Some contractive type mappings and their applications to difference equations

We discuss a class of discrete dynamic systems in a complete metric space (M, d) defined by mappings which satisfy various types of contractive type conditions. Their variants in ordered Banach space are investigated and applied to solve the global asymptotic stability of the equilibriums of some discrete dynamic systems.     

Functions of finite fractional variation and their applications to fractional impulsive equations

We introduce a notion of a function of finite fractional variation and characterize such functions together with their weak σ-additive fractional derivatives. Next, we use these functions to study differential equations of fractional order, containing a σ-additive term—we prove existence and uniqueness of a solution as well as derive a Cauchy formula for the solution. We apply these results to impulsive equations, i.e. equations containing the Dirac measures.     

Implicit operator differential equations and applications to electrodynamics

A cylindrical waveguide with layered dispersive medium is considered. An implicit operator differential equation of second order is obtained by separating variables in the corresponding boundary‐value problem for Maxwell's equations. Existence and uniqueness theorems are proved and explicit formulae for solutions are given. The spectral theory of operators and operator sheaves are used. Copyright © 2000 John Wiley & Sons, Ltd.     

Decomposition of refinable spaces and applications to operator equations

This paper presents a brief survey of several recent applications of multilevel techniques, in particular, in connection with the solution of periodic pseudodifferential equations. It is pointed out that these applications naturally lead to certain decompositions of refinable spaces which are induced by a class of linear projectors. Then recent results on the construction of such nonorthogonal wavelets are reviewed and extended to the particular needs of the present context.     

Linear difference equations mod 2 with applications to nonlinear difference equations1

Known results for linear difference equations mod 2 with T-periodic solutions are extended and compiled for applications to the semicycle analysis of nonlinear difference equations. For the calculation of T, four methods are presented. A further application concerns rational functions in the field of integers mod 2.     

Regularization of linear difference equations with analytic coefficients and their applications

We propose a regularization method for four-element linear difference equations with analytic coefficients. We study these equations in the class of functions which are holomorphic in the complex plane with a cruciform cut and vanish at infinity. We give several examples illustrating the dependence of the solvability properties of equations on the choice of periodic coefficients. We describe various applications.     

A sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations

This paper is concerned with an operator equation Ax+Bx+Cx=x on ordered Banach spaces, where A is an increasing α-concave operator, B is an increasing sub-homogeneous operator and C is a homogeneous operator. The existence and uniqueness of its positive solutions is obtained by using the properties of cones and a fixed point theorem for increasing general β-concave operators. As applications, we utilize the fixed point theorems obtained in this paper to study the existence and uniqueness of positive solutions for two classes nonlinear problems which include fourth-order two-point boundary value problems for elastic beam equations and elliptic value problems for Lane-Emden-Fowler equations.     

Algebras with general commutation relations and their applications. II. Unitary-nonlinear operator equations

Various aspects of the calculus of functions of ordered self-adjoint operators are considered. Passage to the commutative limit in the case of general nonlinear commutation relations is studied. An asymptotic solution of the Cauchy problem and asymptotically self-similar solutions are constructed for unitary-nonlinear operator equations. Asymptotic solutions are found for the Hartree equation with Coulomb interaction.Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. 13, pp. 145–267, 1979.     

Estimate of solutions for differential and difference functional equations with applications to difference methods

The theorems on the estimate of solutions for nonlinear second-order partial differential functional equations mainly of parabolic type with Dirichlet’s condition and for the suitable explicit finite difference functional schemes are proved. The proofs are based on the comparison technique. The convergent difference method given is considered without an assumption of the global generalized Perron condition on the functional variable but with local one in some sense only. It is a consequence of our estimate theorems. The functional dependence is of the Volterra type.     

Adjoint equations and their applications

Proceedings of the Steklov Institute of Mathematics -     

Factorization of a convolution operator and inverse problems for difference equations

Translated from Matematicheskie Zametki, Vol. 48, No. 5, pp. 97–108, November, 1990.     

Bounded solutions of one class of nonlinear operator difference equations

Sufficient conditions for existence of bounded solutions of operator difference equations with quadratic nonlinearity are given.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 7, pp. 890–896, July, 1995.     

Meromorphic functions sharing a set with applications to difference equations

This paper is devoted to proving some uniqueness type results for an entire function f(z) that shares a common set with its shift f(z+c) or its difference operator Δcf. We also give some applications to solutions of non-linear difference equations related to a conjecture proposed by C.C. Yang.     

Fixed point theorems for multifunctions with applications to discontinuous operator and differential equations

In this paper we study existence of minimal and maximal fixed points for increasing multivalued functions in ordered topological vector spaces, and apply the obtained results to operator equations and quasilinear elliptic boundary value problems involving discontinuous nonlinearities.