Some properties of root functions of ordinary second-order differential operators in the absence of the carleman condition

On a finite interval G of the real line, we consider the root functions of an ordinary second-order differential operator without any boundary conditions for the case in which the imaginary part of the spectral parameter is unbounded.We refine the estimates for the C-and L _p -norms of a root function and its first derivative on a compact set contained in the interior of G for the case in which the Carleman condition fails.A sufficient condition is obtained for the root functions of an ordinary second-order differential operator to have the Bessel property, assuming that the Carleman condition fails. We show that, under certain conditions, this problem can be reduced to analyzing the Bessel property of systems of exponentials.     

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 the curve of critical exponents for nonlinear elliptic problems in the case of a zero mass

For semilinear elliptic equations ?Δu = λ|u| ^p?2 u?|u| ^q?2 u, boundary value problems in bounded and unbounded domains are considered. In the plane of exponents p × q, the so-called curves of critical exponents are defined that divide this plane into domains with qualitatively different properties of the boundary value problems and the corresponding parabolic equations. New solvability conditions for boundary value problems, conditions for the stability and instability of stationary solutions, and conditions for the existence of global solutions to parabolic equations are found.     

Infinite horizon backward doubly stochastic differential equations with non-degenerate terminal functions and their stationary property

In this paper we consider infinite horizon backward doubly stochastic differential equations (BDSDEs for short) coupled with forward stochastic differential equations, whose terminal functions are non-degenerate. For such kind of BDSDEs, we study the existence and uniqueness of their solutions taking values in weighted L ^p (dx)?L ²(dx) space (p ≥ 2), and obtain the stationary property for the solutions.     

Non-Monotonic Behavior of One-Signed Solutions of Quasilinear Differential Equations

We consider a class of the second-order quasilinear differential equations. By deriving relations between certain types of monotonic solutions of the quasilinear equation and corresponding reciprocal half-linear equation on a finite interval (a, b), we obtain criteria for all solutions of the main equation, which do not change sign in (a, b), to be non-monotonic in (a, b). This work is also extended to a perturbed half-linear equation as well as to the half-line \((a,\infty )\).     

Pathwise approximation of stochastic differential equations on domains: higher order convergence rates without global Lipschitz coefficients

We study the approximation of stochastic differential equations on domains. For this, we introduce modified Itô–Taylor schemes, which preserve approximately the boundary domain of the equation under consideration. Assuming the existence of a unique non-exploding solution, we show that the modified Itô–Taylor scheme of order γ has pathwise convergence order γ ? ε for arbitrary ε > 0 as long as the coefficients of the equation are sufficiently differentiable. In particular, no global Lipschitz conditions for the coefficients and their derivatives are required. This applies for example to the so called square root diffusions.     

Well-posed solvability of an analytic Cauchy problem in spaces with an integral metric

We study the complex Cauchy problem for a system of linear differential equations in a class of analytic functions with an integral metric. For the case in which L_p is a weighted Lebesgue space, we obtain necessary and sufficient conditions for the local solvability of the problem.     

Local analysis of a polynomial system and its perturbations

On the real (x, y)-plane, we consider an autonomous system of differential equations whose right-hand sides are polynomials of special form in x and y and a perturbed system obtained from the former by varying the coefficients in the class of functions of (x, y) satisfying the Lipschitz condition. We study the behavior of trajectories of the system in a neighborhood of the isolated equilibrium point O = (0, 0). For the main (polynomial) system, we find all possible types of arrangement of the trajectories in a neighborhood of O. For the case in which the system has TO-curves, we give coefficient criteria for each of the possible types of the point O and study conditions under which the type is preserved in the perturbed system.     

Analysis of some numerical methods on layer adapted meshes for singularly perturbed quasilinear systems

We consider a coupled system of first-order singularly perturbed quasilinear differential equations with given initial conditions. The leading term of each equation is multiplied by a distinct small positive parameter, which induces overlapping layers. The quasilinear system is discretized by using first and second order accurate finite difference schemes for which we derive general error estimates in the discrete maximum norm. As consequences of these error estimates we establish nodal convergence of O((N ^?1 lnN) ^p ),p=1,2, on the Shishkin mesh and O(N ^?p ),p=1,2, on the Bakhvalov mesh, where N is the number of mesh intervals and the convergence is robust in all of the parameters. Numerical computations are included which confirm the theoretical results.     

To the theory of the Dirichlet and Neumann problems for strongly elliptic systems in Lipschitz domains

For strongly elliptic Systems with Douglis-Nirenberg structure, we investigate the regularity of variational solutions to the Dirichlet and Neumann problems in a bounded Lipschitz domain. The solutions of the problems with homogeneous boundary conditions are originally defined in the simplest L ₂-Sobolev spaces H ^σ . The regularity results are obtained in the potential spaces H _p ^σ and Besov spaces B _p ^σ . In the case of second-order Systems, the author’s results obtained a year ago are strengthened. The Dirichlet problem with nonhomogeneous boundary conditions is considered with the use of Whitney arrays.     

The Injective Leavitt Complex

For a finite quiver Q without sinks, we consider the corresponding finite dimensional algebra A with radical square zero. We construct an explicit compact generator for the homotopy category of acyclic complexes of injective A-modules. We call such a generator the injective Leavitt complex of Q. This terminology is justified by the following result: the differential graded endomorphism algebra of the injective Leavitt complex of Q is quasi-isomorphic to the Leavitt path algebra of Q. Here, the Leavitt path algebra is naturally \(\mathbb {Z}\)-graded and viewed as a differential graded algebra with trivial differential.     

Optimal boundary control by a displacement in <Emphasis Type="Italic">W</Emphasis>′<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> of a string with free end

We consider an optimal boundary control of a string with free end by a displacement of the other end in W′ _p (Q, T). For p ≠ 2, we prove that the optimal control depends on the initial and terminal conditions nonlinearly.     

Convergent and Asymptotic Methods for Second-order Difference Equations with a Large Parameter

We consider the second-order linear difference equation \(y(n+2)-2a y(n+1)-\Lambda ^2 y(n)=g(n)y(n)+f(n)y(n+1)\), where \(\Lambda \) is a large complex parameter, \(a\ge 0\) and g and f are sequences of complex numbers. Two methods are proposed to find the asymptotic behavior for large \(\vert \Lambda \vert \) of the solutions of this equation: (i) an iterative method based on a fixed point method and (ii) a discrete version of Olver’s method for second-order linear differential equations. Both methods provide an asymptotic expansion of every solution of this equation. The expansion given by the first method is also convergent and may be applied to nonlinear problems. Bounds for the remainders are also given. We illustrate the accuracy of both methods for the modified Bessel functions and the associated Legendre functions of the first kind.     

Asymptotic expansions of periodic solutions of ordinary differential equations with large high-frequency terms

We consider the problem on the periodic solutions of a system of ordinary differential equations of arbitrary order n containing terms oscillating at a frequency ω ? 1 with coefficients of the order of ω ^n/2. For this problem, we construct the averaged (limit) problem and justify the averaging method as well as another efficient algorithm for constructing the complete asymptotics of the solution.     

The convolution method for fourier expansions. The case of generalized transmission conditions of crack (screen) type in piecewise inhomogeneous media

We consider boundary value problems for the equation ? _x (K ? _x ?) + KL[?] = 0 in the space R ⁿ with generalized transmission conditions of the type of a strongly permeable crack or a weakly permeable screen on the surface x = 0. (Here L is an arbitrary linear differential operator with respect to the variables y ₁, …, y _n?1.) The coefficients K(x) > 0 are monotone functions of certain classes in the regions separated by the screen x = 0. The desired solution has arbitrary given singular points and satisfies a sufficiently weak condition at infinity.We derive formulas expressing the solutions of the above-mentioned problems in the form of simple quadratures via the solutions of the considered equation with a constant coefficient K for given singular points in the absence of a crack or a screen. In particular, the obtained formulas permit one to solve boundary value problems with generalized transmission conditions for equations with functional piecewise continuous coefficients in the framework of the theory of harmonic functions.     

A Riemann-Hilbert Problem for the Moisil-Teodorescu System

In a bounded domain with smooth boundary in ?³ we consider the stationary Maxwell equations for a function u with values in ?³ subject to a nonhomogeneous condition (u, v)_x = u₀ on the boundary, where v is a given vector field and u₀ a function on the boundary. We specify this problem within the framework of the Riemann-Hilbert boundary value problems for the Moisil-Teodorescu system. This latter is proved to satisfy the Shapiro-Lopaniskij condition if an only if the vector v is at no point tangent to the boundary. The Riemann-Hilbert problem for the Moisil-Teodorescu system fails to possess an adjoint boundary value problem with respect to the Green formula, which satisfies the Shapiro-Lopatinskij condition. We develop the construction of Green formula to get a proper concept of adjoint boundary value problem.     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Poisson Semigroup Associated with Elliptic Systems

We study the infinitesimal generator of the Poisson semigroup in L ^p associated with homogeneous, second-order, strongly elliptic systems with constant complex coefficients in the upper-half space, which is proved to be the Dirichlet-to-Normal mapping in this setting. Also, its domain is identified as the linear subspace of the L ^p -based Sobolev space of order one on the boundary of the upper-half space consisting of functions for which the Regularity problem is solvable. Moreover, for a class of systems containing the Lamé system, as well as all second-order, scalar elliptic operators, with constant complex coefficients, the action of the infinitesimal generator is explicitly described in terms of singular integral operators whose kernels involve first-order derivatives of the canonical fundamental solution of the given system. Furthermore, arbitrary powers of the infinitesimal generator of the said Poisson semigroup are also described in terms of higher order Sobolev spaces and a higher order Regularity problem for the system in question. Finally, we indicate how our techniques may be adapted to treat the case of higher order systems in graph Lipschitz domains.     

Difference Schemes on Nonuniform Grids for the Two-Dimensional Convection–Diffusion Equation

New second-order accurate monotone difference schemes on nonuniform spatial grids for two-dimensional stationary and nonstationary convection–diffusion equations are proposed. The monotonicity and stability of the solutions of the computational methods with respect to the boundary conditions, the initial condition, and the right-hand side are proved. Two-sided and corresponding a priori estimates are obtained in the grid norm of C. The convergence of the proposed algorithms to the solution of the original differential problem with the second order is proved.     

Statistical characteristics of continuous functions and statistically weakly invariant sets of controllable system

We continue the investigation of expansion of a concept of invariance for sets which consists in studying statistically invariant sets with respect to control systems and differential inclusions. We consider the statistical characteristics of continuous functions: Upper and lower relative frequency of containing for graph of a function in a given set. We obtain conditions under which statistical characteristics of two various asymptotical equivalent functions coincide; then by the value of one of them it is possible to calculate the value of another one. We adduce the equality for finding relative frequencies of hitting functions the given set in the case when the distance from the graph of one of functions to the given set is a periodic function. A consequence of these statements are conditions of statistically weak invariance of a set with respect to controlled system. For some almost periodic functions we obtain the formulas by which we can calculate the mean values and the statistical characteristics. We also consider the following problem. Let the number λ₀ ∈ [0, 1] be given. It is necessary to find the value c(λ₀) such that the upper solution z(t) of the Cauchy problem does not exceed c(λ₀) with the relative frequency being equal λ₀. Depending on statement of the problem, a value z(t) can be interpreted as the size of population, energy of a particle, concentration of substance, size of manufacture or the price of production.     

Global continuation of forced oscillations of retarded motion equations on manifolds

We investigate T-periodic parametrized retarded functional motion equations on (possibly) noncompact manifolds; that is, constrained second order retarded functional differential equations. For such equations we prove a global continuation result for T-periodic solutions. The approach is topological and is based on the degree theory for tangent vector fields as well as on the fixed point index theory.Our main theorem is a generalization to the case of retarded equations of an analogous result obtained by the last two authors for second order differential equations on manifolds. As corollaries we derive a Rabinowitz-type global bifurcation result and a Mawhin-type continuation principle. Finally, we deduce the existence of forced oscillations for the retarded spherical pendulum under general assumptions.