On generalized solutions of differential equations with operator coefficients

We prove a theorem on the smoothness of generalized solutions of differential equations with operator coefficients. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 715–720, May, 2006.     

On holomorphic solutions of some boundary-value problems for second-order elliptic operator differential equations

In the class of holomorphic vector functions, we determine the conditions of solvability of the boundaryvalue problem for a class of second-order operator differential equations expressed in terms of the operator coefficients appearing both in the equation and in the boundary condition.     

On analytic solutions of operator differential equations

We find conditions on a closed operator A in a Banach space that are necessary and sufficient for the existence of solutions of a differential equation y′(t) = Ay(t), t ∈[0,∞),in the classes of entire vector functions with given order of growth and type. We present criteria for the denseness of classes of this sort in the set of all solutions. These criteria enable one to prove the existence of a solution of the Cauchy problem for the equation under consideration in the class of analytic vector functions and to justify the convergence of the approximate method of power series. In the special case where A is a differential operator, the problem of applicability of this method was first formulated by Weierstrass. Conditions under which this method is applicable were found by Kovalevskaya.     

On solutions of linear differential equations with entire coefficients

The first result of the paper concerns the effect of perturbation of the entire coefficients of certain linear differential equations on the oscillation of the solutions. Subsequent results involve the separation of the zeros of a Bank-Laine function.     

On elliptic solutions of nonlinear ordinary differential equations

The problem of constructing and classifying exact elliptic solutions of autonomous nonlinear ordinary differential equations is studied. An algorithm for finding elliptic solutions in explicit form is presented.     

Fundamental systems for differential equations with operator coefficients

Fundamental systems for linear differential equations with operator coefficients are investigated, the concept of a Wronskian for such equations is introduced, and a method for constructing a differential equation from its fundamental system is proposed.Translated from Matematicheskie Zametki, Vol. 8, No. 6, pp. 777–782, December, 1970.We must thank V. É. Lyants for his help and the referee for his valuable advice.     

On the solutions of linear differential equations with singular coefficients

Equations of the form E? = Ax + u, with E and A square matrices and E singular, are considered. The controversy that exists in the literature concerning the solutions of such equations is investigated. Solutions are arrived at through an application of singular perturbation theory.     

On algebroid solutions of linear differential equations with algebroid coefficients

We establish a relationship between the growth rates of algebroid solutions and coefficients of nth-order linear differential equations.     

On the existence of solutions of operator differential equations

We consider nonlinear equations of parabolic type in reflexive Banach spaces. We present sufficient conditions for the existence of solutions of these equations. We use methods for the investigation of problems with operators of pseudomonotone (on a subspace) type. In addition, a sufficient criterion in the Sobolev space L _p(0, T; W_p¹()L₂ (0, T; L₂()) is considered for the case where an operator introduced with the use of functional coefficients belongs to a given class. We also show that it is possible to weaken the classical condition of coerciveness.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 6, pp. 837–850, June, 2004.     

On elliptic equations with piecewise constant coefficients

In this work we prove an existence and uniqueness theorem for solutions in ²(R^t) of second order elliptic equations with coefficients that are constant on the half-spaces R+ⁿ and R ?ⁿ