Periodic solutions for a differential inclusion problem involving the p(t)-Laplacian

In the present paper, we consider the nonlinear periodic systems involving variable exponent driven by p(t)-Laplacian with a locally Lipschitz nonlinearity. Our arguments combine the variational principle for locally Lipschitz functions with the properties of the generalized Lebesgue-Sobolev space. Applying the non-smooth critical point theory, we establish some new existence results.

On Lipschitz Continuity of Projections onto Polyhedral Moving Sets

In Hilbert space setting we prove local lipchitzness of projections onto parametric polyhedral sets represented as solutions to systems of inequalities and equations with parameters appearing both in left- and right-hand sides of the constraints. In deriving main results we assume that data are locally Lipschitz functions of parameter and the relaxed constant rank constraint qualification condition is satisfied.

One the Lagrange multipliers rule in problems with the equality type constraints, given by quasi-differentiable functions

In recent years, there has been a steadily growing interest in the study of extremal problems with parameters that do not satisfy the standard smoothness assumptions. This is due to both theoretical needs and important practical applications in economics, technology, physics, and other sciences. Rough objects naturally arise in a several areas of systems analysis, nonlinear mechanics, and control processes. In the theory of extremal problems, the main interest is the behavior of functions in the vicinity of points where a local extremum is attained. The local behavior of nonsmooth functions is described by subgradients, which are analogs of the derivative of differentiable functions. Using the concepts of subdifferential and subgradient F. Clarke proved the Lagrange multiplier rule in mathematical programming problems with constraints of the type of equalities and inequalities defined by locally Lipschitz functions. However, there are subclasses of locally Lipschitz functions, the simplest examples of which show that the necessary conditions for an extremum obtained by F. Clarke are rather crude and do not allow one to discard obviously non-optimal points. Such a subclass of nonsmooth functions is the subspace of quasi-differentiable functions. In this article, using the Eckland variational principle, we obtain the Lagrange multiplier rule in terms of quasi-differentials. It is shown by examples that this condition is stronger than the necessary condition of F. Clarke.

Infinitely Many Solutions for a Non-homogeneous Differential Inclusion with Lack of Compactness

In this paper, we consider the following class of differential inclusion problems in {\mathbb{R}^{N}} involving the {p(x)} -Laplacian: -\Delta_{p(x)}u+V(x)\lvert u\rvert^{p(x)-2}u\in a(x)\partial F(x,u)\quad\text{% in}\ \mathbb{R}^{N}. We are concerned with a multiplicity property, and our arguments combine the variational principle for locally Lipschitz functions with the properties of the generalized Lebesgue–Sobolev space. Applying the nonsmooth symmetric mountain pass lemma and the fountain theorem, we establish conditions such that the associated energy functional possesses infinitely many critical points, and then we obtain infinitely many solutions.