On difference Riccati equations and second order linear difference equations

In this paper, we treat difference Riccati equations and second order linear difference equations in the complex plane. We give surveys of basic properties of these equations which are analogues in the differential case. We are concerned with the growth and value distributions of transcendental meromorphic solutions of these equations. Some examples are given.     

On finite discrete operators and equations

For discrete operators generated by Calderon–Zygmund kernels we consider certain finite-dimensional approximations. We compare an initial discrete operator with its finite-dimensional analogue and obtain invertibility conditions and approximation rate. The error estimate for approximate finite-dimensional solution of corresponding equation is given also for a special right-hand side. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary value problems for difference equations with causal operators

By using the monotone iterative method, some new results are established for nonlinear boundary conditions of difference problems with causal operators. We formulate sufficient conditions under which such problems have extremal solutions. Difference inequalities with causal operators are also discussed. Two examples are added to illustrate the results.     

On proving stability of multidimensional difference schemes

A method is presented for checking the stability of amplification matrices, which are polynomials in two or more matrices that do not commute with each other.     

On difference of two monotone operators

In this paper, we introduce and consider the problem of finding zeroes of difference of two monotone operators in a Hilbert space. Using the resolvent operator technique, we show that this problem is equivalent to the fixed point problem. This equivalence is used to suggest and analyze an iterative method for finding a zero of difference of two monotone operators. We also discuss the convergence of the iterative method under suitable conditions. Our method of proof is very simple as compared with other techniques.     

On some integral properties of multidimensional Hilbert operators

In the present paper, we study the integral properties of multidimensional Hilbert transforms.     

On nonlinear difference and differential equations

A generalization of the logarithmic norm to nonlinear operators, the Dahlquist constant is introduced as a useful tool for the estimation and analysis of error propagation in general nonlinear first-order ODE's. It is a counterpart to the Lipschitz constant which has similar applications to difference equations. While Lipschitz constants can also be used for ODE's, estimates based on the Dahlquist constant always give sharper results.The analogy between difference and differential equations is investigated, and some existence and uniqueness results for nonlinear (algebraic) equations are given. We finally apply the formalism to the implicit Euler method, deriving a rigorous global error bound for stiff nonlinear problems.Dedicated to my teacher and friend, Professor Germund Dahlquist, on the occasion of his 60th birthday.     

Recursion operators,conservation laws,and integrability conditions for difference equations

We attempt to propose an algebraic approach to the theory of integrable difference equations. We define the concept of a recursion operator for difference equations and show that it generates an infinite sequence of symmetries and canonical conservation laws for a difference equation. As in the case of partial differential equations, these canonical densities can serve as integrability conditions for difference equations. We obtain the recursion operators for the Viallet equation and all the Adler-Bobenko-Suris equations.     

Well-posedness of the Cauchy problem for multidimensional difference equations in rational cones

The Cauchy problem is studied for a multidimensional difference equation in a class of functions defined at the integer points of a rational cone. We give an easy-to-check condition on the coefficients of the characteristic polynomial of the equation sufficient for solvability of the problem. A multidimensional analog of the condition ensuring stability of the Cauchy problem is stated on using the notion of amoeba of an algebraic hypersurface.     

On global periodicity of difference equations

In this research, we consider a difference equation of order k?2 of the following form:

     

On oscillatory third order difference equations

In this paper, sufficient conditions have been obtained for oscillation/nonoscillation of a class of homogeneous/nonhomogeneous linear difference equations of third order.     

On an algebra of multidimensional integral operators with homogeneous-difference kernels

A Banach algebra generated by multidimensional integral operators with homogeneous-difference kernels is considered. A symbolic calculus is constructed for this algebra and, in terms of this calculus, necessary and sufficient conditions for the invertibility of an operator are obtained.     

On compactness of the difference of composition operators

Let φ and ψ be analytic self-maps of the unit disc, and denote by C_φ and C_ψ the induced composition operators. The compactness and weak compactness of the difference T=C_φ−C_ψ are studied on H^p spaces of the unit disc and L^p spaces of the unit circle. It is shown that the compactness of T on H^p is independent of p∈[1,∞). The compactness of T on L¹ and M (the space of complex measures) is characterized, and examples of φ and ψ are constructed such that T is compact on H¹ but non-compact on L¹. Other given results deal with L^∞, weakly compact counterparts of the previous results, and a conjecture of J.E. Shapiro.