‎In this paper we study the existence of nontrivial‎ solutions for a variational inequality on the half-line‎. ‎Our‎ ‎approach is based on the non-smooth critical point theory‎ ‎for Szulkin-type functionals.

In this paper we prove a generalized version of the Ekeland variational principle, which is a common generalization of Zhong variational principle and Borwein Preiss Variational principle. Therefore in a particular case, from this variational principle we get a Zhong type variational principle, and a Borwein-Preiss variational principle. As a consequence, we obtain a Caristi type fixed point th...

Variational principle for probabilistic learning Yet another justification More simplification of updates for mean-field family Examples Dirichlet Process Mixture On minimization of divergence measures Energy minimization justifications Variational learning with exponential family Mean parametrization and marginal polytopes Convex dualities The log-partition function and conjugate duality Belie...

Abstract In this paper, we consider stochastic optimal control of systems driven by differential equations with irregular drift coefficient. We establish a necessary and sufficient maximum principle. To achieve this, first derive an explicit representation the variation process (in Sobolev sense) controlled diffusion. Since coefficient is not smooth, given in terms local time state process. The...

In this paper we study a non-homogeneous eigenvalue problem involving variable growth conditions and a sign-changing potential. We prove that any λ > 0 sufficiently small is an eigenvalue of the nonhomogeneous eigenvalue problem { −div(a(|∇u|)∇u) = λV(x)|u|q(x)−2u, in Ω, u = 0, on ∂Ω. The proofs of the main results are based on Ekeland’s variational principle.

We consider the nonlinear eigenvalue problem −div ( |∇u|∇u ) = λ|u|u in Ω, u = 0 on ∂Ω, where Ω is a bounded open set in R with smooth boundary and p, q are continuous functions on Ω such that 1 < infΩ q < infΩ p < supΩ q, supΩ p < N , and q(x) < Np(x)/ (N − p(x)) for all x ∈ Ω. The main result of this paper establishes that any λ > 0 sufficiently small is an eigenvalue of the above nonhomogene...