in critical case in some sense. [ABSTRACT FROM AUTHOR]

Abstract: Let be a Hilbert space. Consider on a sequence of nonexpansive mappings with common fixed points, an equilibrium function , a contraction with coefficient and a strongly positive linear bounded operator with coefficient . Let . We define a suitable Mann type algorithm which strongly converges to the unique solution of the minimization problem , where is a potential function for and is the intersection of the equilibrium points and the common fixed points of the sequence . [Copyright &y& Elsevier]

Abstract: In this paper we consider the Levitin–Polyak well-posedness of variational inequalities. We derive a characterization of the Levitin–Polyak well-posedness by considering the size of Levitin–Polyak approximating solution sets of variational inequalities. We also show that the Levitin–Polyak well-posedness of variational inequalities is closely related to the Levitin–Polyak well-posedness of minimization problems and fixed point problems. Finally, we prove that under suitable conditions, the Levitin–Polyak well-posedness of a variational inequality is equivalent to the uniqueness and existence of its solution. [Copyright &y& Elsevier]

Abstract: In this paper we use the penalty approach in order to study minimization problems with mixed constraints in Banach spaces. A penalty function is said to have the generalized exact penalty property if there is a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. In this paper we show that the generalized exact penalty property holds and is stable under perturbations of objective functions, constraint functions and the right-hand side of constraints. [Copyright &y& Elsevier]

Abstract: Let be a Hilbert space. Consider on a sequence of nonexpansive mappings with common fixed points, a finite family of equilibrium functions , a contraction with coefficient and a strongly positive linear bounded operator with coefficient . Let . Assuming there are common equilibrium points of the family which are also fixed points for , we define a suitable sequence which strongly converges to the unique such point which also satisfies the variational inequality for all the in the intersection of the equilibrium points and the common fixed points of the sequence . [Copyright &y& Elsevier]

We solve explicitly a certain minimization problem for probability measures in one dimension involving an interaction energy that arises in the modeling of aggregation phenomena. We show that in a certain regime minimizers are absolutely continuous with an unbounded density, thereby settling a question that was left open in previous works. [ABSTRACT FROM AUTHOR]

We consider the functional ∫ Ω g (∇ u + X ∗) d L 2 n where g is convex and X ∗ (x , y) = 2 (− y , x) and we study the minimizers in BV (Ω) of the associated Dirichlet problem. We prove that, under the bounded slope condition on the boundary datum, and suitable conditions on g , there exists a unique minimizer which is also Lipschitz continuous. The assumptions on g allow to consider both the case with superlinear growth and the one with linear growth. Moreover neither uniform ellipticity nor smoothness of g are assumed. [ABSTRACT FROM AUTHOR]

Abstract: Let be a real Hilbert space. Consider the iterative sequence where is some constant, is a given contractive mapping, is a strongly positive bounded linear operator on and is the -mapping generated by an infinite countable family of nonexpansive mappings and such that the common fixed points set . Under very mild conditions on the parameters, we prove that converges strongly to where is the unique solution in of the following variational inequality: which is the optimality condition for the minimization problem [Copyright &y& Elsevier]

Abstract: In this paper we use the penalty approach in order to study two constrained minimization problems in Hilbert spaces. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we establish a sufficient condition for the exact penalty property. [Copyright &y& Elsevier]

Abstract: In this paper we study mathematical programming problems with mixed smooth constraints in a Banach space and show that most of the problems (in the Baire category sense) are well-posed. Our result is a generalization of a result of A.D. Ioffe et al. [A variational principle for problems with functional constraints, SIAM J. Optim. 12 (2001) 461–478] obtained for finite dimensional Banach spaces. It is also an extension of our recent result [A.J. Zaslavski, Generic well-posedness of minimization problems with mixed continuous constraints, Nonlinear Anal., doi:10.1016/j.na.2005.10.032] which was obtained for mathematical programming problems with continuous constraints. [Copyright &y& Elsevier]

Abstract: In this paper we study mathematical programming problems with mixed constraints in a Banach space and show that most of the problems (in the Baire category sense) are well-posed. Our result is a generalization of a result of Ioffe et al. [SIAM J. Optim. 12 (2001) 461–478] obtained for finite-dimensional Banach spaces. [Copyright &y& Elsevier]

Let X be a real separable strictly convex Banach space and G a nonempty closed subset of X. Let K(X) (resp. Kb(X)) denote the family of all nonempty boundedly compact (resp. compact) convex subsets of X endowed with the Hρ-topology (resp. the Hausdorff distance), KG(X) (resp. KGb(X)) the closure of the set {A∈K(X): A∩G=} (resp. {A∈Kb(X): A∩G=}), and V(G) (resp. Vb(G)) the family of A∈KG(X) (resp. A∈KGb(X)) such that the minimization problem min(A,G) fails to be well-posed. It is proved that for most (in the sense of the Baire category) closed subsets (resp. bounded closed subsets) G of X, V(G) (resp. Vb(G)) is everywhere uncountable in KG(X) (resp. KGb(X)). A similar result for the mutually furthest point problem is also given. [Copyright &y& Elsevier]