An abstract functional differential equation and a related nonlinear Volterra equation

We study the existence, uniqueness, regularity and dependence upon data of solutions of the abstract functional differential equation 1 $$\frac{{du}}{{dt}} + Au \ni G(u) (0 \leqq t \leqq T), u(0) = x,$$ , whereT>0 is arbitrary,A is a givenm-accretive operator in a real Banach spaceX, and \(G:C([0,T]; \overline {D(A)} ) \to L^1 (0, T; X)\) is a given mapping. This study provides simple proofs of generalizations of results by several authors concerning the nonlinear Volterra equation 2 $$u(t) + b * Au(t) \ni F(t) (0 \leqq t \leqq T),$$ , for the case in which X is a real Hilbert space. In (2) the kernelb is real, absolutely continuous on [0,T],b*g(t)=∫ ₀ ¹ (t?s)g(s)ds, andf∈W ^1,1(0,T;X).     

Unramified forcing preserving the law of double negation

Shoenfield's unramified version of Cohen's forcing is defined in two stages: one which does not preserve double negation and the other which modifies the former so as to preserve double negation. Here we express the unramified forcing, which preserves double negation, in a single stage. Surprisingly enough, the corresponding definition of forcing for equality acquires a rather simple form. In [2] forcing ∥- is expressed in terms of strong forcing \( \Vdash * \) viap∥-Q iffp \( \Vdash * \) ¬ ¬Q for every formulaQ ofZF set theory and every elementp of a partially ordered set (P, ≦). In its turn,p \( \Vdash * \) Q is defined by the following five clauses: (1) $$p \Vdash * a \in biff(\exists c)(\exists q \geqq p)((c,q) \in b \wedge p \Vdash * a = c)$$ (2) $$\begin{gathered} p \Vdash * a \ne biff(\exists c)(\exists q \geqq p)(((c,q) \in a \wedge p \Vdash * c \notin b) \hfill \\ ((c,q) \in b \wedge p \Vdash * c \notin a)) \hfill \\ \end{gathered} $$ (3) $$p \Vdash * \neg Qiff(\forall q)(q \leqq p \to \neg (q \Vdash * Q))$$ (4) $$p \Vdash * (Q \vee S)iff(p \Vdash * Q) \vee (p \Vdash * S)$$ (5) $$p \Vdash * (\exists x)Q(x)iff(\exists b)(p \Vdash * Q(b))$$ .     

Existence of a solution of a difference scheme for one variational problem

The problem of minimizing the functional (A) $${}_a\smallint ^b \varphi (x,y,y',y'')dx$$ under the conditions (B) $$y(a) = a_0 ,y'(a) = a_1 ,y(b) = b_0 ,y'(b) = b_1$$ is replaced by the problem of finding the vector (y₁,y₂,...,y_n?1) on which the sum (C) $$\sum\limits_{\kappa = 0}^n {C_\kappa \varphi (x_\kappa ,y_\kappa ,\left. {\frac{{y_{\kappa + 1} - y_\kappa }}{h},\frac{{y_{\kappa + 1} - 2y_\kappa + y_{\kappa + 1} )}}{{h^2 }}} \right)}$$ takes a minimal value. Under certain conditions on ? andC _k it is proved that a solution exists for the difference scheme constructed. The method of differentiation with respect to a parameter is used for the proof.     

Total controllability to affine manifolds of control systems

A system is totallyG-controllable if every pointx ₀ of the state spaceE ⁿ can be steered to the targetG in finite time and can be held inG forever afterward. Sufficient conditions are developed for the totalG-controllability of the linear system (a) $$\dot x(t) = A(t)x(t) + B(t)u(t)$$ and its perturbation (b) $$\dot x(t) = A(t)x(t) + B(t)u(t) + F(t,x(t),u(t)),$$ where the targetG is an affine manifold inE ⁿ. We state conditions on the perturbation functionF which guarantee that, if (a) is totallyG-controllable, then so is (b). These conditions onF are natural and are obtained by solving a system of nonlinear integral equations by the Leray-Schauder fixed-point theorem.     

Convergent approximations in parabolic variational inequalities II: Hamilton-jacobi inequalities

In this paper we consider two-sided parabolic inequalities of the form (li) $$\psi _1 \leqslant u \leqslant \psi _2 , in{\mathbf{ }}Q;$$ (lii) $$\left[ { - \frac{{\partial u}}{{\partial t}} + A(t)u + H(x,t,u,Du)} \right]e \geqslant 0, in{\mathbf{ }}Q,$$ for alle in the convex support cone of the solution given by $$K(u) = \left\{ {\lambda (\upsilon - u):\psi _1 \leqslant \upsilon \leqslant \psi _2 ,\lambda > 0} \right\}{\mathbf{ }};$$ (liii) $$\left. {\frac{{\partial u}}{{\partial v}}} \right|_\Sigma = 0, u( \cdot ,T) = \bar u$$ where $$Q = \Omega \times (0,T), \sum = \partial \Omega \times (0,T).$$ Such inequalities arise in the characterization of saddle-point payoffsu in two person differential games with stopping times as strategies. In this case,H is the Hamiltonian in the formulation. A numerical scheme for approximatingu is obtained by the continuous time, piecewise linear, Galerkin approximation of a so-called penalized equation. A rate of convergence tou of orderO(h ^1/2) is demonstrated in theL ²(0,T; H ¹(Ω)) norm, whereh is the maximum diameter of a given triangulation.     

On weak solutions of linear coercive integral equations in spacesL 2(X; ℋ k )

Пусть (X,A, μ) - полное про странство с σ-конечно й мерой, и пусть \(\overline {\mu \times \mu } \) . - замык ание меры μ×μ. Пусть далееg: X×X→C - квадратично интегрируемая функц ия по мере \(\overline {\mu \times \mu } \) . Рассматривается лин ейное интегральное у равнение (слабого) типа (1) (1) $$u(t) + A(\mathop \smallint \limits_x g(t,s)u(s)d\mu ) = f(t)\Pi .B.B\,X,$$ гдеА - максимальное р асширение L _k ^′ (в простр анстве ХëрмандераH ₁=B₂к) соотв ествующего линейного (псевдодиф ференциального) опер атораL: S→S; иS обозначает класс Щварца функций Rⁿ→-C. Уст анавливается сущест вование (слабых) решений (1) при н екотором условии коэрпитивно сти на оператор (2) (2) $$(L\Psi )(t) = \Psi (t) + \int\limits_x {g(t,s)L(\Psi (s))d\mu ,} $$ где Ψ принадлежит про странстувуD(Х, S) всех конечно-значных функ ций изX→S. Далее, изучается обобщенна я обратимость максим ального расширения оператора L. Наконец, пр иводится некоторое алгебраическое усло вие, обеспечивающее коэрцитивность L.     

Finite-difference method for solving the first boundary-value problem for a second-order nonlinear ordinary differential equation with a divergent principal part

The approximation is studied of the first boundary-value problem for the equation (1) $$- \frac{d}{{dx}}K(x,\frac{{du}}{{dx}}) + f(x,u) = 0,0< x< 1,$$ with boundary conditions (2) $$u(0) = u(1) = 0$$ by difference boundary-value problems of form (3) $$- \left[ {a(x,w_{\bar x} )} \right]_x + \varphi (x,w) = 0,x \in w_r ,$$ (4) $$w(0) = w(1) = 0.$$ Theorems are established on the solvability of problem (3), (4). Theorems are proved on uniform convergence and on the order of uniform convergence. Here, as usual, boundedness is not assumed, but just the summability of the corresponding derivatives of the solutions of problem (1), (2). Also considered are singular boundary-value problems of form (1), (2), where uniform convergence with order h is proved under assumption of piecewise absolute continuity of the functionf(x,u(x)).     

Eventual uniform asymptotic stability for stochastic differential equation based on semimartingale with spatial parameter

Given a stochastic differential equation based on semimartingale with spatial parameter (1) $$\varphi _t = x_0 + \int_{t_0 }^t {F(\varphi _s ,ds) } on t \geqslant t_0 $$ and it perturbed system (2) $$\psi _t = x_0 + \int_{t_0 }^t {F\left( {\psi \alpha _s , ds} \right)} + \int_{t_0 }^t {G\left( {\psi _s , ds} \right)} on t \geqslant t_0 $$ In this paper we give some sufficient conditions under which the eventual uniform asymptotic stability of Eq. (1) is shared by Eq. (2).     

On certain properties of linear summation methods of Fourier series on the classesC(ɛ)

Для линейных методов суммирования рядов Ф урье (1) $$L_n (f;x) = \frac{1}{\pi }\mathop \smallint \limits_{ - \pi }^\pi f(x + t)\left( {\frac{1}{2} + \sum\limits_{k = 1}^n {\lambda _{k,n} } \cos kt} \right)dt$$ на классах $$C(\varepsilon ) = \{ f:E_n (f) \leqq \varepsilon _n ;\forall n \geqq 0\} ,\varepsilon = \{ \varepsilon _n \} _{n = 0.}^\infty \varepsilon _n \downarrow 0,$$ доказываются:

1. оценки для порядка р оста норм ∥{L_n∥, если из вестен порядок приближения операторами (1) некоторого классаС (?) (при этом, если опера торы (1) приближают класс С(е) с наилучшим порядком, то находится точная а симптотика возрастания норм {∥ L_n∥);
2. сравнительные оцен ки порядков приближе ния классовС(?) операторами (1), если известен порядок при ближения ими некотор ого более узкого класса С(?*).

В том случае, когда опе раторы (1) приближают кл асс С(?*) с наилучшим порядком, получаются точные по рядковые оценки для л юбого более широкого класса С(?).     

A maximum principle for time-lag control problems with bounded state

A maximum principle is obtained for control problems involving a constant time lag τ in both the control and state variables. The problem considered is that of minimizing $$I(x) = \int_{t^0 }^{t^1 } {L (t,x(t), x(t - \tau ), u(t), u(t - \tau )) dt} $$ subject to the constraints 1 $$\begin{gathered} \dot x(t) = f(t,x(t),x(t - \tau ),u(t),u(t - \tau )), \hfill \\ x(t) = \phi (t), u(t) = \eta (t), t^0 - \tau \leqslant t \leqslant t^0 , \hfill \\ \end{gathered} $$ 1 $$\psi _\alpha (t,x(t),x(t - \tau )) \leqslant 0,\alpha = 1, \ldots ,m,$$ 1 $$x^i (t^1 ) = X^i ,i = 1, \ldots ,n$$ . The results are obtained using the method of Hestenes.     

On minor forms of Desargues and Pappus

Let Π be a projective plane coordinatized by a ternary ring (R, F). In addition to the two operations + and ·, defined bya+b =F(a,1,b and \(a \cdot b = F(a,b,0)\) , a third operation * is defined by \(a * b = F(1,a,b),\forall a,b \in R\) Several minor forms of the propositions of Desargues and Pappus are introduced in Π and their geometrical properties are developed. Several algebraic results are obtained in connection with these minor forms. For example, the first minor form of DesarguesD ₁ is proved to be equivalent to each of the following algebraic identities in every (R, F): (1) $$a \cdot c = c \cdot a \Rightarrow F(a,c,b) = F(c,a,b),$$ (2) $$a \cdot (1 + b) = a + a \cdot b,$$ (3) $$a * b = a + b$$ (4) $$F(x,m,k) = (x \cdot m) * k,\forall a,b,c,k,m,x \in R.$$ Some more algebraic identities are characterized byD ₂ andD ₃.     

Automorphisms of the tensor product of Abelian and Grassmannian algebras

We consider an algebraB _n,m , over the field R with n+m generators x_i,..., x_n, ξ₁,..., η_m, satisfying the following relations: (1') $$\left[ {x_k ,x_l } \right] \equiv x_k x_l - x_l x_k = 0,[x_k ,\xi _i ] = 0,$$ , (2') $$\left\{ {\xi _i ,\xi _j } \right\} \equiv \xi _i \xi _j + \xi _j \xi _i = 0$$ , where k,l =1, ..., n and i, j=1,..., m. In this algebra we define differentiation, integration, and also a group of automorphisms. We obtain an integration equation invariant with respect to this group, which coincides in the case m=0 with the equation for the change of variables in an integral, an equation whichis well known in ordinary analysis; in the case n=0 our equation coincides with F. A. Berezin's result [1, 3] for integration over a Grassman algebra.     

On the oscillation of the second order neutral delay differential equations

In this paper, we consider the oscillation of the second order neutral delay differential equations[x(t) cx(t-τ)]" p(t)x(t-σ)=0 (1)and obtain some sufficient conditions of the oscillation of (1) for the case c≥0, -1≤c<0 and c<-1.     

Sufficient conditions for absolute asymptotic stability of linear equations in a Banach space

For a linear differential equation of the type (1) $$\frac{{dx}}{{dt}} = A_0 x(t) + A_1 x(t - \Delta _1 ) + ... + A_n x(t - \Delta _n )$$ we establish the followingTHEOREM. If $$\overline {\left| {z_1 } \right| = ...\underline{\underline \cup } \left| z \right|_n = 1\sigma \left( {A_0 + \sum\nolimits_{k = 1}^n {z_k A_k } } \right)} \subset \left\{ {\lambda :\operatorname{Re} \lambda< 0} \right\}$$ then system (1) is absolutely asymptotically stable.     

Orthogonal polynomials with complex-valued weight function,II

In this paper we continue our study of the asymptotic behavior of polynomialsQ _mn(z), m, n ∈N, of degree≤n satisfying the orthogonal relation (*) $$( * )\oint_c {\zeta ^l Q_{mn} (\zeta )} \frac{{f(\zeta )d\zeta }}{{\omega _{m + n} (\zeta )}} = 0,l = 0,...,n - 1,$$ and all its singularities are supposed to be contained in a set \(E \subseteq \hat C\) of capacity zero, ω _m+n (z) is a polynomial of degreem+n+1 with all its zeros contained inV, and the path of integrationC separatesV from the setE. We state and prove results concerning the asymptotic magnitude of the integral in (*) forl=n,n+1,?.     

On strong summahility of Fourier series and differentiability of functions

ПустьS _n (f, x) — суммы Фурье периодической сумми руемой функцииf(x). Доказано, что если фун кцияФ(u), определенная, непрерывная и выпукл ая вверх для u≧0 (Ф(0)=0), удовлетворяет ус ловию (1) $$\int\limits_{ + 0} {\frac{{du}}{{\Phi (u)}}< \infty ,} $$ то имеет место следую щее вложение классов функций (2) $$S(\Phi ) = \left\{ {f:\mathop {\max }\limits_x \sum\limits_{n = o}^\infty \Phi (\left| {f(x) - S_n (fx)} \right|)< \infty } \right\} \subset Lip1,$$ и, более того, при услов ии (1) все функции из кла ссаS(Ф) непрерывно дифферен цируемы, а их производные имеют равномерно сходящие ся ряды Фурье. Установлено также, чт о если функция Ф удовл етворяет условию lim supФ(u/2)/Ф(u)<1, то условие (1) является н е только достаточным, но и необходимым для влож ения (2).     

A mixed boundary-value problem for a hyperbolic-parabolic equation

Let Ω be a bounded domain in the n-dimensional Euclidean space. In the cylindrical domain QT=Ω x [0, T] we consider a hyperbolic-parabolic equation of the form (1) $$Lu = k(x,t)u_{tt} + \sum\nolimits_{i = 1}^n {a_i u_{tx_i } - } \sum\nolimits_{i,j = 1}^n {\tfrac{\partial }{{\partial x_i }}} (a_{ij} (x,t)u_{x_j } ) + \sum\nolimits_{i = 1}^n {t_i u_{x_i } + au_t + cu = f(x,t),} $$ where \(k(x,t) \geqslant 0,a_{ij} = a_{ji} ,\nu |\xi |^2 \leqslant a_{ij} \xi _i \xi _j \leqslant u|\xi |^2 ,\forall \xi \in R^n ,\nu > 0\) . The classical and the “modified” mixed boundary-value problems for Eq. (1) are studied. Under certain conditions on the coefficients of the equation it is proved that these problems have unique solution in the Sobolev spaces W ₂ ¹ (QT) and W ₂ ² (QT).     

Determination of the infinite Jacobi matrix with respect to two-spectra

The inverse problem about two-spectra for the equation (1) $$\begin{gathered} b_0 y_0 + a_0 y_1 = \lambda y_0 , \hfill \\ a_{n - 1} y_{n - 1} + b_n y_n + a_n y_{n + 1} = \lambda y_n \left( {n = 1, 2, 3, ...} \right), \hfill \\ \end{gathered} $$ where {y_n} ₀ ^∞ is the desired solution, λ is a complex parameter and $$a_n > 0, \operatorname{Im} b_n = 0 \left( {n = 0, 1 ,2, ...} \right)$$ is studied. Necessary and sufficient conditions for the solvability of the inverse problem about two-spectrafor Eq. (1) are established and also the procedure of reconstruction of the equation from its two-spectra is indicated.     

On linear summation methods of Fourier series

Получены новые оценк иL-нормы тригонометр ических полиномов $$T_n (t) = \frac{{\lambda _0 }}{2} + \mathop \sum \limits_{k = 1}^n \lambda _k \cos kt$$ в терминах коэффицие нтовλ _k и их разностейΔλ _k=λ _k?λ _k?1: (1) $$\mathop \smallint \limits_{ - \pi }^\pi |T_n (t)|dt \leqq \frac{c}{n}\mathop \sum \limits_{k = 0}^n |\lambda _\kappa | + c\left\{ {x(n,\varphi )\mathop \sum \limits_{k = 0}^n \Delta \lambda _\kappa \mathop \sum \limits_{l = 0}^n \Delta \lambda _l \delta _{\kappa ,l} (\varphi )} \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ,$$ где $$\kappa (n,\varphi ) = \mathop \smallint \limits_{1/n}^\pi [t^2 \varphi (t)]^{ - 1} dt, \delta _{k,1} (\varphi ) = \mathop \smallint \limits_0^\infty \varphi (t)\sin \left( {k + \frac{1}{2}} \right)t \sin \left( {l + \frac{1}{2}} \right)t dt,$$ a ?(t) — произвольная фун кция ≧0, для которой опр еделены соответствующие инт егралы. Из (1) следует, что методы $$\tau _n (f;t) = (N + 1)^{ - 1} \mathop \sum \limits_{k = 0}^{\rm N} S_{[2^{k^\varepsilon } ]} (f;t), n = [2^{N\varepsilon } ],$$ являются регулярным и для всех 0<ε≦1/2. ЗдесьS _m (f, x) частные суммы ряда Фу рье функцииf(x). В статье исследуется многомерный случай. П оказано, что метод суммирования (о бобщенный метод Рисса) с коэффиц иентами $$\lambda _{\kappa ,l} = (R^v - k^\alpha - l^\beta )^\delta R^{ - v\delta } (0 \leqq k^\alpha + l^\beta \leqq R^v ;\alpha \geqq 1,\beta \geqq 1,v< 0)$$ является регулярным, когда δ > 1.     

Lower semicontinuous perturbations of maximal monotone differential inclusions

We prove existence of solutions to 1 $$\dot x \in - Ax + F\left( {t,x} \right),x\left( a \right) = x^0 ,$$ whereA is a maximal monotone operator inR ⁿ andF is a multifunction measurable in (t, x) and l.s.c. inx, satisfying a sublinear growth condition.