‎we consider nonconvex vector optimization problems with variable ordering structures in banach spaces‎. ‎under certain boundedness and continuity properties we present necessary conditions for approximate solutions of these problems‎. ‎using a generic approach to subdifferentials we derive necessary conditions for approximate minimizers and approximately minimal solutions of vector optimizatio...

We study a class of nonhomogeneous elliptic problems with Dirichlet boundary condition and involving the p(x)-Laplace operator and powertype nonlinear terms with variable exponent. The main results of this articles establish sufficient conditions for the existence of nontrivial weak solutions, in relationship with the values of certain real parameters. The proofs combine the Ekeland variational...

Ekeland and Scheinkman (1986) prove the necessity of a standard transversality condition under certain technical conditions. Their result is one of the most powerful on the necessity of a transversality condition currently available in the literature, and their proof involves numerous estimations and relies on Ekeland’s variational principle and Fatou’s lemma. This note relaxes some of their as...

This paper deals with the general optimal control problem for fully coupled forward-backward stochastic differential equations with random jumps FBSDEJs . The control domain is not assumed to be convex, and the control variable appears in both diffusion and jump coefficients of the forward equation. Necessary conditions of Pontraygin’s type for the optimal controls are derived by means of spike...

In this article, we consider the multiplicity of positive solutions for a class of Kirchhoff type problems with concave and convex nonlinearities. Under appropriate assumptions, we prove that the problem has at least two positive solutions, moreover, one of which is a positive ground state solution. Our approach is mainly based on the Nehari manifold, Ekeland variational principle and the theor...

This paper has a two-fold focus on proving that the quasimetric and the weak τ -distance versions of the Ekeland variational principle are equivalent in the sense that one implies the other and on presenting the need of such extensions for possible applications in the formation and break of workers hiring and firing routines. Mathematics Subject Classification (2010): 49J53, 47H10.

Equivalent formulations of some statements on subdifferentials and local e-supports of Ekeland and Lebourg are presented. A certain property of a Banach space concerning the local e-supports is shown to be separably determined. This fact is used to show that a Banach space is trustworthy in the sense of Ioffe if (and only if) it is Asplund.