On Riemann problems for single‐periodic polyanalytic functions

In this article, Riemann‐type boundary‐value problem of single‐periodic polyanalytic functions has been investigated. By the decomposition of single‐periodic polyanalytic functions, the problem is transformed into n equivalent and independent Riemann boundary‐value problems of single‐periodic analytic functions, which has been discussed in details according to two growth orders of functions. Finally, we obtain the explicit expression of the solution and the conditions of solvability for Riemann problem of the single‐periodic polyanalytic functions.     

On modified Hilbert boundary‐value problems of polyanalytic functions

Under the decomposition of polyanalytic functions, two classes of Hilbert‐type boundary‐value problems of polyanalytic functions with different factors have been discussed, and the explicit expression of solution and the condition of solvability have been obtained. Copyright © 2008 John Wiley & Sons, Ltd.     

Riemann boundary value problems for some K-regular functions in clifford analysis

In this paper, we study the R_m (m > 0) Riemann boundary value problems for regular functions, harmonic functions and bi-harmonic functions with values in a universal clifford algebra C(V_n,n). By using Plemelj formula, we get the solutions of R_m (m > 0) Riemann boundary value problems for regular functions. Then transforming the Riemann boundary value problems for harmonic functions and bi-harmonic functions into the Riemann boundary value problems for regular functions, we obtain the solutions of R_m (m > 0) Riemann boundary value problems for harmonic functions and bi-harmonic functions.     

On the analytical solution of the linear‐fractional Riemann problem

The linear‐fractional problem is a generalization of the linear Riemann problem that includes the (non‐linear) factorization problem. In case of normal type it can be equivalently reduced to a family of homogeneous linear vector Riemann problems by space foliation and adequate substitutions. Moreover these are equivalent to systems of non‐homogeneous Toeplitz equations with special data. The reduced problem is solved by matrix factorization in various cases. Procedures for reduction to these cases are exposed. Various modified problems and generalizations are pointed out. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Curvilinear integrals of discontinuous functions over nonrectifiable paths and Riemann boundary‐value problem

We study a generalization of the concept of curvilinear integral for the case where path of integration is nonrectifiable, and integrand has discontinuities. Then we apply that integrals for solving of the Riemann boundary‐value problem for domains with nonrectifiable boundaries and discontinuous boundary data. The research is supported by RFBR (grant 18‐31‐00060) and is performed according to a special programme of the Russian government supporting research at Kazan Federal University. Mathematical Methods in the Applied Sciences journal encourages authors to share the data and other artifacts supporting the results in the paper by archiving it in an appropriate public repository. Authors may provide a data availability statement, including a link to the repository they have used, in order that this statement can be published in their paper. Shared data should be cited.     

On Haseman boundary value problem for a class of metaanalytic functions with different factors on the unit circumference

In this article, the Haseman boundary value problem (bvp) is discussed for metaanalytic functions with different factors on the unit circumference. Through a series of appropriate transformations, we obtain general expressions for the solution and the condition of solvability for the problem. Copyright © 2009 John Wiley & Sons, Ltd.     

A further study on value distribution of the Riemann zeta‐function

The paper concerns the uniqueness problem of Riemann zeta‐function. It is showed that the Riemann zeta‐function is uniquely determined in terms of the preimages of three complex values except possibly a set G with , where G is called an exceptional set.     

实轴上的双解析函数Riemann边值逆问题

提出了一类实轴上的双解析函数Riemann边值逆问题.先消去参变未知函数,再采用易于推广的矩阵形式记法,可把问题转化为两个实轴上的解析函数Riemann边值问题.利用经典的Riemann边值问题理论,讨论了该问题正则型情况的解法,得到了它的可解性定理.     

Riemann boundary value problem for harmonic functions in Clifford analysis

In this article, we mainly deal with the boundary value problem for harmonic function with values in Clifford algebra: where is a Liapunov surface in , the Dirac operator , are unknown functions with values in a universal Clifford algebra Under some assumptions, we show that the boundary value problem is solvable.     

A Dirichlet problem for polyharmonic functions

In this article, the Dirichlet problem of polyharmonic functions is considered. As well the explicit expression of the unique solution to the simple Dirichlet problem for polyharmonic functions is obtained by using the decomposition of polyharmonic functions and turning the problem into an equivalent Riemann boundary value problem for polyanalytic functions, as the approach to find the kernel functions of the solution for the general Dirichlet problem is given. Project supported by NNSF of China.     

Studies on anti‐periodic boundary value problems for two classes of special second order impulsive differential equations

We consider anti‐periodic boundary value problems for two classes of special second order impulsive differential equations. On the basis of several important impulsive differential inequalities, by using the monotone iterative technique coupled with lower and upper solutions, we obtain sufficient conditions to guarantee the existence and uniqueness of solutions for such problems. Further, we give two examples to illustrate our conclusions. Copyright © 2013 John Wiley & Sons, Ltd.     

EXTENSION OF SOLUTIONS TO RIEMANN BOUNDARY VALUE PROBLEMS AND ITS APPLICATION

In this paper,solutions of Riemann boundary value problems with nodes are extended to the case where they may have singularties of high order at the nodes.Moreover, further extension is discussed when the free term of the problem involved also possesses singularities at the nodes.As an application,certain singular integral equation is discussed.     

On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations

In this paper, we show that the method of monotone iterative technique is valid to obtain two monotone sequences that converge uniformly to extremal solutions of first and second order periodic boundary value problems and periodic solutions of functional differential equations. We obtain some new results relative to the lower solution α and upper solution β with α?β.     

Multiplicity results near the principal eigenvalue for boundary‐value problems with periodic nonlinearity

Let us consider the boundary‐value problem where g: ? → ? is a continuous and T ‐periodic function with zero mean value, not identically zero, (λ, a) ∈ ?² and ∈ C [0, π ] with ∫^π ₀ (x) sin x dx = 0. If λ ₁ denotes the first eigenvalue of the associated eigenvalue problem, we prove that if (λ, a) → (λ ₁, 0), then the number of solutions increases to infinity. The proof combines Liapunov–Schmidt reduction together with a careful analysis of the oscillatory behavior of the bifurcation equation. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Schwarz problem for higher‐order complex partial differential equations in the upper half plane

Linear and nonlinear elliptic complex partial differential equations of higher‐order are considered under Schwarz conditions in the upper‐half plane. Firstly, using the integral representations for the solutions of the inhomogeneous polyanalytic equation with Schwarz conditions, a class of integral operators is introduced together with some of their properties. Then, these operators are used to transform the problem for linear equations into singular integral equations. In the case of nonlinear equations such a transformation yields a system of integro‐differential equations. Existence of the solutions of the relevant boundary value problems for linear and nonlinear equations are discussed via Fredholm theory and fixed point theorems, respectively.     

A separation theorem for semicontinuous functions on preordered topological spaces

Suppose that X is a topological space with preorder , and that –g, f are bounded upper semicontinuous functions on X such that g(x) f(y) whenever x y. We consider the question whether there exists a bounded increasing continuous function h on X such that g h f, and obtain an existence theorem that gives necessary and sufficient conditions. This result leads to an extension theorem giving conditions that allow a bounded increasing continuous function defined on an open subset of X to be extended to a function of the same type on X. The application of these results to extremally disconnected locally compact spaces is studied.Received: 26 May 2004     

On the Riemann‐Hilbert boundary value problem for generalized analytic functions in the framework of variable exponent spaces

Let Γ be a simple closed curve that bounds the finite domain D , z =z (ζ )=z (r e ^{i ?} ) be the conformal mapping of the circle {ζ :|ζ |<1} onto the domain D . Furthermore, let the functions A (z ), B (z ) be given on D and U ^{s ,2}(A ;B ;D ) be the set of regular solutions of the equation . We call the Smirnov class E ^{p (t )}(A ;B ;D ) the set of those generalized functions W in D for which where p (t ) is a positive measurable function on Γ. We consider the Riemann‐Hilbert problem: Define a function W (z ) from the class E ^{p (t )}(A ;B ;D ) for which the equality, is fulfilled almost everywhere on Γ. It is assumed that Γ is a piecewise‐smooth curve without external peaks; , p is Log Hölder continuous and the function belongs to the class A (p (t );Γ), which is the generalization of the well‐known Simonenko class A (p ;Γ), where p is constant. The solvability conditions are established, and solutions are constructed.