Abstract: Problems with a spectral parameter in equations and boundary conditions form an important part of spectral theory of linear differential operators. A bibliography of papers in which such problems were considered in connection with specific physical processes can be found in [1, 2]. Boundary value problems for ordinary differential operators with a spectral parameter in boundary conditions were considered in various settings in numerous papers [3–14]. The completeness and the basis property of eigenfunctions of boundary value problems with a spectral parameter in equations and boundary conditions were studied in more detail in [7, 8]. Consider the following boundary value problem for the Dirac system with the same spectral parameter in the equations and the boundary conditions:

Abstract: of hyperbolic equations, where the phase θ and the distortion factor g are given functions of the space variables and time and the function f(θ) describing the wave shape is arbitrary. The examples of the plane wave with θ = x1−ct and g = 1 and the spherical wave with θ = |x|−ct and g = |x|−1, where |x| = (x1 + x2 + x3) , were given in [1] for the wave equation with three space variables x1, x2, and x3. We are interested in simple explicit solutions of the form (1) of the wave equation

Abstract: where T =( S[0;T))[f(x; 0)j x2 g is the parabolic boundary of the cylinderQT , the function g : QT R! R equals the dierence between compositionally measurable functions g2(x;t;u )a nd g1(x;t;u) nondecreasing with respect to u. The continuity of g(x;t;u) with respect to the phase variable u is not assumed. A strong solution of problem (1), (2) is dened as a function u2 W 2;1 1 (QT ) with the zero trace on T which satises Eq. (1) for almost all (x;t)2 QT . An upper (lower) solution of problem (1), (2) is dened as a function u (u )f romW 2;1 1 (QT )w ith a nonnegative (nonpositive) trace on T which satises the inequality Lu(x;t )+ g1 (x;t; u(x;t) ) g2 (x;t; u(x;t)) [respectively, Lu(x;t )+ g1 (x;t;u(x;t)+) g2 (x;t;u(x;t))] almost everywhere on QT . Using the abstract scheme of the method of upper and lower solutions [2], they prove propositions about the existence of strong solutions of problem (1), (2) under the assumption that there exist an upper solution u and a lower solution u of this problem; moreover, u u almost everywhere on QT . In this case, we require that the A1-condition be satised for Eq. (1), namely, there exists a no-more-than countable setfSi ;i 2 Ig of surfaces Si = (x;t;u)2 R n+2 j u = ’i(x;t); (x;t)2 QT ;’ i2 W 2;1 loc;1 (QT );

Abstract: were constructed in [1] for a real integrable function q(x) on [0, `] and for arbitrary n = 1, 2, 3, . . . and m = 0, 1, 2, . . . Here λn is the nth eigenvalue, sn ≡ √ λn, yn(x) is the nth normalized eigenfunction, and λn,m(q), sn,m(q), and yn,m(q, x) are quantities that can be constructed explicitly via the function q(x). In particular, the λn,0 ≡ (nπ/`) are the eigenvalues of the degenerate problem, the yn,0(x) ≡ √ 2/` sin(nπx/`)

Abstract: P+ ∈ L (U), Q+ ∈ L (F), where Γ+ ⊂ C is a closed contour lying in the right half-plane and bounding a domain containing the set σ +(M) = {μ ∈ σ(M) : Reμ > 0}, R μ(M) = (μL−M)−1L, Lμ(M) = L(μL−M)−1. Let the operator M be strongly (L, p)-sectorial [2]. Then there exist projections P ∈ L (U) and Q ∈ L (F); moreover, P+P = PP+ = P+ and Q+Q = QQ+ = Q+. We set P− = P − P+ and Q− = Q−Q+, choose T ∈ R+, and consider Verigin’s problem P−u(0) = u0, P+u(T ) = uT (2)

Abstract: The paper proves existence theorems for the common Lyapunov function of a family of asymptotically stable dynamical systems. The theorems generalize and develop the results announced in [1].