On the Cauchy problem for some abstract nonlinear differential equations

In the present paper, we study the Cauchy problem in a Banach spaceE for an abstract nonlinear differential equation of form $$\frac{{d^2 u}}{{dt^2 }} = - A\frac{{du}}{{dt}} + B(t)u + f(t,W)$$ whereW = (A ₁(t)u,A ₂(t)u,?,A _?(t)u), (A _i (t),i = 1, 2, ?,?), (B(t),t ∈I = [0,b]) are families of closed operators defined on dense sets inE intoE, f is a given abstract nonlinear function onI ×E ^? intoE and ?A is a closed linear operator defined on dense set inE intoE, which generates a semi-group. Further, the existence and uniqueness of the solution of the considered Cauchy problem is studied for a wide class of the families (A _i(t),i = 1, 2, ?,?), (B(t),t ∈I). An application and some properties are also given for the theory of partial diferential equations.     

On a coupled system of abstract differential equations similar to the thermoelasticity equations

The study of a model system of differential equations arising from the dynamical problems of thermo elasticity is continued. The case of shifts of the general type is investigated. We employ the commutant method based on the properties of the operator (A,B)=AB-BA.Translated from Ukrainskii Matematicheskii Zhumal, Vol. 45, No. 8, pp. 1162–1165, August, 1993.     

On a Kneser-type problem for a system of ordinary differential equations

We establish sufficient conditions for the solvability of a Kneser-type problem concerning the existence of a monotonic solution of a system of ordinary differential equations satisfying an initial condition.Translated from Matematicheskie Zametki, Vol. 15, No. 6, pp. 897–906, June, 1974.     

On the two-point boundary-value problem for equations of geodesics

For equations of “geodesic spray” type with continuous coefficients on a complete Riemannian manifold, some interrelations between certain geometric characteristics, the distance between points, and the norm of the right-hand side that guarantee the solvability of the boundary-value problem are found. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 65–70, 2005.     

Nonzero solutions to a two-point boundary-value periodic problem for differential equations with maxima

We prove a theorem on the unique existence of a solution to a nonlinear equation with maxima and demonstrate its continuous dependence on the initial function and the parameter of the problem. We also establish conditions for the existence of a nonzero solution to a two-point boundary-value periodic problem in dependence of both linear and nonlinear terms of the equation.     

On abstract degenerate neutral differential equations

We introduce a new abstract model of functional differential equations, which we call abstract degenerate neutral differential equations, and we study the existence of strict solutions. The class of problems and the technical approach introduced in this paper allow us to generalize and extend recent results on abstract neutral differential equations. Some examples on nonlinear partial neutral differential equations are presented.     

Nonlocal two-point problem for partial differential equations with linearly dependent coefficients

In the Cartesian product of the time segment and a spatial multidimensional torus, we investigate a problem with nonlocal two-point boundary conditions for a typeless partial differential equation of the second order with constant linearly dependent coefficients. Conditions for the unique solvability of the problem in the scale of Sobolev spaces are established. Metric theorems on estimates from below for small denominators on linear manifolds are proved.     

On a new class of abstract neutral differential equations

In this work we introduce a new class of abstract neutral differential equations and study the existence of mild and strict solutions. Some concrete applications involving partial neutral differential equations are considered.     

On a class of abstract neutral functional differential equations

By using the theory of semigroups of growth α, we discuss the existence of mild solutions for a class of abstract neutral functional differential equations. A concrete application to partial neutral functional differential equations is considered.     

On bounded solutions of abstract differential equations

Summary We construct nontrivial bounded solutions of an abstract evolution equationu'(t)=Au(t) (—∞<t<∞) whereA generates a (C ₀) semigroup of operators {T _t;t≥0} such thatT _t converges strongly to zero ast→∞.

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Riassunto Si costruiscono soluzioni limitate non triviali di una equazioneu'(t)=Au(t) (—∞<t<∞) doveA è il generatore di un semigruppo di operatori {T _t;t≥0} tale cheT _t converge fortemente verso 0 pert→∞.

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Supported by National Science Foundation grant GP-12722.     

The central two-point connection problem of Heun's class of differential equations

Boundary-value problems of ordinary, linear, homogeneous second-order differential equations belong to the most important and thus well-investigated problems in mathematical physics. This statement is true only as long asirregular singularities of the differential equation at hand are not involved. If singular points of irregular type enter the problem one will hardly find a systematic investigation of such a topic from a practical point of view. This paper is devoted to an outline of an approach to boundary-value problems of the class of Heun's differential equation when irregular singularities may be located at the endpoints of the relevant interval. We present an approach to the central two-point connection problem for all of these equations in a quite uniform manner. The essential point is an investigation of the Birkhoff sets of irregular difference equations, which, on the one hand, gives a detailed insight into the structure of the singularities of the underlying differential equation and, on the other hand, yields the basis of quite convenient algorithms for numerical investigations of the boundary values.The text published here is the author's original text with slight editorial changes.Institut für Theoretische und Angewandte Physik, Universität Stuttgart. Published in Russian in Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 3, pp. 360–368, December, 1994.