In this paper we consider the first order difference equation in a Banach space $\Delta x_n={\sum }_{i=0}^{\infty }{a}_n^{i}f\left(x_{n+i}\right)$. We show that this equation has a solution asymptotically equal to a. As an application of our result we study the difference equation $\Delta x_n={\sum }_{i=0}^{\infty }{a}_n^{i}g\left(x_{n+i}\right)+{\sum }_{i=0}^{\infty }{b}_n^{i}h\left(x_{n+i}\right)+y_n$ and give conditions when this equation has solutions. In this note we extend the results from [8,9]. For example, in [9] the function f is a real Lipschitz function. We suppose that f has values in a Banach space and satisfies some conditions with respect to the measure of noncompactness...

The paper considers the static Maxwell system for a Lipschitz domain with perfectly conducting boundary. Electric and magnetic permeability ε and μ are allowed to be monotone and Lipschitz continuous functions of the electromagnetic field. The existence theory is developed in the framework of the theory of monotone operators.

Operator version of the Stokeslet method in the theory of creeping flow is suggested. The approach is analogous to the zero-range potential one in quantum mechanics and is based on the theory of self-adjoint operator extensions in the space L2 and in the Pontryagin?s space with an indefinite metric. The problem of Stokes flow in two channels connected through a small opening is considered in the framework of this approach. The case of a periodic system of small openings is studied too. The picture...