We consider continuous dependence of solutions on the right hand side for a semilinear operator equation Lx = ∇G(x), where L: D(L) ⊂ Y → Y (Y a Hilbert space) is self-adjoint and positive definite and G:Y → Y is a convex functional with superquadratic growth. As applications we derive some stability results and dependence on a functional parameter for a fourth order Dirichlet problem. Applications to P.D.E. are also given.
We prove an estimate of the kind , where , is the scattering amplitude related to the compactly supported potential at a fixed energy level const., , and is a suitably defined norm.
We investigate problems connected to the stability of the well-known Pohoˇzaev obstruction. We generalize results which were obtained in the minimizing setting by Brezis and Nirenberg [2] and more recently in the radial situation by Brezis and Willem [3].
By means of the fixed-point methods and the properties of the -pseudo almost periodic functions, we prove the existence, uniqueness, and exponential stability of the -pseudo almost periodic solutions for some models of recurrent neural networks with mixed delays and time-varying coefficients, where is a positive measure. A numerical example is given to illustrate our main results.
In this paper we consider the boundary value problem of some nonlinear Kirchhoff-type equation with dissipation. We also estimate the total energy of the system over any time interval with a tolerance level . The amplitude of such vibrations is bounded subject to some restrictions on the uncertain disturbing force . After constructing suitable Lyapunov functional, uniform decay of solutions is established by means of an exponential energy decay estimate when the uncertain disturbances are insignificant....