Linear and nonlinear nonlocal boundary value problems for differential-operator equations

This study focuses on nonlocal boundary value problems (BVPs) for linear and nonlinear elliptic differential-operator equations (DOEs) that are defined in Banach-valued function spaces. The considered domain is a region with varying bound and depends on a certain parameter. Some conditions that guarantee the maximal L_p -regularity and Fredholmness of linear BVPs, uniformly with respect to this parameter, are presented. This fact implies that the appropriate differential operator is a generator of an analytic semigroup. Then, by using these results, the existence, uniqueness and maximal smoothness of solutions of nonlocal BVPs for nonlinear DOEs are shown. These results are applied to nonlocal BVPs for regular elliptic partial differential equations, finite and infinite systems of differential equations on cylindrical domains, in order to obtain the algebraic conditions that guarantee the same properties.     

The boundary value problems of the full coupled theory of poroelasticity for materials with double porosity

In this paper the full coupled quasi-static theory of poroelasticity for materials with double porosity is considered. The basic boundary value problems (BVPs) of the steady vibrations are investigated. The uniqueness theorems of the internal BVPs of steady vibrations are proved. The basic properties of elastopotentials are established. The existence of regular solutions of the BVPs by means of the boundary integral equations method and the theory of singular integral equations is proved. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

External boundary value problems in the quasi static theory of elasticity for triple porosity materials

In this paper the quasi static linear theory of elasticity for materials with triple porosity is considered. Basic external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness and existence theorems for regular (classical) solutions of these BVPs are established. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Systems of Boundary Value Problems for Differential Inclusions

Herein we consider the existence of solutions to second-order, two-point boundary value problems （BVPs） for systems of ordinary differential inclusions. Some new Bernstein-Nagumo conditions are presented that ensure a priori bounds on the derivative of solutions to the differential inclusion. These a priori bound results are then applied, in conjunction with appropriate topological methods, to prove some new existence theorems for solutions to systems of BVPs for differential inclusions. The new conditions allow of the treatment of systems of BVPs without growth restrictions.     

Potential method in the steady vibrations problems of the theory of thermoviscoelasticity for Kelvin-Voigt materials with voids

In this paper the linear theory of thermoviscoelasticity for Kelvin-Voigt materials with voids is considered. The basic internal and external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness and existence theorems for classical solutions of the above mentioned BVPs are proved by using the potential method (boundary integral equation method) and the theory of singular integral equations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lie symmetries of nonlinear boundary value problems

Nonlinear boundary value problems (BVPs) by means of the classical Lie symmetry method are studied. A new definition of Lie invariance for BVPs is proposed by the generalization of existing those on much wider class of BVPs. A class of two-dimensional nonlinear boundary value problems, modeling the process of melting and evaporation of metals, is studied in details. Using the definition proposed, all possible Lie symmetries and the relevant reductions (with physical meaning) to BVPs for ordinary differential equations are constructed. An example how to construct exact solution of the problem with correctly-specified coefficients is presented and compared with the results of numerical simulations published earlier.     

Dynamical problems of the theory of elasticity for solids with double porosity

In this paper the dynamical theory of elasticity for solids with double porosity is presented. The single-layer and double-layer potentials are constructed and basic properties are established. The uniqueness theorems of the internal and external boundary value problems (BVPs) of steady vibrations are proved. The existence theorems of classical solution of the external BVPs by means of the boundary integral method and the theory of multidimensional singular integral equations are proved. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

General Transmission Problems in the Theory of Elastic Oscillations of Anisotropic Bodies (Basic Interface Problems)

Here we discuss three-dimensional so-called basic and mixed boundary value problems (BVP) for steady state oscillations of piecewise homogeneous anisotropic bodies imbedded into an infinite elastic continuum. Uniqueness is shown with the help of generalized Sommerfeld–Kupradze radiation conditions, while existence follows for arbitrary values of the oscillation parameter by the reduction of the original interface transmission BVPs to equivalent uniquely solvable boundary integral or pseudodifferential equations on the interfaces. For the basic BVPs, we show classical regularity and, in addition for the mixed BVPs that the solutions are Hölder continuous with exponent α ∈ (0, 1/2) in the neighbourhood of the curves of discontinuity of the boundary and transmission conditions.     

Analysis of segregated boundary‐domain integral equations for variable‐coefficient problems with cracks

Segregated direct boundary‐domain integral equation (BDIE) systems associated with mixed, Dirichlet and Neumann boundary value problems (BVPs) for a scalar “Laplace” PDE with variable coefficient are formulated and analyzed for domains with interior cuts (cracks). The main results established in the paper are the BDIE equivalence to the original BVPs and invertibility of the BDIE operators in the corresponding Sobolev spaces. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

External boundary value problems in the quasi static theory of viscoelasticity for Kelvin-Voigt materials with double porosity

In the present paper the linear quasi static theory of viscoelasticity for Kelvin-Voigt materials with double porosity is considered. The basic external boundary value problems (BVPs) of steady vibrations in this theory are formulated. The uniqueness and existence theorems for regular (classical) solutions of the BVPs are proved by using of the potential method (boundary integral equations method) and the theory of singular integral equations. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic boundary value problems of the second-order: Bernstein–Nagumo conditions and solvability

This article analyzes the solvability of second-order, nonlinear dynamic boundary value problems (BVPs) on time scales. New Bernstein–Nagumo conditions are developed that guarantee an a priori bound on the delta derivative of potential solutions to the BVPs under consideration. Topological methods are then employed to gain solvability.     

Solutions of BVPs in the fully coupled theory of elasticity for the space with double porosity and spherical cavity

In this paper, the fully coupled theory of elasticity for solids with double porosity is considered. The explicit solutions of the basic boundary value problems (BVPs) in the fully coupled linear equilibrium theory of elasticity for the space with double porosity and spherical cavity are constructed. The solutions of these BVPs are represented by means of absolutely and uniformly convergent series. Copyright © 2015 John Wiley & Sons, Ltd.     

Boundary value problems in the theory of thermoporoelasticity for materials with double porosity

In this paper the linear quasi-static theory of thermoelasticity for solids with double porosity is considered. The system of equations of this theory is based on the equilibrium equations for solids with double porosity, conservation of fluid mass, constitutive equations, Darcy's law for materials with double porosity and Fourier's law for heat conduction. The basic internal and external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness and existence theorems for classical solutions of the above mentioned BVPs are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary value problems of the theory of viscoelasticity for Kelvin-Voight materials

In this paper, the classical Kelvin-Voight model of the linear theory of viscoelasticity is considered and the following results are obtained: the fundamental solution of the equation of steady vibrations is constructed, the basic properties of plane waves and elastopotentials are established, the uniqueness theorem of the internal and external boundary value problems (BVPs) are proved, the existence theorems for classical solutions of the BVPs by means of the potential method and the theory of two-dimensional singular integral equations are proved. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

双锥不动点定理及其在非线性边值问题中的应用

本文研究了具可变号非线性项的非线性边值问题的正解存在性,推广了 Krasnoselskii 不动点定理,得到了新的锥上不动点定理,并应用这些定理给出这类边值问题正解的存在性．     

BVPS FOR DIFFERENTIAL OPERATORS WITH CHARACTERISTIC DEGENERATE SURFACES

The present paper is devoted to the BVPs of Lopatinski's type for differentialoperators of higher order and mixed type.A sign function,on which the formulation ofBVPs and regularity of their solutions rely heavily,is associated with characteristicdegenerate surface.The behavior of finite index for such kind of BVPs is found.     

Homotopy perturbation-reproducing kernel method for nonlinear systems of second order boundary value problems

In this paper, based on homotopy perturbation method (HPM) and reproducing kernel method (RKM), a new method is presented for solving nonlinear systems of second order boundary value problems (BVPs). HPM is based on the use of traditional perturbation method and homotopy technique. The HPM can reduce a nonlinear problem to a sequence of linear problems and generate a rapid convergent series solution in most cases. RKM is also an analytical technique, which can solve powerfully linear BVPs. Homotopy perturbation-reproducing kernel method (HP-RKM) combines advantages of these two methods and therefore can be used to solve efficiently systems of nonlinear BVPs. Three numerical examples are presented to illustrate the strength of the method.     

POSITIVE SOLUTION TO FOURTH-ORDER IMPULSIVE DIFFERENTIAL EQUATIONS WITH P-LAPLACIAN

In order to study three-point BVPs for fourth-order impulsive differential equation of the form(\phi_p(u'(t)))'- f(t,u(t))=0, t≠ t_i,△ u(t_i)=-I_i(u(t_i)), i=1, 2, ..., k,△u'(t_i)=-L_i(u(t_i)), i=1, 2, ..., k,(\star)with the following boundary conditionsu'(0)=u(1)=0, u'(0)=0=u'(1)-\phi_q(α)u'(η),the authors translate the fourth-order impulsive differential equations with p-Laplacian (\star) into three-point BVPs for second-order differential equation without impulses and two-point BVPs for second-order impulsive differential equation by a variable transform. Based on it, existence theorems of positive solutions for the boundary value problems (\star) are obtained.     

一类半导体方程混合初边值问题解的渐近性

考虑在磁场影响下一类半导体方程的混合边值问题，在一定条件下，利用Ｍｏｓｅｒ技巧和先验估计，证明了瞬态解的存在性和平衡解的存在唯一性，经过一系列估计，证明了瞬态解的渐近性。     

Solution of Poisson's equation with global,local and nonlocal boundary conditions

Boundary value problems (BVPs) for partial differential equations are common in mathematical physics. The differential equation is often considered in simple and symmetric regions, such as a circle, cube, cylinder, etc., with global and separable boundary conditions. In this paper and other works of the authors, a general method is used for the investigation of BVPs which is more powerful than existing methods, so that BVPs investigated by the method can be considered in anti-symmetric and arbitrary regions surrounded by smooth curves and surfaces. Moreover boundary conditions can be local, non-local and global. The BVP is expanded in a convex and bounded region D in a plane. First, by generalized solution of the adjoint of the Poisson equation, the necessary boundary conditions are obtained. The BVP is then reduced to the second kind of Fredholm integral equation with regularized singularities.