Periodic solutions of nonlinear differential equations with pulse influence in a banach space

Rank conditions for control of linear pulse systems are established. An example of control synthesis in a problem for linear pulse systems is given.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 3, pp. 370–380, March, 1995.This work was partially supported by the Ukrainian State Committee on Science and Technology.     

On the sequences of zeros of holomorphic solutions of linear second-order differential equations

We find necessary and sufficient conditions under which a finite or infinite sequence of complex numbers is the sequence of zeros of a holomorphic solution of the linear differential equation f″ + a ₀ f = 0 with a meromorphic coefficient a ₀ that has second-order poles. In addition, we present a criterion for all solutions of second-order linear equations to be meromorphic.     

On holomorphic families of subspaces of a banach space

In this paper we study holomorphic families of subspaces of a Banach space, which are parametrized by some analytic space. We consider the question of the existence of a holomorphic complement for a given family. It turns out that such a complement does always exist if the basic space is a Stein space, provided for each fixed value of the parameter the subspaces of the family are complemented. From this follows, in particular, the existence of a holomorphic left inverse of a holomorphic operator function, provided such a left inverse exists for each fixed value of the parameter. The latter result gives a positive answer to a question formulated by I.C. Gohberg in a personal conversation with the author. Note of the Editor: This paper was originally published in "Mat. Issled. (Kishinev) 5, vyp 4 (18) (1970), 153–165. The editor is grateful to K. Clancey, W. Kaballo and G. Ph. A. Thijsse for preparing the translation from Russian. Note of the Editor: After the appearance of the original version of the above paper the author informed the editor of Mat. Issled (see Mat. Issled. (Kishinev) 6, vyp (19) (1971), 180) that Corollaries 2 and 3 of Theorem 2 are contained in the papers of G.R. Allan, J. London Math. Soc. 42 (1967), 463–470 and 509–513.     

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.     

Nonlinear perturbation of linear evolution equations in a banach space

Summary Let X be a Banach space and {A(t)|t ε [0, T]} a family of closed linear, densely defined m-accretive operators in X. This paper is concerned with the additive perturbation of {A(t)|t ε [0, T]} by a continuous family of nonlinear accretive operators {B(t)|t ε [0, T]}. Namely solutions are provided for the integral equation u(t, τ, x) = W(t, τ)x − W(t, s)B(s) · · u(s, τ, x)ds, u(τ, τ, x) = x where W(t, s) is the linear evolution operator associated with the linear differential equation v'(t, s, x) + A(t)v(t, s, x) = 0, v(s, s, x) = x. Entrata in Redazione il 30 luglio 1975.     

On some properties of solutions of second-order linear functional differential equations

The properties of solutions of the equationu″(t) =p ₁(t)u(τ₁(t)) +p ₂(t)u′(τ₂(t)) are investigated wherep _i :a, + ∞[→R (i=1,2) are locally summable functions τ₁ :a, + ∞[→R is a measurable function, and τ₂ :a, + ∞[→R is a nondecreasing locally absolutely continuous function. Moreover, τ _i (t) ≥t (i = 1,2),p ₁(t)≥0,p ₂ ² (t) ≤ (4 - ɛ)τ ₂ ^′ (t)p ₁(t), ɛ =const > 0 and . In particular, it is proved that solutions whose derivatives are square integrable on [α,+∞] form a one-dimensional linear space and for any such solution to vanish at infinity it is necessary and sufficient that .     

Solving linear equations in a vector space over a finite field

We study the maximum possible size of a subset in a vector space over a finite field which contains no solution of a given linear equation (or a system of linear equations). This is a finite field version of Ruzsa's work [7].     

On continuous dependence of solutions of linear differential equations on a parameter in a Banach space

We prove the reduction theorem for linear differential equations in a Banach space in the case where the convergence is strong. This result is used to obtain the necessary and sufficient conditions for continuous dependence of solutions on a parameter. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 2, pp. 275–280, February, 1999.     

Asymptotics of solutions of a system of linear inhomogeneous singularly perturbed differential equations

Solutions with asymptotics in integral and fractional powers of the parameter ? are constructed for the vector differential equation $$\varepsilon ^h \dot X = A(t,\varepsilon ) X + \varepsilon ^{\alpha _1 } p(t,\varepsilon ) \exp \left( {\varepsilon ^{ - h} \int\limits_0^t {\lambda (\tau )d\tau } } \right)$$ in the case of resonance and multiple spectrum of the limit matrix. $$\varepsilon ^h \dot X = A(t,\varepsilon ) X + \varepsilon ^{\alpha _1 } p(t,\varepsilon ) \exp \left( {\varepsilon ^{ - h} \int\limits_0^t {\lambda (\tau )d\tau } } \right)$$      

Stabilization of solutions of linear differential equations in Hilbert space

Conditions, less stringent than those known at present, are found for the stabilization of a solution of a linear differential equation of the form (du/dt) + A(t) u =f(t) in Hilbert space to a solution of the operational equation Ax =f, where A is a positive self-adjoint operator. Some regularization algorithms (in A. N. Tikhonov's sense) for this equation are investigated.Translated from Matematicheskie Zametki, Vol. 9, No. 4, pp. 415–420, April, 1971.I wish to thank Ya. I. Al'ber, O. A. Liskovts, and A. M. Il'in for their advice and useful comments.     

Phragmén-Lindelöf theorem for solutions of elliptic differential equations in a banach space

For a second-order elliptic differential equation considered on a semiaxis in a Banach space, we show that if the order of growth of its solution at infinity is not higher than the exponential order, then this solution tends exponentially to zero at infinity. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 5, pp. 650–657, May, 2007.     

On the estimation of solutions for some linear systems of differential equations

In this paper we estimate the solutions of homogeneous linear system of differential equations with unbounded coefficients on the real lineR. We also give a necessary and sufficient condition in order that the linear differential operator with unbounded coefficients has a bounded inverse in the scalar case.     

一类高阶线性微分方程解的增长性

研究一类高阶线性微分方程f(k)+Hk-1f(k-1)+…+H1f'+ H0f=0解的性质,其中Hj=Aj1(z)ePj1(z)+Aj2(z)ePj2(z)(j=0,1,…,k-1),Pjq(q=1,2)是n次复系数多项式,Ajq(z)是级小于n的整函数,当Pjq首项系数的主幅角不全相等时,得到这类方程的超越解有无穷级且超级为n.