ОБ ОДНОЙ СКАЛЯРНОЙ ФОРМЕ ДВУМЕРНОЙ ЗАДАЧИ ШВАРЦА И ЕЕ ПРИМЕНЕНИЯХ

The Schwarz problem for J-analytic functions in an ellipse is considered. In this case, the matrix J is assumed to be two-dimensional with different eigenvalues located above the real axis. The Schwarz problem is reduced to an equivalent boundary value problem for the scalar functional equation depending on the real parameter l. This parameter is determined by the Jordan basis of the matrix J. An analysis of the functional equation was performed. It is shown that for l∈[0,1], the solution of the Schwarz problem with matrix J exists uniquely in the Hölder classes in an arbitrary ellipse.

Functional equations in three classes of entire functions related to sine type functions are considered. These functional equations occur in inverse Sturm–Liouville problems with eigenvalue parameter dependent boundary conditions. In the first class \(\mathcal {A}\), which consists of certain pairs of entire functions (P, Q) with leading terms \(-\lambda \sin \lambda a\) and \(\cos \lambda a\), respectively, it is shown that, for each \(\eta \in \mathbb {R}\), the functional equation \(Q(\lambda )Y(\lambda )-P(\lambda )Z(\lambda )=\eta \) has a unique solution (Y, Z) in \(\mathcal {A}\). As a consequence, the functional equation \(Q(\lambda )S_1(\lambda )-P(\lambda )S_0(\lambda )=1 \) has a unique solution \((S_1,S_0)\) in a class of functions with leading terms \(\cos \lambda a\) and \(\lambda ^{-1}\sin \lambda a\), respectively. A third class of families \(\Phi _\alpha \), \(\alpha >0\), \(\alpha \not =1\), consists of entire functions with leading terms \(-\lambda \sin \lambda a+i\lambda \alpha \cos \lambda a\). It is shown that for each \(\phi \in \Phi _\alpha \) and \(\Delta \in \mathbb {R}\), the functional equation \(\phi (\lambda )X(-\lambda )-\phi (-\lambda )X(\lambda )=2i\alpha \lambda \Delta \) has a unique solution \(X\in \Phi _\alpha \).

This paper explains a connection between the famous Poncelet problem, the Abel equation, a new moment problem, problems of solution existence and uniqueness of boundary value problems for the string equation and some others in short presentation. We prove a connection of above problem with one problem of Goursat-type for a hyperbolic-like equation of fourth order. We consider a fourth-order differential equation with two multiple characteristics and parameters in the lower terms and a Goursat-type homogeneous boundary value problem with two real boundary conditions in some space of real entire functions. It is proven that this problem has a nontrivial solution in some space if and only if a parameter belongs denumerable dense set of real number.

This paper presents results on some boundary value problems for holomorphic functions of several complex variables in polydomains. The Cauchy kernel is one of the significant tools for solving the Riemann and the Riemann-Hilbert boundary value problems for holomorphic functions as well as for establishment of the connection between them. For polydomains, the Cauchy kernel is modified in such a way that it corresponds to a certain symmetry of the boundary values of holomorphic functions in polydomains. This symmetry is lost if the classical counterpart of the one-dimensional form of the Cauchy kernel is applied. The general integral representation formulas for the functions, holomorphic in polydomains, the solvability conditions and the solutions of the corresponding Schwarz problems are given explicitly. A necessary and sufficient condition for the boundary values of a holomorphic function for arbitrary polydomains is given and an exact, yet compact way of notation for holomorphic functions in arbitrary polydomains is introduced and applied. The Riemann jump problem and the Riemann-Hilbert problem are solved for holomorphic functions of several complex variables with the unit torus as the jump manifold. The higher-dimensional Plemelj-Sokhotzki formula for holomorphic functions in polydomains is established. The canonical functions of the Riemann problem for polydomains are represented and applied in order to construct solutions for both of the homogeneous and inhomogeneous problems. For all three boundary value problems, well-posed formulations are given which does not demand more solvability conditions than in the one variable case. The connection between the Riemann and the Riemann-Hilbert problem for polydomains is proven. Thus contrary to earlier research the results are similar to the respective ones for just one variable.

In Chap. 17, n-th-order functional differential equations are considered. In order to create a concept of nonoscillation for functional differential equations which will be an analogue of nonoscillation theory for ordinary differential equations, we define a nonoscillation interval for the n-th-order homogeneous functional differential equation as interval, where each nontrivial solution can have at most n−1 zeros, counting every zero with its multiplicity. The study of many classical questions of the qualitative theory of n-th-order equations such as existence and uniqueness of solutions of the interpolation boundary value problems, positivity or corresponding regular behavior of their Green’s functions, maximum principles and stability, was considered in the general framework based on the notion of nonoscillation intervals of corresponding linear functional differential equations. It is demonstrated that if the Wronskian of the fundamental system is not equal to zero at the initial point of the interval, then the nonoscillation interval exists for functional differential equations; this is true for a particular case of the n-th-order functional differential equations with Volterra operators (delay equations). For second-order functional differential equations, nonvanishing Wronskian is equivalent to the Sturm theorem on separation of zeros (between two adjacent zeros of any nontrivial solution there is a zero of every other solution). The analogue of the classical Mammana theorem, establishing necessary and sufficient nonoscillation conditions through nonvanishing normal chains of the Wronskians, is proven for n-th-order functional differential equations.

Equivalence, linearization, and decomposition of holomorphic operator functions

The problem of existence and uniqueness of solutions defined on the whole real line and satisfying given initial point data for general abstract linear functional differential equations is considered. The equation is not assumed to be of the delay type. The essence of the method presented here consists in the representation of a solution in the form analogous to the variation of constants formula known for linear ordinary differential equations. It is shown that such an approach can be effectively applied to the problem of existence and uniqueness of solutions satisfying an exponential growth estimate, provided that the deviation of the argument is sufficiently small. The proofs are based on the Banach fixed point principle. Detailed comparison and discussion of the hypotheses ensuring the existence and uniqueness of solutions are presented.

This paper studies an inversion of functional equation for Bell man's infinite-stage allocation process. Obtaining a sufficient condition under which Bellman's (main) functional equation has a unique invertible solution, we propose additive and recursive-inverse func tional equations which have a common solution. It is proved that the additive-inverse functional equation has a unique solution in a moderate class of functions. The solution is obtained from two successive approximation methods.

In this paper, we investigate a general multivariable functional equation. We prove, using the fixed-point method, the generalized Hyers–Ulam stability of this equation in Banach spaces. In this way, we obtain sufficient conditions for the stability of a wide class of functional equations and control functions. We also show, using examples, how some additional assumptions imposed on the function when examining the Hyers–Ulam stability of a functional equation affect the size of the approximating constant and limit the number of considered solutions for this equation. The functional equation studied in this paper has symmetric coefficients (with precision up to the sign), and it is a generalization of an equation characterizing n-quadratic functions, as well as many other functional equations with symmetric coefficients: for example, the multi-Cauchy equation and the multi-Jensen equation. Our results generalize many known outcomes.

It is proved that a general functional equation of the Riemann type with multiple gamma factors has non-trivial solutions in the space of generalized Dirichlet series. Moreover, for a fixed functional equation, the space of such solutions has uncountable basis. The proof is based on Hecke's theory of Dirichlet series associated with modular forms for the groups $G(\lambda)$. This is in constrast with the situation in the extended Selberg class where there exist functional equations without non-trivial solutions. Presumably this holds for non-integer degrees $d$, but up to date was confirmed only for $0 \leqslant d \lt 5/3$. In the case of $d = 0$ or $d = 1$ the space of solutions belonging to the extended Selberg class, through non-trivial, is finite dimensional.

Decomposer and associative functional equations

A function with complex values and at every point of the specific domain contains a derivative is commonly known as analytic functions which is also referred as holomorphic functions. We begin by interpreting \(A\) as the class for all holomorphic functions \(L(\xi)\) with a Taylor series expansion written in the form: \[L(\xi) = \xi + \sum_{i=2}^{\infty} x_i \xi^i\] where \(x_i \in \mathbb{C}\) and \(\xi \in D\). \(D\) is the open unit disk where \(D = \{\xi : \xi \in \mathbb{C}, |\xi| < 1 \}\). Furthermore, we suggest the subclass of \(A\) that is univalent in \(D\) represented as \(S\). It is commonly known that the main subclasses of class \(S\) are the class of starlike functions and the class of convex functions. To develop and analyze the Fekete-Szegö problems, the theory of geometric function contributes significantly to this. Moreover, the frequent use of \(q\)-calculus as a general direction of research among mathematicians has caught our attention. In this research, we attained the initial coefficients, \(x_2\) and \(x_3\), and the upper bound for the functional \(|x_3 - \nu x_2^2|\) of functions \(L\) in the two new subclasses that are introduced by involving the Sălăgeăn \(q\)-differential operator, \(M_q^\eta L(\xi)\) and the definition of subordination.

We consider Riemann-Hilbert boundary value problems (for short RHBVPs) with variable coefficients for axially symmetric monogenic functions defined in axial symmetric domains. This is done by constructing a method to reduce the RHBVPs for axially symmetric monogenic functions defined in four-dimensional axial symmetric domains into the RHBVPs for analytic functions defined over the complex plane. Then we derive solutions to the corresponding Schwarz problem. Finally, we generalize the results obtained to null-solutions of $(\mathcal{D}-\alpha)\phi=0$ , $\alpha\in\mathbb{R}$ , where $\mathbb{R}$ denotes the field of real numbers.

The class of operator-valued functions which are homogeneous of degree one, holomorphic in the open right polyhalfplane, have positive semi-definite real parts there and take selfadjoint operator values at real points, and its subclass consisting of functions representable in the form of Schur complement of a block of a linear pencil of operators with positive semidefinite operator coefficients, are investigated. The latter subclass is a generalization of the class of characteristic matrix functions of passive 2n-poles considered as functions of impedances of its elements, which was introduced by M. F. Bessmertnyi. Several equivalent characterizations of the generalized Bessmertnyi class are given, and its intimate connection with the Agler-Schur class of holomorphic contractive operator-valued functions on the unit polydisk is established.

As a model to provide a hands-on, elementary understanding of ‘vortex dynamics’, we introduce a piecewise linear non-invertible map called a twisted baker map. We show that the set of hyperbolic repelling periodic points with complex conjugate eigenvalues and that without complex conjugate eigenvalues are simultaneously dense in the phase space. We also show that these two sets equidistribute with respect to the normalised Lebesgue measure, in spite of a non-uniformity in their Lyapunov exponents.