By developing a method of Lieb's compactness Theorem with refined ernergy estimation, we overcome the lack of translation invariance and establish existence of ground states. Our unified approach handles both subcritical and doubly critical cases under nonsymmetric perturbations.
In this paper, we address a frictionless contact problem involving a viscoelastic body and an obstacle. The process is assumed to be quasi-static. The contact is modeled with normal compliance and unilateral stress, incorporating the Signorini contact condition. We derive a variational formulation of the model and demonstrate the existence and uniqueness of a weak solution using the theory of variational inequalities and fixed point arguments. By converting the initial contact problem into a minimization framework, we utilize a deep neural network to approximate the solution and address the minimization challenge. Finally, we provide numerical results that showcase the method's efficiency and precision.
The primary objective of this paper is to achieve strong convergence toward a zero of a maximally monotone operator in a Hilbert space. To this end, we first propose a continuous dynamic system that builds upon the framework presented by Bot & cedil; and Nguyen (2025), while remaining distinct from it. Specifically, we apply a positive linear operator to the velocity term x(center dot)(t). This modification enables the adaptation of a suitable proximal algorithm when applying the proposed system to the context of the composite minimization problem f (x) +g(Ax), where f and g are convex functions and A is a linear operator, and then to convex minimization problems under linear constraints, via a primal-dual algorithm by Chambolle and Pock (2011) for saddle points of the associated Lagrangian. The continuous model is analyzed for a single-valued operator M, allowing us to illustrate-through an appropriate discretizationthe corresponding proximal point algorithm for a set-valued operator and to provide a consistent proof for the suitable convergence rates. The algorithmic contribution of this work is particularly significant, as it establishes strong convergence to the minimum norm of zeros of M without requiring additional conditions on its maximal monotonicity, a result that remains absent from more recent literature. We then provide strong convergence to the minimum norm solution, and also the same rate of convergence for values and constraints for the composite minimization problem and convex minimization under linear constraints.
In this paper, we study the following nonlocal problem {( a-lambda integral(Omega) |del u|(2) dx Delta u+ f(u) = 0, in Omega, u = 0, on partial derivative Omega, where Omega subset of R-N (N >= 1) is a bounded smooth domain, a and lambda are positive parameters, and f is of subcritical or critical growth. By using the method of invariant sets for the descending flow, perturbation technique and necessary estimates, we establish the existence of infinitely many sign-changing solutions.
Semivectorial bilevel optimization problems, particularly those where the lower-level problem is solved up to efficiency, have attracted significant attention in optimization theory. While previous research in this area has often focused on problems with continuously differentiable functions, many real-world applications of bilevel optimization feature non-smooth functions. Motivated by this observation, the current study aims to propose new necessary optimality conditions for semivectorial bilevel programs under weaker regularity assumptions that allow for non-differentiability. Specifically, we revisit the Charnes-Cooper scalarization technique and present tailored optimality results for problems where the data satisfy only local Lipschitz continuity. Through this generalization, our results provide a more flexible theoretical framework applicable to a broader class of non-smooth optimization models arising in practical semivectorial bilevel programming contexts.
