We consider a mixed problem with Dirichlet and integral conditions for a second-order hyperbolic equation with the Bessel operator. The existence, uniqueness, and continuous dependence of a strongly generalized solution are proved. The proof is based on an a priori estimate established in weighted Sobolev spaces and on the density of the range of the operator corresponding to the abstract formulation of the considered problem.

We deal with the exact determination of eigenfrequencies of a beam with intermediate elastic constraints and generally restrained ends. It is the purpose of this paper to use the calculus of variations to obtain the equations of motion and the natural boundary conditions, and particularly those at the intermediate constraints. Numerical values for the first five natural frequencies are presented in a tabular form for a wide range of values of the restraint parameters. Several particular cases are presented and some of these cases have been compared with those available in the literature.

We prove the theoretical convergence of a short-step, approximate path-following, interior-point primal-dual algorithm for semidefinite programs based on the Gauss-Newton direction obtained from minimizing the norm of the perturbed optimality conditions. This is the first proof of convergence for the Gauss-Newton direction in this context. It assumes strict complementarity and uniqueness of the optimal solution as well as an estimate of the smallest singular value of the Jacobian.

We consider a nonconvex variational problem for which the set of admissible functions consists of all Lipschitz functions located between two fixed obstacles. It turns out that the value of the minimization problem at hand is equal to zero when the obstacles do not touch each other; otherwise, it might be positive. The results are obtained through the construction of suitable minimizing sequences. Interpolating these minimizing sequences in some discrete space, a numerical analysis is then carried out.