In this paper we extend the definition of K-positive definite operators from linear to Fréchet differentiable operators. Under this setting, we derive from the inverse function theorem a local existence and approximation results corresponding to those of Theorems 1 and 2 of the authors [8], in an arbitrary real Banach space. Furthermore, an asymptotically K-positive definite operator is introdu...

The classical Landau-Lifshitz equation with a damping term has been derived from the time evolution of a quantum mechanical wave function under the assumption of a non-Hermitian Hamilton operator. Further, the trajectory of a classical spin (S) has been compared with the expectation value of the spin operator (Ŝ). A good agreement between classical and quantum mechanical trajectories can be fou...

For an infinite-dimensional linear quadratic control problem in Hilbert space, approximation of the solution of the algebraic Riccati operator equation in the strong operator topology is considered under conditions weaker than uniform exponential stability of the approximating systems. As an application, strong convergence of the approximating Riccati operators in case of a previously developed...

For the nonlocal T-periodic Gross–Pitaevsky operator, formal solutions of the Floquet problem asymptotic in small parameter , → 0, up to O(3/2) have been constructed. The quasi-energy spectral series found correspond to the closed phase trajectories of the Hamilton–Ehrenfest system which are stable in the linear approximation. The monodromy operator of this equation has been constructed to with...

In this article we analyze a fully discrete numerical approximation to a time dependent fractional order diffusion equation which contains a non-local, quadratic non-linearity. The analysis is performed for a general fractional order diffusion operator. The non-linear term studied is a product of the unknown function and a convolution operator of order 0. Convergence of the approximation and a ...

In this paper we construct the scattering operator for the forced non-linear Schrödinger equation with a potential on the half-line. Moreover, in the case where the force is zero, and the solutions satisfy the homogeneous Dirichlet boundary condition at zero, we prove that the scattering operator determines uniquely the potential and the non-linearity and we give a method for the reconstruction...

The unique solvability in the sense of classical solutions for nonlinear inverse problems to differential equations, solved oldest Dzhrbashyan–Nersesyan fractional derivative, is studied. linear part equation contains a bounded operator, continuous operator that depends on lower-order derivatives, and an unknown element. problem given by equation, special initial value conditions lower overdete...

We consider the flow of Burgers' equation on an open set (small) functions in $L^2([0,1])$. derive explicitly Koopman decomposition flow. identify frequencies and coefficients this as eigenvalues eigenfunctionals operator. prove convergence for $t>0$ small Cauchy data, up to $t=0$ regular data. The $t=0$} leads a `completeness' property basis modes. construct all modes eigenfunctionals, includi...

This paper furnishes two classes of methods for calculating the best approximate solution of an operator equation in Banach spaces, where the operator is bounded, linear and has closed range. The best approximate solution can be calculated by an iterative method in Banach spaces stated in terms of an operator parameter. Specifying the parameter yields some new and some old iterative techniques....