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.     

A system of differential equations with an analytic nonlinearity

Mathematical Notes -     

Linear system of differential equations with turning point

A system of linear differential equations with small parameter as a coefficient of a part of derivatives is reduced to the canonical form and the properties of the transformation matrix are investigated.     

On p-superlinear equations with a nonhomogeneous differential operator

We consider a nonlinear elliptic equation driven by a nonhomogeneous differential operator and with a (p?1)-superlinear Carathéodory reaction. Our formulation incorporates as a special case equations monitored by the p-Laplacian. Using variational methods coupled with truncation techniques and comparison principles, we show that the problem has at least five nontrivial smooth solutions.     

Infinite-order differential equations with analytic solutions

An infinite-order linear differential equation with constant coefficients and characteristic equation of the class [1, 0] is investigated, and a class of solutions is introduced. It is shown that, if the zeros _k = _k+ i _k of the characteristic function satisfy the condition , then all solutions of the class under consideration are analytic functions.Translated from Matematicheskie Zametki, Vol. 10, No. 3, pp. 269–278, September, 1971.     

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.     

TURNING POINT PROBLEMS FOR THE THIRD ORDER DIFFERENTIAL EQUATIONS EXHIBITING RESONANCE

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Asymptotic solutions of a system of differential equations with multiple turning point

Using a transformation matrix, we asymptotically reduce a system of differential equations with a small parameter in the coefficients of a part of derivatives and with multiple turning point to an integrable system of equations.     

System of differential equations with a strong turning point

The uniform asymptotics of a solution of a system of singularly perturbed differential equations with strong turning point is constructed. We study the case where the boundary operator is analytic with respect to a small parameter.     

An analytic form of differential equations

A review by the author of his dissertation presented for the academic degree of Doctor of Physico-Mathematical Sciences. This dissertation was defended on May 13, 1969 before the Academic Committee of the Institute of Applied Mathematics of the Academy of Sciences of the USSR. The official challengers were the following Doctors of Physico-Mathematical Sciences: Professor V. I. Arnol'd, Professor V. A. Pliss, and Professor V. V. Rumyantsev.Translated from Matematicheskie Zametki, Vol. 6, No. 6, pp. 771–778, December, 1969.     

Stochastic differential operator equations with random initial conditions

The iterative solution of the stochastic differential operator equation is considered for the case of accompanying random, rather than zero, initial conditions. The mean solution and the correlation are determined.     

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.